REVIEW ARTICLE Band engineering at interfaces: theory and ... · Band engineering at interfaces:...

J. Phys. D: Appl. Phys. 31 (1998) 1273–1299. Printed in the UK PII: S0022-3727(98)68953-1

REVIEW ARTICLE

Band engineering at interfaces:theory and numerical experiments

M Peressi †, N Binggeli ‡ and A Baldereschi †‡

† Istituto Nazionale di Fisica della Materia (INFM),Dipartimento di Fisica Teorica dell’Universita di Trieste, Strada Costiera 11,I-34014 Trieste, Italy‡ Institut de Physique Appliquee, Ecole Polytechnique Federale de Lausanne,PHB-Ecublens, CH-1015 Lausanne, Switzerland

Received 17 March 1997, in final form 14 November 1997

Abstract. Understanding the mechanisms which determine the band offsets andSchottky barriers at semiconductor contacts and engineering them for specificdevice applications are important theoretical and technological challenges. In thisreview, we present a theoretical approach to the band-line-up problem and discussits application to prototypical systems. The emphasis is on ab initio computationsand on theoretical models derived from first-principles numerical experiments. Anapproach based on linear-response-theory concepts allows a general description ofthe band alignment for various classes of semiconductor contacts and predicts theeffects of various bulk and interfacial perturbations on the band discontinuities.

1. Introduction

Semiconductor–semiconductor and metal–semiconductorinterfaces play a crucial role in modern electronic andoptoelectronic devices. The transport properties inheterojunction devices are controlled by the electronicband profiles at the interfaces, more specifically by thevalence and conduction discontinuities that accommodatethe difference in bandgap between the materials, namely,the valence and conductionband offsets(VBO and CBO)in the case of semiconductor heterojunctions and the p- andn-typeSchottky barriers(φp andφn) in the case of metal–semiconductor contacts (see figure 1).

Extensive theoretical and experimental work hastargetted the problem of the interface band alignment [1–8].However, it is only in the last decade that the physicalmechanisms which give rise to the band alignment atsemiconductor heterojunctions have begun to be revealed[1–4] and that the connection between band offsets andSchottky barriers has been put on a firmer basis [1, 2, 9, 10].Today we are still far from a complete understanding of thefactors which control the band alignment, especially in thecase of Schottky barriers.

It is not our purpose to give a general overview onthis broad subject. Many review articles and books alreadyexist. In particular, a good introduction to the problemof semiconductor–semiconductor interfaces and a reviewboth of experimental and of theoretical work can be foundin the book edited by Capasso and Margaritondo [1].Fundamental papers on this subject are collected in thebook edited by Margaritondo [2]. More recently, a

review article by Yuet al appeared [3]. For an extensivereview on the problem of band offset engineering, weaddress the reader to the work by Franciosi and Van deWalle [4]. For an introduction and a review on the problemof Schottky barriers we refer the reader to the book byRhoderick and Williams [5] and to the book edited byMonch, which contains fundamental papers on the physicsof metal–semiconductor contacts [6]. A good review ofthe experimental situation in the field of semiconductor–semiconductor and metal–semiconductor interfaces canbe found in the article by Brillson [7]. Finally, arecent assessment of the fundamental and technologicalknowledge on metal–semiconductor contacts is given in thebook edited by Brillson [8].

An important and general issue, which has beenwidely debated in the literature, is that of whether theband discontinuities are essentially determined by the bulkproperties of the constituents, or whether some interface-specific phenomena may affect them in a significant way.In the latter case, the control of the interfaces couldprovide a way to manipulate the band line-ups andtune thetransport properties across the junctions. In semiconductortechnology, one of the most important and widely usedfeatures has been the possibility of intentionally varyingthe electronic properties by alloying and doping. Forsemiconductor contacts a similarly important challenge is tocontrol and modify artificially the band offsets and Schottkybarriers.

We present here a theoretical approach to the problem.Theoretical investigations of the band-line-up problem canbe divided mainly into two classes: (i) fully self-consistent

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Figure 1. Schematic diagrams of the band structures of semiconductor–semiconductor (a) and semiconductor–metal(b) junctions, as a function of the position along the growth direction. Definitions of band offsets (VBO and CBO) and ofSchottky barriers (φn and φp) are shown. Flat bands were represented, because we are focusing on a region which is of theorder of 10 atomic units and the band bending is negligible at this scale.

ab initio calculations, which provide the electronic chargedistribution at the interface and allow one to studythe importance of interface details, such as orientation,abruptness and defects; and (ii) theories other thanab initiocalculations, which can be described as ‘model’ theories,in that they make simplifying and sometimes drasticapproximations in describing the interface, but have theadvantage of being easier to implement. The work wepresent here is in the framework of theab initio approach.

First, we describe a state-of-the-artab initio methodemployed for the study of interfaces (section 2). We thenshow (section 3) that an original approach based on linearresponse theory (LRT) concepts [11–13] underlies andexplains a general trend of semiconductor–semiconductorinterfaces: at lattice-matched isovalent heterojunctions theband offset depends only on the bulk properties of thetwo materials, whereas at heterovalent heterojunctions itcrucially depends on the interface orientation and othermicroscopic details. However, once the atomic structureis known—either experimentally or theoretically by total-energy minimization—the structure-dependent contributionto the band offset can be rigorously calculated fromelementary electrostatics.

Strain effects and the application of linear-responseschemes to lattice-mismatched semiconductor heterojunc-tions are discussed in section 4. In section 5, we focuson epitaxial metal–semiconductor contacts and examine thelink between Schottky barriers and band offsets. In partic-ular, we illustrate how Schottky barrier trends observed insome epitaxial systems can be explained by extending tometal–semiconductor interfaces linear-response-theory con-cepts used in the study of band offsets. The possibility ofartificially modifying band offsets and Schottky barriers isdiscussed in sections 6 and 7. We first discuss changesin band discontinuities produced by alterations of the bulkchemical and structural properties of the materials form-ing the junctions. We then look for interface-specific effectsthat may modify the interface dipole and thus change theline-up of the electronic states across the junctions.

2. The computational approach

2.1. First-principles self-consistent calculations

It is possible nowadays to study theenergetics andthe electronic structure of many-electron systems byperforming fully ab initio computations, that is, by solvingthe quantum-mechanical equations for the system underconsideration without any use of empirical parameters,for a meaningful comparison with experiment or evenfor accurate predictions of quantities not yet accessibleexperimentally. Among the existingab initio schemes, thelocal-density approximation (LDA) to density-functionaltheory (DFT) [14, 15] has proven to yield reliable results,at an acceptable computational cost, on the electronicground-state properties of complex crystalline systems[16–19]. Within DFT, the many-body problem ofinteracting electrons is reduced to a system of single-particle Schrodinger equations [15], which must be solvedself-consistently (SCF, for self-consistent-field). Electron–electron interactions are fully included by adding to theHartree potential an exchange-correlation term, which is afunctional of the charge density. In the LDA, this functional[15] is reduced to a function of the local charge densitywhich has been calculated accurately and interpolated usinga parametrized form [20].

The work on interfaces that we will illustrate in thefollowing sections is based on the pseudopotential method,which is an efficient approach, within the LDA-SCFframework, for dealing with semiconductors and metalsof practical interest for electronic devices. In thepseudopotential approach only the valence electrons, whichare responsible for the formation of the chemical bonds anddetermine the relevant physical properties, are explicitlytreated. The pseudopotential—derived from LDA-SCFcalculations for the isolated atom with an all-electrontechnique—describes the effects of the nucleus and of thecore electrons on the valence electronic states [21]. Forperiodic solids, a plane-wave basis set is generally used to

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expand the single-particle electronic orbitals. Plane wavesup to a certain kinetic energy cut-off are included in thebasis set, whose size is one of the ingredients determiningthe accuracy of the calculations.

The choice of a plane-wave basis set allowsfor a convenient reciprocal space formulation and astraightforward evaluation of the total energy of the system,as well as the forces on the atoms and the macroscopicstresses [22]. These quantities can then be used to relax theatomic structure, allowing one to determine the equilibriumstructural parameters of a bulk crystal or optimize theinterface geometries in a complex superlattice. Integralsover the Brillouin zone (BZ), which are necessary in orderto determine quantities such as the charge density, areperformed by discrete summation over a set of specialk points [23]. Such sets ofk points are representativeof a uniform grid covering the BZ, whose density isalso an ingredient which determines the accuracy of thecalculations. For metallic systems an electronic levelbroadening scheme is generally used to improve theconvergence with respect to thek-point sampling [24].

A numerical uncertainty of a few milli-electronvoltscan be typically achieved in LDA-SCF calculations of theband line-ups by controlling the convergence with respectto several parameters such as the number of plane wavesin the basis set, thek points used and the size of thesupercell describing the junction [13]. Other sources ofuncertainty are the choice of the pseudopotentials and theresulting lattice parameters used in the calculations [12];the global numerical uncertainty in the band alignmentscan be estimated to be of the order of 20-30 meV for fullyconverged calculations.

We would like to point out that, in principle, thesingle-particle eigenvalues obtained from the LDA-SCFcalculations and used to evaluate the bulk band structuresand the interface line-ups, arenot quasi-particle energiesand should be corrected for many-body effects [25–27]which are much larger, in general, than the numericaluncertainty of the LDA-SCF values. Since these correctionsare normally much less important for the valence bandsthan they are for the conduction bands in semiconductors(typically less than 0.3 eV for the valence-band edge and ofthe order of 1 eV for the conduction-band edge [25, 27]),it is convenient to calculate the band alignments for thevalence part (VBO andφp). We note that these many-bodycorrections affect the bulk band structure of the crystalsand tend to cancel out for the VBO at semiconductorheterojunctions. Moreover, since they do not affect thepotential line-up across the interface, which, being afunction of the charge density, can be accurately calculatedwithin a DFT–LDA approach, they have no effect on thedependenceof the band alignment on interface propertiessuch as orientation, chemical composition and abruptness(see section 2.4).

In this review, we will often be dealing with alloys,which are widely used in semiconductor devices. If oneis not interested in the atomic-scale structure of the alloy,simple non-structural theories can be applied. A widelyused approach is thevirtual-crystal approximation(VCA)in which an AxB1−x alloy (or a pseudo-binary AxB1−xC

alloy) is modelled using a single type of atoms on the ABlattice (or AB sublattice): the virtual atom〈AxB1−x〉, whoseatomic pseudopotential is a weighted average of those ofthe ‘true’ A and B atoms.

This VCA approach clearly cannot reproduce thestructural relaxations present in real alloys, not only onthemicroscopicscale (since the bond-length relaxations areneglected), but also on themacroscopicscale. We found,for instance, that the VCA typically overestimates theequilibrium average lattice parameter of the alloys, givingpositive deviations from Vegard’s law (linear interpolationbetween the lattice parameters of the constituents) [28].The limits of the VCA in describing the structuralproperties of the alloys are even more serious in thecase of ordered alloys and superlattices, for which theinternal relaxations can give rise to anisotropic macroscopicstrain. Nonetheless, the VCA is quite satisfactory indescribing the electronic properties of the alloys, since thevalence electrons are quite delocalized and experience apotential which is an average of the potentials individuallyoriginating from the real atoms. This is particularlytrue for the pseudo-binary alloys (such as Ga1−xAl xAsand Ga1−x InxAs), with which we will be mostly dealing.These considerations, together with the absence of directand precise information, in most cases, about the actualmorphology of the alloy and their interfaces, justify the useof a simple non-structural theory such as the VCA in thepresent context. A quantitative discussion of the effects ofthe VCA on the band line-up can be found in [29] for theInP/Ga1−x InxAs interface.

In the remainder of section 2, we present techniquesspecific to the study of interfaces and to the determinationof band alignments; the last part is devoted to an approachbased on LRT and implemented within the LDA-SCFframework, used to explain general trends of band offsetsand interpret some properties of Schottky barriers.

2.2. Supercell and macroscopic average techniques

Interfaces can be studied using periodically repeatedsupercells, which allow for a convenient reciprocal spaceformulation of the problem, which is otherwise not possiblebecause of the loss of translational symmetry. Supercellsare actually more suitable to describe superlattices ratherthan isolated interfaces. However, insofar as the band-structure alignments are concerned, experience has shownthat the relevant effects due to the presence of a neutralinterface are confined to a small region and the bulkfeatures of the charge density are completely recoveredwithin a few atomic units far from the interface. Thisimplies that the relevant interface features can be studiedusing supercells with a reasonably small number of atoms(a few atomic planes of each material). Typically thesupercells contain two interfaces which are equivalent interms of stoichiometry and geometry (see figure 2 forsome prototypes for semiconductor heterojunctions), inorder to avoid electric fields due to unbalanced charges.The isolated interface configuration is well represented,provided that the adjacent interfaces are sufficientlyseparated that they do not interact.

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Figure 2. Typical supercells used to describe GaAs/AlAs (001), (110) and (111) abrupt heterojunctions (from the top to thebottom).

The supercell self-consistent calculations provide theelectronic charge density distribution and the correspondingelectrostatic potential. Figure 3(a) shows the contourplots of the electronic valence charge density of theGaAs/AlAs(001) interface simulated by a 3+3 superlatticein three different atomic planes containing the growth axis.Since the geometry is periodic in the planes parallel to theinterface (the(x, y) planes), the first obvious simplificationis to consider planar averages as a function of thez coordinate only:

f (z) = 1

S

∫S

f (x, y, z)dx dy. (1)

From the three-dimensional electronic charge densitywe get the one-dimensional charge densityn(z) andelectrostatic potentialSV (z) shown in figure 3(b). Thisexhibits two distinct, albeit very similar, periodic functionsin the two bulk materials, which smoothly join across theinterface. Since the system is a lattice-matched one, theperiod a of n(z) and SV (z) is the same on both side ofthe interface and, in this particular case, equal toa0/2,wherea0 is the bulk lattice parameter. The effect of theinterface is related to thedifferencebetween these periodicfunctions. Such a difference, which is barely visible infigure 3(b), can be enhanced by getting rid of the bulk-like oscillations using themacroscopic averagetechnique

[11, 30]. The macroscopic average is a basic conceptin classical electromagnetism [31]; for any microscopicquantityf (micro)(r) one can define a macroscopic averagef (macro)(r):

f (macro)(r) =∫w(r − r′)f (micro)(r′) dr′ (2)

wherew(r) is a properly chosen filter function dependingon the geometry and on the characteristic length scale of theproblem. The application is straightforward for interfacesbetween two isostructural lattice-matched materials, wherew is a material-independent quantity. Acting directly onthe planar average, the filter function can be chosen simplyas

w(z) = 1

a2(a

2− |z|

)where2 is the one-dimensional step function, giving

f (z) = 1

a

∫ z+a/2

z−a/2f (z′) dz′. (3)

The results for the GaAs/AlAs(100) charge and potentialare shown in figure 3(c). The macroscopically averagedquantities exhibit no microscopic oscillations on eitherside of the interface and one recovers the constant [32]macroscopic limit in the two bulks. Conversely, deviationsfrom the macroscopic value indicate the interface region

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Figure 3. Contour plots of the self-consistent electron density distribution (a) for a GaAs/AlAs(001) heterojunction overvarious atomic planes containing the growth axis and centred on the interface anion. Planar averages (n(z ) and V (z ))(b) and macroscopic averages ( ¯n(z ) and ¯V (z )) (c) of the electron density and of the electrostatic potential along the growthdirection are shown.

and allow one to define the ‘interface dipole’ withoutreferring to arbitrary ‘ideal’ reference configurations.

The macroscopic average technique can also be appliedto an interface between two materials A and B withdifferent periodicities because of lattice mismatch or evenstructural differences, the latter being the case of metal–semiconductor contacts. In order to recover macroscopic

features in the bulk regions of both materials, one has tofilter twice, using the functionswA andwB appropriate toeach material in turn. This double filtering can be recast interms of the single filter function

w(r) =∫wA(r − r′)wB(r′) dr′ (4)

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which explicitly exhibits commutativity. We notice that,even ifwA andwB are localized as much as possible, thedouble filtering reduces the resolution with respect to thesingle filtering and, for the special casewA = wB , the useof w is not equivalent to the use ofwA. Different choices ofthe filter functions will give profiles with different details,but the macroscopic physics, which contains the relevantinformation, is the same.

2.3. Potential line-ups

The long-range character of the Coulomb interaction makesthe average electrostatic potential〈V 〉 of an infinite systemin general ill defined [11, 33]. Consequently, the potentialline-up across the interface between two semi-infinite solidscannot be simply calculated as the difference between bulkquantities; rather, it depends in principle on the detailedstructure of the interface. This makes the problem ofband alignment at interfaces difficult and it is in principlenecessary to calculate accurately the interface chargedistribution and the corresponding electrostatic potential.The difference between themacroscopic averagesof theelectrostatic potential in the two bulk regions is preciselythe electrostatic potential line-up,1V . The macroscopicaverage commutes with the spatial differentiation whichoccurs in the Poisson equation and the potential line-upis thus exactly related to the dipole moment of the chargeprofile:

1V = 4πe2∫zρ(z) dz (5)

where ρ is the total (ionic plus electronic) chargedensity which averages to zero in the bulk-like regions.Equation (5) allows one to define in an unambiguous waythe concept of theinterface dipole for any surface orinterface.

Formally, the average potential in afinite systemis related to the long-wavelength limit of the chargedistribution:

〈V 〉 = limq→0

V (q) = limq→0

4πe2

q2ρ(q). (6)

For systems whose charge densityρ can be decomposedinto localized atomic-like distributions

ρ(r) =∑R

ρloc(r −R) (7)

where the charge distributionsρloc at the lattice sitesRare neutral, carrying no dipole or quadrupole (that is,ρloc(q) ' αq2 + O(q3)), the limit in equation (6) existsand the average potential is a well defined constant also inthe infinite crystal. This means that, for solids composedof rigid neutral building blocksρloc, neither the potentialdrop across the surface of a semi-infinite sample northe line-up at an interface depends on the details of thesurface or interface structure (namely the orientation andabruptness). For such systems, the potential line-up issimply the difference between the average potentials of thetwo infinite solids calculated from equation (6) and verifiesthe transitivity relationship1V (A/C) = 1V (A/B) +

1V (B/C), which characterizes line-ups controlled by bulkproperties.

This suggests that the potential line-up problem may bereaddressed by studying the properties of single buildingblocksρloc. This idea will be further developed within thelinear-response formulation of section 2.5.

2.4. Band offsets and Schottky barriers

From a theoretical point of view, the band offset (Schottkybarrier) is conveniently split into two contributions:VBO = 1Ev+1V (φp = 1Ep+1V ). The band structureterm 1Ev (1Ep) is the difference between the relevantvalence band edges in the two materials (between theFermi level of the metal and the valence-band edge of thesemiconductor for a metal–semiconductor contact), whenthe single-particle eigenvalues are measured with respectto the average electrostatic potential in the correspondingbulk crystal. The band-structure term is characteristicof the individual bulks. This term can be obtainedfrom standard bulk band-structure calculations for eachcrystal and displays, by definition, transitivity. This isnot the case for the electrostatic potential line-up1V ,which can, in principle, depend on structural and chemicaldetails of the interface. According to the above definition,microscopic quantum effects, such as many-body effects onthe quasiparticle spectra [25–27], are all embedded in theband structure term1Ev (1Ep).

We would like to emphasize that the partition of theVBO (φp) into a potential line-up and a band term isnot unique. 1V must contain the line-up of thelong-range electrostaticpotential generated by the electronic andionic charge distributions. All the quantities related to theshort-range local components of the potential (exchangecorrelation, the difference between the local part of thepseudopotential and an ionic point-charge potential) arebulk quantities and can be arbitrarily included in one ofthe two terms. In the work reported here, they are includedin the band structure term [34].

In supercell calculations, the band offset or Schottkybarrier can also be evaluated directly from the local densityof states (LDOS)N(ε, z) defined as

N(ε, z) =∑k,n

¯ρk,n(z)δ(ε − εk) (8)

where the sum runs over the bandsn and the wavevectorskof the BZ of the supercell,ρk,n(r) = |ψk,n(r)|2 andψk,n(r)is the electronic wavefunction. Far from the interface, oneach side of the junction, the LDOSN(ε, z) converges tothe bulk density of states of the corresponding crystal. Theband offset or Schottky barrier can be obtained thus fromthe difference between the band edges of the LDOS on thetwo sides, far from the junction. The LDOS, however,requires supercell computations with a high number ofk points and a large energy cut-off compared with thoseneeded to determine the charge density and the potentialline-up. In addition, larger supercells have to be used,since the LDOS has a spatial convergence to the bulkfeatures slower than that of the charge density. As aresult, the LDOS approach is less convenient, in general,than the potential-line-up approach to determine the banddiscontinuities.

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2.5. Linear-response theory

Dealing with inhomogeneous systems with ‘similar’components, which is the case for lattice-matchedsemiconductor–semiconductor heterojunctions or alloys,one often recognizes that the ‘inhomogeneities’, which aredue to thedifferencebetween the two bulks, are minutewith respect to the typical bulk variations; indeed, theyare one or two orders of magnitude smaller. For instance,figure 3 shows that thedifference between the chargedistributions of the AlAs and GaAs bulks (of the order of10−1 electrons per zincblende cell in the profile along the(001) direction), which determines the interface dipole, isvery small with respect to the bulk-like oscillations (aboutfive electrons per zincblende cell; see also figure 5 in [11]).This suggests the use of a low-order perturbation theorytreating the differences as a small perturbation with respectto an appropriate reference periodic system.

The reference system for studying the junctionor the alloy between a given pair of lattice-matchedsemiconductors could be one of the two bulks itself, but inorder to minimize errors due to the neglect of higher-orderterms, the optimal choice is thevirtual crystal. Consideringfor definiteness an interface or a 50%-compositional alloybetween two binary semiconductors A1C1 and A2C2,the virtual crystal is constituted by virtual anions andcations, which are, in terms of pseudopotentials,〈vA,C〉 =12(vA1,C1 + vA2,C2). The perturbation which builds up theactual system (interface or alloy) amounts to replacingvirtual ions by physical ones, in a given pattern. The bareperturbation is described by

1V(r) =∑R

[σR1vC(r −R)+σR−δ1vA(r−R−δ)] (9)

where1vA,C = 12(vA1,C1 − vA2,C2), δ is the position of

the anion in the unit cell, whereas the cation is set at theorigin, andσR is an Ising-like variable indicating whetherthe lattice siteR is occupied by ions of type 1 or 2.This transformation is equivalent to the implantation ofsubstitutional impurities in the virtual crystal. Wheneverthe single perturbing potential1vA,C is weak enoughto induce a localized electronic response1nA,C linearin the perturbation, the charge-density response to thewhole perturbation1V may be decomposed into localizedresponses to the single substitutions. To linear order in theperturbation thetotal charge density of the real (interfaceor alloy) system is thus

ρ(r) ≈ ρvirt (r)+∑R

[σR1ρC(r −R)+ σR−δ1ρA(r −R− δ)] (10)

whereρvirt is the charge distribution of the virtual crystaland the1ρ terms are thetotal (bare plus electronic) chargesinduced by the single substitutions.

The electronic—and hence also the total—chargedensity induced by the isolated substitution has thefull point symmetry of the substitutional site, the bareperturbation being spherically symmetrical. In the caseof the elemental or binary cubic semiconductors withTd symmetry considered here,1ρA,C has no dipole or

quadrupole moment and its long-wavelength behaviour issimply

1ρA,C(q) = QA,C + AA,Cq2+O(q3) (11)

whereQA,C =∫1ρA,C(r) dr is the net displaced charge

and AA,C = 16

∫r21ρA,C(r) dr is its second spherical

moment. For more details about the formulation, weaddress the reader to [11–13].

For the interface potential-line-up problem, the changein the electrostatic potential induced by the perturbationwhich brings the reference system into the real interface isthe relevant quantity, since the reference system itself doesnot display, by construction, any potential line-up. TheFourier transform of the electrostatic potential generatedby the whole perturbation (using equation (10)) is

1V (q) = 1

N�

4πe2

q2

(1ρC(q)

∑R

σR eiq·R

+1ρA(q)∑R

σR−δ eiq·(R−δ))

(12)

whereN is the number of cation (or anion) sites in thecrystal and� is the volume of the unit cell. Formally, thepotential drop across the interface is the difference betweenthe changes induced by the perturbation in theaverageCoulomb potentials on the two sides of the junction, whichis related to the long-wavelength behaviour of1V (q) in thetwo bulk regions. Recognizing thatσR = σR−δ = +1 onone side andσR = σR−δ = −1 on the other, the potentialline-up reads

1V = limq→0

(1V (q)|σ=+1−1V (q)|σ=−1) (13)

which is ultimately related to the long-wavelengthbehaviour of the isolated1ρA,C(q) in equation (11). Thefollowing section describes the application of the LRTapproach to various prototypical cases of lattice-matchedsemiconductor–semiconductor interfaces.

3. Lattice-matched semiconductor interfaces

3.1. Isovalent semiconductor heterojunctions

The GaAs/AlAs heterojunction is the simplest andmost studied among thelattice-matched common-anionheterojunctions. The appropriate reference crystal in thiscase is the virtual crystal〈Ga1/2Al 1/2〉As. The isolatedsubstitution, being isovalent, is neutral; therefore, theinduced potential drop across the interface is only due tothe electronic charge and, according to equations (11)–(13),is

1V = 8πe2AC

�= 4πe2

3�

∫r21nC(r) dr (14)

independently of the interface orientation and abruptness.We stress that the latter property derives from the chargeneutrality of the perturbation and is therefore valid also forother isovalent interfaces.

The response1nC(r) can be determined througha direct approach, from the difference between twoindependent supercell calculations of the self-consistent

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Figure 4. The electron-density response of the 〈Ga1/2Al1/2〉As virtual crystal to a single 〈Ga1/2Al1/2〉 : Ga substitution in a32-atom BCC cell. We present a contour plot of the linear term 1n (1)(r) in the (110) plane (full/broken lines indicate apositive/negative response) (a); thick dotted, broken and full lines are the intersections of the plane respectively with atwo-atom FCC, 16-atom FCC, and 32-atom BCC Wigner–Seitz cell centred on the impurity. Radial spherical averages of thelinear (full line) and quadratic (broken line) terms of the density response are shown (b). The average radius of the two-atomFCC cell and the nearest and next-nearest neighbour distances are also indicated.

charge density for the perturbed system with an impurityand for the unperturbed host material. The result isshown in figure 4 for the virtual crystal〈Ga1/2Al 1/2〉As,in which the central cation is substituted by a pure Ga: thecalculations are performed using a 32-atom body-centred-cubic (BCC) supercell. Figure 4 suggests that, in thepresent case, even a smaller supercell would be largeenough to describe the isolated substitution. The mostsignificant electronic rearrangement is localized within abulk face-centred-cubic (FCC) cell and occurs along thedirections of the chemical bonds. The comparison of theon-site linear and quadratic terms of the charge responsesupports the validity of the linear approximation, since thequadratic term is negligible.

The validity of the LRT also in the decomposition ofthe total response into localized responses can be seen fromthe charge and potential profiles in figure 5. The top panels

show the response of the virtual crystal to the substitutionof an entire plane of cations by Ga along the (001), (110)and (111) directions; in the other panels the total responseto all the substitutions needed to obtain the GaAs/AlAsinterface (namely the difference between the LDA-SCFGaAs/AlAs charge density and that of the virtual crystal) iscompared with the corresponding superposition of the linearresponses to the single planar substitutions: the two profilesare not distinguishable on this scale, thus indicating that theinter-site higher-order terms due to the interference of theresponses are negligible. Within LRT, the potential line-upat the GaAs/AlAs interface can be expressed in terms ofthe response1nC(z) of the charge density to the singleGa-plane substitution as

1V = 4πe2

a

∫z21nC(z) dz (15)

wherea is the distance between consecutive cation planes.

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Figure 5. Planar averages of the response of the electron density (full line) of the virtual crystal 〈Ga1/2Al1/2〉As to a singleplanar X = 〈Ga1/2Al1/2〉 ⇒Ga substitution and the corresponding electrostatic potential (broken line) (top panels). In thebottom panels we show the response to the whole perturbation leading to the GaAs/AlAs heterojunctions. Results from fullyLDA-SCF calculations and from LRT (superposition of linear responses upon single planar substitutions) are notdistinguishable on this scale. From the left- to the right-hand side: (001), (110) and (111) orientations.

The bulk-like nature of the electrostatic potentialline-up—and consequently of the VBO—in GaAs/AlAs,predicted by LRT, is confirmed by accurate LDA-SCFsupercell calculations performed for the three maincrystallographic orientations (001), (110) and (111) witha sharp interface [12, 30] and, in the (001) case, also for anon-abrupt interface containing a mixed cationic plane, withequal concentrations of Ga and Al atoms [35]. Althoughthe interface dipoles have a different shape (see figure 6,full lines), the electrostatic potential line-up is the same—within the numerical accuracy of our calculations (seesection 2.1)—in the four cases examined and coincides withthe LRT prediction to within 0.02 eV. The total LDA-SCFband offset is VBO≈ 0.45±0.02 eV. This value is obtainedneglecting many-body and relativistic effects which can beaddeda posteriori and which amount to≈ 0.1± 0.02 eV[26, 27] and≈ 0.03± 0.01 eV [36] respectively.

The many-body effects have been evaluated usingthe difference between the corrections to the LDA bulkband-edge energies of GaAs (−0.07 eV) and AlAs(−0.18 eV), determined in [26, 27], where the quasiparticleband structures were obtained by calculating the self-energy operator within the so-called GW approximation.Incidentally, the calculated values of the many-bodycorrection to the VBO turn out to be within 0.2 eVfor a large number of heterojunctions between group IVand/or III–V semiconductors [27]. The relativistic effectshave been evaluated by adding to the two bulk LDA-SCFvalence band edges the spin–orbit correction (1

310), using

the experimental value of the spin–orbit splitting10. Weemphasize here that the many-body and relativistic effectsenter only in the bulk band structure term, according to thepresent scheme, and therefore, whatever their values maybe, these effects do not affect, in principle, the conclusionson the independence of the VBO from the interface details.Adding many-body and relativistic corrections, the resultingfinal estimate for the VBO at the GaAs/AlAs interfaceis thus≈0.58± 0.06 eV, which compares well with theexperimental values which are in the range 0.45–0.55 eV[3].

The LRT approach can be extended to the more generalcase ofno-common-ionheterojunctions A1C1/A2C2, suchas InAs/GaSb [37, 38] and InP/Ga0.47In0.53As [29]. We onlystress here the new features with respect to the simpler caseof GaAs/AlAs and we address the reader to the originalworks for the details. In the spirit of LRT, the VBOcan be obtained by calculating separately the anion andcation contributions to the potential line-up, consideringthe A1C/A2C and AC1/AC2 interfaces, with the latticeparameter taken equal to the common lattice parameter ofthe two real constituents. Within the limits of the LRT,anionic and cationic contributions turn out to be additiveand both are independent of orientation.

Another new feature with respect to the GaAs/AlAscase is the existence of two abrupt inequivalent interfacesin the polar directions, corresponding to the differentterminations A1–C2 and A2–C1 and characterized bythe presence of differentinterfacial strains. In fact,

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Figure 6. Macroscopic averages of the electron density (full line) and the corresponding electrostatic potential (broken line)at a GaAs/AlAs heterojunction. From the topmost to the lowest panel: (001) abrupt, (001) non-abrupt with a mixed cationicplane, (110) abrupt and (111) abrupt.

in the general case of no-common-ion lattice-matchedheterojunctions, the lattice-matching conditions almostalways result from a balance between the differences ofthe cationic and anionic core radii, which may differconsiderably in the two semiconductors. As a consequence,an importantmicroscopic interfacial strainmay developwithin a few interplanar distances from the interface dueto the individually different bond lengths that will beestablished there.

For instance, in the case of the InAs/GaSb (001)interface, the In–Sb (Ga–As) interface interplanar distanceis elongated (contracted) by about 14% with respectto the common value which is present in the bulkInAs and GaSb regions [37, 38]. In the case ofInP/Ga0.47In0.53As, the interface strain for the two abruptinequivalent interfaces amounts to about 4% [29]. Thestrain profile can be found from accurate LDA-SCFsupercell calculations by minimizing the total energy

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with respect to the atomic positions and macroscopicstrain. It is, however, particularly convenient andinstructive to use a model based on LRT concepts,which allows one to obtain thestrain profile along thegrowth direction for any given composition profile ofthe junction [37, 38], and to describe in a physicallysound and accurate way the effects of such a strain onthe VBO in terms of displacements ofeffective charges[29, 37, 38]. Remarkably, LRT explains why, despite thefact that the interfacial strain can vary with the interfacecomposition, the band offset is almost unchanged. This isin agreement with experimental data and full LDA-SCFsupercell calculations for InP/Ga0.47In0.53As(100) [39],which show that, although anion intermixing at the junctioncan reduce the interface strain by 3%, it has virtually noeffect on the band offset provided that the minimal-energystructure is used for each interface composition.

3.2. Heterovalent semiconductor heterojunctions

The simplest and most studied case among heterovalentsemiconductor heterojunctions is Ge/GaAs. In thiscase, the appropriate virtual crystal is a fictitious〈Ge1/2Ga1/2〉〈Ge1/2As1/2〉 zincblende, whose cation hasvalence charge 3.5, while the anion has valence charge4.5 [11, 40, 41]. The localized perturbations leading tothe real ions carry a bare charge±e/2. Therefore withinLRT the total charge induced by these perturbations is±e/2〈ε〉, where〈ε〉 is the dielectric constant of the virtualcrystal. According to equation (11) the correspondingFourier transform is

1ρA,C(q) = ± 1

2〈ε〉 + AA,Cq2+O(q3). (16)

In the spirit of LRT, the potential line-up can be splitinto two contributions:

1V = 1Vhetero +1Viso (17)

due to the constant and quadratic terms in equation (16).The latter is by construction purely electronic and it istherefore the only term present in the case of isovalentinterfaces. According to the previous discussion,1Visois independent of interface details.1Vhetero contains boththe ionic and the corresponding electronic contributionand is formally equivalent to the line-up generated byan assembly ofpoint chargesof absolute valuee/2〈ε〉.This term depends on the atomic structure of the interface(orientation, abruptness, relaxation and so on). However,once the structure is known—either experimentally orby independent theoretical calculations—1Vhetero can beevaluated from simple electrostatics.

In the (110) direction, the virtual crystal is made ofatomic planes with one cation and one anion per unit surfacecell. The perturbation leading from the virtual crystalto an idealabrupt interface is therefore neutral in eachplane parallel to the interface, so that1Vhetero vanishesand1V (110)abrupt = 1Viso. The corresponding offset forthe Ge/GaAs(110) interface is VBO= +0.60 eV (Ge ishigher).

In the (001) direction, the virtual crystal is made of analternating stack of cationic and anionic planes, carryingsurface charge densities of 3.5 and 4.5 electrons per unitsurface cell. At variance with the (110) case, ideallyabrupt interfaces along this direction would be charged andhence thermodynamically unstable, [42–45]. The simplestneutral interfaces one can envision are terminated by onemixed plane. There are two such inequivalent interfaceswhere the mixed plane is〈As1/2Ge1/2〉 or 〈Ga1/2Ge1/2〉 (seefigures 7(a) and (b)). The point-charge contribution to thepotential line-up is1Vhetero = ±πe2/(2a0〈ε〉), wherea0 isthe lattice parameter of the virtual crystal.1Vhetero is equalin magnitude and opposite in sign for the two interfaces,so that theaverage1V predicted by LRT is simply1Viso.The supercell LDA-SCF calculations give VBO equal to+0.88 eV and+0.28 eV respectively [40, 41]. We notethat indeed the average of these two offsets nearly equalsthe value for the (110) abrupt interface. Other authors [46]have already observed this fact by comparing the resultsobtained for the VBO at the (110) and (001) interfaces withdifferent terminations, but without giving any rationale.

In general, atomic interdiffusion in the real samplesmay occur over several atomic planes across the interface,depending on the growth conditions. Topredict theactual composition profile is beyond the possibilities ofthe present theoretical approach, not just because of theheavy computational effort required to study many differentconfigurations, but since it would not be sufficient tocompare the final total energies without taking into accountthermodynamic and kinetic factors in the formation ofthe interface. We therefore limit our discussion hereto the role of the interface morphology in the possiblechanges induced in the band discontinuities, by examininga few specific configurations. Supposing that the atomicinterdiffusion occurs overtwo atomic planes, but withoutformation of antisites, the composition profile would be. . .Ge–Ge–〈GexAs1−x〉–〈Gey Ga1−y〉–As–Ga–As. . .. Chargeneutrality requires thaty = x − 0.5. Using LRT,1Vhetero(x) is [2πe2/(a0〈ε〉)](x − 0.75) [13, 35, 47] andits maximum possible variation withx is πe2/(a0〈ε〉) ≈0.8 eV [48]. In the particular cases in whichx =0.5 and x = 1, the simple atomic intermixings overa single plane are recovered. Whenx = 0.75 theinterface, besides being neutral, does not carry any ionicdipole (1Vhetero = 0). Analogous considerations applyfor the complementary composition profile. . .Ge–Ge–〈GeyGa1−y〉–〈Gex As1−x〉–Ga–As–Ga. . .. Therefore, twoinequivalent neutral interfaces with no ionic dipole arepossible: the first is terminated by one〈Ge3/4As1/4〉 and one〈Ge1/4Ga3/4〉 plane in sequence (figure 7(d)); the other hasthe complementary pattern〈Ge3/4Ga1/4〉 and 〈Ge1/4As3/4〉(figure 7(c)). LRT predicts for these interfaces a VBO equalto the one for the (110) abrupt interface (0.60 eV); supercellLDA-SCF calculations indicate deviations from linearityof about±0.02 eV, giving VBO= 0.62 eV for the firstconfiguration and 0.58 eV for the second [49].

Atomic interdiffusion can be responsible for differentvalues of the VBO even at (110) interfaces (see for instancefigures 7(e) and (f)). Such a possibility has been suggestedby experiments showing that there is a dependence of

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As

Ga

Ge

(110)

(001)

f)

a) b)

c) d)

e)

Figure 7. Atomic configurations for some selected neutralnon-abrupt Ge/GaAs interface morphologies in variousorientations: (001) with one, (a) and (b), and two, (c) and(d), mixed planes; and (110) with 25% of Ga–Ge (e) andAs–Ge (f) swaps. The broken line shows the position of thegeometrical interface.

the VBO on the growth sequence (non-commutativity) atGe/GaAs, GaAs/Ge and ZnSe/Ge, Ge/ZnSe(110) interfaces[3]. In terms of the present LRT scheme, this is explainedin terms of non-vanishing ionic dipoles (1Vhetero 6= 0) thatare established because of swaps between atoms of differentvalences, causing the average ionic charge in planes parallelto the interface to deviate from the constant value of theabrupt case. Systems such as Ge/GaAs, ZnSe/Ge andSi/GaP have been studied in detail [13, 35, 50] and thepredictions of LRT are found to be in excellent agreementwith the full supercell LDA-SCF calculations performed forsome selected configurations, not including (for consistence

with the LRT scheme) structural relaxations. In theheterovalent heterojunctions structural relaxations are notnegligible in general, as in the case of no-common-ionisovalent heterojunctions that we have already discussed;their effect on the VBO can be typically of the orderof 0.1 eV (see for instance [51] for the ZnSe/Ge(110)interface). We do not discuss these effects further here, bothbecause they depend on the atomic-scale configuration ofthe interface (which is actually unknown) and because theeffects of the various interface orientations and terminationsare more relevant. We would like to stress here that thetheoretically predicted effects of the atomic intermixingare compatible with the observed non-commutativity atheterovalent interfaces, but a more detailed comparisonbetween theory and experiment would be meaningless inthe absence of detailed information about the actual atomic-scale interface configuration.

A transitivity relationship may occur, at leastapproximately, for isovalent interfaces, due to the bulk-like character of1Viso. We notice, however, that, iflinearity holds separately for the systems A/B, B/C andA/C, this does not necessarily imply an exact transitivityrelationship, since there is no unique optimal referencesystem which minimizes higher-order terms. One cannotfind three purely isovalent lattice-matched systems toverify transitivity unambiguously: the best case could be(InAs, GaSb, AlSb), but the mismatch between InAs andAlSb is about 1.25%, which is not completely negligible.Conversely, examples of three or even four lattice-matchedsystems including also heterovalent interfaces do exist,for example (Si, GaP, AlP) and (GaAs, AlAs, Ge, ZnSe), buttransitivity must be tested only by comparing interfaceswith equal orientations and equivalent composition profiles.In fact, 1Vhetero strongly depends on these details andtherefore, by its very nature, does not display transitivity.

A comparative study of the interfaces for the system(GaAs, AlAs, Ge) is reported in detail in [12] and weonly summarize the results here. The comparison offull LDA-SCF calculations performed for geometricallyand stoichiometricallyequivalent interfaces has shownthat there are violations of the transitivity rule up to0.09 eV, namely ones that are small but definitelyexceed the computational error, which have to beascribed to higher-order response terms. Conversely,the noticeable lack of transitivity (with deviations of theorder of tenths of an electron-volt rather than hundredths)which is experimentally observed also in other casesinvolving heterovalent interfaces has to be ascribed to theestablishment ofinequivalentinterface configurations at thevarious heterojunctions.

For (GaAs, Ge, AlAs)(001) and (ZnSe, Ge, GaAs)(001)accurate comparative measurements of VBO [47, 49]indicate that there are deviations from the transitivityrule of about 0.25 and 0.4 eV respectively, which aredefinitely beyond the experimental resolution. Consideringthe possibility of an atomic intermixing over two atomicplanes, variations in1Vhetero of up to about 1 eV arepredicted by LRT for the various interface configurationsfor GaAs/Ge (as discussed above), AlAs/Ge [49] andZnSe/GaAs(001) interfaces [47, 52], with small differences

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because of the different dielectric constants involved [48].For the ZnSe/Ge(001) interface, the possible maximumvariation of 1Vhetero is even larger and amounts to2πe2/(a0〈ε〉) ≈ 1.3 eV, since the difference in the chemicalvalence between Zn(Se) and Ge atoms is twice that ofthe other systems. Therefore, the experimentally observedlarge deviations from transitivity can be explained in termsof the formation of different inequivalent interfaces. Insofaras their microscopic morphology is concerned, some insightcan be gained from theoretical predictions, although manydifferent interface configurations may correspond to thesame offset.

At the end of this subsection, two main featuresare to be stressed. First, heterovalent interfaces are theideal candidates astunable heterojunctions: this point isaddressed in section 6. Second, the sensitivity of the bandoffsets to the microscopic structure of the interface makesit possible to find configurations having very differentline-ups, which makes their measurement rather suitablefor characterization purposes; however, any comparisonbetween experiments and calculations must be made with ahigh degree of caution, since it is also possible to find verydifferent configurations having the same line-up.

4. Strained (Lattice-mismatched) semiconductorinterfaces

Inducing strain in one or both semiconductors is awidely used method for engineering band discontinuitiesin quantum-well structures. In comparison with the case ofthe lattice-matched heterojunctions, much less theoreticalwork has been done on the strained systems.

In the lattice-mismatched heterojunctions, the stressstate of the two materials can be varied by changing thecomposition (and, hence, the in-plane lattice parameter) ofthe substrate, thus affecting the VBO through the differentdependences of the band-edge energies on the strain(deformation potentials) [53, 54]. In pseudomorphicallygrown superlattices, considered here, the lattice mismatchbetween a cubic substrate and the epilayer is accommodatedby an appropriate strain along the growth direction,corresponding to a lattice constanta⊥ which essentiallydepends on the elastic properties of the epilayer. Fromthe experimental point of view, the thickness of thepseudomorphic epilayer has to be kept small enough toavoid misfit dislocations.

The macroscopic theory of elasticity provides a firsthint for the determination of the structure. Following [53]and considering here only the (001) growth direction, onehas

aepi

‖ = asubs‖ ≡ asubs

aepi

⊥ = aepi[

1− 2

(c12

c11

)epiεepi

‖

](18)

εepi

‖ =aepi

‖aepi− 1 ε

epi

⊥ =aepi

⊥aepi− 1

whereε is the strain tensor andcij are the elastic constantsof the bulk epilayer; the labelsubs refers to the cubicsubstrate andepi refers to the strained epilayer. More

generally, the possibility of inducing strain in both materialsconstituting the heterojunction must be considered, whenthe substrate governing the pseudomorphic growth of theheterostructure has an intermediate lattice parameter.

The macroscopic theory of elasticity predicts theinterplanar distances accurately enough in the bulk regions,namely, midway between the two interfaces or two to threeatomic planes away from them, whereas it fails in theinterface region where the interplanar distances sensitivelydepend both on the substrate composition and on theinterface termination [55] and also, to a lesser extent, onthe period of the superlattice [28]. The exact determinationof the equilibrium structure can be achieved by lookingfor those atomic positions and values of the tetragonaldeformationc/a of the supercell which make the forcesacting on the atoms and the macroscopic stress vanish.This can be achieved by total energy minimization or byusing a model based on LRT [28]. The latter, which issimilar to the one used for studying the interfacial strainprofile at isovalent lattice-matched heterojunctions [37], hasbeen applied to Si/Ge superlattices and has given resultsin excellent agreement with the fully relaxed LDA-SCFcalculations.

Neglecting relativistic effects, the top of the valenceband at the0 point in a bulk unstrained semiconductoris threefold degenerate. A uniaxial (001)-oriented strainlowers the crystal symmetry from Td to D2d , thus splittingthe valence-band edge into a singlet and a doublet. Inthe case of a tensile (compressive) strain, the singletis below (above) the doublet. Taking into account thespin degeneracy, the valence-band-edge manifold includessix states, which, in the absence of strain, are split byspin–orbit interaction into a quadruplet and a doublet, thesplit-off band (so). Moving away from the zone centre, thequadruplet is split into a pair of doublets: the heavy-hole(hh) and the light-hole (lh) bands. The above degeneracyis further lowered by a (001) uniaxial strain, which splitsthe hh and lh levels at0. The split states are founda posteriori by adding the spin–orbit effects to the resultsof non-relativistic LDA-SCF calculations: their positionswith respect to the weighted averageEavev of the valence-band edge (namely the band edge calculated neglecting bothspin–orbit and strain effects) are [53]

Ehhv = Eavev + 1310− 1

2δE001 (19)

Elh,sov = Eavev − 1610+ 1

4δE001

± 12[12

0+10 δE001+ 94(δE001)

2]1/2

where the+ (−) sign refers tolh (so). 10 is the spin–orbitterm, δE001 is, in absolute value, equal to23 of the totalseparation between the two topmost states of the valence-band manifold calculatedwithout the spin–orbit interaction;its sign is negative for an elongation, positive otherwise.This implies that uniaxial tensile strain shifts thehh bandabove thelh band, whereas uniaxial compressive strainshifts the lh band above thehh band. The averageEavev , calculated with respect to the reference electrostaticpotential, is subject only to shifts due to the hydrostaticcomponent of the strain, corresponding to a relative volumechange1�/�:

Eavev = E0v +D(bs)

v

1�

�(20)

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Figure 8. The variations of the GaAs and GaP topmost(E top

v , full line) and average (E avev , broken line) split

valence-top edge manifolds with the substrate in thepseudomorphic GaAs/GaP(001) heterojunction; thedifferences 1E top

v and 1E avev are the corresponding

contributions of the band structure to the VBO. (Fromunpublished work by Di Ventra and Peressi.)

whereD(bs)v is the ‘band-structure’ term [54] in theabsolute

deformation potential[53, 54] andE0v is the degenerate

valence-band edge of the cubic unstrained material.Eavev

can also be calculated as the threefold-degenerate topvalence state of the material in a cubic configuration withthe same volume as the strained material, that is, with aneffectivecubic lattice constanta = (a2

‖a⊥)1/3 [56]. We

stress thatD(bs)v is not the absolute deformation potential,

which, in principle, is well defined in terms of bulkproperties only for non-polar lattices and for uniaxial strain[54].

The variation of the VBO with strain includes also thevariation of the potential line-up1V and it is related tothe total absolute deformation potential. It is instructiveto study the VBO at various prototype lattice-mismatchedA/B heterojunctions as a function of the substrate latticeparametera‖ in the rangeaA ≤ a‖ ≤ aB . For all thesystems studied [55–59] the largest contribution to the strainvariation of the VBO originates from the band structureterm1Ev rather than from the interface-dependent potentialline-up1V , which varies very little witha‖. The variationof the VBO with strain is therefore mainly abulk effect.Furthermore, its variation is small when it is calculatedfrom theaveragesof the valence-band manifolds (VBOave)and relevant when it is calculated between thetopmost(VBOtop) split valence states. In fact, in all cases studiedthe variation of the band structure term measured from theaverages (1Eavev ) is one order of magnitude smaller thanthat measured from the topmost states (1E

topv ). Therefore,

in general the variation of the VBO with strain is mostlydue to the strain-inducedsplittings δE001 (equation (19))of the valence-band manifold rather than from theshifts(equation (20)) related to1�/�. In figure 8 the variationsof the band structure terms1Etopv and 1Eavev with thesubstrate are shown for the case of GaAs/GaP.

4.1. Isovalent interfaces

The simplest case of lattice-mismatched heterojunctions isSi/Ge [56], which is an example of anisovalent homopolar

interface. The offset VBOave is about 0.44 eV (Ge ishigher) for the configuration corresponding to a substratemade of 50%–50% alloy, with a tunability of about 0.06 eVingoing from a substrate of pure Ge to one of pure Si. ForVBOtop, on the contrary, the corresponding tunability overthe whole range is one order of magnitude larger, about0.5 eV. The relationship between the strain variation of theVBO and theabsolute deformation potentialof the twobulks has been discussed in [56] for the Si/Ge system.

Several common-ion lattice-mismatched heterojunc-tions have been investigated in detail, such as GaAs/InAs[57], GaAs/GaP [58] and ZnS/ZnSe [59]. We address thereader to the original works for details and we summarizehere a result of common validity for these systems. Sincethe potential line-up is the most computationally expensivepart in the LDA-SCF calculation of the VBO, it is of fun-damental and practical interest to understand and possiblypredict without much effort its variations with the substrate,despite the fact that they are small. It has recently beenshown that, in the case of pseudomorphic growth conditionsand with precise limits of validity which are discussed indetail in [58], once1V is known for a given substrate (saya0s ), its value1V ′ for another substratea′s can be predicted

by the simple scaling law

1V ′ ' (1− 2ε‖)1V (21)

whereε‖ = a′s/a0s − 1. For the systems mentioned above,

the predictions of equation (21) are in excellent agreementwith the full LDA-SCF calculations.

4.2. Heterovalent interfaces

With respect to the isovalent case,heterovalent lattice-mismatched interfacesoffer greater flexibility in terms ofthe tunability of the VBO, thanks to the peculiar non-bulk character of the band alignment between heterovalentmaterials. The Si/GaAs heterojunction has been studied[55] in a few selected configurations corresponding topseudomorphic growth along the (001) orientation todiscuss the effects on the VBO ofstrain (that is, ofdifferent substrates) and ofchemistry(that is, of differentinterface terminations). Among the several possible stableinterface morphologies, the simplest terminations whichgive rise to neutral interfaces and were considered are thosewith only one mixed atomic plane X= 〈Si1/2As1/2〉 orY = 〈Si1/2Ga1/2〉. The mixed planes at the interface weredescribed by the VCA. Particularly noticeable in this systemare the relaxations of the bond lengths in the interfaceregion (see figure 9 and the original paper [55] for details)which are predicted by total energy minimization. Wealso found that the potential line-up is more sensitive tothe atomic positions at the interface than it is in isovalentsystems, such as InP/Ga0.47In0.53As [29], Si/Ge [56] andInAs/GaAs [57].

In summary, both chemistry and microscopic strainhave sizeable effects. The final results for the Si/GaAs(001)VBO, with spin–orbit effects included, measured betweenthe topmost valence states for the various morphologiesconsidered in this work are as follows: +0.54,

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Figure 9. Upper panel: interplanar distances in theSi4/(GaAs)5(001) superlattices for different substrates (Siand GaAs) and different interface terminations. X and Yindicate the virtual atoms X = 〈Si1/2As1/2〉 andY = 〈Si1/2Ga1/2〉 respectively. Open triangles refer to thepredictions of the macroscopic theory of elasticity; closedsymbols to fully relaxed supercell LDA-SCF calculations.Lower panel: a schematic indication of the displacements ofthe interface plane in the relaxed configurations (from [55]).

−0.15, +0.10 and −0.55 eV for the As-termination–Si-substrate, Ga-termination–Si-substrate, As-termination–GaAs-substrate and Ga-termination–GaAs-substrate cases,respectively, where the positive/negative sign indicates thatthe topmost GaAs valence state is higher/lower than thetopmost Si valence state, with a maximum tunability dueto the combined effects of strain and chemistry of the orderof 1.1 eV.

5. Epitaxial metal–semiconductor contacts

When dealing with metal–semiconductor systems, one isconfronted with the tremendous diversity and complexity oftheir interface morphologies. It had been recognized earlyon, however, that the Schottky barrier height generally hasa much stronger dependence on the semiconductor materialthan it does on the metal type and the contact fabricationmethod, for junctions of practical interest [7]. The origin ofthis Fermi-level pinning, which contributes to determiningthe value of the Schottky barrier achieved in practice, is

still controversial. It has most often been ascribed either tointrinsic metal-induced gap states (MIGS) [9, 10, 60] or toextrinsic gap states arising from defects near the interface[61–64].

Progress in epitaxial growth has made possiblethe fabrication of a number of stable epitaxial metal–semiconductor structures. These include selected elementalmetals on GaAs and GaAlAs, the well known epitaxialcontacts formed by transition metal disilicides on Si [65]and various monocrystalline arsenide-based metal contactson III–V semiconductors [66]. Such systems, which maybe viewed as the metal–semiconductor counterparts of thelattice-matched semiconductor heterostructures, provide anideally simple starting point to examine from first principlesthe mechanisms determining band alignment at metal–semiconductor interfaces. Among the most studied systemsare the Si–silicide interfaces [67–69] and the Al and Aucontacts on GaAs and GaAlAs [64, 70–73].

As an example, we discuss here the case of theAl/Ga1−xAl xAs(100) junctions. These junctions, involvinga simple metal and the lattice-matched Ga1−xAl xAs alloys,have been the focus of many recent experimental studiesand are also well suited to investigate theoretically theconnection between Schottky barriers and heterojunctionband offsets. Experimentally, there is indeed evidenceof a correlation between band offsets and Schottky-barriertrends. The transitivity relationship

φp(A/C)− φp(A/B) = VBO(B/C) (22)

between a metal A and two semiconductors B and C hasbeen verified typically to within 0.2 eV for a number ofmetal–semiconductor systems [1]. These include Al, Moand CoGa contacts on Ga1−xAl xAs alloys, for instance, andAu contacts on elemental semiconductors and a number ofIII–V compounds and alloys [1, 66, 74, 75]. It should bementioned, however, that significantly larger deviations (ofthe order of 1 eV) were recently reported for especiallyengineered interfaces; this aspect will be discussed insection 7.

The transitivity property of the Schottky barriers hasgenerally been interpreted either in terms of a MIGS-relatedpinning of the Fermi level to a canonical charge neutralitylevel [9, 10] or in terms of pinning to some native defectlevel of the semiconductor [76]. Several authors suggestedthat studying the dependence on hydrostatic pressure of theSchottky barrier may help in discriminating between thedifferent models and could identify the defects possiblyresponsible for the pinning [77, 78]. In the case of theAl/Ga1−xAl xAs junctions, the dependence of the Schottkybarriers both on the alloy composition and on hydrostaticpressure has been examined experimentally [78, 79]. Onthe basis of qualitative arguments concerning the pressuredependence of the neutrality level, the measured pressuretrends of the Schottky barriers were interpreted as beingin conflict with intrinsic pinning mechanisms [78, 79]and as evidence of pinning due to defects with bondingcharacter, thus excluding as candidates native defects suchas the As antisites and EL2 centres, which commonlyare associated with the pinning [61]. Similar conclusionsagainst MIGS-related mechanisms had been proposed

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M Peressi et al

Figure 10. The epitaxial geometry and supercell used for the Al/GaAlAs(100) junction with the anion-terminated (type I) orcation-terminated (type II) GaAlAs(100) interface.

earlier in studies of the alloy-composition dependence ofthe Al/GaAlAs Schottky barriers [79].

We will illustrate here the results of recent LDA-SCFcalculations for defect-free Al/GaAlAs junctions, whichshow that the experimental alloy composition and pressuretrends of the Schottky barriers can be fully accountedfor without postulating defect-induced Fermi-level pinning[73, 80]. Specifically, the trends can be explained byapplying to ideal (defect-free) interfaces LRT schemessimilar to those employed in the study of band offsets.The supercells used to model the Al/Ga1−xAl xAs(100)junctions are illustrated in figure 10. The Al and GaAlAslattice parameters satisfy the epitaxial conditionaAl ≈(1/√

2)aGaAlAs , which implies that Al(100) may be grownepitaxially on Ga1−xAl xAs(100) with the Al overlayerrotated by 45◦ about the Ga1−xAl xAs [100] axis. Sincethe Al/GaAlAs(100) junction is polar, one can envisagetwo types of inequivalent abrupt interfaces with anion(type I) and cation (type II) GaAlAs(100) termination,respectively. The corresponding interfacial geometries,which are represented in figure 10, were optimized withrespect to various translational configurations of the metaloverlayer relative to the semiconductor surface [73].

Figure 11 shows the Schottky barrier heightsφp(Al/Ga1−xAl xAs) obtained from the LDA-SCF calcula-tions for the two types of interfaces illustrated in figure 10.The VCA was employed in these calculations to take intoaccount the alloying on the cation sublattice. The the-oretical values include the effect of the spin–orbit split-ting of the semiconductor’s valence-band edge, which wasderived from experiment, as well as many-body correctionsto the single-particle eigenstates. The many-body correc-tions of [71] were used for the GaAs valence-band edgeand the Al Fermi level, yielding a correction of 0.22 eVto φp(Al/GaAs). For φp(Al/AlAs), the many-body cor-rection was included using the correction toφp(Al/GaAs)and the difference in the quasiparticle band-edge energiesof GaAs and AlAs (0.1 eV) evaluated in [26]. A linear

Figure 11. The composition dependence of theAl/Ga1−x Alx As(100) Schottky barrier height for theanion-terminated (type I) and cation-terminated (type II)interfaces (see figure 10). The symbols show theexperimental data from [74] (M, I(V); ◦, C(V); and �, IPE)and [79] (♦, I(V)). (From [80]).

interpolation between the GaAs and AlAs corrections wasused for 0< x < 1. The resulting Schottky barrier heightsfor the Al/GaAs (Al/AlAs) junction are 0.78 eV (1.34 eV)at the anion-terminated interface and 0.67 eV (1.05 eV) atthe cation-terminated interface, thus showing that there isa non-negligible dependence on the interface termination.

In figure 11, the theoretical values of the Schottky bar-rier height are compared with the experimental current–voltage (I(V)), capacitance–voltage (C(V)) and internal

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photoemission (IPE) data from [74, 79]. Taking into ac-count the estimated theoretical uncertainty of about 0.1 eVin φp and the unknown stoichiometry of the experimen-tal interfaces, good general agreement between the calcu-lated and experimental alloy-composition dependences ofthe Schottky barrier is found. Experimentally, the transi-tivity rule for the Al/GaAs, Al/AlAs and GaAs/AlAs(100)band discontinuities is verified within experimental un-certainty; that is, the differenceφp(AlAs) − φp(GaAs) istypically in the range 0.45–0.6 eV and the band offsetVBO(GaAs/AlAs) is in the range 0.45–0.55 eV [3]. In thecalculations, the transitivity is verified to within 0.03 eVfor the anion–terminated interfaces, whereas a deviation of−0.21 eV is observed for the cation-terminated interfaces[80].

This difference between the two types of junctionscan be understood by comparing their charge densitieswith the charge densities of the bulk semiconductor andmetal crystals. This is done in figure 12(a) in the caseof the As-terminated Al/GaAs interface. Inspection offigure 12(a) shows that the charge density in the junctiondiffers from that of the bulk constituents only in a verylocalized region, which mostly includes the semiconductorplane closest to the metal. This is a general feature alsoobserved in the other Al/GaAlAs(100) junctions. Thedifference in the transitivity properties of the anion- andcation-terminated interfaces can thus be understood fromthe fact that, in the As-terminated junctions, as opposed tothe cation-terminated junctions, the chemical compositionof the semiconductor plane closest to the metal is notmodified by the alloy compositionx.

Unlike the charge density, which fully recovers itsbulk character within about a monolayer from the metal,the electronic states reproduce the bulk semiconductordensity of states only several atomic layers away from theinterface. This is illustrated in figure 12(b), which showsthe contribution to the charge density of the electronicstates with energies above the GaAs valence band edge,namely the MIGS contribution to the charge density. TheMIGS decay within the semiconductor with a decay lengthof about 3A. Figure 12 shows that these states tend tore-establish the bulk semiconductor charge density in thevicinity of the interface. This behaviour of the MIGSis related to the local charge neutrality emphasized, inparticular, by Tersoff in his model of the charge neutralitylevel [9].

Tersoff’s model is based on a metallic screening bythe MIGS of local interfacial perturbations and predictsSchottky barrier heights which are a bulk propertyof the semiconductor. In contrast to this model,however, the different Schottky barriers obtained from theab initio calculations for the cation- and anion-terminatedAl/GaAlAs(100) interfaces, in figure 11, demonstrate thatthe absolute value of the Schottky barrier isnot a bulkproperty. This is also consistent with the results of severalfirst-principles investigations of metal/Si [68, 69, 81] andmetal/GaAs [64, 70–72] junctions showing that there is asignificant dependence of the Schottky barrier height onthe details of the atomic structure at the interface. Furtherevidence of the incomplete screening by the MIGS of local

interfacial perturbations will be presented also in section 7in the context of Schottky barrier tuning.

The results in figure 11, nevertheless, strongly suggestthat the variation of the Schottky barrier with thesemiconductor’s alloy composition might be dominated bythe same bulk mechanisms as those that control band offsetsat isovalent semiconductor heterojunctions. In order toclarify this behaviour and explain the observed transitivityproperties of the Schottky barriers, we have used a linear-response-theory approach similar to that employed in thestudy of band offsets [73, 80]. To this end, the Al/GaAsand Al/AlAs interfaces are considered as perturbationswith respect to a reference junction formed by Al andthe 〈Ga1/2Al 1/2〉As virtual crystal. The charge density ofthe Al/GaAs (Al/AlAs) junction is then constructed fromthe linear superposition of the charge-density responses tosingle-atomic–plane substitutions transforming the virtual〈Ga1/2Al 1/2〉 cations into the real Ga (Al) ions.

An ideal linear superposition of the responses1nGa(Al)bulk

of the charge density of bulk〈Ga1/2Al 1/2〉As semiconductor(figure 13, upper panels; see also figure 5, centraluppermost panel) in the semi-infinite semiconductor regionwould produce a variation in the potential line-up

1vbulk = −4πe2

a0

∫z21n

Ga(Al)bulk (z) dz

across the interface (see equation (15),a = a0/2and the additional factor12 accounts for the fact that,in the metal/semiconductor system, the substitutions areperformed only on one side of the junction), whichis a bulk quantity and satisfies the transitivity rule.In the LDA-SCF calculations the additional bulk termsyield 1vbulk(Al/AlAs) − 1vbulk(Al/GaAs) = 0.41 eV =1V (AlAs/GaAs).

The actual response1nGa(Al)i (z) of the Al/〈Ga1/2Al 1/2〉As (100) junction to the substitution of theith cation planefrom the interface coincides with the bulk semiconductor’sresponse1nGa(Al)bulk (z) for large i. The responses1nGa1 (z)

and 1nAl1 (z) are represented in figure 13 for theAs-terminated junction (lower panels). Figure 13 showsthat even1nGa(Al)1 (z) is similar to 1nGa(Al)bulk (z), exceptfor the presence of a small dipolar asymmetry due tothe proximity of the metal. Assuming thus a linearsuperposition of the responses1nGa(Al)i , the deviation fromthe bulk trend is given by

∑i d

Ga(Al)i , where

dGa(Al)i = 4πe2

∫z1n

Ga(Al)i (z) dz

are the microscopic dipoles induced by the asymmetrical1n

Ga(Al)i (z) for substitutions near the junction (see

figure 13, di ≈ 0 for i > 3). The calculated interface-specific correction to the transitivity rule obtained with thisapproach:

∑i (d

Ali −dGai ) is−0.02 eV for the As-terminated

interface and about−0.2 eV for the cation-terminatedinterface [80], in very good agreement with the LDA-SCFresults of figure 11.

This atomic-scale study of the electronic response tothe substitutions allows one thus to explain, in terms ofthe microscopic properties of the charge density in the

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0.0

0.5

−15 −10 −5 0 5z (A)

6

7

8

9

10

Ch

arge

den

sity

(ele

ctro

ns/c

ell)

0.0

0.5

−15 −10 −5 0 5z (A)

6

7

8

9

10(a) (b)

Figure 12. The planar average of the charge density in the Al/GaAs(100) As-terminated junction (full line). Comparison withthe planar-averaged charge densities of the two bulk materials (dotted line) forming the junction (a). Contribution of the MIGSto the total charge density (b); the dotted line indicates the contribution of the states with energies below the GaAs valenceband edge. The lower panels show the macroscopic average of the difference between the interface and bulk charges (a),which depends on the arbitrary choice of the interface, and the charge density contribution of the MIGS (b). (From [80].)

junctions, the observed correlation between the band-offsetand Schottky-barrier trends. The results show that thedependence of the line-up on the alloy composition canbe decomposed into two contributions. A first (dominant)bulk contribution depends only on the semiconductormaterial and is related to the charge-density building blocksemphasized in the LRT description of the band offsets. Thiscontribution is always present and analogous to the bulkterm1Viso (see equation (17)) in the line-up of heterovalentsemiconductor heterojunctions. The second contributionis interface related, accounts for the deviation from thetransitivity rule and derives from local dipoles producedby changes in the alloy composition within mainly thelast semiconductor plane in contact with the metal. Forthe polar Al/GaAlAs(100) junctions, this contribution isrelevant thus only in the case of the cation-terminatedjunctions.

It should be noted in this connection and whencomparing with experiment that a Ga–Al exchange reactionis known to occur in the Al/GaAs(100) junctions (driven bythe large heat of formation of AlAs), yielding excess Al onthe cation sites near the interface [82]. When the LDA-SCFcalculations are performed for Al/GaAlAs(100) junctionscontaining one (or several) AlAs bilayers at the interface,the deviations from the transitivity rule become very small,of the order of 10−3 eV [73], both for the cation- and forthe anion-terminated junction, in close agreement with theexperimental data in figure 11.

The effect of hydrostatic pressure on the Schottkybarrier of the defect-free Al/GaAlAs(100) junctions hasbeen investigated using the sameab initio techniques[73]. Consistently with the experimental data, the pressure

dependence of the p-type barrier was found to be negligibleand that of the n-type barriers to coincide with that of thesemiconductor band gap, both for direct-and for indirect-gap Ga1−xAl xAs. Similarly to the alloy-composition trend,the pressure dependence of the Schottky barriers couldbe explained in terms of bulk properties derived fromLRT. In the case of the pressure dependence, the relevantbulk quantities are the absolute band-edge deformationpotentials and more precisely those of the semiconductor,because the absolute deformation potential of the metal’sFermi level is a vanishing quantity. The experimentalpressure trends of the p-type and n-type barriers, that hadpreviously been interpreted as being inconsistent with anintrinsic pinning mechanism and as evidence of a pinningto defects with bonding character, were shown instead,in [73], simply to derive from the appropriate band-edgedeformation potential of bulk GaAlAs.

Contrary thus to some recent proposals [74, 78, 79],both the pressure and the alloy-composition dependences ofthe Al/GaAlAs Schottky barrier could be fully accountedfor without invoking a defect-induced Fermi-level pinning.The composition and pressure dependences of the barrierand the transitivity property of the Al/GaAs, Al/AlAsand GaAs/AlAs band discontinuities could be consistentlyexplained by extending to metal–semiconductor interfacesthe LRT schemes employed in the study of band offsets.These studies of the epitaxial Al/GaAlAs system show,however, that, although the pressure and compositionvariations of the barrier are dominated by the samebulk mechanisms as those that determine band offsetsat isovalent semiconductor heterojunctions, the absolutevalue of the Schottky barrier depends on interface-specific

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-0.08

-0.04

0.00

0.04

0.08

∆n (

elec

tron

s pe

r un

it ce

ll)

GaGa1/2Al1/2As Ga1/2Al1/2As-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

pote

ntia

l (eV

)

AlGa1/2Al1/2As Ga1/2Al1/2As

-0.08

-0.04

0.00

0.04

0.08

∆n (

elec

tron

s pe

r un

it ce

ll)

d1(Ga)

GaGa1/2Al1/2As Al-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

pote

ntia

l (eV

)

d1(Al)

AlGa1/2Al1/2As Al

Figure 13. Charge (full line) and potential (broken line) profiles induced by a single Al or Ga plane substitution in theAl1/2Ga1/2As(100) semiconductor homojunction (top panels) and within the semiconductor near the interface in theAl/Al1/2Ga1/2As(100) heterojunction (bottom panels). The charges and potentials were macroscopically averaged with a filterhaving the periodicity of the metal. (From [73].)

features such as the semiconductor termination and isnot abulk property. In this respect, the treatment of the Schottkybarrier examined here presents some analogies with that ofthe band offset at heterovalent polar interfaces. Finally, wealso note that we focused on rather ideal epitaxial systemsand our conclusion concerning the relative importance ofdefect-related versus intrinsic mechanisms in determiningthe barrier trends need not apply to other, more complex(for example, involving more reactive metals or transitionmetals) metal–semiconductor systems.

6. Band offset tuning

6.1. Tuning with ‘bulk’ strain and composition

In section 3, we showed that, for lattice-matchedisovalent semiconductor heterojunctions, the VBO is mostlydetermined by the bulk properties of the constituents.Interface details such as orientation and stoichiometry playa very minor role in determining the VBO, even in the case

of no-common-ion heterojunctions, for which the atomicinterdiffusion can considerably change the composition-induced interfacial strain.

Using only isovalent materials, the only way totune the offset is to act on thebulk rather than onthe interface. This can be done withbulk strain (seesection 4) or with alloying. The variation of the VBOas a function of the alloy composition has been studiedfor the InxGa1−xAs/InyAl 1−yAs(001) heterojunction [83].The small lattice mismatch between GaAs and AlAs can beneglected and the system can be considered lattice matchedwhen x = y, without introducing an appreciable errorinto the calculations. As we discussed in section 2.1,the calculated equilibrium lattice parametersatheq(x) exhibitpositive deviations from the experimentally observed linearbehaviour when the alloys are described using VCA. Suchdeviations disappear if the alloy is described usingtrue(rather thanvirtual) atoms. It is therefore reasonable touse Vegard’s law for the lattice parameter and calculate theelectronic structure using the VCA.

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Figure 14. The VBO at an In1−x Gax As/In1−x Alx As(001)heterojunction as a function of the alloy composition x(inset). The error bars indicate the range of theexperimental data. The main panel emphasizes thedeviation from the linear interpolation between the two endpoints. Circles are from supercell LDA-SCF calculations of[83]; the full line is a quadratic fit, given as a guide to theeye; and the diamond is from similar pseudopotentialcalculations from [84].

The VBO varies with composition from 0 eV forx = 1 to 0.58 eV for x = 0, for the case ofthe GaAs/AlAs interface. Spin–orbit splitting [36] andmany-body corrections to the valence band maximum areincluded in these values. For the many–body correctionan interpolation between the values calculated by Zhuand Louie (table XIV of [27]) for the end-point materialsand the 50%–50% alloys was used. The VBO(x) hasa significant negative bowing with respect to the linearinterpolation between the two end points (see figure 14).For x = 0.53, when the system is lattice matchedto InP, the calculated VBO is 0.19 eV, in the rangeof the experimental measurements [3] and in agreementwith other pseudopotential calculations predicting the samedeviations from linearity [84]. The effects of a possibleordering in the alloys have been estimated by calculatingthe offset for the limiting case of maximum ordering,namely for the interface between (InAs)1(GaAs)1 and(InAs)1(AlAs)1(001) superlattices. Taking into account alsothe internal distortions, the total effect on the VBO is ofthe order of 50 meV in the direction of partially reducingthe bowing.

We would like to stress that, even though the VBO islinear in the perturbation1V (equation (9)) leading fromthe optimal virtual crystal to the interface, this does notimply a linear behaviour of the VBO withx. In additionto the bulk band structure term, also the potential-line-upterm, following the present scheme, can exhibit a bowingwith x. For the potential line-up, in particular, this canbe understood from two different points of view. First, by

considering a unique reference crystal for all the interfaces,independently of the alloy composition; the perturbationson the two sides of the interfaces are, in general, not equalin magnitude and opposite in sign (in contrast to1V)and theon-site quadratic terms of the response thereforedo not necessarily cancel out in the potential line-up.On the other hand, if one considers instead the optimalreference virtual crystal for each composition, it depends onx (C(x) = 〈 InxGa(1−x)/2Al (1−x)/2〉); the potential line-up istherefore not linear with the composition since, in principle,both the response function and the perturbation recoveringthe actual interface (1vC = (1−x)(vGa−vAl)/2) depend onx. Similar arguments would predict a possible non-linearityin the VBO even at GaAs/Ga1−xAl xAs interfaces; however,due to the similarity of the dielectric properties and latticeparameters of GaAs and AlAs, one should expect a verysmall effect in this case, which is indeed experimentallyobserved [3].

6.2. Tuning with interlayers

The peculiarity of heterovalent interfaces leads naturallyto a practical way of modifying the offset at isovalentheterojunctions such as GaAs/AlAs, or even of creatingan offset at a homojunction. Let us examine first the caseof homojunctions and consider the following (001) growthsequence:. . . As–Ga–As–〈Ge1/2Ga1/2〉–〈Ge1/2As1/2〉–Ga–As–Ga. . .. Ideally, this sequence of atomic planes canbe obtained from bulk GaAs in two steps: creating firsta 〈Ga1/2Ge1/2〉-terminated GaAs/Ge interface and thentransforming back the Ge half space to GaAs with a〈Ga1/2Ge1/2〉-terminated GaAs/Ge interface shifted by oneinterplanar spacing,a0/4. These are precisely the twoinequivalent GaAs/Ge interfaces discussed in section 3.2;therefore, using the difference between the VBOs of thesetwo inequivalent interfaces a net potential drop1V =πe2/(a0〈ε〉) is predicted for the above transformation. Thispotential drop is the same as that which would result froma microscopic capacitor[85] whose plates are placed at adistancea0/4, carry a surface chargeσ = e/a2

0 and arefilled with a material whose dielectric constant is the sameas that of the virtual crystal. The above sequence of atomicplanes can also be thought of as due to the transfer of aproton per atomic pair from the As to the Ga planes [44].

Within LRT this viewpoint is easily generalized toarbitrary concentrations of Ge in a pair of consecutivecompensated GaAs planes,〈GexGa1−x〉〈GexAs1−x〉 (thisensures local charge neutrality). In this case, one has1V (x) = 2πe2x/(a0〈ε〉(x)), where〈ε〉(x) is the effectivedielectric constant of the reference system whose optimalchoice is a bulk alloy having the same compositionas the doped region between the two plates of themicroscopic capacitor. The corresponding dielectricconstant is〈ε〉−1(x) = (1 − x)ε−1

h + xε−1i [86], where

h indicates the host material, GaAs, andi the interlayer,Ge. Following this reasoning, the behaviour of the VBOat small doping (x → 0) depends on the host material,whereas at high doping (x → 1) it is dominated bythe electrostatic screening of the dopant. The excellentagreement between the simple predictions of LRT and

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full supercell LDA-SCF calculations [86] confirms thesoundness of the physical picture underlying this LRTapproach. Analogous considerations hold for other polarorientations, such as (111), and also for compensatedheterovalent interlayers embedded within a group IVelement bulk. In fact, the existence of a measurablepotential drop across a GaAs interlayer embedded in Gealong (111) has been detected experimentally [87].

The above results can be generalized to the heterojunc-tions, for example by doping a GaAs/AlAs interface withultra-thin layers of Si or Ge. In this case, the band offset isthe sum of anintrinsic term, VBOI, plus a doping contri-bution. If the Si dopant atoms are assumed to be uniformlydistributed over two consecutive atomic layers, the VBO issimply

VBO(x) = VBOI ± πe2

ax[(1− x)ε−1

h + xε−1i ] (23)

where ε−1h is, in this case, the average of the inverse

dielectric constants of GaAs and AlAs. In figure 15,the predictions of LRT are compared with full supercellLDA-SCF calculations [86] and experimental data [88] forthe GaAs/Si/AlAs VBO, as a function of the Si coveragewhich is 2x (measured in atomic monolayers, so thatx = 1corresponds to a full Si bilayer). The supercell calculationswere performed using the VCA forx < 1 to describe thetwo consecutive doped atomic layers at the interface. Theresults compare well with experiment [88] up to a coverage2x ≈ 0.5, whereas a substantial disagreement appears forhigher coverages. Such a disagreement between theoryand experiments should be ascribed to the simple pictureof dopants confined to two atomic planes, rather thanto numerical inaccuracies of the calculations such as theneglect of microscopic relaxation of the atomic positions,which is a very small effect (according to the test performedfor the limit x = 1, for which it should be more relevant).

The model of the microscopic capacitor is adequate forthin interlayers, but it cannot be extended to the thick-coverage limit. It would predict a dipole monotonicallyincreasing with coverage, which cannot be reconciledwith the energetic stability of the junction; furthermore,in the limit of two isolated interfaces, each of themmust be individually neutral, as previously discussed.Experimental measurements [49] in AlAs/Ge/GaAs(001)and GaAs/Ge/AlAs(001) single-quantum-well structureswith thick Ge interlayers (2–16 monolayers) show that theband offset between GaAs and AlAs across the interlayeris independent of its thickness in the range investigatedand that it is different from that directly measured atthe GaAs/AlAs interface; this suggests that two neutraland inequivalent interfaces are established even at suchthicknesses. This situation is analogous to the lack oftransitivity that is observed in the case of isolated III–V/Ge,ZnSe/III–V and ZnSe/Ge interfaces, which we discussed insection 3.2.

7. Schottky barrier tuning

7.1. Bulk composition and strain effects

We have seen that Schottky barrier changes as large as0.5 eV could be induced in the prototypical epitaxial

0.0 0.5 1.0 1.5 2.0 Si coverage

-1.0

0.0

1.0

2.0

VB

O (

eV) AlAs/Si/GaAs

GaAs/Si/AlAsLRTexperiment

supercell, idealsupercell, relax

Figure 15. The VBO at GaAs/AlAs heterojunctions as afunction of the coverage 2x of a Si interlayer. Circles:LDA-SCF supercell calculations with ideal unrelaxedzincblende atomic positions; squares, experimental resultsfrom [88]; and full line, predictions of LRT. Trianglesindicate results of calculations performed allowing a fullmicroscopic relaxation of the atomic positions, due to thelattice mismatch between Si and GaAs/AlAs. (From [86].)

Al/Ga1−xAl xAs(100) system considered in section 5, byaltering the alloy composition of the semiconductor. Thebarrier modification was shown to be dominated by thesame bulk mechanism as that responsible for determiningthe value of the band offsets at isovalent semiconductorheterojunctions. Similar LDA-SCF calculations have beencarried out also for abrupt (110)-oriented Al/Ga1−xAl xAsjunctions. A comparative study of the Schottky barriers forthe (100) and (110) orientations has been reported in [73].The study shows that although the values of the barriersfor the abrupt (110) junctions are 0.1–0.4 eV lower thanthose in the (100) junctions and depend on the interfacegeometry, for a given (meta)stable interface geometry, the(110) junctions exhibit the same (bulk) barrier variationwith the alloy composition as do the (100) As-terminatedjunctions.

This trend with the alloy composition is not sub-stantially affected by deviations from stoichiometry whichmay occur in the interface region. We have seen (section 5)that localized isovalent substitutions such as Ga:Al affectthe line-up only when they are performed within a fewatomic layers far from the interface and their effect onthe line-up does not exceed 0.1 eV for a deviation instoichiometry1x = 0.5. In addition, as will be shownin the next section, the ‘screening’ describing the responseof MIGS to localized dipolar perturbations produced byheterovalent substitutions at the interface is significantlylarger than that in a semiconductor heterojunction. Inthe case of a heterovalent Si bilayer at the Al/GaAs(100)interface, the electronic response reduces the local interfacedipole induced by the bilayer by a factor of about 50. For agiven metal and a fixed interface geometry one may expect

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thus to obtain reasonable estimates for Schottky barriertrends with semiconductor alloying by using band-offsetdata and the transitivity rule (equation (22)).

Experimentally, the transitivity rule works surprisinglywell, in general, for junctions (even non-epitaxial ones)involving covalent III–V compounds or alloys, and metalswhich are neither highly reactive nor transition metals [1].The bulk trend is observed, most often, in junctions usedfor transport measurements which have been annealed forfabrication of the contacts. This strong bulk dependenceon the semiconductor’s composition can thus be exploitedin practice to modify Schottky barriers substantially.

For a fixed interface geometry, the variation of thebarrier with metal composition is much smaller, in general(for covalent semiconductors, see below), than the changeachievable through modification of the semiconductor’scomposition. It should be noted that, because the Fermilevels align across any metal–metal junction, for a givensemiconductor any change in the barrier with the metalcomposition leads to a deviation from the transitivity ruleand reflects thus the non-bulk nature of the Schottky barrier.Using the epitaxial geometry illustrated in figure 10 (forthe type I junction), LDA-SCF calculations similar tothose presented for the variation with alloy composition ofthe Al/GaAlAs(100) Schottky barrier have been performedalso to probe the influence of a change in the chemicalcomposition of the metal [89]. Replacing the Al atoms inAl/GaAs(100) junctions by other simple metal atoms, eitherisovalent such as Ga or heterovalent such as P, modifies thebarrier by 0.1 eV or less. These barrier changes arefullyestablished by modifyingonly the composition of the first1–2 metallic layers near the interface. This is in strongcontrast to the mostly transitive behaviour observed for thesemiconductor alloy composition and further illustrates thenon-bulk character of the dependence of the barrier on themetal’s composition.

Ga and P both have a higher electronegativity thandoes Al (the difference of electronegativity with respectto Al is 1X = 0.1 and 0.6 for Ga and P, respectively, onPauling’s scale). For the junctions in which Ga (P) hasbeen used instead of Al the calculated value of the p-typeSchottky barrier is 0.05 eV (0.10 eV) lower than that of thereference Al/GaAs(100) system. This trend is in qualitativeagreement with the well known empirical ruleφn(M, s) =S(s)XM + φ0(s), where XM is the electronegativity ofthe metal (M) (which is also related to the metal’s workfunction,8M , by the empirical relation18M = A1XM ,A ≈ 2.3 eV), and S(s) and φ0(s) are constants for agiven semiconductor (s) [90]. From a compilation ofexperimental data,S(s) is found to be about 0.1 (usingelectron-volt units) for the most covalent semiconductorssuch GaAs and Si, and increases with the semiconductor’sionicity to values as large as 0.7 for ZnSe and 1.2 for ZnS[91].

The effect of the semiconductor’s ionicity on theS(s)parameter has been investigated by Louieet al [92]. Inpioneering studies of metal–semiconductor systems usingthe LDA-SCF pseudopotential approach, those authorsexamined the electronic structure of prototypical group IV,III–V and II–VI semiconductors (Si, GaAs, ZnSe, and

0.30.40.50.60.70.80.91.01.11.2

φ p (e

V)

0 0.5 1 1.5 2Si coverage (ML)

As-terminated

Ga-terminated

Figure 16. The Schottky barrier height of Al/Si/GaAs(100)diodes as a function of the Si coverage. The full circles arethe results of the self-consistent calculations; the opentriangles correspond to the experimental data of [96] for Sigrown under As (M) and Al (O) fluxes. The full lines givethe predictions of the capacitor model (see the text,equation (25)); the broken lines indicate the modelpredictions at low coverage in the linear regime. (From[97].)

ZnS) of increasing ionicity in contact with a jellium. Onthe basis of their results, they could explain the trend ofthe phenomenologicalS parameter in terms of the atomic-scale properties of the MIGS. The central quantities are thedecay length of the MIGS (δ) and their surface densities ofstates at the interface defined asDs =

∫ 0−∞N(εF , z

′) dz′,where N(εF , z) is the LDOS at the Fermi energy (seeequation (8)),z = 0 indicates the position of the interfaceand−∞ a position well inside the semiconductor wherethe MIGS vanish. Bothδ andDs were shown to decreasesignificantly with the semiconductor’s ionicity (by a factorof two on going from GaAs or Si to ZnS). The analysis ofthe S term is based on the following relation:

S = A/(1+ 4πe2Dsδeff ) (24)

whereδeff = t ′s + δ/εs is an effective distance for chargetransfer between the metal surface and the MIGS;t ′s =0.5 A is the typical screening length used for the metaland εs ≈ 2 determines screening in the semiconductor.Equation (24) was obtained by following an approachdeveloped by Cowley and Sze [93] and Heine [60], in whichthe response1v to a change in the metal work function,18M = A1XM , describing the change in the Schottkybarrier 1φn = A1XM + 1v, is modelled by a dipolarterm 1v = −4πe21qδeff with 1q = Ds1φn. Such adipole describes charge transfer from the metal surface tothe MIGS, which occurs over a distanceδeff . The charge1q = Ds1φn is that induced (to the first order) by a changein the Fermi level1φn into a MIGS reservoir characterizedby a surface charge per unit energyDs . The results obtainedby Louie et al [92] using this model and the computedvalues forDs and δ yield a very good description of theexperimental trend and values of theS parameter.

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0.00

0.01

0.02

0.03

0.04

0.05

0.06

Ds(

z) (A

−2 eV

−1)

−10 −5 0z (A)

Loc

al d

ensi

ty o

f sta

tes

(arb

. uni

ts)

−2 −1 0 1 2ε (eV)

εF

I

II

III

IV

V

IIIIIIIVV

θ = 2 ML

θ = 0 MLφp

Figure 17. The surface density of states at the Fermi energy without Si and with two monolayers of Si at z = 0 (see the text).The full line (broken line) corresponds to the As (Ga)-terminated interface. The inset shows the local density of states withinthe semiconductor’s band gap at increasing distances from the interface. The local densities are evaluated at the centre ofthe regions shown in the upper part of the figure. (From [97].)

The effects of selected perturbations of the atomicstructure on the prototypical epitaxial Al/GaAs(100) systemhave also been investigated using LDA-SCF calculations(for the As-terminated configuration in figure 10) [94]. Theinfluence of microscopic distortions on the metal side of thejunction (displacements of atomic planes near the interface)and of macroscopic uniaxial strains on the metal were foundto have very little effect on the Schottky barrier. Forinstance, a 3% uniaxial bulk strain or a 3% displacementof the Al layer at the interface modifies the barrier by only0.01 eV. This is in contrast to the observed sensitivity ofthe barrier to the details of the atomic structure on thesemiconductor side of the junction. These results could berationalized in terms of the vanishing absolute deformationpotential of the metal’s Fermi level and the properties ofthe dynamical effective charges in the junctions [94].

7.2. Tuning with interlayers

Considerable changes in the Schottky barrier height ofmetal contacts to covalent semiconductors, such as Siand GaAs, have been achieved in recent experiments,by controlling the chemical and/or structural propertiesof the interface [95, 96]. In particular, Schottky barrierswith a 1 eV wide tunability in Al/GaAs(100) junctionsengineered with ultra-thin (0–2 monolayers) Si interlayerswere reported [96]. We will discuss here the effect of

heterovalent interlayers in abrupt Al/GaAs(100) junctionsand show that such a tuning can be explained in termsof microscopic mechanisms similar to those illustrated forsemiconductor heterojunctions. This discussion is basedon the results of recent LDA-SCF calculations for theAl/Si/GaAs(100) system presented in [97].

The epitaxial type-I and type-II geometries, shownin figure 10, were used to model the initial (undoped)Al/GaAs(100) junctions. Si interlayers with coveragesθ in the range 0–2 monolayers were then introduced byreplacing by Si an equal number of Ga and As atoms in theGaAs bilayer closest to the metal. This doping produces abare ionic dipole1Vb = ±πe2θ/a0 (see section 6.2), wherea0 is the GaAs lattice parameter, which tends to increase(decrease) the p-type Schottky barrier at the As-terminated(Ga-terminated) interface. The magnitude of this dipole isabout 16 eV forθ = 2. Such a dipole will be significantlyreduced by electronic screening at the metal–semiconductorjunction. This can be inferred from figure 16, in which theSchottky barriers obtained from the LDA-SCF calculationsare displayed as a function of the Si coverage together withthe measured [96] Al/Si/GaAs barriers. The theoreticalvalues include spin–orbit and many-body corrections tothe LDA band-structure terms, evaluated in [71]. Forboth terminations, the dipole produced by the Si, whichis responsible for the change in the barrier1φp, has beenreduced from 16 to 0.35 eV forθ = 2, which corresponds to

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M Peressi et al

0.8

0.9

1.0

1.1

1.2

φ p (e

V)

10

15

20

25

30

35

ε eff

−10 −5 0z (A)

ε(GaAs)

8

Figure 18. The Schottky barrier (filled squares, left-hand scale) and effective screening (filled diamonds, right-hand scale) asfunctions of the interlayer position of Si within the semiconductor. The symbols give the results of self-consistent calculationsfor a Si coverage θ = 0.5. Their size indicates the estimated inaccuracy of the numerical results. The full lines correspond tothe predictions of equation (25) in the linear-coverage regime. (From [97].)

an effective screeningεeff = 1Vb/1φp of about 50. Thisyields a difference of about 0.8 eV between the barriersof the As-terminated and Ga-terminated interfaces withtwo monolayers of Si, in very good agreement with theexperimental data in figure 10.

The non-linear dependence ofφp(θ) in figure 16 showsthat εeff depends on the Si coverage:εeff varies fromabout 30 at smallθ to about 50 atθ = 2. Wehave seen (section 6.2) that the screening of a dipolecreated by Si interlayers analogously grown at GaAs/GaAshomojunctions could be explained in terms of the dielectricconstant of GaAs at low coverages and of the dielectricconstant of Si at a coverage of about two monolayers. Hereεeff is 3–5 times larger than the dielectric constants of GaAsand Si. There is thus a drastic enhancement of the screeningrelated to the presence of the metal.

The screening of such a Si dipole layer in the metal–semiconductor junction at a positionz from the metal andfor a coverage 0≤ θ ≤ 2 can be described by a microscopiccapacitor model [97]:

εeff (θ, z) = ε(GaAs)∞ + 4πe2Ds(z)δ

+θ2{ε(Si)∞ − ε(GaAs)∞ + 4πe2[D′s(z)−Ds(z)]δ} (25)

which is a generalization of the model previouslyintroduced (with Ds = D′s = 0) to describe theband-offset modifications produced by the Si interlayersin the GaAs homojunction (withε(GaAs)∞ and ε(Si)∞ theelectronic dielectric constants of GaAs and Si) [98]. Theadditional susceptibilityχM = (εeff |Ds,D′s − εeff |0,0)/(4π)is explained, using equation (25), in terms of the MIGS(see section 7.1) and more precisely in terms of their decaylengthδ and surface density at the Fermi energy without Si,

Ds(z), and with two monolayers of Si,D′s(z). The surfacedensity of states is evaluated at the position of the dipolez, namely,Ds(z) =

∫ z−∞N(εF , z

′) dz′. Figure 17 showsthe surface densities of statesDs(z) andD′s(z) of the As-and Ga-terminated interfaces, obtained from the LDA-SCFcalculations. The surface densities of states for the twoterminations are very similar and decay exponentially inthe semiconductor with essentially the same decay length(δ ≈ 3 A).

Using equation (25), one finds that at low Si coverage(θ → 0), the contribution of the MIGS to the potentialline-up induced by a small change1v in the Fermi-levelposition is−4πχM1v = −4πe2Ds(z)1vδ, consistentlywith the result expected from simple electrostatic arguments[93]. At higher Si coverage (0< θ ≤ 2), the last term onthe right-hand side of equation (25) includes the gradualchange in the susceptibility due to the modifications of thehost semiconductor and MIGS with the Si doping. Usingthe capacitor model in equation (25) and the values of theparameters obtained from theab initio calculations, onefinds the coverage dependence indicated by the full linein figure 16, which compares well with the self-consistentresults. It should be emphasized that most of the non-linearity in φp(θ) is due to the large difference betweenD′s(0) andDs(0) in equation (25); the difference betweenε(GaAs)∞ and ε(Si)∞ plays only a minor role. Furthermover,

from equation (25) the symmetry in the barrier variations ofthe two interfaces, in figure 16, can be understood from thesimilarity of theirDs(0) (andD′s(0)) values (see figure 17).

The correlation between the local-dipole screeningand the spatial decay of the MIGS, which is explicit inequation (25), has also been probed by moving the Sidipole layer away from the interface into the semiconductor.

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Given the exponential decay ofDs(z) in the semiconductor,εeff (θ = 0, z) should exponentially converge toε(GaAs)∞with a similar decay length. The results forεeff (z) obtainedfrom the LDA-SCF calculations for a coverageθ = 0.5are displayed in figure 18 together with the correspondingSchottky barrier values. The exponential decay length ofεeff (z) towards ε(GaAs)∞ is similar to that ofDs(z) andthe results closely follow the dependence predicted byequation (25) in the linear-coverage regime. The very goodagreement, in figure 18, between the predictions of themodel and the results of theab initio calculations furtherconfirm the soundness of the MIGS-based description usedto explain the screening of the local interface dipoles at themetal–semiconductor junction.

8. Conclusions

Considerable progress in understanding and controlling theband line-up at semiconductor contacts has been achievedin recent years by combined experimental and theoreticalefforts. Progress for metal–semiconductor junctionshas lagged somewhat behind that for semiconductorheterojunctions due to the higher complexity of thecorresponding atomic and electronic structures. Newdevelopments in computational physics have made possibleaccurateab initio calculations of the electronic structure ofsemiconductor contacts and the complexity of the systemswhich can be examined is steadily increasing. Thesecomputations can be used to study the effects of variousmicroscopic features in numerical experiments and to derivemodels which, within clearly defined limits of applicability,retain the same accuracy as the calculations from whichthey are derived.

In the present review we illustrated the basic pointsof a state-of-the-art theoretical approach which allowedus not only to compute band discontinuities for thevarious classes of heterojunctions but also to obtaininsight into the atomic-scale mechanisms which determinethe band line-ups and to interpret and predict theirtrends.

There remains a number of unresolved issues, whichstill limit the predictive capability of theoretical schemes.The most important concerns the mechanisms responsiblefor the actual atomic-scale arrangement at the interfacesand their kinetic versus thermodynamic character. A betterunderstanding of these mechanisms and the possibility ofpredicting the morphology of the epitaxial structures whichare actually established would clearly improve our abilityto engineer interface parameters. Another important relatedissue concerns the presence of localized interface states inheterojunctions, their origin and their role in determiningthe electronic characteristics of the junctions. These issuesare being addressed experimentally and should receive moretheoretical attention in the near future.

Acknowledgments

Two of us (M P and A B) would like to express specialthanks to their friends and colleagues S Baroni andR Resta: most of the work presented here for semiconductor

heterojunctions and the LRT method comes from the longand fruitful collaboration with them. We would like also tothank all our coworkers over the years, in particular J Bardi,C Berthod, L Colombo, M Di Ventra, but also F Favot,B Montanari, E Molinari, K Mader, M Lazzouni, N Titand A Valente. We also acknowledge valuable discussionswith A Franciosi, L Sorba, F Beltram, P Giannozzi,S De Gironcoli and A Dal Corso. The work at the EPFLwas supported by the Swiss National Science foundationunder grant 20-47065.96. The work in Trieste was fundedin part by the Consiglio Nazionale delle Ricerche of Italyunder the programme Progetto Coordinato Calcolo Paralleloin Materia Condensata.

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