TUNNELING CONTACTS FOR A DISSERTATION Roy Thesis.pdf · coupled with a tunneling model for spin...

TUNNELING CONTACTS FOR

NOVEL SEMICONDUCTOR DEVICES

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Arunanshu Mohan Roy

June 2012

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/fn491vz2263

© 2012 by Arunanshu Mohan Roy. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/fn491vz2263

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Krishna Saraswat, Primary Adviser


Yoshio Nishi


Dmitri Nikonov

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Preface

While the continued scaling of transistors faces severe limitations, novel devices in-

corporating new materials, new transport mechanisms and new state variables are

emerging as a strong contender for future logic and memory. This thesis describes

the modeling and simulation of three such devices.

The first of these is the metal-insulator-semiconductor (MIS) contact to achieve low

contact resistance. A contact resistance simulator for MIS structures is developed

and quantitative predictions are made of the achievable contact resistance using dif-

ferent insulator materials. TiO2 is predicted to be a suitable interface material. The

second device is the spin MOSFET using ferromagnetic source and drain contacts.

An efficient framework is developed to simulate spin transport in semiconductors

coupled with a tunneling model for spin injecting contacts. The important effects

of electric fields, the voltage dependence of magnetoresistance (MR) and the effect

of parameters such as tunnel oxide thickness, semiconductor channel length and

doping density on MR are investigated. The third device is the magnetic tunnel

junction which uses spin selective MgO tunnel barriers. An extended Huckel theory

(EHT) atomistic simulator coupled with non-equilibrium Greens function (NEGF)

formalism for transport is implemented. Through these EHT-NEGF simulations

the reduction of magnetoresistance due to Fermi level pinning is demonstrated. An

approximate approach for the simulation of CoFe alloy electrodes is developed.

iv

Acknowledgements

I would like to thank my advisor Prof. Krishna Saraswat for being extremely sup-

portive and helpful. He encouraged me to choose my own research directions and

helped me in every possible way to achieve my goals. I have gained immensely from

his insights and experience.

Dr. Dmitri Nikonov has been an excellent mentor. He introduced me to the fascinat-

ing field of spintronics and I have benefitted greatly from his technical expertise and

constant encouragement. I wish to thank Prof. Yoshio Nishi for his helpful advice

and for being a part of my thesis committee. I admire his energy and professionalism.

Much of this work would not be possible without my collaborators. I would espe-

cially like to thank Jason Lin for being a great co-worker. I have thoroughly enjoyed

working with him in the lab and on the whiteboard. Our exciting discussions have

helped to keep me motivated even when research was moving slowly. I have learnt

much from the experimental expertise of Donkoun Lee. I have enjoyed discussing and

simulating the interesting devices that he fabricated. More recently, working with

Donguk Nam on the germanium membrane project has also been very exciting. Over

the years, I have also benefitted greatly from all my discussions with the members of

the Saraswat group. I would like to thank them for all their advice and suggestions.

I would like to thank the SNF community and staff for all their help during the time

that I spent in the laboratory.

Intel Corp. has played an important role in my time at Stanford. They have sup-

ported my research and I have gained significant experience from my two internships

with Intel. I would like to thank Ian Young for a wonderful summer that I spent

v

learning about atomistic simulations with Components Research in Hillsboro, Ore-

gon. I would like to thank Pranav Kalavade for his friendly advice and for teaching

me so much about flash memory during my internship with the Flash memory group

at Santa Clara, California. I also wish to acknowledge Sasikanth Manipatruni for

insightful discussions while I was at Hillsboro. His enthusiasm for technology and

physics was very inspiring.

I wish to acknowledge a School of Engineering fellowship, an Intel PhD fellowship

and financial support for my project through Intel Corporation that has made my

stay at Stanford possible. I would like to thank Gail Chun-Creech for helping out

very efficiently with all administrative issues.

I have been fortunate to have many friends at Stanford. I would like to thank them

for all the good times we have shared in the last five years. I would like to thank Aai,

Baba, Salil, Aditi and Surabhi for all their love and support and the happiness that

they bring to Sukhada and me. I would like to thank Ananya for always being such

a caring and concerned sister. I am also very happy and thankful to have Hrishikesh

as a part of our family. No words are sufficient to thank Mama and Baba for all that

they have done for me. Their love and encouragement has been an important part

of all my achievements. I have many wonderful memories from my time at Stanford

but the ones I cherish the most are thanks to my lovely wife Sukhada. I am very

lucky to have Sukhada in my life and will remember Stanford most for all the fun

that we have had here together.

vi

Contents

Preface iv

Acknowledgements v

1 Introduction 1

2 Metal-Insulator-Semiconductor contacts 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Fermi level pinning and contact resistance . . . . . . . . . . . . . . . 6

2.3 MIS contacts for contact resistance reduction . . . . . . . . . . . . . . 9

2.3.1 Experimental work on MIS contacts . . . . . . . . . . . . . . . 10

2.3.2 Mechanisms for contact resistance reduction . . . . . . . . . . 11

2.4 Modeling of specific contact resistivity for MIS structures . . . . . . . 12

2.4.1 Model for the tunneling I-V characteristics . . . . . . . . . . . 12

2.4.2 Models for contact resistance reduction . . . . . . . . . . . . . 14

2.5 Choice of insulator materials . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Effects of fixed charge in MIS contacts . . . . . . . . . . . . . . . . . 19

2.6.1 Fixed charge density required for contact resistance reduction 21

2.6.2 Comparison of insulator materials . . . . . . . . . . . . . . . . 22

2.6.3 Effect of semiconductor doping . . . . . . . . . . . . . . . . . 23

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

vii

3 Semiconductor spin transport 26

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Semiconductor spintronics . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Spintronic devices based on semiconductors . . . . . . . . . . 27

3.2.2 Prior work on spin transport . . . . . . . . . . . . . . . . . . . 29

3.2.3 Experimental advances in semiconductor spintronics . . . . . . 33

3.3 Spin transport using the spin diffusion model . . . . . . . . . . . . . . 34

3.3.1 Spin diffusion model . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.2 Model for spin injection . . . . . . . . . . . . . . . . . . . . . 37

3.3.3 Interface resistance parameters near zero voltage . . . . . . . . 39

3.3.4 Voltage dependence of MR . . . . . . . . . . . . . . . . . . . . 44

3.3.5 Effect of material parameters on magnetoresistance . . . . . . 46

3.4 Spin drift-diffusion and electric field effects . . . . . . . . . . . . . . . 49

3.4.1 Spin drift-diffusion model . . . . . . . . . . . . . . . . . . . . 52

3.4.2 Ferromagnet-semiconductor interfaces . . . . . . . . . . . . . . 53

3.4.3 Magnetoresistance in a FM-Semiconductor-FM structure . . . 54

3.4.4 Model for spin injection . . . . . . . . . . . . . . . . . . . . . 57

3.4.5 Voltage dependence of MR . . . . . . . . . . . . . . . . . . . . 58

3.4.6 Effect of device parameters . . . . . . . . . . . . . . . . . . . . 60

3.5 Simulation of spin MOSFETs . . . . . . . . . . . . . . . . . . . . . . 60

3.5.1 Spin drift-diffusion model in long channel MOSFETs . . . . . 62

3.5.2 Pao-Sah double integral for MOSFET characteristics . . . . . 64

3.5.3 MR characteristics in spin MOSFETs . . . . . . . . . . . . . . 66

3.6 Challenges for spin MOSFETs . . . . . . . . . . . . . . . . . . . . . . 67

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.8 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Magnetic tunnel junctions 70

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

viii

4.2.1 Extended Huckel Theory . . . . . . . . . . . . . . . . . . . . . 71

4.2.2 NEGF transport formalism . . . . . . . . . . . . . . . . . . . 73

4.3 Bandstructures of bulk materials . . . . . . . . . . . . . . . . . . . . 74

4.4 Simulation of Fe-MgO-Fe MTJs . . . . . . . . . . . . . . . . . . . . . 80

4.5 Effect of Fermi level pinning on TMR . . . . . . . . . . . . . . . . . . 83

4.6 Approximate approach for alloy electrodes . . . . . . . . . . . . . . . 85

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.8 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 Conclusions 90

A Tunneling transport 92

A.1 Tsu-Esaki formalism for tunneling . . . . . . . . . . . . . . . . . . . . 92

A.2 Transfer matrix formalism for tunneling . . . . . . . . . . . . . . . . . 95

B Spin transport 99

B.1 Transfer matrices for spin diffusion . . . . . . . . . . . . . . . . . . . 99

B.2 Transfer matrices for spin drift-diffusion . . . . . . . . . . . . . . . . 103

C Linear combination of atomic orbitals 106

C.1 Slater type orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

C.2 LCAO tight binding . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Bibliography 111

ix

List of Tables

2.1 Experimental work on MIS contacts . . . . . . . . . . . . . . . . . . . 10

2.2 Contact resistance reduction mechanisms . . . . . . . . . . . . . . . . 13

2.3 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Slater type orbital parameters for EHT . . . . . . . . . . . . . . . . . 75

x

List of Figures

1.1 The contact resistance becomes a limiting factor for scaled devices

with low channel resistance. Metal-insulator-semiconductor contacts

can help solve the problem of contact resistance. . . . . . . . . . . . . 2

1.2 Spin injection, spin transport and spin detection are critical processes

for semiconductor spintronic devices such as the spinFET. . . . . . . 3

1.3 The magnetic tunnel junction is a fast non-volatile memory device

which also scales well. . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Ideal Schottky barrier formed at a metal-semiconductor interface is

determined by the difference in metal and semiconductor work-functions. 7

2.2 Relation between Schottky barrier height and pinning parameter S.

In an ideal unpinned junction, the Schottky barrier height is a linear

function of the metal work-function with slope = 1. For a perfectly

pinned contact, the Schottky barrier height is independent of the metal

work-function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Fermi level pinning in Ge. The Schottky barrier height is nearly inde-

pendent of the metal work-function. The line marked (a) shows the

expected barrier height for an ideal junction. Figure from Dimoulas

et al. [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 MIS band diagrams for various mechanisms of contact resistance re-

duction. The Schottky barrier reduction can be caused by interface

dipoles, interface charge or bulk charge in the insulator. . . . . . . . . 16

xi

2.5 Band alignment of different insulator materials with semiconductor

materials Si, Ge and GaAs. TiO2 and SrTiO3 have very low band

offsets to the conduction bands in these semiconductors making them

ideal candidates for contact resistance reduction. . . . . . . . . . . . . 17

2.6 Low barrier height materials promise lower minimum contact resis-

tance and better process margins. . . . . . . . . . . . . . . . . . . . . 18

2.7 Variation of specific contact resistivity with pinning factor and in-

sulator thickness using different insulators. Only TiO2 can achieve

the sufficiently low specific contact resistivity required for transistor

contacts since it does not add any significant tunneling resistance . . 20

2.8 Variation of specific contact resistivity with Al2O3 thickness for dif-

ferent amounts of fixed charge in the insulator. The semiconductor

is n-type Ge with a doping density of 1019cm−3. The estimated fixed

charge density required for reduction of contact resistance is approx-

imately 3 × 1020cm−3 for bulk charge and 3 × 1013cm−2 for interface

charge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Variation of specific contact resistivity for different tunnel barrier ma-

terials with fixed charge. The semiconductor is n-type Ge with doping

density 1019cm−3. The rate of reduction of contact resistance with in-

sulator thickness depends on the fixed charge density, the insulator

dielectric constant and the electron barrier height of the insulator. . . 22

2.10 Variation of specific contact resistivity due to fixed charge as a function

of Ge doping density. The tunnel barrier material used here is Si3N4. 23

3.1 Schematic diagram of the Datta-Das spin transistor . . . . . . . . . . 28

3.2 Schematic diagram of the Sugahara-Tanaka spin transistor . . . . . . 29

xii

3.3 Effect of spin accumulation when electrons flow from a ferromagnetic

contact into a non-magnetic material. Significant current spin polar-

ization can be achieved for injection into non-magnetic metals but the

spin injection is severely diminished in non-magnetic semiconductors.

Here µ↑ and µ↓ are the electrochemical potentials for majority and mi-

nority electrons respectively while J↑ and J↓ are the current densities

for majority and minority electrons respectively. After Fert et al. [25]. 30

3.4 The effect of interface resistance area product (r∗b ) on the magnetore-

sistance signal seen for the device shown in the inset. The MR plotted

on the Y axis is equal to the difference in resistance between the anti-

parallel and parallel configurations (∆R) divided by the resistance in

the parallel configuration (Rp). An optimum range of contact resistiv-

ity exists where high magnetoresistance is achieved. Smaller channel

lengths lead to higher magnetoresistance due to reduced spin relax-

ation. After Fert et al. [26]. . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Electric fields lead to asymmetry in semiconductor spin transport. A

shorter upstream spin diffusion length and a longer downstream spin

diffusion length is expected due to effect of electrons drifting in the

electric field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Schematic diagram of semiconductor spin injection and detection de-

vice. When a voltage is applied to the device, one junction is forward

biased and the other is reverse biased . . . . . . . . . . . . . . . . . . 34

3.7 The exchange splitting between the electrons in the majority and mi-

nority bands of the ferromagnet gives rise to a spin dependent contact

resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.8 Simulated currents for majority and minority bands. The structure

used is metal (φm = 4 eV, Ef = 2.2 eV,∆ = 2.2 eV) Al2O3 (8

Angstrom) Si (N- type doping 1016 cm−3) . . . . . . . . . . . . . . . 39

xiii

3.9 Dependence of spin dependent interface resistance area product pa-

rameters r∗b and γ on the Si n-type doping density for different oxides

of thickness 1 nm. Ferromagnet work function φm = 4.0 eV. . . . . . 41

3.10 Dependence of interface resistance area product parameters r∗b and γ

on the oxide thickness for different oxides on Si with n-type doping of

1017cm−3. Ferromagnet work function φm = 4.0 eV. . . . . . . . . . . 42

3.11 Dependence of spin dependent interface resistance area product pa-

rameters r∗b and γ on the metal work function for different oxides of

thickness 1 nm and Si n-type doping 1017cm−3 . . . . . . . . . . . . . 43

3.12 Voltage dependence of resistance area product for device in Fig. 1.

The parallel configuration has a lower resistance than the anti-parallel

configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.13 Voltage dependence of magnetoresistance for device in Fig. 1. The

interface resistance and spin selectivity is calculated using the current

voltage characteristics shown in Fig. 3.8. Length of the semiconductor

channel is 200 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.14 Voltage dependence of interface resistance area product for simulations

in Fig. 3.8. Reverse biased junctions show a substantial increase in

interface resistance area product with applied voltage . . . . . . . . . 46

3.15 Voltage dependence of spin selectivity of the interfaces shows that the

forward and reverse biased junction have very different spin selectivity

which varies significantly with the applied voltage . . . . . . . . . . . 46

3.16 Voltage dependence of magnetoresistance for simulations of different

values of n-type doping. Oxide thickness (Al2O3) = 1nm and metal

work function φm= 4 eV . . . . . . . . . . . . . . . . . . . . . . . . . 47


values of oxide (Al2O3) thickness. N-type doping = 1017cm−3 and

metal work function φm = 4 eV . . . . . . . . . . . . . . . . . . . . . 47

xiv


values of metal work function. N-type doping = 1017 cm−3 and oxide

thickness (Al2O3) = 1nm . . . . . . . . . . . . . . . . . . . . . . . . 48

3.19 Parallel resistance area product (Ωm2) variation using Al2O3. Ferro-

magnet work function = 4 eV. Increasing the oxide thickness increases

the resistance area product. The color-bar denotes the resistance area

product of the device. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.20 Magnetoresistance variation using Al2O3 (a), SiO2 (b) and HfO2 (c).

Ferromagnet work function = 4 eV. Increasing the resistance area

product reduces the MR. Higher doping values need lower oxide thick-

ness. The color-bar denotes the MR values obtained. . . . . . . . . . 50

3.21 Magnetoresistance variation using Al2O3 (a), SiO2 (b) and HfO2 (c).

Ferromagnet work function = 3.8 eV. A low work function gives higher

MR due to a lower resistance area product and higher spin selectivity

at the interfaces. The color-bar denotes the MR values obtained . . . 51

3.22 Relationship between spin accumulation and electro-chemical split-

ting. For small spin accumulations, this can be approximated by a

linear function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.23 Effect of interface RA product on MR ratio for a 2 µm semiconductor

channel calculated using nonlinear boundary conditions and the spin

drift diffusion model. Parameters used for this simulation are lsf,FM =

38 nm, ρ∗FM = 7.5× 10−8Ωm and βFM = 0.46 so that the resistivity of

the ferromagnet is 2ρ∗FM(1−βFM) for majority electrons and 2ρ∗FM(1+

βFM) for minority electrons, lsf,Semi = 2µm, µn = 550cm2V−1s−1, Dn =

14.3cm2s−1, total electron density 2n0 = 5×1015cm−3, T = 300K and

interface spin selectivity γ = 0.5 based on [26, 66] and approximate

silicon parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

xv

3.24 Spin accumulation and current spin polarization at semiconductor-

ferromagnet interfaces x=0µm and x=2µm in parallel (P) and anti-

parallel (AP) cases. Plots for semiconductor electric field of 103 V/m,

104 V/m and 105 V/m. . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.25 Spin selective contact parameters r∗b (resistance area product) and

γ (spin selectivity) as a function of voltage and semiconductor spin

accumulation for a ferromagnet-insulator-semiconductor contact. . . . 58

3.26 Schematic diagram of semiconductor spin injection and detection de-

vice studied using the spin drift-diffusion formalism. . . . . . . . . . . 58

3.27 Voltage dependence of MR and corresponding spin accumulation and

current spin polarization at the FM-semiconductor interfaces. . . . . 59

3.28 Variation of MR ratio with different semiconductor channel lengths.

Parameters used in this simulation are Nd = 1017cm−3, tox = 1 nm,

lsf = 2µm, µn = 550cm2V−1s−1, Dn = 14.3cm2s−1, T = 300 K. . . . . 61

3.29 Variation of MR ratio with different oxide thickness. Parameters used

in this simulation are Nd = 1017cm−3, lsemi = 1 µm, lsf = 2µm, µn =

550cm2V−1s−1, Dn = 14.3cm2s−1, T = 300 K. . . . . . . . . . . . . . 61

3.30 Variation of MR ratio with different semiconductor doping density.

Parameters used in this simulation are tox = 1 nm, lsemi = 1 µm,

lsf = 2µm, µn = 550cm2V−1s−1, Dn = 14.3cm2s−1, T = 300 K. . . . . 62

3.31 The structure shown is a schematic of the spin MOSFET device that is

simulated by accounting for the effect of the gate field and the varying

electric field and electron density along the inverted semiconductor

channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.32 Id-Vd and MR characteristics for a spin MOSFET device. . . . . . . 67

4.1 Bandstructure of majority spin electrons for a bulk Fe crystal in a bcc

lattice. The parameters used in the extended Huckel theory calcula-

tion are reported in Table 4.1. The symmetry of the bands near the

Fermi energy is indicated. . . . . . . . . . . . . . . . . . . . . . . . . 76

xvi

4.2 Bandstructure of minority spin electrons for a bulk Fe crystal in a bcc

lattice. The parameters used in the extended Huckel theory calcula-

tion are reported in Table 4.1. The symmetry of the bands near the

Fermi energy is indicated. . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Bandstructure of majority spin electrons for a bulk Co crystal in a

hcp lattice. The parameters used in the extended Huckel theory cal-

culation are reported in Table 4.1 . . . . . . . . . . . . . . . . . . . . 78

4.4 Bandstructure of minority spin electrons for a bulk Co crystal in a

hcp lattice. The parameters used in the extended Huckel theory cal-

culation are reported in Table 4.1 . . . . . . . . . . . . . . . . . . . . 78

4.5 Bandstructure of bulk MgO crystal in an fcc lattice. The parameters

used in the extended Huckel theory calculation are reported in Table.

Note that these parameters allow us to simulate MgO with a bandgap

of 7.7 eV which is close to the experimentally observed value . . . . . 79

4.6 Schematic diagram of a magnetic tunnel junction consisting of Fe elec-

trodes and MgO tunnel barrier. The corresponding atomic positions

used for this device are also presented. . . . . . . . . . . . . . . . . . 80

4.7 Transmisson probability at E = 0 over parallel k-space (V=0) . . . . 81

4.8 Transmission probability at fixed k parallel values (chosen from pre-

vious figure) over energy range simulated (V = 0) . . . . . . . . . . . 82

4.9 Current voltage characteristics for Fe-MgO-Fe structure with 4 MgO

layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.10 MR as a function of applied voltage for Fe-MgO-Fe MTJ with 4 MgO

layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.11 The MR vs voltage characteristics simulated for Fe-MgO-Fe MTJ us-

ing a bandgap of 7.7 eV for MgO and a barrier height of 3.7 eV . . . 85

4.12 The MR vs voltage characteristics simulated for Fe-MgO-Fe MTJ us-

ing a bandgap of 7.7 eV for MgO and a reduced barrier height of 0.3

eV to account for Fermi level pinning at the interface . . . . . . . . . 86

xvii

4.13 The effect of barrier height on the parallel and anti-parallel resistance

of an Fe-MgO-Fe MTJ. The barrier height plays an important role in

controlling both the MTJ resistance and the MR achieved. . . . . . . 86

4.14 The Hamiltonian and Overlap matrices for alloy electrodes are calcu-

lated as a weighted sum of the Hamiltonian and Overlap matrices of

the 4 possible configurations shown in this figure. The weight for each

matrix is given by the probability of the corresponding configuration. 87

4.15 The dependence of MR on votage applied to the an MTJ with alloy

electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.16 The dependence of MR on votage applied to the an MTJ with alloy

electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.1 Typical band diagram for a metal-insulator-semiconductor contact.

Parabolic bands are assumed in the metal and semiconductor and

Schrodinger equation is solved using this potential profile to calculate

electron transmission probabilities as a function of energy. . . . . . . 93

A.2 Transfer matrix formalism for tunneling in MIS structures. . . . . . . 95

C.1 Radial part of Slater type orbitals . . . . . . . . . . . . . . . . . . . . 107

C.2 Angular part of Slater type orbitals . . . . . . . . . . . . . . . . . . . 108

xviii

Chapter 1

Introduction

The last thirty years have seen an exponential improvement in the performance

of computers. Faster, smaller and cheaper computers have revolutionized every

aspect of life. The main driving force behind this tremendous achievement has

been the shrinking of the basic building block of digital circuits, the metal-oxide-

semiconductor(MOS) transistor. Such an exponential improvement of computational

power was predicted by Gordon Moore in 1965 [59]. His observation that the areal

density of transistors doubles every two years is now popularly known as Moore’s

law. Traditional scaling of transistors has for many years enabled the semiconductor

industry to keep up the predictions of Moore’s law. However, further shrinking of

transistors is becoming extremely difficult. Also, the benefits of scaling down the

conventional transistor are diminishing. This thesis explores the use of tunneling

transport in novel devices that might enable the semiconductor industry to further

improve the performance of integrated circuits and keep up with the demands of

Moore’s law.

The first of these concepts is the metal-insulator-semiconductor (MIS) contact. The

MOSFET is essentially a digital switch which can be in a low resistance ON state

or a high resistance OFF state. Improving these transistors involves reducing the

ON state resistance and increasing the OFF state resistance. As shown in Fig. 1.1,

1

CHAPTER 1. INTRODUCTION 2

Figure 1.1: The contact resistance becomes a limiting factor for scaled devices withlow channel resistance. Metal-insulator-semiconductor contacts can help solve theproblem of contact resistance.

the total resistance of a transistor can be approximately thought of as a series com-

bination of a contact resistance at the source end, the channel resistance and the

contact resistance at the drain end. While the use of high mobility channel materials

like germanium and III-V semiconductors promises lower channel resistance, these

materials also suffer from a large contact resistance at the source and drain ends.

An interesting solution to this problem proposed by Connelly et al. [14] involves the

insertion of ultra-thin tunneling barriers between the metal and semiconductor. In

chapter 2 MIS contacts are modeled and simulated to understand the limitations and

feasibility of using this approach in scaled transistors.

The second part of the this thesis studies a very similar device except with the

source and drain metal replaced by ferromagnetic materials. This device is called

the spin field effect transistor (spinFET). In addition to having gate control over the

channel in this device, the relative orientation of the source and drain electrodes can

also control the current through the transistor. This enables low leakage devices,

reconfigurable circuits and transistors with an integrated bit of memory. As shown

in Fig. 1.2, the important processes involved in this device are the injection of spin

polarized electrons from the ferromagnetic source into the channel, the transport of

spin polarized electrons across the channel and the detection of electron spin at the


Figure 1.2: Spin injection, spin transport and spin detection are critical processesfor semiconductor spintronic devices such as the spinFET.

Figure 1.3: The magnetic tunnel junction is a fast non-volatile memory device whichalso scales well.

drain electrode. Chapter 3 describes an efficient formalism to study these processes

that enables the simulation of spin MOSFETs.

The third part of this thesis studies a device similar to the spinFET which is

a memory device. This device depicted in Fig. 1.3 is called the magnetic tunnel

junction (MTJ) and consists of two ferromagnetic electrodes sandwiching a thin

tunneling barrier. This device is very promising due to its high speed of operation,

non-volatility and scalability. Chapter 4 studies the use of crystalline MgO tunnel

barriers, a major breakthrough in this area, that has lead to significant improvements

in MTJs.

A summary of the main results and conclusions is provided in chapter 5 followed


by appendices containing more details on simulation models and techniques.

Chapter 2

Metal-Insulator-Semiconductor

contacts

2.1 Introduction

Metal-Insulator-Semiconductor (MIS) contacts have recently emerged as a promising

solution to the problem contact resistance in semiconductor devices. In many semi-

conductors such as n-Ge, n-GaAs, n-GaSb, the effect of metal Fermi level pinning

leads to the formation of large Schottky barriers. This results in a large contact

resistance which can severely affect the performance of semiconductor devices. The

use of a thin dielectric tunneling barrier at the metal semiconductor interface has

been experimentally demonstrated to reduce the semiconductor Schottky barrier and

hence the contact resistance. However, using a tunnel barrier to alleviate Fermi level

pinning also leads to the addition of tunneling resistance at the contact. In order

to optimize these MIS contacts, it is essential to understand the trade-off between

the added tunneling resistance and the reduced Schottky barrier resistance. It is

also important to understand the physical mechanisms underlying the reduction of

contact resistance.

5

CHAPTER 2. METAL-INSULATOR-SEMICONDUCTOR CONTACTS 6

This chapter presents a theoretical study of the specific contact resistivity of MIS con-

tacts. The effect of Fermi level pinning in metal-semiconductor contacts is reviewed.

Prior results on contact resistance reduction using MIS contacts are described and

the proposed mechanisms for this effect are discussed. A theoretical model for the

specific contact resistivity on MIS contacts is presented and implemented to enable

the simulation of MIS contacts. Using this simulator, different dielectric materials

are studied in order to understand the feasibility and limitations of this technique in

making good Ohmic contacts to n-type Ge. The effect of tunnel barrier fixed charge

is studied to evaluate it as a possible mechanism for contact resistance reduction. Fi-

nally, the main results of this chapter are summarized and future directions for work

in this area are proposed. The results in this chapter are based on our published

work [71, 73].

2.2 Fermi level pinning and contact resistance

In an ideal metal-semiconductor contact, the height of the Schottky barrier in the

semiconductor is equal to the difference of the metal and semiconductor work-

functions. This was first proposed by Schottky and Mott [77]. According to this

theory, a high work-function metal will produce a large Schottky barrier in the semi-

conductor whereas a low work-function metal results in a small Schottky barrier

as depicted in Fig. 2.1. The ability to control the Schottky barrier height is ex-

tremely important in semiconductor devices. A large Schottky barrier corresponds

to a large resistance at the metal semiconductor contact which can severely limit the

performance of scaled transistors. The approximate relationship between the contact

resistance and the Schottky barrier height is given by (2.1).

ρc = ρc0exp

[

KφbN

]

(2.1)


(a) Schottky barrier with high work-function metal

(b) Schottky barrier with low work-function metal

Figure 2.1: Ideal Schottky barrier formed at a metal-semiconductor interface is de-termined by the difference in metal and semiconductor work-functions.


where φb is the Schottky barrier height, N is the active dopant concentration in the

semiconductor and ρc0 and K are constants.

According to the Schottky and Mott theory, it should be possible to reduce this

contact resistance by using a low work-function metal. However, it is observed exper-

imentally that the behaviour of metal-semiconductor contacts is not predicted very

well by this theory. The metal Fermi level is typically pinned close to a fixed energy

level with respect to the bandgap of the semiconductor. This makes the Schottky

barrier height nearly independent of the metal work-function. The extent of this

non-ideality is described by a pinning parameter S. As shown in Fig. 2.2, in an ideal

contact, S = 1 and the Schottky barrier height is modulated perfectly by the metal

work-function. In a perfectly pinned contact with S = 0, the Schottky barrier height

is independent of the metal work-function. In practice, the metal-semiconductors

tend to show small values of S indicating that the Schottky barrier height and hence

the contact resistance cannot be easily controlled. Germanium is a promising mate-

Figure 2.2: Relation between Schottky barrier height and pinning parameter S. Inan ideal unpinned junction, the Schottky barrier height is a linear function of themetal work-function with slope = 1. For a perfectly pinned contact, the Schottkybarrier height is independent of the metal work-function.


rial to replace silicon in scaled transistors due to its high electron and hole mobility.

However, Fermi level pinning of metals near the valence band in n-Ge leads to large

Schottky barriers and poor contacts [22, 62]. As shown in Fig. 2.3, the Schottky

barrier height in n-Ge is pinned close to 0.6 eV. This leads to a large series resistance

in NMOS transistors based on Ge. This problem is particularly difficult to solve in

n-Ge because it is difficult to achieve a high active dopant concentration. Similar

problems arise in materials such as GaAs, InGaAs and GaSb.

Figure 2.3: Fermi level pinning in Ge. The Schottky barrier height is nearly inde-pendent of the metal work-function. The line marked (a) shows the expected barrierheight for an ideal junction. Figure from Dimoulas et al. [22].

2.3 MIS contacts for contact resistance reduction

Metal-insulator-semiconductor contacts present an attractive solution to the problem

of Fermi level pinning and contact resistance. This section reviews experimental

work in this field prior to our contributions. Possible mechanisms responsible for the

reduction of contact resistance are also discussed.


2.3.1 Experimental work on MIS contacts

The use of tunnel barrier contacts to reduce the contact resistance was first demon-

strated by Connelly et al. [14] in silicon. A reduction in contact resistance to n-Si

was seen using a thin silicon nitride tunneling barrier. Nishimura et al. [63] demon-

strated similar results for n-type Ge using a germanium nitride tunneling barrier at

the interface. The first results on III-V semiconductor were demonstrated by Jenny

Hu et al. [37] using silicon nitride on gallium arsenide. Following these pioneering

experiments, many more materials and deposition approaches for the tunnel barri-

ers have been reported by researchers. Tunnel barrier contacts need to balance out

Table 2.1: Experimental work on MIS contacts

Semiconductor Interface layer References

Silicon Si3N4 Connelly et al. [13, 14]

AlOx|SiOx,LaOx|SiOx Coss et al. [15]

AlOx|SiOx Coss et al. [16, 17]

Germanium Ge3N4 Lieten et al. [50]

SiOxNy Kobayashi et al. [48]

GeOx, AlOx Nishimura et al. [63]

MgO Lee et al. [49], Zhou et al. [96]

TiO2 Lin et al. [51]

Gallium Arsenide SiN Hu et al. [37]

Al2O3 Hu et al. [39]

HfO2,TiO2,ZrO2 Hu et al. [40]

HfO2|TiO2 Hu et al. [40]

Indium Gallium Arsenide Al2O3 Hu et al. [39]

Gallium Antimonide TiO2 Ze et al. [93]

the competing effects of Schottky barrier reduction and tunneling resistance from

the barrier itself to achieve the minimum specific contact resistivity. Experiments

have shown that the use of a tunneling barrier for depinning can reduce the contact


resistivity by several orders of magnitude and there exists an optimum dielectric

thickness where the contact resistivity is lowest. It is not clear however that the

lowering of the contact resistivity by this method is sufficient to make good contacts

to transistors. In the following chapters, tunneling models are used to evaluate the

specific contact resistivity for various interface materials. Through this study, TiO2

is predicted to be a suitable interface material for contact resistance reduction. Fol-

lowing this prediction, good MIS contacts using TiO2 have also been demonstrated

on Ge [51], GaAs [40] and GaSb [93]. A summary of experimentally demonstrated

MIS structures is presented in Table 2.1.

2.3.2 Mechanisms for contact resistance reduction

While the physics of contact resistance reduction in MIS structures is still not well

understood, several mechanisms have been proposed to explain this effect. Two

theories that have received significant attention are the metal induced gap states

(MIGS) [62] and interface dipole formation [85, 90] mechanisms. The MIGS model

proposes that the electron wave-function in the metal penetrates the semiconductor

bandgap forming electron states. These states in the bandgap can result in Fermi

level pinning. The insertion of a thin tunnel barrier attenuates the electron wave-

function, thereby preventing Fermi level pinning. The dipole model proposes that

electric dipoles can form at abrupt material interfaces. The potential drop across

these dipoles can modify the Schottky barrier height in the contact. These dipoles

can exist at every interface in the structure such as the metal insulator (M-I) in-

terface, the insulator-semiconductor (I-S) interface or even insulator-insulator (I-I)

interfaces where more than one insulator layer is used. More recently, fixed charge

has also been proposed to play a role in contact resistance reduction. Significant

bulk and interface fixed charge density has been observed in atomic layer deposition

of insulator films [79]. Forming gas annealing of tunnel barriers which is known

to reduce fixed charge density causes the contact resistance to increase [38]. The

modulation of Schottky barrier height and contact resistance may be a combined


effect of fixed charge, interface dipoles and other mechanisms. Fermi level depinning

is a complex effect and a sound theoretical understanding of each individual mech-

anism is invaluable while analyzing experimental data. Table 2.2 summarizes the

salient characteristics of different pinning mechanisms and expected experimental

observations for each mechanism. In this work, the effects of fixed charge on contact

resistance have been examined in detail for the first time.

2.4 Modeling of specific contact resistivity for MIS

structures

The total resistance of a metal-insulator-semiconductor (MIS) junction includes the

tunneling barrier and the Schottky barrier which appear at the junctions of metals

and semiconductors. The specific contact resistivity of these tunnel barrier based

contacts is studied using the Tsu-Esaki tunneling model [84] as explained in the

following subsection. The specific contact resistivity of the MIS junction and is

defined as

ρc =

∂V

∂J

V=0

(2.2)

where V is the voltage applied and J is the current density.

2.4.1 Model for the tunneling I-V characteristics

A simple model for the MIS band structure consists of energy bands with a parabolic

E-k relationship in both the metal and the semiconductor. Using the Tsu-Esaki

model [84], the current density through the interface is given by (2.3).

Jtot =4πmeffq

h3

∫ Emax

Emin

η(εz)Ntot(εz) dεz, (2.3)

where meff is the density of states effective mass, h is Planck’s constant, η is the

electron tunneling probability, q is the charge of an electron, εz is the electron energy


Table 2.2: Contact resistance reduction mechanismsMechanism Description Expected observation

MIGS MIGS are mitigatedby the attenuationof the metal work-function in the in-sulator layer. Re-duction of surfacestate density leadsto Fermi level unpin-ning.

Higher electron tunnel barriershould produce greater wavefunction attenuation and bet-ter Fermi level depinning. Pin-ning factor S should improvewhile maintaining a linear re-lationship between metal work-function and Schottky barrierheight.

Interfacedipole

Dipole at the insu-lator - semiconduc-tor or metal - insula-tor interface causes apotential drop whichreduces the semicon-ductor Schottky bar-rier.

Difficult to distinguish the twodipoles. A linear relationship be-tween metal work-function andSchottky barrier height is not es-sential and the Schottky barriermay be reduced below the idealSchottky barrier height.

Fixed charge Bulk and interfacefixed charges in theinsulator layer causea potential drop thatreduces the semicon-ductor Schottky bar-rier

Fixed charge can be character-ized through capacitance mea-surements on thicker insulatorlayers.

Surface passi-vation

Passivation of surfacestates leads to depin-ning of Fermi level.

Pinning factor S should improvewhile maintaining a linear re-lationship between metal work-function and Schottky barrierheight. Surface state densitycan be estimated through capaci-tance and conductance measure-ments using thicker dielectrics.


in the direction perpendicular to the interface and Ntot is a supply function which

depends on the Fermi levels in metal and semiconductor. Under the assumptions of

parabolic bands, Fermi-Dirac statistics for electrons in the conduction bands, and

transmission probability independent of the parallel wave vector at the interface, the

supply function Ntot is given by (2.4)

Ntot = (kbT )ln

1 + exp(

εfm−εzkbT

)

1 + exp(

εfs−εzkbT

)

, (2.4)

where εfm is the Fermi energy in the metal, εfs is the Fermi energy in the semicon-

ductor, kb is the Boltzmann constant and T is the absolute temperature. A more

detailed derivation of the current density and supply function expressions are in-

cluded in Appendix A.1. To calculate the tunneling probability for a given applied

voltage, the depletion region in the semiconductor is determined by solving the 1D

Poisson equation using equilibrium charge concentrations consistent with the band

bending. The value of the tunneling transmission probability η(εz) is then obtained

using the transfer matrix formalism for tunneling problems as described in Appendix

A.2.

2.4.2 Models for contact resistance reduction

As discussed previously, proposed mechanisms for Fermi level depinning in MIS struc-

tures include dipole formation at the metal-insulator interface [90], interface charge

due to the creation of metal induced gap states (MIGS) [62], dipole formation at

the semiconductor-insulator interface [85] and fixed charge in the interface layer [38].

Thus in general, the difference in metal work function (φm) and semiconductor work

function (φs) is equal to the sum of potential drops across the metal-insulator dipole

(Vdip,M−I), the insulator (Vins), the insulator-semiconductor dipole (Vdip,I−S) and the

semiconductor Schottky barrier (VSB).

φm − φs = Vdip,M−I + VIns + Vdip,I−S + VSB (2.5)


The potential profile of the MIS structure depends on the mechanisms responsible

for the change in Schottky barrier height as seen in Fig. 2.4. A large Schottky

barrier is typically present in metal contacts to n-type Ge due to Fermi level pinning

near the valence band of Ge. A dipole at the metal insulator interface produces an

apparent change in the metal work function and can reduce the Schottky barrier

height. Similarly, the insulator-semiconductor dipole produces an apparent change

in the dielectric tunnel barrier height and also reduces the Schottky barrier. The

discontinuities in the vacuum level in Fig 2.4 result from the dipoles present at

the interfaces. If an insulator containing positive fixed charge is inserted at the

interface, assuming sufficient passivation of surface states, the potential drop across

the insulator results in a reduction of the Schottky barrier. Though the contact

resistance depends on the exact cause for Fermi pinning, all models will show very

similar trends for the ultra thin dielectric barriers used here. When the tunnel barrier

is thin, the tunneling resistance added by the insulator is not large enough to offset

the benefit of Schottky barrier reduction, hence a decrease in contact resistance is

seen. As the tunnel barrier becomes thicker, the added tunneling resistance increases

exponentially and dominates the total contact resistance.

The simulation parameters used for different insulators (dielectric constant ε,

bandgap Eg, tunnel barrier height to conduction band electrons in n-Ge and tun-

neling effective mass me) are reported in Table 2.3 along with references for these

values.

Table 2.3: Simulation parameters

Insulator ε Eg(eV ) Barrier height (eV) me

Al2O3 [69, 70] 9 8.8 2.8 0.2

GeO2 [65] 5.9 4.3 0.6 0.7

HfO2 [69, 70] 25 6.0 1.68 0.2

Si3N4 [69, 70] 7 5.3 1.9 0.2

TiO2 [70] 80 3.5 0.08 0.3


Figure 2.4: MIS band diagrams for various mechanisms of contact resistance reduc-tion. The Schottky barrier reduction can be caused by interface dipoles, interfacecharge or bulk charge in the insulator.


2.5 Choice of insulator materials

As discussed in the previous sections, several insulator materials show the property

of contact resistance reduction in MIS structures. In order to narrow the search

space for ideal interface materials, it is essential to understand the limitations on the

best achievable contact resistivity for different interface materials. Fig. 2.5 shows

the band alignment of various insulators with the semiconductor materials Si, Ge

and GaAs. The major limiting factor in MIS contacts is the tunneling resistance

added by the insulator layers. In fact, in this section, it is estimated that for most

insulator materials, the minimum achievable contact resistivity is not low enough for

scaled transistor applications. However, materials such as TiO2 and SrTiO3 present

a very low tunneling resistance to electrons and can help achieve very low contact

resistivity.

Figure 2.5: Band alignment of different insulator materials with semiconductor mate-rials Si, Ge and GaAs. TiO2 and SrTiO3 have very low band offsets to the conductionbands in these semiconductors making them ideal candidates for contact resistancereduction.

As depicted in Fig. 2.6, the benefits of using a low barrier height material are two


fold. Not only is it possible to achieve a much lower minimum contact resistance,

but the increase in contact resistance with increasing interface layer thickness is also

much more gradual. This provides much better process margins for the interface

layer thickness.

Figure 2.6: Low barrier height materials promise lower minimum contact resistanceand better process margins.

In the following simulations, a metal work function of 4.1 eV and dielectric barrier

heights as reported in Table 2.3 are used. The MIGS model is used to account for

the change in Schottky barrier height and the new Schottky barrier height achieved

is described by the pinning factor S [90].

φm,eff = φcnl + S(φm − φcnl) (2.6)

Here φm,eff is the effective metal work-function after pinning, φcnl is the charge

neutrality level at the semiconductor interface and φm is the actual metal work-

function with respect to the vacuum level. Thus S = 0 implies complete Fermi

pinning near the charge neutrality point while S = 1 implies a fully depinned con-

tact. For these simulations the charge neutrality level (CNL) used is 0.03 eV above

the valence band in germanium. Using the model described above, different insula-

tors are evaluated for the purpose of Fermi depinning in MIS structures on n-Ge. A


metal with a low work-function of 4.1 eV has been used in these simulations since

the contact resistance for n-type germanium is investigated. In order to effectively

reduce the pinning, at least 5A of insulator are required in the structure. Therefore

we study the specific contact resistivity value variation for insulator thickness in the

range 5A to 20A. For each value of insulator thickness the specific contact resistivity

is calculated for a range of Schottky barrier heights that may be achieved. Large

Schottky barriers (low pinning factors) and high insulator thicknesses both increase

the specific contact resistivity. The best contact resistivity is achieved when a low

Schottky barrier height (high pinning factor) results using a low insulator thickness.

For all simulations a heavily doped n-type Ge semiconductor with doping density

1019 cm−2 is used. From Fig. 2.7a, Fig. 2.7b, Fig. 2.7c, Fig. 2.7d it is seen that for

insulators Al2O3, GeO2, HfO2, Si3N4, even 5A of insulator thickness adds substantial

tunneling resistance to the contact. Low values of specific contact resistivity near

10−7Ωcm2 are hard to achieve. However for TiO2, there is no barrier to the conduc-

tion band electrons and increasing the oxide thickness does not cause a significant

change in the specific contact resistivity as seen in Fig. 2.7e. If a pinning factor of

0.8 is achieved using TiO2, good contacts can be made to n-Ge.

2.6 Effects of fixed charge in MIS contacts

Though the fixed charge may be arbitrarily distributed in the insulator, the two

special cases of uniformly distributed bulk charge and interface fixed charge at the

insulator-semiconductor interface have been studied. The metal is assumed to have

a work function of 4.6 eV, forming a Schottky barrier of 0.47 eV with n-type Ge,

which has an electron affinity of 4.13 eV. This is close to the Schottky barriers seen

in pinned contacts to Ge [22]. It is also assumed that no other pinning mechanisms

(surface states or dipoles) are present. Thus, the effects of the insulator fixed charge

on contact resistance can be isolated.


(a) ρc variation using Al2O3 insulator (b) ρc variation using GeO2 insulator

(c) ρc variation using HfO2 insulator (d) ρc variation using Si3N4 insulator

(e) ρc variation using TiO2 insulator

Figure 2.7: Variation of specific contact resistivity with pinning factor and insula-tor thickness using different insulators. Only TiO2 can achieve the sufficiently lowspecific contact resistivity required for transistor contacts since it does not add anysignificant tunneling resistance


2.6.1 Fixed charge density required for contact resistance

reduction

(a) ρc variation due to bulk fixed charge (b) ρc variation due to interface fixed charge

Figure 2.8: Variation of specific contact resistivity with Al2O3 thickness for differentamounts of fixed charge in the insulator. The semiconductor is n-type Ge with a dop-ing density of 1019cm−3. The estimated fixed charge density required for reductionof contact resistance is approximately 3×1020cm−3 for bulk charge and 3×1013cm−2

for interface charge.

As seen in Fig. 2.8a, a relatively high bulk fixed charge density of 3×1020cm−3 is

needed to produce a significant reduction in contact resistance. The large tunneling

resistance added by the Al2O3 layer limits the minimum achievable ρc to be greater

than 10−4Ωcm2. Similarly, when fixed charge is present at the interface, as shown

in Fig. 2.8b, a charge density of about 3 × 1013cm−2 is required. At the lowest

simulated bulk and interface charge, minimum ρc is attained at a Schottky barrier

height of approximately 0.2 eV. At the highest simulated bulk and interface fixed

charge values, the Schottky barrier is slightly negative so that all the resistance is

contributed by the tunnel barrier. In general, the insulator layer contains both bulk

and interface fixed charges and the combined effect of the two charges manifests as

a change in contact resistance. Also, the depinning effect falls fast with decreasing

charge density. For bulk fixed charge density below 1019cm−3 and interface fixed

charge density below 1012cm−2, ρc increases exponentially with insulator thickness.


2.6.2 Comparison of insulator materials

(a) ρc variation for different insulators withbulk fixed charge density of 3× 1020cm−3

(b) ρc variation for different insulatorswith interface fixed charge density of 3 ×1013cm−2

Figure 2.9: Variation of specific contact resistivity for different tunnel barrier materi-als with fixed charge. The semiconductor is n-type Ge with doping density 1019cm−3.The rate of reduction of contact resistance with insulator thickness depends on thefixed charge density, the insulator dielectric constant and the electron barrier heightof the insulator.

The Fermi level unpinning effect due to fixed charge in different insulator ma-

terials is depicted in Fig. 2.9. The potential drop across the insulator is inversely

proportional to its dielectric constant. Thus we see that the insulators with a high

dielectric constant (HfO2, TiO2) show a slow reduction in contact resistance with in-

creasing thickness while Si3N4 and Al2O3 show a rapid reduction. The barrier height

of the insulators is the other important factor which controls both the initial drop in

contact resistance as well as the exponential increase for high insulator thicknesses.

A high barrier height is detrimental for contact resistance reduction. Among the

insulators investigated here, Si3N4 appears to be best suited for contact resistance

reduction due to fixed charge. In general, the best interface material would possess

the qualities of low electron barrier height, low dielectric constant and high fixed

charge density.


2.6.3 Effect of semiconductor doping

Several experiments on Fermi level unpinning have been reported using lightly doped

semiconductor substrates [40, 48]. However, for technologically relevant contacts, it

is essential to understand the benefits of unpinning when the semiconductor is doped

heavily. Fig. 2.10 shows the effect of the semiconductor doping on the reduction of

contact resistance due to the fixed charge mechanism. At a low doping density of

1018cm−3 the improvement in ρc can be more than 3 orders of magnitude. However,

the drastic initial change in ρc due to Schottky barrier modulation is much reduced

when the semiconductor is heavily doped to 1020cm−3. In this case the ρc reduces

approximately by a factor of 50. The limiting factor becomes the tunneling resistance

of the insulator.

(a) ρc variation using Si3N4 insulator withbulk fixed charge of 3× 1020cm−3

(b) ρc variation using Si3N4 insulator withinterface fixed charge of 3× 1013cm−2

Figure 2.10: Variation of specific contact resistivity due to fixed charge as a functionof Ge doping density. The tunnel barrier material used here is Si3N4.

2.7 Summary

We demonstrate that MIS structures used for Fermi level depinning can have a sig-

nificant tunneling resistance. This leads to large values of specific contact resistivity


making this approach ineffective in producing good contacts on n-Ge. Different in-

sulator materials have been studied and we see that only insulators with very low

barrier to conduction band electrons such as TiO2 can provide the low specific con-

tact resistivity required while depinning the Fermi level. Insulators with substantial

barrier to conduction band electrons will have a large tunneling resistance thereby

producing poor contacts even if they can achieve good Fermi level depinning. It is

important to experimentally study insulators which will not add significant tunneling

resistance since they can help to make good contacts to n-Ge if a Fermi depinning

effect is seen.

The possible mechanism of contact resistance reduction due to positive fixed

charge in the tunnel barrier contacts to n-type Ge has been simulated. These nu-

merical calculations indicate that a high bulk fixed charge density of the order of

1020cm−3 or a high interface fixed charge density of the order of 1013cm−2 is required

for this mechanism to cause significant contact resistance reduction. Since fixed

charge in the tunnel barriers is easy to characterize, this work provides experimen-

talists with useful estimates while analyzing data on Fermi level unpinning. For this

mechanism, apart from charge density, the insulator dielectric constant and electron

tunneling barrier height are identified as the important parameters affecting contact

resistance. The effect of semiconductor doping density on contact resistance reduc-

tion is also studied. The effects of fixed charge combined with effects of interface

dipoles can provide a more complete model for contact resistance reduction.

2.8 Future work

The idea of MIS contacts has seen significant research in recent years however sev-

eral questions remain unanswered. From a theoretical standpoint, it is still not clear

what physical mechanisms are responsible for contact resistance reduction. Experi-

mentally, this area is suffers from a very large design space in the choice of materials,

deposition techniques and deposition conditions. In the future, there are several


directions that need to be pursued. It is necessary to evaluate all the available

experimental data against the suggested physical mechanisms to see if a consistent

explanation can be found for all the results. Certain materials like InAs pin the metal

Fermi energy inside the conduction band and are a very interesting material to study

as an interface layer for contact resistance reduction. The characterization of MIS

structures also has several open questions such as the high frequency characteris-

tics of MIS contacts and the characterization of surface state density and interface

dipoles in these structures. It is however clear that the use of interface layers for

contact resistance modulation is bound to play an important role especially in scaled

finFETs where silicidation of contacts is not a feasible solution or in high mobility

channel materials where Fermi level pinning is a severe problem. Thus, more detailed

theoretical and experimental studies are certainly called for in this area.

Chapter 3

Semiconductor spin transport

3.1 Introduction

Semiconductor spintronics is one area where MIS contacts are an essential part of

device design. The ability to tune the interface contact resistivity of spin injecting

MIS contacts is critical to obtain efficient spin injection from ferromagnetic elec-

trodes into semiconductors. This chapter discusses the use of MIS contact modeling

in the context of semiconductor spintronics by combining it with spin transport

models in semiconductors. The area of semiconductor spintronics is first briefly re-

viewed discussing various proposals of semiconductor spintronic devices, prior work

on semiconductor spin transport and some experimental progress in this field. The

spin diffusion model is described and a transfer matrix formalism is developed for

spin diffusion in multi-layer structures. This is combined with the MIS contact mod-

eling from the previous chapter to study important voltage dependent properties of

a spin injection and detection structure. The effect of electric field in spin trans-

port is investigated in the following section and the transfer matrix formalism is

extended to include these effects. The spin drift-diffusion transfer matrix formalism

is then combined with the Pao-Sah model for MOSFETs to enable the simulation

of long channel spin MOSFETs. A summary of the major results and directions for

26

CHAPTER 3. SEMICONDUCTOR SPIN TRANSPORT 27

future work are also included. The results in this chapter are based on our published

work [72, 74, 75].

3.2 Semiconductor spintronics

Injection of spin polarized current into semiconductors has garnered great interest

recently with the development of spintronics [88] and proposals of semiconductor

spintronic devices. Such devices are considered as one of emerging logic technologies

which have the potential to work alongside electronic transistors and fulfill applica-

tion specific functions [1]. A critical review of different concepts in semiconductor

spintronics is presented in [9,41,61]. Since the pioneering proposal of the spin transis-

tor by Datta and Das [20], several experiments have confirmed the possibility of spin

injection into semiconductors and significant efforts are being made to realize useful

semiconductor spintronic devices. This section briefly reviews some of the different

proposals for semiconductor spintronic devices, theoretical work on spin transport in

semiconductors and experimental advances in semiconductor spintronics.

3.2.1 Spintronic devices based on semiconductors

Several ideas have been proposed for spintronic devices based on semiconductors.

The processes common to most of these devices are the injection of spin polarized

electrons from a ferromagnetic electrode into a semiconductor, transport of spin

polarized electrons through the semiconductor and the detection of spin polarized

electrons at another contact. This section will briefly review two such devices, the

Datta-Das spin transistor and the Sugahara-Tanaka spinFET. Several other pro-

posals for device concepts include magnetic bipolar transistors [24, 29], transit-time

spin field-effect transistors [5], unipolar spin diodes [28], spin dependent resonant

tunneling devices [10], field programmable gate arrays using spin transistors [83],

reconfigurable spin based logic using non-local signals [21] and all spin logic [7].


Figure 3.1: Schematic diagram of the Datta-Das spin transistor

Datta-Das spin transistor

The Datta-Das spin transistor has two ferromagnetic electrodes for the source and

drain [20]. Electrons injected into the semiconductor channel from the source are spin

polarized and in the absence of significant spin relaxation in the channel, they arrive

at the drain electrode with the spin polarization of the source electrode. The drain

electrode is polarized parallel to the source electrode and these electrons thus find

several available states in the drain electrode and a large current flows through the

device corresponding to an ON state. The electron spin polarization can however

be controlled using an external gate field through the spin orbit coupling in the

semiconductor. This effect is called the Rashba effect or the Dresselhaus effect [88].

Thus, using the gate voltage, it is possible to orient the electrons spins at the drain

end in the opposite direction of the drain electrode. These electrons are now minority

spin electrons for the drain electrode and do not have a large available density of

states leading to a lower current corresponding to an OFF state. These processes in

the Datta-Das spin transistor are depicted in Fig. 3.1. In the ideal case where half

metallic ferromagnetic electrodes are used, excellent spin injection and detection is

achieved along with low spin relaxation good control of electron spin in the channel,

this device promises a very high ON-OFF current ratio.


Figure 3.2: Schematic diagram of the Sugahara-Tanaka spin transistor

Sugahara-Tanaka spinFET

The Sugahara-Tanaka spinFET also consists of a field effect transistor with a ferro-

magnetic source and a ferromagnetic drain electrode [81]. In this case, the device can

operate like a regular MOSFET. However this device provides an additional control

knob in the form of the source and drain magnetization. When the source and drain

electrodes are polarized in the same direction (parallel configuration), a large current

flows through the device. When polarized in opposite directions (anti-parallel con-

figuration), a much lower current flows through the device. This is once again due

to lower density of states near the Fermi energy for minority electrons. A schematic

diagram showing the operation of this device is presented in Fig. 3.2. The addi-

tional magnetization control can help achieve much lower off state leakage currents,

reconfigurable logic as well as having a memory device integrated into a transistor.

This device also works best under the conditions of half metallic ferromagnetic elec-

trodes, efficient spin injection and detection and minimum spin relaxation in the

semiconductor channel.

3.2.2 Prior work on spin transport

Transport of spin polarized electrons in ferromagnetic and non-magnetic materials

has been an important field of study since the initial experimental work on spin in-

jection [46] and giant magnetoresistance (GMR) [6,8]. In spintronic device based on


Figure 3.3: Effect of spin accumulation when electrons flow from a ferromagneticcontact into a non-magnetic material. Significant current spin polarization can beachieved for injection into non-magnetic metals but the spin injection is severelydiminished in non-magnetic semiconductors. Here µ↑ and µ↓ are the electrochemicalpotentials for majority and minority electrons respectively while J↑ and J↓ are thecurrent densities for majority and minority electrons respectively. After Fert etal. [25].

spin transport in metals such as the multi-layer GMR devices, spin transport occurs

in the diffusion regime. This was studied by Valet and Fert [86] by deriving the

one dimensional spin diffusion model from the Boltzmann transport equation using

suitable approximations. Following the proposal of the Datta-Das transistor [20],

research on spin transport in semiconductors was pursued both theoretically and

experimentally. It was discovered that magnetoresistance (MR) devices based on

spin injection and detection through semiconductors suffer from the problem of con-

ductivity mismatch between the ferromagnetic spin injector and the semiconductor

layer [76].


As shown in Fig. 3.3 [25], a significant current spin polarization can be achieved

at the interface of a ferromagnetic metal and a non-magnetic metal. However, when

current is injected from a ferromagnetic contact into a semiconductor, the spin injec-

tion achieved is very small. This is due to the large difference in the conductivity of

the ferromagnetic metal and the semiconductor. As shown in Fig. 3.3, when current

flows from a ferromagnetic metal into a non-magnetic material, a spin accumulation

is generated at the interface. This spin accumulation decays with a characteristic

length called the spin diffusion length and must remain unchanged across the inter-

face. The current spin polarization must also remain unchanged across the interface.

Due to the large conductivity of the ferromagnetic metal, a small spin accumulation

can lead to a large current spin polarization, however, a small spin accumulation can

only support a small current spin polarization in the semiconductor. A consistent

solution is achieved only for a very small spin injection and current spin polarization

at the interface. This effect is discussed in more detail by Schmidt et al. [76].

Spin injection from a ferromagnetic metal into a semiconductor is one of the basic

processes essential for semiconductor spintronic devices. It was shown that by using

a suitable spin dependent interface resistance the problem of conductivity mismatch

can be overcome [26,67]. By applying the spin diffusion model, Fert et al. [26] showed

that the spin selective interface resistance must be well tuned to lie in an optimum

range in order to see a significant magnetoresistance signal from a semiconductor

spin injection and detection device. This device is shown in the inset of Fig. 3.4

with the effect of interface contact resistance on the achieved MR. The spin diffusion

model will be discussed in more detail in the following sections. The dependence

of spin injection and magnetoresistance on the height of the Schottky barrier and

doping of the semiconductor has also been discussed in [57].

The spin diffusion model has been widely used to study spin transport in multi-

layered structures of metals and semiconductors [26, 74, 86]. However, Flatte et

al. [92] showed that electric field effects cannot be neglected in semiconductor spin

transport. The effects of an electric field on spin transport is depicted in Fig. 3.5. In


Figure 3.4: The effect of interface resistance area product (r∗b ) on the magnetoresis-tance signal seen for the device shown in the inset. The MR plotted on the Y axis isequal to the difference in resistance between the anti-parallel and parallel configura-tions (∆R) divided by the resistance in the parallel configuration (Rp). An optimumrange of contact resistivity exists where high magnetoresistance is achieved. Smallerchannel lengths lead to higher magnetoresistance due to reduced spin relaxation.After Fert et al. [26].

the absence of a significant electric field, spin accumulation decays at the same rate in

all directions. However, when a sufficient electric field is present in the semiconductor,

the electrons can drift further in the direction opposite to the electric field before

losing spin information. Thus, the spin transport now becomes characterized by an

upstream and a downstream spin diffusion length. The issues of spin transport and

spin relaxation have also been studied by Zutic et al. [88]. More recently, quantum

transport simulation of a 10 nm channel spinFET via the Keldysh non-equilibrium

Green’s function (NEGF) method has been performed [55].


Figure 3.5: Electric fields lead to asymmetry in semiconductor spin transport. Ashorter upstream spin diffusion length and a longer downstream spin diffusion lengthis expected due to effect of electrons drifting in the electric field.

3.2.3 Experimental advances in semiconductor spintronics

In addition to the theoretical proposals and studies, significant effort has been made

to experimentally characterize spin injection and spin transport in semiconductors.

Experiments on GaAs have been based on optical and electrical detection [27,54,64,

97] while those on Si have utilized all-electrical effects such as non-local measurements

[87], use of hot electron spin injectors [4] or optical detection [47] via spin-polarized

fluorescence. Spin injection into semiconductors using tunnel barriers and Schottky

barriers at a ferromagnet-semiconductor interface has been demonstrated [18,32,45,

60]. Recently several breakthroughs were made in this field with room temperature

spin injection and detection in Si [19] using a three terminal structure. Following

this work, room temperature spin injection was also demonstrated using non-local

measurements [82]. Germanium is yet another interesting semiconductor material

with high spin orbit coupling system which has been investigated [43, 52, 78]. While

the ultimate aim for spintronic logic devices is to replace CMOS devices beyond

the 8 nm node, the current experiments still study much larger devices where spin

transport is dominated by drift diffusion processes and cannot be assumed to lie


Figure 3.6: Schematic diagram of semiconductor spin injection and detection device.When a voltage is applied to the device, one junction is forward biased and the otheris reverse biased

in the ballistic regime. The following sections in this chapter describe results on

semiconductor spin transport in this regime and develop an efficient transfer matrix

formalism for the same.

3.3 Spin transport using the spin diffusion model

A typical semiconductor spin injection and detection device consisting of a semicon-

ductor layer sandwiched between two ferromagnetic layers is shown in Fig. 3.6. The

two ferromagnets can be magnetized parallel or anti-parallel (or, in general, at an

angle to each other). Magnetoresistance (MR) is defined as the following ratio of

resistances, RP and RAP , in the above two cases respectively.

MR =RAP − RP

RP(3.1)

Magnetoresistance depends on the conduction of the carriers with spin ”up” (along

the magnetization of the source) and spin ”down” (opposite to it) as well as spin

relaxation, which causes spin-flip, i.e., transitions between the spin orientations.

Tunnel barriers serve as spin dependent resistance elements and thereby increase

MR. The total resistance of the interface is combined from the tunneling and the

Schottky barriers which appear at the junctions of metals and semiconductors.


In this section, spin-polarized carrier transport in the semiconductor is simu-

lated based on the one dimensional spin diffusion approach proposed by Fert and

co-workers [26, 86]. While in [26] the resistance and spin selectivity of the barrier

were set constant and independent of voltage, a more rigorous treatment of tunneling

through the barriers is introduced here using the previously described work on MIS

contacts. The spin dependent interface resistance is one of the most crucial parame-

ters in spin diffusion simulations as it controls non-equilibrium spin polarization and

hence magnetoresistance. Using this approach, the voltage dependence of the overall

magnetoresistance is obtained for the first time. Non-trivial features of the voltage

dependence also arise from the fact that the resistances of the reverse and forward

biased Schottky barriers are different. The dependence of MR on the parameters of

the tunneling barriers and the semiconductor is also studied.

3.3.1 Spin diffusion model

Using the approach of Valet and Fert [86], a macroscopic model for spin diffusion in

a multilayer stack of materials is characterized by spin diffusion equations. A short

review of the mathematical model used is provided here.

The spin diffusion is described in terms of the difference in electrochemical po-

tentials for electrons with spin +12and spin −1

2by (3.2). The subscripts used refer to

electrons with spin +12and spin −1

2. Thus µ+ and µ− designate the electrochemical

potentials and σ+ and σ− designate the conductivity for the carriers with spins ”up”

and ”down”, respectively and ∆µ = µ+−µ−2

is used to describe spin diffusion. The

current continuity condition is expressed by (3.3).

∂2∆µ

∂z2=

∆µ

l2sf(3.2)

∂2(σ+µ+ + σ−µ−)

∂z2= 0 (3.3)


The conductivity itself is modeled using a spin-dependent parameter β and a resis-

tivity parameter ρ∗ using (3.4).

ρ↑(↓) =1

σ↑(↓)= 2ρ∗[1− (+)β] (3.4)

The resistivity ρ↑ and ρ↓ refer to the resistivity for the majority and minority electrons

respectively in the material. This notation ↑ (↓) is also used to label other variables

describing majority(minority) electrons. The parameter lsf is called the spin flip

length; it is characteristic of the material and is related to the spin relaxation time

τsf and the density of carriers as follows (for non-degenerate semiconductors) [26].

kb is the Boltzmann constant and T is the absolute temperature.

lsf =

√

kBTτsf2ne2ρ∗

(3.5)

A general solution to this set of equations can be written as follows where J is the

current density and z is the distance in the direction perpendicular to the material

interfaces.

µ↑ = (1− β2)qρ∗Jz +K1 + (1 + β)

[

K2exp

(

z

lsf

)

+K3exp

(

−zlsf

)]

(3.6)

µ↓ = (1− β2)qρ∗Jz +K1 − (1− β)

[

K2exp

(

z

lsf

)

+K3exp

(

−zlsf

)]

(3.7)

J↑ = (1− β)J

2+

1

2qρ∗lsf

[

K2exp

(

z

lsf

)

−K3exp

(

−zlsf

)]

(3.8)

J↓ = (1 + β)J

2− 1

2qρ∗lsf

[

K2exp

(

z

lsf

)

−K3exp

(

−zlsf

)]

(3.9)

Note that K1, K2, K3 are constants for every layer and we need to choose appro-

priate values for these constant that satisfy boundary conditions. For non-magnetic

material layers, the same equations apply but β = 0.

Under the assumption that current continuity is maintained and there is no spin


flipping at an interface, the discontinuity in the electrochemical potential of majority

spin electrons ∆µ↑ = µ↑(z+0 )− µ↑(z

−0 ) and minority spin electrons ∆µ↓ = µ↓(z

+0 ) −

µ↓(z−0 ) at interface z = z0 is modeled using an interface resistance area product

parameter r∗b and spin selectivity parameter γ.

∆µ↑(z0) = 2qr∗b (1− γ)J↑ (3.10)

∆µ↓(z0) = 2qr∗b(1 + γ)J↓ (3.11)

The above equations are further expressed in terms of material parameters relating

the coefficients of the solution K1, K2, K3 in adjacent layers. A transfer matrix

formalism is developed here which permits writing down the solution easily even

for a stack of multiple layers, without being encumbered by complicated algebraic

expressions. The derived transfer matrices are outlined in Appendix B.1. This

transfer matrix formalism is suitable for studying multilayer devices with interfaces

and also semiconductors with graded doping where the spin diffusion properties vary

with semiconductor doping density.

3.3.2 Model for spin injection

As shown in Fig. 3.7 a simple model for a ferromagnet band structure consists of

majority and minority bands which are split due to exchange interaction. In this

figure, ∆ is the exchange interaction energy splitting, Ef is the Fermi energy of the

ferromagnetic metal, φm is the metal work function, χs is the semiconductor electron

affinity and χox is the electron affinity of the tunnel barrier.

The current for each band is calculated by treating the majority and minority

bands independently. Using the Tsu-Esaki model [84] as detailed in the previous

chapter, the current through the interface for the majority/minority spin band is

given by (3.12/3.13).

J↑ =2πmeff↑q

h3

∫ Emax↑

Emin↑

η↑(εz)Ntot(εz) dεz, (3.12)


Figure 3.7: The exchange splitting between the electrons in the majority and minoritybands of the ferromagnet gives rise to a spin dependent contact resistance

J↓ =2πmeff↓q

h3

∫ Emax↑

Emin↓

η↓(εz)Ntot(εz) dεz, (3.13)

where meff↑/↓ is a density of states effective mass for majority/minority electrons,

h is Planck’s constant, η↑/↓ is the electron tunneling probability for majority/minority

states, εz is the electron energy perpendicular to the interface and Ntot is a supply

function which depends on the Fermi energy in ferromagnetic metal and semiconduc-

tor. Emin↑/↓ is the minimum energy where electron states are present on both sides

of the barrier. Emax↑/↓ this case is practically limited by the Fermi-Dirac statistics

governing electron state occupancy which causes the current contribution of electron

states to fall very rapidly for energies above the Fermi energy level. In order to

reduce the number of simulation parameters, it is assumed that the density of states

effective mass and the electron tunneling probability do not depend on the spin of

the electron. This is reasonable for amorphous tunneling barriers that do not have

special spin filtering properties. Using the assumptions of parabolic bands, Fermi-

Dirac statistics for electrons in the conduction bands, and transmission probability

independent of the parallel wave vector at the interface, the supply function Ntot is

given by (3.14). This is the same as described in the previous chapter for the case

of MIS contacts.


−1 −0.5 0 0.5 110

4

106

108

1010

Voltage (V)

Cur

rent

den

sity

(A

/m2 )

Majority band currentMinority band current

Figure 3.8: Simulated currents for majority and minority bands. The structure usedis metal (φm = 4 eV, Ef = 2.2 eV,∆ = 2.2 eV) Al2O3 (8 Angstrom) Si (N- typedoping 1016 cm−3)

Ntot = (kbT )ln

1 + exp(

εfm−εzkbT

)

1 + exp(

εfs−εzkbT

)

, (3.14)

Here εfm is the Fermi energy in the metal and εfs is the Fermi energy in the semi-

conductor. The tunneling probability for a given applied voltage is calculated by

solving the 1D Poisson equation and using the transfer matrix formalism for tunnel-

ing problems as discussed for MIS contacts previously. Fig. 3.8 shows the calculated

majority and minority currents for spin injection into n-type Si with an Al2O3 tunnel

barrier. The interface resistance depends upon the tunnel barrier thickness and the

semiconductor Schottky barrier. The spin dependence of the tunneling currents is

mainly due to the lack of minority states near the Fermi energy in the ferromagnet.

3.3.3 Interface resistance parameters near zero voltage

At low voltages, both the forward and the reverse biased junctions will see a very

small potential drop and will have the same value of r∗b and γ defining the interface

spin dependent resistance at the two interfaces. Following the methodology described


Table 3.1: Simulation parameters

Insulator ε Eg(eV ) χox − χs (eV) me

Al2O3 10 8.8 2.6 0.2

HfO2 20 5.8 1.5 0.2

SiO2 3.9 9 3.1 0.5

in the previous section the spin dependent interface resistance parameters near V =

0 are determined using the Tsu-Esaki model for structures with different doping

densities, oxide thicknesses, oxide material and metal work functions. Some results

are presented here showing the variation of the spin dependent interface resistance

parameters and their relation to the MR of the entire device is discussed. For these

simulations the ferromagnetic metal has parameters ∆ = 2.2 eV and Ef = 2.2 eV. The

ferromagnet is thus almost half metallic. The minority spin electron states only start

at the Fermi energy level. The majority spin electron states present below the Fermi

energy thus contribute to an additional current which gives rise to spin injection. The

simulation parameters used for different insulators (dielectric constant ε, bandgap Eg,

barrier height to conduction band electrons in n-Si (χox−χs) and tunneling effective

mass me) are reported in Table 3.1. Important simulation parameters used for Si

are relative permittivity εs = 11.9, electron affinity χs = 4.05 eV. Other simulation

parameters specific to simulations are reported with the corresponding figures. All

simulations are for temperature T = 300K.

Dependence on doping density

As seen in Fig. 3.9, changing the doping density does not affect the resistance area

product parameter r∗b significantly but the spin selectivity parameter γ increases at

higher values of doping density. The variation in r∗b for different oxides is due to

different barrier heights of the oxides to conduction band electrons. Spin injection

at the ferromagnet contact is due to the additional majority spin electron density of

states present below the Fermi energy level. The faster is the decay of transmission


(a) (b)

Figure 3.9: Dependence of spin dependent interface resistance area product param-eters r∗b and γ on the Si n-type doping density for different oxides of thickness 1 nm.Ferromagnet work function φm = 4.0 eV.

probability as we go to lower energies, the lower is the spin selectivity of the bar-

rier. Thus a tunnel barrier with a higher tunneling effective mass will lead to lower

spin selectivity since the transmission probability attenuates faster inside it and the

relative contribution of the additional majority spin states is lower. Thus we can

see that SiO2 has a lower spin selectivity than Al2O3 and HfO2. Barrier height also

plays a role in determining spin selectivity however the effect is less significant than

that of tunneling effective mass as we can see that Al2O3 and HfO2 have similar

spin selectivity. At higher doping densities, the Schottky barrier is reduced thereby

increasing the transmission probability at energies near the additional majority spin

electron states. This leads to improved spin selectivity at higher doping densities.

Dependence on Oxide Thickness

Increasing the oxide thickness leads to an exponential increase in the parameter r∗b

as shown in Fig. 3.10a due to the increase in tunneling resistance. Thus r∗b increases

rapidly and for large oxide thicknesses will lie outside the favorable region for spin

injection and detection. Increasing the oxide thickness also leads to a fall in the spin

selectivity of the barrier as shown in Fig. 3.10b. As the oxide thickness increases, the


(a) (b)

Figure 3.10: Dependence of interface resistance area product parameters r∗b and γon the oxide thickness for different oxides on Si with n-type doping of 1017cm−3.Ferromagnet work function φm = 4.0 eV.

transmission probability falls faster as we go to lower electron energies. As explained

in the previous subsection, this leads to a lower relative contribution of the additional

majority spin electron states and results in a lower spin selectivity. This indicates

that thin oxides are required for good MR characteristics. It is important to note

that this prediction does not hold for crystalline MgO where the spin selectivity

increases with oxide thickness. This is because the simple tunneling model used

here is insufficient to describe the tunneling through a crystalline MgO barrier. It is

important to take into account the detailed band structure of the metal and MgO

while calculating tunneling currents in this case because the electrons will couple

to different states in the insulator depending on the symmetry of the majority and

minority electron bands. This issue is discussed in more detail in the following

chapter. Practically at very low thickness too the spin selectivity of the oxide barrier

is reduced due to defects in the thin oxide film and spin flipping at the interface can

no longer be neglected.


(a) (b)

Figure 3.11: Dependence of spin dependent interface resistance area product param-eters r∗b and γ on the metal work function for different oxides of thickness 1 nm andSi n-type doping 1017cm−3

Dependence on Metal Work Function

Increasing the metal work function increases the tunnel barrier height as well as

the Schottky barrier height in the semiconductor leading to an increase in the r∗b

parameter as shown in Fig. 3.11a. However since the tunnel barriers used in these

devices are very thin, the contribution of the Schottky barrier to the increasing r∗b

parameter is most significant. As seen in Fig. 3.11a, since Si has a work function of

4.05 eV there is an exponential increase in r∗b for metal work functions above that

due to the increasing Schottky barrier. As seen in Fig. 3.11b the spin selectivity

γ also falls when the metal work function is greater than the semiconductor work

function due to the increased Schottky barrier resistance. The increased Schottky

barrier strongly attenuates the transmission probability for the additional majority

spin electron states. Fermi pinning of the metal work function has been neglected

here. As discussed in the previous chapter, incorporating a thin insulator layer

between the metal and the semiconductor can alleviate the problem of Fermi pinning

[14,48] and this can help produce better spin injecting contacts. In materials such as

germanium where the metal Fermi energy level gets pinned near the valence band,


most ferromagnetic metals will form a contact with a large Schottky barrier in the

semiconductor. This situation is similar to having an unpinned contact with a high

metal work function and leads to higher contact resistivity and lower spin injection

efficiency.

3.3.4 Voltage dependence of MR

Since previously published simulations [26, 86] used a constant spin dependent in-

terface resistance, they failed to capture the effect of the voltage dependence of a

tunnel barrier interface resistance and also the fact that I-V characteristics of a tun-

nel barrier are not necessarily symmetric for positive and negative voltages. The

spin-dependent interface resistance has been modeled in terms of constant parame-

ters r∗b (interface resistance area product) and γ (spin selectivity). Thus the interface

resistance area product is given by 2r∗b (1−γ) for the majority electrons and 2r∗b (1+γ)

for the minority electrons.

In this section, the computed I-V characteristics for majority and minority bands

are used self-consistently with the majority and minority electrochemical potentials

in the spin diffusion simulations to compute r∗b and γ (a voltage and spin dependent

interface resistance for each interface). As shown in Fig. 3.12 and Fig. 3.13, the

resistance area product of these devices increases with applied voltage and the MR of

these devices is maximal when the potential drop across the structure nears zero volt.

This is similar to the experimentally observed characteristics of magnetic tunnel

junctions [36]. Figs. 3.14 and 3.15 show that both interface resistance area product

and the spin selectivity of the forward and reverse biased contacts are very different

and they vary substantially with voltage applied to the device. This is because the

Schottky barrier resistances are intrinsically voltage dependent. To see how the

properties of the spin injection and detection device affect the voltage dependence

of MR, devices with different semiconductor doping densities, oxide thickness and

metal work function are studied. In Fig. 3.16 it is seen that as the semiconductor

doping density is increased, the maximum MR seen near zero volts first increases and


Figure 3.12: Voltage dependence of resistance area product for device in Fig. 1. Theparallel configuration has a lower resistance than the anti-parallel configuration

−1.5 −1 −0.5 0 0.5 1 1.50

10

20

30

40

50

60

Voltage applied (V)

Mag

neto

resi

stan

ce r

atio

(%

)

Figure 3.13: Voltage dependence of magnetoresistance for device in Fig. 1. Theinterface resistance and spin selectivity is calculated using the current voltage char-acteristics shown in Fig. 3.8. Length of the semiconductor channel is 200 nm.

then decreases again. In Fig. 3.17 it is seen that the oxide thickness also affects the

maximum MR and there exists an optimum value of oxide thickness where the MR

is maximal. Fig. 3.18 shows that a lower value of metal work function will invariably

lead to better MR characteristics and the MR value can saturate near a high value

even when applied voltage is large.


Figure 3.14: Voltage dependence of interface resistance area product for simulationsin Fig. 3.8. Reverse biased junctions show a substantial increase in interface resis-tance area product with applied voltage

Figure 3.15: Voltage dependence of spin selectivity of the interfaces shows that theforward and reverse biased junction have very different spin selectivity which variessignificantly with the applied voltage

3.3.5 Effect of material parameters on magnetoresistance

In this section, using the MR at zero volts as a metric, the effect of parameters such as

oxide thickness, semiconductor doping and ferromagnet band structure is studied. As

seen in the previous section, the device properties need to be optimized to get the best

MR characteristics. The spin selectivity of tunnel barrier is affected most strongly by


Figure 3.16: Voltage dependence of magnetoresistance for simulations of differentvalues of n-type doping. Oxide thickness (Al2O3) = 1nm and metal work functionφm= 4 eV

Figure 3.17: Voltage dependence of magnetoresistance for simulations of different val-ues of oxide (Al2O3) thickness. N-type doping = 1017cm−3 and metal work functionφm = 4 eV


Figure 3.18: Voltage dependence of magnetoresistance for simulations of differentvalues of metal work function. N-type doping = 1017 cm−3 and oxide thickness(Al2O3) = 1nm

the ferromagnet band structure. As we approach a half metallic ferromagnet with the

minority conduction band minimum at or above the Fermi level, the spin selectivity

improves greatly since the Fermi distribution ensures that minority currents are

low. The semiconductor doping determines the required interface contact resistance

area product. The resistance area product itself is determined by the metal work

function, the oxide thickness and the semiconductor doping. The effect of these

parameters on MR is studied. In Fig. 3.19 it is observed that the RA product for

the entire device is a strong function of oxide thickness and also depends on the

doping density of the semiconductor. In Figs. 3.20 and 3.21 the variation of MR

at zero volts with oxide thickness, semiconductor doping and metal work function is

shown. Different tunnel barrier materials Al2O3, SiO2 and HfO2 have been simulated.

The semiconductor is n-type Si and the ferromagnet is nearly half metallic with the

bottom of the minority conduction band coinciding with the Fermi energy. A lower

metal work function increases the MR significantly and for a given work function

the plots show the optimal values of doping and oxide thickness. Among the oxides


simulated here, HfO2 appears to show the best MR characteristics while SiO2 shows

poor MR characteristics due to the high barrier height it has for electrons.

Figure 3.19: Parallel resistance area product (Ωm2) variation using Al2O3. Ferro-magnet work function = 4 eV. Increasing the oxide thickness increases the resistancearea product. The color-bar denotes the resistance area product of the device.

3.4 Spin drift-diffusion and electric field effects

The previous section discussed the application of the spin diffusion model to study

a semiconductor spin injection and detection structure. In this section, the spin

drift-diffusion model is used to study the same structure and analyze the important

effects of electric fields in semiconductor spin transport. A transfer matrix method

for simulating spin injection into semiconductors in the case of high electric fields

is developed. The nonlinear relationship between electron spin density and electro-

chemical potential splitting, the effect of electric field on spin diffusion lengths and

spin polarized drift current are accounted for. Using this approach, high magnetore-

sistance (MR) at high electric fields is predicted. As in the previous section, the effect

of device parameters on the MR achieved is studied while also accounting for electric

field effects. The spin drift-diffusion model is first reviewed briefly. Certain special

considerations regarding the boundary conditions at metal-semiconductor interfaces


(a) (b)

(c)

Figure 3.20: Magnetoresistance variation using Al2O3 (a), SiO2 (b) and HfO2 (c).Ferromagnet work function = 4 eV. Increasing the resistance area product reducesthe MR. Higher doping values need lower oxide thickness. The color-bar denotes theMR values obtained.


(a) (b)

(c)

Figure 3.21: Magnetoresistance variation using Al2O3 (a), SiO2 (b) and HfO2 (c).Ferromagnet work function = 3.8 eV. A low work function gives higher MR due toa lower resistance area product and higher spin selectivity at the interfaces. Thecolor-bar denotes the MR values obtained


are discussed. This is followed by simulation results on MR obtained from the spin

drift-diffusion transfer matrix approach.

3.4.1 Spin drift-diffusion model

The effect of electric fields in semiconductors can be taken into account by solving a

spin drift-diffusion equation in terms of electron densities (n+ and n−) rather than

eletrochemical potentials [92].

∂2(n+ − n−)

∂x2+µE

D

∂(n+ − n−)

∂x− (n+ − n−)

l2sf= 0 (3.15)

where E is the electric field, µ is the electron mobility, D is the diffusion coefficient

for electrons and lsf is the spin flip length in the semiconductor. The expressions for

current density can then be calculated as follows

J+(−) = σ+(−)E + qD∂n+(−)

∂x(3.16)

In terms of variables ∆n = (n+ − n−)/2, n0 = (n+ + n−)/2 and ∆J = J+ − J−,

assuming that n0 remains constant within the layer, the general solution of (3.15) is

given by

∆n

n0= Aexp

(−xl1

)

+Bexp(−xl2

)

(3.17)

l1 =

µE

2D+

(

µE

2D

)2

+1

l2sf

1

2

−1

(3.18)

l2 =

µE

2D−

(

µE

2D

)2

+1

l2sf

1

2

−1

(3.19)

Using (3.16) and (3.17) the current spin polarization can be written in terms of


the constants A and B as

∆J

J=

2qn0

J

[

Am1exp(−xl1

)

+Bm2exp(−xl2

)]

(3.20)

m1 = µE − D

l1(3.21)

m1 = µE − D

l2(3.22)

3.4.2 Ferromagnet-semiconductor interfaces

As discussed above, spin transport in ferromagnet (FM) layers is solved using the

spin diffusion model based on electrochemical potentials. At a metal-semiconductor

interface, the electrochemical potential splitting (∆µ) must be converted into a spin

accumulation (∆n/n0). This is a non-linear relation described by (3.23).

2∆µ

kbT= ln

1 + ∆nn0

1− ∆nn0

(3.23)

A first order Taylor expansion can be used to linearize the relation thereby changing

the spin drift-diffusion problem into a set of linear equations which can be solved

using transfer matrices.∆µ

kbT≈ ∆n

n0(3.24)

As seen in Fig. 3.22, this approximation is valid in the regime of low spin injection

(less than 40%). For higher spin accumulation, simple numerical techniques can be

used to account for the non-linear boundary conditions while still using transfer

matrices to solve spin transport within the semiconductor and metal layers.


Figure 3.22: Relationship between spin accumulation and electro-chemical splitting.For small spin accumulations, this can be approximated by a linear function.

3.4.3 Magnetoresistance in a FM-Semiconductor-FM struc-

ture

A simulator is developed based on the models described in the previous section and

the transfer matrices detailed in Appendix B.2. The standard spin injection and

detection structure shown in the inset of Fig. 3.23 is studied using this simulator.

The magnetoresistance ratio is defined as before (3.1) in terms of the resistance in

the anti-parallel and parallel states, Rap and Rp respectively. These are calculated

from the potential drop across the device at a current determined by the electric field

in the semiconductor. The spin diffusion model predicts an optimum range of the

interface resistance area (RA) product for which significant MR is observable [26].

For a low electric field (103V/m) our results are consistent with this prediction as

shown in Fig. 3.23. However, for large electric field, it is predicted that the MR ratio

saturates at a non-zero value with increasing interface RA product. An increase in

the maximum MR ratio is also seen with increasing electric field. This effect of

a saturating MR ratio is due to drift dominated spin transport and saturation of

spin accumulation (∆n/n0) at high electric fields. At large values of interface RA

product (r∗b), the MR ratio is determined mainly by current spin polarization (∆J/J)


Figure 3.23: Effect of interface RA product on MR ratio for a 2 µm semiconductorchannel calculated using nonlinear boundary conditions and the spin drift diffusionmodel. Parameters used for this simulation are lsf,FM = 38 nm, ρ∗FM = 7.5× 10−8Ωmand βFM = 0.46 so that the resistivity of the ferromagnet is 2ρ∗FM(1 − βFM) formajority electrons and 2ρ∗FM(1 + βFM) for minority electrons, lsf,Semi = 2µm, µn =550cm2V−1s−1, Dn = 14.3cm2s−1, total electron density 2n0 = 5 × 1015cm−3, T =300K and interface spin selectivity γ = 0.5 based on [26,66] and approximate siliconparameters.


at the interfaces. As seen in Fig. 3.24, at low electric field, there is an optimum

range where the magnitude of interface ∆J/J for the parallel and anti-parallel cases

is different. The symmetry in ∆n/n0 at the two interfaces implies that the spin

transport is diffusion dominated. At higher electric fields, the current density in

Figure 3.24: Spin accumulation and current spin polarization at semiconductor-ferromagnet interfaces x=0µm and x=2µm in parallel (P) and anti-parallel (AP)cases. Plots for semiconductor electric field of 103 V/m, 104 V/m and 105 V/m.

the device is proportionally increased leading to higher spin injection and extraction

at the contacts. Thus we see a higher interface ∆n/n0 for the parallel case in Fig.

3.24. The transition from diffusion to drift dominated spin transport is evident in the

difference of interface ∆n/n0 for the anti-parallel case. Since the magnitude of ∆n/n0

is limited to a maximum value of 100%, spin diffusion currents also saturate once the

maximum value of ∆n/n0 is reached. The corresponding reduction of ∆J/J at the

left interface (x=0µm) increases the resistance of the anti-parallel state since more

current is forced through a higher resistance path at the interface. This increases

the MR ratio. The effects of high spin injection and drift dominated spin transport

are more prominent at a higher electric field of 105 V/m as seen in Fig. 3.24. In

both the parallel and anti-parallel cases, ∆n/n0 saturates as spin injection increases.


The spin transport here is almost completely dominated by the drift component and

∆J/J is negative at the left interface, causing MR to saturate at even higher values.

3.4.4 Model for spin injection

At the semiconductor interface, there can exist a non-negligible electron spin accu-

mulation (∆n/n0). This leads to a splitting of the electrochemical potential (∆µ)

which needs to be accounted for in current calculations. The relation between the

electrochemical potential splitting and the spin accumulation is described by (3.23).

The typical characteristics of these spin injecting contacts are depicted here in terms

of the parameters r∗b and γ, the resistance area product and spin selectivity of the

contact respectively. Using these parameters, the interface resistance is modeled as

2qr∗b (1− γ) for majority electrons and 2qr∗b (1 + γ) for minority electrons where q is

the magnitude of electronic charge [74]. Using the same approximate band-structure

as shown in Fig. 3.7 in the previous section, but also accounting for the effect of

this semiconductor spin accumulation, the spin injection properties of the contacts

are calculated. In this simulation, the tunnel barrier is chosen to be aluminum oxide

and the semiconductor is silicon. The simulation parameters used are φm = 4 eV,

Ef = 2.2 eV, ∆ = 2 eV, semiconductor doping Nd = 1017cm−3, oxide thickness tox =

1 nm, oxide relative permittivity εox = 9, oxide tunneling effective mass mox = 0.2,

χs − χox = 2.6 eV. As seen in Fig. 3.25, when the contact junction is reverse biased

(negative voltage), the spin selectivity of the contact does not change significantly

with the semiconductor spin accumulation. The large depletion region tunnel barrier

in the semiconductor prevents changes in electron density near the semiconductor

conduction band from affecting the current significantly. However when the contact

is forward biased (positive voltage), the spin selectivity of the contact is strongly

dependent on the semiconductor spin accumulation due to the absence of the de-

pletion region tunnel barrier. The resistance area product of the junction is also

controlled by the semiconductor depletion region. r∗b decreases when the junction is

forward biased and initially increases when reverse biased. For a large reverse bias,


Figure 3.25: Spin selective contact parameters r∗b (resistance area product) and γ(spin selectivity) as a function of voltage and semiconductor spin accumulation fora ferromagnet-insulator-semiconductor contact.

Figure 3.26: Schematic diagram of semiconductor spin injection and detection devicestudied using the spin drift-diffusion formalism.

electrons from the ferromagnet can tunnel directly through tunnel barrier into the

semiconductor conduction band, leading to a reduction in r∗b .

3.4.5 Voltage dependence of MR

The model for spin injecting contacts from the previous section and the spin drift

diffusion model using transfer matrices [72, 92] is combined to study MR in a FM-

semiconductor-FM structure shown in Fig. 3.26. This is the same device structure

that was discussed in the previous section on spin diffusion. The spin transport

through the contacts and the semiconductor channel is solved self-consistently so


Figure 3.27: Voltage dependence of MR and corresponding spin accumulation andcurrent spin polarization at the FM-semiconductor interfaces.

that the spin accumulation and current spin polarization at the FM-semiconductor

interfaces are consistent with both the contact characteristics and spin transport

characteristics of the semiconductor region. This enables us to study the voltage

dependence of MR and the interface properties of spin accumulation and current

spin polarization. Parameters used in this simulation are Nd = 1017cm−3, tox = 1

nm, lsemi = 1 µm, lsf = 2µm, µn = 550cm2V−1s−1, Dn = 14.3cm2s−1, T = 300 K.

As seen in Fig. 3.27, the MR in the FM-Semiconductor-FM structure is predicted

to reach and maximum value and then decrease with increasing voltage. This is due

to the effect of electric field in the semiconductor as well as the voltage dependence

of contact characteristics. In the FM-semiconductor-FM structure, one of the spin

injecting junctions is always forward biased while the other is reverse biased. As the

voltage across the device increase, the total resistance of the device also increases


due to the reverse biased junction. This leads to a fall in the MR ratio of the device.

For low voltages, the spin transport in the semiconductor is diffusion based and can

be seen in the fact that in the anti-parallel configuration (P), the sign of ∆J/J at the

two interfaces is opposite whereas it is the same in the parallel configuration (AP).

However, when the spin transport becomes drift dominated, ∆J/J at the forward

biased junction changes sign. At the reverse biased junction, at high electric fields,

the ∆n/n0 and ∆J/J is controlled by the spin selectivity of the contact leading to

a saturation near 0.5 for both P and AP cases. At the forward biased junction, the

∆n/n0 changes sign with increasing voltage due to the increased spin injection at

the reverse biased junction and constraints imposed by the forward biased junction.

3.4.6 Effect of device parameters

As seen in Fig. 3.28, Fig. 3.29 and Fig. 3.30, for different device parameters such

as semiconductor channel length, tunnel oxide thickness and semiconductor doping

density, the general nature of the voltage dependence of MR remains the same. For

shorter channel lengths, the increased MR is a result of the reduced spin relaxation

in the channel. We can see from the dependence on tunnel oxide thickness that

there is an optimum thickness at which the highest MR ratio is achieved. Hence this

parameter needs to be carefully adjusted in device design The effect of increasing

doping density is to also increase the MR ratio due to improved spin selectivity of

contacts at higher doping densities.

3.5 Simulation of spin MOSFETs

The previous section discussed the transfer matrix formalism for spin drift diffusion

and applied it to study a simple semiconductor spin injection and detection structure.

The spin MOSFET consists of essentially the same structure along with a gate that

controls the inversion channel in the semiconductor. This section describes how the

spin drift-diffusion transfer matrix formalism can be applied to spin MOSFETs.


Figure 3.28: Variation of MR ratio with different semiconductor channel lengths.Parameters used in this simulation are Nd = 1017cm−3, tox = 1 nm, lsf = 2µm, µn

= 550cm2V−1s−1, Dn = 14.3cm2s−1, T = 300 K.

Figure 3.29: Variation of MR ratio with different oxide thickness. Parameters usedin this simulation are Nd = 1017cm−3, lsemi = 1 µm, lsf = 2µm, µn = 550cm2V−1s−1,Dn = 14.3cm2s−1, T = 300 K.


Figure 3.30: Variation of MR ratio with different semiconductor doping density.Parameters used in this simulation are tox = 1 nm, lsemi = 1 µm, lsf = 2µm, µn =550cm2V−1s−1, Dn = 14.3cm2s−1, T = 300 K.

The cross-section of a spin MOSFET structure is depicted in Fig. 3.31. In

addition to the gate control in this device, the relative orientation of the source

and drain ferromagnetic electrodes can also control the current through this device.

When the ferromagnetic electrodes are aligned in the parallel configuration, a lower

contact resistance is obtained as compared to the anti-parallel configuration.

3.5.1 Spin drift-diffusion model in long channel MOSFETs

In order to find the MR and current-voltage characteristics of the spin MOSFET

device, we need a consistent solution for the charge and spin transport in the semi-

conductor channel. Since the semiconductor is a non-magnetic material, the channel

resistance is independent of the electron spin. Also, the inversion charge determined

by the Poisson solution at a particular gate voltage is independent of the spin of the

electrons. The MR in the spin MOSFET appears from the spin dependent contact


Figure 3.31: The structure shown is a schematic of the spin MOSFET device that issimulated by accounting for the effect of the gate field and the varying electric fieldand electron density along the inverted semiconductor channel.

resistance at the MIS contacts at the source and drain. The consistent solution to

charge and spin transport in the spin MOSFET can be computed using the following

steps.

1. Find the inversion charge density, channel resistance and lateral electric field in

the semiconductor channel as a function of the gate voltage and source-drain

voltage.

2. Divide the semiconductor channel into several layers and apply spin drift-

diffusion transfer matrix formalism to solve for spin transport in the parallel

and anti-parallel configurations. Every layer is characterized by the electron

density and electric field calculated previously.

3. The spin dependent contact resistance is determined by the spin accumulation

calculated from the spin transport solution and the current density calculated

from the charge transport solution.

The following assumptions are made to simplify the calculations

1. The width of the contacts is assumed to be much larger than the channel

thickness. This enables us to apply the one dimensional spin drift-diffusion


model to spin transport. The current density through the contacts is inversely

proportional to the width of the contact. Current crowding effects are neglected

in these simulations however, they will be important for scaled devices.

2. The current in the device is assumed to flow only in the direction along the

channel from the source to the drain. No current flows in the direction from

the semiconductor channel to the gate. Due to this reasonable assumption, the

spin accumulation can be assumed to be constant through the depth of the

inversion channel. For the purpose of spin transport calculations, the inversion

channel can thus be assumed to have a constant charge density across the

channel thickness such that the total charge remains the same.

3.5.2 Pao-Sah double integral for MOSFET characteristics

The Pao-Sah double integral is used to calculate the variation of inversion charge

density , the variation of the lateral electric field and the channel resistance from the

source end to the drain end as a function of the gate voltage and the source-drain

voltage. As described in the previous subsection, starting with this spin independent

solution, the spin drift-diffusion transfer matrix approach can be used to calculate

the spin accumulation and current spin polarization in the transistor channel as well

as the spin dependent contact resistances. This enables the calculation of the MR

characteristics of a long channel spin MOSFET. This subsection will briefly review

the salient features of the Pao-Sah double integral.

The Pao-Sah double integral is an accurate approach to computing the current-

voltage characteristics of long channel MOSFETs. This technique accounts for both

the drift and diffusion currents in the transistor channel. It also correctly accounts

for the inversion charge density across the depth of the channel. The gradual channel

approximation is made while deriving the Pao-Sah double integral. This means that

the vertical electric field from the gate to the transistor channel is considered to be

much larger than any lateral electric fields form the source to the drain. Such an


approximation is only valid for long channel MOSFETs. Thus the inversion charge

density can be calculated by only considering vertical electric fields from the gate to

the channel and calculating the one dimensional Poisson solution. The electron and

hole concentration in the semiconductor is calculated using Boltzmann statistics and

the position of the quasi Fermi levels.

Ids =W

L

µn√2qn0βLD

∫ Vds

0

∫ ψs

φb

βψ − βV

F (βψ, V, n0/p0)dψdV (3.25)

whereW is the transistor width, L is the transistor length, µn is the electron mobility

in the channel, q is the magnitude of electron charge, β = q/kT , n0 and p0 are the

equilibrium electron and hole concentrations in the semiconductor, Vds is the source

drain voltage and ψs is the band bending in the semiconductor for a channel voltage

V . The function in the denominator of the integral is defined in (3.26).

F (βψ, V, n0/p0) =

[

(e−βψ − βψ − 1) +n0

p0e−βV (eβψ − βψeβV − 1)

]1

2

(3.26)

The band bending at the oxide-semiconductor interface ψs is controlled by the applied

gate voltage. The relationship between ψs and Vgs is described by (3.27). Using

this equation along with (3.25), the Id-Vd characteristics of the MOSFET can be

calculated as a function of the gate voltage.

Vgs = ψs + Vfb +√2kT

qLD

εsCox

F (βψs, V, n0/p0) (3.27)

For the spin MOSFET calculations, we also require the electron density and electric

field variation along the semiconductor channel from source to drain. The channel

electron density is calculated by integrating the inversion charge over the depth of

the channel and is given by (3.28).

n(y) =∫ ψs

φb

n0exp(βψ − βV (y))

F (βψ, V, n0/p0)dψ (3.28)


The lateral electric field in the channel is calculated from the potential variation

along the channel. Once the current Ids is calculated, the potential at any point

y in the channel can be calculated by repeating the integral only till that point as

described in (3.29).

Ids =W

y

µn√2qn0βLD

∫ Vy

0

∫ ψs

φb

βψ − βV

F (βψ, V, n0/p0)dψdV (3.29)

3.5.3 MR characteristics in spin MOSFETs

Using the information on the variation of electric field, electron density and inversion

layer thickness along the channel computed using the previously described Pao-Sah

double integral, the spin drift diffusion transfer matrices can be applied to study spin

MOSFETs. The spin transport simulations are performed by dividing the channel

into several layers of different electric field, electron density and thickness and using

the transfer matrix approach. The current through the MOSFET is determined by

the potential drop from source to drain, however in this device, the tunnel barrier

and FM contact add an additional spin dependent resistance. As described in the

previous section, the contact characteristics and the MOSFET spin transport are

solved self consistently to arrive at the IV characteristics in P and AP cases. These

are shown in Fig. 3.32. The spin MOSFET has source drain contacts which consist

of a ferromagnet and a tunnel barrier as described previously. The tunnel barrier is 1

nm Al2O3 and the ferromagnet parameters are φm = 4 eV, Ef = 2.2 eV, ∆ = 2 eV as

described in Fig. 3.7. The source drain doping density is 1019 cm−3 and determines

the contact characteristics. The channel is 1 µm long and 100 µm wide and the

gate oxide is 20 nm SiO2. The contacts have a width of 500 nm. The calculated

MR ratio shows an increase at low voltages but then decreases to zero once the

MOSFET current saturates. This is because, in saturation, the channel resistance

of the MOSFET continues to increase while the spin dependent contact resistance

at the contacts remains unchanged thereby reducing the effect of the spin dependent

contact resistance. This simulation demonstrates the application of transfer matrix


Figure 3.32: Id-Vd and MR characteristics for a spin MOSFET device.

formalism for spin transport to experimentally relevant devices such as the spin

MOSFET.

3.6 Challenges for spin MOSFETs

Based on simulation insights and experimental advances in semiconductor spin in-

jection and detection, the following observations have been made for spin MOSFET

devices

• The use of tunnel barriers between the ferromagnetic metal and semiconductor

at the contacts plays a very important role. The spin selective resistance ob-

tained at the contacts is the main component of the spin dependent resistance

in long channel devices.

• The tunnel barriers also prevent diffusion of ferromagnetic atoms into the semi-

conductor spin injection regions. This prevents the high spin relaxation that

can be caused by randomly oriented spins of ferromagnetic atoms in the semi-

conductor. This issue can be avoided by using low temperature processes such

as electroplating.


• The tunnel barriers at the contacts add significant contact resistance in the

ON state of the transistor. This is detrimental to low power operation of the

device.

• The long channel MOSFET is essentially a transistor with an integrated mem-

ory element. In a scaled device, where the spin selective barrier is the Schottky

barrier which can be modulated by a gate field, better transistor characteris-

tics can be obtained by using ferromagnetic source drain. However, introducing

additional insulator tunnel barriers such as Al2O3 or MgO will add significant

series resistance to such a device.

• The success of the spinFET is strongly tied to being able to achieve close to

100% spin injection and detection efficiency. It is essential to find half-metallic

ferromagnets compatible with semiconductor processes for this purpose.

• Although spin injection and detection have been demostrated at room tem-

perature in a non-local configuration, significant local MR signal in a FM-

semiconductor-FM structure at room temperature is still elusive.

• Given the move of the industry to 3D finFET devices, the deposition and

control of magnetization of electrodes becomes difficult leading to additional

challenges.

• Spin torque switching of electrodes is a slow process and this leads to the

requirement of new architectures.

While the spinFET device still faces several challenges, semiconductor spintronics

continues to progress with improving spin injection and detection efficiencies [44]

and the prediction of new half-metallic ferromagnetic materials [2].


3.7 Summary

In this chapter, it is demonstrated that the voltage dependence of interface resistance

captured by modeling MIS contacts significantly affects the resulting MR in a semi-

conductor spin injection and detection structure. The voltage dependence of MR is

moderate and comparable to that in magnetic tunnel junctions. An efficient transfer

matrix formalism is developed for spin diffusion and extended to spin drift-diffusion

as well. New insights into semiconductor spin transport are obtained by accounting

for electric field effects. These are particularly relevant to technologically important

spintronic devices such as the spin field effect transistor. Using the transfer matrix

approach presented, the solution of the Poisson and current continuity equations can

be decoupled from the spin drift diffusion equations and solved easily. This enables

the modeling of spin transport in long channel semiconductor devices. The MR ratio

for spin MOSFETs is calculated and it is shown that the spin dependent contacts

play a crucial role in these devices. The models and implementation are both general

and efficient and can serve as useful tool for experimental research in this area.

3.8 Future work

At the time that these models were developed, working experimental devices such as

a spin MOSFET were yet to be demonstrated. The models and simulation techniques

described in this section are easily applicable to a variety of situations where spin

transport is combined with tunneling through MIS contacts. In the future when

working spin MOSFETs are successfully fabricated, these models can be used to

gain physical insights and optimize the design of such devices.

Chapter 4

Magnetic tunnel junctions

This chapter describes another novel device dependent on electron tunneling through

this insulator layers. This device is a promising memory device called the magnetic

tunnel junction. Like the devices discussed in the previous chapter, this is also a

spintronic device and benefits from the special spin selective tunneling properties of

MgO tunneling barriers.

4.1 Introduction

During the last decade, the field of spintronics has seen significant development and

achieved commercial success. Giant magnetoresistance (GMR) sensors have made it

possible to push hard disk drives to ever higher areal densities of data storage. Yet

another device, the magnetic tunnel junction (MTJ) is already commercially available

in magnetic random access memory (MRAM) [30] devices and the development of

spin torque transfer MRAM promises yet another revolution in the field of solid

state memory [36]. MTJ devices consist of a thin tunnel barrier sandwiched between

two magnetic layers. These devices exhibit the tunneling magnetoresistance (TMR)

effect which can be harnessed to detect the relative orientation of magnetization in

the magnetic layers of the MTJ. Initial experiments reported MTJs with modest

70

CHAPTER 4. MAGNETIC TUNNEL JUNCTIONS 71

magnetoresistance (MR) ratio of 50-70% using amorphous Al2O3 tunnel barriers

[31,58,89]. However, the prediction of spin selective crystalline MgO tunnel barriers

[56, 95] has lead to experimental demonstrations of TMR exceeding 600% at room

temperature [23,33,42]. The special spin filtering properties of crystalline MgO arise

due to its special band symmetries [95]. Such effects cannot be captured using simple

simulation approaches such as effective mass based tunneling models. An accurate

description of the band structure of the crystalline multi-layer MTJ structures is

required to properly investigate the MR ratio and spin filtering effect in these devices.

In this the extended Huckel theory approach [35] to electronic structure calculation

and non-equilibrium green’s function (NEGF) formalism for transport are used to

study MR in these devices.The important effects of Fermi level pinning on MR ratio

are studied and a simple approach to study the effect of alloy electrodes in MTJs is

proposed.

4.2 Methodology

The special spin filtering properties of crystalline MgO tunnel barriers make it es-

sential to accurately include the electronic structure properties of the MTJs. The

extended Huckel theory approach is an efficient formalism for calculating the elec-

tronic structure of the MTJ devices [11, 12]. The transport through these devices is

then calculated using the NEGF formalism [3] to predict important properties such

as MR and resistance area (RA)product of the MTJ devices.

4.2.1 Extended Huckel Theory

The extended Huckel theory (EHT) is a tight binding approach to electronic structure

calculations. It has the advantage of being much less computationally intensive than

ab-initio techniques and uses fewer parameters than other tight binding approaches.

The EHT formalism takes as an input, the spatial distribution of the atoms in the

device and the parameters which specify the non-orthogonal Slater type orbital basis


functions for these atoms. The atomic spatial distribution is described by the relative

position of atoms in a unit cell of the structure and the lattice vectors which describe

the spatial repetition of the unit cell to describe the complete device structure. Using

the non-orthogonal double zeta Slater type orbitals as basis functions, orbital overlaps

can be calculated. More information on Slater type orbitals is included in Appendix

C.1. Knowing this information, the overlap matrices corresponding to the orbital

overlaps between the origin unit cell and another unit cells within the cut-off radius

are computed.

The Hamiltonian matrices corresponding to the same two unit cells are then

calculated as follows

Hii = Eii (4.1)

The off diagonal terms are calculated using the extended Huckel theory approxima-

tion

Hij = KEHTSij(Eii + Ejj)/2 (4.2)

These EHT calculations have been performed in our own MATLAB based imple-

mentation and have been checked against previously published results. The linear

potential drop across the tunnel oxide can also be included in the EHT formalism.

In order to do this, potential matrices are calculated as follows

Uij = Sij(Uii + Ujj)/2 (4.3)

where Uii and Ujj are the values of the Laplace potential at the spatial locations of

the atom associated with orbital i and the atom associated with orbital j respec-

tively. The matrices computed in this manner can then be used to study electron

transmission probability for different energy and momentum values using the NEGF

formalism as outlined in the next section.


4.2.2 NEGF transport formalism

Using the Hamiltonian and Overlap matrices generated by the EHT formalism, the

NEGF formalism can be used to compute transmission probabilities. A multi-layered

device with infinite cross-section is assumed for simplicity, hence the transmission

probabilities are computed for all−→k|| values in the transverse Brillouin zone. The

Hamiltonian, Overlap and Potential matrices corresponding to a particular−→k|| value

can be calculated using the following expressions.

S(−→k||) =

∑

−→rS(−→r )ei

−→k||.

−→r (4.4)

H(−→k||) =

∑

−→rH(−→r )ei

−→k||.

−→r (4.5)

U(−→k||) =

∑

−→rU(−→r )ei

−→k||.

−→r (4.6)

Using these matrices, the Green’s function at any−→k|| and energy (E) can be calculated

as

G(E, V,−→k||) = [(E + i0+)S(

−→k||)−H(

−→k||)− U(V,

−→k||)− Σl(E,

−→k||)− Σr(E,

−→k||)] (4.7)

where Σl and Σr are the surface Green’s functions corresponding to the semi-infinite

electrodes. The transmission probability at any particular energy, applied voltage

and parallel momentum vector is given by

T (E, V,−→k||) = trace(ΓlGΓrG

†) (4.8)

This can then be integrated over the entire transverse Brillouin zone to get the total

transmission probability at any energy and applied voltage.

T (E, V ) =1

4π2

∫

T (E, V,−→k||)d

−→k|| (4.9)


For our simulations, the transmission probability is computed at a limited number of−→k|| points in the transverse Brillouin zone and used to approximate the above integral.

The current density through the device is calculated by multiplying the transmission

probability with the supply function and integrating over energy. The transmission

probability is a function of the relative orientation of the magnetic electrodes thereby

giving us different current densities for the parallel and anti-parallel configurations.

J(V ) =q

h

∫

T (E, V )[f(E − µl)− f(E − µr)]dE (4.10)

In the above equation, q is the magnitude of electron charge and h is the Planck’s

constant. The supply function itself depends on the Fermi function (f) in the two

contacts and has significant values only within a few kT of the Fermi energies in the

two contacts. This enables us to limit the energy range in our integration for current

density. Having calculated the current density as a function of the voltage applied to

the tunnel junction, we can compute the tunneling resistance and magnetoresistance

ratio (MR) for the simulated magnetic tunnel junctions.

4.3 Bandstructures of bulk materials

Using the EHT formalism, the bandstructures of the materials that are relevant to

our devices i.e. iron, cobalt and magnesium oxide are computed. It has been veri-

fied that the correct bandstructures are obtained using the EHT formalism and the

parameters listed in Table 4.1 [11, 12]. Since the EHT parameters are local in na-

ture, assuming good transferability, the same parameters can be applied to the case

of multi-layered MTJ structures that we intend to study. These bandstructures

can be used to explain the special spin filtering effects in Fe-MgO-Fe tunnel junc-

tions. The bulk bandstructure describes the allowed electron energies at every value

of crystal momentum. The electron wave-function in the periodic potential of the

infinite crystal satisfies is called a Bloch wave. According to Bloch’s theorem, these

Bloch waves which are the energy eigenfunctions for the system can be written as


Table 4.1: Slater type orbital parameters for EHT

Atom Orbital Eonsite ζ1 c1 ζ2 c2

Fe majority

4s -9.55432 1.48840 0.58920

4p -6.82472 1.25262 0.59596

3d -11.91792 1.48912 0.23968 3.34830 0.84348

Fe minority

4s -9.69388 1.48840 0.58920

4p -7.02276 1.25262 0.59596

3d -10.36973 1.48912 0.32297 3.34830 0.77965

Co majority

4s -10.19191 1.65982 0.72843

4p -6.96120 1.38108 0.67675

3d -12.46976 1.48620 0.19807 3.24397 0.86949

Co minority

4s -10.11838 1.65982 0.72843

4p -6.78064 1.38108 0.67675

3d -11.06492 1.48620 0.25019 3.24397 0.83083

Mg

3s -10.5280 1.53391 0.61270

3p -7.1170 0.97969 0.61385

O

2s -28.6240 3.22378 1.00000

2p -28.6240 2.03171 1.00000


Figure 4.1: Bandstructure of majority spin electrons for a bulk Fe crystal in a bcclattice. The parameters used in the extended Huckel theory calculation are reportedin Table 4.1. The symmetry of the bands near the Fermi energy is indicated.

the product of a plane wave envelope function and a periodic Bloch function that has

the same periodicity as the potential. The periodic Bloch function can be modeled

as a linear combination of the atomic orbitals in the crystal as described in Appendix

C.2. The symmetry of these periodic Bloch functions is then determined by the sym-

metry of its constituent atomic orbitals. ∆1 symmetry Bloch states are comprised

of s, pz and d3z2−r2 orbitals. Similarly, ∆5 Bloch states have symmetry consistent

with px, py, dxz and dyz atomic orbitals. The ∆2 and ∆2′ symmetries come from the

remaining d and f orbitals. The rate of decay of the electron wave-function through

the crystalline MgO tunneling barrier is determined the wave-function symmetry.

The MgO tunnel barrier presents the lowest resistance to ∆1 symmetry Bloch waves.

∆5 symmetry Bloch waves decay at a faster rate in the MgO layer and the ∆2 and

∆2′ Bloch waves decay at still faster rates. Due to the nature of the Fermi-Dirac

distribution of electrons, electron transport is dominated by energies within a few

kT of the Fermi energy level. In the case of Fe, the majority spin bands contain ∆1,


Figure 4.2: Bandstructure of minority spin electrons for a bulk Fe crystal in a bcclattice. The parameters used in the extended Huckel theory calculation are reportedin Table 4.1. The symmetry of the bands near the Fermi energy is indicated.

∆5 and ∆2′ symmetry Bloch waves near the Fermi energy. The minority spin bands

contain ∆5, ∆2 and ∆2′ states near the Fermi energy level. Thus, in the parallel

condition, ∆1 Bloch states of majority electron dominate the total electron trans-

port since Bloch states with this symmetry are present in both electrodes. In the

anti-parallel case, the ∆1 symmetry states are present in only one of the two elec-

trodes. The electron wave-function decays further in the other electrode and hence

a much lower current contribution is obtained. The MgO barrier presents a much

larger tunneling resistance for all other Bloch states of symmetry ∆5, ∆2 and ∆2′

and they do not contribute significantly to the total current. Thus significantly lower

resistance offered by MgO to the ∆1 states and the absence of Bloch states with this

symmetry for the minority spin electrons results in the special spin selective resis-

tance of crystalline MgO tunnel barriers. As shown in Fig. 4.3 and Fig. 4.4, the

bandstructures of cobalt are also well reproduced using the EHT formalism. These

cobalt parameters are used in later sections for the study of alloy electrodes. The


Figure 4.3: Bandstructure of majority spin electrons for a bulk Co crystal in a hcplattice. The parameters used in the extended Huckel theory calculation are reportedin Table 4.1

Figure 4.4: Bandstructure of minority spin electrons for a bulk Co crystal in a hcplattice. The parameters used in the extended Huckel theory calculation are reportedin Table 4.1


Figure 4.5: Bandstructure of bulk MgO crystal in an fcc lattice. The parametersused in the extended Huckel theory calculation are reported in Table. Note thatthese parameters allow us to simulate MgO with a bandgap of 7.7 eV which is closeto the experimentally observed value


bandstructure of MgO is depicted in Fig. 4.5. Due to the less number of parameters

used in the EHT formalism, it is possible to fit parameters to obtain a bandstructure

corresponding to a bandgap of 7.7 eV. The importance of being able to adjust this

bandgap has been discussed by Raza et al. [12].

4.4 Simulation of Fe-MgO-Fe MTJs

In this section, some results on the simulation of Fe-MgO-Fe MTJs are presented.

The atomic positions in the device are represented in Fig. 4.6 and are taken from [34].

Using the approach described above, the electronic structure and electron transmis-

sion probabilities through the device are calculated. The MR and RA products for

Fe-MgO-Fe tunnel junctions can then be computed. The MgO parameters used in

this simulation correspond to a realistic bandgap of 7.7 eV as shown in Fig. 4.5,

leading to better estimates of MR ratio [12]. As seen in Fig. 4.7, the transmission

Figure 4.6: Schematic diagram of a magnetic tunnel junction consisting of Fe elec-trodes and MgO tunnel barrier. The corresponding atomic positions used for thisdevice are also presented.

probability is a complex function of the parallel momentum vector which depends on

the relative magnetization of the electrodes. This result further emphasizes the need

to correctly account for electronic structure of the MTJ devices. The transmission

probability integrated over k space is shown as a function of energy in Fig. 4.8. The

applied voltage in this case is zero. The transmission peaks for the majority and


Figure 4.7: Transmisson probability at E = 0 over parallel k-space (V=0)


minority bands in the parallel case as well as the in the anti-parallel case are effects

captured only due to the atomistic treatment of electronic structure. Simpler models

fail to capture these important effects. As seen from the current-voltage character-

Figure 4.8: Transmission probability at fixed k parallel values (chosen from previousfigure) over energy range simulated (V = 0)

istics and the corresponding extracted MR characteristics, the MR value decays as

the applied voltage is increased. This is also observed in experimental devices [94].

The predicted maximum MR in this case is however much higher than in experi-

mental data. The simulation is performed for a very idealized device with no defects

and this can contribute to the large predicted MR. Some other effects such as Fermi


level pinning can also affect the MR ratio of real devices and this is discussed in the

following section.

Figure 4.9: Current voltage characteristics for Fe-MgO-Fe structure with 4 MgOlayers.

4.5 Effect of Fermi level pinning on TMR

Fermi level pinning leads to band alignments different from those theoretically ex-

pected from the work-function of the metal and the electron affinity of the tunnel

barrier material. In Fe-MgO-Fe magnetic tunnel junctions, the theoretical band

alignment gives us a barrier height of 3.8 eV. However in experiments it has been

observed that the effective barrier height is close to 0.4 eV [94]. This reduced barrier

height results in a much reduced tunneling resistance observed experimentally. It

has been observed that the annealing of magnetic tunnel junctions leads to a reduc-

tion in the effective MgO barrier height. This may be due to the diffusion of metal

atoms into the MgO leading to the formation of conducting channels [53]. Another


Figure 4.10: MR as a function of applied voltage for Fe-MgO-Fe MTJ with 4 MgOlayers.

possible explanation for the reduced tunneling barrier is the formation of dipoles at

the metal-dielectric interface [91]. Within the EHT formalism, the band alignment

can be adjusted by modifying the Hamiltonian matrix as follows.

Hij = Sij(KEHTEii +KEHTEjj + Eoffset,i + Eoffset,j)/2 (4.11)

where the values of Eoffset for the different atomic orbitals are chosen such that

the required band-alignment can be obtained for bulk band-structures [68]. This

modification of the Hamiltonian leads to a shifting of all the energy bands by the

specified energy offset. As seen in Fig. 4.11, the MR ratios expected are much

higher than those usually reported experimentally. When a smaller barrier height

is simulated as shown in Fig. 4.12, the expected MR is much reduced indicating

that this may be an important factor contributing to the low experimental MR seen.

The simulated junction resistance in the parallel and anti-parallel configurations is

depicted in Fig. 4.13. We see that the smaller barrier height leads to lower junction


resistance and a slower rate of increase of resistance with increasing oxide thickness

as expected.

Figure 4.11: The MR vs voltage characteristics simulated for Fe-MgO-Fe MTJ usinga bandgap of 7.7 eV for MgO and a barrier height of 3.7 eV

4.6 Approximate approach for alloy electrodes

The simulation of alloys in atomistic simulations is very difficult due to inherently

non-periodic nature of the structures. Here an attempt is made to approximate the

alloy electrodes used experimentally in MTJs through a simple approach. The Hamil-

tonian and Overlap matrices as a weighted sum of the corresponding matrices for the

different possible atomic configurations as shown in Fig. 4.14. The weights are equal

to the probability of different configurations given any two atomic positions. Using

this approach, MTJs with CoFe alloy electrodes with different alloy compositions

have been simulated.These results are shown in Fig. 4.15 and Fig. 4.16. It is seen

that the MR vs. applied voltage characteristics predicted for the two configurations


Figure 4.12: The MR vs voltage characteristics simulated for Fe-MgO-Fe MTJ usinga bandgap of 7.7 eV for MgO and a reduced barrier height of 0.3 eV to account forFermi level pinning at the interface

Figure 4.13: The effect of barrier height on the parallel and anti-parallel resistance ofan Fe-MgO-Fe MTJ. The barrier height plays an important role in controlling boththe MTJ resistance and the MR achieved.


are significantly different. For these simulations, the bandgap used for the MgO is

7.7 ev and the pinned barrier height of 0.3 eV is used. In Fig. 4.15 and Fig. 4.16, it

Figure 4.14: The Hamiltonian and Overlap matrices for alloy electrodes are calcu-lated as a weighted sum of the Hamiltonian and Overlap matrices of the 4 possibleconfigurations shown in this figure. The weight for each matrix is given by theprobability of the corresponding configuration.

is observed that the presence of small fractions of Co does not play a very important

role in the MR achieved. As discussed previously, the large magnetoresistance seen

in Fe-MgO-Fe junctions arises from the half metallic nature of the ∆1 bands in the

Fe electrodes. These simulations indicate that the electronic structure changes due

to the addition of Co do not play a significant role in controlling the MR ratio. These

simulations also suggest that the use of Co in the electrodes can help us achieve a

more voltage independent magnetoresistance as seen in Fig. 4.15. However, it is

important to keep in mind that this simulation only explores the effect of electronic

band-structure and does not account for other effects such as a change in the inter-

face structure leading to a different tunnel barrier height or improved crystallinity

of experimental devices. Such effects can also be responsible for the improved MR

ratio seen in CoFe based MTJs.

4.7 Summary

This chapter describes a study of the MR ratio in MTJs using an atomistic approach

based on the extended Huckel theory and NEGF formalism. This atomistic approach

can capture the special spin filtering properties of MgO tunnel barriers. Additionally,


Figure 4.15: The dependence of MR on votage applied to the an MTJ with alloyelectrodes

Figure 4.16: The dependence of MR on votage applied to the an MTJ with alloyelectrodes


it is also much less computationally intensive than other ab-initio techniques. A

simulator based on these techniques is developed, validated and used to study MR

in MTJs. It is proposed that the effect of Fermi level pinning might explain the low

MR ratios seen in experimental devices. An approximate method to account for alloy

electrodes is also proposed and the MR ratio achieved when using alloy electrodes is

studied.

4.8 Future work

The general simulator developed is a very general purpose and capable tool for elec-

tronic structure studies. The EHT simulator can be applied to a variety of problems

ranging from carbon nanotube, graphene, to surface states in silicon or other semi-

conductors. The alloy approach may be used for approximate studies on emerging

materials such as GeSn. Other situations where a careful calculation of tunneling

transmission is required such as band to band tunneling (BTBT) can also be studied

using the approach described in this chapter. These mechanisms are important for

promising new devices such as the tunne FET. In the area of spin injection too,

further work on the spin injection into semiconductors through MgO barriers is pos-

sible. One example is germanium where a perfectly crystalline stack of Fe-MgO can

be achieved on a Ge substrate. The low computational power and fast calculations

that can be achieved through this approach make it a very useful tool for atomistic

simulations.

Chapter 5

Conclusions

This thesis has described the use to tunneling transport in three types of novel de-

vices. The study of MIS contacts has provided useful insights on the mechanisms

responsible for contact resistance reduction by the introduction of tunnel barriers.

The search space for suitable materials has been narrowed by studying the added

tunneling resistance using different insulators. TiO2 has been recommended as a suit-

able interface material, following which several experiments have verified the benefits

of using TiO2. MIS contacts are one example of the general idea of using interface

layers for contact resistance reduction. Contact resistance will be a very serious

issue as devices continue to scale. The use of techniques such as those described

here will inevitably become an important part of device design. The second type of

device studied was the spinFET where tunneling plays a critical role. An efficient

formalism is developed for spin transport int he diffusion and drift-diffusion regime.

For the first time, the voltage dependence of the spin injecting tunneling contacts

were accounted for and the characteristics of a long channel spinFET were simulated.

Through these simulations, several limitations and design considerations of the spin-

FET have emerged. The spinFET is one possible idea to extend the functionality

of the basic MOS transistor by using electron spin information. While several issues

90

CHAPTER 5. CONCLUSIONS 91

remain to be solved before semiconductor spintronics can become a commercial suc-

cess, spintronic memory devices are likely to be commercially available soon. This

promising memory device, the magnetic tunnel junction, is studied using extended

Huckel theory based atomistic simulations. The special spin filtering properties of

the MgO tunneling layer were captured and important effects of Fermi level pinning

and the use of alloyed electrodes were studied indicating reasons for the low experi-

mentally observed MR in these devices as well as providing a useful tool to optimize

alloy electrode compositions. The simulators and techniques developed in this thesis

are general in nature and can be applied to a wide variety for tunneling simulations,

spin transport and atomistic simulations.

Appendix A

Tunneling transport

A.1 Tsu-Esaki formalism for tunneling

The Tsu-Esaki model is used to compute tunneling currents using an approximate

band-structure as shown in Fig. A.1.

The following assumptions are made within this model

1. An effective mass model with parabolic band-structure is used for all the mate-

rials. A lumped effective mass (meff) is used for the contacts. Thus the electron

energy (ε) and crystal momentum (k) in the material are related as follows.

ε =h2k2

2meff=h2(k2x + k2y + k2z)

2meff(A.1)

2. The specular transmission approximation is made. Within this approxima-

tion, the tunneling probability is a function of only the electron momentum

perpendicular to the tunnel barrier interface.

3. The electron momentum parallel to the interface is conserved in the tunneling

process.

Under the assumptions listed above, the differential element of the tunneling

92

APPENDIX A. TUNNELING TRANSPORT 93

Figure A.1: Typical band diagram for a metal-insulator-semiconductor contact.Parabolic bands are assumed in the metal and semiconductor and Schrodinger equa-tion is solved using this potential profile to calculate electron transmission probabil-ities as a function of energy.

current at a particular electron energy (ε) can be written as

dJm→s = qη(kz)vzgm(kz)fm(ε)1− fs(ε)dkz (A.2)

dJs→m = qη(kz)vzgs(kz)fs(ε)1− fm(ε)dkz (A.3)

g(kx, ky, kz) =1

4π3(A.4)

Here dJm→s is the differential current from the metal to the semiconductor and

dJs→m is the differential current from the semiconductor to the metal. vz is the elec-

tron velocity in the z-direction, perpendicular to the interface. η(kz) is the tunneling

probability which depends only on the electron momentum perpendicular to the in-

terface. fm(ε) and fs(ε) represent occupancy probability at the energy ε as described


by the Fermi-Dirac distribution in the metal and semiconductor respectively.

Using (A.1), the integration variables are changed from crystal momentum to

energy.

dJm→s =4πmeffq

h3

∫ Emax

Emin

η(εz) dεz

∫ ∞

0fm(ε)1− fs(ε) dερ (A.5)

dJs→m =4πmeffq

h3

∫ Emax

Emin

η(εz) dεz

∫ ∞

0fs(ε)1− fm(ε) dερ (A.6)

In order to circumvent a numerical integration in two variables, the supply func-

tions N(εz) can we calculated

dJm→s =4πmeffq

h3

∫ Emax

Emin

η(εz)Nm→s(εz) dεz (A.7)

dJs→m =4πmeffq

h3

∫ Emax

Emin

η(εz)Ns→m(εz) dεz (A.8)

The following expressions for the supply functions are derived analytically

Nm→s(εz) = kbT

exp(

εz−εfskbT

)

exp(

εz−εfmkbT

)

− exp(

εz−εfskbT

)

ln

1 + exp(

εfs−εzkbT

)

1 + exp(

εfm−εzkbT

)

(A.9)

Ns→m(εz) = kbT

exp(

εz−εfmkbT

)

exp(

εz−εfskbT

)

− exp(

εz−εfmkbT

)

ln

1 + exp(

εfm−εzkbT

)

1 + exp(

εfs−εzkbT

)

(A.10)

The net current from the metal to the semiconductor is obtained by subtracting

(A.8) from (A.7).

dJtot =4πmeffq

h3

∫ Emax

Emin

η(εz)Ntot(εz) dεz (A.11)


where the supply function Ntot(εz) is given by

Ntot(εz) = Nm→s(εz)−Ns→m(εz) (A.12)

For a given applied voltage, the solution of the Poisson equation gives us the

shape of the tunneling barrier. The value of the tunneling transmission probability

is calculated using the transfer matrix formalism for tunneling problems which is

described in the following section. The expressions derived here are quite general

and can also be adapted in a straightforward fashion to metal-oxide-metal junctions

for spin dependent tunneling currents.

A.2 Transfer matrix formalism for tunneling

The transfer matrix formalism is a convenient approach to solve the Schrodinger

equation in one dimension for an arbitrary potential barrier. The potential barrier is

divided into several layers and approximated by a piecewise linear function as shown

in Fig. A.2.

Figure A.2: Transfer matrix formalism for tunneling in MIS structures.

Since the potential is assumed to be constant within a particular layer, the solu-

tion of the Schrodinger equation within a specific layer j is given by


Ψj(z) = Ajexp(ikjz) + Bjexp(−ikjz) (A.13)

where kj is the wave number which depends on effective mass mj in layer j and

the potential Vj is layer j as follows

kj =

√

2mj(E − Vj)

h(A.14)

The boundary conditions at an interface between layers present at z are

Ψj(z−) = Ψj+1(z

+) (A.15)

1

mj

dΨj(z−)

dz=

1

mj+1

dΨj+1(z+)

dz(A.16)

Using these equations, the layer and interface transfer matrices are derived. These

matrices describe the change in the electron wave function Ψ(z) and its derivative

with respect to z across a layer or interface respectively.

Ψj(z)

Ψ′

j(z)

=

eikjz e−ikjz

ikjeikjz −ikje−ikjz

Aj

Bj

(A.17)

The boundary conditions can be expressed in the transfer matrix formalism as

follows

Ψj+1(0)

Ψ′

j+1(0)

=

1 0

0 mj+1/mj

Ψj(tj)

Ψ′

j(tj)

(A.18)

Ψj+1(0)

Ψ′

j+1(0)

=

1 1

ikj+1 −ikj+1

Aj+1

Bj+1

(A.19)

Aj+1

Bj+1

=1

−2ikj+1

−ikj+1 −1

−ikj+1 1

1 0

0 mj+1/mj

Ψj(tj)

Ψ′

j(tj)

(A.20)


Aj+1

Bj+1

=1

−2ikj+1

−ikj+1 −mj+1/mj

−ikj+1 mj+1/mj

Ψj(tj)

Ψ′

j(tj)

(A.21)

Aj+1

Bj+1

=1

−2ikj+1

−ikj+1−mj+1

mj

−ikj+1mj+1

mj

eikjtj e−ikjtj

ikjeikjtj −ikje−ikjtj

Aj

Bj

(A.22)

Aj+1

Bj+1

=1

2

1−mj+1

mj

1mj+1

mj

1 1

− kjkj+1

kjkj+1

eikjtj 0

0 e−ikjtj

Aj

Bj

(A.23)

Aj+1

Bj+1

=1

2

1 + kjmj+1

kj+1mj1− kjmj+1

kj+1mj

1− kjmj+1

kj+1mj1 +

kjmj+1

kj+1mj

eikjtj 0

0 e−ikjtj

Aj

Bj

(A.24)

This describes the change in the layer constants Aj and Bj and is a product of two

transfer matrices. The first matrix Dj+1 is called the boundary matrix and describes

the change in the constants across a layer boundary from layer jto layer j + 1.

Dj+1 =1

2

1 + kjmj+1

kj+1mj1− kjmj+1

kj+1mj

1− kjmj+1

kj+1mj1 +

kjmj+1

kj+1mj

(A.25)

The second matrix Pj is the layer matrix which describes the change in wave-

function phase across the layer j.

Pj =

eikjtj 0

0 e−ikjtj

(A.26)

By repeating this process for all the layers involved in the calculation and multi-

plying the transfer matrices, we can compute the constants in the final layer n.


An

Bn

= DnPn−1Dn−1Pn−2...D2P1D1

A1

B1

(A.27)

Thus, using the transfer matrix approach, we can solve the Schrodinger equation

in one dimension by discretizing the potential barrier. To calculate the tunneling

probability at this specific energy E = h2k2/2m we assume that there is no reflected

wave beyond the nth layer

An

0

=

T11(E) T12(E)

T21(E) T22(E)

A1

B1

(A.28)

We can now calculate the transmission probability as

η(E) = (|A1|2 − |B1|2)/|A1|2 = 1− |T21|2/|T22|2 (A.29)

Appendix B

Spin transport

B.1 Transfer matrices for spin diffusion

We define the following variables for use in the transfer matrix formalism.

µavg(z) =µ+(z) + µ−(z)

2(B.1)

∆µ(z) =µ+(z)− µ−(z)

2(B.2)

∆µJ(z) = qρ∗lsf [J+(z)− J−(z)] (B.3)

µJ(z) = qρ∗lsf [J+(z) + J−(z)] (B.4)

µ(z) =

µavg(z)

∆µ(z)

∆µJ(z)

µJ(z)

(B.5)

The constants ρ∗ and lsf depend on the material of the layer. Note that ∆µJ and

µJ are obtained from currents and have the same dimensions as the electrochemi-

cal potential. The new variables as defined above may be written in terms of the

constants K1, K2 and K3 for each layer as follows.

99

APPENDIX B. SPIN TRANSPORT 100

µavg(z) = K1 +K2βexp

(

z

lsf

)

+K3βexp

(

−zlsf

)

+ (1− β2)

(

z

lsf

)

µJ (B.6)

∆µ(z) = K2exp

(

z

lsf

)

+K3exp

(

−zlsf

)

(B.7)

∆µJ(z) = K2exp

(

z

lsf

)

−K3exp

(

−zlsf

)

− βµJ (B.8)

K =

K1

K2

K3

qρ∗lsfJ

(B.9)

We can now see how these variables change from the start of one layer to the end

of the same layer. An extra variable has been defined even though there are only 3

unknown constants so that the transfer matrix like formalism can be maintained.

µ(z) = TK→µ(z)K (B.10)

TK→µ(z) =

1 βez

lsf βe−zlsf

(1−β2)zlsf

0 ez

lsf e−zlsf 0

0 ez

lsf −e−zlsf −β

0 0 0 1

(B.11)

Also at the beginning of the layer (z = 0)

µ(0) = TK→µ(0)K (B.12)

K = T−1K→µ

(0)µ(0) (B.13)

µ(z) = TK→µ(z)T−1K→µ

(0)µ(0) (B.14)


Tlay =

1 βcosh zlsf

βsinh zlsf

β2sinh zlsf

−β +(1− β2) zlsf

0 cosh zlsf

sinh zlsf

βsinh zlsf

0 sinh zlsf

cosh zlsf

βcosh zlsf

−β0 0 0 1

(B.15)

Now that we have a relation between our defined variables at the beginning and

end of the layer, we focus on the boundary conditions at the interfaces. Under the

assumption that current continuity is maintained and there is no spin flipping at

an interface, we have the following boundary conditions at an interface written in a

matrix form that will help in implementation.

µJ(z) =

µ↑(z)

µ↓(z)

J↑(z)

J↓(z)

(B.16)

µJ(z+0 ) =

1 0 2qr∗b (1− γ) 0

0 1 0 2qr∗b (1 + γ)

0 0 1 0

0 0 0 1

µJ(z−0 ) (B.17)

These boundary variables are related to our previously defined variables as follows

µ(z) =

12

12

0 012

−12

0 0

0 0 qρ∗(z)l∗sf (z) −qρ∗(z)l∗sf (z)0 0 qρ∗(z)l∗sf (z) qρ∗(z)l∗sf (z)

µJ(z) (B.18)


Using these relations, we get the transfer matrix at the boundary

µ(z+0 ) = Tintµ(z−0 ) (B.19)

Tint =

1 0−γr∗

b

ρ∗(z−0)lsf (z

−0)

r∗b

ρ∗(z−0)lsf (z

−0)

0 1r∗b

ρ∗(z−0)lsf (z

−0)

−γr∗b

ρ∗(z−0)lsf (z

−0)

0 0ρ∗(z+

0)l∗sf

(z+0)

ρ∗(z−0)l∗sf

(z−0)

0

0 0 0ρ∗(z+

0)l∗sf

(z+0)

ρ∗(z−0)l∗sf

(z−0)

(B.20)

We can now use Eqs. (B.11) and (B.15) to find the change in the system variables

from the first interface to the last interface. If we have semi-infinite ferromagnetic

materials at the first and last interface then just before the first interface

µavg

∆µ

∆µJ

µJ

=

βK2

K2

−βqρ∗lsfJ +K2

qρ∗lsfJ

(B.21)

Beyond the last interface

µavg

∆µ

∆µJ

µJ

=

K1 + βK3

K3

−βqρ∗lsfJ −K3

qρ∗lsfJ

(B.22)

Since we have 3 unknowns here and we have 3 independent equations that we

will get relating the system variables before the first interface to the system variables

beyond the last interface, we can solve for all 3 and thus know the polarization of

the current in any layer.


B.2 Transfer matrices for spin drift-diffusion

This appendix briefly describes the spin diffusion transfer matrix formalism devel-

oped previously [74] and presents the transfer matrices developed for spin drift-

diffusion in this work. The relevant layer and interface matrices for both models as

well as interface matrices coupling the spin diffusion model in metals to the spin drift-

diffusion model in semiconductors are derived. The spin diffusion transfer matrices

are defined for the following variables.

µ(x) =

µavg(x)

∆µ(x)

∆µJ(x)

µJ(x)

=

µ+(x) + µ−(x)/2µ+(x)− µ−(x)/2

qρ∗lsfJ+(x)− J−(x)qρ∗lsfJ+(x) + J−(x)

(B.23)

The constants ρ∗ and lsf depend on the material of the layer. The derived layer and

interface transfer matrices are

Tlay =

1 βcosh xlsf

βsinh xlsf

β2sinh xlsf

−β +(1− β2) xlsf

0 cosh xlsf

sinh xlsf

βsinh xlsf

0 sinh xlsf

cosh xlsf

βcosh xlsf

−β0 0 0 1

(B.24)

Tint =

1 0−γr∗

b

ρ∗(x−0)lsf (x

−0)

r∗b

ρ∗(x−0)lsf (x

−0)

0 1r∗b

ρ∗(x−0)lsf (x

−0)

−γr∗b

ρ∗(x−0)lsf (x

−0)

0 0ρ∗(x+

0)l∗sf

(x+0)

ρ∗(x−0)l∗sf

(x−0)

0

0 0 0ρ∗(x+

0)l∗sf

(x+0)

ρ∗(x−0)l∗sf

(x−0)

(B.25)


The transfer matrices for spin drift-diffusion are more conveniently derived in terms

of the following variables.

X(x) =

φ(x)

∆n(x)/n0

∆J(x)/J

J

(B.26)

Each semiconductor layer is specified by parameters E the electric field, 2n0 the total

conduction band electron density, µ the electron mobility, D the electron diffusion

coefficient, lsf the zero field spin flip length in semiconductor, ts the layer thickness

and J the current flowing through the layer. Using the general solutions described

in (3.17) and (3.20) the layer transfer matrix is derived to be

Tlay =

1 0 0 −ExJ

0 T 2,2s T 2,3

s 0

0 T 3,2s T 3,3

s 0

0 0 0 1

(B.27)

where

T 2,2s =

l1l2D(l2 − l1)

m2e−x/l1 −m1e

−x2/l2

T 2,3s =

l1l2D(l2 − l1)

−J2qn0

e−x/l1 − e−x/l2

T 3,2s =

l1l2D(l2 − l1)

2qn0m1m2

J

e−x/l1 − e−x/l2

T 3,3s =

l1l2D(l2 − l1)

−m1e−x/l1 +m2e

−x2/l2

Since none of the variables in (B.26) will change at a semiconductor-semiconductor

interface in absence of any sheet charge or spin relaxation, the interface transfer

matrices are the identity matrices. However at the metal semiconductor interfaces the

variables change from electrochemical potentials to electron densities. As described


previously, the conversion between ∆µ and ∆n/n0 can be linearized. In that case,

the interface matrices are derived to be

Tm→s =

1q

0−γr∗

b

qρ∗lsf

r∗b

qρ∗lsf

0 1kbT

r∗b

kbTρ∗lsf

−γr∗b

kbTρ∗lsf

0 0 1qρ∗l∗

sfJ

0

0 0 0 1qρ∗l∗

sf

(B.28)

when going from a metal layer to semiconductor layer and

Ts→m =

q 0 −γr∗bqJ r∗bq

0 kbT r∗bqJ −γr∗bq0 0 qρ∗l∗sfJ 0

0 0 0 qρ∗l∗sf

(B.29)

when going from a semiconductor layer to a metal layer. Here ρ∗ and lsf are the

parameters of the metal layer. This completes the transfer matrix formalism for

including a semiconductor layer with electric field effects taken into account. By

applying suitable boundary conditions at first and last layers we can now easily solve

the spin transport problem in multilayer 1D structures.

Appendix C

Linear combination of atomic

orbitals

C.1 Slater type orbitals

Slater type orbitals (STO) are an approximation to atomic orbitals that are used

in the linear combination of atomic orbitals (LCAO) approach. Unlike the atomic

orbitals, the radial part of the STOs do not have any zero crossings. They do

however possess the property of decaying exponentially as a function of distance

from the center of the atom. The simpler analytical form for the STOs make it

easier to compute overlap integrals between different atomic orbitals making them

a natural choice for LCAO tight binding approaches such as the extended Huckel

theory. The radial part of the STO is characterized by parameters n which is the

principal quantum number for the orbital and ζ which describes the rate of decays

of electron density away from the center of the atom.

R(r, n, ζ) = (2ζ)n(

2ζ

(2n)!

)1/2

rn−1e−ζr (C.1)

Fig. C.1 shows some example plots for the radial part of STOs The spherical har-

106

APPENDIX C. LINEAR COMBINATION OF ATOMIC ORBITALS 107

Figure C.1: Radial part of Slater type orbitals

monics which are a function of the azimuthal quantum number l and the magnetic

quantum number m form the angular part of the STO.

Y ml (θ, φ) =

[

(2l + 1)

4π

(l −m)!

(l +m)!

]1/2

Pml (cosθ)eimφ (C.2)

where the associated Legendre polynomials Pml (x) are given by

Pml (x) =

(−1)m

2ll!(1− x2)m/2

dl+m

dxl+m(x2 − 1)l (C.3)

P−ml = (−1)m

(l −m)!

(l +m)!Pml (C.4)


Figure C.2: Angular part of Slater type orbitals

Fig. C.2 shows some example plots for the angular part of STOs in Cartesian coor-

dinates.

The double zeta STOs are linear combinations of two STOs with the same quan-

tum numbers (n, l,m) but different ζ parameters. These double zeta type STOs need

to be normalized appropriately.

C.2 LCAO tight binding

The linear combination of atomic orbitals (LCAO) approach to electronic structure

calculations [80] is briefly reviewed in this appendix. Consider a crystal structure


consisting of a periodic arrangement of different atoms in space. The LCAO approach

approximates the electron wave-function in the crystal by a linear combination of

the atomic orbitals of constituent atoms. The atomic orbitals corresponding to an

atom are eigen functions of the atomic Hamiltonian. In (C.5), n specifies a particular

orbital, b specifies the type of atom and−→Ri specifies the position in space relative to

the location i of the atom.

[

−h22m

∇2 + Ui,b

]

∣

∣

∣nb−→Rid

⟩

= εn,b∣

∣

∣nb−→Ri

⟩

(C.5)

When there are several unit cells in the lattice, a Block sum over all the cells provides

a suitable basis function which

• ia a linear combination of atomic Hamiltonian eigen functions

• satisfies translational symmetry in the periodic lattice.

The expression for this Bloch sum is presented in (C.6).

∣

∣

∣nb−→k⟩

=1√N

∑

−→Ri

ei−→k .(

−→Ri+

−→νb)∣

∣

∣nb−→Ri

⟩

(C.6)

The leading term 1√N

is the normalization constant. This Bloch sum introduces the

parameter−→k which is called the crystal momentum. νbis the translation with the

unit cell that describes the position of the atom b. The electron density of this wave-

function satisfies the periodicity of the crystal lattice. The next step involves finding

a suitable linear combination of these Bloch waves that is an eigen function of the

Hamiltonian of the crystal.

∣

∣

∣

−→k λ

⟩

=∑

n,b

Cn,b,λ∣

∣

∣nb−→k⟩

(C.7)

The parameter λ corresponds to a particular band in the electronic band structure.

By substituting this linear combination of Bloch functions into the Schrodinger equa-

tion for the system, the constants Cn,b,λ corresponding to the eigen functions of the


system Hamiltonian can be determined.

[

H − ε(−→k , λ)

] ∣

∣

∣

−→k λ

⟩

= 0 (C.8)

∑

n,b

Cn,b,λ[

H − ε(−→k , λ)

] ∣

∣

∣nb−→k⟩

= 0 (C.9)

The Slater type orbitals which are used as the atomic orbital basis satisfy the property

of being mutually orthogonal.

⟨

md−→k∣

∣

∣ nb−→k⟩

= δm,nδd,b (C.10)

The constants Cn,b,λ can be thus determined by solving a system of equations shown

in (C.12). These are obtained by multiplying and integrating with the other atomic

orbitals involved which is a standard technique in linear algebra and quantum me-

chanics.∑

n,b

Cn,b,λ⟨

md−→k∣

∣

∣

[

H − ε(−→k , λ)

] ∣

∣

∣nb−→k⟩

= 0 (C.11)

∑

n,b

Cn,b,λ[⟨

md−→k∣

∣

∣H∣

∣

∣nb−→k⟩

− ε(−→k , λ)

⟨

md−→k∣

∣

∣ nb−→k⟩]

= 0 (C.12)

The matrix solving this system of equations leads us to finding the eigen energies

corresponding to every value of crystal momentum and provides information on the

electronic structure.

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