ELECTRONIC PROPERTIES OF IDEAL AND ......Metal/semiconductor (MS) interfaces are present in...

ELECTRONIC PROPERTIES OF IDEAL AND

ENGINEERED METAL/SEMICONDUCTOR

INTERFACES

THESE N 1914 (1998)

PRESENTEE AU DEPARTEMENT DE PHYSIQUE

ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE

POUR L’OBTENTION DU GRADE DE DOCTEUR ES SCIENCES

PAR

Christophe BERTHOD

Ingenieur physicien diplome EPFLoriginaire d’Orsieres (VS)

acceptee sur proposition du jury:

Prof. A. Baldereschi, directeur de these

Dr N. Binggeli, rapporteur

Dr B. Delley, rapporteur

Prof. E. Kapon, rapporteur

Prof. R. Resta, rapporteur

Lausanne, EPFL

1998

Version abregee

La technologie des dispositifs electroniques a semiconducteurs repose en grande

partie sur des jonctions metal-semiconducteur, dont la caracteristique electro-

nique la plus importante est la hauteur de la barriere de Schottky qui s’etablit

a l’interface. Cette these a pour objectifs de determiner l’influence qu’exercent

certains parametres structurels et electroniques des jonctions sur la barriere de

Schottky, par une etude numerique ab initio, et d’explorer quelques procedes qui

peuvent etre utilises pour modifier artificiellement cette barriere.

Pour les jonctions polaires et abruptes Al/GaAs, Al/AlAs et Al/ZnSe (100), nous

avons etudie la relation entre la barriere de Schottky et la composition chimique

du semiconducteur, la terminaison de sa surface, ainsi que sa reconstruction. Afin

de comprendre le comportement de la barriere dans ces differents systemes, nous

avons analyse la densite de charge des jonctions a l’echelle atomique et developpe

un modele qui decrit avec precision la variation de la barriere avec la composition

chimique du semiconducteur. L’effet de la terminaison sur la barriere de Schottky

s’explique par la presence d’un dipole a l’interface, constitue d’une part par la

charge ecrantee de la surface polaire du semiconducteur et d’autre part par une

charge image a la surface du metal. Grace aux calculs ab initio, nous avons pu

montrer comment le concept classique de charge image, valable pour des charges

situees a grande distance du metal, peut etre etendu a des distances de l’ordre

des dimensions atomiques.

Nous avons egalement considere l’effet de couches atomiques heterovalentes in-

troduites dans les jonctions Al/GaAs (100). Plusieurs experiences recentes ont

montre que la barriere de Schottky de ces jonctions peut etre considerablement

modifiee en deposant a l’interface des couches minces d’atomes de valence IV.

Pour comprendre le mecanisme qui provoque ces modifications, nous avons etudie

les proprietes structurelles et electroniques de jonctions Al/Si/GaAs (100) con-

tenant de 0 a 6 monocouches de silicium. Les resultats de nos calculs ab initio

reproduisent fidelement la variation de la barriere, en fonction de l’epaisseur de

la couche de Si, et permettent de l’expliquer en terme d’un dipole local induit par

le Si a l’interface. Nous avons en outre propose un modele qui permet de predire,

grace a des parametres physiques simples, les modifications de la barriere de

Schottky produites par ces couches a l’interface.

Abstract

Semiconductor device technology largely relies on metal/semiconductor junctions,

whose most important electronic parameter is the Schottky barrier height. The

purpose of this thesis is to determine by first-principle computational studies the

influence on the Schottky barrier of various structural and electronic parameters

of the junctions, and explore mechanisms which can be used to artificially modify

the barrier heights.

We have investigated the Schottky barrier heights of abrupt polar Al/GaAs,

Al/AlAs, and Al/ZnSe (100) junctions, and their dependence on the semicon-

ductor chemical composition as well as surface termination and reconstruction.

The resulting trends of the Schottky barrier have been understood based on the

atomic-scale properties of the charge densities in the junctions, and a model has

been developed, which provides a simple, yet accurate, description of the barrier

height dependence on the semiconductor chemical composition. We explained the

effect of the surface termination in terms of a dipole produced by the screened

charge of the polar semiconductor surface and its image charge at the metal sur-

face. In this connection, our first-principle computations showed how the classical

image charge concept, valid for charges placed at large distances from the metal,

extends to atomic-scale distances.

We have also addressed the effect of heterovalent atomic interface layers in engi-

neered Al/GaAs (100) junctions. Recent experiments showed that considerable

changes can be achieved in the Al/GaAs (100) Schottky barrier height by growing

thin layers of group-IV atoms at the interface. In order to understand the mech-

anism responsible for this tuning, we have studied the structural and electronic

properties of Al/Si/GaAs (100) junctions containing up to 6 monolayers of Si.

Our ab initio results provide an accurate description of the Schottky barrier mod-

ifications as a function of the Si interlayer thickness, and explain the tuning in

terms of Si-induced local-interface dipoles. A model has been derived to predict

the barrier modifications produced by the interlayers in terms of simple physical

parameters.

Contents

Introduction 5

1 Phenomenology and models of Schottky barriers 7

1.1 Electronic transport in metal/semiconductor junctions . . . . . . 7

1.1.1 Rectifying contacts . . . . . . . . . . . . . . . . . . . . . . 8

1.1.2 Ohmic contacts . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Experimental measurement of Schottky barrier heights . . . . . . 13

1.2.1 I–V and C–V measurements . . . . . . . . . . . . . . . . . 13

1.2.2 Photoelectric and photoemission measurements . . . . . . 13

1.2.3 Laterally resolved measurements . . . . . . . . . . . . . . . 14

1.3 Theories of Schottky barrier formation . . . . . . . . . . . . . . . 14

1.3.1 Semiempirical models . . . . . . . . . . . . . . . . . . . . . 15

1.3.2 Fermi-level pinning by surface or interface states . . . . . . 19

1.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Ab-initio methods 23

2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.1 Density Functional Theory . . . . . . . . . . . . . . . . . . 23

2.1.2 Solution of the Kohn-Sham equations . . . . . . . . . . . . 26

2.2 Application to metal/semiconductor contacts . . . . . . . . . . . . 30

2.2.1 Supercells . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 Interface Brillouin zone and level broadening . . . . . . . . 31

2.2.3 Calculation of the Schottky barrier height . . . . . . . . . 31

2.2.4 Other electronic properties . . . . . . . . . . . . . . . . . . 34

2.2.5 Many-body and spin-orbit corrections on the Schottky barrier 36

2.2.6 Numerical parameters and precision . . . . . . . . . . . . . 37

3 The abrupt epitaxial Al/GaAs (100) interface 39

3.1 Interface atomic structure . . . . . . . . . . . . . . . . . . . . . . 39

2 Contents

3.2 Electronic charge density . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Comparison with the bulk densities . . . . . . . . . . . . . 41

3.2.2 Discussion of the chemical bonds . . . . . . . . . . . . . . 43

3.3 Schottky barrier height . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Electronic states . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.1 Interface band structure . . . . . . . . . . . . . . . . . . . 45

3.4.2 Local density of states . . . . . . . . . . . . . . . . . . . . 47

3.4.3 Metal-induced-gap states . . . . . . . . . . . . . . . . . . . 48

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Schottky barrier tuning in Al/Si/GaAs (100) junctions 57

4.1 Al/Si/GaAs junctions with 0–2 Si ML . . . . . . . . . . . . . . . 59

4.1.1 Si-induced local dipole . . . . . . . . . . . . . . . . . . . . 59

4.1.2 Schottky barrier modification . . . . . . . . . . . . . . . . 61

4.1.3 Effect of atomic relaxations . . . . . . . . . . . . . . . . . 62

4.1.4 Si-induced local density of states . . . . . . . . . . . . . . 68

4.1.5 Interface formation energy . . . . . . . . . . . . . . . . . . 70

4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.1 Screening of local dipoles at the interface . . . . . . . . . . 74

4.2.2 Inhomogeneous screening near the junction . . . . . . . . . 76

4.2.3 Application to Al/Ge/GaAs (100) contacts . . . . . . . . . 77

4.3 Al/Si/GaAs junctions with 2–6 Si ML . . . . . . . . . . . . . . . 79

4.3.1 Schottky barrier modification . . . . . . . . . . . . . . . . 80

4.3.2 Effect of resonant interface states . . . . . . . . . . . . . . 80

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Ionicity, surface properties, and Schottky barrier heights 89

5.1 Abrupt Al contacts to Ge, GaAs, AlAs, and ZnSe (100) . . . . . . 90

5.2 Interpretation and models . . . . . . . . . . . . . . . . . . . . . . 92

5.2.1 General trend with the semiconductor composition . . . . 92

5.2.2 Effect of surface termination . . . . . . . . . . . . . . . . . 93

5.3 Effect of the surface reconstruction in the Al/ZnSe (100) junction 99

5.3.1 Results for model c(2× 2), 2× 1, and 1× 1 interface con-

figurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3.2 Se-induced local dipole . . . . . . . . . . . . . . . . . . . . 102

Conclusion 105

Contents 3

Appendix A. Technical aspects 107

A.1 Non-linear core correction (NLCC) . . . . . . . . . . . . . . . . . 107

A.2 Brillouin-zone sampling . . . . . . . . . . . . . . . . . . . . . . . . 108

A.3 Parameters for the pseudopotentials . . . . . . . . . . . . . . . . . 109

A.4 Bulk-related parameters . . . . . . . . . . . . . . . . . . . . . . . 110

A.5 Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Appendix B. Models 113

B.1 Average barrier height at the polar (100) interface . . . . . . . . . 113

B.2 Effective screening of a surface charge . . . . . . . . . . . . . . . . 116

Bibliography 119

Introduction

Metal/semiconductor (MS) interfaces are present in virtually all electronic de-

vices, either as ohmic or as rectifying contacts. A single parameter characterizes

the transport properties of these interfaces, namely the Schottky barrier height

(SBH) [1]. The SBH measures the difference between the Fermi energy of the

metal and the band edge of the semiconductor majority carriers at the junction.

In spite of the fundamental importance of the SBH, the mechanisms which control

the Schottky barrier formation are still far from being fully understood [2, 3].

Until recently, most measurements of SBH’s at metal contacts to covalent semi-

conductors indicated a relatively weak dependence of the barrier height on the

metal used or on the contact fabrication method [3]. This behavior had been

generally attributed to a pinning of the Fermi level by “interface” states, either

intrinsic metal-induced gap states or extrinsic gap states associated with native

defects near the interface [4]. More recent experiments, however, provided evi-

dence of a much stronger dependence of the SBH on metallization and surface-

or interface-specific properties [3]. In particular, very recently, there have been

reports of considerable barrier changes in metal/Si and metal/GaAs junctions ob-

tained by altering the structural properties and/or chemical composition of the

interface [5–11]. These developments challenge our understanding of Schottky

barrier formation, and have stimulated new experimental activities targeted to

engineering Schottky barriers by means of interfacial perturbations, with the aim

of improving the performances or modulating the characteristics of MS devices.

Progress in the field of computational physics have recently made possible first-

principle investigations of systems as complex as MS interfaces, for some model

interfacial atomic configurations. These ab initio approaches are an ideal tool

to probe the dependence of the Schottky barriers on interface-specific properties

[14–18]. In this thesis, we use first-principle calculations to investigate the elec-

tronic structure of some ideally abrupt and engineered MS interfaces. We study

the dependence of the SBH on the structural and electronic properties of the in-

terface, and explore mechanisms which can be used to control the barrier heights.

6 Introduction

Based on the first-principle results, we develop models to explain these effects,

which are then tested by numerical experiments.

A large part of the thesis deals with the Al/GaAs junction. The Al and GaAs

materials possess a good lattice matching, which gives rise experimentally to

quasi-epitaxial Al/GaAs (100) interfaces. We use the Al/GaAs (100) junction

as a prototype system to study the electronic structure of a MS junction at the

atomic scale, and discuss the limits of semiempirical models of Schottky barrier

formation. We also consider related lattice matched systems, such as Al/Ge,

Al/AlAs, and Al/ZnSe (100), and probe the effect of the increasing semiconductor

ionicity and surface termination of the semiconductor on the SBH. The effect of

the semiconductor-surface reconstruction on the SBH is also examined in the case

of the Al/ZnSe (100) interface. As prototype engineered system, we consider the

Al/Si/GaAs (100) junction. Recent experiments have shown that drastic changes

in the Al/GaAs (100) Schottky barrier could be achieved by depositing ultrathin

Si interlayers at the interface [8–11]. Conflicting empirical models have been

proposed to explain these barrier variations [10–13]. We will show that first-

principle calculations can help solving this controversy.

The thesis is organized as follows. Chapter 1 gives a brief introduction to the

field of MS contacts and discusses basic models of SBH formation. Chapter 2 de-

scribes the ab initio methods and the scheme employed in this work for computing

the electronic properties of MS interfaces. In Chapter 3, we present our results

for the Al/GaAs (100) junction and for some related lattice-matched systems,

and analyze these results in connection with existing theories of SBH forma-

tion. Chapter 4 is devoted to the engineered Al/Si/GaAs (100) junction and

to the mechanism responsible for the large barrier modifications observed ex-

perimentally. In Chapter 5, we compare ideal Al/Ge, Al/GaAs, Al/AlAs, and

Al/ZnSe (100) junctions, and explain the effects of the semiconductor-surface ter-

mination and of the change in semiconductor ionicity on the SBH. In Chapter 5,

we also study the effect of different reconstructions of the ZnSe (100) surface on

the Al/ZnSe SBH. Atomic Hartree units are used throughout the thesis.

Chapter 1

Phenomenology and models of

Schottky barriers

Since the rectifying behavior of metal/semiconductor (MS) contacts was discov-

ered, more than hundred years ago, much theoretical effort has been devoted to

interpreting the electrical measurements in these systems, such as the I–V and

C–V characteristics, and predicting the value of the parameters which enter the

semiempirical models developed in this context. The most important of these

parameters is the Schottky barrier height (SBH). The semiempirical transport

models are widely used, and generally provide a good description of the exper-

imental data. The physical mechanisms behind the formation of the Schottky

barrier, however, remain a matter of debate. Many conflicting models have been

proposed. In this chapter, we first give a brief review of the semiempirical models

employed to interpret the transport measurements in MS junctions, and also a

short description of the experimental techniques used to measure the SBH. A

comprehensive review can be found in the book by Rhoderick and Williams [19].

In the last part of this chapter, we review some of the basic models of SBH

formation and discuss their general properties and implications.

1.1 Electronic transport in metal/semiconduc-

tor junctions

The models which are used to describe the electronic transport at a MS contact

rely on a macroscopic representation of the junction; the latter is considered as a

juxtaposition of two homogeneous, abruptly truncated, bulk materials, and the

plane where the electronic properties discontinuously change defines the interface.

8 Phenomenology and models of Schottky barriers Chap. 1

Within each bulk, the transport is treated in a semiclassical manner. In this way,

the problem becomes one-dimensional, and relatively similar to the problem of

the p-n junction. Although difficult to reconcile with the intuitive image one

could have in mind, when thinking of a MS contact on the atomic scale, this

picture of a flat abrupt interface is generally justified in transport theories, by

the large wavelength of conduction electrons (∼ 100 A) relative to the atomic

dimensions [19].

1.1.1 Rectifying contacts

Figure 1.1(a) shows the electronic energy-band profile of a metal/n-type semi-

conductor rectifying contact at equilibrium, i.e., without external bias. Only the

main features of the local density of states are drawn, namely the Fermi level

(FL) and the semiconductor valence-band maximum (VBM) and conduction-

band minimum (CBM), as a function of the coordinate normal to the interface.

The central quantity is the n-type Schottky barrier height φn, which is the differ-

ence between the CBM and the FL at the interface. For a p-type semiconductor,

the key quantity is the difference between the FL and the VBM, called the p-type

Schottky barrier height, and denoted φp. The band discontinuities φn and φp

are intrinsic properties of the MS interface, which do not depend, in principle,

on the applied bias or semiconductor doping.1 They are related by the Schottky

rule [20]:

φp + φn = Eg, (1.1)

where Eg is the semiconductor bandgap [see Fig. 1.3(a)]. The lowering of the

SBH indicated in Fig. 1.1, ∆φn, results from the attractive image force acting on

electrons close to the metal surface (for a detailed discussion, see Ref. [19]).

The bending of the bands in the semiconductor, in Fig. 1.1, is due to a positive

space charge of uncompensated donor ions near the interface. In the bulk semicon-

ductor, the uniform charge of the ionized donors is compensated by the density

of electrons in the conduction band. Closer to the metal, instead, the density

of electrons in the conduction band decreases exponentially with the difference

Ec − ζs, where Ec is the semiconductor conduction-band edge and ζs is the elec-

trochemical potential2 in the bulk semiconductor. The resulting region depleted

1For junctions with an insulating interlayer between the metal and the semiconductor, theSBH has a weak intrinsic dependence on the doping in the semiconductor [19].

2The electrochemical potential is the derivative of the local free energy with respect to thelocal number of electrons.

Sect. 1.1 Electronic transport in metal/semiconductor junctions 9

cE

sε

mε

svmv

w

Eg

vE

nξsζ

φn

ζm

∆φn

(a)

cE

vE

sζφn

ζm

∆φn

qV

I

e−

(b)

Figure 1.1 Energy diagram of a metal/n-type semiconductor contact with a largepositive Schottky barrier height, φn, at thermodynamic equilibrium (a), and underforward bias V (b); q is the absolute value of the electron charge. Eg, Ev, and Ec arethe bandgap, and the valence- and conduction-band extrema, respectively; ζm and ζs

are the electrochemical potentials in the metal and in the semiconductor (ζm ≡ ζs ≡ EF

at zero temperature and zero bias); vm and vs are the average electrostatic potentials;εm and εs are the chemical potentials, measured with respect to vm and vs; ξn is theposition of the conduction-band edge in the bulk semiconductor, measured from theFermi level; ∆φn indicates the image-force lowering of the barrier.

of electrons close to the interface, called the depletion zone, has a width [1]

w =

√εs (φn − ξn − qV − kT )

2πe2ND

, (1.2)

where εs is the semiconductor dielectric constant, ξn the position of the CBM in

the bulk semiconductor measured from the FL, ND the density of donors, V the

bias, q the absolute value of the electron charge, and kT the thermal energy. The

band bending in the depletion zone, which acts as a potential barrier for electrons,

is nearly quadratic, as can be found by self-consistently solving Poisson’s equation

with a uniform charge of ionized donors. For a p-type semiconductor, the interface

region is depleted of holes. The bands are bent in the opposite direction, and

the quantities φp, ξp, and NA replace φn, ξn, and ND in Eq. (1.2), where ξp is the

difference between the FL and the VBM in the bulk semiconductor and NA the

density of acceptors. The depletion zone (whose typical width is 100–1000 A)

acts as a bias-dependent series resistance, and is responsible for the rectifying

behavior of the contact.


The energy-band profile under a positive bias is illustrated in Fig. 1.1(b). As first

suggested by Schottky [21], the voltage drop develops entirely in the depletion

zone, because of its higher resistivity, and changes the shape of the barrier. In the

bulk semiconductor, the electrons acquire an additional energy qV , so that the

number of them which are able to surmount the barrier increases exponentially

with V , and so does the current as a function of voltage. When qV becomes

comparable to the SBH, the depletion zone disappears (flat-band regime), and

the contact becomes approximately ohmic (see section 1.1.2). If a negative bias

is applied, the width of the depletion zone increases and the potential barrier

becomes higher for electrons in the semiconductor, so that less and less of them

can reach the metal. However, the electrons in the metal which have a thermal

energy sufficient to overcome the SBH φn are accelerated by the electric field in

the depletion zone and give a small negative contribution to the current.

The two main mechanisms controlling the transport properties of electrons in

MS junctions are (i) the drift and diffusion in the depletion zone and (ii) the

thermionic emission of electrons at the interface. The current is limited by the

less efficient of these two processes. In the forties, the mechanism (i) has been

studied by Schottky and Spenke [22], and the mechanism (ii) by Bethe [23]. Both

analysises lead to an I–V characteristic of the form

I = I0

[exp

(qV

kT

)− 1

], (1.3)

but the “saturation currents” I0 are different. In particular, the mechanism (i)

gives rise to a |V | 12 dependence of I0, while the mechanism (ii) results in a sat-

uration current independent of voltage. When I0 is independent of voltage, the

contact is called an ideal rectifier. In the sixties, Crowell and Sze [24] included

both processes (i) and (ii) into a single approach, and derived the following

expression for I0,

I0 = SA∗∗T 2 exp

(−φn −∆φn

kT

), (1.4)

where S is the contact area and A∗∗ is the modified Richardson constant,3 which

takes into account the relative efficiency of the mechanisms (i) and (ii), as well

as other secondary effects, namely the quantum mechanical tunneling of elec-

trons through the barrier, and the backscattering of electrons via optical phonon

excitations (see Ref. [24]).

Because of the effect of secondary transport processes, and also because of the

voltage dependence of ∆φn, the experimental I0 [derived from Eq. (1.3)] often

3A∗∗ is not strictly a constant, but its dependence on the bias is normally negligible over awide range of biases [1].

Sect. 1.1 Electronic transport in metal/semiconductor junctions 11

shows some dependence on the applied voltage. Although this dependence can

have quite different origins, one usually describes it by a single parameter n

defined as [19]1

n= 1 +

kT

q

d

dVln

[I

exp(qV/kT )− 1

], (1.5)

and which is evaluated in the regime qV & 3kT . The parameter n is called the

ideality factor, and is unity for an ideal rectifier. The voltage dependence of ∆φn

alone can increase the ideality factor up to 1.02–1.04 [19]. Introducing Eq. (1.5)

into Eqs. (1.3) and (1.4) one obtains

I = I0 exp

(qV

nkT

) [1− exp

(−qV

kT

)], (1.6)

where the zero bias value of ∆φn should be used for I0 in Eq. (1.4). Eqs. (1.6) and

(1.4) are the basic formulas used to analyze I–V data in most transport studies.

If an ac signal is superimposed to the dc voltage V , the depletion zone essentially

plays the same role as an insulator between two plates of a capacitor, and opposite

charges are induced on the metal surface and in the semiconductor. In the case of

uniform doping and neglecting the effect of minority carriers and possible localized

interface states, the capacitance of a MS junction is given by [1]:

C

S≈ q2εs

4πe2w= q2

√εsND

8πe2 (φn − ξn − qV − kT ). (1.7)

A detailed analysis of the effects of non-uniform doping, minority carriers, and

interface states on the capacitance can be found in Ref. [19].

1.1.2 Ohmic contacts

In many practical applications, the MS junctions are specifically used for their

rectification properties [1]. However, most often, the metallic contacts are just

used as current supplies for the active part of a device. In this case, one wishes

the junctions to be ohmic. A situation in which the contact is ideally ohmic,

independently of the doping properties of the bulk semiconductor, is when the

SBH vanishes or is negative (more precisely, smaller than ξn). Figure 1.2 shows the

energy-band profile of a metal/n-type semiconductor ohmic contact with a small

negative SBH. As for the rectifier, a rearrangement of the conduction electrons

occurs at the interface. Here, however, some electrons flow from the metal into

the semiconductor because of the negative SBH, and accumulate at the interface

instead of receding from it, overcompensating the charge of the ionized donors (see


Ref. [25]). Since the density Nc of electrons in the conduction band is typically

100 to 1000 times larger than the density of impurities, the accumulation zone is

much thinner than the depletion zone, as can be estimated by replacing ND by Nc

in Eq. (1.2). When the junction is biased, the potential difference is distributed

over the whole semiconductor side, and carriers can freely flow across the interface

owing to the absence of a potential barrier.

MS systems which exhibit negative or vanishing SBH, however, are not very

common. In device technology, “ohmic contacts” are defined more generally as

contacts which have a negligible resistivity relative to the resistivity of the bulk

semiconductor. Neglecting the image-force lowering and any field dependence of

the SBH, the contact resistivity Rc at zero bias, obtained from Eq. (1.6), reads

Rc =kT/q

SA∗∗T 2exp

(φn

kT

). (1.8)

This relation is valid for moderate values of ND. At high doping densities, the

barrier width is small and electron tunneling starts to dominate the contact resis-

tance. In the tunneling regime, the amplitude of the electron wave functions at

the interface behaves as exp(−κw), where ~2κ2/2m∗ ≈ φn for electrons with ener-

gies near the Fermi energy; m∗ is the electron effective mass in the semiconductor.

The resistivity is inversely proportional to this amplitude [1]:

Rc ∼ exp

[√εsm∗

πe2~2

(φn√ND

)]. (1.9)

Most ohmic contacts are thus fabricated by doping so heavily the region closest to

the interface, in a rectifying junction, (n+ layer in a metal/n-type contact or p+

layer in a metal/p-type contact) that the depletion zone is very thin and tunneling

dominates. In some cases, however, this procedure can meet intrinsic limitations

in the doping concentrations or introduce complications in the behavior of the

contacts when the doping is too high [26]. For wide-gap semiconductors, in

particular, the doping limitations can seriously inhibit the fabrication of ohmic

contacts.

The exponentials in Eqs. (1.8) and (1.9) have different physical origins — the

Fermi-Dirac statistics in the former case; the decay of tunneling electronic wave

functions in the latter case — but they both show that Rc is more sensitive to

changes in φn than to changes in ND. SBH engineering thus provides, in principle,

a more efficient method to realize ohmic contacts than interface doping.

Sect. 1.2 Experimental measurement of Schottky barrier heights 13

(b)

qV

I

e− cE

vE

sζ

ζm

(a)

cE

Eg

vE

nξ

sζζm

φn

Figure 1.2 Energy diagram of a metal/n-type semiconductor ohmic contact witha small negative barrier height, at thermodynamic equilibrium (a), and under forwardbias (b).

1.2 Experimental measurement of Schottky bar-

rier heights

1.2.1 I–V and C–V measurements

The current-voltage relationships in Eqs. (1.4) and (1.6) involve four unknown

quantities: the effective Richardson constant A∗∗, the SBH φn, the image-force

lowering ∆φn, and the ideality factor n. Reliable analytical expressions have

been derived for ∆φn (see Ref. [1]). Using Eq. (1.6), I0 and n can be determined

from measurements of I as a function of V , by plotting ln [I/ (1− exp(−qV/kT ))]

against V . If the measurements are repeated at different temperatures, the plot

of ln [I0/T2] as a function of 1/T provides A∗∗ and φn−∆φn. On the other hand,

if the concentration of impurities is uniform in the semiconductor, measuring the

capacitance under reverse bias and plotting 1/C2 as a function of V gives directly

φn through Eq. (1.7).

1.2.2 Photoelectric and photoemission measurements

When a MS junction is at thermodynamic equilibrium, the net current flowing in

an external circuit vanishes. However, if a sufficiently energetic light is directed

toward the metal, some electrons near the FL are excited over the barrier, are

then accelerated in the depletion zone, and produce a current in the circuit. This

is the principle of operation of solar cells. By measuring the current as a function

of the photon energy, one directly obtains an estimate of φn. This technique is

known as internal photoemission (IPE).


One can also study the energy distribution of electrons excited from the interface

region into the vacuum by higher energy photons. This method can only probe

electrons close to the sample surface, and is applied to MS interfaces with a very

thin (∼ 20 A) metal overlayer. In favorable cases, an ultraviolet photoelectron

spectrum (UPS) can already show sufficiently defined features corresponding to

the FL in the metal and the VBM in the semiconductor, providing the p-type

barrier φp = EF − Ev. With more energetic photon beams, such as synchrotron

radiation (SRPES) or X-rays (XPS), one can probe core electrons, whose location

in energy can be more precisely determined than the VBM edge hidden in the

continuum of metallic bands. The measurement then proceeds in two steps. First,

the binding energy Eb of the core level with respect to the VBM is measured on

the free semiconductor surface. Then, a metallic overlayer is deposited, and a

new spectrum is taken. The energy difference between the core level and the

threshold energy, corresponding to electrons at the FL, is equal to Eb + φp.

1.2.3 Laterally resolved measurements

Both the conventional electrical and photoemission techniques probe large lateral

regions of the MS interface as compared to the atomic dimensions. Therefore,

they provide a spatially averaged value of the SBH. At real MS contacts, lateral

inhomogeneities in the interface structure may exist, and the effect of these inho-

mogeneities on the SBH has only recently been considered [2, 27, 28]. Nowadays,

different methods can be implemented in order to gain information about the

SBH on the nanometer scale.

Ballistic-electron-emission microscopy (BEEM) takes advantage of an STM tip

for injecting ballistic electrons into the metal — on top of the semiconductor,

which acts as a collector — while monitoring the potential difference between

the tip and the metal. When this difference becomes larger than the SBH, a

sharp increase of the current flowing from the metal into the semiconductor is

detected [29], leading to a direct determination of φn. Lateral resolution can now

also be achieved with photoemission techniques [30], but with a lower resolution

than with the BEEM technique.

1.3 Theories of Schottky barrier formation

As we saw in the preceding section, the SBH (φn or φp) plays a central role: it

controls the whole energy band profile at the junction, and thus the electrical

characteristics of the contact. However, the issue of what are the physical pa-

Sect. 1.3 Theories of Schottky barrier formation 15

rameters and mechanisms which determine the value of the SBH remains open.

We will not undertake an exhaustive discussion of all existing models, but only

describe briefly the most representative ones. Furthermore, we do not attempt to

analyze these theories in the light of experimental results, since the most sophis-

ticated of these models often provide similar predictions for the systems to which

they are applicable, and experiments can rarely discriminate between them. We

just mention some experimental facts when needed. Refs. [19] and [31] contain

historical and critical reviews of the subject, and fundamental papers are collected

in the book edited by Monch [4].

1.3.1 Semiempirical models

Mott: the non-interacting surfaces. — The simplest approach, due to Mott [32],

to the SBH problem, assumes that the charge distribution at a MS contact is

identical to the superposition of the charge distributions of two isolated surfaces.

Thus, the work done by an electron crossing the interface is equivalent to the work

done by first escaping from the metal surface into the vacuum, gaining an energy

equal to the work function φm, and then entering the semiconductor surface,

loosing an energy equal to the electronic affinity χs, as depicted in Fig. 1.3(b).

The resulting n-type SBH is simply

φn = φm − χs, (1.10)

which is known as the Schottky-Mott rule. According to this rule, a linear re-

lationship exists between the SBH and the metal work function, and the slope

parameter, γ = dφn/dφm, is equal to 1. The early experimental measurements

of φn and φm involved substantial uncertainties. Nevertheless, a linear fit to the

data showed that the slope is clearly much smaller than 1, except for the most

ionic semiconductors. In particular, for the covalent semiconductors Si and Ge,

γ was found to be closer to 0 than to 1 [33].

Bardeen: the effect of surface states. — An explanation for this behavior was

given by Bardeen [34], who demonstrated that a relatively small density of lo-

calized states at the semiconductor surface can prevent the SBH from varying,

when metals with different work functions are used as contacts on a given semi-

conductor. His model constitutes the prototype of most subsequent theories, and

is illustrated schematically in Fig. 1.3(c). A dipole layer of amplitude ∆ is estab-

lished at the interface, because of the presence of opposite surface-state charges

separated by a distance d, like in a capacitor, and corrects the band alignment so

that the SBH is no longer given by Eq. (1.10). Bardeen attributed the dipole to


localized states which are assumed to pre-exist on the free semiconductor surface

with energies uniformly distributed in the bandgap. On the free semiconductor

surface, these states are occupied up to a level φ0 (measured from the CBM) for

the surface to be electrically neutral. When the contact is formed, the semicon-

ductor surface carries a charge proportional to the density of surface states and to

the difference φn− φ0. This charge is compensated by an opposite charge on the

metal surface, and by a small modification of the charge in the depletion zone.

The SBH is then determined by requiring the overall neutrality, and reads [35]

φn ≈ γ (φm − χs) + (1− γ) φ0,

γ =1

1 + 4πe2Di d/εi

. (1.11)

The slope γ is now related to the dielectric constant εi and thickness d of the

region separating the positive and negative charges, and to the density Di of the

surface states, but has no dependence upon the metal characteristics. At high

density Di, φn becomes identical to φ0, independently of the metal used: the FL

is said to be pinned by the surface states.

In this approach, the semiconductor surface states must be sufficiently isolated

from the metal surface in order not to interact with the continuum of metal states

and loose their localized character. This condition is expected to be fulfilled for

MS contacts obtained by chemical etching or cleavage in air, since they generally

involve a thin insulating interlayer at the interface. For this type of contacts,

Bardeen’s approach provides an explanation for the observed low values of γ for

the most covalent semiconductors: with d ≈ 10 A, εi ≈ 1, a density of sur-

face states Di = 5 × 10−3 eV−1 A−2 is indeed sufficient to make γ as small as

0.1. Using Eq. (1.11), the experimental values of γ were shown to be compat-

ible with surface-state densities in the range 10−3 to 10−2 eV−1 A−2 for many

semiconductors [35, 36]. The idea of Fermi-level pinning by surface (or interface)

states is the basis of most models of Schottky barrier formation and has stimu-

lated many investigations aiming at determining the nature and density of these

states (see section 1.3.2).

Heine: the metal-induced-gap states. — The work of Heine [37] was motivated

by the observation that the values of the parameter γ are basically the same for

interfaces with and without interlayer. Instead of surface states, Heine empha-

sizes the role of metal-induced-gap states (MIGS).4 At any solid/solid interface,

there are three kinds of electronic states: those which propagate on both sides

of the junction, those which propagate in one material and decay in the other

4This denomination is due to Louie et al. [39].


(a)

φn

φpEg

EF

m s

χsφm

(b)

φn

m s

Vacuum level

EF

χs

φm

(c)

m s

φn

d

+ −

φ0

∆

i∋EF

Figure 1.3 (a) The Schottky rule: φp + φn = Eg, expressing the fact that φp andφn are intrinsic properties of the MS interface. (b) The model of Mott: φn = φm − χs,for the n-type SBH. (c) The principle of Bardeen’s model: because of the presence ofinterface states, a dipole layer (of amplitude ∆) is established at the interface, whichtends to cancel out any variation of the SBH φn due to a change in φm−χs with respectto a canonical value φ0.

material, and those which decay in both materials and are localized at the in-

terface (see p. 45). The MIGS belong to the second group; they have energies

within the semiconductor fundamental gap, and decay exponentially on the semi-

conductor side of the junction. The tails of the MIGS can “store” charge in the

semiconductor, at some distance from the metal surface, as surface states would.

The corresponding expression for the slope γ has the same form as in Bardeen’s

model, but Di is replaced by the surface density of MIGS at the FL, Ds(EF),

and the interlayer thickness d is replaced by an effective distance, related to the

decay length δs of the MIGS tails, and to the Thomas-Fermi screening length δm

in the metal:

γ =1

1 + 4πe2Ds(EF) (δs/εs + δm). (1.12)

Within Heine’s theory, γ is not a property of the semiconductor only, since Ds(EF)

is roughly proportional to the density of states D at the FL in the metal. For

simple metals, D ∼ r−1s and φm ∼ rs [38]. Therefore, one may expect that Ds(EF)

behaves like φ−1m , leading to a non-linear relation between φn and φm. The MIGS

properties have been the focus of many subsequent theoretical studies [39–45], and

became central ingredients in the interpretation of various experimental trends

of the SBH’s [46–52].

The SBH and the semiconductor ionicity. — In the attempt to relate φn to

a property of the bulk metal rather than to a property of its surface, Kurtin,

McGill, and Mead [53] introduced the slope parameter S = dφn/dXm, where


Xm is the electronegativity of the atoms forming the (elemental) metal.5 From

a compilation of experimental data for many semiconductors, they correlated

S with the semiconductor bond ionicity, as measured by the electronegativity

difference ∆X = Xanion − Xcation. Their results exhibit a sharp covalent-ionic

transition around ∆X ≈ 0.8 and saturation of the S parameter at S = 1 for ionic

systems. Schluter [54] reexamined the experimental data, found large scattering

in the values, but no evidence of a sharp transition or saturation at S = 1, when

least-square fits were performed. Using the empirical linear relationship [55]:

φn = AXm + B, with A = 2.27 eV and B = 0.34 eV, he found, instead, good

agreement between the experimental data and the MIGS-based model [37, 39]:

S =A

1 + 4πe2Ds(EF) (δs/εs + δm), (1.13)

for the cubic semiconductors for which Ds and δs had been computed [39]. For

the mostly covalent semiconductors, such as Ge, Si, GaAs, and GaP, S is in the

range 0.1–0.4 eV, whereas for ionic systems, such as Al2O3 and SiO2, S is in the

range 1–2 eV.

The SBH and the interface chemical bonds. — It has also been argued that it

is the formation of chemical bonds across the interface which determines the

SBH. Andrews and Phillips [56], for example, correlated the SBH at transition-

metal/silicon interfaces with the heat of formation −∆Hf of the transition-metal

silicide which forms at the contact. The idea is that the charge transfer determin-

ing the SBH is controlled by the degree of hybridization between the sp3 Si states

and the transition-metal d states, a hybridization which in turn is proportional

to −∆Hf . In the same spirit, Brillson proposed to analyze the slope parameter S

in terms of the heat of formation of the semiconductor [57], since it is a measure

of the chemical reactivity at the interface.

The charge neutrality level. — An innovative proposal was put forth by Tejedor,

Flores, and Louis [43]. They pointed out that the semiconductor actually be-

haves like a metal in the interface region, because of the continuum of MIGS in

the bandgap. A charge neutrality level (CNL) of the semiconductor is introduced,

5The work function of a metal involves an electrostatic contribution specifically attached tothe metal/vacuum interface, the so called surface dipole, which is due to the spreading of thewave functions into the vacuum region, and which should be strongly reduced at a MS interface.The “bulk” contribution to φm can be related to the atom electronegativity Xm. Correlatingthe SBH with Xm rather than with φm amounts to eliminating the undesirable contribution ofthe surface dipole.


similar to the Fermi level of the metal, below which the MIGS must be occupied

for the interface region to be locally neutral. Any deviation from neutrality will

then be screened over a very short distance. In Ref. [43], the CNL is calculated

within a jellium/2-band-semiconductor model (leading to a CNL at midgap po-

sition), and an expression similar to Eq. (1.11) is derived for the SBH.6 In the

same spirit, Tersoff [58] suggests that the CNL is located close to the branch point

EB of the bulk semiconductor band structure (see Ref. [58]). Unlike the FL, EB

is an orientation-dependent quantity. In the case of perfect screening, the CNL

lines up with the metal FL, and the SBH takes the same value for all metals on a

given semiconductor. If the MIGS density is not sufficiently high, the screening

is incomplete, and the SBH changes from one metal to another, in a way which

depends on the semiconductor dielectric constant [59]. Tersoff proposed an easy-

to-use approximation for the branch point [59], which gives in the limit of perfect

screening:

φn = Eg −EB ≈ Eg − 1

2

(Ei

g −∆so

3

), (1.14)

where Eig is the minimum indirect gap and ∆so is the valence-band edge spin-

orbit splitting. Other models for EB, based on tight-binding parameters, also

exist [60].

It should be noted that, although simple and elegant, the concept of a bulk

semiconductor CNL — as it is generally used — which lines up with the metal

FL, considerably oversimplifies the SBH problem. First, the CNL is a property

of the interface electronic structure, which results from the complex interaction

between the metal and the semiconductor (see the discussion on the Al/Si and

Na/Si systems in Ref. [43]). Secondly, the principle of lining up the metal FL

and the semiconductor CNL relies on the assumption of metallic-like screening

on both sides of the interface, which is not very realistic, as we will see, on the

semiconductor side (see also Ref. [61]).

1.3.2 Fermi-level pinning by surface or interface states

After the work of Bardeen, it was soon recognized that intrinsic surface states

of the semiconductor could not satisfactorily explain the observed pinning of the

FL at metallic contacts to covalent semiconductors. In particular, it is known

that there are no such states in the bandgap at the (110) surface of GaAs [62],

but metal contacts to GaAs (110) still exhibit FL pinning. Therefore, extrinsic

localized states were suggested. For III-V compounds, Spicer and coworkers [63]

6The model of Ref. [43] includes an additional surface-dipole correction, DJ , to φm − χs.


proposed that the heat of adsorption of metal atoms on the surface of compound

semiconductors, followed by the segregation of anions in the metal, causes the

formation of antisite defects at the interface. The cation-on-anion antisites give

rise to acceptor states in the bandgap, which are believed to pin the FL. It has

been shown that a density of about 1014 defects states per cm2 per eV is necessary

to achieve FL pinning [64]. There appears to be, however, no direct evidence of

such high density of antisite states [65]. Other mechanisms of defect formation

at MS interfaces have also been proposed [66]. For a recent critical analysis of

“interface-state models”, see Ref. [67].

It has also been argued [68] that at contacts to III-V and II-VI semiconductors ex-

cess anions are present at the interface, and that the Schottky-Mott rule remains

valid for these systems, provided that one replaces the metal work function φm

in Eq. (1.10) by an effective anion work function. More recently, the formation of

the SBH at very low metallic coverage has been studied experimentally [69, 70].

Such studies generally lead to the conclusion that the Schottky barrier formation

results from several different and complementary mechanisms.7

1.3.3 Discussion

As mentioned earlier, the Schottky-Mott rule is approximately valid for contacts

to the most ionic semiconductors, but it fails in the case of covalent IV-IV or

nearly covalent III-V semiconductors, where φn turns out to be rather insensitive,

in general, to the metal used and to the contact fabrication method, in junctions

of practical interest. These phenomena receive a natural explanation in terms of

local neutrality and screening by “gap states” of any displacement of the FL at the

interface.8 The mechanism of this “screening” is explicit in the models of Bardeen

and Heine, where any modification of the FL due to a change of φm charges the

surface states or MIGS, and gives rise to a dipole layer acting against the FL

change. In this context, the slope parameter γ measures the ability of surface

states or MIGS to “screen” a displacement of the FL in the semiconductor gap

7The conclusions, however, of photoemission studies performed before 1990 should be recon-sidered in view of the photovoltaic effect [71].

8The term screening is often used in the MS literature in reference to the fact that theinterface electronic structure tends to inhibit any movement of the FL in the semiconductorbandgap induced by a change of the metal work function φm. However, a change of φm cannotbe identified with an external perturbation, since it involves the response of the electrons atthe free metal surface. In the following, we reserve the term screening for the response of theelectrons to an external perturbation (a change of ion cores, for instance), and we use quoteswhen “screening” is used in reference to other types of perturbations such as changes of thework function.


induced by a change in the work function. If γ = 0, the “screening” is complete,

the FL is pinned, and the SBH will not reflect any change of the metal work

function.

The questions then arising are where in the bandgap does this FL pinning oc-

cur and is the pinning position determined solely by the bulk properties of the

semiconductor and the metal or can it be changed by altering interface-specific

features. According to Bardeen, the pinning position is a property of the semicon-

ductor surface, while following Tersoff’s model, it is an intrinsic property of the

bulk semiconductor band structure. In the former case, a change in the atomic

structure of the surface would result in a different pinning energy, while if the SBH

is strictly a bulk property, there is no way to modify it by altering the properties of

the surface. As will be shown in this work, changes of the semiconductor-surface

chemical composition at the Al/GaAs (100) contact induce significant changes

of the SBH, showing that the latter clearly is not just a bulk property. The

conclusion that, for a given metal contact on a covalent semiconductor, the SBH

does depend on the atomic structure of the semiconductor surface (orientation,

reconstruction, or surface termination) has been reached by many authors, both

on experimental [72–76] and theoretical [14–17, 76–81] grounds. While opening a

promising line of research on SBH tuning, these observations complicate seriously

the search for simple models of Schottky barrier formation, since the inclusion of

the interface atomic structure in the models seems unavoidable.

The semiempirical approaches described in section 1.3.1 attempt to model the

electronic charge and the density of states at the interface, based on free sur-

faces (Mott), defect states (Bardeen and other defect models), MIGS (Heine,

Tejedor-Flores-Louis, Tersoff), or charge transfer within chemical bonds across

the interface (Andrews-Phillips, Brillson). These models all focus on a particular

aspect of the interface density of states, providing a simple picture of the SBH

formation and neglecting other features of the spectrum. At variance, ab initio

calculations can provide a full quantum-mechanical description of the interface

charge density and local density of states, and allow systematic studies of the

effects of changes in the interface atomic structure on the electronic structure.

These calculations can thus be used to assess the relative importance of different

mechanisms in determining the SBH.

Given the complexity and variety of the atomic structures of MS contacts, and

the sensitivity of the electronic structure to atomic relaxations, chemical bonding,

defects, etc., it seems unlikely that a simple model covering the entire issue of

SBH formation could emerge. Instead, a step-by-step investigation of the problem

based on first-principle methods, starting from abrupt, defect-free interfaces, and


progressively introducing perturbations at the interface, can help in identifying

relevant physical mechanisms, and provide a firmer basis for modeling Schottky

barrier properties.

Chapter 2

Ab-initio methods

The questions raised by the models outlined in section 1.3.1, and the limited

success they have met in predicting the SBH, show the necessity to adopt a more

rigorous quantum-mechanical description of the interface electronic properties,

which treats the microscopic structure of the contact and the electronic structure

of the constituent materials on the same footing. In this chapter, we describe the

ab initio methods that we have used, and show how the SBH and other electronic

properties of the MS interfaces can be derived from ab initio calculations.

2.1 Theory

2.1.1 Density Functional Theory

The precise prediction of the ground-state properties of many-electron systems

has become possible with the advent of Density Functional Theory (DFT) [82]

and the derivation of the Kohn-Sham equations [83]. DFT generalizes the early

attempt by Thomas and Fermi to directly relate the ground-state density n(r)

of N interacting electrons to the external potential u(r) in which these electrons

evolve, without calculating the N -particle correlated ground-state wave function

Ψ. It is the merit of the Hohenberg-Kohn (HK) theorem [82], with its various

extensions [84–86], to prove the existence of a one-to-one correspondence between

the ground-state density n(r) and the external potential u(r). Moreover, HK

demonstrate that the ground state itself is uniquely defined by the density, and

hence the ground-state expectation value of any observable can, in principle, be

evaluated as a functional of the density alone. In particular, the N -electron

ground-state kinetic energy, 〈T 〉 = 〈Ψ | T |Ψ〉 = 〈Ψ[n] | T |Ψ[n]〉 = T [n], and the

24 Ab-initio methods Chap. 2

electron-electron Coulomb energy, 〈W 〉 = W [n], are two universal functionals1

of the density. Following HK, given an external potential u, the corresponding

ground-state density, n, and energy, E, of the interacting N -electron system

minimize the functional

Eu[n] = T [n] + W [n] +

∫u(r) n(r) dr, (2.1)

namely

E = Eu[n] < Eu[n′] ∀ n′ 6= n, (2.2)

where n′ is the ground-state density of the N electrons in any other potential

u′(r) 6= u(r) + cst. E and n can thus be determined, in principle, from the

variational condition

δ

δn(r)

Eu[n]− ζ

∫n(r) dr

= 0, (2.3)

where the second term in the curly brackets ensures the normalization of n(r)

and ζ is a Lagrange multiplier. The HK theorem solely proves the existence of

Eu[n] and its variational property with respect to the ground-state density, with-

out giving explicitly T [n] and W [n]. Unfortunately, the functional Eu[n] is not

known exactly; the major difficulty lies in the representation of the kinetic-energy

functional T [n]. The most successful approach introduced to circumvent this dif-

ficulty has been proposed by Kohn and Sham [83], who exploited an independent-

electron scheme to evaluate T [n].

The basic idea of the Kohn-Sham (KS) approach is to associate to the system of N

interacting electrons evolving in an external potential u(r) and characterized by

a ground-state density n(r), an auxiliary system of N non-interacting electrons,

evolving in an effective external potential us(r), and having the same ground-state

density n(r). On this basis, KS showed that n(r) can be obtained by solving a

set of one-particle Schrodinger equations (the Kohn-Sham equations):− ~2

2m∇2 + us(r)

ϕi(r) = εi ϕi(r), (2.4)

where the whole complexity of the many-particle problem is hidden in the density-

dependent one-particle effective potential us(r) (whose explicit form will be given

later). The exact ground-state density is obtained in terms of the N lowest-energy

solutions of Eq. (2.4) from:

n(r) =

N∑i=1

|ϕi(r)|2. (2.5)

1In the sense that these functionals are independent of the external potential u(r).

Sect. 2.1 Theory 25

The central assertion behind the KS approach is that an exact mapping of the

interacting problem onto a non-interacting problem is always possible. The valid-

ity of this assertion has been investigated in detail in Ref. [84]. We will follow the

authors of Ref. [84], and assume, for all external potentials u(r), the existence

of an effective potential us(r), which yields the exact density of the interact-

ing electrons as the solution of the non-interacting-electron problem. If us(r) is

given, the ground state of the independent-electron system is the Slater deter-

minant constructed from the N lowest-energy eigenstates of the single-particle

Hamiltonian in Eq. (2.4),2 and the ground-state density is given by Eq. (2.5).

Assuming that us(r) is given, one can apply the HK theorem to the system of

independent electrons: the ground-state density of the non-interacting electrons

minimizes the energy functional Eus [n] = Ts[n] +∫

us(r) n(r) dr, where Ts[n] is

the universal kinetic-energy functional for the independent electrons. Thus, the

ground-state density verifies the variational equation

δTs[n]

δn(r)+ us(r)− ζs = 0, (2.6)

where ζs is a Lagrange multiplier. On the other hand, the HK energy functional

of the interacting electrons can be rewritten as

Eu[n] = Ts[n] +

∫ u(r) + 1

2vH(r)

n(r) dr + Exc[n], (2.7)

where vH(r) = e2∫ n(r′) dr′

|r−r′| is the classical Hartree potential, and the remaining

term, Exc, is called the exchange-correlation (xc) energy functional and reads:

Exc[n] = T [n] − Ts[n] + W [n] − 12

∫vH(r) n(r) dr. The variational equation,

Eq. (2.3), for the ground-state density of the interacting electrons becomes

δTs[n]

δn(r)+ u(r) + vH(r) +

δExc[n]

δn(r)− ζ = 0. (2.8)

Since, by hypothesis, the ground-state densities are identical for the interacting

and non-interacting electrons, a unique n verifies at the same time Eqs. (2.6) and

(2.8), and δTs[n]δn(r)

can be eliminated between the two equations.3 Neglecting the

trivial constant, the KS effective potential is thus

us(r) = u(r) + vH(r) +δExc[n]

δn(r). (2.9)

2This is true only if the ground state of the independent-electron system is not degenerate.The more general case of a degenerate ground state is treated in Ref. [84].

3From a rigorous point of view, the functional Ts[n] and its functional derivatives are definedonly for non-interacting-electron densities. The applicability of Eqs. (2.6)–(2.8) to interacting-electron densities requires a proof that Ts[n] and δTs[n]/δn(r) exist also for this class of densities.This proof is the gist in demonstrating the existence of us(r), as suggested by Eq. (2.6) [84].


Formally, the xc energy involves both kinetic and Coulomb contributions. Practi-

cally, however, the functional forms of T −Ts and W are unknown, and Exc must

be approximated. In their original paper [83], KS proposed the local-density ap-

proximation (LDA) to Exc, which remains the most widely used at present in

solid-state computations. In the LDA, the functional form of Exc[n] is simply:

ELDAxc [n] =

∫vh

xc

(η = n(r)

)n(r) dr, (2.10)

where vhxc(η) is the xc energy per particle, for a homogeneous electron gas of

density η. Very accurate quantum-Monte-Carlo calculations have been performed

to evaluate the function vhxc(η) numerically. In this work, we use the results for

vhxc(η) of Ceperley and Alder [87], as parameterized by Perdew and Zunger [88].

Within the LDA, the xc potential is then

δELDAxc [n]

δn(r)≡ vLDA

xc (r) =

[d

dη

(η vh

xc(η))]

n(r)

. (2.11)

Eqs. (2.9) and (2.11) show that the KS effective potential, us, depends on n

through the Hartree and xc terms. Therefore, the independent-electron problem,

Eqs. (2.4) and (2.5), must be solved self-consistently by an iterative procedure,

starting with an initial guess for n(r). Once the self-consistent density and or-

bitals are known, the LDA ground-state energy follows from Eqs. (2.7) and (2.10),

ELDA = Ts[n] +

∫ u(r) + 1

2vH(r) + vh

xc(n(r))

n(r) dr, (2.12)

where the kinetic energy of the fictitious independent electrons is

Ts[n] = − ~2

2m

N∑i=1

〈ϕi | ∇2 |ϕi〉. (2.13)

The independent-electron scheme developed by Kohn and Sham has the advan-

tage of giving accurate ground-state energies, thanks to a good description of

the kinetic energy. Physically, the orbital picture remains the most convenient

way to discuss and interpret electronic phenomena in solids; it is therefore very

satisfactory that the LDA leads to a local mean-field potential, vLDAxc (r), which

accounts for most of the exchange and correlation effects. The LDA is exact only

in the limit of a constant density, but has proven to be a good approximation

even for strongly inhomogeneous systems such as atoms and molecules [84].

2.1.2 Solution of the Kohn-Sham equations

The plane-wave expansion. — For the practical implementation of the KS scheme,

it is often convenient to use periodic-boundary conditions (PBC) [89]. Such

Sect. 2.1 Theory 27

boundary conditions are natural for crystalline systems and also convenient for

other condensed-matter systems. The Bloch wave functions may then be ex-

panded in a plane-wave basis, 〈r|k + G〉 = 1√Ωc

exp i (k + G) r, where the

principal quantum number k, which expresses the translational invariance of the

Hamiltonian, lies in the first Brillouin zone (BZ), and G is a vector of the recip-

rocal space. Ωc is the volume of the unit cell. For each k, the diagonalization of

the Hamiltonian matrix, HGG′(k) = 〈k+G | − ~2

2m∇2 +us |k+G′〉, yields the KS

states ϕki(r), where i is the band index. These states are used to recalculate n(r)

and us(r) until the self-consistency is achieved. Using PBC, Eq. (2.5) becomes

(in the thermodynamic limit),

n(r) =Ωc

(2π)3

∫BZ

dk

∞∑i=1

fki |ϕki(r)|2. (2.14)

The occupation numbers fki account for the spin degeneracy: fki = 2, i 6 N2

and

fki = 0, i > N2, where N is the number of electrons in the unit cell. Knowing the

self-consistent density, one can calculate the total energy,

Etot = ELDA + Elattice, (2.15)

where ELDA is given by Eq. (2.12), and Elattice stands for the electrostatic energy of

the lattice. In infinite periodic systems, the total energy involves three divergent

terms, namely the positive nucleus-nucleus and electron-electron energies, and

the negative electron-nucleus energy. The proper summation of these terms can

be done, if the unit cell is neutral and has no dipole moment [90], and yields a

well defined total energy per unit cell.

In numerical calculations, the plane-wave expansion of the wave functions ϕki(r)

has to be truncated, and the BZ integration in Eq. (2.14) has to be discretized.

The truncation of the plane-wave expansion is characterized by a kinetic-energy

cutoff Ecut; only plane waves |k + G〉 with

~2

2m|k + G|2 6 Ecut (2.16)

are included in the basis set.4 For the discrete BZ summation, a set of special

k points is employed, which are generated in the spirit of the Baldereschi-point

technique [91]. In this work, we use the BZ sampling proposed by Monkhorst and

Pack [92].

4Since the density is given by the square modulus of the wave functions, the effective cutoffin the Fourier expansion of the density is 4Ecut.


The pseudopotential approximation. — To account for the rapid oscillations of the

wave functions close to the nuclei, which result from the orthogonality constraint

between the wave functions, a very high cutoff is needed. Unfortunately, the size

of the Hamiltonian matrix, and hence the computer time and memory required

for the calculation, increase rapidly with the number of coefficients retained in

the plane-wave expansion. In addition, for a given cutoff energy, the number of

these coefficients increases like the third power of the unit-cell size. Therefore,

calculations involving large unit cells become impractical. This difficulty can be

overcome using ab initio pseudopotentials [93]. The basic idea is to freeze the core

orbitals in their atomic configuration, and to replace the Coulomb potentials of

the nuclei by effective potentials, which yield the correct wave functions for the

valence electrons, outside a small “core radius” rc. This approximation is justified

because the core electrons are not involved in the chemical bonding between

atoms, so that the core states do not change significantly with the chemical

environment of the atom. The pseudopotential methodology in solid-state physics

has been reviewed by Pickett [94].

There is no unique prescription to determine the pseudopotentials, and this ar-

bitrariness has been exploited to construct pseudopotentials with different char-

acteristics. We use the ab initio pseudopotentials of Troullier and Martins [95]

and of Bachelet, Hamann, and Schluter [96]. These pseudopotentials have the

following properties: (i) the atomic pseudo-wave functions exactly match the

true all-electron wave functions5 outside a core radius rc; (ii) the pseudo-wave

functions are nodeless; (iii) the amount of charge enclosed within rc is the same

for the true and pseudo wave functions (norm conservation). The properties (i)

and (iii) imply that: (iv) the eigenvalues are identical for the two wave func-

tions. Moreover, the Troullier-Martins pseudopotentials are “soft”, i.e., they can

be accurately represented with a relatively small number of plane waves. These

pseudopotentials are semi-local — local in the radial coordinate and non-local

in the angular coordinates — meaning that each angular-momentum component

of the pseudo wave function sees a different potential; to calculate their matrix

elements in the plane-wave representation, one has to evaluate a separate integral

for each couple (G, G′) of reciprocal-lattice vectors. Kleinman and Bylander [97]

pointed out that the number of these integrals can be significantly reduced, if

one splits the pseudopotentials uips of each atom i into a local part and a fully

non-local part (non-local also in the radial coordinate). The pseudopotentials

5By “true all-electron wave functions”, we mean the single-electron wave functions calculatedwithin the KS scheme, taking all core electrons into account.

Sect. 2.1 Theory 29

therefore read:

uips = ui

loc +lmax∑l=0

l∑m=−l

|ξilm〉Ei

l 〈ξilm|, (2.17)

where the local potential uiloc(r) is spherical, regular for r < rc, and behaves

as uiloc(r) = −e2Zi

rfor r > rc; the nonlocal projector is defined according to

〈r|ξilm〉 = ξi

l(r) Ylm(θ, φ), where the functions ξil (r) vanish rapidly for r > rc and

Ylm(θ, φ) are the spherical harmonics. Here, Zi is the number of valence electrons

for the atom i. The pseudopotentials in the Kleinman-Bylander form can be

factorized in a plane-wave representation, and solely one integral per G vector is

required. Since the HK theorem, and consequently the KS equations, have been

established only for local external potentials, the theory must be generalized if one

uses non-local pseudopotentials. This point is discussed in Ref. [94]. The general

expression for the LDA total energy, appropriate for non-local pseudopotentials,

can be found in Refs. [98, 99].

From now on, the density n(r) must be understood as the density of the (pseudo)

valence electrons. For the atoms which have a full d shell, such as the Zn and

Ga atoms, the d electrons are generally included in the core. In certain cases,

this leads to some errors in the calculated structural and electronic properties of

the solids [100, 101]. One can partially correct these errors, while keeping the d

electrons in the ionic core, by means of a non-linear core correction (NLCC) [102],

which we shortly describe in Appendix A. For some calculations, we use virtual

atoms, which mimic the average electronic properties of compounds or alloys; they

are obtained from the weighted average of the pseudopotentials of two atoms A

and B, and they can have a fractional number of valence electrons. We denote a

virtual atom by the symbol 〈A1−ϑBϑ〉, where ϑ (1−ϑ) is the weight of the B (A)

pseudopotential. The relevant parameters for the pseudopotentials employed in

this study are given together in Appendix A.

Forces and stresses. — In the calculation of the electronic ground-state density,

the positions of the (pseudo) ions and the shape and volume of the unit cell are

frozen in a given static configuration. However, if this configuration does not cor-

respond to an equilibrium structure, the ions are subject to non-vanishing forces

and/or the unit cell is subject to finite stresses. If the constraints on the ions and

unit cell are relaxed, and variational degrees of freedom are given to the atomic

positions and unit-cell geometry, the structure will evolve toward a configuration

where the stresses and forces vanish. The stress tensor σαβ is a property of the

electronic ground state, and it can be evaluated within DFT [103]. By defini-

tion, the force F i on the atom at the position Ri is the negative derivative of

the total energy Etot with respect to Ri. The appropriate expressions for the


forces, in the pseudopotential approach, are derived in Ref. [99]. To determine

the lowest-energy configuration, we incorporate the forces in a gradient proce-

dure, to minimize the total energy with respect to the ionic positions. In infinite

periodic systems, the fixed geometry of the elementary cell can still induce finite

stresses, even if no force acts on the atoms. Therefore, one has to optimize the

cell parameters as well, in order to determine the equilibrium structure.

2.2 Application to metal/semiconductor contacts

The band bending at a MS interface, which may be used to define the interface

region in transport theories, is generally established over a distance which is of

the order of 102 to 103 A. The SBH, however, is an intrinsic interface property,

independent of the band bending, and is established over a much shorter length

scale, of the order of 10 A. This fact reduces drastically the number of atomic

layers we need to consider in our ab initio calculations to study the electronic

structure of the MS junction.

2.2.1 Supercells

In order to take advantage of the reciprocal-space and plane-wave formalisms,

we represent the MS junction by a sequence of metal and semiconductor slabs,

which are repeated periodically in the direction normal to the interface (thereafter

the x direction). In the (y, z) planes, the atoms are also arranged on a periodic

pattern, whose lateral size depends on the atomic structure at the interface.

We built thus a superlattice with an artificial 3-dimensional periodicity, and the

band-structure and total-energy methods developed for pure bulk crystals can

be applied therefore to the interface problem without any modification. Fig. 2.1

shows a typical superlattice, with 3 planes of an fcc metal alternating with 5

Metal Semiconductor Metal

Figure 2.1 The super-cell technique. An artificialsolid with a 3-dimensional pe-riodicity is constructed to de-scribe the MS junction. Onlythe highlighted atoms are in-cluded in the supercell calcu-lation, the others being equi-valent by symmetry.

Sect. 2.2 Application to metal/semiconductor contacts 31

planes of a zinc-blende semiconductor in the (100) direction. The box underlines

the elementary cell, which is referred to as supercell. In each supercell, there are

two MS interfaces, which must be sufficiently distant from each other in order not

to interact. This is not the case, in particular, of the 3+5 supercell illustrated in

Fig. 2.1. In this work, we use supercells with 7 to 13 planes of metal and 13 to

21 planes of semiconductor, depending on the properties to be investigated. For

most calculations, and unless otherwise specified, we use the 7 + 13 supercell.

2.2.2 Interface Brillouin zone and level broadening

The width of the BZ in the kx direction is 2π/L, where L is the total length

of the supercell. In the limit of a large L, which corresponds to the real MS

junction, 2π/L vanishes and the BZ is therefore 2-dimensional. Fig. 2.2 shows

the direct and reciprocal lattices in the (y, z) and (ky, kz) planes, respectively, for

the geometry of Fig. 2.1. Because of the underlying zinc-blende structure, the

infinite MS junction has only a 2-fold symmetry in the (y, z) plane. However, as

can be seen from Fig. 2.1, there are (non-symmorphic) 4-fold improper rotation

symmetries (S4) in the supercell, due to the artificial periodicity in the x direction,

which are absent in the infinite junction, and lead to a smaller irreducible BZ [see

Fig. 2.2(b)]. Making use of the symmetry, the integration in Eq. (2.14) can be

restricted to the irreducible BZ. Our typical sets of special k points for the discrete

summation over the irreducible BZ are collected in Appendix A.

As a whole, the supercell is metallic; it has a complicated Fermi surface, and a

very dense sampling of the BZ is needed, in principle, in order to obtain a precise

determination of the supercell Fermi level, EF. The convergence with respect to

the number of k points can be improved significantly by allowing for a Gaussian

occupation of the energy levels [104, 105]:

fki =2

∆√

2π

∫ EF

−∞exp

−1

2

(ε− εki

∆

)2

dε, (2.18)

where the factor 2 accounts for the spin degeneracy. We use a typical Gaussian

broadening width ∆ = 0.2 eV. With this value, the error on the total energy due

to the Gaussian broadening scheme [106] is less than 0.02 eV/atom.

2.2.3 Calculation of the Schottky barrier height

The calculation scheme for the SBH φp is illustrated in Fig. 2.3, where we present

the calculated energy diagram for the band alignment at the Al/GaAs (100) MS


bybz

as

2π

Γ

K

JJ’

as

ayaz

a m

ayaz

(a)

am

2π(b) (c)

Γ

JJ’

K

xk

Figure 2.2 (a) Basis vectors, ay and az, of the direct lattice in the plane parallel tothe interface. The gray (white) circles indicate the sites of the metal (semiconductor) fcclattices. The two lattices have been shifted for clarity. The dashed squares show the fcccubic cells in the metal and in the semiconductor, respectively, and the gray square is theelementary cell. am and as are the lattice parameters for the metal and semiconductor,respectively. (b) The corresponding basis vectors of the reciprocal space, by and bz,and the 2-dimensional BZ (gray square) with its irreducible edge (hatched). The pointsof high symmetry are also indicated. (c) Position of the irreducible edge of the 2-dimensional BZ, in the whole BZ of the semiconductor fcc lattice.

contact. We note that a sketch of the energy diagram in the presence of band

bending was also presented in Fig. 1.1(a). The band bending associated with the

depletion zone in the semiconductor, as represented in Fig. 1.1, becomes negligible

on the scale of Fig. 2.3, which involves only a few atomic layers on both sides of

the junction. In Fig. 2.3, the average electrostatic potential (to be defined more

precisely below), denoted V (x), takes two different constant values in the bulk

metal and semiconductor materials, and the difference between these two values,

∆V , is called the electrostatic-potential lineup. According to Fig. 2.3, the p-type

Schottky barrier height can be written as

φp = ∆Ep + ∆V, (2.19)

where the band term, ∆Ep = εAl−εGaAs, is the difference between the Fermi level

in the metal and the valence-band maximum (VBM) in the semiconductor, each

measured with respect to the average electrostatic potential in the corresponding

crystal. This term can be obtained from standard bulk band-structure calcula-


Figure 2.3 Band alignment atthe Al/GaAs (100) junction. Onboth sides of the interface, themacroscopic average V (x) of theelectrostatic potential tends to con-stant values, which are taken as theenergy references to align the bandenergies, εAl and εGaAs, and to de-termine the p-type Schottky barrierheight φp. The potential lineup,∆V , and the GaAs bandgap, Eg, arealso indicated.

0

1

2

3

En

ergy

(eV

)

7

8

9

φp

∆V

εGaAsεAl

V(x)

E g

Al As Ga

tions for each of the two materials forming the junction, and is independent of

the interface structure and chemistry. The potential lineup ∆V contains all the

interface-specific features, and is an outcome of the supercell calculation.

We apply the planar and macroscopic average techniques [107, 108] to compute

∆V . The physical quantities f(r) of interest — the electrostatic potential V (r)

and the charge density, which are related by Poisson’s equation — are periodic in

the (y, z) planes. To discuss the x-dependence of such quantities, we introduce

the planar average f(x) = 1S

∫S

f(r) dy dz, which is not a periodic function of

x at the interface, but reproduces the bulk periodic features of the metal and

semiconductor far from the interface. To get rid of the atomic-scale oscillations

of f(x) in the two bulk materials, and to highlight the interface-specific features,

we follow Ref. [107] and define the macroscopic average of f as

f(x) =

∫f(x′) g(x′ − x) dx′, (2.20)

where g(x) is a suitable filter function. For MS junctions, which are normally

lattice-mismatched, the periods Lm and Ls of f(x) in the bulk metal and semi-

conductor, respectively, are different, and g(x) must be a convolution of two

filter functions gm(x) and gs(x) appropriate for each material. Using the filter

functions gm, s(x) = 1/Lm, s Θ(

12Lm, s − |x|

), where Θ(x) is the step function, the

macroscopic average of f becomes:

f(x) =1

LmLs

∫ x+ 12Lm

x− 12Lm

dx′∫ x′+ 1

2Ls

x′− 12Ls

dx′′ f(x′′). (2.21)

Applied to the total electrostatic potential V (r), the macroscopic average tech-

nique yields the average electrostatic potential V (x), which provides the energy


reference in the metal and semiconductor materials to align the bulk energies

(Fig. 2.3). If xm and xs are the centers of the metal and semiconductor slabs, the

potential lineup reads: ∆V = V (xm)− V (xs).

In this work, we use as reference potential for the lineup the total electrostatic

potential (ion-point charge plus Hartree):

V (r) = e2∑

i

−Zi

|r −Ri| + e2

∫n(r′) dr′

|r − r′|= vion(r) + vH(r), (2.22)

where the Ri’s are the ionic positions. We note, however, that other conventions

about ∆V and ∆Ep can be found in the literature. For instance, the total local

potential, Vloc(r) =∑

i uiloc(|r −Ri|) + vH(r) + vLDA

xc (r), could also be used as

reference potential to define the potential lineup and the corresponding band-

structure terms. The difference between V (r) and Vloc(r) contains only short-

range terms, which have a well defined average in the separated bulks. These

terms can therefore be included either in the potential lineup or in the band-

structure term. The inclusion of these terms in ∆V or in ∆Ep is actually a

matter of taste; the only constraint is that all long-range contributions of vion

and vH must be included in ∆V [109].

2.2.4 Other electronic properties

The local density of states. — The details of the electronic structure at MS

interfaces are best discussed in terms of the local density of states (LDOS), which

combines the energetic and spatial resolutions. In the language of the PBC, the

LDOS is formally defined as

D(E; r) =2Ωc

(2π)3

∫BZ

dk∞∑i=1

|ϕki(r)|2 δ (E − εki) , (2.23)

where the factor 2 is for the spin. For practical calculations, the integration over

the constant-energy surface, picked out by the δ function in Eq. (2.23), is obtained

by a discrete BZ summation, and by replacing the δ function by a Gaussian of

width ∆. The planar and macroscopic averages can then be applied to D. If

the supercell size is sufficient, the macroscopic average of the LDOS, evaluated

at the center of the metal slab, D(E; xm), and semiconductor slab, D(E; xs),

reproduces the bulk densities of states (DOS) of the metal and semiconductor,

respectively. In the supercell calculation, these two bulk DOS are automatically

aligned one with respect to the other. This provides an alternate way of determin-

ing the SBH, i.e., one can extract the position Ev of the VBM from the average


LDOS D(E; xs) in the semiconductor slab, and knowing the position of the su-

percell Fermi energy EF, one can then obtain the p-type SBH as EF − Ev. This

second method, however, is generally less reliable than the first method described

in section 2.2.3, both because the determination of the VBM from D(E; xs) is

somewhat approximate, and because the convergence of EF with supercell size,

BZ summation, and kinetic-energy cutoff is slower than that of the potential

lineup ∆V .

From the local density of states, one can also compute the surface density of

states (see, e.g., Heine’s model, p. 17)

Ds(E; r) =

∫ ∞

x

D(E; x′, y, z) dx′, (2.24)

for energies within the semiconductor gap, where D(E; x′, y, z) vanishes expo-

nentially for large x′. In our supercell calculations, the upper limit of the integral

for Ds is taken as xs, i.e., the middle of the semiconductor slab.

The interface formation energy. — To compare the relative stability of different

MS interfaces, the relevant quantity is the formation energy, defined as

εf = G−∑

α

nαµα ≈ Etot −∑

α

nαµα, (2.25)

where G = Etot+PV −TS is the Gibbs free energy, nα is the number of atoms and

µα the chemical potential for each atomic specie α in the system. By definition,

the chemical potential µα is the derivative of the Gibbs free energy with respect to

the number of atoms nα. As we are considering here condensed matter systems,

the pressure term PV is completely negligible for the pressures we are interested

in (ambient or low pressure), and is discarded. We also ignore the temperature

dependence of the free energy. The temperature-dependent terms tend to cancel

out in the free-energy difference, between condensed matter phases, which yields

εf , and their contribution is expected to be relatively small [110] (of the order or

less than kT per surface atom). In this study we will thus use the T = 0 values of

G (Etot) and µα. When applying the formula (2.25) to the supercell of Fig. 2.1,

one must take care of the facts that there are two equivalent interfaces in the

supercell, and that the numbers of anions and cations in the semiconductor are

different.6 For example, if there are nm elemental-metal atoms, ns cations and

ns + 1 anions in the supercell, the interface formation energy is:

εf =1

2(Etot − nmµm − nsµs − µanion) , (2.26)

6In the (100) direction, it is necessary to have an odd number of atomic planes in thesemiconductor slab, otherwise the two interfaces in each supercell are not equivalent.


where µm (µs) is the metal (semiconductor) chemical potential, which is equal, at

equilibrium, to the total energy per atom (anion-cation pair) of the bulk phase,

and µanion is the chemical potential of the anions, which can only vary between

well defined limits to be discussed further in section 4.1.5.

2.2.5 Many-body and spin-orbit corrections on the Schot-

tky barrier

The electrostatic-potential lineup ∆V is a ground-state property of the MS junc-

tion, which depends only on the position and nature of the atoms, and on the

electronic density; therefore, it can be determined exactly, in principle, in the KS

approach. The only intrinsic uncertainty on ∆V , in our calculations, comes from

the LDA approximation to the xc potential. The situation is different for the

band term ∆Ep in Eq. (2.19). The single-particle energies εki of the fictitious KS

non-interacting electrons are intrinsically different from the quasiparticle energies

measured experimentally, although the numerical discrepancy might be occasion-

ally very small in the valence bands. This problem leads to a systematic error

in the band term, which we calculate using the bulk KS energies rather than the

true quasiparticle energies [111]. In addition, for real semiconductors, the energy

bands are split at the VBM because of the spin-orbit interaction. Since we neglect

this interaction in our calculations based on scalar relativistic pseudopotentials, a

correction must be added to the calculated VBM of the semiconductor, εs. This

correction is simply +∆so

3, where ∆so is the total spin-orbit splitting, which we

take from experiment. For a meaningful comparison of our calculated SBH’s with

experimental data, ∆Ep must include the quasiparticle and spin-orbit corrections;

the corrected band term is thus

∆Ep = ∆EKSp + ∆εm −∆εs − ∆so

3, (2.27)

where ∆EKSp is the KS band term, and ∆εm(s) is the difference between the

quasiparticle and KS energies in the metal (semiconductor).

For metals, it has been demonstrated that the exact KS Fermi energy and the

quasiparticle Fermi energy coincide at zero temperature [84]. Within the LDA,

and even for systems in which the correlation plays an important role, the KS

Fermi energy usually agrees very well with experiment [84, 112]. For example,

calculations of the work functions of various Al surfaces, performed using the same

method and the same pseudopotentials as we do and neglecting the quasiparticle

correction on εm, yield values which agree with the experimental data to within

a few tenths of meV [113]. When using Eq. (2.27), we will therefore skip the


quasiparticle correction ∆εm on the metal FL. Techniques have been developed

in order to calculate precise quasiparticle energies in semiconductors within many-

body perturbation theory [114]. The typical uncertainty on the calculated ∆εs

is 0.1 eV. When available in the literature, we take the correction ∆εs from such

calculations.

2.2.6 Numerical parameters and precision

To evaluate the SBH of the Al/GaAs junctions, we generally use a 7+13 supercell

(7 planes of metal and 13 planes of semiconductor), a (2, 6, 6) k-points grid (see

Appendix A) with a Gaussian broadening width ∆ = 0.2 eV, and a 10 Ry cutoff

energy. Extensive convergence tests indicate that our numerical uncertainty on

the SBH, with this set of parameters, is ∼ 0.04 eV (Appendix A). Unless other-

wise specified, we have used the above parameters for the calculations reported in

the next chapters. Typical exceptions are as follows. While the 7 + 13 supercell

and the (2, 6, 6) grid turn out to be sufficient to converge the potential lineup

within 0.04 eV, a larger semiconductor slab and more k points in the irreductible

BZ are needed to obtain a converged LDOS. Therefore, to evaluate the local and

surface densities of states, we first calculate the self-consistent density in a 7+21

supercell with the (2, 6, 6) grid, and this density is then used to compute the KS

states on a finer (2, 12, 12) grid. Moreover, we have increased the cutoff up to

20 Ry for calculating the lattice relaxations, the interface formation energy, and

also when a detailed representation of the charge density is desired. In Chapter 5,

all the calculations for the systems involving ZnSe have been performed with a

20 Ry cutoff. The parameters used to compute the bulk band-structure energies

εs and εm are reported in Appendix A.

Concerning the sensitivity to the type of pseudopotentials, various theoretical

results reported in the literature for band offsets at semiconductor heterojunc-

tions [109] show typical variations of 0.05 eV when different pseudopotentials

are used. For metal work functions, however, the sensitivity can be of the order

of 0.1 eV [115]. For MS junctions, we therefore expect an uncertainty of about

0.1 eV due to the pseudopotentials.

Chapter 3

The abrupt epitaxial

Al/GaAs (100) interface

The existence of a good lattice matching between Al and GaAs, that gives rise

experimentally to quasi-epitaxial Al/GaAs (100) junctions [73], allows us to study

from first principles the electronic properties of a fully developed metal/semi-

conductor interface [81]. In this chapter, we describe our calculations for the ideal

(abrupt) epitaxial Al/GaAs (100) junction, and analyze our results on the charge

density, SBH, and LDOS in some details. We pay particular attention to the

MIGS, which will be a central ingredient for our discussions on Schottky barrier

tuning in Chapter 4. In the last section, we discuss our results in connection with

existing theories of SBH formation.

3.1 Interface atomic structure

Fig. 3.1(a) shows the epitaxial alignment of Al on GaAs (100) corresponding to

the lattice-matching condition aAl = aGaAs/√

2. The Al [100] direction is paral-

lel to the GaAs [100] axis, and the whole Al fcc lattice is rotated 45 about its

[100] axis with respect to the GaAs substrate. Experimentally, and also in our

calculations, the equilibrium lattice constant of Al is found to be slightly larger

(1%) than aGaAs/√

2. This results in a small compressive strain in the Al in-plane

lattice constant, which is accommodated by an elongation (∼ 3%) of the Al over-

layer, under pseudomorphic conditions. The polar Al/GaAs (100) interface offers

two inequivalent configurations, either with an As- or Ga-terminated GaAs sur-

face, as shown in Fig. 3.1(b). In the following, we will refer to the As-terminated

interface as the interface I and to the Ga-terminated one as the interface II. In

40 The abrupt epitaxial Al/GaAs (100) interface Chap. 3

AsGaAl(S)

Al(I)

I

II

(a) (b)

aGaAs

aAl

d[1

00]

[100

]

Figure 3.1 (a) Epitaxial alignment of Al on GaAs (100). (b) Atomic structure ofthe Al/GaAs (100) interface. The GaAs surface is terminated either by an As plane(I), or a Ga plane (II). In the Al overlayer, the atoms occupy either substitutional sitesin the continuation of the GaAs structure (Al(S)), or interstitial sites (Al(I)).

these heterostructures, the Al overlayer can be viewed as a superposition of two

sublattices. In one of the sublattices the Al atoms occupy substitutional sites in

the tetragonally distorted continuation of the bulk semiconductor structure (Al(S)

atoms), whereas in the other sublattice the Al atoms occupy interstitial (tetrahe-

dral) sites of the zincblende structure (Al(I) atoms), as indicated in Fig. 3.1(b).

Fig. 3.1(b) represents the central part of our Al/GaAs (100) supercells. The sur-

face of the supercell, in the plane parallel to the interface, is S = 12a2

GaAs; the

supercell thus contains one atom per plane in the GaAs slab, and two atoms per

plane in the Al slab. For the geometrical parameters of the supercell, we used the

calculated equilibrium lattice parameters of bulk GaAs and Al, aGaAs = 5.55 A

and aAl = 3.97 A, and evaluated the tetragonal distortion of the Al overlayer

(εx ≈ 3%) following macroscopic elasticity theory and using the calculated elas-

tic constants of Al, C11 = 120 GPa and C12 = 70 GPa. The interplanar distance

at the interface, d, was taken as the average between the interplanar distances

in the semiconductor and in the metal slabs: d = 1.72 A. Fig. 3.2 shows the

variation of the total energy when the distance d is varied, keeping all other ge-

ometrical parameters fixed. Under these conditions, the minimum energy occurs

at a distance d slightly larger (by ∼ 0.3 A) than the average distance d. We

found out, however, from fully relaxed calculations, that the average interplanar

distance d is the appropriate initial value which should be used, together with the

bulk interplanar distances in the GaAs and Al slabs, in order to correctly repro-

duce (to within 0.05 A) the equilibrium lattice parameter of the whole supercell

Sect. 3.2 Electronic charge density 41

0 2 4 6 8 10 12 14

−2.0

−1.5

−1.0

−0.5

0.0

II

I

d (Å)

(eV

)E

tot

21

Figure 3.2 Variation of the total energy as a function of the interplanar distanced at the interface, for the junctions I and II. The energy (per supercell) is divided bytwo since there are two interfaces in each supercell. The dotted line corresponds to theaverage between the theoretical Al and GaAs interplanar distances (taking into accountthe strain in Al), namely d = 1.72 A. The zero of energy has been fixed arbitrarily.

along the growth axis, when the atomic positions at the interface are relaxed (see

p. 63).

3.2 Electronic charge density

3.2.1 Comparison with the bulk densities

In Fig. 3.3(a), we compare the planar average of the electron density in the junc-

tion I to the planar-averaged densities of the bulk Al and GaAs crystals. We have

arbitrarily truncated the bulk densities halfway between the last Al and first GaAs

planes. Inspection of the figure indicates that the electron density in the junction

reproduces the bulk densities almost up to the last plane in the metal as well as

in the semiconductor. We also show in the figure (lower panel) the macroscopic

average of the difference between the interface and bulk charge densities. With

respect to the truncated bulks, there is an accumulation of electronic charge at

the interface, which exactly compensates the excess of ionic charge (not shown in

the figure), so that the interface is globally neutral. From the figure, one sees that

the region at the interface where the density of the junction significantly differs

from its bulk counterpart has a typical width of ∼ 5 A. The potential lineup ∆V

must therefore also be established over this length scale. Individual electronic

states, instead, generally extend over a wider region, before they recover a bulk-

like behavior. This is apparent in Fig. 3.3(b), where we show the total charge


density in the junction, and the contribution to the charge density of the states

with energies below the GaAs valence-band edge. The macroscopic average of

the difference, which corresponds to the electron density due to occupied MIGS,

is also displayed. These states yield an important contribution to the density in

the first 3 to 4 semiconductor planes, in such a way that they restore locally the

charge neutrality, as postulated by Tersoff [58]. The typical length involved here

is ∼ 10 A. The same type of behavior is observed in the junction II. The spatial

dependence of the LDOS and the decay of the MIGS in the semiconductor will

be discussed in more details in section 3.4.

0.0

0.5

1050−5

6

7

8

9

10

0.0

0.5

1050−5

6

7

8

9

10

∆n(x

)

(a) (b)

n(x

) (

Ω−1

)

Al As Ga Al As Ga

x (Å) x (Å)

Figure 3.3 Planar average of the valence-electron density in the Al/GaAs (100)As-terminated junction (solid line). (a) Comparison with the planar-averaged densitiesof the two bulk materials (dotted line) forming the junction. The lower panel showsthe macroscopic average of the difference between the interface and bulk charges, whichsomewhat depends on the choice of the interface plane, but vanishes rapidly on bothsides of the interface. (b) Contribution of the MIGS to the total electron density;the dotted line indicates the contribution of the states with energies below the GaAsvalence-band edge. The lower panel shows the contribution of the MIGS to the electrondensity. Ω is the unit-cell volume of bulk GaAs. The charge density in the junctionwas calculated in a 7 + 21 supercell.

Sect. 3.2 Electronic charge density 43

3.2.2 Discussion of the chemical bonds

In Fig. 3.4, we show contour plots of the total valence charge density in the abrupt

junctions I and II. To blow up the charge transfer responsible for the chemical

bonding in the junctions, we also show the differential charge density, obtained by

subtracting from the total density the superposition of atomic charge densities.

At the interface I, the Al(S) atom occupies a Ga site in the continuation of the

GaAs lattice, and this explains why it is strongly bonded to the interface As atom.

The bond, however, is weaker than in the bulk GaAs, because of the spreading

of the bonding charge toward the Al(I) atom. At the interface II, the Al(S) atom

may be viewed as an heterovalent substitution at an As site in the continuation

of the GaAs lattice. On the other hand, the Ga atom at the interface may also

be viewed as an isovalent substitution at an Al site in the continuation of the Al

fcc lattice. The Ga–Al(S) bond is therefore expected to be weaker than a covalent

28 28 812

8

826 28 2826

8

6

0

4

0

30

5 6

0

2

I II

[110] [110]

[001

][0

01]

As

Ga

(I)Al

(S)Al As

Ga(I)

Al

(S)Al

[001

][0

01]

[1−ε, 1+ε, 0] [1−ε, 1+ε, 0]

Figure 3.4 Total (upper panels) and differential (lower panels) electronic valencecharge densities in the Al/GaAs (100) I and II junctions. The differential charge densityis the difference between the self-consistent charge density in the junction and thesuperposition of atomic charge densities. In the semiconductor side of the junction,the density is plotted in a (110) plane, which contains the bonding chains. In themetal side, the plane is slightly rotated (6.2 degrees) about the semiconductor [001]axis, in order to follow the As–Al or Ga–Al interfacial bonds. The rotation angle isnot zero because the interplanar spacing d at the interface is larger than in the bulksemiconductor. The two planes intersect on the [001] axis which contains the first atomin GaAs. The density is expressed in electrons per bulk GaAs unit cell. A cutoff of20 Ry was used in these calculations.


bond in GaAs and stronger than a metallic bond is Al, consistently with the

intensity and directionality of the bonding features in Fig. 3.4. We anticipate

the discussion of the atomic relaxations (section 4.1.3), to say that these different

bond strengths will be reflected in the atomic displacements at the junctions I and

II. In particular, both the As–Al(I) distance in the junction I and the Ga–Al(I)

distance in the junction II are increased in the relaxed configuration, but the

increase is twice as large in the junction I than in the junction II. As can be seen

in the lower panels of Fig. 3.4, the first bond in the GaAs is slightly weaker than in

the rest of the GaAs slab, due to the proximity of the metal. This effect will also

be reflected in an elongation of the first GaAs bond in the relaxed configuration.

3.3 Schottky barrier height

In Fig. 3.5, we show the macroscopic average of the electrostatic potential V (x),

and the potential lineup ∆V , in the junctions I and II. As anticipated in the

previous section, the lineup is established over a narrow region at the interface

(∼ 5 A), which corresponds to the region where the density differs from the bulk

densities. We note that this is in contrast to the case of an abrupt polar semi-

conductor heterojunction, such as the unreconstructed Ge/GaAs (100) interface,

where the interface is charged [107], and V (x) does not converge to a constant

value on each side of the junction. Such charged interfaces are unstable and recon-

struct so as to cancel the macroscopic electric fields in the bulk materials [116].

At the abrupt polar Al/GaAs (100) interface, instead, the metal neutralizes the

excess charge, and no reconstruction is needed, in principle, to insure charge neu-

trality. As Fig. 2.3 shows, the potential lineups of the junctions I and II differ

by 0.1 eV, ∆V being larger in the junction I. Since all the other terms entering

the calculation of the SBH are bulk terms, which are identical for both junctions,

the 0.1 eV difference is directly reflected in the values of the SBH. The calculated

bulk energies used to evaluate φp from Eq. (2.19) can be found in Appendix A.

The LDA barrier heights (without quasiparticle and spin-orbit corrections) are

φLDAp, I = 0.61 eV and φLDA

p, II = 0.51 eV for the junctions I and II, respectively. In

Chapter 5, we will show that, at polar MS interfaces, the SBH is in general larger

for the anion-terminated junction than for the cation-terminated junction.

Our calculated SBH’s are somewhat smaller than other values reported in the lit-

erature. Dandrea and Duke [15] obtained φp = 0.75 eV for the junction I (struc-

ture 3-B in Ref. [15]), including the spin-orbit correction (−0.11 eV). Without

this correction, their LDA barrier height is thus φLDAp = 0.86 eV, i.e., 0.25 eV

larger than ours. They considered a geometry very similar to the one we use,

Sect. 3.4 Electronic states 45

Figure 3.5 Macroscopicaverage of the electrostaticpotential V (x), and po-tential lineup ∆V , in theAs-terminated (I) and Ga-terminated (II) Al/GaAs(100) junctions.

−5 0 5

0

1

2

3

I

II

V(x

) (

eV)

∆V = −2.29 eV

∆V = −2.39 eV

x (Å)

so that we attribute the difference to the different pseudopotentials and smaller

cutoff they took for the supercell calculation.1 Recently, Ruini et al. [17] reported

φLDAp = 0.74 eV for the junction I (also using different pseudopotentials and a

geometry similar to ours), in better agreement with our result.

A detailed comparison of our SBH’s with the available experimental data will be

presented in Chapter 4, taking into account atomic relaxation at the interface and

quasiparticle and spin-orbit corrections on the bulk energies. In the following, we

investigate the electronic structure of the interface, and the behavior of the LDOS

and MIGS. The discussion focuses on the junction I.

3.4 Electronic states

3.4.1 Interface band structure

In Fig. 3.6, we present the Kohn-Sham energy bands and the probability density

of selected states of the As-terminated Al/GaAs (100) junction, as obtained from

calculations at kx = π2L

in the 7 + 21 supercell.2 The bands are plotted along

the high-symmetry lines of the 2-dimensional BZ (see Fig. 2.2). The shaded

areas correspond to the bulk Al and GaAs band structures, projected along the

[100] direction. The projected bulk band structures have been aligned using the

calculated value of the SBH φLDAp . The alignment of the supercell and Al bulk

band structures was done using the difference between the average electrostatic

potentials in the supercell and in the Al slab. Three types of electronic states

1Kerker pseudopotentials were employed in Ref. [15]. The authors use a 9 Ry cutoff, andindicate a change of the SBH by 0.1 eV in going from 9 to 15 Ry, but the direction of thischange is not mentioned.

2All of the results presented in section 3.4 are based on calculations in a 7 + 21 supercell.


can coexist at any isolated interface [117, 118]. The states whose energy falls

within the projected bands of the two constituent materials form the first group.

At some distance from the interface, these states behave like Bloch states of the

two bulks. The states in the region 3 of the figure (Al sp states in the metal, Ga

p states in the semiconductor) belong to this group. The second group contains

the states whose energy lies within a gap region in the projected band structure

of one of the two materials. These states cannot propagate in this material and

decay exponentially on one side of the junction. This is the case, e.g., of the

states in the region 2 (sp3 states in GaAs), which vanish in the metal, and of

the states in the region 4 (Al p states), which decay in the semiconductor. The

states with energy in a gap region of both projected band structures constitute

the third group. These states are localized in the interface region. The state in

the region 1, for instance, belongs to this group. This is an s state of the interface

Ga atom, which feels a more attractive potential than the other Ga s states in

the semiconductor, and is shifted down in energy below the Al bands.

Γ J K Γ−14

−12

−10

−8

−6

−4

−2

0

2

4GaAsAl

1

2

3

4

4321

Met

alSe

mic

on

du

cto

r

[001] [001] [001] [001]

[110

]

EF

E (

eV)

As

Ga

Al Al

[1−ε

, 1+ε

, 0]

Figure 3.6 Energy bands and probability density of selected states of the As-terminated Al/GaAs (100) interface, calculated in the 7 + 21 supercell at kx = 0.25 bx.The shaded areas show the projected bulk band structures of Al and GaAs.


3.4.2 Local density of states

The LDOS, D(E; r), defined by Eq. (2.23), and its planar average, D(E; x),

allow one to visualize the spatial behavior of the states of the MS junction in

particular energy windows. D(E; x) is represented in Fig. 3.7(a) as a continuous

function of E and x. The important features of the LDOS are highlighted, namely

the Fermi level EF, and the VBM and CBM in the semiconductor. We also show

the average electrostatic potential V (x) (see Fig. 2.3 for comparison). Averaging

D(E; x) in each of the regions 1 to 7 of the supercell shown at the bottom of

Fig. 3.7(a), one obtains the curves 1 to 7 shown in Fig. 3.7(b). These curves

also represent the macroscopic average of the LDOS, D(E; xi), evaluated at the

positions xi, i = 1, . . . , 7, at the center of the seven regions. Inspection of the

band structure in Fig. 3.6 facilitates the interpretation of the LDOS features in

Fig. 3.7.

−10

0

EF

E (

eV)

D(E; x)

x [100]

7654321

(a) (b)

2

−4

−2

−6

−8

−12

−14 −12 −10 −8 −6 −4 −2 20

1

2

3

4

5

6

7

8

E (eV)

1

2

3

4

5

6

7

0

EF

VB

MV(x)

CBM

VBM

(eV

Ω

)

−1−1

D(E

; x ) i

Al As Ga

Figure 3.7 Planar average of the local density of states at the As-terminatedAl/GaAs (100) interface. (a) The LDOS as a continuous function of the energy E

and the x coordinate. The black regions correspond to zero density of states and thebright regions to a high density of states. The FL and the average electrostatic poten-tial (solid lines) and the VBM and CBM (dashed lines) are also indicated. The zero ofenergy is at the VBM, which is 0.61 eV below EF. (b) LDOS averaged in the regions1 to 7. The curves are shifted for clarity. Ω is the unit-cell volume of bulk GaAs.


A quick look at Fig. 3.7(a) reveals the main characteristics of the LDOS. On

the GaAs side, the whole electronic structure is bent downwards in energy in the

region of the first As atom. This is clearly visible for the 4s states of the As atoms

(−12 to −10 eV), which are well localized on the As sites and follow relatively

closely the average electrostatic potential. These states lie below the Al valence

bands (see also Fig. 3.6) and do not propagate in the metal. At the interface,

the As 4s state gives rise to a band of localized states visible between −12 and

−11 eV in Fig. 3.6.3 More importantly, Figs. 3.7(a) and (b) show that D(E; x)

does not vanish for E in the GaAs fundamental bandgap (0 to 1.2 eV) and x in

the regions 3 to 5, due to the MIGS. The finite LDOS in the bandgap, associated

with the MIGS, is best seen in Fig. 3.7(b), and it decays in going from region 3 to

region 5. In regions 6 and 7, the LDOS reproduces the density of states of bulk

GaAs. On the metal side, the bulk Al density of states is recovered in region 1,

and the distance between the regions 1 and 6 is of the order of 10 A, consistently

with the length scale reported in the discussion of Fig. 3.3(b).

The position of the FL with respect to the average electrostatic potential in Al,

in Fig. 3.7(a), has been derived from bulk band structure calculations for Al. The

result agree within 0.05 eV with the Fermi energy evaluated from the supercell

calculation. We have used the calculated SBH, φp = 0.61 eV, to locate the VBM

with respect to EF. It can be seen, in Fig. 3.7(b), that the VBM defined in this

way corresponds very well to the valence-band edge of the bulk GaAs density of

states in region 7. A calculated bandgap-width of 1.2 eV has been used to draw

the CBM in Fig. 3.7(a).4

3.4.3 Metal-induced-gap states

In the models of SBH formation based on MIGS, the surface density of states

Ds(E; r), defined by Eq. (2.24), and its macroscopic average Ds(E; x), play

an important role. The quantity Ds(EF; x) δEF corresponds to the amount of

3In Fig. 3.6, the band of localized As 4s states appears as a double band, which is split alongthe Γ–J and J–K directions. In the real MS junction (with semi-infinite metal and semiconductorslabs), there is only one interface As atom, and one band extending on the whole 2-dimensionalBZ. In the supercell, however, the BZ is folded because of non-symmorphic S4 symmetries, asdiscussed in section 2.2.2. The Γ–J′ and J′–K lines are folded onto the Γ–J and J–K lines,giving two distinct branches, and the Γ–K direction is folded onto itself, giving two degeneratebranches.

4With our pseudopotentials, the fully converged LDA gap at the Γ point is 0.8 eV for bulkGaAs. In the supercell calculations, however, we avoid the Γ point in the BZ summation (seeAppendix A) and we use a lower kinetic-energy cutoff, so that the actual gap in the supercellLDOS is ∼ 1.2 eV. The experimental gap at zero temperature is 1.5 eV.


x (Å)

(a) (b)

0 5 10 150.00

0.02

0.04

0.06

0.08

0.10x 0 x 1 x 2 x 3 x 9

E = E 2

E = E 1

E = E F

0.0 0.5 1.00.00

0.02

0.04

0.06

0.08

0.10

E (eV)

EF

x = x0

x = x 1

x = x2

x = x3

x = x9

E1

E2VB

M

Al As Ga

D (

E; x

)−1

−2s

(eV

Å

)

D (

E; x

)−1

−2s

(eV

Å

)

Figure 3.8 Surface density of states associated with the MIGS. (a) Decay of thesurface density of states in the semiconductor, for different energies in the gap. (b)Variation of the surface density of states in the bandgap, for several positions in thesemiconductor.

charge per unit surface induced (to the first order) in the semiconductor, beyond

a position x, by a small displacement δEF of the Fermi level. In Heine’s model,

for example, this induced charge is compensated by an opposite charge located

near the metal surface, and the two charges are separated by an effective distance

which depends on the decay length δs of the MIGS. This gives rise to a dipole layer

which tends to reduce the FL displacement (see p. 17). An accurate knowledge

of Ds(E, x) and of the MIGS decay length is important thus, not only to discuss

models of Schottky barrier formation, but also, as will be shown in Chapter 4, to

understand the behavior of engineered metal/semiconductor junctions.

The calculated Ds(E; x), for the As-terminated Al/GaAs (100) junction, is dis-

played in Fig. 3.8. In Fig. 3.8(a), Ds is represented as a function of x, for differ-

ent energies in the GaAs bandgap. These energies are indicated in Fig. 3.8(b).

In Fig. 3.8(b), Ds is plotted as a function of E, for the x values indicated in

Fig. 3.8(a). The supercell Fermi level, EF, is also shown. The VBM (zero of

energy) has been fixed 0.61 eV below EF, as in Fig. 3.7.

Fig. 3.8(a) shows the rapid decay of Ds in the semiconductor. The decay length,

which is of the order of 2 to 3 A, can be seen to depend somewhat on the energy.


We will come back to this point later in this section. Close to the metal surface,

the surface density of states at the FL is of the order of 5 × 10−2 eV−1 A−2,

i.e., sufficiently large to induce FL pinning within Bardeen and Heine’s theories.

It is worth noticing that our calculated Ds(E; x) involves, by definition, all the

states with energies in the semiconductor bandgap, and would also account for

possible intrinsic localized or resonant states present close to the interface, such

as dangling-bond states. However, there are no such states in the fundamental

bandgap at the abrupt Al/GaAs (100) interface.

As a function of E, Ds exhibits well defined features, which reflect mainly struc-

tures of the bulk Al density of states. The maximum at E = E2, in particular, is

related to a peak in the Al density of states, which can be identified in Fig. 3.7(b),

just above the FL, in the supercell region 1. Fig. 3.8(b) shows that the Fermi

energy is located in an energy region where Ds(E; x) is approximatively constant.

We address now the decay length of the MIGS LDOS. To characterize the decay

of the MIGS, we define an energy- and position-dependent wave number κ(E; x)

by the relation

D(E; x + δx) = D(E; x) e−2κ(E; x)δx. (3.1a)

We emphasize that we now talk about the LDOS D, and not about the surface

density of states Ds. If D(E; x) is an exact exponential function of x, then κ is

independent of x. Expanding Eq. (3.1a) to the first order in δx, and taking the

limit δx → 0, yields

κ(E; x) = −1

2

∂D(E; x)/∂x

D(E; x), (3.1b)

showing that κ is simply half the derivative of − ln D with respect to x. We

display κ(E; x) in Fig. 3.9(a), as a function of x, for three energies indicated in

Fig. 3.9(b). Since Eq. (3.1b) is not well conditioned numerically for D(E; x) → 0,

we do not compute κ(E; x) in the central part of the semiconductor slab, where

the values of D(E; x) are very small. Moreover, in this region, the spatial behavior

of the evanescent MIGS is affected by the periodic-boundary conditions in the

supercell. In the metal, D(E; x) tends to the density of states of bulk Al, which

is independent of x; therefore κ must approach zero in the metal. In the region of

the first semiconductor planes, κ(E; x) shows approximately a plateau, with some

large fluctuations. This shows that D(E; x) is not a pure exponential function

of x in the semiconductor. For the sake of simplicity, however, and in view of

model theories, it is convenient to describe the spatial decay of the MIGS density

of states, at a given energy, with a single parameter: the decay length δs, or the


0 5 10 150.0

0.1

0.2

0.3

(a) (b)

−1(Å

)

κ (E

; x)

0.0 0.5 1.0E (eV)

EF

E1 E2

E = E F

E = E 2

E = E 1

0.0

0.1

0.2

0.3

8

2

3

45

10

x 0

b

Eq. (3.4)Eq. (3.3)

VB

M

2.5

Al As Ga

−1(Å

)

κ (

E)

s

(Å)

δ (E

)s

x (Å)

Figure 3.9 (a) MIGS energy- and position-dependent wave number κ(E; x), ob-tained from Eq. (3.1b), as a function of x for different energies in the GaAs bandgap(see text). (b) Decay length of the MIGS, as defined by Eqs. (3.3) and (3.4). The leftaxis corresponds to κs and the right axis to δs = 1/(2κs).

corresponding wave number κs. The two quantities are related by

δs(E) =1

2 κs(E). (3.2)

There is no unique procedure to define δs or κs. Here, we will compare two

different ways of defining these important quantities. The first approach is to

average κ(E; x) between the positions x0 and x0 + b shown in Fig. 3.9(a):

κs(E) =1

b

∫ x0+b

x0

κ(E; x) dx. (3.3)

This is equivalent to fitting the calculated D to an exponential function. For the

fit, it is important to exclude the regions where D does not have an exponential

behavior, namely x < x0 and x > x0 + b. The result obtained using the values

of x0 and b indicated in Fig. 3.9(a) is presented in Fig. 3.9(b). Obviously, the

precise choice of x0 and b is somewhat arbitrary. One can also define the decay

length δs as the “center of mass” of the charge contained in the MIGS, beyond

a position x0 in the semiconductor, and measured with respect to this position,

namely

δs(E) =1

Ds(E; x0)

∫ ∞

x0

D(E; x) (x− x0) dx. (3.4)


Here also, the choice of x0 is somewhat arbitrary. To perform the integral in

Eq. (3.4), the upper limit was taken at the center of the semiconductor slab.

Using the same x0 in both definitions of κs (δs), we obtain very similar results,

in general, as shown in Fig. 3.9(b). When x0 is chosen between the last Al

plane and the sixth atomic plane in GaAs (third plane of Ga atoms from the

interface), the value of δs obtained from Eq. (3.4) changes by at most 20% over

the whole bandgap. At the FL, we obtain a decay length of 2.4 A. This result is

similar to the value of 2.6 A obtained by Ruini from ab initio calculations for the

Al/GaAs (100) As-terminated junction [119], and the value of 2.8 A reported by

Louie et al. [39] from calculations for the Al/GaAs (110) interface describing the

metal by a jellium with the density of Al.5

3.5 Discussion

Ideal metal/GaAs contacts and deviations from Schottky-Mott rule. — Experi-

mentally, the SBH’s of metal contacts to GaAs do not verify the Schottky-Mott

rule, Eq. (1.10). As most contacts are far from abrupt and epitaxial, one may

wonder to which extent ideal, defect-free, contacts also violate the Schottky-Mott

rule. We have addressed this issue from a purely theoretical point of view, by

examining a fictitious metal/GaAs contact obtained by replacing the Al atoms

of the Al/GaAs (100) junction by Phosphorus atoms, keeping all structural pa-

rameters unchanged. The fictitious metal formed in this way (denoted simply

P) has a higher electronegativity than Al, and thus a higher work function. We

calculated the work functions of the Al and P (100) surfaces by removing the

GaAs slab from the supercell. In the vacuum, the average electrostatic potential

V tends to a constant value yielding the reference vacuum level [see Fig. 1.3(b)].

The work function φm was then derived from the potential lineup using the Al

and P bulk band-structure terms, in full analogy to the procedure used for the

SBH φp. The results are φAl = 4.45 eV and φP = 5.15 eV for the work functions

of Al and P, respectively.

Knowing the change of φm in going from Al to P, and the SBH calculated for the

Al/GaAs and P/GaAs junctions, we may evaluate the γ parameter as γ = − ∆φp

∆φm

(see section 1.3.1). We find γ = 0.13 for the interface I and γ = 0.07 for the

5In Ref. [39], δs is defined by the relation D(EF; x0 + δs)/D(EF; x0) = 1/e, where x0 isthe edge of the jellium slab. The main shortcoming of this definition is the use of the planaraverage D, which does not decay monotonically in the semiconductor, but reflects the atomic-scale oscillations of the LDOS. The decay length calculated in this way thus strongly dependson the choice of x0.

Sect. 3.5 Discussion 53

interface II. The calculated γ values show that the SBH’s of our abrupt epitaxial

metal/GaAs contacts strongly deviate from the Schottky-Mott rule (γ = 1). They

roughly agree, instead, with the model of Heine, Eq. (1.12), using the values of

Ds and δs given in section 3.4.3 (γ = 0.05 and 0.06 for the interfaces I and II,

respectively).

To compare the calculated dependence of the SBH on the metal contact with

experimental data, we also computed the S parameter. The electronegativity

difference between Al and P is ∆Xm = 0.6 on Pauling’s scale, and we may

evaluate the slope parameter as S = − ∆φp

∆Xm. The results are S = 0.15 eV for the

interface I and S = 0.08 eV for the interface II, in agreement with the value of

0.15±0.05 eV reported from a compilation of experimental data for metal/GaAs

contacts [54].

Other ideal metal/semiconductor contacts. — In order to compare the general

behavior of the SBH in metal/GaAs junctions and in other metal/semiconductor

systems, we have repeated the calculations for the Al and P contacts using

three different semiconductors lattice-matched to GaAs, namely, Ge, AlAs, and

ZnSe. All the structural parameters were kept unchanged with respect to the

metal/GaAs junctions. The supercell calculations were performed with a cutoff

energy of 10 Ry, except for the metal/ZnSe junctions, where a 20 Ry cutoff was

used.6 The calculated values of φLDAp , γ, and S are reported in the first four

columns of Table 3.1. In the case of the metal/Ge interface, the calculated φLDAp

is independent of the metal used, within our numerical accuracy, and γ and S

vanish. This is consistent with the very low experimental value S = 0.08± 0.04

reported in Ref. [54]. For the polar junctions, φLDAp is systematically larger for the

interface I than for the interface II, and the difference φLDAp, I −φLDA

p, II increases with

increasing semiconductor ionicity (from top to bottom in Table 3.1). For a given

semiconductor, the difference φLDAp, I −φLDA

p, II is smaller in the P/semiconductor junc-

tions than in the Al/semiconductor junctions. These properties will be discussed

in detail in Chapter 5. For all polar interfaces, we observe significant differences

between the S parameters of the interfaces I and II, indicating that the slope

parameter is not an intrinsic property of the semiconductor, as usually assumed,

but also depends on the interface structure. Once averaged over the interfaces I

and II, the S parameters are in good general agreement with the available exper-

imental values S = 0.15± 0.05 eV for GaAs and S = 0.66± 0.1 eV for ZnSe [54].

To our knowledge, no experimental value of S has been reported for AlAs.

6The Zn 3d orbitals were treated as core states in these calculations.


Semiempirical models. — From the results in Table 3.1, one sees that the SBH can

vary significantly with the atomic structure of the interface, as illustrated by the

∼ 0.4 eV difference between the SBH of the Al/ZnSe (100) I and II interfaces.

These results show the limitations of any model of SBH formation based on

intrinsic properties of the bulk constituents of the junction. In particular, our

results do not agree well with the idea of a FL pinning at a canonical level of the

semiconductor bulk band structure [58].

We have also evaluated the γ and S parameters for the MS systems of Table 3.1

using the models of Heine and Louie et al. [Eqs. (1.12) and (1.13)] and the values

of the surface density of states, Ds(EF; x0), and decay length of the MIGS, δs(EF),

obtained from the ab initio calculations for the Al/semiconductor junctions. The

values of Ds and δs are shown in columns 5 and 6 of Table 3.1. The surface

densities of states are close to 0.05 eV−1 A−2 for all the systems considered, and

do not show a strong dependence on the semiconductor ionicity. For a given

semiconductor, the decay length δs is similar at the interfaces I and II, and

decreases as the semiconductor ionicity increases. To evaluate γ and S from

the models, we have used εs = 2 and δm = 0.5 A for all systems, as in Ref. [54]

(see also Ref. [120]). The models give satisfactory results for the most covalent

semiconductors, Ge and GaAs, but fail in describing the correct increase in γ

and S as the semiconductor ionicity increases. Furthermore, they are unable to

reproduce the differences in γ and S between the interfaces I and II.

Comparison with a simple model for semiconductor heterojunctions. — In the

case of lattice-matched non-polar (110) semiconductor heterojunctions, it has

been shown that a model electronic density obtained from the superposition of

atomic charge densities provides a relatively good approximation to evaluate the

electrostatic-potential lineup ∆V (to within ∼ 0.2 eV) [121].7 As a consequence,

the valence-band offset (VBO) of these heterojunctions can be estimated from

a difference between quantities which are intrinsic to each constituent material

without any need to compute the self-consistent electron density at the interface.

Physically, this result implies that the VBO is not sensitive to interface-specific

properties. As we have already pointed out, the SBH at MS interfaces is sensitive

to the interface atomic structure, so that we do not expect this model to be

accurate for MS interfaces. Here, we show that the lineup ∆Vac, obtained from

the superposition of atomic charges, differs by as much as several eV’s from the

7In the approach of Ref. [121], the model charge density is used only to compute theelectrostatic-potential lineup. To evaluate the band discontinuities, the positions of the bandswith respect to the average electrostatic potential are still determined from self-consistent cal-culations for the bulk materials.


self-consistent lineup ∆V at MS interfaces, and is therefore not of great help

to study SBH’s. At the Al/GaAs (100) interface, for instance, the self-consistent

calculation yields ∆V ≈ −2.3 eV (see Fig. 3.5), while the superposition of atomic

charges gives ∆Vac = −4.1 eV, which is 1.8 eV off. For the Al/Ge and Al/AlAs

interfaces, the discrepancy is similar, 1.7 and 1.5 eV, respectively, and it becomes

even worse for Al/ZnSe: 3.3 eV. We attribute these large differences to the fact

that the model atomic density is an especially poor approximation for the charge

density of a metal and is unable to describe the charge transfer associated with

the formation of a MS interface.

Table 3.1 Schottky barrier height φLDAp , and slope parameters γ and S, for the

ideal Al/semiconductor and P/semiconductor contacts, as obtained from the ab initiocalculation (col. 1 to 4), and from Eqs. (1.12) and (1.13) (col. 7 and 8), using εs = 2,δm = 0.5 A, and the parameters determined ab initio for the Al/semiconductor contacts(col. 5 and 6). The calculated work functions of Al and P are φAl = 4.45 eV and φP =5.15 eV, respectively. In the calculation of Ds and δs [Eq. (3.4)], x0 is fixed halfwaybetween the last metal and first semiconductor planes [position x0 in Fig. 3.8(a)]. φLDA

p

is calculated in the 7+13 supercell; Ds and δs are calculated in the 7+21 supercell. Thecalculated values φLDA

p do not include quasiparticle and spin-orbit splitting correctionson the semiconductor valence-band edge.

φLDAp (eV) γ S (eV) Ds(EF; x0) δs(EF) γ S (eV)

Al P − ∆φp

∆φm− ∆φp

∆Xm(eV−1 A−2) (A) Eq. (1.12) Eq. (1.13)

Ge 0.04 0.04 0.00 0.00 6.3×10−2 3.3 0.04 0.09

I 0.61 0.52 0.13 0.15 5.6×10−2 2.4 0.05 0.11GaAs

II 0.51 0.46 0.07 0.08 4.6×10−2 2.5 0.06 0.14

I 1.09 0.95 0.20 0.23 6.6×10−2 2.0 0.05 0.11AlAs

II 0.80 0.74 0.09 0.10 5.4×10−2 2.1 0.06 0.14

I 1.82 1.37 0.64 0.75 3.6×10−2 1.8 0.10 0.23ZnSe

II 1.46 1.34 0.17 0.20 4.6×10−2 1.8 0.08 0.18


Summary

Using the ab initio approach described in Chapter 2, we have studied the elec-

tronic structure of abrupt epitaxial Al/GaAs junctions. The comparison of the

electronic properties at the interface and in the constituent bulk materials al-

lowed us to extract two length scales, representative of the region where (i) the

interface electronic density differs from the bulk densities (∼ 5 A) and (ii) the

interface LDOS differs from the bulk DOS (∼ 10 A). The Schottky barrier height

was shown to depend on the termination (As or Ga) of the GaAs (100) sur-

face. We discussed the interfacial bonds, the electronic states of the junction,

and in particular the MIGS, for which we obtained a surface density of states

of ∼ 0.05 eV−1 A−2 and a decay length of ∼ 2.4 A. We have determined the

modifications of the SBH induced by changes in the metal and semiconductor

chemical composition, for a fixed interfacial geometry. The corresponding SBH

variations were found to be only roughly reproduced by existing models.

Chapter 4

Schottky barrier tuning in

Al/Si/GaAs (100) junctions

There is considerable interest in identifying methods to tailor the SBH at MS con-

tacts. The performances of many devices based on metal/semiconductor junctions

could be considerably improved if their SBH could be changed. Recently, large

modifications of the SBH at molecular-beam epitaxially grown metal/n-GaAs

(100) interfaces have been achieved by depositing thin (6–60 A) Si interlayers un-

der an excess group-III or group-V atomic flux, prior to contact fabrication [8–11].

The origin of these variations has not yet been established. Several empirical mod-

els based on the macroscopic properties of bulk Si have been proposed to explain

different sets of experimental results [10–13]. In these approaches, the modifica-

tion of the SBH is due to the band bending in the Si layer, which is monitored

by changing the Si thickness and the type and density of impurities in the inter-

layer. The models differ in their assumptions regarding (i) the value of the band

discontinuity at the Al/Si and Si/GaAs interfaces, and (ii) the doping-induced

band bending in the Si layer.

More recently, however, Cantile et al. [8] demonstrated that the modifications of

the Al/Si/GaAs (100) barrier are already established for Si coverages less than 2

monolayers (ML), thus indicating that the tuning has a more microscopic origin.

The barrier heights of Ref. [8] are reported in Fig. 4.1 as a function of the Si

coverage ϑ expressed in monolayers (2 ML correspond to ≈ 2.6 A). When the Si

interlayers are grown under ultra-high vacuum (UHV) conditions, the SBH’s show

some variations from one sample to the other, but there is no systematic barrier

modification. When the interlayers are deposited under As (Al) flux, instead, one

observes a monotonic increase (decrease) of φp with Si coverage. In particular,

for ϑ = 1 ML, the SBH obtained under the As-flux and Al-flux conditions differ

58 Schottky barrier tuning in Al/Si/GaAs (100) junctions Chap. 4

by ∼ 0.5 eV, and for ϑ = 2 ML they differ by ∼ 0.8 eV. Electrical I–V and C–V

measurements [8], as well as internal photoemission measurements [122] of the

Al/Si/n-GaAs SBH have also been performed, and show the same trends as the

XPS data.

In the case of III-V/III-V semiconductor heterojunctions, a microscopic inter-

face dipole model introduced by Harrison [123] successfully explained band-offset

variations induced by ultrathin Si or Ge interlayers [124]. For MS junctions, the

complexity of the structure and the lack of information on the inhomogeneous

screening near the interface hindered the application of similar microscopic mod-

els to the analysis of Schottky barrier tuning. The strength of the screening of

interface dipoles by MIGS is a key issue in the theory of Schottky barrier forma-

tion (see section 1.3.1), and a deeper understanding of screening on the atomic

scale is important for developing models with truly predictive capabilities on

barrier modifications via interfacial perturbations.

In this chapter, we tackle the problem of the Al/Si/GaAs (100) Schottky barrier

tuning by means of ab initio calculations. We first present a study of the effect of

ultrathin (submonolayer to 2 ML thick) Si interlayers on the electronic structure

of Al/GaAs (100) junctions [125]. Our results show that microscopic interface

dipoles generated by replacing anion-cation pairs by Si pairs at As- and Ga-

terminated Al/GaAs (100) interfaces can successfully explain the experimental

Schottky barrier tuning. We then address the relative stability of the As- and

ϑ (ML)0.0 0.5 1.0 1.5 2.0

0.4

0.6

0.8

1.0

1.2

φ

(eV

)p

As flux (Ga 3d)

Al flux (Ga 3d)

UHV (Ga 3d)

Al flux (As 3d)

UHV (As 3d)

Figure 4.1 Schottky barrier height at engineered Al/Si/GaAs (100) interfaces, asmeasured by XPS [8]. The Si interlayers are grown in UHV conditions, under As flux,or under Al flux. φp is derived using either the energetic position of the Ga 3d core-levelpeak or that of the As 3d core-level peak, as indicated in parenthesis.

Sect. 4.1 Al/Si/GaAs junctions with 0–2 Si ML 59

Ga-terminated Al/Si/GaAs (100) junctions, and the role the excess fluxes may

play in the formation of these interfaces [126]. Based on our atomic-scale study

of the screening of the Si-induced local dipoles, we also derive a model which

explains both the experimental data and the first-principle results in terms of

simple physical parameters. Finally, we examine Al/Si/GaAs (100) junctions

with thicker Si interlayers (4–6 ML), and provide a possible explanation for the

observed barrier saturation in this coverage regime.

4.1 Al/Si/GaAs junctions with 0–2 Si ML

4.1.1 Si-induced local dipole

We introduced Si interlayers with coverage in the 0–2 ML range at the abrupt

Al/GaAs (100) I and II junctions illustrated in Fig. 3.1, by replacing with Si

atoms an equal number of Ga and As atoms in the two planes closest to the

metal. We thus assume fully self-compensated Si dopants distributed over two

layers, and simulate the doping for coverage 0 < ϑ < 2 ML with the virtual-crystal

approximation, using 〈Ga1−ϑ2Siϑ

2〉 and 〈As1−ϑ

2Siϑ

2〉 pseudo-ions. The formation of

the microscopic dipole is illustrated in Fig. 4.2. Si doping of adjacent cation and

anion planes in GaAs can be viewed as a proton transfer between substituted

As and Ga atoms [123]. For a given coverage ϑ, the bare dipole layer associated

with these point charges produces a discontinuity of the electrostatic potential,

∆Ub = ± πe2

aGaAsϑ. As a result, the potential lineup ∆V in Eq. (2.19) is modi-

fied, and the FL in the metal is shifted with respect to the energy bands in the

semiconductor. The actual change of ∆V is not just given by the bare disconti-

nuity ∆Ub but by the screened discontinuity ∆U = ∆Ub/εeff , where the effective

screening parameter εeff accounts for the response of the electrons to the bare

perturbation. Thus, since interfacial perturbations such as the Si interlayer do

not affect the bulk term ∆Ep in Eq. (2.19), φp increases (decreases) by ∆U at the

As- (Ga-)terminated interface. As we will see, εeff depends on the Si coverage ϑ,

and we can write the SBH of the Al/ϑ Si ML/GaAs I and II interfaces as

φI, IIp (ϑ) = φI, II

p, 0 ± πe2

aGaAs

ϑ

εeff(ϑ), (4.1)

where φI, IIp, 0 is the SBH of the undoped I and II interfaces, and the minus sign is

for the interface II. In Eq. (4.1), we have implicitly supposed that the effective

screening parameter is the same for the junctions I and II.

In Fig. 4.3, we display the calculated charges and potentials induced by 1 ML


ϑ ϑAs 1− Si2 2

Ga AsAl Ga AsAsAlAl(100)

Ga As

Al

As

AlAl

As Ga

Al

Ga

AlAl

GaGaAl Ga AsAsAlAl(100)

+−

−+

∆U > 0

∆U < 0

I (As-terminated)

II (Ga-terminated)

∆Ub

∆Ub

ϑ ϑGa 1− Si2 2

ϑϑAs 1− Si2 2

ϑ ϑGa 1− Si2 2

E v

FE

E v

FE

E v

FE

E vFE

Figure 4.2 Si-induced microscopic dipole at the Al/GaAs (100) interface. The dipolelayer increases φp in the junction I and decreases φp in the junction II.

of Si in the junction I. Fig. 4.3(a) shows the planar averages of the ionic charge

density — two opposite delta functions on the Si-doped planes — and of the

electronic density. The electronic contribution is obtained by subtracting from

the total electronic density of the Al/1 Si ML/GaAs interface, the density of

the undoped Al/GaAs interface calculated with the same parameters (supercell

size, cutoff, etc.1). The ionic perturbation is symmetric with respect to the plane

separating the two doped layers, but the electronic response is not, because of the

presence of the metal on one side of the interlayer. Still, the induced electronic

charge is well localized at the interface, and exhibits a clear dipole shape. In

Fig. 4.3(b), we plot the macroscopic averages of the potentials associated with

the ionic and electronic dipoles. The thick line represents the sum of the ionic

and electronic potentials, i.e., the net effect on the SBH (scaled by a factor

of 10). At 1 ML coverage, we have ∆Ub = 8.15 eV; the electronic contribution,

obtained from the self-consistent calculation, reduces the potential discontinuity

to ∆U = 0.23 eV. This corresponds to an effective screening εeff(ϑ = 1) ≈ 35 at

the junction I (the same value is obtained at the junction II). This large screening

is related to the high polarizability of the MIGS, as will be shown in section 4.2.

1Unless otherwise specified, the standard parameters of the calculations are as in Chapter 3:supercell size, 7 + 13; k-points grid: (2, 6, 6); cutoff energy: 10 Ry.


−6

−4

−2

0

2

4

6

Po

ten

tial

(eV

)

−0.8

−0.4

0.0

0.4

0.8

Den

sity

(el

ectr

on

s/Ω

)

As

As1/2Si1/2

GaGa1/2Si1/2

Al As

As1/2Si1/2

GaGa1/2Si1/2Al

x 10

∆Ub

∆U

(a) (b)

ϑ = 1 ML

Figure 4.3 (a) Planar averages of the ionic (dotted lines) and electronic (solid line)charge densities induced in the Al/Si/GaAs (100) junction I by a Si bilayer, at 1 MLcoverage. (b) Macroscopic average of the corresponding ionic and electronic potentials.The thick line represents the total electrostatic potential scaled by a factor of 10. Thebare, ∆Ub, and screened, ∆U , discontinuities across the Si interlayer are also indicated.

4.1.2 Schottky barrier modification

In Fig. 4.4, we present our results for the SBH of the As- and Ga-terminated

Al/Si/GaAs interfaces as a function of Si coverage, together with the photoe-

mission measurements of Ref. [8] for Si grown under excess As or Al flux. The

theoretical results were obtained neglecting atomic relaxation at the interfaces.

The relaxation, however, does not affect significantly the SBH (as will be seen

in section 4.1.3). The theoretical values of φp, calculated within the DFT-LDA

framework, should be corrected for quasiparticle and spin-orbit effects. These

corrections concern the bulk term ∆Ep in Eq. (2.19), and are identical for all

Al/Si/GaAs junctions. For GaAs, we use the experimental spin-orbit splitting

∆so = 0.34 eV [127] and the quasiparticle correction ∆εGaAs = −0.36 eV calcu-

lated by Charlesworth et al. [14]. As discussed in section 2.2.5, we neglect the

quasiparticle correction on the metal Fermi energy, ∆εAl. The resulting estimated

correction on φLDAp , as given by Eq. (2.27), is +0.25±0.1 eV, taking into account

the uncertainty on ∆εGaAs. In Fig. 4.4, we therefore rigidly shifted the LDA

values φLDAp by +0.25 eV.

At zero Si coverage, in Fig. 4.4, we recover the difference of 0.1 eV between

the SBH’s of the Al/GaAs I and II interfaces presented in Chapter 3. This

difference increases with increasing ϑ, and becomes 0.82 eV for ϑ = 2 ML, in

excellent agreement with the experimental data. The agreement between theory

and experiment in Fig. 4.4 supports the contention by Cantile et al. [8] that an


0.0 0.5 1.0 1.5 2.0

0.4

0.6

0.8

1.0

1.2

ϑ (ML)

φ

(eV

)p

I (As-terminated)

II (Ga-terminated)

As flux

Al flux

From Ref. [8]

Figure 4.4 Schottky barrier height of Al/Si/GaAs (100) diodes as a function ofSi coverage. The solid circles are the results of the LDA self-consistent calculations(shifted by +0.25 eV), the triangles correspond to the photoemission data of Ref. [8] forSi grown under As and Al fluxes (see also Fig. 4.1). The solid lines give the predictionsof our model, Eqs. (4.1) and (4.7); the dashed lines indicate the model predictions atlow coverage in the linear regime.

excess anion (cation) flux may tend to favor an anion-cation (cation-anion) site

sequence for the Si layers at the GaAs (100) surface. Although the atomic-scale

morphologies of the buried interfaces are difficult to probe experimentally, the

results of Fig. 4.4 show that microscopic dipoles obtained by heterovalent doping

at the interface can account for the experimental trend, and allow for an efficient

tuning of the SBH at polar MS contacts.

4.1.3 Effect of atomic relaxations

To determine the atomic relaxations at the Al/GaAs and Al/2 Si ML/GaAs I

and II interfaces, we first considered MS structures with a thin metal overlayer in

contact with vacuum. Specifically, our starting configurations for the relaxation

runs were generated by removing 5 atomic layers from the middle of the Al slab

in the 13 + 13 supercells of the unrelaxed structures. The atomic configurations

were then fully relaxed by minimizing the total energy with respect to the atomic

positions. All the supercell relaxation calculations were performed using a 20 Ry

energy cutoff. Due to the presence of the vacuum, the metallic overlayer and the

Si interlayer could freely relax along the growth direction, and release thus any

residual stress σxx. To examine the properties of the fully developed junctions,


we transferred then the equilibrium interplanar distances to a new supercell in-

cluding a full Al slab (13 + 13 superlattice). The interplanar spacings for the

additional layers at the center of the Al slab were set to their bulk value (taking

into account the strain in Al). We then let this structure relax again to allow for

small readjustments in the metal. The final forces in all of the structures were

smaller than 0.09 eV A−1.

In Fig. 4.5, we display the equilibrium interplanar distances as obtained in the

Al/GaAs and Al/2 Si ML/GaAs junctions with the thin metallic overlayer (four

Al layers), and in the corresponding fully developed junctions (13+13 supercell).

As there are two inequivalent Al atoms in each Al (100) plane, which exhibit

some buckling near the interface, we report on the metal side of the junction in

Fig. 4.5 the largest and the smallest distances between the sublayers in adjacent

planes.

For all of the systems examined here, the distance obtained from the ab initio

calculations for the separation between the central layers of the Al and GaAs

slabs is identical, to within 0.05 A, to the ideal distance derived from macro-

scopic elasticity theory. This is somewhat unexpected given the large relaxations

occurring near the junction (more than 10% of the initial spacing in some cases),

and the different nature of the materials forming the contact. Except for small

readjustments on the Al side of the junction, the relaxations are very similar in

the systems with the thin Al overlayer and in the corresponding 13+13 superlat-

tices. The relaxations obtained for the superlattices without the Si interlayer are

also consistent with the results of a previous calculation for the Al/GaAs (100)

junction [15].

Fig. 4.5 shows that the bulk GaAs spacing is recovered beyond the second (for

Al/GaAs) to third (for Al/Si/GaAs) semiconductor layer from the junction,

whereas in the metal the bulk (strained) interlayer spacing is recovered only

beyond the fourth Al plane from the junction. The relaxations taking place in

the metal are qualitatively similar in the four structures shown in Fig. 4.5. The

buckling of the first Al layer at the junction reflects the establishment of two

types of interfacial bonds: a predominantly covalent (short) bond between the

Al(S) and the atoms of the semiconductor surface layer, and a metallic (long)

bond between the Al(I) and the semiconductor. These two types of bonds can

be distinguished in the charge-density plots of the relaxed interfaces presented

in Fig. 4.6. The covalent character of the Al(S) bond is clearly stronger in the

junctions I relative to the junctions II, as reflected by the larger bonding charge

density (Fig. 4.6) and the shorter bond length (Fig. 4.5) established in this case.

The Al(I) metallic bond is, instead, stronger for the junctions II, and this results


1.4

1.6

1.8

2.0

2.2

1.4

1.6

1.8

2.0

2.2

1.4

1.6

1.8

2.0

2.2

1.4

1.6

1.8

2.0

2.2

Al GaAsSi

As-terminated2 Si ML

I

Al AsGaSi

Ga-terminated2 Si ML

II

Ga-terminated0 Si ML

Al AsGa

II

As-terminated0 Si ML

Al As Ga

I

Inte

rpla

nar

sp

acin

g (

Å)

Inte

rpla

nar

sp

acin

g (

Å)

Figure 4.5 Equilibrium interplanar spacings for the Al/GaAs (100) andAl/2 Si ML/GaAs (100) junctions. Open squares correspond to the system with athin Al overlayer in contact with vacuum, and solid squares correspond to the 13 + 13superlattice (see text). The sequence of atomic layers is indicated at the bottom ofeach graph. Each Al (100) plane contains two inequivalent sublayers; the largest andsmallest distances between sublayers in adjacent planes are reported on the metal sideof the junction. The solid lines show the bulk interplanar spacings. The value of theSi–Si interlayer distance, as obtained from macroscopic elasticity theory for Si pseudo-morphically strained to GaAs, is also indicated (short line).


26 2826 2828

0

4 64

0 0

66

12 2826 2828

0

26

0

5

0

66

Al/GaAs I Al/GaAs II

As Ga

(I)Al

(S)Al AsGa(S)

Al

(I)Al

1828

222828 18

2222

2828

0

4

0

67 6

0

6

0

366

0

6

0

6

Al/Si/GaAs I Al/Si/GaAs II

Si SiAs Ga

Figure 4.6 Valence-electron density at the relaxed Al/GaAs and Al/2 Si ML/GaAs(100) interfaces. The upper panels show the total valence-electron density, and thelower panels show the bonding (differential) charge. The planes of the plot are thesame as in Fig. 3.4, except for small adjustments taking into account the atomic dis-placements. Specifically, in each part of the graph, the plane is defined by the threeatoms represented. Left parts: interface bonds; central parts: first two bonds in thesemiconductor side of the junction; right parts: bonding in the middle of the semicon-ductor slab. The density is expressed in electrons per bulk GaAs unit cell. A 20 Rycutoff was used in these calculations.


in a smaller buckling of the first Al layer in the junctions II. Comparison of the

charge-density plots in the relaxed and unrelaxed Al/GaAs junctions (Figs. 3.4

and 4.6) shows a strengthening of the covalent As–Al(S) bond at the expense of

the metallic As–Al(I) bond after relaxation at the interface I, while no visible

change in the charge-density features occurs in the region of the Ga–Al(S) and

Ga–Al(I) bonds at the interface II.

Without the Si interlayer, the As (Ga) surface plane in the junction I (II) moves

outwards by about 0.10 A (0.08 A) relative to the bulk GaAs. There is thus

a non-negligible weakening of the covalent bonding between the first two GaAs

atomic layers due to the formation of the metallic bonds across the interface, as

can be seen in Fig. 4.6. When the Si interlayers are present, the Si–As spacing

increases by 3.5% in the junction I, whereas the Si–Ga spacing decreases by

4% in the junction II relative to the bulk Ga–As spacing. Given the similar

atomic radii of the atomic species involved here, we attribute the increase in the

Si–As interplanar distance to the formation of a donor type of bond at the polar

As-terminated Si/GaAs (100) interface. This produces an excess of electronic

charge on the Si–As bond, and an elongation of the resulting oversaturated bond.

Inversely, the Si–Ga bond is contracted due to the formation of an acceptor type

of bond at the Ga-terminated Si/GaAs interface. In the relaxed geometry, the

bonding charges are essentially identical on the Si–As and Si–Ga bonds (Fig. 4.6).

We will see in section 4.3.2 that the donor and acceptor bonds of the Si/GaAs

interface give rise to well defined resonant interface states in the Al/Si/GaAs

junctions including more than 2 ML of Si.

In the Al/Si/GaAs I and II junctions, the Si–Si interplanar spacing remains ap-

proximatively equal (within 0.02 A) to the spacing in bulk GaAs, and is therefore

larger than the equilibrium spacing in bulk Si (a⊥Si = 101% aSi for the interface I,

and 102% aSi for the interface II). Based on macroscopic elasticity theory (MET),

one would have expected the Si–Si interplanar distance to decrease with respect to

the spacing in bulk Si. In fact, from ab initio calculations for bulk Si coherently

strained to GaAs we find a⊥Si = 98% aSi = 95% aGaAs. For a two-monolayer-

thick Si overlayer grown on GaAs (100), the angle resolved x-ray photoelectron

diffraction measurements by Chambers and Loebs [128] indicate an even larger

contraction: a⊥Si = 94% aGaAs. The reversed trend we find for the Si–Si spacing in

the Al/Si/GaAs junctions results from the presence of the metal that weakens the

covalent bonding between the two semiconductor layers closest to the Al surface

(as we also observed in the case of the Al/GaAs junctions).2 We performed simi-

2We have investigated the effect of the Ga core states on the structural properties of theGa-terminated junctions using the non-linear core correction (NLCC). The NLCC is found toincrease the GaAs theoretical lattice parameter from 5.55 A to 5.59 A, and to change the lattice


Table 4.1 Schottky barrier φp, forthe Al/Si/GaAs (100) I and II inter-faces, calculated in the ideal and re-laxed configurations. Results are re-ported for Si coverages ϑ = 0 and2 ML. The LDA Schottky barriers wererigidly shifted by +0.25 eV. The ex-perimental photoemission data fromRef. [8] are also reported. For the ϑ = 0coverage, we reported the experimen-tal values measured at ϑ = 0.2 ML.The calculations were performed usinga 20 Ry cutoff and the 13+13 supercell.

Interface φp (eV)

Type ϑ (ML) Ideal Relaxed Expt.

I 0 0.90 0.90 0.87–0.89

I 2 1.23 1.24 1.15–1.26

II 0 0.80 0.70 0.54–0.61

II 2 0.42 0.37 0.37–0.46

lar calculations replacing the metal by GaAs, i.e., in a GaAs/2 Si ML/GaAs (100)

heterostructure coherently strained to GaAs, and in this case a contraction by

5% (7%) of the Si–Si interlayer spacing is obtained with respect to the value in

bulk Si (GaAs), consistently with the trend expected from MET and with the

results by Chambers and Loebs for the GaAs/2 Si ML/vacuum system.

The calculated values of the Schottky barriers for the different junctions are re-

ported in Table 4.1, both with and without lattice relaxation.3 The Schottky

barriers are only weakly affected by the relaxation. This can be understood in

terms of the effective charges at the interface, as first pointed out by Ruini et

al. [17]. On the metal side of the junction, the longitudinal dynamical charges

vanish at the Al/GaAs interface, and the large atomic displacements in Al do

not affect the SBH. On the semiconductor side, the atomic displacements are

much smaller. The largest changes are in the Al/GaAs I and II junctions, and

concern the outward relaxation of the last semiconductor plane. Although effec-

tive charges are non-vanishing in the semiconductor, they are much smaller near

the interface than in bulk GaAs [17]. This explains the relatively small effect of

the microscopic dipoles induced by the individual atomic displacements in these

junctions. For more ionic semiconductors, however, such as ZnSe, the longitu-

dinal effective charges are larger, and the Schottky barrier height is much more

sensitive to atomic relaxation, as will be seen in Chapter 5.

relaxation in the Al/GaAs and Al/2 Si ML/GaAs junctions by at most 1%. With the NLCC,the Si–Ga distance in the Al/2 Si ML/GaAs junction is 3% smaller than in bulk GaAs (4%without NLCC), and the Si–Si distance is equal to 1.36 A (1.38 A without NLCC), i.e., 3%larger than the MET prediction (as found without the NLCC).

3The values of φp for the ideal interfaces in Table 4.1 are slightly different from the valuesplotted in Fig. 4.4. This is due to the use of different supercells and cutoff energies in the twocalculations. The largest difference is 0.04 eV.


Most transport measurements performed on Al/GaAs contacts give values of φp

between 0.6 eV and 0.7 eV [73, 74, 129–131], in good agreement with our calcu-

lated SBH, φp = 0.7 eV, for the relaxed Ga-terminated interface. Different con-

clusions have been reached regarding the effect of the GaAs-surface stoichiometry

on the SBH. Some transport studies, including Refs. [73] and [74], find a small

(∼ 0.1 eV) difference between the SBH measured in junctions made with As-rich

and Ga-rich surfaces (As-rich leading to higher φp), while in other studies, such

as Refs. [129] and [131], no difference is found between the As- and Ga-rich inter-

faces. The XPS results of Cantile et al. [8] for Al/GaAs junctions fabricated in

UHV conditions at ϑ = 0 are consistent with the values of 0.6 eV to 0.7 eV deter-

mined by transport measurements (see Fig. 4.1). We note, however, that other

XPS experiments lead to a wider range of SBH values for the Al/GaAs (100)

interface [132].

The large calculated and measured SBH’s at ϑ = 2 ML in Table. 4.1 are very

close to the barrier heights measured by Grant and Waldrop [9] in Al/Si/GaAs

junctions including 14 A (≈ 10 ML) of Si grown under As flux: φp = 1.23 eV.

This suggests that the maximum increase in φp achievable with Si interlayers of

increasing thickness, grown under As flux, is of the order of 0.4 eV. Indeed, Costa

et al. [10] and Koyanagi et al. [11] measured a 0.4 eV (0.3 eV) increase in φp for

60 A (10 A) thick Si interlayers grown under As flux. In the case of Si interlayers

grown without As flux or under a Ga flux, a 0.1–0.2 eV decrease in φp is reported

in Refs. [10] and [11] for Si thicknesses ranging from 6 to 60 A, corresponding to

values of φp between 0.4 eV and 0.5 eV. These values are similar to our ab initio

results for the interface II — and to the low p-type SBH’s measured by Cantile

et al. — at a coverage ϑ = 2 ML. Most of the Schottky barrier tuning observed

experimentally in Al/Si/GaAs junctions is thus already present in the very low

coverage regime.

4.1.4 Si-induced local density of states

In Fig. 4.7, we show the local density of states in the relaxed junctions, with

and without the Si interlayer. To compute the LDOS, we have transferred the

equilibrium positions of the atoms in the interface region, as obtained in the

13 + 13 supercell, into a 7 + 21 supercell, and let the structure relax again, using

a (2, 8, 8) k-points mesh, and a 10 Ry cutoff. The average LDOS has been

computed in the regions 1 to 7 of the supercell, which are indicated schematically

at the bottom of Fig. 4.7. On the semiconductor side of the junction, the LDOS

recovers the bulk GaAs density of states features in region 6, i.e., beyond the

seventh semiconductor layer from the metal. On the metal side, instead, the bulk


−14 −12 −10 −8 −6 −4 −2 0 20

1

2

3

4

5

6

7

8

E (eV)

2 Si ML EF

1

2

3

4

5

6

7

−14 −12 −10 −8 −6 −4 −2 0 20

1

2

3

4

5

6

7

8

E (eV)

2 Si ML EF

1

2

3

4

5

6

7

I (As-terminated)

II (Ga-terminated)

−14 −12 −10 −8 −6 −4 −2 0 20

1

2

3

4

5

6

7

8

E (eV)

0 Si ML EF

1

2

3

4

5

6

7

LDO

S (e

V

Ω

)−1

−1

−14 −12 −10 −8 −6 −4 −2 0 20

1

2

3

4

5

6

7

8

E (eV)

0 Si ML EF

1

2

3

4

5

6

7

LDO

S (e

V

Ω

)−1

−1

x [100]

7654321

Al GaAs

7654321

Al GaAsSi

x [100]

Figure 4.7 Local density of states in the As- and Ga-terminated Al/GaAs (100)(left) and Al/2 Si ML/GaAs (100) (right) relaxed junctions. Each curve corresponds toa particular region of the supercell, indicated at the bottom of the figure. The shadedcurves correspond to the supercell region including the Si bilayer. In each graph, thevalence-band edge (zero of energy) has been aligned with respect to the supercell FL,EF, using the calculated φLDA

p (without the +0.25 eV correction). Ω is the unit-cellvolume of bulk GaAs.


Al density of states is essentially recovered beyond the second Al plane from the

junction (region 1), despite the large atomic displacements. When the Si bilayer

is introduced, the bulk Al LDOS is shifted by ≈ 0.4 eV to higher (lower) energies

with respect to the bulk GaAs LDOS in the junction I (II), as a result of the

Si-induced local interface dipole.

The results in Fig. 4.7 show that the Si interlayer substantially changes the LDOS

in the region 3. Without the Si interlayer, the tail of the metal wave functions

give rise, in this region, to a finite LDOS of about 0.5 eV−1Ω−1 for energies in the

optical bandgap of GaAs. In the presence of the Si bilayer, this LDOS increases

to about 0.7 eV−1Ω−1 in the GaAs-gap region. This large LDOS near the Fermi

energy in the engineered junctions is responsible for a large screening of the local

dipole, and plays an important role in determining the actual barrier modification

with the Si bilayer. Fig. 4.7 also shows that the interlayer induces a large density

of states in the energy region of the GaAs valence gap (−10 to −7 eV). This new

LDOS structure, in region 3, derives mainly from the s states of the substitutional

Si atoms on the As sites. Due to the weaker ionic potential of Si with respect to

that of As, these substitutions push the anion s states at higher energy relative

to the bulk As s-like feature in GaAs, inducing resonant interface states in the

GaAs valence gap.

4.1.5 Interface formation energy

In this section, we address the role the excess anionic or cationic fluxes may play

in the formation of the engineered Al/Si/GaAs interfaces. The growth process

per se involves complex kinetic mechanisms whose description is far beyond the

scope of the present study. Here we will concentrate on the energetics of the

As- and Ga-terminated interfaces, and show that the experimental trend can

already be accounted for based on a thermodynamic description. We focus on

the limiting cases of Si coverages ϑ = 0 and 2 ML, and we examine the relative

stability of these prototype systems under different experimental conditions. The

calculations were done in the 13 + 13 supercell, using a 20 Ry cutoff.

To discuss the relative stability of the As- and Ga-terminated Al/GaAs (100) and

Al/Si/GaAs (100) heterostructures, which contain a different number of As and

Ga atoms, it is necessary to compute their formation energy εf , as a function of

the atomic chemical potentials. The As and Ga chemical potentials are related by

the condition µAs +µGa = µGaAs — where µGaAs is the total energy of bulk GaAs

per Ga-As pair — but their difference is a free variable. Taking into account

the fact that the number of As and Ga atoms differ by one unit in all of our


supercells, the formation energies of the interfaces I and II, respectively, may be

recast as [see Eq. (2.26)]:

εIf(µAs) =

1

2

(EI

tot − µAs

)(4.2a)

εIIf (µAs) =

1

2

(EII

tot − µGaAs + µAs

), (4.2b)

where Etot = Etot−nGaAsµGaAs−nSiµSi−nAlµAl, and nGaAs stands for the number

of Ga-As pairs contained in the supercell. The chemical potentials µSi and µAl

do not affect the difference between the formation energies of the type I and

II interfaces. In our calculations we set the value of µSi to that of the bulk Si

chemical potential, calculated at the Si theoretical equilibrium lattice parameter

(see Appendix A). Our formation energies for Al/Si/GaAs include therefore also

the strain energy of the Si interlayer. At equilibrium, the Al chemical potential is

determined by the total energy per atom of the bulk (strained) Al metal. The As

(Ga) chemical potential cannot exceed the chemical potential of bulk As (Ga), as

bulk As (Ga) would form in the system. Therefore, choosing µAs as independent

variable, the range of values of the As chemical potential can be restricted to [110]

µbulkAs + ∆Hf 6 µAs 6 µbulk

As , (4.3)

where µbulkAs is the chemical potential of the bulk rhombohedral phase of As,

∆Hf = µGaAs − µbulkAs − µbulk

Ga is the heat of formation of GaAs, and µbulkGa is

the chemical potential of the bulk orthorhombic phase of Ga. We evaluated the

bulk chemical potentials using for the rhombohedral As phase the angle between

the unit-cell vectors α = 54.8 [133] and the theoretical equilibrium lattice pa-

rameter a = 3.70 A (aexp. = 3.80 A). For the Ga orthorhombic structure we

used the cell ratios b/a = 1.70, c/a = 1.00 [133], and the equilibrium lattice

parameter a = 4.41 A (aexp. = 4.52 A). All internal structural parameters were

fully relaxed to determine the equilibrium lattice parameters and the correspond-

ing chemical potentials.4 From our calculations, we obtain µbulkAs = −173.84 eV,

µbulkGa = −61.53 eV, µGaAs = −236.17 eV, and hence ∆Hf = −0.80 eV. Our cal-

culated heat of formation, ∆Hf , compares well with the experimental value of

−0.85 eV [134], and with earlier ab initio calculations [110]. For the formation

energies, we expect a similar uncertainty of about 0.1 eV per atom on the abso-

lute values of εf ; for relative values, we expect a better accuracy of the order of

0.01 eV/atom.

4The calculations for the As (Ga) metallic phase were performed with 116 (75) inequivalentk points in the Brillouin zone. With our choice of k-points grids, the densities of points in thereciprocal space were similar for the As and Ga phases.


III

0 Si ML

2 Si ML

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Fo

rmat

ion

En

ergy

(eV

)

µAsmaxµAs

min

−0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.0

µAs − µAsbulk (eV)

Figure 4.8 Al/GaAs (100)and Al/2 Si ML/GaAs (100)formation energies (per semi-conductor surface atom) forthe interfaces I and II. Theformation energies are dis-played in the physically al-lowed range of values of the Aschemical potential (see text).

In Fig. 4.8 we show the Al/GaAs and Al/2 Si ML/GaAs formation energies, for

the I and II relaxed junctions, as a function of the As chemical potential. The

formation energies are displayed within the allowed range of values of µAs, as

imposed by the inequality (4.3). The formation energies decrease (increase) with

µAs for the junction I (II), according to Eqs. (4.2a–4.2b). The values of εIf and εII

f

computed at µAs = µbulkAs and µAs = µbulk

As + ∆Hf , respectively, are also reported

in Table 4.2 for the relaxed and unrelaxed interfaces. The relaxation lowers the

formation energies by roughly the same amount for the interfaces I and II, i.e.,

by 0.2 eV for a Si coverage ϑ = 0 ML and by 0.1 eV for ϑ = 2 ML.

Fig. 4.8 shows that there is a reversal in the relative stability of interfaces I and

II, which occurs within the experimentally accessible range of the As (or Ga)

chemical potential. This is true for both coverages, ϑ = 0 and 2 ML, and we

therefore expect this trend to hold also for intermediate coverages 0 < ϑ < 2 ML.

This result yields thus a possible explanation, consistent with the contentions

of Ref. [8], for the effect of the excess cation and anion fluxes on the interface

atomic structure, and hence on the Schottky barrier. Under an excess As flux,

Sect. 4.2 Model 73

Table 4.2 Al/Si/GaAs (100) forma-tion energy per surface atom, εf , for theAs-terminated (I) and Ga-terminated(II) interfaces. Results are reported forSi coverages ϑ = 0 and 2 monolayers(ML). εf is evaluated using for the Aschemical potential µAs = µbulk

As for thejunctions I and µAs = µbulk

As + ∆Hf forthe junctions II (see text).

Interface εf (eV)

Type ϑ (ML) Ideal Relaxed

I 0 0.79 0.58

I 2 0.98 0.88

II 0 0.80 0.68

II 2 0.97 0.90

i.e., at high µAs, the interface I is favored with the anion-cation sequence for the

dipole layer, resulting in the large p-type barriers. At sufficiently high cation flux,

instead, µAs is low, and the interface termination with the reversed sequence for

the local dipole becomes more stable, producing the small p-type barriers. This

explanation for the establishment of the tunable Al/Si/GaAs Schottky diodes

is further supported by the quantitative agreement between the computed and

experimental Schottky barriers in Fig. 4.4.

Recent experiments indicated a higher thermal stability of Al/2 Si ML/GaAs

junctions with low p-type barriers (interfaces II) as compared to those with high

p-type barriers (interfaces I) [135]. Although the trend of our T = 0 formation

energies in Fig. 4.8 indicates an increase in the stability of the interfaces II rel-

ative to the interfaces I with increasing Si coverage, the lowest energies we find

are essentially the same (within our numerical accuracy) for the two types of

interfaces at ϑ = 2 ML. In fact, above 300C, the temperature dependence of the

chemical potentials and of the formation energies is expected to become impor-

tant. Arsenic is known to desorb from GaAs surfaces in vacuum upon annealed

above 300C, producing increasingly Ga-rich surfaces [136]. A similar behavior

for the Al/Si/GaAs junctions may be responsible for the observed trend after

annealing at 450C. It should be stressed, however, that formation energies alone

may be insufficient to explain the different thermal stability of the two types of

junctions as the kinetics of the degradation mechanisms may also be important.

4.2 Model

In the previous sections, we have seen that the ab initio calculations for the

Al/Si/GaAs (100) interface correctly reproduce the SBH changes observed ex-

perimentally. The lattice relaxation has a small effect on the calculated φp’s,

and the Si interlayers induce important changes of the LDOS in the interface


region. In the following, we investigate the screening of the Si-induced dipole,

and its relation with the density of states at the interface. We present a model

which describes the effective screening, and we discuss some numerical experi-

ments intended to test its validity. The atomic relaxations are neglected in the

model.

4.2.1 Screening of local dipoles at the interface

The effective screening εeff(ϑ), defined in Eq. (4.1), changes from about 30 at

low Si coverages (ϑ ≈ 0) to about 50 at ϑ = 2. This large variation in εeff

accounts for the non-linear coverage dependence of φp in Fig. 4.4. The screening

of the dipoles created by analogous Si interlayers inserted in a GaAs/GaAs (100)

homojunction has been studied by Peressi et al. [124] and explained in terms

of the GaAs dielectric constant (at low Si coverages) and Si dielectric constant

(at ∼ 2 ML Si coverage). Here εeff is 3 to 5 times larger than the GaAs and Si

dielectric constants. There is therefore a drastic enhancement of the screening

related to the presence of the metal.

At low Si coverage (ϑ ≈ 0), we may describe the variation ∆U of the electrostatic-

potential lineup within a linear-response formulation by

∆Ub ≡ εeff(ϑ = 0) ∆U = (1 + 4πχs + 4πχm) ∆U, (4.4)

where χs = (εs∞ − 1)/4π is the electronic dielectric susceptibility of the semicon-

ductor, and χm accounts for the presence of the metal on the other side of the

junction. The additional susceptibility χm can be related to the MIGS surface

density of states, Ds(EF; x), at the position x of the Si bilayer, based on simple

electrostatic arguments. If the FL is shifted by a small amount ∆U , a charge

Ds(EF; x) ∆U is induced (to the first order) in the MIGS tails, on the semicon-

ductor side of the bilayer. The distance between the bilayer position and the

center of gravity of this induced charge is of the order of the decay length of the

MIGS at the Fermi energy, δs(EF) (see p. 50). An equal and opposite charge

is induced on the metal side of the bilayer, and the two charges give rise to a

potential discontinuity −4πe2Ds(EF; x) ∆U δ, in a direction opposite to the FL

displacement ∆U . Here, δ corresponds to the distance between the positive and

negative charges induced on both sides of the Si bilayer. From Eq. (4.4), one sees

that the MIGS contribution to the change ∆U is −4πχm∆U , which, based on

the above arguments, can be identified with −4πe2Ds(EF; x) ∆U δ. The effective

susceptibility describing the MIGS response to a dipole at a position x is thus

χm = e2Ds(EF; x) δ. (4.5)

Sect. 4.2 Model 75

In the following, we will use for Ds(EF, x) the values obtained from the first-

principle calculations, and treat δ as a parameter, whose value should be between

δs and 2δs, based on the above analysis. In principle, δ could depend on the type

of interface (I or II), and also on the Si coverage ϑ (at high coverage). We have

seen in Chapter 3, however, that δs is basically the same for the Al/GaAs I and II

junctions; to make the model as simple as possible, we will use for both interfaces

and for all coverages ϑ a single value, δ = 3.2 A, which best agrees with our

data on the SBH. This value is relatively similar to the decay length δs ≈ 2.4 A

obtained for the Al/GaAs junctions (p. 50).

In Fig. 4.9 we present the surface density of states Ds(EF; x) for the As- and

Ga-terminated interfaces without Si, and also with 2 Si ML, calculated in the

7 + 21 supercell. The figure shows that the Ds for the two terminations are very

similar. At the position of the Si dipole layer (x = xd) we find in the junctions

without Si Ds(EF; xd) ≈ 0.027 eV−1A−2. At low Si coverage (linear regime for

∆U), the effective screening is thus

εeff(ϑ = 0) = εGaAs∞ + 4πe2Ds(EF; xd) δ ≈ 28, (4.6)

corresponding to the dashed lines in Fig. 4.4, in excellent agreement with the

experiment and first-principle results.5

At higher Si coverage (ϑ ≈ 2), Eq. (4.4) can still be used to describe the macro-

scopic screening of the Si bilayer, provided the gradual changes in the susceptibil-

ities χm and χs are taken into account. For a Si bilayer in the GaAs/GaAs (100)

homojunction, the change in the host susceptibility χs with the Si coverage can

be described by:6 χs(ϑ) = χGaAs + ϑ2

[χSi − χGaAs] [124]. Along the same line,

we should include the change in χm induced by the modification of the MIGS

with Si doping: χm(ϑ) = e2Ds(EF; x) δ + ϑ2e2

[D′

s(E′F; x)−Ds(EF; x)

]δ, where

D′s(E

′F; x) is the surface density of states at the Fermi energy in presence of 2 ML

of Si (see Fig. 4.9). The effective screening for a dipole layer at a position x and

for a coverage 0 6 ϑ 6 2 is thus:

εeff(ϑ, x) = εGaAs∞ + 4πe2Ds(EF; x) δ +

+ϑ

2

εSi∞ − εGaAs

∞ + 4πe2[D′

s(E′F; x)−Ds(EF; x)

]δ

. (4.7)

5We calculated εGaAs∞ and εSi

∞ by inserting Si bilayers with coverages ϑ = 0.2 and ϑ = 0.5 atthe GaAs/GaAs homojunction, and using the capacitor model of Ref. [124]. We find εGaAs∞ =12.4, εSi

∞ = 13.9; the experimental values are 10.9 and 11.4 for GaAs and Si, respectively.6In Ref. [124], a weighted average of the inverse dielectric constants was used to interpolate

the susceptibility at fractional coverages 0 < ϑ < 2. The choice of the average, however, haslittle effect on the present results since the two dielectric constants are close to each other.


ϑ = 2 ML

ϑ = 0 ML

x (Å)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1050

I (As-terminated)

II (Ga-terminated)

(eV

Å

)

−1−2

D (

E ;

x)s

F

x dFigure 4.9 Surface den-sity of states at the Fermienergy in the Al/GaAs (100)junctions without Si and with2 ML of Si at the interface.The solid (dashed) line corre-sponds to the As- (Ga)-termi-nated interface.

The perturbed surface density of states at the position of the Si dipole layer

(Fig. 4.9) is D′s(E

′F; xd) ≈ 0.050 eV−1A−2. Using this value in Eq. (4.7), we

obtain the coverage dependence indicated by the solid line in Fig. 4.4, in excellent

agreement with the results of the self-consistent calculations. We would like to

point out that most of the non-linearity in φp(ϑ) is due to the large difference

between D′s and Ds in Eq. (4.7); the difference between εGaAs

∞ and εSi∞ plays only

a minor role. Finally, based on Eq. (4.7), the symmetry in the barrier variations

of the two interfaces in Fig. 4.4 can be understood from the similarity of their

(Ds, D′s) values (Fig. 4.9).

4.2.2 Inhomogeneous screening near the junction

To probe the correlation between the local-dipole screening and the spatial decay

of the MIGS, which is explicit in Eq. (4.7), we moved the Si dipole layer away from

the interface to a position x > 0 into the semiconductor. Given the exponential

decay of Ds(EF; x) in the semiconductor, we expect εeff(ϑ ≈ 0, x) to exponentially

converge to εGaAs∞ with a similar decay length. We investigated the change in the

screening as a function of the position x of the Si dipole layer for a coverage ϑ =

0.5. We selected this coverage to minimize the numerical uncertainty on εeff within

the linear regime [ϑ = 0 in Eq. (4.7)]. The results for εeff obtained with first-

principle calculations are displayed in Fig. 4.10 together with the corresponding

Schottky barrier values. The exponential decay length of εeff(x) towards εGaAs∞ in

Fig. 4.10 is similar to the decay length of Ds(EF; x) in Fig. 4.9. The solid line

indicates the position dependence predicted by Eq. (4.7) in the linear coverage

regime. The very good agreement, in Fig. 4.10, between the predictions of the

Sect. 4.2 Model 77

Figure 4.10 Schottky barrier (filledsquares, right scale) and effectivescreening (filled diamonds, left scale)as a function of the Si dipole layer po-sition within the semiconductor, in theAl/Si/GaAs (100) junction I. The sym-bols give the results of self-consistentcalculations for a Si coverage ϑ =0.5 ML. The solid lines correspond tothe prediction of Eq. (4.7) in the linear-coverage regime (ϑ = 0). The calcula-tions were done in the 7+ 21 supercell.

x (Å)

0.8

0.9

1.0

1.1

1.2

φ p(e

V)

10

15

20

25

30

35

1050ef

f∋GaAs∋∞

model and the results of the ab initio calculations further confirms the soundness

of the picture here proposed to explain the screening of local interface dipoles at

MS junctions.

4.2.3 Application to Al/Ge/GaAs (100) contacts

If Ge atoms — or other group-IV atoms — are inserted at the polar Al/GaAs (100)

junction instead of the Si atoms, the same bare dipole is induced, and similar SBH

variations are expected. According to our model, Eq. (4.7), the effective screen-

ing is the same in the linear regime for both the Al/Si/GaAs and Al/Ge/GaAs

diodes. At higher coverages, however, the screening is larger in the Al/Ge/GaAs

system, since εGe∞ > εSi

∞.7 We have repeated the same calculations as for the

Al/Si/GaAs junctions, replacing the Si atoms by Ge atoms, and neglecting the

atomic relaxations at the interface. The results for φp as a function of the Ge cov-

erage are shown in Fig. 4.11. At low coverage, the SBH’s for the Al/Ge/GaAs and

Al/Si/GaAs systems are almost indistinguishable, in agreement with the model

prediction. For higher coverages, the dipole-induced change in φp is smaller for the

Ge interlayer than for the Si interlayer, and the effective screening is thus larger

at the Al/Ge/GaAs interfaces. At ϑ = 2 ML, the SBH’s for the Al/Ge/GaAs I

and II interfaces differ by 0.63 eV (0.82 eV for Al/Si/GaAs), corresponding to a

screening εeff(ϑ = 2) ≈ 63 (48 for Al/Si/GaAs).

7We calculated εGe∞ by inserting Ge bilayers with coverages ϑ = 0.2 and ϑ = 0.5 at the

GaAs/GaAs homojunction, and using the capacitor model of Ref. [124]. We find εGe∞ = 18.6;the experimental value is 15.4.


From the self-consistent calculations, we find D′s(E

′F; xd) ≈ 0.054 eV−1A−2 for

the surface density of states of the Al/2 Ge ML/GaAs I and II interfaces, i.e.,

essentially the same value as for the Al/2 Si ML/GaAs systems (0.050 eV−1A−2).

The enhancement of the effective screening at the Al/Ge/GaAs with respect

to the Al/Si/GaAs junctions, in Fig. 4.11, can be only partially explained by

the larger dielectric constant of Ge with respect to Si. Using the parameter

δ = 3.2 A, as in the Al/Si/GaAs junctions, and εGe∞ = 18.6 in Eq. (4.7), we

obtain the dashed line in Fig. 4.11, which overestimates the SBH changes at high

Ge coverages. This is probably due to some additional non-linear effects, which

could be neglected in the case of the Al/Si/GaAs system, but become important

in the Al/Ge/GaAs junction. In particular, we neglected in Eq. (4.7) the change

in δ with the interlayer atomic coverage. Fig. 4.11 shows that the additional non-

linearities can be accounted for by using a slightly higher value of δ in the model;

the model prediction with δ = 4 A is shown by the dotted line and provides a

better description of SBH values at high Ge coverages.

ϑ (ML)0.0 0.5 1.0 1.5 2.0

0.4

0.6

0.8

1.0

1.2

φ

(eV

)p

I (As-terminated)

II (Ga-terminated)

Al/Ge/GaAs

Al/Si/GaAs

, δ = 3.2 Å

, δ = 4.0 Å

Ge∋∞Ge∋∞

Si∋∞ , δ = 3.2 Å

Figure 4.11 Comparison of the Schottky barrier heights in the Al/Ge/GaAs (100)and Al/Si/GaAs (100) diodes as a function of interlayer atomic coverage. The circlesare the results of the self-consistent calculations (shifted by +0.25 eV). The curves givethe predictions of the model, Eq. (4.7), for the Si interlayer (solid lines) and for theGe interlayers with δ = 3.2 A (dashed lines) and δ = 4.0 A (dotted lines). The straitdashed lines indicate the model predictions in the linear regime with δ = 3.2 A.


4.3 Al/Si/GaAs junctions with 2–6 Si ML

The Si-induced-dipole picture, proposed here to interpret the SBH changes at

Al/Si/GaAs diodes with up to 2 ML of Si, can be easily extended to thicker

interlayers. From a macroscopic point of view, the ionic charge induced by a

Si interlayer of any width at the Al/GaAs (100) junction is equivalent to two

opposite surface charges on both sides of the Si layer.8 The case of 6 ML of Si at

the As-terminated junction is illustrated in Fig. 4.12. The macroscopic average

of the induced ionic density shows two opposite charges in Fig. 4.12(a), which

lead to the ionic dipole field in the Si layer in Fig. 4.12(b). The bare potential

difference induced by the ionic charge is given by ∆Ub = ± πe2

aGaAsϑ, as in the

regime 0 6 ϑ 6 2 ML. With respect to the unperturbed Al/GaAs junction, an

electronic dipole is induced between the Al/Si and the Si/GaAs interfaces, and

it reduces the bare discontinuity to ∆U , as shown in Fig. 4.12(b). For the 6 ML

case, we have ∆Ub = 48.9 eV; the electronic contribution reduces the potential

discontinuity to ∆U = 0.69 eV. In the spirit of Eq. (4.1), this corresponds to an

effective screening εeff(ϑ = 6) ≈ 70, much larger than the value of ∼ 50 obtained

in the Al/2 Si ML/GaAs junctions.

−5 0 5 10−1.0

−0.5

0.0

0.5

1.0

Den

sity

(el

ectr

on

s/Ω

)

As GaAl

(a)

Si

x (Å)

−20

−10

0

10

20

−5 0 5 10

Po

ten

tial

(eV

)

x 10

∆Ub

∆U

(b)

As GaAl Si

x (Å)

Figure 4.12 (a) Macroscopic averages of the ionic (dotted lines) and electronic (solidlines) charge densities induced by 6 ML of Si at the As-terminated Al/GaAs (100)junction. (b) Macroscopic average of the corresponding ionic and electronic potentials.The thick line represents the total electrostatic potential scaled by a factor of 10. Thebare, ∆Ub, and screened, ∆U , discontinuities across the Si interlayer are also indicated.

8We must assume that the Si atoms are always distributed over an even number of GaAsplanes, to ensure that the Si interlayer does not induce a macroscopic electric field.


4.3.1 Schottky barrier modification

We have calculated the modification of the Al/GaAs SBH induced by 2, 4 and

6 ML of Si inserted at the As- and Ga-terminated junctions, using the 7 + 21

supercell. The results are reported in Table 4.3. In the case of the junction I,

the p-type SBH increases by 0.32 eV when 2 ML are inserted at the Al/GaAs

interface, but only by 0.26 eV in going from 2 ML to 4 ML, and by 0.11 eV in

going from 4 ML to 6 ML, indicating a saturation of the SBH change for in-

creasing ϑ. The same conclusion can be drawn in the case of the junction II.

These findings are consistent with the experimental measurements [137], which

indicate a saturation of the SBH modification for Si thicknesses ϑ & 4 ML. The

SBH variation is expected to saturate when the FL reaches the edges of the

GaAs bandgap. This mechanism could influence our SBH’s for ϑ > 4 ML, be-

cause the LDA-DFT bandgap of GaAs is smaller than the experimental bandgap.

Experimentally, however, the smallest and largest p-type SBH’s achieved in the

engineered Al/Si/GaAs junctions are 0.2 eV and 1.2 eV. Thus, the experimental

SBH’s remain well within the room-temperature GaAs bandgap (1.42 eV). The

observed saturation could be due to outdiffusion of the Si atoms in the Al over-

layer. Although this mechanism probably plays an important role, especially at

large coverages, we will present, in the next section, another possible explana-

tion for the saturation of the SBH’s in the range 4–6 ML, based on the effect of

resonant interface states in the GaAs bandgap.

4.3.2 Effect of resonant interface states

Evidence for resonant interface states. — The pioneering ab initio study of the

electronic structure at a semiconductor heterojunction, by Baraff et al. [117],

revealed the existence of interface states in the fundamental bandgap at the Ga-

terminated GaAs/Ge (100) heterojunction. This finding was later confirmed by

other authors [138], and analogous states were reported lower in the valence

bands at the GaAs/Ge (110) non-polar interface [118]. Recently, interface states

were also found at the coherently strained Si/GaAs (110) interface [139], and a

comparison of the Si/GaAs (110) heterojunction, with the GaAs/Si/GaAs (110)

and GaAs/Si/GaAs (100) engineered homojunctions, with 1 and 2 ML thick

Si interlayers, led to a general understanding of the origin of these interface

states [140].

At the Al/Si/GaAs (100) junction, the same interface states are expected to de-

velop at the Si/GaAs interface, for sufficiently thick Si interlayers. At low Si

coverage, however, they interact with the continuum of metal states and become


Table 4.3 Modification of the Al/GaAsSchottky barrier, ∆U , induced by 2, 4,and 6 ML of Si at the As-terminated (I),and Ga-terminated (II) interfaces. Thereference Al/GaAs SBH’s calculated inthe 7 + 21 supercell are φI

p = 0.90 eV andφII

p = 0.83 eV for the interfaces I and II,respectively (including a +0.25 eV quasi-particle and spin-orbit correction).

Si coverage ∆U (eV)

ϑ (ML) I II

2 0.32 −0.35

4 0.58 −0.49

6 0.69 −0.61

resonant. In order to follow the localization of the resonant interface states as a

function of the Si coverage, we have calculated the LDOS at the Al/Si/GaAs (100)

I and II interfaces, with 2, 4, and 6 ML of Si, using the 7 + 21 supercell. The

results are displayed in Fig. 4.13. The LDOS is represented in the energy region

of the GaAs fundamental bandgap. The supercell FL, EF, and the VBM (zero of

energy), are also indicated. For both the As- and Ga-terminated interfaces, the

figure shows the formation of Si/GaAs interface states, which are clearly visible al-

ready for 4 Si ML. In the junctions I, the resonant interface states (energies ∼ EI)

derive from the oversaturated Si–As bond (donor bond); these states are pushed

down from the conduction band into the bandgap. In the junctions II, the inter-

face states (energies ∼ EII) derive from the undersaturated Si–Ga bond (acceptor

bond); they are pushed up from the valence band into the bandgap.

In Fig. 4.14(a), we display the bands of resonant interface states at the Al/6 Si ML/

GaAs (100) I and II junctions in the 2-dimensional BZ. The bands have been

aligned with respect to the bulk GaAs projected band structure using the calcu-

lated difference between the average electrostatic potential in the supercell and

in the GaAs slab. The band of resonant interface states in the junction I (II) is

responsible for the structure EI (EII) in Fig. 4.13. In Fig. 4.14(b), we represent

the probability density of both resonant states at the Γ point. In the junction I,

the state includes predominantly antibonding-like sp3 hybrids of the Si and also,

to a lesser extent, of the As atoms at the interface. This state is related to the

bulk conduction bands. However, because of the more attractive ionic potential

in the region between the Si and As planes, with respect to the ionic potential

in bulk Si and GaAs, this state is lowered below the Si and GaAs CBM’s. The

inverse mechanism occurs in the junction II. The ionic potential is repulsive in the

region between the Si and Ga planes, and a state at Γ composed of 3p bonding

orbitals of the interfacial Si and Ga atoms is pushed from the top of the valence

band to higher energies. The p-bonding character of the resulting interface state

is clearly visible in Fig. 4.14(b).


I II

−1

0

1

2Al As GaSi

2 ML

VBM

E F

E (

eV)

−1

0

1

2Al Ga AsSi

2 ML

VBM

E F

E (

eV)

−1

0

1

2Al As GaSi

4 ML

VBM

E F

E (

eV)

−1

0

1

2Al Ga AsSi

4 ML

VBM

E F

E (

eV)

−1

0

1

2Al As GaSi

6 ML

VBM

E F

x [100]

E (

eV) E I

−1

0

1

2Al Ga AsSi

6 ML

VBM

E F

E (

eV)

x [100]

E II

Figure 4.13 Local density of states, D(E; x), in the Al/Si/GaAs (100) I and IIjunctions, with 2, 4, and 6 ML of Si. The dark (bright) regions correspond to low(high) densities. We have used the calculated SBH to place the valence-band maximum(VBM) with respect to the supercell Fermi level EF. EI (EII) indicates the energyposition of the center of the bright spot associated with the Si–As (Si–Ga) resonantinterface states.


(a)

Γ J K Γ

−4

−2

0

2

4

E (

eV)

Γ J K Γ

−4

−2

0

2

4

[100]

[001

]

SiAs

Al

0 5 100.0

0.5

1.0 Al Si As

x (Å)

−2−3

( X 1

0

Å

)|ϕ

(x)|2

(b)

0 5 100.0

0.5

1.0Al Si Ga

x (Å)

[100]

[001

]

SiGa

Al

I II

(c)

0.0 0.5 1.0 1.5E (eV)

E F

Dis

(E)

−2−1

(X 1

0

eV

Å

)−2

0

2

4

6

8

10

E I

0.0 0.5 1.0 1.5E (eV)

E F

0

2

4

6

8

10

E II

Figure 4.14 Resonant interface states in the Al/6 Si ML/GaAs (100) junctions I(left) and II (right). (a) Dispersion of the resonant interface states (solid line); the grayarea corresponds to the projected bulk band structure of GaAs. (b) Probability densityof the resonant states at the Γ point. In the contour plots, the plane is defined by theSi–As (I) and Si–Ga (II) bonds at the Si/GaAs interface. The dotted circles and linesindicate atoms and bonds which are out of the plane. The planar average of the densityis shown below the contour plots. (c) Surface density of states Dis in the energy regionof the fundamental bandgap. Dis is obtained by integrating the LDOS in the spatialregion defined by the second Si plane from the Si/GaAs interface (indicated by thedashed line in the panels (b)) and the middle of the GaAs slab (in the 7+21 supercell).The zero of energy is at the VBM. The FL for both junctions is also indicated.


Looking at the planar average of the density in Fig. 4.14(b), one can see the

resonant nature of the Si–As and Si–Ga interface states, which are coupled with

the continuum of metallic states across the Si interlayer. Because of their energy

location in the GaAs bandgap, these states may play an important role in the

transport properties of the engineered junctions, by acting as recombination cen-

ters, or facilitating the tunneling of electrons through the barrier. From the band

dispersion in Fig. 4.14(a), one sees that the peak of the LDOS at the energy EI

(EII) in the junction I (II) in Fig. 4.13 comes mainly from the states close to the

Γ point, and in the vicinity of the Γ–J line. In the rest of the BZ, the resonant

states are well above the CBM (junction I) or well below the VBM (junction II).

The density of interface states in the bandgap, Dis(E), can be calculated according

to: Dis(E) =∫ xs

x0D(E; x) dx, where xs is the center of the GaAs slab, and

x0 defines a plane between the Al surface and the interface states. Rigorously,

this definition may be applied only if the LDOS vanishes at x0 (and xs), and

if the whole charge of the interface states is confined between x0 and xs. This

is not possible in our case since the states are resonant. However, in order to

estimate the density of interface states, we have arbitrarily chosen the position

x0 as indicated by the dashed vertical lines in Fig. 4.14(b). The resulting density

of states is displayed in Fig. 4.14(c) for the junctions I and II. The zero of energy

is at the VBM, as in Fig. 4.13, and the FL is also indicated for each junction.

In the case of the junction I, the main peak at ∼ 1.2 eV comes from the flat-

band region between Γ and J in Fig. 4.14(a). On the low energy side of the

main peak, one can distinguish an additional shoulder, associated with the J

point. In the junction II, the density of states shows two well separated peaks

at ∼ 0.05 eV and ∼ 0.4 eV, which correspond to the regions near the J and Γ

points, respectively. At the Fermi energy, we find a density of resonant interface

states Dis(EF) ≈ 0.06 eV−1 A−2 for the junctions I and II.

Model for the SBH behavior. — One can extend the linear-response approach

developed in section 4.2, to describe the saturation of the SBH change in terms

of an effective screening of the bare discontinuity ∆Ub by the resonant states. In

analogy with Eq. (4.4), we write

∆Ub = (1 + 4πχs + 4πχm + 4πχis) ∆U, (4.8)

where χis is an effective polarizability which describes the effect of the resonant

interface states. At a given Si coverage ϑ, the change of the SBH with respect

to the unperturbed Al/GaAs interface is ∆U(ϑ); thus, the position of the FL in

the GaAs bandgap is EF(ϑ) = E0F + ∆U(ϑ), where E0

F is the FL at the unper-

turbed interface. Assuming that Dis(E) does not depend on the coverage ϑ, the


surface charge induced in the Si/GaAs interface states by the FL displacement

is σis(∆U) =∫ E0

F+∆U

E0F

Dis(E) dE. This charge is compensated by an opposite

charge on the metal surface, and a potential difference −4πe2σis(∆U) l is in-

duced between the metal surface and the Si/GaAs interface, where l ≈ 14aGaAsϑ

is the width of the Si interlayer. This potential difference can be identified

with the contribution of the interface states to the total discontinuity ∆U , i.e.,

−4πe2σis(∆U) l = −4πχis∆U ; the derivative with respect to ∆U yields:

χis = e2Dis(EF(ϑ)) 14aGaAsϑ. (4.9)

For large Si coverages ϑ, the term 1 + 4πχs + 4πχm is negligible with respect

to 4πχis in Eq. (4.8). Since ∆Ub and χis are both linear in ϑ (neglecting the ϑ-

dependence of the interface-state density in Eq. (4.9)), the screened discontinuity

∆U should saturate as ϑ increases:

∆U(ϑ →∞) =∆Ub

4πχis

= ± 1

Dis a2GaAs

. (4.10)

In Eq. (4.10), Dis should be evaluated at EF(ϑ → ∞) and the minus sign is

for the junction II. At ϑ = 6 ML, the density of interface states at the FL is

of the order of 0.06 eV−1 A−2. Using this value for Dis in Eq. (4.10), we obtain

∆U(ϑ →∞) = ±0.54 eV, in good qualitative agreement with the calculated SBH

modifications at ϑ = 6 ML (Table 4.3) and with the maximum SBH variation

measured experimentally for Al/Si/GaAs engineered junctions (∼ 0.4 eV).

4.4 Discussion

In our calculations, the SBH variations in Al/Si/GaAs (100) engineered hetero-

junctions are driven by screened microscopic dipoles. These dipoles are due to

the heterovalent nature of the Si substitutions in GaAs; for Si grown on the polar

(100) surface of GaAs, the Si-induced dipole can increase or decrease the SBH,

depending on whether the GaAs surface is terminated by an As or a Ga plane.

We discuss now some other interpretations of the SBH variations in Al/Si/GaAs

contacts, which are based, instead, on the macroscopic properties of degenerate

bulk Si.

Nathan and coworkers [10] proposed a simple heterostructure model to explain

their results in Al/Si/GaAs junctions with 6 to 100 A thick Si interlayers. The

band alignment in the heterostructure was obtained by solving Poisson’s equation

in the junction with the following assumptions: (i) the position of the FL with

respect to the Si VBM at the Al/Si interface is fixed, and is the same as in the


fully developed Al/Si MS junction, i.e., φAl/Sip ≈ 0.4 eV; (ii) the valence- and

conduction-band discontinuities at the Si/GaAs interface are ∆ESi/GaAsv ≈ 0.0 eV

and ∆ESi/GaAsc ≈ 0.3 eV; (iii) the equilibrium bulk band structure of Si can be

used to describe the Si layer. For thin interlayers (< 10 A) they found that the

band bending in the Si is negligible, even for impurity concentrations as high as

1020 cm−3 (the usual doping level of n+-Si and p+-Si is about 5×1019 cm−3 [141]).

At low Si coverage, the model thus predicts a SBH φp ≈ φAl/Sip + ∆E

Si/GaAsv =

0.4 eV. For thicker interlayers (60 A) the model SBH becomes sensitive to the

doping concentration in the Si layer. A 0.7 eV increase in φp is predicted for

a n-doped interlayer with ND − NA = 5 × 1019 cm−3, and a 0.2 eV decrease

in φp is obtained for a p-doped interlayer with NA − ND = 1019 cm−3, where

ND and NA are the donor and acceptor concentrations in the Si, respectively.

When the Si coverage is further increased, the band bending in the Si is such

that the FL enters the Si valence band (p-doped Si) or conduction band (n-doped

Si) at the Si/GaAs interface, and the SBH no longer changes as a function of

the interlayer thickness. Therefore, the minimum and maximum p-type SBH’s

predicted for thick n+- or p+-Si interlayers are φminp = ∆E

Si/GaAsv = 0.0 eV and

φmaxp = EGaAs

g − ∆ESi/GaAsc = 1.1 eV. To explain their experimental results,

the authors of Ref. [10] assume a doping level of 5× 1019 cm−3 in the interlayer.

Under these conditions, the extremal values of the SBH are obtained for interlayer

thicknesses of the order of 60–80 A.

The model of Ref. [10] was refined by the Hasegawa group [11], in order to include

the effect of interlayer relaxation, when the Si interlayer thickness is larger than

the critical thickness (∼ 10–14 A). This new model takes into account the presence

of a continuum of defect-induced states in the bandgap of the relaxed Si, and

predicts a slower saturation of the SBH with Si thickness, as compared to the

model of Nathan and coworkers. However, the minimum and maximum p-type

SBH’s obtained for large Si coverage are still dictated by the relative position of

the Si and GaAs bands at the Si/GaAs interface, and identical to those of Nathan

and coworkers, i.e., 0.0 6 φp 6 1.1 eV.

Experimentally, the lowest values of φp reported for Al/Si/GaAs junctions are

close to 0.2 eV, in disagreement with the above limit of 0.0 eV. Moreover, at

low Si coverage, the results of Waldrop and Grant [9] and Cantile et al. [8] show

important variations of the SBH while the above theories predict no SBH changes

for thin interlayers. These two problems were pointed out by Chen, Mohammad,

and Morkoc [141]. These authors proposed a revised band-structure model, which

incorporates the effect of the strain on the Si bandgap, and considers the FL as

unpinned at the Al/Si interface. The bulk Si bandgap is known [142] to decrease

Sect. 4.4 Summary 87

by 40% when Si (100) is coherently strained to the GaAs lattice parameter, as a

result of the tetragonal contraction εxx(Si) ≈ −3%. In the model of Chen et al.,

the Si interlayer contracts according to macroscopic elasticity theory, and has a

bandgap ESig = 0.7 eV. If the FL is unpinned at the Al/Si interface, the SBH

can vary in the range ∆ESi/GaAsv 6 φp 6 EGaAs

g −∆ESi/GaAsc . Using ∆E

Si/GaAsv =

0.4 eV [141], one obtains ∆ESi/GaAsc = 0.3 eV with a Si bandgap of 0.7 eV, and

the range of possible SBH’s becomes 0.4 6 φp 6 1.1 eV, in good qualitative

agreement with the experimental data, 0.2 6 φexp.p 6 1.2 eV.

Our results for the coherently strained Al/2 Si ML/GaAs (100) system, however,

call into question the applicability of bulk Si concepts in the explanation of the

tuning. All of the models we described in this section rely on the existence of

a bulk Si bandgap in the interlayer. Moreover, based on macroscopic elasticity

theory, the model of Chen et al. postulates a 3% contraction of the Si–Si inter-

planar distance at low coverage. We find, however, that none of the above holds

for the coherently strained Al/Si/GaAs (100) system. Our ab initio results show

that there is no bandgap in the region of the Si interlayer (Fig. 4.7), and that

the Si interplanar distance elongates in the 0–2 ML coverage regime, when the

tuning takes place.

Summary

We have investigated from first principles the structural and electronic proper-

ties of the Al/Si/GaAs (100) junctions with Ga-terminated and As-terminated

GaAs (100) interfaces. The Schottky barriers calculated for Si coverages 0 6 ϑ 62 ML using the virtual-crystal approximation reproduce the experimental data,

and support the interpretation that the measured SBH variations result from

local interface dipoles produced by the heterovalent Si interlayer. The interfa-

cial atomic relaxation was examined, and explained in terms of covalent- versus

metallic-type of bonds established at the interface. For the Al/2 Si ML/GaAs

(100) junctions, we found that, in contrast to the trend expected from macroscopic

elasticity theory, and at variance with the situation in GaAs/2 Si ML/GaAs (100)

heterostructures, the Si–Si interplanar distance is increased relative to the spac-

ing in bulk Si. This behavior is due to an Al-induced weakening of the covalent

bonding in the interlayer. Moreover, based on the energetics of the systems, we

showed that the relative stability of the As- and Ga-terminated junctions changes

within the experimentally accessible range of the As and Ga chemical potentials.

We showed that this change is consistent with the Schottky barriers observed ex-

perimentally, and could explain the role of the excess cation and anion fluxes in


the establishment of the relevant interface configurations. We also investigated

the screening of the local dipoles induced by the interlayers, and presented a

model explaining the effective screening in terms of the semiconductor dielectric

constant and the local interface density of states at the Fermi energy.

Furthermore, we studied the Si/GaAs resonant interface states in Al/Si/GaAs

junctions with 4 to 6 monolayers of Si, and presented a possible explanation of

the saturation of the SBH changes as a function of the Si coverage, in term of

the density of resonant interface states in the GaAs bandgap. Finally, we dis-

cussed empirical band-structure models recently proposed to explain the tunable

Al/Si/GaAs Schottky barriers. Our ab initio results for the coherently strained

Al/2 Si ML/GaAs (100) system showed that the macroscopic properties of bulk

Si postulated in these models are inappropriate to describe the electronic and

atomic structures of the interlayer, and hence the tuning established at low Si

coverage (0–2 ML).

Our theoretical results suggest a couple of new experiments, which could help

settling the issue of the Al/Si/GaAs SBH tuning. First, the spatial correlation,

indicated by the ab initio calculations, between the local dipole screening and

the MIGS density of states, calls for an experimental confirmation. This could be

done, in principle, by growing Al/GaAs/Si/GaAs (100) epitaxial structures, and

measuring the SBH as a function of both the GaAs and Si interlayers thicknesses.

The control, however, of the atomic-scale structure of such contacts is a consider-

able experimental challenge. Secondly, spectroscopic studies of the Al/Si/GaAs

junction could allow one to confirm or invalidate the existence of the resonant

states obtained theoretically at the Si/GaAs interface.

Chapter 5

Ionicity, surface properties, and

Schottky barrier heights

The problem of the SBH formation has been traditionally investigated by study-

ing the dependence of the SBH on the characteristics of the metal used in the MS

junction. In this chapter, we approach the problem from a different point-of-view,

looking at the dependence of the SBH on the chemical composition and surface

properties of the semiconductor, for a given metal. In Fig. 5.1, we report some

experimental results for the Al/semiconductor SBH, which illustrate our motiva-

tions. The figure shows a general increase of the SBH with the semiconductor

ionicity, and some variations due to interface-specific features related to struc-

tural and/or chemical properties of the semiconductor surface. Such variations

tend to increase with the semiconductor ionicity. Experimentally, the investiga-

tion of the dependence on interface-specific features is challenging, because of the

technical difficulty of controlling the atomic structure of the interfaces. Ab initio

calculations performed for model atomic geometries can provide complementary

Figure 5.1 Experimental values of thep-type Schottky barrier height in Al/se-miconductor junctions, for various initialreconstructions or chemical compositionsof the semiconductor surface. On thehorizontal axis, the semiconductors areclassified by order of increasing ionicity.The values are taken from Ref. [143] forAl/Ge (111), Ref. [74] for Al/GaAs (100),and Ref. [76] for Al/ZnSe (100).

c (2 x 2)

(2 x 1)

(1 x 1)

φ p (

eV)

As-rich

Ga-rich

0.0

0.2

0.4

0.6

0.8

1.8

2.0

2.2

GaAs ZnSeGe

Se-rich

Zn-rich

90 Ionicity, surface properties, and Schottky barrier heights Chap. 5

information and help in understanding some of these trends. In particular, for

metal contacts to wide-bandgap semiconductors which generally exhibit large

Schottky barriers, there is presently much interest, in connection with the prob-

lem of establishing ohmic contacts, to understand and control the SBH variations

with the semiconductor surface properties.

In this chapter, we first give an explanation for the general trend of the SBH with

the chemical composition of the bulk semiconductor. We then study the effect

of changes in the interface morphology. We will discuss in particular the effect

of the different terminations (anion or cation) of the semiconductor surface in

the abrupt polar Al/GaAs, Al/AlAs, and Al/ZnSe (100) junctions. In the last

section, we examine the variations of the SBH due to different reconstructions of

the semiconductor surface in Al/ZnSe (100) junctions [76].

5.1 Abrupt Al contacts to Ge, GaAs, AlAs, and

ZnSe (100)

In Chapter 3, we briefly presented our calculated SBH values φLDAp for the abrupt

Al contacts to the lattice matched semiconductors Ge, GaAs, AlAs, and ZnSe

(Table 3.1). These values were introduced to discuss semiempirical models of

Schottky barrier formation. To compare these values to experiment, we must

add the quasiparticle and spin-orbit corrections discussed in section 2.2.5. These

corrections are given in Table 5.1. For Al/GaAs, we reported the value of

+0.25 eV discussed in Chapter 4, which corresponds to a quasiparticle correction

∆εGaAs = −0.36 eV on the GaAs VBM [14].1 For the Ge (AlAs) quasiparticle cor-

rections, we use the correction for GaAs, and the difference between the Ge (AlAs)

and GaAs corrections evaluated in Ref. [144], i.e., ∆εGe − ∆εGaAs = +0.09 eV

(∆εAlAs − ∆εGaAs = −0.11 eV).2 With the experimental spin-orbit splittings

∆Geso = 0.30 eV and ∆AlAs

so = 0.28 eV, the total corrections on φLDAp for the Al/Ge

and Al/AlAs junctions are 0.17 and 0.36 eV, respectively. The quasiparticle cor-

rections to the band structure of ZnSe have been evaluated in Ref. [146]. As

1According to the discussion in section 2.2.5, we neglect the quasiparticle correction ∆εm

on the metal Fermi energy in Eq. (2.27).2The GW calculations for Ge and AlAs in Ref. [144] have been performed using the von

Barth-Hedin form of the xc potential. The quasiparticle correction to the LDA bandgap ismore or less independent of the choice of the xc potential, but the distribution of this correctionbetween the valence and conduction bands depends on this choice [145]. Since we employ theCeperley-Alder xc potential in our calculations, we prefer to use the correction of Ref. [14]for GaAs, based on this potential, and only the difference between the Ge, AlAs, and GaAscorrections of Ref. [144].

Sect. 5.1 Abrupt Al contacts to Ge, GaAs, AlAs, and ZnSe (100) 91

Table 5.1 Estimated quasi-particle and spin-orbit correc-tions to φLDA

p for differentsemiconductors. The calcu-lated SBH’s including thesecorrections are shown in thelast two columns. All num-bers are in eV.

Semi- Estimated φp

conductor correction I II

Ge 0.17 0.21 0.21

GaAs 0.25 0.86 0.76

AlAs 0.36 1.45 1.16

ZnSe 0.36 2.18 1.82

the LDA bandgap in our calculations and in Ref. [146] are different, due to the

different pseudopotentials employed, we take the valence-band-edge correction of

Ref. [146] and scale it by the ratio of the difference between the LDA and GW

bandgap in the two calculations. The resulting estimate for ∆εZnSe is −0.50 eV.

With the experimental spin-orbit splitting ∆ZnSeso = 0.43 eV, the total correction

on the Al/ZnSe SBH is 0.36 eV.

The resulting SBH’s for the abrupt Al/Ge, Al/GaAs, Al/AlAs and Al/ZnSe (100)

I and II interfaces, including the estimated corrections, are shown in Table 5.1,

and displayed in Fig. 5.2 together with the typical ranges of experimental values.

The general agreement between theory and experiment in Fig. 5.2 indicates that

our calculations for ideal MS structures capture the general trend of the SBH with

the semiconductor chemical composition. We stress that a detailed comparison

with the experimental values is not justified at this stage, given the simplicity

of the structures considered (abrupt interfaces with no atomic relaxation). For

Al/ZnSe, we will see that the inclusion of the appropriate reconstructions and

relaxation brings the theoretical results in close agreement with the experimental

values in Fig. 5.2. We also note that the calculated change in the SBH due to

the semiconductor-surface termination, φIp−φII

p , increases with the semiconductor

Figure 5.2 Schottky barrierheight at Al/semiconductor (100)contacts. The circles show the cal-culated SBH’s for the ideal junc-tions, corrected for quasiparticleand spin-orbit effects. The shadedregions show typical ranges of ex-perimental values, based on thedata from Refs. [1, 143] for Al/Ge,Refs. [129, 73] for Al/GaAs,Refs. [147, 74] for Al/AlAs, andRef. [76] for Al/ZnSe. GaAs AlAs ZnSeGe

(eV

)φ p

0.0

0.5

1.0

1.5

2.0(Anion terminated)(Cation terminated)(Al / IV−IV)

Range of experimental values

φp

φp

φp

I

II


ionicity.

5.2 Interpretation and models

5.2.1 General trend with the semiconductor composition

In this section, we focus on the behavior of the average SBH

φp =1

2

(φI

p + φIIp

). (5.1)

The “splitting”, φIp − φII

p , due to the semiconductor-surface termination, will be

discussed in the following section. Our purpose here is to show that the variation

of φp with the semiconductor ionicity, from the group-IV materials to the III-V

and II-VI compounds, is controlled essentially by the same bulk mechanisms that

determine the band offsets at non-polar semiconductor heterojunctions [107].

In the case of non-polar, or isovalent, lattice-matched semiconductor heterojunc-

tions, a linear-response-theory approach, which treats the interface as a pertur-

bation with respect to an appropriate bulk reference system (the virtual crystal),

showed that the potential lineups, and therefore the band offsets, depend only on

the bulk properties of the semiconductor constituents [107]. Based on a similar

type of approach, the following model may be established (see Appendix B):

φmodp = φp [Al/〈s〉 (100)] + VBO [〈s〉/s (110)] (5.2)

for the average SBH of the interfaces I and II between Al and the III-V or II-

VI semiconductor s (s = GaAs, AlAs, ZnSe). The first term on the right-hand

side of Eq. (5.2) is the SBH at the (100) interface between Al and the virtual

group-IV crystal, denoted 〈s〉, which is obtained by averaging the anion and

cation pseudopotentials of the III-V or II-VI compound. The second term is the

valence-band offset of the non polar 〈s〉/s (110) heterojunction.

To derive Eq. (5.2), the basic approximation is to built the charge density of the

Al/s I and II interfaces, starting from the reference Al/〈s〉 system, by adding

a linear superposition of the charge densities induced in the crystal 〈s〉 by the

single anion and cation substitutions that transform 〈s〉 into s. The Al/〈s〉 (100)

junction is an appropriate reference system, which minimizes the deviations of

φp from φmodp in Eq. (5.2); these deviations vanish to the first order in the sub-

stitutions which transform Al/〈s〉 into the Al/s I and II junctions. In Fig. 5.3,

we compare graphically the results of the model, Eq. (5.2), with the calculated

Sect. 5.2 Interpretation and models 93

Figure 5.3 Comparison ofthe average SBH φLDA

p at theAl/s (100) I and II interfaceswith the results of the model,Eq. (5.2). The horizontal barshows the average SBH, andthe small dot indicate the re-sult of the model, i.e., the sumof the SBH at the Al/〈s〉 (100)junction (gray circles) and theVBO at the 〈s〉/s (110) inter-face (double arrows). GaAs AlAs ZnSeGe

0.0

0.5

1.0

1.5

2.0

⟨ GaA

s⟩ /

GaA

s

⟨ AlA

s⟩ /

AlA

s ⟨ Zn

Se⟩ /

Zn

Se

(

eV)

φ pLDA

φp [ Al/ (100) ]⟨s⟩LDA

φpLDA φp

mod

average SBH of the Al/s I and II interfaces.3 The valence-band offsets of the

〈GaAs〉/GaAs, 〈AlAs〉/AlAs, and 〈ZnSe〉/ZnSe (110) heterojunctions have been

computed using supercells containing 8 planes of each material in the ideal (un-

relaxed) lattice-matched geometry. The same energy cutoffs and k-points grids

were used as in the calculations of the corresponding Schottky barriers. The SBH

of the Al/〈s〉 (100) junctions were obtained using the same parameters as for the

Al/s (100) I and II junctions. The results in Fig. 5.3 show that Eq. (5.2) provides

an accurate description of the average SBH φp. The SBH’s at the Al/〈s〉 junctions

are all small, due to the small bandgaps of the virtual crystals (E〈s〉g < 0.4 eV),

and similar to the LDA SBH at the Al/Ge (100) interface (0.04 eV). The figure

shows that the VBO’s, which are bulk-related quantities, give the most important

contribution to the average barrier heights, and are behind the general increase

of the Schottky barrier from the IV-IV to the III-V and to the II-VI materials.

5.2.2 Effect of surface termination

In this section, we show how one can understand the difference φIp−φII

p due to the

semiconductor-surface termination — and in particular the fact that the SBH is

systematically higher for the anion than for the cation termination — in terms of

surface-charge and image-charge effects. The mechanism is illustrated in Fig. 5.4.

With respect to the Al/〈s〉 interface, the ionic charge distributions of the inter-

faces I and II are obtained by substituting an anion (charge +σ) on each anionic

site and a cation (charge −σ) on each cationic site [Fig. 5.4(a)]. For the III-V

and II-VI compounds, we have σ = 1S

and σ = 2S, respectively, with S = a2

s/2

3The quasiparticle and spin-orbit corrections are not included in these calculations. Thesecontributions are, by definition, bulk terms which verify Eq. (5.2).


Al + | ∆U |

Imagecharge

Surfacecharge

Al − | ∆U |

Imagecharge

Surfacecharge

Al IV Al IV

+σ

−σ

+σ

−σ

Al IV

+σ

−σ

+σ

−σ

Al IV

I II

(a)

(b)

(c)

Surface charge

Surface charge

Figure 5.4 (a) Planar average of the difference in ionic charge density between theanion- (cation-) terminated Al/s (100) interface, and the Al/〈s〉 (100) interface; σ = 1for the semiconductors s = (GaAs, AlAs), and σ = 2 for s = ZnSe. (b) Macroscopicaverage of the ionic charge density difference. (c) Positive (negative) discontinuityestablished at the interface I (II) by a positive (negative) surface charge and its imagecharge on the metal surface.

the unit-cell surface in the (100) plane. The arrows in Fig. 5.4(a) represent op-

posite delta functions on each anionic and cationic (100) plane, corresponding to

the planar average of the ionic charge density. The macroscopic average of this

ionic charge density is represented in Fig. 5.4(b). In the bulk semiconductor, the

macroscopic average eliminates the atomic-scale oscillations of the planar charge;

at the interface, however, a positive (negative) charge density subsists in the junc-

tion I (II). This macroscopic charge has a density ρ = 2σ/as, and it extends over

a distance as/4 between the last Al plane and the first semiconductor plane. It

is therefore equivalent to a surface charge of density σ = 12σ.

Classically, a plane of charge in a semiconductor is reduced by the macroscopic

dielectric constant ε∞ of the host material. In the presence of a metal, the

screened surface charge is neutralized by an image charge on the metal surface,

as shown in Fig. 5.4(c), and a potential difference is established between the two


(a)

−5 0 5 10

Den

sity

(el

ectr

on

s/Ω

)

As GaAlGa0.95Si0.05

x 10

x (Å)

Imagecharge

σ = 0.05/S

−5 0 5 10−0.3

−0.2

−0.1

0.0

Po

ten

tial

(eV

)

∆U

x (Å)

l

(b)

−0.06

−0.04

−0.02

0.00

0.02

0.04

Figure 5.5 (a) Macroscopic average of the electronic (thin solid line) and ionic(dotted line) charge densities induced by a plane of 〈Ga0.95Si0.05〉 virtual ions inAl/GaAs (100). A Gaussian filter function has been used for the macroscopic aver-age. The thick solid line is the sum of the electronic and ionic densities, scaled by afactor of 10. (b) Macroscopic average of the corresponding induced electrostatic poten-tial. The screened discontinuity ∆U , and the distance l between the plane of chargeand the metal surface are also indicated.

charges. If σ is the density of the surface charge in the semiconductor and l is

the distance separating the surface charge from the metal surface, the potential

difference is, classically, ∆U = 4πe2 σ l/ε∞. As can be seen from Fig. 5.4(c), in

the junction I, the dipole lowers the average potential energy in the semiconductor

with respect to its value in the metal, increasing the SBH φp; inversely the surface

charge decreases φp in the junction II. This mechanism thus provides a qualitative

explanation for the difference between the SBH’s of the interfaces I and II.

In Fig. 5.5, we plot the charge densities and the potential induced by a surface

charge in the semiconductor, as obtained from the ab initio calculation in the

Al/GaAs (100) junction I. The bare surface charge has been artificially introduced

by replacing the Ga ions of the sixth semiconductor plane from the metal by

〈Ga0.95Si0.05〉 virtual ions, leading to a bare surface-charge density σ = 0.05/S.

Fig. 5.5(a) shows the macroscopic average of the ionic and electronic densities. In

order to blow up the features related to the image charge, in the sum of the ionic

and electronic densities, we have used a Gaussian filter function, with full width

at half maximum 12aGaAs, in the macroscopic average procedure in Eq. (2.20).

The resulting total induced charge density in Fig. 5.5(a) clearly shows the image

charge close to the Al surface. In Fig. 5.5(b), we show the macroscopic average of

the induced electrostatic potential. The potential discontinuity, ∆U = 0.24 eV,


corresponds to the net effect on the barrier height. The distance between the

plane of charge in the semiconductor and the surface of the metal (as evaluated

from the jellium edge) is l = 7.63 A. Classically, one would expect a potential

discontinuity of 0.36 eV, using the theoretical dielectric constant εGaAs∞ = 12.4.

This result somewhat overestimates the total barrier change, and suggests that

the classical image-charge model should be revisited when the surface charge is

introduced at atomic-scale distances from the metal surface.

Following the approach employed in Chapter 4 to explain the screening of micro-

scopic dipoles close to the interface, we introduce an effective screening parame-

ter, εeff(σ, x), to describe the screening of a bare surface charge σ at a position

x. Using the classical expression for the potential discontinuity and the effective

screening parameter, we may write:

∆U =4πe2 σ |x− x0|

εeff(σ, x), (5.3)

where x0 is the position of the “center of gravity” of the image charge. The choice

of x0 in Eq. (5.3) is somewhat arbitrary; in the following, we will use the position

halfway between the last Al and first semiconductor plane (x0 ≡ 0 in Fig. 5.5).

From the value ∆U = 0.24 eV [Fig. 5.5(b)], we obtain εeff ≈ 18 for σ = 0.05/S.

This effective screening is similar to the value εeff ≈ 15 obtained in the linear

regime for a dipole at the same position in the semiconductor (see Fig. 4.10).

We have also calculated the SBH changes induced by surface charges of varying

magnitude, placed on the first semiconductor plane in the Al/GaAs, Al/AlAs,

and Al/ZnSe junctions I and II. At the interface I (II), a surface charge of density

−σ (+σ) was introduced by replacing the anion A (cation C) of the first semicon-

ductor plane by the virtual ion 〈A1−σS2

CσS2〉 (〈C1−σS

2AσS

2〉). The resulting changes

∆U of the SBH are shown in Fig. 5.6. In the case of the anion-terminated inter-

faces, the SBH φp decreases (∆U < 0) almost linearly for surface-charge densities

−1/S < σ < 0. According to our previous discussion, the macroscopic average of

the difference between the ionic potentials in the junctions II and I is equivalent

to a surface charge σ = − 1S

at the interface for the III-V semiconductors and

σ = − 2S

for the II-VI semiconductors. Therefore, the modification of the SBH in

the junctions I for σ = − 1S

(− 2S) should be of the order of the difference φII

p − φIp

between the SBH of the cation- and anion-terminated junctions, for the III-V

(II-VI) semiconductors. Similarly, the change of the SBH in the junctions II in-

duced by a surface charge σ = + 1S

(+ 2S) should be close to φI

p−φIIp . Our ab initio

results show, however, that the responses of the two interfaces are not linear and

differ in magnitude for such surface charges. Therefore, we take the average ∆U

between the discontinuities induced in the junctions I and II as our estimate for


−1.0 −0.5 0.0 0.5 1.0−0.4

−0.3

−0.2

−0.1

0.0

σ (electrons/S)

∆U (

eV)

Anion-terminated

Cation-terminated

Al/ZnSeAl/AlAsAl/GaAs

σ

σ

0.0

0.1

0.2

0.3

0.4I II

Figure 5.6 Schottky barrier modification ∆U induced by a bare surface charge σ

on the first semiconductor plane in the Al/s (100) I and II junctions. S = a2s/2 is the

unit-cell surface in the (100) planes.

the difference φIp − φII

p . The results are shown in Table 5.2. For Al/ZnSe, we

reported the calculated SBH changes for σ = ± 2S

(not shown in Fig. 5.6). The

average values ∆U describe well the calculated difference φIp − φII

p , and also the

increase of φIp − φII

p when the semiconductor ionicity increases.

The effective screening for small surface charges close to the interface (σ . 0.1/S

in Fig. 5.6), evaluated from Eq. (5.3), is of the order of 50. In order to probe the

inhomogeneous nature of this screening, we have introduced a small test charge

σ = ±0.05/S in the As-terminated Al/GaAs junction, at different distances from

the metal surface. This has been done by replacing As (Ga) ions by virtual

〈As0.95Si0.05〉 (〈Ga0.95Si0.05〉) anions (cations). In Fig. 5.7, we show the induced

discontinuities |∆U | as a function of the position of the surface charge. The

effective screening calculated from Eq. (5.3) is also shown. The screening param-

Table 5.2 Comparison of theaverage SBH change ∆U inducedby surface charges σ = ± 1

S

(GaAs, AlAs) and σ = ± 2S

(ZnSe) at the interface, with thedifference φI

p − φIIp between the

SBH of the anion- and cation-terminated Al/s (100) junctions(see text). The last column showsthe results of the model Eq. (5.5).All numbers are in eV.

s |∆U | ∆U φIp − φII

p Eq.

I II (5.5)

GaAs 0.18 0.06 0.12 0.10 0.16

AlAs 0.27 0.19 0.23 0.29 0.18

ZnSe 0.68 0.24 0.46 0.36 0.57

98 Ionicity, surface properties, and Schottky barrier heights Chap. 5| ∆

U |

(eV

)

x (Å)

eff

∋

GaAs∋∞0 2 4 6 8

0.00

0.05

0.10

0.15

0.20

0.25

10

20

30

40

50

60 Figure 5.7 Schottky barrier modifi-cation |∆U | (filled squares, left scale)and effective screening (filled dia-monds, right scale) as a function of thesurface charge position within the semi-conductor. The symbols give the re-sults of self-consistent calculations fora surface charge σ = 0.05/S in theAl/GaAs junction. The solid lines cor-respond to the prediction of Eqs. (5.3)and (5.4), with δi = 1.9 A. The calcula-tions were done in the 7+ 21 supercell.

eter decreases from ∼ 50 at the interface to ∼ 20 at the position of the sixth

semiconductor plane.

We have found that the inhomogeneous screening near the junction can be de-

scribed by a simple electrostatic model, which takes into account the effect of

the MIGS. The details of the derivation are given in Appendix B. The resulting

expression for the effective screening of a small charge at the position x > x0 in

the semiconductor is:

εeff(σ = 0, x) = εs∞ + 4πe2Ds(EF; x0) δi δs/ (x− x0) , (5.4)

where δi is an effective distance of the order of the decay length δs of the MIGS at

the Fermi energy. The second term on the right-hand side of Eq. (5.4) accounts

for the MIGS-related quantum contribution to the screening of the test charge.

Eq. (5.4) correctly reproduces the classical image-charge model when x−x0 δs,

and the complete screening of a charge on the metal surface for x = x0. The

results of Eqs. (5.3) and (5.4), with δi = 1.9 A, compare very well with the ab

initio calculations (see Fig. 5.7).

Eq. (5.4) may be used to obtain an estimate for the difference between the SBH’s

of the interfaces I and II. As we have seen, this difference is of the order of the SBH

change induced by a surface charge |σ| = 1/S (2/S) on the first semiconductor

plane for the III-V (II-VI) semiconductors, i.e., at a distance x− x0 = d/2 from

the metal, where d = 1.72 A is the interplanar distance at the interface (see

Fig. 3.1). The resulting estimate, ∆φp, for the difference φIp − φII

p is thus

∆φp =2πe2|σ| d

εs∞ + 8πe2Ds(EF; x0) δi δs/d. (5.5)

Sect. 5.3 Effect of the surface reconstruction in the . . . 99

To apply Eq. (5.5), we use the values of the surface densities of states Ds(EF; x0)

and of the MIGS decay length δs given in Table 3.1 for the Al/GaAs, Al/AlAs

and Al/ZnSe ideal junctions, and we take the same parameter δi = 1.9 A for all

systems. As Ds and δs are different for the interfaces I and II, the values of ∆φp

calculated from these parameters differ somewhat, and we report in Table 5.2 the

average between the values of ∆φp calculated for the two interfaces. The model

gives the correct trend and order of magnitude for the difference between the

SBH’s. The discrepancies, however, between the model predictions and the ab

initio values increase with the ionicity of the semiconductor. This is attributed

to the use of the effective screening in the linear regime, Eq. (5.4), to describe

the strongly non-linear SBH changes in Fig. 5.6, and also to the use of a unique

value of δi for the three systems.

5.3 Effect of the surface reconstruction in the

Al/ZnSe (100) junction

ZnSe has attracted much attention in connection with the development of blue-

green optoelectronic devices. For such applications, the difficulty of fabricating

low resistance ohmic contacts to p-ZnSe is a major obstacle. Indeed, all stan-

dard metallic contacts to p-ZnSe are rectifying, and the highest p-doping levels

achieved to date do not allow for an efficient hole tunneling through the depletion

zone. Therefore, the investigation of other possible schemes for obtaining low-

barrier contacts is desirable. Recent photoemission measurements of the SBH at

Al/ZnSe (100) contacts have shown that the SBH is sensitive to the initial recon-

struction of the ZnSe surface prior to metal deposition [76]. Relatively similar

values of the barriers heights were found for interfaces fabricated on Zn-stabilized

c(2× 2) and Se-dimerized 2× 1 surfaces, while a substantially lower value of the

p-type barrier (0.25 eV lower) was observed for Al grown on the Se-rich 1 × 1

surface. These values were displayed in Fig. 5.1 to illustrate the SBH trend with

the semiconductor ionicity.

The c(2 × 2) and 2 × 1 reconstructions of the ZnSe (100) surface are observed

during ZnSe MBE in Zn-rich and Se-rich conditions, respectively [148–150]. The

c(2× 2) reconstruction is ascribed to a surface terminated by half a monolayer of

Zn atoms on a complete monolayer of Se, i.e., to an ordered array of Zn vacancies

within the outermost layer of Zn atoms. The 2×1 reconstruction corresponds to a

complete dimerized monolayer of Se. These reconstructions have also been studied

by first-principle methods [151], and were shown to be the most stable among the


structures examined. The 1× 1 reconstruction is observed only after desorption

of a Se cap layer, and annealing at ∼ 200C [76, 149, 150], and is attributed to a

Se-rich surface containing an excess Se layer on top of a full monolayer of Se. In

Ref. [76], an excess Se coverage of about 0.5 ML was measured for the samples

exhibiting the 1 × 1 reconstruction, by comparing the relative intensities of the

Zn-3d and Se-3d photoemissions.

To investigate the microscopic mechanisms that may account for the change in

the Schottky barrier with the surface reconstruction, we have performed first-

principle calculations for a series of model configurations of the Al/ZnSe interface.

The effect of the Zn 3d electrons was taken into account using the NLCC for the

Zn ion. The calculations were performed at the theoretical lattice parameter of

ZnSe, including the NLCC, namely aZnSe = 5.46 A. With respect to the Al/GaAs

interface, the tetragonal distortion of the Al overlayer was slightly larger (4%

instead of 3%) due to the smaller theoretical lattice parameter of ZnSe with

respect to GaAs (see Appendix A). We used 7 + 13 supercells, a 20 Ry cutoff,

and the local atomic structure at the interfaces was fully relaxed.

5.3.1 Results for model c(2 × 2), 2 × 1, and 1 × 1 inter-

face configurations

The starting interface configurations prior to atomic relaxation are schematically

illustrated on the right-hand side of Fig. 5.8. We selected simple configurations

corresponding to ideal continuations of the bulk semiconductor while taking into

account the initial composition of the starting surface. For Al/ZnSe fabricated

on the c(2 × 2) surface, we positioned Al atoms at the Zn vacancy sites of the

outermost semiconductor layer (configuration A in Fig. 5.8). We used real ions to

describe the Zn0.5Al0.5 composition of the c(2× 2) surface, so that the resulting

supercell contains 2 atoms per plane in the semiconductor and 4 atoms per plane

in the metal. For Al/ZnSe fabricated on the 2 × 1 surface, we terminated the

semiconductor with a full layer of Se atoms at the ideal bulk positions (configu-

ration B). For Al/ZnSe fabricated on the 1× 1 surface, we used a virtual crystal

approach to terminate the semiconductor with a 50% Se–50% Al atomic layer

(configuration C). The resulting supercell thus contains one atom per plane in

ZnSe, and a plane of 〈Se0.5Al0.5〉 virtual ions at the interface.

In Fig. 5.8, we show the macroscopic average of the electrostatic potential V (x)

and the potential lineups across the relaxed junctions. Since a decrease in ∆V

corresponds to an identical decrease in φp [see Eq. (2.19)], the calculations predict

a 0.05 eV decrease in φp in going from the relaxed configuration A to the relaxed

Sect. 5.3 Effect of the surface reconstruction in the . . . 101

Figure 5.8 Right: Start-ing interface configurations em-ployed in the supercell calcula-tions. Configuration A involvesAl atoms positioned at the Zn va-cancy sites of the c(2×2) surface.Configuration B involves a ZnSesurface terminated by a full Semonolayer. Configuration C in-volves a ZnSe surface terminatedby a 50% Se–50% Al atomic layeron top of a full Se monolayer.Left: Macroscopic average of theelectrostatic potential V (x) andpotential lineup ∆V across therelaxed junctions. Relaxation isgraphically illustrated at the bot-tom for each atomic plane. Dou-ble atomic symbols denote in-equivalent relaxation at differ-ent sites. The calculated valuesof φp, including the +0.36 eVquasiparticle and spin-orbit cor-rection, are also shown.

0

1

2

3

4

B

pφ = 1.94 eV

∆V =− 2.75 eV

4

0

1

2

3

C

φp = 1.38 eV

∆V =− 3.31 eV

0

1

2

3

4

A

φp = 1.99 eV

50−5

∆V =− 2.70 eV

Se Zn Al 0.5Se 0.5Al

V(x

) (

eV)

x (Å)

V(x

) (

eV)

V(x

) (

eV)

configuration B, and a further decrease in φp by 0.56 eV in going from the relaxed

configuration B to the relaxed configuration C. The direction and order of mag-

nitude of the predicted shifts are consistent with those observed experimentally,

suggesting that although the model configurations employed may not describe

the detail of the actual atomic reconstructions, they do capture the basic elec-

trostatic trend as a function of interface composition. As discussed previously,

the quasiparticle and spin-orbit corrections amount to a shift of the calculated

LDA barrier heights by +0.36 eV. With this shift, we obtain φp = 1.99, 1.94,

and 1.38 eV for the relaxed configurations A, B, and C, respectively. Consider-

ing the uncertainty of ∼ 0.1 eV on the quasiparticle correction, our results for

the configurations A and B are in good agreement with the experimental values

φp = 2.15±0.06 eV and 2.11±0.06 eV measured for the c(2×2) and 2×1 recon-

structions, respectively [76]. The case of the configuration C, which corresponds

to a Se-rich 1× 1 interface, is discussed below.

We emphasize that atomic relaxation at the Al/ZnSe (100) interfaces is substan-


tial, and has an important effect (of the order of 0.5–1 eV) on the Schottky barrier

height, especially for the Se-rich configuration C. In this case, from the initial con-

figuration C we found that the Se–〈Al0.5Se0.5〉 interplanar spacing at the interface

increased by 40% after convergence, and became comparable to the Al0.5Zn0.5–Al

and Se–Al interplanar distances of configurations A and B, respectively. This

large relaxation in configuration C reflects the increased metallic character of the

bonds between the Se and the 〈Al0.5Se0.5〉 atomic layer.

5.3.2 Se-induced local dipole

The mechanism responsible for the large reduction in φp for the interface fabri-

cated on the 1× 1 surface is illustrated in Fig. 5.9, where we plot the difference

in the electronic charge distribution and in the electrostatic potential calculated

between an interface terminated with two full Se monolayers and a single Se

monolayer. The latter corresponds to the configuration B, while the former cor-

responds to the limiting case of a type-C configuration with an excess Se coverage

ϑ → 1. Specifically, our starting configuration for the 1 Se ML system was de-

rived from the relaxed configuration B, by removing the buckling of the Al layers

at the interface (visible in Fig. 5.8), and the 2 Se ML system was obtained by

replacing the two Al atoms at the interface by a single Se atom in the same plane,

and in the continuation of the ZnSe lattice.

With respect to the 1 Se ML case, charge transfer from both the metal and the

Se-terminated semiconductor to the excess Se atoms at the interface is clearly

visible in Fig. 5.9, and the asymmetry in the charge transfer gives rise to a

well defined dipole field across the interface. The dipole-induced change in the

electrostatic-potential lineup is roughly proportional to the Se excess coverage ϑ

on the Se-terminated semiconductor surface for 0 6 ϑ 6 1. Indeed, we find a

0.95 eV decrease in φp for ϑ = 1 (Fig. 5.9), to be compared with the value of

0.56 calculated for ϑ = 0.5 (Fig. 5.8). For the contacts based on the 1× 1 Se-rich

surface, the Se coverage has been estimated in Ref. [76] to be ϑ = 0.41±0.18 ML,

considering the relative intensities of the Zn-3d and Se-3d photoemission and

simple atomistic models. From our results for the 1 and 0.5 ML cases, we may

expect a decrease of φp by ∼ 0.4 eV for a Se coverage of 0.4 ML, and a resulting p-

type SBH of ∼ 1.54 eV, to be compared with the value of 1.91±0.06 eV reported

in Ref. [76].

Recently, Chen et al. [150] measured by photoemission a 0.25 eV reduction in the

p-type barrier for Au/ZnSe (100) junctions by introducing a 2–3 ML Se interlayer

between the metal and the semiconductor, and proposed an electronegativity-

Sect. 5.3 Summary 103

Figure 5.9 Difference in the chargedistribution (top) and average electro-static potential (bottom) calculated be-tween an interface terminated with twofull Se monolayers, and a single Se mono-layer. The change of the SBH, ∆U =−0.95 eV, is also indicated.

0

1

2

−1

−2

10505

4

2

0 ∆U

Se ZnAl

Al:Se

U(x

) (

eV)

x (Å)

∆n (

elec

tro

ns/

Ω)

based interpretation of the barrier reduction. The similar 0.25 eV barrier reduc-

tion in the presence of vastly different electronegativity variations ∆X (∆X =

0.94 for Al-Se versus 0.01 for Au-Se in Pauling’s scale), and the expected satura-

tion of the dipole in Fig. 5.9 for ϑ > 1, call into question the general applicability

of such a simple electronegativity-based approach. Our results suggest that lat-

tice relaxation plays an important role in determining the Schottky barrier of

these metal/wide-gap-semiconductor systems, and should be taken into account

to improve upon electronegativity-based estimates of the effect of the Se-induced

interface dipole.

Summary

Using a first-principle pseudopotential approach, we have investigated the Schot-

tky barrier heights of abrupt, lattice-matched, Al/Ge, Al/GaAs, Al/AlAs, and

Al/ZnSe (100) junctions, and their dependence on the semiconductor chemical

composition as well as surface termination and reconstruction. A model was pre-

sented, which provides a simple, yet accurate, description of the barrier-height

variations with the chemical composition of the semiconductor. The larger bar-

rier values for the anion termination than those for the cation termination were

explained in terms of the screened charge of the polar semiconductor surface and

its image charge at the metal surface. Atomic-scale computations showed how

the classical image charge concept, valid for charges placed at large distances

from the metal, extends to distances less than the decay length of the metal-

induced-gap states. The lower p-type barrier heights observed experimentally in

Al/ZnSe (100) junctions for reconstructions exhibiting an excess Se coverage were

explained in terms of a Se-induced local interface dipole.

Conclusion

We have studied several metal/semiconductor interfaces, using the ab initio pseu-

dopotential method. This method takes into account the detailed atomic and

electronic structures of the interface, and is necessarily limited to relatively sim-

ple atomic geometries. Nevertheless, the few ideal and engineered interfaces we

have investigated, although structurally simple, were sufficiently rich to reveal

the sensitivity of the Schottky barrier height to different microscopic properties

of the interface, and explain some of the experimental trends.

Our studies of abrupt polar Al/GaAs, Al/AlAs, and Al/ZnSe (100) junctions

showed the influence on the Schottky barrier of the semiconductor surface ter-

mination and reconstruction, as well as of the atomic relaxation at the interface.

The impact of surface- or interface-specific properties on the SBH has been found

to increase significantly with the semiconductor ionicity. In particular, for the

Al/ZnSe (100) system, our calculations, in agreement with experiment, indicate

that the SBH can be modified substantially by exploiting different ZnSe (100)

surface reconstructions. Furthermore, in this ionic system, atomic relaxation at

the interface plays a major role in determining the actual value of the SBH.

The sensitivity of the SBH to microscopic interface features can also be exploited

by artificially changing the chemical composition of the interface. This can be

done experimentally by growing thin heterovalent atomic-interface layers. In this

thesis, we have studied the Al/Si/GaAs (100) heterostructure with ultrathin Si

interlayers (0–6ML), as a prototype engineered junction. The heterovalent nature

of the Si interlayers leads to a microscopic dipole, whose magnitude and orienta-

tion can be modified by varying the Si coverage and the GaAs surface termination.

Our ab initio results show that this mechanism can indeed quantitatively explain

the tuning observed experimentally as a function of the Si coverage.

The sensitivity of the SBH to microscopic interface features also reveals the lim-

its of the currently accepted semiempirical models of Schottky barrier. Such

model theories generally neglect the effects of the microscopic interfacial mor-

phology. This is due in part to the complexity of the actual atomic structure of

106 Conclusion

most MS contacts, and also to the relatively limited information on the detailed

atomic-scale geometry of buried interfaces. Based on our ab initio studies, we

have derived models which explicitly include the effects of the interface atomic

structure. These models reproduce our numerical results on the effects of various

microscopic properties on the SBH, and retain, within specific ranges of appli-

cability, the same accuracy as the ab initio calculations. These models show,

in particular, that while the average SBH at the anion- and cation-terminated

Al/semiconductor (100) interfaces can be explained mainly in terms of a bulk

property of the semiconductor, the difference between the two barrier heights

results from a microscopic dipole due to the screened charge of the polar semi-

conductor surface and its image charge at the metal surface. To construct such

surface-charge screening model, we had to extend the classical image charge con-

cept, valid only in the limit of charges placed at large distances from the metal,

to distances less than the decay length of the metal-induced-gap states. To inter-

pret the SBH tuning in Al/Si/GaAs (100) junctions, we also proposed a model

based on a linear-response theory scheme, which explains the screening of the

Si-induced local dipole in terms of the semiconductor dielectric constant and the

local interface density of states at the Fermi energy.

The models derived from the results of our first-principle calculations highlight

general features in the control of Schottky barrier height via interfacial pertur-

bations. Such models can be extended to other systems and different types of

interfacial engineering [6, 7, 152, 153]. The heterovalent-interlayer method can

be used, in principle, to change the SBH of any polar MS interface. This method

would be of particular interest for metal/wide-gap-semiconductor interfaces, in

connection with the problem of obtaining ohmic contacts. For these systems,

however, we expect some important differences with respect to the prototype

metal/small-gap-semiconductor systems we considered. In particular, given the

high ionicity of the semiconductor, we expect atomic relaxation to play a critical

role in metal/wide-gap-semiconductor systems. Furthermore, the models based

on linear-response schemes and developed in this work to describe the screening

of interfacial perturbations in systems such as Al/GaAs will have to be modified

for the metal/wide-gap-semiconductor interfaces, as non-linear effects are known

to play an important role in the wide-gap materials.

Appendix A

Technical aspects

A.1 Non-linear core correction (NLCC)

In this discussion, we omit for simplicity the angular-momentum dependence of

the pseudopotentials and concentrate on the central idea of the NLCC. Schemat-

ically, if u is the Coulomb potential of a nucleus and u0s is the corresponding

self-consistent KS potential for the isolated atom, the pseudopotential of the ion

(the atom without the valence electrons) is normally obtained by “unscreening”

the self-consistent KS potential:

ups = u0s − vH[nv]− vxc[nv], (A.1)

where vH[nv] and vxc[nv] are the Hartree and exchange-correlation potentials as-

sociated with the valence electrons. On the other hand, from Eq. (2.9), we have

u0s = u + vH[nc + nv] + vxc[nc + nv], (A.2)

where nc is the core charge. Comparing Eqs. (A.1) and (A.2), we obtain

ups = u + vH[nc] + vxc[nc + nv]− vxc[nv] , (A.3)

since vH[nc + nv] = vH[nc] + vH[nv]. When the valence and core charges overlap,

however, vxc[nc + nv] 6= vxc[nc] + vxc[nv], and the term in brackets, in Eq. (A.3)

depends on nv. In the presence of a large overlap, therefore, although adequate for

the isolated atom, the pseudopotential ups would not provide a good description

of the ion in a chemical environment where the self-consistent valence charge n′vdiffers from the valence charge nv of the isolated atom. In order to avoid this

lack of transferability, one can unscreen u0s using [102]:

uNLCCps = u0

s − vH[nv]− vxc[nc + nv]

= u + vH[nc]. (A.4)

108 Appendix A

With this approach, the self-consistent KS potential in the new chemical envi-

ronment reads [see Eq. (2.9)]:

us = uNLCCps + vH[n′v] + vxc[nc + n′v]. (A.5)

Since the core charge is quite peaked on the nucleus, the evaluation of the xc

potential in Eq. (A.5) requires, in principle, large components in a plane-wave

expansion. An accurate description, however, of the core charge is important

only where the core and valence charges have a strong overlap. Therefore, a

partial core correction has been proposed in Ref. [102], where the core charge is

described by a smooth function inside a certain radius. We use that approach in

our calculations.

A.2 Brillouin-zone sampling

We approximate the integrations over the irreducible part of the BZ of Fig. 2.2(b)

by a discrete sum over the points of a Monkhorst-Pack grid [92]. Each point ki

has a weight w(ki) and the integral becomes Ωc

(2π)3

∫g(k) dk → ∑

i w(ki) g(ki). In

order to avoid the Γ point, which is not very representative, the grid is centered

at a position ∆s away from the origin. A Monkhorst-Pack grid is specified by the

number of points Nx, Ny, and Nz in the kx, ky, and kz directions, respectively,

in the entire BZ. The figure A.1 shows the irreducible points of the (2, 6, 6) and

(2, 12, 12) grids used in most of our calculations. Because the supercell has a

finite length along the growth axis x, the BZ has a non-vanishing extension in

the kx direction. This extension is small, however, and can be reduced by a factor

x y z(N = 2, N = 6, N = 6) x y z(N = 2, N = 12, N = 12)

bybzK

JJ’

Γ1/12

3/12

5/12

1/92/9

weight:

bybzK

JJ’

Γ1/24

3/24

5/24

7/24

9/24

11/24

1/362/36

weight:

k = 0.25 bx x

∆s =(bx

2N x

by

2N y

bz

2N z), ,

Figure A.1 The (2, 6, 6) and (2, 12, 12) Monkhorst-Pack grids. The (2, 6, 6) gridgives 6 points in the irreducible BZ, while the (2, 12, 12) grid gives 21 points. Theweights and coordinates of the points are also indicated. All the points are located atkx = 0.25 bx.

Appendix A 109

of two because of time reversal symmetry. A single point along bx is used in our

supercell calculations.

A.3 Parameters for the pseudopotentials

Our pseudopotentials were constructed using either the procedure of Troullier

and Martins [95] or the Bachelet, Hamann, and Schluter [96] approach with the

parameters of Stumpf, Gonze, and Scheffler [154]. The pseudopotentials were then

cast into the Kleinman-Bylander non-local form [97]. We considered unpolarized

ground-state electronic configurations, and used the Ceperley-Alder exchange-

correlation functional [87]. For Ga and Zn, the pseudopotentials were generated

with and without the NLCC. For all pseudopotentials, we checked that no ghost

states [155] were present.

Table A.1 Parameters for the pseudopotentials used in this work. Number ofvalence electrons (col. 2). Ground-state electronic configuration (col. 3). Type ofpseudopotential (col. 4): Troullier-Martins (TM) or Bachelet-Hamann-Schluter (BHS);the star means that relativistic atomic calculations have been performed. Cutoff radiiin a0 for the components s, p, d, and f (col. 5–8). Angular momentum of the localpotential used in the Kleinman-Bylander procedure (col. 9).

Ion Valence Configuration Type rc(s) rc(p) rc(d) rc(f) KB

Al 3 3s2 3p1 TM 2.20 2.20 2.20 – p

Al 3 3s2 3p1 BHS 1.16 1.83 1.27 – p

Si 4 3s2 3p2 TM 2.10 2.10 2.10 – p

P 5 3s2 3p3 TM 2.20 2.20 2.20 – p

Zn 2 4s32 4p

14 4d

14 BHS∗ 1.15 1.36 2.38 – s

Ga 3 4s2 4p1 TM∗ 2.20 2.20 2.70 2.70 f

Ge 4 4s2 4p2 TM∗ 2.20 2.20 2.70 2.70 f

As 5 4s2 4p3 TM∗ 2.20 2.20 2.70 2.70 f

Se 6 4s2 4p4 BHS∗ 0.94 1.04 1.19 – d

110 Appendix A

A.4 Bulk-related parameters

Tables A.2 and A.3 show the calculated LDA band-structure terms εs and εm

used in this study. The quasiparticle and spin-orbit corrections are not included.

The band-structure terms were calculated with a 20 Ry cutoff, using a (8, 8, 8)

Monkhorst-Pack grid for the semiconductors and a (12, 12, 12) grid with a 0.1 eV

electronic level broadening for Al and P.

Table A.4 shows the calculated LDA bandgaps and dielectric constants for the

semiconductors considered in this study. The experimental values are also re-

ported (T = 0 values for the bandgaps and lattice constants). The theoretical val-

ues of ε∞ were obtained by inserting Si or Ge bilayers with coverages ϑ = 0.2 ML

and ϑ = 0.5 ML at the (100) GaAs/GaAs, AlAs/AlAs, or ZnSe/ZnSe homojunc-

tion, and using the capacitor model of Ref. [124].

The experimental bandgaps at room temperature are 0.66, 1.42, 2.16, and 2.70 eV

for Ge, GaAs, AlAs, and ZnSe, respectively. The experimental spin-orbit split-

tings ∆so are 0.30, 0.34, 0.28, and 0.43 eV for Ge, GaAs, AlAs, and ZnSe, respec-

tively [127].

Table A.2 Energy εs of the valence-band maximum measured with respect to theaverage electrostatic potential in the crystal, for the semiconductor materials consideredin this work. The virtual IV-IV crystals are indicated in brackets. Unless specifiedin note, the calculation has been performed using the theoretical equilibrium latticeparameter of GaAs, as obtained without the NLCC for the Ga atom.

Ge GaAs AlAs ZnSea) ZnSeb) 〈GaAs〉〈AlAs〉〈ZnSe〉b)

a (A) 5.55 5.55 5.55 5.46 5.55 5.55 5.55 5.55

εs (eV) 6.01 5.21 5.15 4.04 3.17 5.85 6.02 5.09

a) Calculation done at the equilibrium lattice parameter of ZnSe obtained with the NLCC for the Zn atom.b) Calculation done without the NLCC for the Zn atom.

Appendix A 111

Table A.3 Fermi energy εm measured with respect to theaverage electrostatic potential in the crystal, for the met-als considered in this work. The metals are pseudomorphi-cally strained to match the GaAs or ZnSe theoretical latticeparameter in the (y, z) plane parallel to the interface (seeFig. 3.1). The theoretical (experimental) lattice parameterof bulk Al is 3.97 A (4.05 A).

a‖ (A) a⊥ (A) εm (eV)

Al on GaAs 3.92 4.12 8.11

Al on ZnSe 3.86 4.15 8.37

P on GaAs 3.92 4.12 9.58

Table A.4 Properties of the bulk materials considered in this work. a is the latticeparameter, Eg the bandgap, and ε∞ the macroscopic dielectric constant. Except forZnSe, the theoretical lattice parameter indicated in the first column (and used in thecalculation) is the lattice parameter of GaAs as obtained without the NLCC for the Gaatom. The theoretical macroscopic dielectric constant, ε∞, has been calculated usingthe capacitor model of Ref. [124].

a (A) Eg (eV) ε∞

The. Exp. The. Exp. The. Exp.

Ge 5.55 5.65 0.44 1.0 18.6 15.4

Si 5.55 5.43 0.62 1.2 13.9 11.4

GaAs 5.55 5.65 0.81 1.5 12.4 10.9

AlAs 5.55 5.65 1.27 2.2 9.1 8.2

ZnSe 5.46a) 5.65 2.22 2.9 6.2 6.3

a) Equilibrium lattice parameter calculated with the NLCC for the Zn atom.

112 Appendix A

A.5 Convergence tests

Fig. A.2 shows our convergence tests for the SBH of the Al/GaAs (100) As-

terminated junction. The LDA value, without quasiparticle and spin-orbit cor-

rections is represented. The figure shows that all calculated SBH’s are between

0.61 and 0.65 eV, except for the very special cases with the (1, 1, 1) and (1, 3, 3)

k-point grids, or a cutoff smaller than 10 Ry. Changing the k-points grid from

(2, 6, 6) to (2, 8, 8) or (2, 12, 12) has a small effect (. 0.01 eV) on the SBH.

There is no systematic variation of the calculated SBH with the supercell size.

In most of our SBH calculations, we use the 7 + 13 supercell and a 10 Ry cutoff.

The resulting SBH is very close to the value obtained with a 20 Ry cutoff, in the

7 + 13 and 7 + 21 supercells, but it differs by 0.04 eV from the value obtained

with a 20 Ry cutoff and the 13 + 13 supercell. From these results, we conclude

that our numerical uncertainty on the SBH, when we use the 7+13 supercell and

a 10 Ry cutoff, is ∼ 0.04 eV.

0.50

0.55

0.60

0.65

0.70

0.751.45

4 8 2010E cut (Ry)

10 20 20

(1, 1, 1) (1, 3, 3) (2, 6, 6) (2, 8, 8) (2, 12, 12)

3 +

5

7 +

5

7 +

9 7 + 13

Supercell size

7 + 21

7 +

17

3 +

13

5 +

13

9 +

13

13 +

13

φ p LD

A(e

V)

Figure A.2 Convergence tests for the Schottky barrier height of the Al/GaAs (100)As-terminated junction. The supercell size can be read on the horizontal axis in theform nm + ns, where nm is the number of Al planes, and ns the number of GaAsplanes. When a cutoff different from 10 Ry is used, it is indicated on the secondaryhorizontal axis. The size and color of the symbols correspond to different k-pointsgrids, as indicated below the graph.

Appendix B

Models

B.1 Average barrier height at the polar (100)

interface

In order to explain the behavior of the average SBH φp with the semiconductor

composition, we extend to MS contacts a linear-response-theory approach [107],

which is commonly used to study band offsets at semiconductor heterojunctions.

The present analysis is also an extension to heterovalent semiconductors of the

approach outlined in Ref. [16] to model Schottky barrier changes with the alloy

composition in metal/III-V-semiconductor junctions.

The potential lineup corresponding to the average SBH φp = 12

(φI

p + φIIp

)is

∆V =1

2

(∆V I + ∆V II

), (B.1)

where ∆V I(II) is the potential lineup for the interface I(II). We will derive the

following model for the average lineup:

∆V mod = ∆V [Al/〈s〉 (100)] + ∆V [〈s〉/s (110)] . (B.2)

The first term on the right-hand side of Eq. (B.2) is the potential lineup at the

(100) interface between Al and the virtual valence-IV crystal, denoted 〈s〉, which

is constructed by averaging the anion and cation pseudopotentials of the III-V or

II-VI compound s (s = GaAs, AlAs, ZnSe). The second term is the lineup at the

non-polar (110) interface between the virtual crystal 〈s〉 and the corresponding

semiconductor s. To derive Eq. (B.2), we write the self-consistent electrostatic

potentials at the Al/s (100) I and II junctions as

V I(II)(r) = V0(r) + VI(II)1 (r), (B.3)

114 Appendix B

where V0(r) is the electrostatic potential at the Al/〈s〉 (100) junction. The aver-

age lineup ∆V can be expressed, according to Eq. (B.3), as

∆V = ∆V0 + ∆V1, (B.4)

where ∆V0 ≡ ∆V [Al/〈s〉 (100)] is the potential lineup for the Al/〈s〉 system, and

∆V1 is the lineup of the potential

V1(r) =1

2

[V I

1 (r) + V II1 (r)

]. (B.5)

The potential V I1 (V II

1 ) is the self-consistent electrostatic potential induced in the

Al/〈s〉 (100) junction by the ionic perturbation which transforms the Al/〈s〉 sys-

tem into the anion- (cation-) terminated Al/s system. Thus, V I1 and V II

1 have

long-range contributions associated with each heterovalent anion and cation sub-

stitution in the IV-IV virtual crystal. These long-range terms cancel out in the

average in Eq. (B.5), since all anion (cation) substitutions in V I1 are compensated

by a cation (anion) substitution associated with the same site in V II1 . The average

potential V1 has therefore a well defined macroscopic average in the semiconduc-

tor, which is equal to ∆V1, as V1(r) vansihes in the metal. One may estimate this

contribution using a perturbation approach neglecting inter-site interactions, be-

cause of the short-ranged nature of the potentials associated with each individual

site. V1 may thus be approximated by the superposition of the potentials induced

by isolated anion and cation substitutions in the bulk virtual crystal [109].

Let va(c)(r) be the self-consistent electrostatic potential induced by a single anion

(cation) substitution, at the origin, in the infinite virtual crystal 〈s〉. Because of

the heterovalent nature of the substitution, va(c) has a long-range component, and

may be recast as va(c)(r) = ∓ 1〈ε〉r + δva(c)(r), where 〈ε〉 is the dielectric constant

of the virtual crystal 〈s〉, and δva(c) is short ranged. Thanks to the cancellation

of the opposite long-range components of the anions and cations, V1(r) becomes

V mod1 (r) =

1

2

∑i

[δva(r −Ri) + δvc(r −Ri)] , (B.6)

where the Ri’s are the ionic positions (anionic and cationic positions) on the semi-

conductor side of the junction (x > 0). To evaluate the macroscopic average of

V mod1 , we introduce the quantities Pa(c) = 4πe2

∫δva(c)(r) dr, so that 1

ω(Pa + Pc)

is the change in the average potential of the virtual crystal when anions and

cations are substituted in the half space x > 0, with a density of one substitution

per volume ω. Since the sum in Eq. (B.6) extends over all sites of the diamond

crystal 〈s〉, the specific volume for each substitution is Ω/2, where Ω = a3s/4 is

Appendix B 115

Table B.1 Comparison of the average potential lineup ∆V

at the Al/s (100) I and II interfaces with the prediction ofthe model: ∆V mod = ∆V [Al/〈s〉 (100)] + ∆V [〈s〉/s (110)].All numbers are in eV.

s ∆V ∆V mod ∆VAl/〈s〉 (100) 〈s〉/s (110)

GaAs −2.19 −0.12 −2.31 −2.34

AlAs −1.97 −0.01 −1.98 −2.01

ZnSe −2.85 −0.38 −3.23 −3.30

the unit-cell volume of the zinc-blende III-V or II-VI crystal s. Therefore, we

have

∆V mod1 =

1

2

(Pa

Ω/2+

Pc

Ω/2

)=

Pa + Pc

Ω

= ∆V ′ [〈s〉/s (110)] , (B.7)

where ∆V ′ [〈s〉/s (110)] is the lineup at the non-polar 〈s〉/s (110) interface, built

from a superposition of the charge-density responses to the single atomic substi-

tutions transforming the virtual crystal 〈s〉 into the 〈s〉/s (110) junction [107].

The precise calculation of ∆V ′ [〈s〉/s (110)] in Eq. (B.7) is not possible with the

supercell technique, because of the long-range Coulombian tails of the potentials

induced by single heterovalent substitutions in the virtual crystal. However, the

exact lineup at the non-polar (110) interface between 〈s〉 and s, ∆V [〈s〉/s (110)],

can be easily calculated within the supercell approach. The difference between

∆V and ∆V ′ at heterovalent semiconductor interfaces can be estimated to be

typically less than 0.1 eV for the IV-IV/III-V junctions, and of the order of 0.1 eV

for the IV-IV/II-VI junctions, considering the deviations from the transitivity

rule.1 We therefore replace ∆V ′ [〈s〉/s (110)] by ∆V [〈s〉/s (110)], which leads to

Eq. (B.2).

The prediction of the model, Eq. (B.2), are compared to the average potential

1The band offsets at the (110) interfaces between three semiconductors A, B, and C verifythe transitivity rule if the quantity T (A, B, C) = VBO [A/B (110)] + VBO [B/C (110)] +VBO [C/A (110)] vanishes. Considering all the VBO’s at the (110) interfaces between the〈s〉 and s crystals, with s = (Ge, GaAs, AlAs, ZnSe), our calculations show that the typicaldeviation from the transitivity rule is less than 0.1 eV for the IV-IV/III-V junctions and of theorder of 0.1 eV for the IV-IV/II-VI junctions.

116 Appendix B

lineup at the Al/s (100) I and II interfaces in Table B.1. This table shows that

the agreement between ∆V mod and the calculated ∆V at the Al/s (100) I and II

junctions is very good, the discrepancy being 2% or less in all cases.2

Introducing the band energies in Eq. (B.2), we can write the estimate, φmodp , for

the average SBH φp as

φmodp = φp [Al/〈s〉 (100)] + VBO [〈s〉/s (110)]. (5.2)

The first term is the SBH at the Al/〈s〉 (100) junction, and the second term is

the valence-band offset (VBO) at the 〈s〉/s (110) junction.

B.2 Effective screening of a surface charge

Classically, a plane of ionic charge σ in a semiconductor is reduced by the di-

electric screening to σ/ε∞, and gives rise to a long-range potential on both sides

of the charged plane. If the semiconductor is in contact with a metal, the net

charge σ/ε∞ in the semiconductor is exactly compensated by an opposite surface

charge close to the metal surface. The resulting electrostatic potential is iden-

tical to the potential in a capacitor, where the metal surface and the plane of

charge in the semiconductor play the role of the capacitor plates. Thus, if x0

and x are the positions of the image and surface charge, respectively, the bare

potential difference between x0 and x is ∆Ub = 4πe2 σ (x− x0), and the screened

discontinuity is ∆U = ∆Ub/ε∞. This classical picture is expected to hold for

x − x0 δs, where δs is the decay length of the MIGS in the semiconductor. If

x − x0 ≈ δs, however, the potential difference ∆U is established in the region of

the MIGS tails, and a surface charge σ0i ≈ −Ds(EF; x0) ∆U is induced (to the

first order) in the semiconductor. To model the surface charge induced in the

semiconductor by a bare charge σ at any position x > x0, we assume that the

electrostatic potential varies linearly between x0 and x, even when x − x0 is of

the order of δs; if x − x0 = c δs, c > 1, a fraction 1/c of the potential difference

∆U is induced in the region of the MIGS tails, and we may thus assume that a

charge σ0i /c is induced in the semiconductor. Therefore, we model the surface

charge induced in the MIGS tails by

σi = −Ds(EF; x0)δs

x− x0

∆U, (B.8)

2The numbers reported in Table B.1 depend on the choice of the pseudopotentials. Thedifference between ∆V and ∆V mod, however, is in principle pseudopotential independent.

Appendix B 117

and we consider that the “center of gravity” of this charge is located at a position

xi in the semiconductor. Following the approach of section 4.2, we have ∆Ub =

(1 + 4πχs + 4πχm) ∆U , where 1 + 4πχs = ε∞, and −4πχm ∆U describes the

contribution of the MIGS to the screened discontinuity. This contribution can

be written as −4πχm ∆U = 4πe2 σi δi, where δi = xi − x0. In the linear regime

(σ → 0), the effective screening εeff = ∆Ub/∆U is thus

εeff(σ = 0, x) = ε∞ + 4πe2Ds(EF; x0) δi δs/ (x− x0). (5.4)

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Remerciements

Je remercie le Professeur Baldereschi de m’avoir accueilli dans son equipe et

d’avoir dirige cette these tout en me laissant une grande independance dans mon

activite de recherche.

Je remercie Dr Nadia Binggeli qui m’a initie a la physique des interfaces et aux

methodes de calcul ab initio. Je lui suis egalement reconnaissant de sa disponi-

bilite tout au long de ces quatre annees, des nombreux eclaircissements et conseils

qu’elle m’a prodigues et du soin qu’elle a apporte a la correction du manuscrit.

Je remercie Dr Delley, Prof. Kapon, et Prof. Resta d’avoir accepte la charge

d’expert.

Je remercie tous les collegues de l’IPA et de l’IRRMA, en particulier Dr Klaus

Maschke, Dr Pablo Fernandez, Dr Massimiliano Di Ventra, Julien Bardi, Jerome

Burki, Caspar Fall, Fabio Favot et Vincent Musolino de leur amitie et des nom-

breuses et stimulantes discussions que nous avons eues.

Enfin, de tout cœur merci a ma femme, mes parents, mes proches et mes amis.

Lausanne, le 6 novembre 1998 Christophe Berthod

Curriculum vitæ

Nom et prenom Berthod Christophe

Date de naissance 24 octobre 1968

Lieu d’origine Orsieres (VS)

Etat civil Marie

Formation

1983–1988 College de l’Abbaye, Saint-Maurice

juin 1988 Maturite type A, Latin-Grec

1990–1994 Etudes de physique, EPFL

mars 1994 Diplome d’ingenieur physicien EPFL

des avril 1994 Assistant a l’Institut de Physique Appliquee, EPFL

ELECTRONIC PROPERTIES OF IDEAL AND ......Metal/semiconductor (MS) interfaces are present in...

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