EFFECTIVE MODELS FOR OPTICAL PROPERTIES: A STUDY ON … · EFFECTIVE MODELS FOR OPTICAL PROPERTIES: A STUDY ON 1D, 2D, AND 3D MATERIALS Juan Cuauhtemoc Salazar Gonzalez´ Doctor of

EFFECTIVE MODELS FOR OPTICAL PROPERTIES:A STUDY ON 1D, 2D, AND 3D MATERIALS

by

Juan Cuauhtemoc Salazar Gonzalez

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

c© Copyright 2018 by Juan Cuauhtemoc Salazar Gonzalez

Abstract

EFFECTIVE MODELS FOR OPTICAL PROPERTIES:

A STUDY ON 1D, 2D, AND 3D MATERIALS

Juan Cuauhtemoc Salazar Gonzalez

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2018

In this thesis I employ effective Hamiltonian models to study the electronic structure of

materials. With these models I study charge- and spin-injection induced by optical absorption

processes, and current-injection induced by quantum interference processes between different

orders of absorption. I study the optical response of narrow stripes (“nanoribbons”) of mono-

layer graphene, monolayers of tin atoms (stanene), and alloys of AlGaAs.

First I focus on graphene nanoribbons with zigzag shapes along their lengths, along which

strongly localized (“edge”) states exist at the Fermi level of undoped samples. I present re-

sults for different chemical potentials, showing that edge states are responsible for the main

contribution to the optical response at low photon energies (< 1 eV).

Next I study stanene, a monolayer of tin atoms arranged in a buckled honeycomb lattice

with Dirac-like cones in its bandstructure. The spin-orbit coupling in stanene is significant

and leads to a small band gap opening of about 90 meV. I present a scheme to extract an

effective Hamiltonian model starting from an ab-initio calculation. I keep track of the quality

of the approximations by a careful analysis of the electronic energies and the states obtained

with this effective model. Using this model I study the one- and two-photon absorption, and

spin-injection by circularly polarized light.

Finally, I investigate the optical coherent control of charge currents by two- and three-

photon absorption (“2+3”) processes in alloys of AlαGa1−αAs . An important feature of this

material is that its electronic bandstructure can be tailored to photon energies of interest. Com-

ii

pared to lower orders of interference, such as the 1+2 scheme, in 2+3 coherent control the

laser intensities required for maximal effects are larger, but the optical response is richer, the

number of optical coefficients is larger, the interference processes occur in smaller regions in

the Brillouin zone, and the electronic swarm velocities are higher.

iii

A mis padres, Rosario y Juan;

a mis hermanos, Xochitl, Francisco y Xicotencatl;

a mis tos, Teresita† y Raul†;

y con especial gratitud, a mi esposa Alenita.

To my parents, Rosario and Juan;

to my siblings, Xochitl, Francisco, and Xicotencatl;

to my aunt Teresita† and my uncle Raul†;

and with special gratitude, to my wife Alenita.

iv

Acknowledgements

My most sincere acknowledgements and gratitude go to Prof. John Sipe for his dedicated

and professional PhD supervision. I feel fortunate to have been his PhD student. His high

standards for scientific research and for writing were invaluable for my PhD project. His great

capacity and curiosity for science, his humbleness, and his innate passion to teach and supervise

students and postdocs will always be an example for my career, and a motivation to be a better

professional. I also thank his wife, Maggie Grisdale, for having invited me to their home on

multiple occasions during these years. Their warm hospitality is much appreciated. Special

thanks to my PhD Examination Committee: Professors Sajeev John, Young-June Kim, Daniel

James, and Robin Marjoribanks. Very special thanks go to my external examiner, Prof. Tami

Pereg-Barnea, for her detailed appraisal of my thesis, particularly for having done so during

her sabbatical time.

Next, I would like to thank Jin-Luo Cheng and Rodrigo A. Muniz, two former excellent

postdocs in the group, with whom I had the opportunity to collaborate and from whom I have

learned a lot. I wish you, your wives, and children the best of all in every aspect of life. I also

thank former and current members of the group with whom I had the opportunity to share time.

Special thanks to Julien Rioux for his computational training in my first year; in alphabetic

order, special thanks to Daniel Travo, Federico Gomez-Duque, Zachary Vernon, and Zaheen

Sadeq. My gratitude also goes to Steven Butterworth, Julian Comanean, and Gregory Wu, from

Physics Computing Services for their computational support and for sharing with me tricks of

trade. Thanks also to Krystyna Biel, Teresa Baptista, and Janet Blakely for their kind and

professional help on administrative matters; thanks to Lisa Fannin and Rory McKeown who

were exceptionally kind for processing my final admin work during the last week of December

2017.

I would also like to thank several institutions that supported me in one way or another

during my PhD. The University of Toronto School of Graduate Studies provided a PhD/Masters

scholarship. The Mexico’s Consejo Nacional de Ciencia y Tecnologia (CONACYT) provided

a partial stipend support during the initial years of my PhD. The supercomputing centre Scinet

v

guided me to learn and implement modern practices in scientific software development: special

thanks to the Analysts Ramses van Zon, Marcelo Ponce, and Erik Spence. The Julich Research

Centre hosted me for three weeks during the 45th IFF Spring School and the Fall 2014 workshop

on DFT codes. Special thanks here also to my parents in law, Ivana and Uli, for helping me with

accommodation during these academic stays in Julich. I thank the members of the ABINIT,

Quantum-Espresso, and Wannier90 User Forums, and the entire Free Software Community.

Along this line, I would like to thank Tonatiuh Rangel for his multiple computational advices

on the ABINIT code. Special thanks to the staff and peers at the ashtanga studio Downward

Dog Yoga Centre and to the kind members of the Mindfulness Practice Community sangha

in Roncesvalles: yoga and Zen meditation have become daily practices over the last years

that have reshaped my life for the better. Thanks also to the staff at University of Toronto’s

MacIntosh Sport Medicine Clinic for the world-class physiotherapy that allowed me to go

back to the running track after 15 years of disability.

In the personal arena, I would like to thank my entire family for their support during this

and my previous education. Indeed, I would like to dedicate this work to my parents, Rosario

Gonzalez and Juan Salazar; to my siblings Xochitl, Francisco, and Xicotencatl; to my aunt

and uncle, Teresa† and Raul† Gonzalez, who assisted my parents in raising me up; and to my

wife, Alena Drieschova: I thank all of you for your support and love, for all your material and

emotional sacrifices, and for taking care of each other in my absence.

My most fine gratitude and acknowledgements to my wife; Alenita: you have played a

crucial role in the recent years of my life and during almost my entire PhD degree. Your love,

your warm, playful, and caring company, your acute and frank comments and suggestions,

and your emotional and economic support in the last years were fundamental to complete my

PhD. Thank you also for having introduced me to mindfulness meditation, ashtanga yoga, and

South-Asian cooking. Thank you for helping me to speak English!

Special thanks to former romantic partners who besides sharing part of their lives with me

were also supportive during my studies. Specifically, thanks to Alma G.C. and Cristina Y.T. for

their economic support in the last stages of my undergrad and master’s degrees. Thank you so

vi

much Cristina Y.T. for the flight ticket to Canada to start my PhD and for the economic support

that allowed me to aid my aunt Teresa† during her final illness throughout the course of my

first year in Toronto; my aunt was like a mother for me and your opportune help will always be

present in myself.

Among the people that I had the opportunity to meet outside the Sipe group, I would like

to thank Maria To and Andres Covarrubias for their friendship and support. Andres’s example

prompted me to buy my ever first bike in Kensington market, and within a few weeks I became

a year-round daily cyclist. Having grown up in Mexico made those winter rides in Toronto

a complete adventure and my Humber River rides will certainly be a recollection of my PhD

years. I would also like to express my delight for my continued friendship with former room-

mates, Violet McCrady, Arsalan Ahmad, and Jan van der Tempel: besides your friendship and

company, I also appreciate the number of viewpoints and activities you introduced me to. My

appreciation also goes to Hazem Daoud, for his humble friendship and his help with admin

work while I was out of town. All you enriched my time in Canada and I look forward to a

lifetime friendship.

I would like to thank all the academics who gave me advice, motivation, and guidance

to start and complete my PhD. Special thanks to five Mexican scientists who motivated me

to pursue my PhD abroad, in spite of logistic restrictions and opposite suggestions I faced at

the time: my sincere thanks to Benjamin M. Fregoso (KSU), Tonatiuh Rangel (UC Berke-

ley), Salvador Venegas-Andraca (ITESM-Mexico), Marco Lanzagorta (US NRL), and Luis

Orozco (JQI-UMD); their career examples and their advices during my thought process on

grad schools played an opportune role. Thanks to Chandra Veer Singh (Materials Science,

UofT) and to Paolo Bientinesi and Edoardo DiNapoli (RWTH-Aachen) for their comments

and career advices. My special gratitude to Elizabeth C. for her frank and acute comments and

suggestions. Finally, my sincere apologies to anyone that I have unconsciously omitted.

vii

Contents

Terms and abbreviations xiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The electronic structure problem in condensed matter . . . . . . . . . . . . . . 2

1.3 Effective models: tight-binding and k · p methods . . . . . . . . . . . . . . . . 8

1.4 Effect of the dimensionality of materials . . . . . . . . . . . . . . . . . . . . . 11

1.5 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Coherent control of current injection in zigzag graphene nanoribbons 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Velocity matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Coherent injection and control . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.2 First-order absorption process . . . . . . . . . . . . . . . . . . . . . . 31

2.3.3 Second-order absorption processes . . . . . . . . . . . . . . . . . . . . 33

2.3.4 Current injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5 Limits of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.5.1 Graphene sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

viii

2.5.2 Zigzag nanoribbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 An Effective Model for the Electronic and Optical Properties of Stanene 55

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Method for deriving effective models . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Effective model for stanene . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.1 First-principles ground state of stanene . . . . . . . . . . . . . . . . . 61

3.3.2 Evaluation of the effective model: accuracy of the approximation . . . 66

3.3.3 Hamiltonian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3.4 Accuracy of the energies . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Linear and non-linear optical properties . . . . . . . . . . . . . . . . . . . . . 74

3.5 Tight binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


4 Coherent Control of Two- and Three-photon Absorption in AlGaAs 83

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2 Optical injection rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3 Electronic model of AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4 Coherent control with two- and three-photon absorption in AlGaAs . . . . . . . 93

4.4.1 Carrier injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4.2 Current injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


5 Conclusions 103

A Nonzero injection coefficient components of zincblende lattices 109

Bibliography 112

ix

List of Figures

1.1 Optical responses considered in this thesis . . . . . . . . . . . . . . . . . . . . 13

2.1 Lattice structure of the zigzag nanoribbons (ZZNR) . . . . . . . . . . . . . . . 19

2.2 Bandstructure of ZZNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Conventional and ERS coherent control schematics . . . . . . . . . . . . . . . 27

2.4 One-photon absorption of ZZNR . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Two-photon absorption of ZZNR . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Stimulated Raman Scattering of ZZNR . . . . . . . . . . . . . . . . . . . . . . 39

2.7 Current injection on ZZNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.8 Electronic swarm velocity of ZZNR . . . . . . . . . . . . . . . . . . . . . . . 43

2.9 One-photon absorption on doped ZZNR . . . . . . . . . . . . . . . . . . . . . 46

2.10 Two-photon absorption on doped ZZNR . . . . . . . . . . . . . . . . . . . . . 47

2.11 Stimulated Raman scattering on doped ZZNR . . . . . . . . . . . . . . . . . . 48

2.12 Net current injection on ZNR . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1 Hexagonal lattice of stanene . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Structural relaxation of stanene . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Charge density of stanene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Bandstructure of stanene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5 Figure of merit of the singular value decomposition . . . . . . . . . . . . . . . 67

3.6 Effective-model and ab-initio bandstructures of stanene . . . . . . . . . . . . . 72

3.7 Effective-model and ab-initio band warping of stanene . . . . . . . . . . . . . 73

x

3.8 Linear absorption and optical conductivity of stanene . . . . . . . . . . . . . . 74

3.9 Spin-density injection of stanene . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.10 Two-photon absorption of stanene . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1 Depiction of the 2+3 QuIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Bandstructure of AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3 Dielectric function of AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4 Two-photon absorption of AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . 94

4.5 Interference of 2- and 3-photon absorption of AlGaAs . . . . . . . . . . . . . . 94

4.6 Interference of 2- and 3-photon absorption of AlGaAs . . . . . . . . . . . . . . 96

4.7 Current injection by 2+3 QuIC on AlGaAs . . . . . . . . . . . . . . . . . . . . 96

4.8 Current injection by 2+3 QuIC on AlGaAs as a function of polarization angle . 97

4.9 Swarm velocity of AlGaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

xi

List of Tables

1.1 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Velocity matrix elements of zigzag graphene nanoribbons (ZGNR) . . . . . . . 26

2.2 Onset energies for the lowest energy transitions of ZGNR. . . . . . . . . . . . 33

3.1 Parameter values for an effective model of stanene . . . . . . . . . . . . . . . . 70

xii

Terms and abbreviations

1PA One-Photon Absorption.

2PA Two-Photon Absorption.

3PA Three-Photon Absorption.

BZ Brillouin Zone.

DFT Density-Functional Theory.

DOS Density of States.

ERS Electronic Raman Scattering.

JDOS Joint Density of States.

LCAO Linear Combination of Atomic Orbitals.

LDA Local Density Approximation.

ONCVP Optimized Norm Conserving Vanderbilt Pseudopotentials.

QSHE Quantum Spin Hall Effect.

QuIC Quantum Interference Control.

SVD Singular Value Decomposition.

xiii

TBM Tight-Binding Models.

TMD Transition Metal Dichalcogenide.

VME Velocity Matrix Elements.

ZGNR Zigzag Graphene Nanoribbons.

xiv

Chapter 1

Introduction

1.1 Motivation

One of the most important thrusts of modern technology is miniaturization, as evidenced by the

introduction of the term “nanotechnology” in the late part of the past century to refer to devices

around or below 100 nanometers in size. As this miniaturization continues, technologists and

scientists are faced with new opportunities and challenges. The tinier a device becomes the

more scientists are faced with quantum phenomena such as quantum confinement, quantum

tunnelling, and novel electronic-structure properties, such as electronic dispersion relations

with Dirac cones and topological properties. Such issues represent current areas of research

for the design, control, and operation of devices at nanoscale dimensions.

Traditionally, research in semiconductor physics has focused on silicon and its alloys with

germanium, gallium-arsenide and its alloys with aluminium, and the effect of defects such

as vacancies or dopants, in these materials. The pristine versions of silicon and GaAs have

small bandgaps — 1.11 eV and 1.43 eV, respectively — and exciton binding energies on the

order of a few meV. These semiconductors are usually studied in bulk or in structures where the

electrons and holes are at most weakly confined, such as quantum wells and quantum dots. The

electrical and optical properties of these “conventional” materials are now well characterized,

and in particular, the use of phase properties of one or more short optical pulses to generate and

1

Chapter 1. Introduction 2

control carrier population and carrier currents in semiconductors – referred as coherent control–

have been extensively studied both theoretical and experimentally here at Toronto [1–9].

The subject of this thesis is theoretical and computational studies of the optical properties

and coherent control scenarios in novel semiconductors that depart from this pattern of tra-

ditional materials. Novel materials are always interesting because they offer the possibility

of finding new technical applications. Researchers are also attracted to them because novel

materials can display new phenomena, and therefore can be used to test current theoretical de-

scriptions. If they are unable to explain the observations, then these materials are also useful as

a guide to develop new approaches to understand the behaviour of matter. In the next two sec-

tions I outline the electronic structure problem in condensed matter and two of the most widely

used effective models in this field of study. Then I finalize this chapter with an overview of the

work presented in this thesis.

1.2 The electronic structure problem in condensed matter

A system of M electrons and N nuclei in a solid obey the Schrodinger equation,

i~∂

∂tΨ(X,x, t) = H(X,x, t) Ψ(X,x, t), (1.1)

where H and Ψ are the system’s Hamiltonian and wavefunction, which are functions of the

spatial and spin coordinates of all nuclei and all electrons,

X ≡X1,X2, . . . ,XN

, Xi ≡ (Ri,Σi), (1.2)

x ≡ x1, x2, . . . , xM , xi ≡ (ri, si), (1.3)

where Xi and xi are shorthand notations to refer to the spatial and spin coordinates of nuclei

and electrons. The Hamiltonian includes the kinetic (K) and potential (U) energies from all


electrons (elec) and all nuclei (nuc), respectively; that is,

H =[Knuc +Unuc−nuc

]+

[Kelec +Uelec−elec

]+Uelec−nuc, (1.4)

where the last term describes the coupling between electrons and nuclei. The effect of external

electromagnetic fields and relativistic corrections will be outlined below, after I describe some

simplifications to this all-electron Hamiltonian.

Ignoring time dependence and taking Mn as the nuclei masses, m as the electron mass, Zn

as the charge numbers of the nuclei, e = −|e| as the electron charge, ~ as the reduced Plank

constant, and ∇2j as the Laplacian operator indicating derivatives with respect to nucleus or

electron spatial coordinates, the explicit expressions for the kinetic energy terms are

Knuc = −

N∑n=1

~2∇2n

2Mnand Kelec = −

M∑i=1

~2∇2i

2m, (1.5a,b)

while those for the repulsive nucleus-nucleus and electron-electron Coulomb potentials are

Unuc−nuc =12

N,N∑n,ν

e2

4πε0

ZnZν|Rn − Rν|

, and Uelec−elec =12

M,M∑i, j

e2

4πε0

1∣∣∣ri − r j

∣∣∣ , (1.6a,b)

where ε0 is the permittivity of vacuum. Finally, the (attractive) Coulomb coupling term between

the electronic and nuclear parts is

Uelec−nuc = −

N∑n=1

M∑i=1

e2

4πε0

Zn

|Rn − ri|. (1.7)

Since the proton is about 1,800 times heavier than the electron, in most situations in con-

densed matter physics it is considered that the electrons respond instantaneously to changes in

the nuclei coordinates. If one is only interested in the electronic bandstructure properties (as

I am in this thesis) and not in the phonon properties, it is common to assume that the nuclei

are fixed; this assumption is known as the clamped-ion approximation. As such, Knuc can be

assumed to vanish andUnuc−nuc becomes a constant term. In practice, all coordinates of nuclei


become just parameters and the Schrodinger equation can be solved for each fixed set X.

Although Eq. (1.4) is substantially simplified with these assumptions, practical solutions

still require a more drastic approximation, known as the single-particle (independent-particle)

approximation [10–13] or simply as the mean-field approximation [14]. In this scheme, the

assumption is that each electron moves in an average potential field Veff(xi) created by all the

other electrons and all nuclei. Then the total Hamiltonian is the sum of one-electron Hamilto-

nians,

H(x, t) =∑

i

Hi(xi, t), Hi(xi, t) = −~2∇2

i

2m+ Veff(xi, t) (1.8a,b)

In summary, the single-particle potential in Eq. (1.8b) Veff contains all the electron-electron

and all the electron-nuclei interactions [14], which are assumed “averaged”; furthermore, Veff

has also all the symmetries of the system. Since choosing an appropriate average potential

Veff is still a hard problem, the solutions usually follow a self-consistent approach that starts

with a reasonable guess for ψi and Veff; if a first-principles approach is sought, then appropriate

exchange-correlation terms in Veff are needed; the later itself is an entire area of active research.

In general terms, externally applied fields in the system are included in the treatment as

follows [15]: externally applied electric and magnetic fields are accounted by introducing

scalar and vector potentials. If time-dependent electromagnetic fields are present (i.e., a laser

field), these can be included in the treatment by applying the minimal coupling prescription,

p → p − eA(r, t), and perturbation theory. Relativistic spin-orbit interactions are described by

the Dirac equation and are considered relevant only for medium to heavy atoms [15]. How-

ever, one usually follows a simpler approach, as I do in this thesis, by adding a Pauli term to

the single-particle Hamiltonian in Eq. (1.8b),

HSO =~

4m2c4

[∇Veff × p

]· σ (1.9)

where p is the momentum operator of a single electron, ~ is the reduced Planck constant, m is

the bare electron mass, c is the speed of light, σ is the dimensionless spin operator σ = 2~−1S,


expressed as a vector of Pauli matrices, σ = (σx, σy, σz). With these considerations, the single-

particle Schrodinger equation is

i~∂

∂tψ(x, t) = H(x, t) ψ(x, t), H(x, t) = −

~2∇2

2m+ Veff(x, t) + HSO, (1.10a,b)

where the particle subindex has been removed and we recall that x ≡ r, s, where r and s refer

to the position and the spin-index of the electron, respectively.

Among the fundamental interests in solving Eq. 1.10 is to determine the energy-wavevector

dispersion relation followed by electrons, i.e., the bandstructure. The electronic structure of the

system is determined by minimizing the total energy with the restriction of a normalized wave-

function. In most cases this is an intractable problem, unless the system contains a few atoms

and few electrons, on the order of ten each. However, a number of effective methods have been

developed to solve Eq. 1.10, all of them based on symmetry considerations and/or approxima-

tions that reduce the number of degrees of freedom and allow us to obtain accurate solutions.

In a crystal semiconductor, for example, the solution to Eq. 1.10 can be further simplified by

considering all the translational, rotational, and reflection symmetries of the crystal.

Among the initial steps to simplify Eq. 1.10 is the consideration that electrons in filled

orbitals are mostly localized around the nuclei, and consequently the former can be considered

as united with the later, forming ionic cores; these electrons are referred as core electrons.

The remaining electrons in unfilled orbitals are referred as valence electrons, and are the ones

involved in chemical bonding, and very importantly, are the responsible for the electrical and

optical properties of a solid. As a consequence of this consideration, we can take the sums over

electronic indices to run over the valence electrons only, with the atomic numbers Z modified

accordingly.

The aforementioned classification of the different approaches to solve Eq. 1.10 is not rigid

and modern methodologies include a mix of these. Among the fundamental, first-principles

approaches there is the density-functional theory (DFT), both in time-dependent and time-

independent frameworks. Partially due to the complexity involved in devising time-dependent


exchange and correlation potentials, the time-independent (static) DFT has led the progress

in ab-initio DFT methods. In this thesis time-independent DFT methods are used in Ch. 3

as a starting point to develop an effective model. As such, in the following outline of DFT I

restrict to the static case. The reader interested in the time-dependent framework can consult

Refs. [16, 17].

The time-independent DFT is based on the original ideas of Hohenberg, Kohn and Sham

[18, 19]: for every interacting electron system under the influence of an external potential Vext,

there is a local potential, the Kohn-Sham energy potential VKS, that leads to a charge density

equal to that of the interacting system. As such, the (single-particle) Kohn-Sham Equations

(one for each electron) are

HKS[n(x)

]ψi(x) = Ei ψi(x), (1.11)

HKS[n(x)

]= −~2∇2

2m+Vnuc(x) + VH

[n(x)

]+ Vxc

[n(x)

]︸︷︷︸Kohn-Sham potential

, (1.12)

where

Vnuc(x) = −

N∑n=1

Zne2

|Xn − x|, VH

[n(x)

]=

∫d3x′

e2 n(x′)|x − x′|

, n(x) =

M∑j=1

∣∣∣ψ j(x)∣∣∣2, (1.13)

are the nuclear potential, the Hartree (Coulomb) potential and the charge density due to all

the single-electron wavefunctions; the sums run over all the N nuclei and all the M electrons.

The Vxc[n(r)

]term is the exchange-correlation potential that takes into account all correlation

effects, including the Hartree-exchange terms. Designing good exchange-correlation function-

als is perhaps the most challenging part of implementing transferable1 implementations of the

DFT formalism.

1In the language of pseudopotentials and functionals for DFT, transferability refers to the capability of apseudopotential or a exchange-correlation functional to produce good physical descriptions for a (wide) range ofmaterials and physical conditions. For example, a transferable pseudopotential for carbon should provide physicalresults for an isolated carbon, a carbon nanotube, for graphene, graphite, and diamond.


DFT is a vast and developing theory and a further description of it falls outside the scope of

this outline. The reader interested in further details may consult the following: Giustino [20]

for an introductory and modern textbook at the undergrad/graduate level, and the two-volume

series by Martin, Reining, and Ceperly [11, 15], for an advanced research-level description.

I would like to close this Section by mentioning that another class of methods for elec-

tronic structure calculations is based on effective Hamiltonian approaches. Among these are

the tight-binding method and the k · p method (a concise description of both is found in Yu-

Cardona [14], Ch. 2). In general terms, both methods start from different initial assumptions.

Tight-binding methods start from the assumption that electrons are tightly bound to atoms. In

a solid crystal, the atomic separation is comparable to the lattice constant, neighbour electronic

wavefunctions overlap, and some electronic states become delocalized and resemble nearly-

free electron states, hence they are referred as conduction states. The remaining states remain

bounded to the atoms and constitute the core and valence states.

In the other hand, the class of k · p−like methods can be derived from a different initial

assumption, referred as the mean-field approximation, where electrons are assumed to expe-

rience the same average potential V(r). k · p-like methods are widely used in semiconductor

optics because they are based on an extrapolation in terms of (1) energy gaps at a reference

qref point in the BZ and (2) the corresponding oscillator strengths (optical matrix elements) of

the transitions between states at such qref. This bandstructure extrapolation is extended over

a region around qref; the size of which depends on the number of known states at qref. If a

sufficient large number of states at qref is employed, then the entire BZ bandstructure can be in

principle computed.

Modern state-of-the-art methodologies have evolved from both sides of this spectrum.

From the mean-field side we have modern DFT methods based on pseudopotentials and plane-

wave basis sets; from a more atomistic approach, there are modern DFT implementations that

employ basis sets of localized orbital-like functions, like Gaussian or Wannier-like functions.

With the latter one can compute electronic structure properties of materials with unit cells con-

taining hundreds of atoms, in some implementations with an accuracy referred as “planewave


precision” [21].

1.3 Effective models: tight-binding and k · p methods

Tight-Binding approximation: an atomistic approach

The tight-binding approximation [10, 14, 22, 23], also known as linear combination of atomic

orbitals (LCAO), starts with the ansatz that crystalline solids are build up from an assembly of

atoms located on a lattice and that electrons are tightly bound to atoms. Ignoring the spin degree

of freedom, using the single-particle approximation, and considering one atom per lattice site,

and one electron per atom, the trial wavefunction at coordinate r is expressed as

Ψk(r) =∑

j

ck, j φ(r − R j), (1.14)

where k is the crystal momentum, R j are lattice-site vectors, and φ is an atomic orbital of

appropriate “orbital character” (i.e., s, px, py, . . .). Since we assume one atom per site and

one electron per atom, then there is only one atomic orbital φ at each lattice point. The trial

wavefunction Ψk(r) satisfies

H Ψk(r) = EkΨk(r). (1.15)

Enforcing the Bloch condition Ψk(r) = ei k·r unk(r), with unk(r + R) = unk(r), the tight-binding

wavefunction transforms to

Ψk(r) =∑

j

eik·R j φ(r − R j). (1.16)

If there are two or more electrons per unit cell, then the trial wavefunction must be chosen

accordingly. Consider for example monolayer graphene, which has two atoms per unit cell, and

each atom resides on two distinct sub-lattices, A and B. Take δ` as the vectors that connect the


sites ` =A, B

with the underlying Bravais lattice. Since translations by δ` are not symmetry

operations, then each sublattice must be treated explicitly. The low-energy bandstructure of

graphene is well described by considering only one electron per atom, hence we can write the

tight-binding trial wavefunction as [24]

Ψk(r) = ak ψ(A)k (r) + bk ψ

(B)k (r), (1.17)

which satisfies H Ψk(r) = EkΨk(r) and where

ψ(`)k (r) =

∑j

eik·R j φ(`)(r − R j + δ`), for ` =A, B

. (1.18)

The next step is to compute the total energy, 〈Ψk| Ek |Ψk〉. Two further approximations are

usually made: (1) only “on-site” and few “nearest-neighbours” energy terms have significant

contributions to the total energy, and (2) overlap integrals are small compared to unity.

A point to stress in spite of the latter approximation, is that electrons in this model are

not confined to a given atomic site. Indeed, electrons are mobile throughout the entire crys-

tal, since electrons described by a Bloch state (e.g., in this example enforced in Eq. (1.16))

have electronic velocities given by v(k) = ~−1∂kEk. In Chapter 3 I employ the tight-binding

method to identify structural parameters in an effective model developed from a first-principles

calculation.

The k · p approximation: a continuous model

The k · p approximation [14, 25] is a continuous semi-empirical model that extrapolates the

band structure of materials from a set of known states. It is based on time-independent degen-

erate perturbation theory, and resembles a Taylor expansion up to second order terms of the

electronic energy as a function of the crystal momentum. The starting assumption is that the

independent-particle approximation holds, and that the electronic states are known at a band

extrema, located at k0. That is, the set En(k0) is known, where n is a band index. Consider a


non-degenerate case, and a simple Hamiltonian

H =p2

2m0+ V(r), (1.19)

that satisfies the single-particle Schrodinger equation, Hψnk(r) = Enkψnk(r). Employing the

Bloch theorem for periodic crystals, and assuming that k0 is at the origin of the Brillouin zone,

we replace ψnk(r)→ eik·r unk(r) and the Schrodinger equation takes the form [14],

[p2

2m0+~2k · p

m0+~2k2

2m0+ V(r)

]unk(r) = En(k) unk(r). (1.20)

Here p = −i~∇ is the momentum operator, and V(r) is a periodic potential, V(r) = V(r + R),

unk(r) = unk(r + R) are the periodic parts of the Bloch wavefunctions, and k is the crystal

momentum. Assuming that we know the solutionsunk0(r), En(k0)

at the origin k0 = (0, 0, 0),

Eq. (1.20) reduces to

[p2

2m0+ V(r)

]unk0(r) = En(k0) unk0(r), (1.21)

then we can take the second and third terms in Eq. (1.20) as perturbations to the Hamiltonian

in Eq. (1.21). From standard perturbation theory, we have

En(k) = En,k0 +~2k2

2m0+~2

m20

∑n′,n

∣∣∣〈unk0 |k · p |un′k0〉∣∣∣2

Enk0 − En′k0

. (1.22)

This is the basic equation of the k · p method, from which the electronic energies in a range

of k space are expressed in terms of the known energiesEn(k0)

and the momentum matrix

elements appearing in the third term. Such matrix elements are known as the optical matrix ele-

ments, and are commonly measured in optical experiments from the determination of oscillator

strengths [14]. The k ·p method is routinely used to obtain analytical expressions and effective

masses at high symmetry points [14, 25]. As clearly seen from Eq. 1.22, the precision of the

bandstructure En(k) depends on the number of basis functions unk0 . Hence, for a sufficiently


large number of basis functions at the expansion point k0, an accurate bandstructure En(k) can

be obtained over the entire Brillouin Zone. A flavour of the k·p method, known as the envelope

function approach [25], will be employed in Chapter 2 when I study narrow strips of mono-

layer graphene. Then in Chapter 3, starting from a first-principles calculation, I will develop

an effective model that resembles a k · p method; however, the scheme will not be restricted

to a second order dependence on the crystal wavevector k. Importantly, the scheme will use

a minimum basis set to describe states on a relevant region of the Brillouin Zone. Finally, in

Chapter 4 I employ a traditional 30-band k · p method to study the AlαGa1−αAs alloy.

1.4 Effect of the dimensionality of materials

This thesis explores the electronic and optical properties of different materials in one- two, and

three-dimensions. As it is generally known, when the size of a material is reduced such that

electronic quantum confinement takes place in one or more dimensions, then the electronic,

optic and many other material properties change drastically [26–28]. From the uncertainty

principle (∆p ∼ ~/∆x), a simple estimate in one dimension suggests that a spatial restriction

leads to an additional energy term associated with motion, ∆E = (∆p)2/(2m) = ~2/(2m(∆x)2);

we refer to this ∆E as the confinement energy [26]. When this confinement energy becomes

larger than the kinetic energy associated with the thermal motion of the particle, then we expect

electronic behaviour that depends on the length of confinement. In a simple approximation

[26],

∆x .

√~2

m kB T, (1.23)

where m is the mass of the electron, kB is the Boltzmann constant, and T is the temperature.

For an electron in a semiconductor, taking m = 0.1 m0, with m0 as the bare electron mass,

and a cold semiconductor at about 20 K, we obtain an estimate of a confinement length of

∆x . 30 nm, at which quantum confinement behaviour is expected.


In general terms, the main consequence of having a material with a dimensionality different

from three is that electrons and holes have a restricted motion along one or more dimensions.

Consequently, the functional form of the density of states (DOS) as a function of the electronic

energy is modified. For instance, charge carriers in a 3D material are free to move in any direc-

tion, hence quantum confinement is absent. Within a parabolic band approximation, the DOS

varies as (E − Egap)1/2. For a 2D material, quantum confinement occurs along one dimension,

and the DOS has a step-wise variation; for a 1D material, confinement occurs in two dimen-

sions, and the DOS varies as (E − Egap)−1/2; finally, for a 0D material (e.g., a quantum dot)

confinement occurs in all three dimensions, and the DOS has the form of a comb.

1.5 Thesis overview

In this thesis I investigate basic electronic and optical properties, as well as optical coherent

control scenarios, of semiconductor materials of different dimensions. All calculations assume

a low saturation regime of excited carriers2 and electronic properties are described within the

single-particle approximation, hence many-body effects are neglected; light-matter interactions

are described with a Fermi golden rule approach.

In Chapter 2 I start by investigating narrow strips of monolayer graphene, commonly known

as graphene nanoribbons. Since the periodic part extends along a single direction, this is con-

sidered a 1D material; hence its density of states displays a rich structure that offers the possi-

bility of having an optical response that varies by orders of magnitude within a small range of

photon energy. Moreover, this material possesses localized states extremely close to the Fermi

level, which are easily controlled by external potentials or adsorbants.

Then in Chapter 3 I investigate the electronic and spin properties of a monolayer of tin

atoms, recently named as “stanene”. In this study I employ an effective Hamiltonian model

extracted from a first-principles calculation. Due to the atomic weight of tin atoms, signifi-

cant spin-orbit coupling is present, which leads to a small band gap opening of about 90 meV.

2Regimes of high density of carriers can be described by the Bloch-Semiconductor Equations. See for instancep. 216 of [59].


2hω

1PA 2PA

hω

1+2CI

2+3

hω

32hω

2hω

hωhω−hω

hω

CIERS

ERS: Electron Raman Scattering

CI: Current Injection

PA: Photon Absorption

hω

Figure 1.1: Optical responses considered in this thesis. Electronic properties are describedin the single-particle approximation (many-body effects are neglected) and the light-matterinteraction is described with a Fermi golden rule approach.

Although performing first-principles calculations is becoming routine, calculating the optical

response of materials with such a small fundamental bandgap requires significant computa-

tional expenditure, since the computation of these properties involve integrals over the Bril-

louin Zone (BZ), and the smaller the bandgap the finer the required numerical partitioning of

the BZ that is necessary to resolve the absorption onset. For this reason I propose an effective

model that is free of experimental input and is based solely on an ab-initio calculation of the

electronic dispersion relations of the material, i.e., on its electronic energy vs. crystal momen-

tum relations. Besides the computation of the optical response, I also present figures of merit

to systematically assess the range of validity of the approximations.

As the last case study, in Chapter 4 I employ a well known effective method, the k · p ap-

proximation, to study electronic carrier and current injection rates induced by optically coher-

ent control techniques on the alloy AlαGa1−αAs . It turns out that the electronic bandstructure

of this alloy is easily modified by varying the relative concentration of aluminium and gallium,

i.e., the α parameter, known as the stoichiometric value of the alloy. Hence, this is a prime

example of bandgap engineering, where I aim to study features of the optical response that

arise and display their most significant structure over certain photon energies of interest. The


This workSystem

Material Method Relevance 1PA 2PA 3PA CI

1DGraphene Envelope

Edge states X X RNC XRibbons Functions

2D Staneneab initio Small gap↓ SOC X X RNC P

k · p-like QSHE, TI

3D AlGaAsTraditional

Tuneable gap

k · p SOC NA X X XDevices

NA: Not ApplicableP: PlannedRNC: Relevant, but Not Considered (here)SOC: (significant) Spin-Orbit CouplingTI: Topological InsulatorQSHE: Quantum Spin Hall Effect

Table 1.1: Materials and methods studied in this thesis.

novelty on this chapter is not the method I employ, but the optical response of the material I

study: to the best of my knowledge, optical coherent control using the interference between

two- and three-photon absorption processes in AlαGa1−αAs have not yet been reported. On

Figure1.1 I present a sketch of the optical processes I study in this thesis and on Table 1.1 I

outline the materials and methods employed for such purpose. Finally, in Chapter 5 I present

the Conclusions of this thesis and describe suitable lines of future work.

Chapter 2

Coherent control of current injection in

zigzag graphene nanoribbons

Abstract

I present Fermi’s golden rule calculations of the optical carrier injection and the coherent con-

trol of current injection in graphene nanoribbons with zigzag geometry, using an envelope

function approach. This system possesses strongly localized states (flat bands) with a large

joint density of states at low photon energies; for ribbons with widths above a few tens of

nanometers, this system also posses large number of (non-flat) states with maxima and minima

close to the Fermi level. Consequently, even with small dopings the occupation of these local-

ized states can be significantly altered. In this Chapter, I calculate the relevant quantities for

coherent control at different chemical potentials, showing the sensitivity of this system to the

occupation of the edge states. I consider coherent control scenarios arising from the interfer-

ence of one-photon absorption at 2~ωwith two-photon absorption at ~ω, and those arising from

the interference of one-photon absorption at ~ω with stimulated electronic Raman scattering

(virtual absorption at 2~ω followed by emission at ~ω). Although at large photon energies these

processes follow an energy-dependence similar to that of 2D graphene, the zigzag nanoribbons

exhibit a richer structure at low photon energies, arising from divergences of the joint density

15

Chapter 2. Coherent control of current injection in zigzag graphene nanoribbons 16

of states and from resonant absorption processes, which can be strongly modified by doping.

As a figure of merit for the injected carrier currents, I calculate the resulting swarm velocities.

Finally, I provide estimates for the limits of validity of our model. A modified version of this

chapter was published in Physical Review B 93, 075442 (2016).

2.1 Introduction

The electronic properties of low-dimensional materials depend strongly on the size and ge-

ometry of the system [29, 30]. For instance, the bandstructure of a monolayer and a stripe

of graphene are significantly different. A stripe of graphene is usually referred as a graphene

nanoribbon, where the boundaries impose novel conditions on the wavefunctions; for a zigzag

graphene nanoribbon (ZGNR), the wavefunction vanishes on a single sublattice, A or B, at

each edge. As shown earlier [25,30,31], in ZGNR, there are confined states that extend across

the width of the system, incorporating states from both sublattices. There is also another class

of states strongly localized at each edge, which incorporate states from either one or the other

sublattice; these states are known as edge states. Although confined states are also found

in other types of ribbons, such as armchair, the edge states are present only for zigzag rib-

bons. These edge and confined states provide many of the novel characteristics seen in ZGNR.

Moreover, the energy of these states can be easily tuned by changing the ribbon width, apply-

ing external fields, and functionalizing the system [32, 33]. Since for an undoped ZGNR the

Fermi level coincides with the flat part of the edge states, tuning the doping level allows to

easily control the contribution of the edge states. Given that a 2D graphene sheet lacks these

localized states, a ZGNR offers the advantage of having optical responses that are easily tune-

able. Over the last few years, a number of studies have reported the special properties of these

localized states [25, 30, 31, 34–37] and recent investigations have described more novel prop-

erties and applications [38–43]. At zero energy they have an important role in the electronic

transport properties of both clean and disordered ZGNR, as Luck et al. [39] (and references

therein) have recently shown using a tight-binding formalism with a transfer-matrix approach.


A detailed review of these localized states in graphene-like systems can be found in Lado et

al. [44]. The optical properties of ZGNR and graphene nano-flakes have been studied from a

number of perspectives [34, 40, 45–50], always showing the strong influence of the edge states

in the dielectric function. First-principles studies of functionalization in graphene ribbons have

shown [32] that the low-energy π electrons at the edges of the ZGNR lead to higher binding

energies as compared with ribbons of different shape edges. Similar studies indicate [33] that

the optical response of functionalized ZGNR depends strongly on the size, shape and location

of the deposited molecule, suggesting functionalization as an effective way of fine-tuning the

electronic and optical properties of ZGNR.

In this Chapter, I investigate the optical injection of carriers and currents in graphene

nanoribbons by means of coherent light fields at ω and 2ω. In general, for arbitrary beams, this

technique is referred as coherent current control. It is based on the fundamental feature that

if the quantum evolution of a system can proceed via several pathways, then the interference

between such pathways can play a determining role in the final state of the system [51, 52]. In

a semiconductor, it is possible to control the injection of carriers [1, 3, 53, 54], spins, electrical

current [9], spin current [6], and even valley current [7], using phase-dependent perturbations,

usually involving coherent beams or pulses of light. In a one-color scheme, the interference is

between transition amplitudes associated with different polarizations [1]. Although carrier in-

jection can be achieved in graphene ribbons with one-color excitation, current injection cannot.

This is due to symmetry considerations, since one-color current injection is characterized by

a third-rank tensor, hence it is only allowed in systems that lack inversion symmetry [1]. Due

to the inversion symmetry in zigzag graphene ribbons, the one-color coherent control process

is forbidden. In a two-color scheme, the interference is between pathways related to photon

absorption processes arising from different phase related beams, one at ω and the other at 2ω.

In this case, current injection is characterized by a fourth-rank tensor; hence it is nonzero for a

ZGNR. In both schemes, the different pathways connect the same initial and final states. Here

our focus is on two-color current injection, and we consider two classes of processes: the first

class arises from the interference of one-photon absorption at 2~ω with two-photon absorption


at ~ω, and the second class arises from the interference of one-photon absorption at ~ω with

stimulated electronic Raman scattering at ~ω. In general, coherent control injection allows for

the placement of electrons and holes in different bands and portions of the Brillouin Zone as

ω is varied. Thus, as I will show, the current injection is very sensitive to the presence of both

confined and edge states. In line with plausible experiments, we consider nanoribbons with a

width on the order of 20 nanometers, which leads to unit cells containing a few hundreds of

atoms. For this reason, we employ an envelope function strategy to calculate the relevant en-

ergies and velocity matrix elements; the rest of the calculation follows a conventional Fermi’s

golden rule approach to calculate the absorption coefficients.

The Chapter is organized as follows. In Sec. 2.2, we describe the model Hamiltonian

employed to describe the wavefunctions, the resulting bandstructure, and the selection rules

for the velocity matrix elements. In Sec. 2.3, we describe the different carrier injection and

current injection coefficients, including the conventional and Raman contributions. In Sec. 2.4,

we revisit these calculations, but for a p-doped system. This allows us to show the significant

change in the signals that can be accomplished by altering the occupation of the edge states. In

Sec. 2.5, we provide an estimate of the limits of validity of the model employed in this Chapter.

Finally, in Sec. 2.6, we present our final discussions and conclusions.

2.2 Theoretical model

2.2.1 Model Hamiltonian

A zigzag graphene nanoribbon (ZGNR) is a strip of monolayer graphene [55, 56] that has

been cut such that the edges along its length have a zigzag shape, as shown in Fig. 2.1. We

take the ribbon to lie in the (xy) plane, with x as the longitudinal direction along which the

ribbon extends over all space; y then identifies the direction across the ribbon, along which the

electron states are confined.

We assume passivated carbon atoms at the longitudinal boundaries, as if hydrogen atoms

were adsorbed [25,40]; this allows the passivation of any dangling edge states and the neutral-


ization of the spin moments at the edges [40]. We take W = a√

3 (2N + 2)/6 as the effective

width, where N is the total number of atoms in the unit cell, a = acc√

3 = 0.246 nm is the

graphene lattice constant, and acc is the carbon-carbon distance (see Fig. 2.1). The edge at

y = a/√

3 is formed by A-atoms, while the edge at y = W − a/√

3 is formed by B-atoms. The

lattice vector is a = ax and the atomic sites are set in terms of the graphene lattice vectors,

a1 = (x −√

3y) a/2 and a2 = (−x −√

3y) a/2. The Dirac points of monolayer graphene are

projected [25] into the one-dimensional Brillouin zone of the ZGNR, [−πa ,πa ), as K = 2π

3a and

K′ = −2π3a . We express the total wavefunctions as linear combinations of atomic orbitals ϕ that

x

y

W

acca

aa1a2

, ,= A-site = B-site

. . .. . .

Figure 2.1: (Color online) Illustration of the lattice structure of a zigzag graphene nanoribbon extendedalong x and confined along y. Passivation atoms and carbon atoms are represented by unfilled and filledcircles, respectively; A (B) sites are colored red (cyan) and the unit cell is represented in grey.

are centered at atomic sites A and B,

Ψ(r) =∑RA

ψA(RA)ϕ(r −RA) +∑RB

ψB(RB)ϕ(r −RB). (2.1)

Then, following Marconcini and Macucci [25], we employ the semi-empirical k · p method

to describe Ψ(r) with a smooth envelope function approach. The coefficients ψA and ψB in


Eq. (2.1) can be written as

ψA(r) = eiK·rFKA (r) + eiK′·rFK′

A (r), (2.2a)

ψB(r) = −eiK·rFKB (r) + eiK′·rFK′

B (r), (2.2b)

where the FK(K′)A(B) (r) are the envelope function components associated with the K(K′) Dirac

point and the orbital at atom A(B)1. In writing Eq. (2.2) we have replaced ψi(Ri) → ψi(r) for

i = A, B, on the basis of two assumptions. First, we assume that atomic orbitals are strongly

localized at their corresponding atom, and second, we assume that the envelope functions are

slow-varying functions of r near the K (K′) Dirac point. These envelope functions satisfy the

Dirac equation,

0 −i∂x − ∂y 0 0

−i∂x + ∂y 0 0 0

0 0 0 −i∂x + ∂y

0 0 −i∂x − ∂y 0

×

FKA (r)

FKB (r)

FK′A (r)

FK′B (r)

=

Eγ

FKA (r)

FKB (r)

FK′A (r)

FK′B (r)

, (2.3)

where γ = (√

3/2) ta, t = 2.70 eV is the nearest-neighbor hopping parameter and vF = γ~−1

is the graphene Fermi velocity. Because of the translational symmetry along x, each envelope

function can be factorized as the product of a propagating plane wave along the length direction

(x), and a function confined along the width direction (y),

FK(r) ≡

FKA (r)

FKB (r)

= eiκx x

ΦKA (y)

ΦKB (y)

, (2.4a)

F K′(r) ≡

FK′A (r)

FK′B (r)

= eiκ′x x

ΦK′A (y)

ΦK′B (y)

, (2.4b)

where κx (κ′x) is the wavevector along the length of the ribbon, measured from the Dirac point

1The hexagonal (“honeycomb”) lattice of graphene is composed by two distinct triangular lattices, A and B.On each sub-lattice all atoms are equivalent.


K (K′). The dangling π orbitals of the carbon atoms at the edges of the ribbons are passivated

with hydrogen atoms; it is then reasonable to assume that the full wavefunction vanishes at the

lattice sites located at the effective edges. This leads to the following boundary conditions for

the confined part of the envelope functions [25],

ΦKB (y = 0) = 0, ΦK

A (y = W) = 0, (2.5a)

ΦK′

B (y = 0) = 0, ΦK′

A (y = W) = 0. (2.5b)

These boundary conditions and the block diagonal form of the matrix in Eq. (2.3) cause the

envelope functions at K to be uncoupled from their counterparts at K′; therefore, they can be

studied separately2. With the use of Eq. (2.4a), the Dirac equation for the K valley is

γ

0 κx − ∂y

κx + ∂y 0

Φ

KA (y)

ΦKB (y)

= E

ΦKA (y)

ΦKB (y)

. (2.6)

The solutions of Eq. (2.6) are of the form [25],

ΦKA (y) =

γ

E

[(κx − K)AeKy + (κx +K)Be−Ky

], (2.7)

ΦKB (y) = AeKy + Be−Ky, (2.8)

where K =√κ2

x − (E/γ)2. Under the boundary conditions (Eq. (2.5a)), this leads to a relation

between the transverse (K) and the longitudinal (κx) wavenumbers,

e−2KW =κx − K

κx +K, (2.9)

which shows that they are coupled for ZGNR. If K is taken to be real, then Eq. (2.9) reduces

to

κx = K coth (WK) , (2.10)

2This is not necessarily the case of other strip geometries, e.g., arm-chair ribbons, p. 568 of [25].


and without loss of generality we assume K to be positive. Equation (2.10) supports two

eigensolutions for κx > W−1, which we label as n = 1 for positive energies and n = −1 for

negative energies; both correspond to states strongly confined at the edges, henceforth referred

as edge states [25],

ΦKA (y) =

−2√

LAedge ζ

edgen sinh

[K edge(W − y)

], (2.11)

ΦKB (y) =

2√

LAedge sinh

[K edgey

], (2.12)

ζedgen = n, for n = ±1, (2.13)

where L is a normalization length along the x direction. We have also set K → K edge, and

Aedge is the usual wavefunction normalization coefficient,

Aedge =

√K edge/2

sinh(2K edgeW) − (2K edgeW), (2.14)

and the eigenenergy is

Eedgen = n γ

√κ2

x − (K edge)2. (2.15)

Equations (2.11)–(2.12) indicate that the edge states occupy both sublattices, and that edge

states from one sublattice are localized at one edge (e.g., for y = W, Eq. (2.11) vanishes and

Eq. (2.12) reaches its maximum; for for y = 0 the situation is reversed).

Conversely, if we consider solutions of Eq. (2.9) with K purely imaginary, of the form iKn

with Kn real, then Eq. (2.9) reduces to

κx = Kn cot (WKn) , (2.16)

where, without loss of generality, we take Kn to be positive. These solutions give states that

extend over the full width of the ribbon, and are known simply as confined states; for these

we set Kn → Kconfn and label them by n = ±1,±2,±3, . . ., starting with ±1 for those with


energies closest to zero. These confined states exist for any real κx, except those with band

index n = ±1, which exist only for κx≤W−1. Hence, the wavevector κx≤W−1 is the point of

the BZ where the edge states couple with the confined states; clearly, the wider the ribbon the

closest this coupling occurs towards the Dirac cones (κx = 0). The dispersion relations of the

confined states with band index n = ±1 connect with that of the edge states; both share the

band index n = ±1 (transition from the red to the blue traces in Fig. 2.3). The confined states

have the form

ΦKA (y) = −i

2√

LAconf

n ζconfn sin

[K conf

n (W − y)], (2.17)

ΦKB (y) = i

2√

LAconf

n sin[K conf

n y], (2.18)

ζconfn = (−1)n+1sgn(n), (2.19)

where

Aconfn =

√K conf

n /2− sin(2K conf

n W) + (2K confn W)

, (2.20)

Econfn = sgn(n) γ

√κ2

x + (K confn )2. (2.21)

We can indicate any of the edge or confined states simply by |nκx〉, where if |n| ≥ 2 the state

is confined, while if |n| = 1 then the state is confined for κx≤W−1, but it is an edge state if

κx > W−1.

Equations (2.15) and (2.21) describe the bandstructure of ZGNR, shown in Figs. 2.2 and

2.3. The edge states are flattened towards the zero energy level for κx > W−1 (Fig. 2.3), whereas

the confined states have a parabolic structure around the Dirac points, with an axis of symmetry

at κx = W−1, except for the two confined states nearest to zero energy, with band index n = ±1

and κx ≤ W−1 (Fig. 2.3). These confined states are associated with the Dirac cones of 2D

graphene. Since the extrema of the confined states occur at κx = W−1 , we can express the band


energies at such value of κx as

E±1(W−1) = ±γW−1, (2.22a)

E±n(W−1) ≈ ±γW−1√

1 + π2 (n − 1/2)2, (2.22b)

for the edge and confined states, respectively. This indicates that the band gap scales as W−1 and

provides an estimate of the photon energy at which the absorption edge occurs with respect to

the ribbon width W. It turns out that the sign functions (ζedge, ζconf) appearing in the expressions

for the wavefunctions at A-sites, ΦKA (y) [Eq. (2.11) for edge states and Eq. (2.17) for confined

states], alternate for consecutive states, being +1 for the first state above zero energy, −1 for the

next up, and so on; the situation is reversed for negative energies. These alternating sign factors

are attributed to the fact that eigenstates of the ZZGR are eigenstates of parity [25, 30, 31, 45].

This sign factor plays an important role in the selection rules of the quantities we calculate.

Therefore we indicate these sign factors on the bandstructure diagram (Figs. 2.2 and 2.3): a

solid line indicates that the confined part of an A-site component of the envelope function

has ζn = +1, whereas a dashed trace means it has ζn = −1. Fig. 2.3 is an amplification of the

bandstructure of ZZGR around the Dirac point, κx = 0. In this figure I have signaled the critical

point κx = W−1 with a vertical gray line. This is the crystal wavevector at which the edge states

couple with the confined states. In computing the bandstructure in the limit of large widths

W, I find that (1) this coupling tends towards the Dirac point and that (2) the energy difference

between contiguos energy bands decreases fast (not shown). This behaviour of the energy

states is in agreement with a simple model of a particle in a box system. Notice that both the

edge and confined states reside on both sublattices (equations (2.11)-(2.12) and (2.17)-(2.18)).


-1

0

1

−

π

a−

2π

3a 0 2π

3a

π

a

K′

K

Energy

[eV]

kx

Edge

Confined

Figure 2.2: (Color online) Zigzag nanoribbon bandstructure with 95 zigzag lines (about 20 nm width).Solid and dashed lines distinguish the polarity of the states. The confined states are shown in red andred-dashed lines, while the edge states are shown in blue and blue-dashed lines. The latter are flattenedtowards zero energy. The different polarities of these edge states is more distinguishable in the insetgiven in Fig. 2.3. The horizontal axis corresponds to the total wavevectors kx, measured from theBrillouin zone center, cf. Fig. 2.3.

2.2.2 Velocity matrix elements

We employ the envelope functions given by Eq. (2.4a) in order to calculate the velocity matrix

elements (VME) that describe the coupling between two states |n, κx〉 and |m, κx〉 as,

vnm(κx) =

∫dr

[FK(r)

]†v

[FK(r)

], (2.23)

where κx is a wavenumber and n, m are band indices. The velocity operator is given by

v = [r,H] /(i~), which, together with the Hamiltonian in Eq. (2.6) for the K valley,

H = γ

0 −i∂x − ∂y

−i∂x + ∂y 0

, (2.24)

leads to v = vF(σx, σy), where σx and σy are the Pauli matrices and vF = γ/~ is graphene’s

Fermi velocity.


Tabl

e2.

1:V

eloc

itym

atri

xel

emen

tsat

the

Dir

acpo

intK

.A

tagi

venκ x

,any

ofth

ese

mat

rix

elem

ents

are

pure

lyre

alor

pure

lyim

agin

ary

(whi

chis

expl

icitl

yin

dica

ted

byth

epr

esen

ce(a

bsen

ce)o

fthe

imag

inar

yun

iti)

.The

corr

espo

ndin

gex

pres

sion

sat

the

othe

rDir

acpo

intK

′ar

eid

entic

al,e

xcep

tth

atth

ex−

com

pone

nts

ofth

em

atri

xel

emen

tsfli

psi

gn;t

hey

-com

pone

nts

ofth

em

atri

xel

emen

tsre

mai

nun

chan

ged.

The

rang

eof

valid

ityfo

rth

isex

pres

sion

sis

give

nin

the

thir

dco

lum

n.Ty

peE

xpre

ssio

nC

ondi

tions

nCon

fvx nm

(κx)

=−

4vF

( ζconf

m+ζco

nfn

) Aco

nfn

Aco

nfm

[ Kconf msi

n(K

conf

nW

)−K

conf

nsi

n(K

conf

mW

)(K

conf

m)2−

(Kco

nfn

)2

]|n|≥

2,|m|≥

2,∀κ x

,or

l|n|≥

2,|m|=

1,κ x<

W−

1 ,or

mC

onf

vy nm(κ

x)=−

i4v F

( ζconf

m−ζco

nfn

) Aco

nfn

Aco

nfm

[ Kconf msi

n(K

conf

nW

)−K

conf

nsi

n(K

conf

mW

)(K

conf

m)2−

(Kco

nfn

)2

]|n|=

1,|m|≥

2,κ x<

W−

1

nEdg

evx nm

(κx)

=−

4vF

( ζedge

m+ζed

gen

) Aed

gen

Aed

gem

[ Kedge

nsi

nh(K

edge

mW

)−K

edge

msi

nh(K

edge

nW

)(K

edge

m)2−

(Ked

gen

)2

] |n|≥

1,|m|≥

1,κ x>

W−

1l

mE

dge

vy nm(κ

x)=−

i4v F

( ζedge

m−ζed

gen

) Aed

gen

Aed

gem

[ Kedge

nsi

nh(K

edge

mW

)−z m

sinh

(Ked

gen

W)

(Ked

gem

)2−

(Ked

gen

)2

]

nCon

fvx nm

(κx)

=i4

v F( ζed

gem

+ζco

nfn

) Aco

nfn

Aed

gem

[ Kconf nsi

nh(K

edge

mW

)−K

edge

msi

n(K

conf

nW

)

(Ked

gem

)2+

(Kco

nfn

)2

] |n|≥

2,|m|=

1,κ x>

W−

1l

mE

dge

vy nm(κ

x)=−

4vF

( ζedge

m−ζco

nfn

) Aco

nfn

Aed

gem

[ Kconf nsi

nh(K

edge

mW

)−K

edge

msi

n(K

conf

nW

)

(Ked

gem

)2+

(Kco

nfn

)2

]

nEdg

el

( Con

f↔

Edg

e) †|n|=

1,|m|≥

2,κ x>

W−

1

mC

onf


-0.4

-0.2

0

0.2

0.4

-0.2 0 W−1 0.2

µ2

µ1

K

2hω

hω

hω

m

CONV

2hω

−hω

hω

ℓ

ERS

Energy[eV]

κx [nm−1]

+1

−1

+2

−2

−3

−4

n

+3

+4

m

nℓ

Figure 2.3: (Color online) Depiction of the conventional coherent control (CC) scheme (set of arrowson the right) and the ERS CC (left arrows). Confined and edge states are shown in red and blue lines,respectively; solid and dashed lines distinguish the polarity of the states (see also Fig. 2.2). The initial(final) state is m (n) and ` is a virtual state. For m = −3, n = 2, ` = −1, the three purple dots alongκx = 0 pinpoint three states at which both the conventional and the ERS current injection are resonant.The upper boundaries of the grey areas depict Fermi levels of µ1 = −0.10 eV and µ2 = −0.20 eV (p-doped system). The horizontal axis corresponds to wavevectors κx measured from the Dirac point K,cf. Fig. 2.2. The vertices of the parabolic (confined) states occur at κx = W−1.

The resulting expressions are given in Table 2.1, and obey the following selection rules:

vxnm(κx) = 0 if ζn , ζm, (2.25a)

vynm(κx) = 0 if ζn = ζm. (2.25b)

We close this section by mentioning that the solutions corresponding to the Dirac point K′ are

analogous to those presented here for K. As shown by Marconcini et al. [25], the wavefunc-

tions for the A sites, Eqs. (2.11) and (2.17), at the K′ differ by a sign factor from those at K.

Moreover, the velocity operator at the K′ has the form v = vF(σx,−σy). This, together with

the properties of the envelope functions at both valleys, causes the x component of the VME at

K′ to have opposite sign of those at K; the y components of the VME are the same near K as

near K′.


2.3 Coherent injection and control

2.3.1 Framework

In this section, we describe the general framework of the two-color coherent control scheme.

As mentioned in the Introduction, the quantum interference is between pathways associated

with photon absorption processes arising from different phase related beams. These pathways

connect the same initial and final states, as shown for the processes in Fig. 2.3, where we

consider the two-color scheme with beams at ω and at 2ω. This figure depicts the two classes

of processes I study in this chapter.

The first, conventional processes, are those where current injection arises due to the in-

terference of one-photon absorption (1PA) at 2~ω and two-photon absorption (2PA) of (two)

photons with energy ~ω [1]; this is depicted with the set of arrows on the right of Fig. 2.3, un-

der the label “CONV”. In the remaining of the discussion, we label variables associated with

conventional processes with a subindex ‘C’.

The second class of processes arise in experiments on narrow band gap or gapless materials,

with ~ω > Eg, where Eg is the energy band gap. Under this condition, current injection can

arise due to the interference of 1PA at ~ω and stimulated electronic Raman scattering (ERS) at

~ω [5]. This ERS is indicated by the set of arrows at 2~ω and ~ω in the left of Fig. 2.3, under

the label “ERS”. We refer to variables associated with this Raman processes with a subindex

‘R’. We mention that in coherent control experiments on typical semiconductors, the beam

frequencies employed are such that ~ω < Eg < 2~ω, and, consequently, the ERS current is

absent because 1PA at ~ω is impossible.

Following van Driel and Sipe [1, 2], we calculate the one- and two-photon carrier injection

and current injection rates due to the interaction with a classical electromagnetic field

E(t) = E(ω)e−iωt +E(2ω)e−2iωt + c.c., (2.26)

in the long wavelength limit, where ω is the fundamental frequency. The interaction between


the electric field and the electron system is accounted by the minimal coupling prescription in

the Hamiltonian of Eq. (2.24); we do the usual replacement p j → p j − eA j(t), for j = (x, y),

with p j = −i~∂ j, and obtain the interaction Hamiltonian that acts as the perturbation,

Hint(t) = −ev ·A(t), (2.27)

where e = −|e| is the electron charge andA(t) is the vector potential associated with the electric

field, E(t) = −∂A(t)/∂t. We treat this problem using standard time-dependent perturbation

theory and Fermi’s golden rule. Since we are interested in 1PA, 2PA and ERS processes, the

unitary evolution operator U(t) is expanded perturbatively up to second order,

U(t) = e−iH0t/~Uint(t) (2.28)

where

Uint(t) =1 + (i~)−1∫ t

−∞

Vint(t1) dt1 + (i~)−2∫ t

−∞

Vint(t1) dt1

∫ t1

−∞

Vint(t2) dt2 + . . . (2.29)

and

Vint(t) = eiH0t/~ Hint(t) e−iH0t/~. (2.30)

Under the perturbation of Eq. (2.27), the evolution of the system’s state∣∣∣Υ〉 is not just the

ground state∣∣∣0〉, but it also contains an amplitude of the excited state |nmκx〉 (this ket corre-

sponds to a state with an electron-hole pair),

∣∣∣Υ(t)〉 = c0(t)|0〉 + cnmκx(t)|nmκx〉 + . . . , (2.31)

where∣∣∣cnmκx(t)

∣∣∣2 is the probability that the system is at∣∣∣nmκx〉; the missing terms in Eq. (2.31)

correspond to higher order excitations, which we neglect in this Chapter. The carrier injection


and the current injection rates are given by

n =1L

∑nmκx

ddt

∣∣∣cnmκx(t)∣∣∣2, (2.32)

Ja =1L

∑nmκx

e[va

nn(κx) − vamm(κx)

] ddt

∣∣∣cnmκx(t)∣∣∣2, (2.33)

respectively, where L is the normalization length introduced below Eq. (2.13). To describe

the optical processes we are interested, we compute cnmκx(t) up to second order (a tutorial

derivation is given by van Driel and Sipe in Ref. [1]; a more recent review is given by Rioux

and Sipe in Ref. [4]). Then, the expressions for these injection rates get the form,

n(1) = ξab(ω)Ea(−ω)Eb(ω), (2.34)

n(2)C = ξabcd

C (ω)Ea(−ω)Eb(−ω)Ec(ω)Ed(ω), (2.35)

n(2)R = ξabcd

R (ω)Ea(−2ω)Eb(−ω)Ec(2ω)Ed(ω), (2.36)

Ja = ηabcd(ω) Eb(−ω)Ec(−ω)Ed(2ω) + c.c., (2.37)

where repeated indexes indicate summation, ω is the fundamental frequency, n(1) and n(2)C(R)

account for the first- and second-order absorption processes, respectively; overall n refers to the

rate of injected carriers per unit length along the ribbon (carriers per unit length per unit time).

The 1PA coefficient is described by a second-order tensor, ξab, while the 2PA and the ERS

absorption coefficients are described by fourth-order tensors, ξabcdC and ξabcd

R , respectively. Here,

Ja includes the electron and hole contributions to the current (charge per unit time), injected

per unit time along the ribbon. The current injection coefficient η(ω) in Eq. (2.37) includes the

conventional and the ERS contributions, i.e., η(ω) = ηC(ω) + ηR(ω). In the following sections,

we give the full expressions for these coefficients. Note that the coefficients can be chosen such

that ξabcdC = ξbacd

C = ξbadcC and ηabcd = ηacbd.


2.3.2 First-order absorption process

We calculate the expressions for the coefficients ξ and η appearing in Eq. (2.34)–(2.37) using

Fermi’s golden rule. For the one-photon absorption coefficient, we obtain

ξab(ω) =4πe2

~2

∑nm

∫fmn(κx)

dκx

2πva

nm(κx) vb∗nm(κx)

ω2nm(κx)

δ(ωnm(κx) − ω), (2.38)

where we have gone from a sum over states to an integral over reciprocal space by L−1 ∑κx→

(2π)−1∫

dκx. In this expression the sum∑

nm runs over all bands, filled and empty (simi-

larly for the other response functions considered here); ωnm(κx) = ~−1Enm(κx) and Enm(κx) =

En(κx) − Em(κx) is the energy difference between two states at a given κx. A factor of two

has been included to account for spin degeneracy, which I do throughout this chapter. The

x−components of the VME at the K and K′ valleys differ just by a sign while the y−components

of the VME are the same. Consequently, since all integrals over reciprocal space include pairs

of VME, the integration over κx can be restricted to a single valley, K, and another factor of

two included to account for the contribution of the K′ valley.

The occupation of the states is described by the Fermi-Dirac distribution. In all of our

integrals over reciprocal space fmn(κx) = fm(κx) − fn(κx), with fn(κx) = [1 + e(En(κx)−µ)/(kBT )]−1 at

temperature T and chemical potential µ. Until the end of Sec. IV, we confine ourselves to zero

temperature, hence fn(κx) = θ(En(κx) − µ), where θ is the Heaviside step function. Because of

the selection rules for the VME, Eq. (2.25), the only nonzero components of the one-photon

coefficient are ξxx and ξyy, which we plot in Fig. 2.4 for a system at zero chemical potential. As

a comparison 3, we include plots of Wξxx2D, where W is the effective width of the ribbon,

ξxx2D(ω) = 2σ0(~ω)−1, (2.39)

and ξxx2D = ξ

yy2D is the 1PA coefficient for a 2D monolayer of graphene [3]; hereσ0 = gsgve2/(16~)

is the universal optical conductivity of graphene, and gs = 2, gv = 2 are the spin and valley

3At large photon energies, the two-photon absorption coefficients for zigzag nanoribbons drop off with thefifth power of the photon energy, as they do for a monolayer of graphene.


degeneracies, respectively. For ZGNR, the main difference between the two 1PA coefficients is

that ξyy diverges at zero photon energy, due to a divergence in the joint density of states (JDOS)

between bands n = +1 and n = −1. In contrast, for such a pair of bands ξxx is identically

zero, due to the VME selection rules. For an undoped ZGNR, ξxx displays its first divergence

at about 0.15 eV, which is the value of the band gap at zero Fermi level, and corresponds to the

onset of the transitions (2,−1) and (1,−2) at κx = W−1; these four states give the initiation en-

ergy for ξxx. In the following we indicate a transition from band m to band n by (n,m); hence,

for zero chemical potential and zero temperature, the possible transitions have m ≤ −1 and

n ≥ 1. In general, the ξxx and ξyy 1PA coefficients possess an infinite number of divergences

that arise due to the infinite number of parabolic bands in the bandstructure. Indeed, the JDOS

between states with band index n and m,

JDOSnm(E) = gsgv

∫dκx δ(E − Enm(κx)), (2.40)

can be shown to diverge as (E − Egapnm )−1/2 for the confined states, and as E−1 for edge states,

where E is the photon energy and Egapnm is the energy band gap between bands n and m. In fre-

quency space, these divergences occur at photon energies E such that E = Egapnm ; in reciprocal

space, they occur at κx points where argument of the delta function has a zero derivative. The

absorption coefficients inherit these JDOS divergences if the associated velocity matrix ele-

ments are nonzero at the κx at which dEnm/dκx = 0. The sensitivity of an experiment to these

divergences would depend on the resolution of the photon energy and on the magnitude of the

velocity matrix elements, as well as on the presence of scattering effects that are not included in

this simple treatment. In every pertaining Figure, we signal the location of these JDOS diver-

gences by small green ticks. An interesting characteristic of ξxx and ξyy is that the divergence at

the initiation energy always involves an edge state (see Table 2.2); this is reasonable, as these

states are involved in the minimum band gap for an undoped system.

As mentioned above, the sum over states runs over all bands, filled and empty, but for a

given photon energy range (e.g. 0 − 0.5 eV, as in Fig. 2.4) the sum requires a finite number of


bands. We refer to this as the “full” response. In order to highlight the contribution of the edge

states, we also compute the response coefficients with a restricted sum over states∑

nm, such

that n or m are ±1, e.g., (n,m) = (1,−1), (1,−2), (2,−1), . . .; we refer to this as the “edge”

contribution and in the appropriate figures we plot it with black-dashed lines. This allows us

to easily identify the contribution to 1PA from states at bands ±1. At low photon energies

such contribution is dominant: for ξxx, all transitions at photon energies ~ω < 0.350 eV are

from or to edge states; for ξyy, all transitions at photon energies ~ω < 0.439 eV are from

or to edge states. Consequently, at low-photon energies the “full” and “edge” contributions

are indistinguishable. This is shown in Fig. 2.4 (see also Table 2.2), where for comparison

we also plot Wξxx2D, where ξxx

2D is the 1PA coefficient of graphene calculated [3] at the same

level of approximation adopted here; it is clear how the presence of the edge states in ZGNR

significantly modifies the 1PA. Finally, we mention that the Dirac delta functions appearing in

all our expressions are treated with an interpolation scheme [57].

Table 2.2: Onset energies for the lowest energy transitions for an undoped cold ZGNR. Tuples (n,m)indicate a transition from band m to band n and every onset energy indicates the position of a JDOSdivergence. The peak number is as indicated in Fig. 2.4.

Peak ξxx ξyy

number E (eV) Transition E (eV) Transition

1 0.149 (2,−1), (1,−2) 0.000 (1,−1)2 0.323 (4,−1), (1,−4) 0.236 (3,−1), (1,−3)3 0.350 (3,−2), (2,−3) 0.410 (5,−1), (1,−5)4 0.498 (6,−1), (1,−6) 0.439 (4,−2), (2,−4)...

......

......

2.3.3 Second-order absorption processes

Conventional process

In this section, we start by considering the second order process related to the absorption of two

photons of energy ~ω, indicated by the rightmost arrows in Fig. 2.3. Carrying the perturbation


0

1

0 0.1 0.2 0.3 0.4 0.5

1 2 3 4

(b)

ξyy(ω

)[

ms−

1V

−2]

Photon energy hω [eV]

0

1

×109

1 2 3 4

(a)

ξxx(ω

)[

ms−

1V

−2]

FullEdgeWξxx

2D

Figure 2.4: (Color online) One photon absorption spectrum for a ZGNR of 95 zigzag lines (about20 nm width). The definitions of the full and edge contributions are given in the last paragraph of Sec.II B. For comparison, we include Wξxx

2D (red dot-dashed curves), where ξxx2D = ξ

yy2D is the 1PA coefficient

for graphene [3], given by Eq. (2.39). The green ticks along the horizontal axis indicate photon energiesat which JDOS divergences occur, which are numbered in concordance with Table 2.2.

calculation up the second order, we obtain the two-photon absorption (2PA) coefficient,

ξabcdC (ω) =

64πe4

~4

∑nm

∫fmn(κx)

dκx

2π

Vab∗C;nm V

cdC;nm

ω4nm(κx)

δ(ωnm(κx) − 2ω), (2.41)

where

Vi j

C;nm ≡ ~∑`

v in` v j

`m + v jn` v i

`m

2E` − En − Em + i βC, (2.42)

which we regard as the effective velocity matrix element (effective VME) for the second order

conventional process (C) process and the sum over ` corresponds to the virtual electron and

virtual hole contributions [1]. Although this sum runs over all bands (filled and empty), a

converged value is obtained for ` = 20 bands for a photon energy range of 0–1 eV.

In Eq. (2.42) we introduce a small constant βC in order to broaden resonant processes


(discussed below). This is done in a phenomenological basis, as a simple way to account for

the finite lifetime of the excited states. Throughout this thesis I set the value of this broadening

constant to the energy equivalent of room temperature, about 26 meV. By varying the value

of βC, I find that values closer to 26 meV provide a good compromise between preserving

the structure of the responce functions and broadening the resonant processes. At a more

fundamental level, deriving the interacting-particle Green function of an electron in a periodic

potential, the imaginary broadening constant βC can be shown to be related to the inverse of

the finite lifetime of the excitation (see, for example, page 615 of [58]).

From the selection rules for the regular VME, Eq. (2.25), we obtain the selection rules for

V,

VxxC;nm = 0 if ζn , ζm, (2.43a)

VyyC;nm = 0 if ζn , ζm, (2.43b)

VxyC;nm = 0 if ζn = ζm, (2.43c)

and from this we identify eight nonzero ξabcdC components, four of them independent, namely

ξxxxxC , ξ

xxyyC =

(ξ

yyxxC

)∗, ξ

xyxyC = ξ

xyyxC = ξ

yxxyC = ξ

yxyxC , and ξ

yyyyC , which we show in Fig. 2.5.

A feature of these coefficients is that the onset of the two-photon absorption signal is at the

minimum band gap between bands (2,−1), except for ξxyxyC , which has its onset at 0 eV; this

follows from the selection rules for the effective VME, which are inherited from the usual

VME, and indicate that the transition (1,−1) is allowed.

As we found for the 1PA coefficients ξab, the 2PA coefficients ξabcdC suffer from divergences,

but for the 2PA coefficients they are of two types: JDOS divergences and effective-VME-

divergences. The latter results when the nominal virtual state lies at the average of the energies

between two transition states, |nκx〉 and |mκx〉, i.e., when (see Eq. (2.42))

E` = (En + Em)/2. (2.44)


Such condition corresponds to a resonant 2PA and an instance where this occurs is indicated

on Fig. 2.3 by the three dots along the vertical line at κx = 0. In Fig. 2.5 we distinguish these

two types of divergences by small vertical lines of different color; a green tick indicates the

presence of a JDOS-divergence, while a red tick indicates the presence of an effective-VME-

divergence. In order to broaden the latter resonances, a small damping constant βC of 20 meV

was introduced in the denominator of Eq. (2.42). This value, which is close to the thermal

energy kBT associated with room temperature, was chosen arbitrarily. A more detailed theory

would be necessary to indicate how these resonances are really broadened; the choice we make

here simply allows us to identify easily where these resonances occur in our calculations. We

mention that the onset of ξxxxxC is due to the transitions (2,−1) and (1,−2), which are free from

resonances because the matrix elements to the intermediate states (one of the edge bands ±1

that would lead to a divergent condition) are forbidden by the selection rules. Therefore, in the

photon energy range 0 to 0.15 eV, the coefficient ξxxxxC is free of resonances.

We present the ξabcdC coefficients in Fig. 2.5, and identify the edge contributions to them

(black-dashed lines). As we found for ξab, for ξabcdC the edge states make a dominant contri-

bution at low photon energies, and are involved at the onset of 2PA. As a comparison 4, in

Fig. 2.5, we include plots of Wξabcd2D , where W is the effective width of the ribbon,

ξxxxx2D (ω) = 8gsgv~e4v2

F(2~ω)−5, (2.45)

and ξxxxx2D = ξ

yyyy2D = ξ

xyxy2D = ξ

xyyx2D = −ξ

xxyy2D are the 2PA coefficients for a 2D monolayer of

graphene [3]; as before, gs = 2 and gv = 2 are the spin and valley degeneracies, respectively.

ERS process

Now we consider another second order process involving light at 2ω and light at ω, stimulated

electronic Raman scattering, which can be characterized as virtual absorption at 2~ω followed

by emission at ~ω; see the left diagram in Fig. 2.3. This process exists in semiconductors

4At large photon energies, the two-photon absorption coefficients for zigzag nanoribbons drop off with thefifth power of the photon energy, as they do for a monolayer of graphene.


0

2

4

6

8

0 0.05 0.1 0.15 0.2 0.25

Wξyyyy

2Dր

(d)

ξyyyy

C(ω

)[

m3s−

1V

−4]

Fundamental photon energy hω [eV]

×10−5

0

1

2

3

4

ւWξxyxy

2D

(c)

ξxyxy

C(ω

)[

m3s−

1V

−4]

×10−4

−6

−4

−2

0

2

4

6

տWξxxyy

2D

(b)

ξxxyy

C(ω

)[

m3s−

1V

−4]

×10−6

0

1

2

3

4

5

6

ւWξxxxx2D

(a)

ξxxxx

C(ω

)[

m3s−

1V

−4]

Full

Edge

×10−6

Figure 2.5: (Color online) Nonzero two-photon absorption coefficients ξabcd

C for aZGNR with 95 zigzag-lines (about 20 nmwidth). The definitions of the full and edgecontributions are given in the last paragraphof Sec. II B. On each panel, we includeWξabcd

2D (red dot-dashed curves), where ξabcd2D

(Eq. (2.45) and text below) is for a graphenesheet [3]. The green (red) ticks along thehorizontal axis indicate the photon energiesat which JDOS divergences (resonances) oc-cur.


when the fundamental photon energy is larger than the band gap, which is always the case for

an undoped ZGNR, because the edge states provide a zero-gap system. Following an earlier

treatment of graphene [5], we find the ERS carrier injection to be

ξabcdR (ω) =

2πe4

~4

∑nm

∫fmn

dκx

2π

V d a ∗R;nmV

b cR;nm

ω4nm

δ(ωnm(κx) − ω), (2.46)

where the effective VME for the ERS process are

Vi j

R;nm ≡ ~∑`

v in`v

j`m

E`n − Enm + iβR+

v jn` v i

`m

E`m + Enm + iβR

. (2.47)

As in Eq. (2.42), βR is a small constant introduced to broaden resonant processes and the sum

over ` runs over all bands (filled and empty), but a converged value is obtained for ` = 30 bands

for a photon energy range of 0–1 eV. The first term in the sum of Eq. (2.47) corresponds to

photo-emission by an electron, and the second to photo-emission by a hole [5]. Note that due

to the different frequencies involved in Eq. (2.36), symmetrization of VijR is unnecessary. The

selection rules forVijR are the same as those forVij

C (Eq. (2.43)); note, however, thatVijR , V

jiR,

although VijR and Vji

R satisfy the same selection rule. From this we identify six nonzero terms

for the ERS carrier injection coefficient, ξxxxxR , ξxyyx

R =(ξ

yxxyR

)∗, ξxxyy

R =(ξ

yyxxR

)∗, ξxyxy

R , ξyxyxR , and

ξyyyyR . As do the conventional coefficients, the ERS coefficients suffer from JDOS and effective-

VME divergences, the later arising whenever

E` = 2En − Em or (2.48a)

E` = 2Em − En (2.48b)

is satisfied. These conditions correspond to resonant processes, when a state is located at an

energy |Enm(κx)| above (below) the final (initial) state n (m). As in Eq. (2.42), a small damping

constant βR of 20 meV was introduced in the denominators of Eq. (2.47). All of these ERS

coefficients present a large number of these resonances, causing ξabcdR to be highly sensitive to

the value of the βR parameter. However, these resonances are of small magnitude for the energy


range chosen for Fig. 2.6, hence they are not apparent. As shown, three of these components

have their onset at zero photon energy, because the symmetry properties of the involved matrix

elements allow for transitions between the two edge states.

0

1

2

0 0.05 0.1 0.15 0.2 0.25

(c)

ξyxyx

R(ω

)[

m3s−

1V

−4]


×10−4

0

1

2

3

4 (b)

ξxyxy

R(ω

)[

m3s−

1V

−4]

×10−5

0

0.5

1

1.5

2(a)

ξxxxx

R(ω

)[

m3s−

1V

−4]

Full

Edge

Wξ′abcd2D

×10−7

0

1

0 0.05 0.1 0.15 0.2 0.25

(f)

ξyyyy

R(ω

)[

m3s−

1V

−4]


×10−6

0

1

2

3 (e)

|ξxyyx

R(ω

)|[

m3s−

1V

−4]

×10−7

0

1

2(d)

ξxxyy

R(ω

)[

m3s−

1V

−4] ×10−5

Figure 2.6: (Color online) ERS carrier injection tensor, as given by Eq. (2.46). The definitions of thefull and edge contributions are given in the last paragraph of Section II.B. Notice that the edge statesplay a dominant contribution to the ERS absorption process, due to the large amount of resonant states.The green (red) ticks along the horizontal axis indicate the photon energies at which JDOS divergences(resonances) occur. The red dot-dashed lines indicate the ERS processes for 2D graphene [5].

2.3.4 Current injection

Injection coefficients

We begin with the expression for ηC, the current injection coefficient characterizing the con-

ventional process. Here the interference between the 2PA at ~ω with 1PA at 2~ω (see the

right diagram in Fig. 2.3) leads to net current injection coefficients (including electron and hole


0

1

2

0 0.05 0.1 0.15 0.2 0.25

(c)

|ηxyyx(ω

)|[

Cm

3s−

2V

−3]


×10−11

0

1

2(b)

|ηxxyy(ω

)|[

Cm

3s−

2V

−3]

×10−11

0

1

2(a)

|ηxxxx(ω

)|[

Cm

3s−

2V

−3]

Full

Edges

Wηabcd

2D

×10−12

Figure 2.7: (Color online) Nonzero net current in-jection coefficients, including the conventional andERS contributions, i.e. η(ω) = ηC(ω) + ηR(ω). Thedefinitions of the full and edge contributions aregiven in the last paragraph of Section II.B. On eachpanel, we include Wηabcd

2D (dot-dashed red curves),where ηabcd

2D (Eq. (2.51) and text below) is for agraphene sheet [5]. The red ticks along the horizon-tal axis indicate the energies at which resonancesoccur; a damping constant of 20 meV is introducedto broaden such resonances. The dips observed inthese coefficients arise due to negative contributionsto the conventional and ERS currents, in turn due tothe shape of the involved matrix elements.

contributions) given by [1]

ηabcdC (ω) =

16iπe4

~3

∑nm

∫fmn

dκx

2π

(va

nn − vamm

)Vbc∗

C;nmvdnm

ω3nm

δ(ωnm(κx) − 2ω). (2.49)

From the selection rules for the regular and the effective VME, Eq. (2.25) and Eq. (2.43), we

identify three nonzero current injection coefficients, ηxxxxC , η

xyyxC , and ηxxyy

C = ηxyxyC . Notice that

for all these tensors the first Cartesian component is x: Due to the confinement of the ribbons

along the y direction (see Fig. 2.1), the current injection can only flow along the x direction,

and all tensor components ηyabcC are zero.

Turning to the expression for ηR, the current injection coefficient characterizing the inter-

ference between the ERS discussed above and the 1PA at ω (see the left diagram in Fig. 2.3),


including both electron and hole contributions we find

ηabcdR (ω) =

2iπe4

~3

∑nm

∫fmn

dκx

2π

(va

nn − vamm

)ω3

nm

[vb∗

nmVcdR;nm + vc∗

nmVbdR;nm

]δ(ωnm(κx) − ω), (2.50)

where VR is given by Eq. (2.47). On the basis of the matrix elements selection rules, we

identify three nonzero ERS current injection coefficients, ηxxxxR , ηxyyx

R , and ηxxyyR = η

xyxyR .

Over the frequency range shown in Fig. 2.7, the conventional and the ERS current injection

coefficients are of the same order, dropping off as the inverse of the third power of the photon

energy, as do the coefficients for graphene [5]. Thus we only plot the total injection coefficients

ηabcd = ηC(ω) + ηR(ω). For comparison, we include plots of Wηabcd2D (with the respective values

of the Cartesian indices), where

ηxxxx2D (ω) = i

34

gsgve4v2F(2~ω)−3, (2.51)

and ηxxxx2D = 3ηxxyy

2D = 3ηxyyx2D are the net current injection coefficients for a 2D monolayer of

graphene [5]; as before, gs = 2 and gv = 2 are the spin and valley degeneracies, respectively.

As we saw for carrier injection, the edge states provide the strongest contribution at the onset

of current injection. Another characteristic of these coefficients is that ηxxxx has its onset at the

band gap between bands (2,−1), while ηxxyy and ηxyyx have their onset at 0 eV. This is due to

the selection rules that the matrix elements involved in both the conventional and ERS process

satisfy, allowing transitions between bands (1,−1). An important characteristic of the current

injection coefficients is that they are free of JDOS divergences, because the diagonal matrix

elements in their respective expressions, Eqs. (2.49) and (2.50), are identically zero at the κx at

which the minimum gap occurs. However, a number of effective VME resonances do exist at

photon energies indicated by the small red ticks in Fig.2.7, such that Eq. (2.44) is satisfied. As

explained before, the magnitude of these resonances is broadened by a small damping constant.

These coefficients are shown in Fig. 2.7, where we present the net current injection arising from

the addition of the conventional and ERS contributions, i.e., η(ω) = ηC(ω) + ηR(ω).


Swarm velocities

The numerical values of the coefficients ξab, ξabcdC(R), and ηabcd

C(R) do not immediately give a sense of

the average velocities with which the electrons and holes are injected. Sometimes an average,

or swarm velocity is introduced to indicate this [1]. In the system considered here, we could

introduce a swarm velocity for both the conventional and ERS processes, according to

V C(R) =1e

JC(R)(ω)

n(1)(Ω) + n(2)C(R)(ω)

, (2.52)

where Ω = 2ω for V C because JC arises from the interference of 1PA at 2ω with 2PA at

ω, while Ω = ω for V R because JR arises from the interference of 1PA at ω with the ERS

described above. Besides describing an average speed that characterizes the injected carriers,

one can consider maximizing Eq. (2.52) by using appropriate phases in the optical beams, and

adjusting the relative amplitudes of the light at ω and 2ω. Considering just the swarm velocity

of the conventional process, such optimization leads to equal 1PA and 2PA, and it follows that

the intensity of the fundamental beam at ω should be about half an order of magnitude larger

that of the beam at 2ω, for a fundamental photon energy of about 0.4 eV. In contrast, the swarm

velocity of the ERS process depends only on the intensity of the beam at 2ω. Further, in trying

to optimize the net swarm velocity, determined by the total current injected divided by the total

carrier density injected, one finds that the beam at 2ω should have an intensity about an order of

magnitude larger than the beam at ω. Since in typical experiments the beam at 2ω is obtained

by second harmonic generation of part of the beam at ω, this would be impractical. Thus we

calculate the conventional and Raman swarm velocities for typical [53] beam intensities of

the fundamental and second harmonic fields, shown in Fig. 2.8. We complement these carrier

velocities with the total average velocity of the injected carriers

V tot =1e

JR(ω) + JC(ω)

n(1)(ω) + n(1)(2ω) + n(2)C (ω) + n(2)

R (ω), (2.53)


0

1

0 0.05 0.1 0.15 0.2 0.25

×1026 (b)

n[(m

·s)

−1]


Conv

ERS

Total

-1012345

×105 (a)

ImV

[ms−

1] Conv swarm

ERS swarmWeighted Average

Figure 2.8: (Color online) Swarm and (weighted) average velocities (top), accompanied by the carrierdensity rates (bottom) along x due to ηxxxx for typical [53] beam intensities of the fundamental andsecond harmonic fields. The average carrier velocities (black-dashed line) for ηxxyy and ηxyxy are of thesame order, but their net components have a smooth onset at zero photon energy.

also evaluated at typical [53] beam intensities. These carrier velocities are shown in Fig. 2.8. As

a reference, at the photon energy of 0.25 eV, the maximum swarm velocity of the conventional

process for a monolayer of graphene is 2.9 × 105 ms−1. Hence the carrier velocities in ZGNR

are comparable to those on a monolayer of graphene, as might be expected.

2.4 Doping

In the previous sections, we investigated the carrier and current injection at zero chemical

potential. Since the dispersion relations of the edge states in ZGNR have a zero band gap and

are flattened for κx > W−1 (Fig. 2.3), those states are always involved at the onset energy of

all of the optical response coefficients studied here. This suggests that doping is an effective

method to alter the population of these two bands and the current that can be injected by

the optical transitions between them. In this section, we revisit the calculations of ξab, ξabcdC(R)

and ηabcd for a negative chemical potential, corresponding to a p-doped system. Besides the


modified contribution from the edge states, we will also see significant modification in the

contributions from other bands, particularly in the region near the K and K′ points, where

doping leads to either a “valley” of filled states (n-doped), or a “hill” of unfilled states (p-

doped); see Fig. 2.3.

We consider two negative Fermi levels, µ1 = −0.1 eV and µ2 = −0.2 eV, which in Fig. 2.3

we indicate by the upper boundaries of the grey areas. The value of −0.1 eV is interesting

because, at this chemical potential, the flat part of band −1 (i.e., the region where κx > W−1, cf.

Fig. 2.3) contains empty states; this condition allows transitions from lower energy bands with

final states in band −1, but also disables transitions from band −1 to upper bands. The second

value, µ = −0.2 eV, is interesting because at this potential a “hill” of unfilled states arises in

the first parabolic band (band −2 in Fig. 2.3) at energies below our nominal value of zero.

We present the results of the calculations of 1PA coefficients for those values of the chem-

ical potential in Fig. 2.9. In an undoped sample, the JDOS divergences in ξxx at low photon

energies are due to the onset of the transitions (2,−1), (1,−2), (4,−1), and (1,−4) (see Table 2.2

and Fig. 2.4). Since all of these transitions involve bands ±1, any nonzero chemical potential

has the capacity to significantly alter the 1PA at these photon energies. For instance, if the

Fermi level is at −0.1 eV, then the flat part of band −1 contains empty states, and the low pho-

ton energy divergences are removed. In addition, at this chemical potential transitions of the

type (−1, n), for n odd and < −1 are permitted. However, the contributions to the 1PA from

these new transitions are of smaller magnitude than the contribution from the (1,−2) transition,

which is unaffected by the −0.1 eV doping. For this reason, the (1,−2) transition remains as

the main contribution to the ξxx coefficient at low photon energies at this chemical potential

(see Fig. 2.9).

At the Fermi level −0.2 eV, the edge states are completely empty, as are the states at the

higher points of band −2 near the K and K′ points. This condition allows transitions of the

type (−2, n), for n even and < −2, and also forbids transitions of the type (n,−2), for n odd and

≥ 1, and κx near the K and K′ points. It is this latter restriction which significantly changes the


ξxx coefficient near its onset. A further decrease in the Fermi level would consistently remove

the divergences in ξxx at low photon energies. All these observations were confirmed with a

band-by-band calculation of ξxx.

The effect of doping the system has a larger influence on the onset energy of ξyy that on

that of ξxx. This is because the JDOS divergences at low photon energies relevant for ξyy are

due to the transitions (1,−1), (3,−1), and (1,−3) (cf. Table 2.2). Therefore, even for small

doping, the large contribution coming from the transitions between the two edge states (bands

±1, κx > W−1) is significantly decreased, and leads to a greater change of the magnitude of

ξyy than of the magnitude of ξxx. A special signature of ξyy for µ = −0.2 eV (dark-violet

signal, Fig. 2.9 b)) is the presence of two narrow peaks at 0.045 and 0.075 eV; the first of these

peaks is due to the (−1,−2) transition, while the second is from the (−2,−3) transition. These

two transitions are active only for those κx states at which the “hill” of band −2 is empty (see

Fig. 2.3). Notably, the transition (−2,−3) brings a new JDOS divergence because it is active

over a range of reciprocal space that includes κx = W−1, where both bands have their maximum

and their energy difference Enm(κx) has a zero derivative (see the discussion below Eq. (2.40)).

In general, all these new transitions involve more JDOS divergences if the range of κx over

which they are active includes the κx at which the band pairs have their maxima or minima. For

instance, the divergences 1–4 in Fig. 2.9 are the same as those in Fig. 2.4 and Table 2.2, but

the divergences 5–6 arise due to the new transitions allowed at nonzero chemical potentials: in

Fig. 2.9 a), at the chemical potential −0.20 eV, the divergence 5 at 0.179 eV is due to the tran-

sition (−2,−4), which is active over a range of κx that includes the κx at which bands (−2,−4)

have their maxima, hence a new JDOS divergence appears. Likewise for ξyy in Fig. 2.9 b)

at µ = −0.20 eV: divergences 5 at 0.089 eV and 6 at 0.268 eV exist because the transitions

(−2,−3) and (−2,−5) are active over regions of reciprocal space that include the κx at which

such bands have their maxima.

In Fig. 2.10, 2.11, and 2.12 we present the nonzero ξabcdC , ξabcd

R , and ηabcd coefficients for

selected nonzero Fermi levels. As was seen for ξab, doping the ZGNR has the effect of modi-

fying the responses around their onset energy, either due to the removal of some transitions, or


due to the appearance of new ones, which in the undoped system were forbidden because the

initial and final states were filled [e.g. (−1,−2) or (−1,−3)]. This shows that doping is an ef-

fective way of modifying the carrier and current injection in ZGNR, where the most significant

changes are due to the removal of density of states at the edge bands.

We close this section by mentioning that we performed finite temperature calculations at

room temperature; this was achieved by implementing a temperature dependence of the Fermi

factors through the Fermi-Dirac distribution. We found that the only significant change is in

that the onset energy of the coefficients ξab, ξabcdC(R), and ηabcd are smaller. However, the mag-

nitudes of the coefficients at energies near the lower onsets are several orders of magnitude

smaller that the magnitudes of the corresponding coefficients at zero temperature near their

energy onsets.

0

2

4

6

0 0.1 0.2 0.3 0.4 0.5

1 2 3 45 6

(b)

ξyy(ω

)[

ms−

1V

−2]


0

1

2

×109

1 2 3 45

(a)

ξxx(ω

)[

ms−

1V

−2]

µ [eV]0

−0.10−0.20

Figure 2.9: (Color online) One photon absorption coefficients as a function of the photon energy forselected Fermi levels corresponding to p-doped samples. The ZGNR has 95 zigzag lines (about 20 nmwidth). For nonzero chemical potentials, some transitions become impossible and some new transitionsarise, possibly leading to new JDOS divergences (e.g. divergences 5 and 6). Divergences 1–4 are thesame as in Fig. 2.4.


0

2

4

6

8

0 0.05 0.1 0.15 0.2 0.25

0

1

2×10−5←−

×10−3−→

(d)

ξyyyy

C(ω

)[

m3s−

1V−4]


0

1

2

3

4

0

2

4

6

8×10−4

(c)

ξxyxy

C(ω

)[

m3s−

1V−4]

0

1

2

3

4

5

0

1

2

3

4

5×10−6

(b)

|ξxxyy

C(ω

)|[

m3s−

1V−4]

0

1

2

3

4

5

6

0

1

2

3

4

5

6×10−6

(a)

ξxxxx

C(ω

)[

m3s−

1V−4] µ [eV]

0

−0.10

−0.20

Figure 2.10: (Color online) Two-photon ab-sorption coefficients for selected Fermi lev-els corresponding to p-doped samples. TheZGNR has 95 zigzag lines (about 20 nmwidth). For panels where two different verti-cal scales are present, i.e. panel (d), the scaleon the left (right) is for undoped (doped)cases (arrows below the factors indicate theordinate for which they apply). A dampingconstant βC = 20 meV was introduced.


0

1

2

0 0.05 0.1 0.15 0.2 0.25

0

1

2

(c)

ξyxyx

R(ω

)[

m3s−

1V−4]


×10−4←−

×10−5−→

0

1

2

3

4

0

1

2×10−5

(b)

ξxyxy

R(ω

)[

m3s−

1V−4]

0

1

2

0

2

4

6

8

10×10−7

(a)ξxxxx

R(ω

)[

m3s−

1V−4]

µ [eV]0

−0.10

−0.20

0

0.5

1

0 0.05 0.1 0.15 0.2 0.25

0

0.5

1

×10−6

(f)

ξyyyy

R(ω

)[

m3s−

1V−4]


0

1

2

3

0

1

2

3×10−7

(e)

|ξxyyx

R(ω

)|[

m3s−

1V−4]

0

1

2

0

1

2

3

4

(d)

ξxxyy

R(ω

)[

m3s−

1V−4]

×10−5←−

×10−6−→

Figure 2.11: (Color online) ERS carrier injection coefficients for selected Fermi levels corresponding top-doped samples. The ZGNR has 95 zigzag lines (about 20 nm width). For panels where two differentvertical scales are present, i.e. (c) and (d), the scale on the left (right) is for undoped (doped) cases(arrows below the factors indicate the ordinate for which they apply). Notice that at µ = −0.20 meVsome resonances are absent, e.g., at 0.15 eV in (a) and (f); this is because, at this Fermi level, the statesat which these resonances are present for the undoped system, now contain empty states. A dampingconstant βC = 20 meV was introduced.

2.5 Limits of the model

The model employed in this Chapter inherits the limits of applicability of time-dependent per-

turbation theory, which is restricted to situations of low electron-hole pair densities [59] (for

high injection densities a density matrix formalism could be employed to study the dynamics).

The regime of validity of the perturbation treatment used here can be estimated: we require the

populated fraction of excited states accessible to a typical Gaussian pulse to be small.


01234567

0 0.05 0.1 0.15 0.2 0.25

01234567

(c)

|ηxyyx(ω

)|[

Cm

3s−

2V−3]


×10−11←−

×10−12−→

0

1

2

0

1

2

(b)

|ηxxyy(ω

)|[

Cm

3s−

2V−3]

×10−10←−

×10−11−→

0

1

2

0

1

2

×10−12 (a)

|ηxxxx(ω

)|[

Cm

3s−

2V−3]

µ [eV]0

−0.10

−0.20

Figure 2.12: (Color online) Net current in-jection tensors (conventional plus ERS con-tributions) for selected Fermi levels corre-sponding to p-doped samples. The ZGNRhas 95 zigzag lines (about 20 nm width).For panels where two different verticalscales are present, i.e. (b) and (c), the scaleon the left (right) is for undoped (doped)cases (arrows below the factors indicate theordinate for which they apply). A dampingconstant of 20 meV was introduced.


2.5.1 Graphene sheet

As a reference, we first consider monolayer graphene. When the electric fields of the optical

beams are all aligned along x, the one- and two-photon injection coefficients for a 2D graphene

sheet are [3] given by Eqs. (2.39) and (2.45). For each of ξxx2D and ξxxxx

2D , we set the number of

carriers injected per unit area to be less than the number of states per unit area accessible to the

optical beam. Then taking the beam intensity as I(ω) = 2ε0c|E(ω)|2, we arrive to

I(2ω) <ε0cαω

2πv2F(∆t)2ξxx

2D(2ω), (2.54)

I2(ω) <(2ε0c)2αω

2πv2F(∆t)2ξxxxx

2D (ω), (2.55)

where α is the time-bandwidth product for the optical beam (which we take as 0.44, typical for

a Gaussian beam), ∆t is the pulse-duration, and vF ≈ 106 m/s is graphene’s Fermi velocity.

2.5.2 Zigzag nanoribbons

The estimate for the nanoribbon case is similar to the graphene sheet, aside from the fact that

the areal ratios become length ratios, i.e. for each one of 1PA and 2PA coefficients we set the

number of carriers injected per unit length to be less than the number of states per unit length

accessible to the optical beam, giving us

I(2ω) <ε0cα

π(∆t)2ξxx(2ω)(|ve| + |vh|), (2.56)

I2C(R)(ω) <

(2ε0c)2 α

π(∆t)2ξxxxxC(R)(ω)(|ve| + |vh|)

, (2.57)

where α and ∆t where defined previously, ve is the velocity of the injected electrons in the

conduction band, given by the matrix element vnn, and vh is the velocity of the holes injected in

the valence band, given by vmm. Equation (2.57) provides the expression for the conventional

(C) and ERS processes (R).

In order to compare the limiting intensities of our model for a graphene sheet and for


ZGNR, we assume a typical pulse duration of 220 fs and beam wavelengths of 3.2µm and

1.6µm for the ω and 2ω beams [53]. Then we identify the states that contribute at these two

wavelengths, and find that, on average, |ve| + |vh| ≈ vF . From Eqs. (2.54) and (2.56), at λ =

1.6 µm,

IGraphene(2ω)IRibbons(2ω)

=ωξxx(2ω)

2vF ξxx2D(2ω)

≈ 2.6, (2.58)

and from Eqs. (2.55) and (2.57), at λ = 3.2 µm,

IGraphene(ω)IRibbons(ω)

=

√ωξxxxx

C (ω)2vF ξ

xxxx2D (ω)

≈ 1.6. (2.59)

Equations (2.58) and (2.59) indicate that the limiting intensities of our model are similar for a

graphene sheet and for a ZGNR, within an order of magnitude.

We find that, under the assumptions made in this section, the estimated limit for the beam

intensities at ω in the ZGNR and the 2D graphene are about two orders of magnitude below

the intensities used in some experiments [53] on 2D graphene, where coherent current injection

was observed. Due to relaxation processes, of course, the number of allowed carrier excitations

below saturation is expected to be higher than our estimates, leading to larger values of the

beam intensities for which a perturbation approach would be valid. Based on the estimates in

Eqs. (2.58) and (2.59), if relaxation processes affect the ribbon samples as effectively as they

do for 2D samples, we can expect coherent control in ZGNR to be observable at the higher

intensities used in 2D graphene experiments.

2.6 Summary and discussion

We have calculated the response coefficients for one- and two-photon charge injection and the

two-color current injection in a graphene zigzag nanoribbon; we use the semi-empirical k · p

method to describe the electron wavefunctions by smooth envelope functions.

The only nonzero one-photon injection coefficients correspond to the case of all-x or all-y


aligned fields, i.e., ξxx and ξyy. These two coefficients possess a rich structure of divergences,

caused by divergences of the joint-density-of-states originating from the infinite set of parabolic

bands present in the zigzag nanoribbon. These two coefficients have distinct selection rules for

the allowed transitions.

The two-photon carrier injection coefficients drop off as the fifth power of the photon en-

ergy at large photon energies, as they do for monolayer graphene. Moreover, these coeffi-

cients possess two classes of divergencies. One corresponds to the joint-density-of-states di-

vergences associated with the parabolic bands. The second class corresponds to divergences

arising from resonant conditions, when the two-photon absorption processes arise from sequen-

tial one-photon absorption processes between real states. In our calculation here we broadened

these resonances phenomenologically, but a more sophisticated treatment of these resonantly

enhanced transitions is an outstanding problem on which we hope this work will encourage

further study. The onset of the signals is determined by the minimum energy band gap and the

selection rules for these coefficients.

We calculated the electron and hole contributions to the conventional and the stimulated

electronic Raman scattering (ERS) current injection processes, finding that the only nonzero

components are associated with current injected along the length of the nanoribbon, as ex-

pected. The behavior of these coefficients as a function of the photon energy follows the be-

havior of 2D graphene [∼ (~ω)−3] at large photon energies, aside of the resonances present in

the ribbons. We have also calculated the so-called swarm velocity of the injected electrons,

which inherits a rich structure as a function of the photon energy due to the details of the struc-

ture of the injection coefficients. All these calculations were presented for a system at zero

Fermi level and zero temperature. However, we also carried finite temperature calculations

and found that, within this model, finite temperatures only account for changes at the onset

of the signals, which are several orders of magnitude smaller than the nominal values at zero

temperature.

Lower bound estimates on the permissible incident intensities for which the calculations

here can be valid were presented. They are similar to those of monolayer graphene, where


coherent current injection has been observed at much higher intensities than these simple es-

timates, which do not take into account the relaxation effects in the excited populations. Thus

experiments to demonstrate coherent current injection in ZGNR seem to us to be in order.

For experiments contemplated for ribbons of different width than those studied here, it is

important to note that simple scaling arguments show that the wider the ribbon, the stronger

the confinement of the energy bands. As shown in the calculations presented in this Chapter,

at low photon energies, the band gap follows a linear relation with respect to the inverse of

the ribbon width. Consequently, increasing the width of the ribbon decreases the energy band

gap between any pair of bands. This in turn shifts the onset energy of the response coefficients

towards zero energy and increases the number of JDOS divergences per photon energy. For

instance, the onset of the response coefficients when light is polarized along the length of the

ribbon is determined by the bangap between bands (1,−2) (see Fig. 2.3). For such pair of

bands, a linear fit shows that the band gap depends on the ribbon width W as Egap1,−2 ≈ aW−1

with a = 2.98 eV · nm. Besides altering the onset energy of the responses, a larger width also

leads to a larger magnitude of the injection coefficients, larger than would be expected simply

on the basis of the increase in material; e.g., a width increase of about 15% doubles the size of

ηxxxx.

As the outstanding signature of the zigzag nanoribbons are the strongly localized edge

states, we have identified their contribution to the carrier- and current-injection processes. In

all cases the edge states always participate in the onset of the signals. This lead us to consider a

second scenario to study these localized states: given that the dispersion relations of these states

are flattened towards zero energy for certain regions in k-space, we re-visited our calculations

considering doped scenarios. We found that that even small doping levels allow for signifi-

cant changes around the onset energy of the signals. This is because the large joint-density

of states present between the edge states is diminished with nonzero chemical potentials. Due

to the relative ease of doping graphene systems, the present work shows that zigzag nanorib-

bons offer an excellent opportunity to investigate scenarios in which electrical currents can be

generated and controlled optically. While more sophisticated treatments of the electron states


and the inclusion of electron-electron interaction [34, 44] will undoubtedly add to the richness

of the injection processes, we hope that the description given here will motivate all-optical

current injection experiments. Although coherent control has been studied and observed on

graphene sheets, zigzag graphene nanoribbons have the advantage of having optical responses

that depend strongly on the geometry and width of the ribbon. Moreover, as shown in the liter-

ature, the localized states present in these ribbons are highly sensitive to external fields, doping

and functionalization. All these characteristics endow graphene zigzag ribbons with a richness

absent in simpler graphene sheets.

Chapter 3

An Effective Model for the Electronic and

Optical Properties of Stanene

Abstract

The existence of several 2D materials with heavy atoms has recently been demonstrated. The

electronic and optical properties of these materials can be accurately computed with numer-

ically intensive density functional theory methods. However, it is desirable to have simple

effective models that can accurately describe these properties at low energies. Here I present

an effective model for stanene that is reliable for electronic and optical properties for photon

energies up to 1.1 eV. For this material, I find that a quadratic model with respect to the lattice

momentum is the best suited for calculations based on the bandstructure, even with respect

to band warping. I also find that splitting the two spin-z subsectors is a good approximation,

which indicates that the lattice buckling can be neglected in calculations based on the band-

structure. I illustrate the applicability of the model by computing the linear optical injection

rates of carrier and spin densities in stanene. Our calculations indicate that an incident circu-

larly polarized optical field only excites electrons with spin that matches its helicity. A modified

version of this chapter was published in Physical Review Materials 1, 054006 (2017).

55

Chapter 3. An EffectiveModel for the Electronic and Optical Properties of Stanene 56

3.1 Introduction

The experimental isolation of single layers of graphene nearly a decade ago has inspired a

search for new 2D materials [60, 61]. Among those recently studied are silicene [62, 63], zinc-

oxide [64], and the transition metal dichalcogenides [65,66]. There is also substantial research

on other elemental 2D materials, including the remaining elemental crystallogens [67–69],

elemental pnictogens, such as nitrogene [70], phosphorene [71], arsenene [72], antimonene

[73,74], and bismuthene [75], as well as members from other families [76,77]. One of the most

interesting materials in this group is stanene, a monolayer of Sn atoms arranged in a buckled

honeycomb lattice. Due to the heavy Sn atoms, the spin-orbit coupling (SOC) is expected to

be strong and to lead to nontrivial topological properties of the bands that make stanene a 2D

topological insulator [78]. The strong SOC is predicted to open band gaps of 88 meV at the

K and K′ points of the Brillouin zone [78, 79], and thus the quantum spin Hall effect, with its

characteristic spin polarized edge modes free of backscattering from non-magnetic impurities,

could in principle be observed at room temperature. Recently, monolayers of stanene have

been epitaxially grown [80], and phase-change laser ablation techniques [81] have been used

to produce few-layer stanene. Experiments probing high photon energy absorption properties

of few-layered stanene have also been reported [82].

While the electronic and optical properties of crystalline materials can be studied with

modern ab initio methods, the numerical task can be challenging. It is thus desirable to have

simple effective models that reliably reproduce the basic properties of materials, at least over

energy ranges of interest. In order to compute electronic and optical properties from an ef-

fective model, it is necessary to know the Hamiltonian and the Lax connection1, which gives

important geometric information about the basis of the quantum states [84] in the model. Two

of the most common types of effective models for crystals are tight-binding and k · p models.

In tight-binding models, the basis of states is defined in terms of a set of Wannier func-

tions that are exponentially localized in space; it is always possible to obtain such a set of

1Notice that in this description I refer to the connections in the Brillouin zone introduced by Melvin Lax [83].They should not be confused with the connections related to Lax pairs introduced by Peter Lax.


functions for a block of electronic bands with vanishing total Chern number that do not cross

others [85, 86]. The Hamiltonian and the Lax connection are respectively expressed in terms

of hopping parameters and dipole matrix elements. The hopping parameters can be inferred

from bandstructure properties, obtained either from experiments or from first-principle calcu-

lations. In contrast, the Lax connection parameters are harder to deduce since they are usually

obtained from electronic and optical properties. When the Wannier functions are well local-

ized, the overlap between them – and consequently the matrix elements for any operator –

can be restricted to only nearest neighbor atomic sites; the model is then usually simple and

has relatively few parameters that need to be inferred. However, if the Wannier functions at

sites further apart have a considerable overlap, the number of free parameters increases signif-

icantly. While this is not a major problem for determining hopping parameters, it leads to a

large number of dipole parameters that are hard to fit.

In k · p models, the basis of states consists of the periodic parts u`q (r) of Bloch wavefunc-

tions ψ`q (r) = eiq·ru`q (r) /√

(2π)D for a set of bands ` at a reference point q in the Brillouin

zone (BZ) of dimension D. Since the basis is independent of the lattice momentum k, the Lax

connection is null for a k · p model, which simplifies the calculation of electronic and optical

properties. However, k · p models also have drawbacks. For instance, the Hamiltonian has

a fixed form that is quadratic in the lattice momentum k, but its free parameters are only as-

sociated with the linear terms in k, as the quadratic term is related to the electron bare mass.

Because of that, the only way to introduce more parameters in the Hamiltonian is to increase

the number of bands in the model, even if the additional bands are irrelevant except for aiding

in the fitting of the band energies of interest. Also, since the periodic functions depend on k,

the basis needs to include the states of several bands at the reference q point in order to span the

state of a single band at other k points in the BZ. Thus k ·p models for the whole Brillouin zone

usually include several bands, but describe only a few of them accurately, a fact that increases

the number of parameters to be inferred. Moreover, the accuracy of the states [84, 87, 88] at

a point k in the BZ decreases with the distance from the reference point q, and since results

are usually reported without a standard measure of the error, it is not possible to know exactly


where the approximation becomes unacceptable.

In this chapter I develop an effective model for stanene that is similar to a k·p model but that

is free of the drawbacks pointed out in the previous paragraph. We keep track of the accuracy

of the eigenstates, and the free parameters of the Hamiltonian are not restricted to the linear

terms in the lattice momentum k. Starting from an ab initio set of wavefunctions, we expand

the eigenstates at a region of the BZ in terms of the states at a reference point q in that region.

For a finite set of bands, this expansion is not unitary, as the basis set is incomplete. In order

to preserve unitarity, we approximate this expansion by a unitary transformation [89] using

a singular value decomposition (SVD), the singular values of which provide a measure of the

accuracy of the eigenstates. This transformation allows the same basis to be used for a region of

the BZ, so the Lax connection is null as desired. A Taylor expansion of the Hamiltonian matrix

written in this basis with respect to the lattice momentum k then gives the free parameters

of our model. For stanene we use three regions in the BZ, around the points K, K′, and

Γ. We obtain an effective model that is accurate for transition energies up to 1.1 eV, with a

quadratic expansion for each reference point. We find that the band warping is well accounted

for by a quadratic model, and that a cubic model does not improve upon it significantly. We

also find that neglecting some small parameters leads to the separation of the spin sectors

in our model; such approximation is accurate within a tolerance corresponding to the room

temperature energy.

To illustrate the applicability of our model, we compute the one-photon injection rate coef-

ficients for carrier and spin densities in stanene. We predict that an incident circularly polarized

optical field with photon energy close to the gap only excites electrons with spins that match

the helicity of the optical field. This result suggests the possibility of employing stanene in

optically-controlled spin pump applications.

The outline of this chapter is as follows: In Sec. 3.2 I present the procedure to obtain the

effective model; in Sec. 3.3.2 I apply it to stanene and analyze the accuracy of the eigenstates

and the eigenenergies, including the band warping. In Sec. 3.4 I use our model to compute

linear optical absorption rates of stanene. I end with a discussion of our results in Sec. 3.6.


3.2 Method for deriving effective models

Bloch’s theorem asserts that the eigenstates ψ`k (r) of a periodic Hamiltonian functionH (r,−i~∇) =

H (r + R,−i~∇), where R is a lattice vector, can be written as

ψ`k (r) =1√

(2π)Deik·ru`k (r) , (3.1)

where u`k (r) = u`k (r + R) are periodic functions. In typical ab initio calculations, a very large

number of basis functions uak (r), which usually consist of plane waves or atomic orbitals, are

used to specify the Bloch HamiltonianH (r,−i~∇ + ~k) by the matrix elements

Habk = 〈uak|Hk|ubk〉

≡ Ω−1uc

∫uc

dr u∗ak (r)H (r,−i~∇ + ~k) ubk (r) , (3.2)

where Ωuc is the volume of the unit cell. The Hamiltonian matrix Hk consisting of these

elements is then diagonalized, and provides the eigenstates and eigenenergies corresponding

to each electronic band ` at the lattice momentum k. We denote the diagonalized matrix by

Hk and its corresponding eigenstates by |u`k〉. If the large set of basis functions in the ab initio

calculation are taken to be the same for different lattice momenta, say q and k, we can compute

the overlap matrix between states,Wk;q, with matrix elements

Wm`k;q = 〈umq|u`k〉 = Ω−1uc

∫uc

dr u∗mq (r) u`k (r) . (3.3)

The overlap matrix allows us to decompose the states u`k (r) at k in terms of those at the

reference point q in the BZ and to use the states umq (r) as a basis for any k point in the region

of the BZ around q. In order to have a simple effective model, it is desirable to include only

a small number of bands in the basis set. However, if only a few functions umq (r) = 〈r | umq〉

are included in the basis, even the states umk (r) = 〈r | umk〉 corresponding to the same block

of bands at other k point in the BZ neighborhood might not be completely spanned by them.


This means that the overlap matrixWk;q might not be unitary when restricted to a small set of

bands. Here we ensure the unitarity of the model by replacingWk;q with a unitary matrix based

on its “singular value” decomposition (SVD). In the remaining of this discussion we drop the

subindex indicating the reference q point in the BZ where it does not lead to confusion. In its

singular form, the overlap matrixWk is written as

Wk = UkΣkV†k , (3.4)

where Uk and Vk are orthogonal matrices, and Σk is a diagonal matrix with its elements as the

singular values. For a complex matrix W, such decomposition always exists, U and V are

unitary (i.e., UU† = I = UU†), and the “singular values” are guaranteed to be positive (for a

proof, see for example, Sec 2.5.2 and 2.5.6 of [90]). IfWk were a unitary matrix, Σk would

be the identity matrix I, thus a simple “unitary approximation” toWk is to replace Σk with the

identity matrix as

Wk → Wk ≡ UkV†k . (3.5)

An obvious measure for the accuracy of this approximation is the difference I− Σk. For each k

in a region around the reference q point in the BZ, the approximate unitary overlap matrix Wk

allows the expansion of the states |u`k〉 in terms of the basis |umq〉 as

|u`k〉 =∑

m

Wm`k∣∣∣umq

⟩. (3.6)

The next step is to use the above equation to write the Hamiltonian matrix Hk for each k in

terms of the states∣∣∣umq

⟩at the reference q point in the BZ. Since |u`k〉 are the eigenstates of the

(diagonalized) Hamiltonian matrix Hk, then

Hm`k = 〈umk| Hk |u`k〉 = δm`E`k. (3.7)


We write the elements of the Hamiltonian matrix for lattice momentum k expressed in the∣∣∣u`q⟩

basis as

Hm`k =⟨umq

∣∣∣Hk∣∣∣u`q⟩ , (3.8)

and using Eq. (3.6), the matrix Hk is related to Hk through the unitary matrix Wk that performs

the change of basis

Hk = WkEkW†

k , (3.9)

where Ek is a diagonal matrix with diagonal elements E`k. Since the basis of states ∣∣∣umq

⟩ is

independent of k, its Lax connection vanishes, ξm`k ≡ i⟨umq

∣∣∣∇k∣∣∣u`q⟩ = 0. Consequently, such

a basis is suitable for expanding the Hamiltonian matrix Hk around q simply as

Hk = Hq + κ · ∇kHk∣∣∣k=q + O

(κ2

)+ O

(κ3

). . . , (3.10)

where κ = k−q. If the basis were dependent on the lattice momentum k, the expansion would

include a correction given by the Lax connection.

In summary, the overlap matrixWk from an ab initio calculation is replaced by its unitary

approximation Wk, the diagonalized Hamiltonian is written in a basis that is independent of the

lattice momentum k, and a Taylor expansion of its matrix elements gives the free parameters

in our model. We now turn to discuss the application of this procedure to stanene.

3.3 Effective model for stanene

3.3.1 First-principles ground state of stanene

We start by obtaining the electronic wavefunctions from a first-principles calculation, in the

framework of Density Functional Theory (DFT) and the Local Density Approximation (LDA),

using the freely available ABINIT code [91, 92]. The wavefunctions are expanded in a ba-

sis of planewaves; the size of the basis is determined by a kinetic-energy cutoff of 653 eV

(≈ 24 Ha), corresponding to 6166 planewaves. The crystal (ionic) potential is modeled us-


z

x b

y

x a

a2

a1

δ1

δ2

δ3

= A-site = B-site

Figure 3.1: Hexagonal lattice of stanene with Sn atoms at A and B sites. The lattice vectorsare denoted by a1 and a2, and we show the unit cell with a gray rhombus. The interatomicdistance projected on the plane is a, and along the vertical direction it is b, so the interatomicdistance is

√a2 + b2.

ing the Optimized Norm-Conserving Vanderbilt Pseudopotentials (ONCVP) [93], which is a

modern alternative for pseudopotential representations that provides a compromise between

computational efficiency and accuracy for the current demands in automated, high-throughput,

computational materials science research2. We take 14 out of the 50 Sn electrons as valence

electrons, and the others are assumed clamped. We converge the ground-state total energy up

to 2.7 meV, leading to a 12 × 12 k-point mesh. Since we simulate the Sn monolayer with a

supercell model, we introduce an interlayer vacuum space of 11.42 Å, such that spurious inter-

layer interactions are negligible; with this amount of vacuum space, the total energy remains

unchanged within 2.7 meV if the vacuum space is incremented.

The relaxation of the atomic positions leaves the atoms at the (x, y) coordinates of a hon-

eycomb lattice, i.e., one Sn atom at (0, 0) and another at (a1 + a2) /3, see Fig. 3.1. The lattice

2Other pseudopotential representations like the “ultra-soft pseudopotentials” [94] or the “projected-augmentedwaves” [95, 96] method are standard alternatives, but they attain efficiency and accuracy at the expense of mathe-matical complexity of the inherent quantities in the pseudopotential method [93].


0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

zcoordinate(A

)

BFGS Iteration

Initial coordinate of site-Bno d-electrons z00

z01

z02

z03

4.5

4.6

4.7

0 2 4 6 8 10 12

Lattice

constan

t(A

)

BFGS Iteration

Initial lattice constantno d-electrons c01

c02

c03

c04

c05

Figure 3.2: Relaxation of the atomic coordinates. The xy coordinates remain at the usual equilibriumpositions (not shown here) for hexagonal 2D lattices with two atoms in the unit cell: in the basis of thereciprocal vectors, site A is at (0, 0) and site-B is at (1/3, 1/3). In contrast, the z coordinates convergeto out-of-plane positions. In this Figure we show the evolution of the z coordinates of the atom atsite-B during the structural relaxation (atoms at sites-A follow the same). Each curve represents theevolution for different starting positions. All of these curves converge to the same final value of 0.418 Å,except when we neglect the d-electrons. Indeed, as shown in both panels, when the d-orbitals of Sn areassumed as core states (black curves), the out-of-plane atomic coordinates and the lattice constant areunderestimated. That is, for stanene, the d-electrons must be considered as semicore states due to thespatial extension of these orbitals (not shown). The structural relaxation of atoms at sites-A is mirrored,with negative values of z, hence the buckling distance is 0.836 Å. The relaxation is done using theABINIT code, employing its Broyden-Fletcher-Goldfarb-Shanno (BFGS) built-in routine.

vectors are a1 = a(3x +

√3y

)/2 and a2 = a

(3x −

√3y

)/2, where a = 2.66 Å is the inter-

atomic distance projected on the plane. In contrast, the relaxation of the z-coordinates leads to

out-of-plane coordinates ±0.418 Å, giving rise to a “buckling distance” of b = 0.836 Å, such

that the interatomic distance is√

a2 + b2. This small buckling has been shown to enhance the

overlap between π and σ orbitals, leading to an equilibrium configuration in materials where

the π-π bonding is relatively weak [78, 97]. The evolution of the relaxation process of the

atomic z-coordinates and the lattice constant a is shown on Fig. 3.2, where we clearly see that,

for stanene, the d-orbitals must be considered as semi-core states, due to their relatively long

spatial extension (not shown). The first nearest neighbor vectors are δ1 = a2 (x +

√3y) + bz,

δ2 = a2 (x −

√3y) + bz and δ3 = −ax + bz. In Fig. 3.1, we show the crystal lattice of stanene.

With these structural parameters we proceed to obtain the key ingredient in density-functional


DFT Charge Density n(x, y, z) (e−/Bohr3)(at z = 0)

-6 -4 -2 0 2 4 6

x (Bohr)

-6

-4

-2

0

2

4

6

y(B

ohr)

0

0.05

0.1

0.15

0.2

0.25

-6-4

-20

24

6 -6-4

-20

24

6

0.120.140.160.180.20.220.240.26

DFT Charge Density n(x, y, z) (e−/Bohr3)(at z = 0)

x (Bohr)

y (Bohr)

0.120.140.160.180.20.220.24

Figure 3.3: ab-initio DFT charge density of stanene. The highest values of charge density occur at thelocations of the atoms, hence these maximal values map the hexagonal lattice of stanene (cf. Fig. 3.1).The tin atoms are modeled with an ONCVPSP pseudopotential [93] (see main text), wich accounts forthe s, p and d electronic orbitals. The inclusion of the d-electrons do have an influence on the finalstructural parameters (lattice constant and atomic coordinates), cf. Fig. 3.2.

theory, the self-consistent electronic charge density n(x, y, z). On Fig. 3.3 I show two represen-

tations of the contour levels of n(x, y, z = 0), where we clearly identify the honeycomb lattice

of stanene. The maxima of n(x, y, z = 0) corresponds to the locations of the tin atoms. Once

the self-consistent charge density is obtained, then the Kohn-Sham Hamiltonian is diagonal-

ized one more time, maintaining the charge density and the Kohn-Sham potential unchanged.

This step is usually referred as the “non-self-consistent” calculation of the bandstructure (elec-

tronic dispersion relations). Such process give us the eigenstates and eigenenergies. The plot

of the eigenenergies along a path that connects crystal wavevectors of high symmetry in the

“irreducible” part of the BZ (IBZ) is referred as the bandstructure. On Fig. 3.4 I plot the band-

structure of stanene along the typical M → Γ → K → M path in the IBZ for a hexagonal


-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

a)

c1

v1

v2

Γ

MK

Energy(eV)

0.7

0.8

0.9

1

b)c1

v1 v2

Σk

q

six-band model

0.7

0.8

0.9

1

M Γ K M

c)c1

v1

Σk

q

four-band model

Figure 3.4: a) The ab initio bandstruc-ture of stanene with the bands includedin our effective model highlighted. Theband gaps at K and K′ have a value of88 meV. At Γ, the minimal transition isat 0.472 eV (transition v1 → c1) and thesecond is at 0.808 eV (transition v2 →

c1). All bands are doubly (spin) degen-erate. The dashed (gray) bands are notdescribed by our model. b) The singularvalues (the elements of the diagonal ma-trix Σk, Eq. (3.4)) with q = Γ as the ref-erence point. The shaded area indicatesthe region where the singular values areall greater than 0.9, and the unitary ap-proximation Σk → I is acceptable. c)Same as b), but for q = K as the refer-ence point.

lattice. All the bands are spin degenerate since stanene has space inversion symmetries. How-

ever, SOC in stanene leads to a gap opening due to a removal of orbital degeneracy at the Dirac

point [78]. As shown in Fig. 3.4, the bandstructure of stanene has gaped Dirac cones at the K

and K′ points with a gap of 0.088 eV. At Γ, the first transition occurs at 0.472 eV and the next

one at 0.808 eV. At M the first transition is at 1.55 eV; hence I ignore that region of the BZ in

the effective model, as the focus of this work is on energies ranges up to 1.1 eV. Our effective

model contains only states with lattice momentum around the K, K′, and Γ points, it includes

six bands around the Γ point and only four bands around K and K′ points.


3.3.2 Evaluation of the effective model: accuracy of the approximation

Once the ab initio wavefunctions are computed, we proceed to obtain the overlap matrix

(Eq. (3.3)) between the periodic functions at the reference point q and the other points in its

neighborhood in the BZ; this is done for each of the regions of interest in the BZ, namely the

regions around the reference points K, K′, and Γ. The overlap matrices of the ab initio wave-

functions can be approximated by unitary matrices based on singular value decompositions

according to Eq. (3.5). In order to determine the region of the BZ where this approximation

is accurate, in Fig. 3.4 we plot the elements of the diagonal matrix Σk (the singular values)

for the reference points K and Γ; the results for the K′ point are similar to those of K. In

Fig. 3.4, we also highlight the regions where each element of Σk is greater than 0.9, which is

taken as our tolerance for the approximation in Eq. (3.5). Notice that the highlighted regions

encompass every point on the BZ where optical transitions with photon energies below 1.1 eV

are possible.

In order to have a measure of the accuracy of the states that is easier to be visualized, we

define a figure of merit

δΣ (k) = n−1√

Tr (Σk − I)2, (3.11)

where n is the number of bands included in the model. In Fig. 3.5 we present the figure of merit

δΣ (k) for the three regions of interest in the BZ. We notice that the error indicated by δΣ (k) is

lower than 5% for large neighborhoods around the reference points.

3.3.3 Hamiltonian matrices

Having established the regions where the approximation of the states is valid, we now turn to

the approximation of the Hamiltonian matrix. We expand the matrix elements of the Hamilto-

nian Hk directly as in Eq. (3.10), and report the results below. Since we use a basis independent

of the lattice momentum for the neighborhood of the BZ around each reference point, the Lax

connection is null for each of these neighborhoods, ξabk = 0.


M

q = K′

q = K

q = Γ

M

q = K′

q = K

q = Γ

M

q = K′

q = K

q = Γ

SVD error isolines1 %2 %3 %4 % Figure 3.5: (Color online) Figure of

merit of the unitary approximation Σk →

I , as defined by Eq. (3.11), for the threeregions of the BZ centered at the ref-erence points q = Γ,K,K′; each qis marked with black dots. The yel-low lines connecting all contiguous pairsof K and K′ points indicate boundariesof the first Brillouin zone (cf. inset ofFig. 3.4, a)).


K and K′ points

The valleys around the K and K′ points are similar in our model, so we present the matrices

associated with each of them together, and use the valley parameter τ = 1 to refer to K and

τ = −1 to refer to K′. At the K and K′ points, the wavefunctions have a predominant character

of pz orbitals located at an atom in the unit cell. So we can use si and σi to respectively

denote the Pauli matrices in the spin and sublattice sectors; here i = 0, x, y, z, as we adopt the

convention of denoting the identity as the zeroth Pauli matrix. In this notation, the Hamiltonian

is written in terms of the matrices si ⊗ σ j.

Up to linear order in the lattice momentum κ = k − q, where q = K,K′, we find

H(1)τκ =∆K (−τsz ⊗ σz + s0 ⊗ σ0) + ζ(1)

K as0 ⊗(κxσx + τκyσy

)− λ(1)

K a(κysx − κxsy

)⊗ σz, (3.12)

where in the second part of the first term (i.e., ∆K s0 ⊗ σ0) we add an energy shift ∆K such

that the top of the valence band is at zero energy. Equation (3.12) is the linear-order term

in the crystal wavevector κ of our effective Hamiltonian for stanene. As we will mention

shortly, the last term in equation (3.12) can be neglected due to the relatively small value of

the parameter λ(1)K . This simplification is in agreement with a simple estimate of the spin-

orbit interaction of a monolayer material, which indicates that the spin polarization of the

material is along the direction perpendicular to the plane; this is expected for mono-layered

materials with mirror symmetry about the plane [98]. The simplified version of equation (3.12)

is in agreement with the Kane-Mele model (equations (2)-(3) in [98]3), which predicts that,

at sufficiently low energies, the spin-orbit effects on mono-layered material with atoms in a

honeycomb lattice lead to the opening of a gap accompanied by the quantum spin Hall effect

(QSHE). More specifically, it is the term (∆Kτ)sz ⊗ σz that leads to a QSHE. Following the

procedure outlined by Kane-Mele [98], the QSHE in stanene could be confirmed by describing

the electronic states with a Hamiltonian model for the entire Brillouin Zone; the QSHE states

3Notice that equation (4) in [98] does not apply in the context of this chapter. Such term refers to a “Rashbacontribution” that arises when the mirror symmetry about the plane is broken, e.g., due to a substrate or a perpen-dicular electric field.


are commonly visualized solving such model for a strip (“ribbon”) geometry. More recently ,

Chamon et al [99] provided a general classification of all gap-opening relativistic masses (such

as the Haldane mass term) of two-dimensional Dirac Hamiltonians that involve spin, valley,

and sub-lattice contributions. The subset of such masses that lead (support) the SQH effect is

in agreement with the Kane-Mele model [98]4

The quadratic terms in κ are

H(2)τκ = − ζ(2)

K a2s0 ⊗

[τκxκyσx +

12

(κ2

x − κ2y

)σy

]− v(2)

K a2 |κ|2 s0 ⊗ σ0 + ϑ(2)K a2τ |κ|2 sz ⊗ σz (3.13)

+ η(2)K a2τ

[(κ2

x − κ2y

)sx − 2κxκysy

]⊗ σz,

where the values of the parameters are shown in Table 3.1. Neglecting the relatively small

parameters λ(1)K and η(2)

K leads to a separation of the spin subsectors, since without them H(1)τk

and H(2)τk do not have terms with sx and sy, the only matrices with cross-spin elements. The spin

separation is expected for lattices without buckling, and it indicates that the lattice buckling

can be neglected in calculations involving k close to the expansion point q.

The parameters v(2)K and ϑ(2)

K can also be neglected, and the three parameters ∆K , ζ(1)K and ζ(2)

K

are the only ones needed for our model to give band energies that match those from DFT within

a tolerance of room temperature energy. We nevertheless report the negligible parameters λ(1)K ,

η(2)K , v(2)

K and ϑ(2)K , because their physical significance can be identified with the help of a pz-

orbital tight-binding model, as we discuss in Section 3.5. Finally, we provide an analytical

expression for the band energies around the K and K′ points obtained from our effective model.

4Notice that Chamon et al [99] denote the Pauli matrices that act on different sublattice and valley states byτ and σ, respectively, whereas in this thesis and in the Kane-Mele model [98] we use we σ and τ for sublatticesand valleys, respectively. Pauli matrices acting on spin states are denoted by s in these three works.


All values in eV

∆K = 0.044 ζ(1)K = 0.67 ζ(2)

K = 0.33

K λ(1)K = 0.03 v(2)

K = 0.03

K′ ϑ(2)K = 0.03

η(2)K = 0.02

EcΓ

= 0.37 ζ(1)Γ1 = 1.23 v(2)

Γc = 0.34

Γ Ev1Γ

= −0.10 ζ(1)Γ2 = 1.16 v(2)

Γ1 = 0.45

Ev2Γ

= −0.44 v(2)Γ2 = 0.34

ζ(2)Γv = 0.35

Table 3.1: Parameter values for the effectivemodel. The upper half of the Table lists the param-eters for the K and K′ valleys (Eqs. (3.12)–(3.13)).The parameters λ(1)

K , η(2)K , v(2)

K and ϑ(2)K can be ne-

glected without significant changes in the band en-ergies. Neglecting the parameters λ(1)

K and η(2)K alone

already leads to a separation of the spin subsectors.The parameter values for the Γ valley (Eqs. (3.15)–(3.16)) are listed in the lower half of this Table; neg-ligible parameters were also omitted.

Neglecting the small parameters mentioned in the previous paragraph, we have

E±τκ = ∆K ±

√∆2

K + X2κ +Y2

κ, (3.14a)

Xκ = aκx

(ζ(1)

K − τζ(2)K aκy

), (3.14b)

Yκ = ζ(1)K aκy −

12τζ(2)

K a2(κ2

x − κ2y

), (3.14c)

where the positive and negative signs of the square root correspond to the conduction and

valence bands respectively.

Γ point

At the Γ point, the wavefunctions cannot be easily associated with a sublattice, but they can

still be identified according to spin, so we continue using si to denote the Pauli matrices acting

on the spin sector of the Hilbert space. Up to linear order in the lattice momentum, here

κ = k − q = k since q = Γ, we find

H(1)Γκ = s0 ⊗

E(c)

Γ0 0

0 E(v1)Γ

0

0 0 E(v2)Γ

+ aκxs0 ⊗

0 ζ(1)

Γ1 ζ(1)Γ2

ζ(1)Γ1 0 0

ζ(1)Γ2 0 0

+ aκysz ⊗

0 −iζ(1)

Γ1 iζ(1)Γ2

iζ(1)Γ1 0 0

−iζ(1)Γ2 0 0

,(3.15)


while the quadratic terms in κ are

H(2)Γκ = 1

2a2 |κ|2 s0 ⊗

v(2)

Γc 0 0

0 −v(2)Γ1 0

0 0 −v(2)Γ2

+ 12a2

(κ2

x − κ2y

)s0 ⊗

0 0 0

0 0 ζ(2)Γv

0 ζ(2)Γv 0

+a2κxκysz ⊗

0 0 0

0 0 iζ(2)Γv

0 −iζ(2)Γv 0

.(3.16)

The values of the parameters are presented in Table 3.1. Here we have omitted negligible

parameters. The parameters reported constitute the minimum set necessary to describe the

energies with an accuracy equivalent to room temperature when compared to the bands from

DFT. Notice that the model for the valley at the Γ point can also be separated in two spin

sectors.

3.3.4 Accuracy of the energies

The accuracy of the Taylor expansion of the Hamiltonian matrices in the previous subsection

can be determined by comparing the band energies obtained from our model with those from

the ab initio calculation. In Fig. 3.6 we present the band energies obtained from models in-

cluding first-, second-, and third-order expansions of the Hamiltonian on the lattice momentum

difference κ; third-order expansions are not discussed further in this work. We also show the

ab initio bands for comparison, and focus on the regions where the approximation for the states

is accurate as discussed in Sec. 3.3.2. From Fig. 3.6, we see that keeping the cubic terms in

the Hamiltonian expansion is unnecessary to reproduce the ab initio band energies around the

Γ point, while for the region around the K point (and equivalently the K′ point) it is actually

detrimental to go beyond the second-order expansion.

A plot of band energies along a simple path through a region of the BZ is not enough to

establish the accuracy of the bands from our model in that entire region. Analyzing the band

warping is a way to ensure that the good agreement displayed in Fig. 3.6 is not coincidental to


-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

M Γ K M

c1

v1

v2

E1

Energy

(eV)

REF1st2nd3rd

E2

E3

E4

E5

Figure 3.6: Comparison of the band energies obtained from the Taylor expansions of theHamiltonian matrices (dashed lines) in Sec. 3.3.3 with those from the ab initio calculation(continuous gray line). The band energies from the effective model are plotted only in theshaded regions, where approximation for the states is accurate as discussed in Sec. 3.3.2.

the directions associated with that plot. In Fig. 3.7 we show isoenergy lines for each relevant

band obtained from our model and those from the ab initio computation. The latter are shown

as pairs of lines that enclose an energy range equivalent to room temperature, which is taken as

our tolerance for energy accuracy. We compare the band warping corresponding to expansions

of the Hamiltonian that are quadratic and cubic on the lattice momentum difference κ; on

Fig. 3.7 we show that the cubic expansion does not improve upon the quadratic one. Thus we

confirm that the quadratic expansion provides the best model for the bandstructure of stanene

for excitation energies up to 1.1 eV.


|KM|4

c1 at E3

Γ

|KM|4

c1 at E5

K

|KM|4

v1 at E2

Γ|KM|

4

v1 at E4

K

|KM|4

v2 at E1

Γ

Figure 3.7: Band warping of the relevant bands around the reference points in the BZ inour model. Red-dashed and green-dotted lines indicate respectively second and third orderexpansions of the Hamiltonian matrices. The thin gray lines are ab initio energy isolines thatenclose a range of energy equivalent to room temperature. To give a sense of proportion weinclude a line segment of length one fourth of the distance KM. The isolevels Ei and bandlabels vi and c1 are as indicated in Fig. 3.6.


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

hω (eV)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

ξxx(ω

)(1/V

2s)×

1016

a)

BZ Region

FullK + K′

Γ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

hω (eV)

0.00

0.05

0.10

0.15

Reσxx(ω

)(m

A/V

)

b)

BZ Region

FullK + K′

Γ

Figure 3.8: Linear optical absorption properties computed with the effective model ofSec. 3.3.2. a) One-photon absorption coefficient ξxx (ω) and b) real part of the optical con-ductivity σxx (ω) of stanene. The contributions from the regions around the K and K′ points(dot-dashed red line) and the Γ point (dashed black line) in the BZ are shown separately, alongwith that from the full BZ (solid blue line); we stress that the “full” signal indeed contains con-tributions from all crystal momenta k around K(K′) and Γ for which one-photon transitionsless than 1.1 eV are possible; consequently, it is equivalent to a full BZ calculation, within thelimits of validity of our model.

3.4 Linear and non-linear optical properties

The optical properties of a crystalline system depend only on the Hamiltonian matrix and the

Lax connection [100]. Since the Lax connection is null in the basis of our model ξabk = 0,

the velocity matrix elements are simply given by v (k) = ~−1∇kH (k). We consider the optical


injection rates of carrier and spin densities, given by

ddt

n = ξab (ω) Ea (ω) Eb (−ω) , (3.17)

ddt

S z = ζzab (ω) Ea (ω) Eb (−ω) , (3.18)

where we use the convention of summing repeated indices, E (t) = E (ω) e−iωt + c.c. is an inci-

dent optical field, and the tensors ξab (ω) and ζzab (ω) are the carrier and spin density injection

coefficients

ξab (ω) =2πe2

~2ω2

∑cv

∫d2k

(2π)2 vacvkvb

vckδ (ω − ωcvk) , (3.19)

ζzab (ω) =2πe2

~2ω2

∑cv

∫d2k

(2π)2

(S z

cc − S zvv)

vacvkvb

vckδ (ω − ωcvk) , (3.20)

where v and c are respectively valence and conduction band indices, e = −|e| is the electron

charge, vacvk are the velocity matrix elements, S z

cc = ±~/2 and S zvv = ∓~/2 are the spin-z matrix

elements of respectively the conduction and valence bands, and ~ωcvk = ~ωck − ~ωvk are band

energy differences. In numerical calculations, we approximate the Dirac delta function in the

above equations by a Lorentzian function with a broadening width of 6 meV.

In Fig. 3.8, we present plots of the linear optical absorption coefficient ξxx (ω) and the

real part of the optical conductivity Reσxx (ω), which are related to each other by ξxx (ω) =

2Reσxx (ω) / (~ω). As the frequency increases, the absorption begins at the band gap energy

0.088 eV due to electronic transitions at the K and K′ valleys in the BZ. The contribution

from Γ has an absorption onset at 0.472 eV, and a second absorption onset at 0.808 eV, when

electronic transitions from the second valence band are allowed.

For photon energies close to the band gap, stanene has the interesting property that circu-

larly polarized light excites mostly electrons with the spin that matches its helicity. Similar

characteristics have been identified and studied in other monolayers, such as silicene [101].

This feature can be seen from our linear model for the K and K′ points in Eq. 3.12, which

can be separated in spin sectors, and the expressions of ξ (ω) and ζ (ω) for a Dirac cone [6, 7].


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

hω (eV)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

h−

1Im

ζzyx(ω

)(1/V

2s)×

1016

Im ζzyx = −Im ζzxy a)

BZ Region

FullK + K′

Γ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

hω (eV)

−1.0

−0.5

0.0

0.5

1.0

Sz/(hn

)

−Im ζzyx

hRe ξxx

b)

BZ Region

FullK + K′

Γ

Figure 3.9: Spin-density injectioncomputed with the effective modelof Sec. 3.3.2. a) Spin densityinjection coefficient ζzyx (ω) and b)Spin polarization of injected carri-ers with circularly polarized light−Im ζzyx (ω) /

[~ξxx (ω)

]for stanene,

with h = 1. The contributions fromthe regions around the K and K′points (dot-dashed red line) and theΓ point (dashed black line) in the BZare shown separately, along with thatfrom the full BZ (solid blue line), inthe sense described in the caption ofFig. 3.8.

For circular polarizations, the light field propagating along the z direction can be written as

E (ω) = Eωph, where h = ±1 is the helicity, and ph = (x + ihy) /√

2. Then the expression for

each spin s = ±1 is

ξhτs (ω) =Θ (ω − 2∆K) e2

8~2ω

(1 + hs

2∆K

ω

)2

, (3.21)

which is independent of valley, and where Θ (x) is the step function, valued as zero or unity

if x < 0 or x > 0, respectively. From Eq. (3.21) we see that the spin polarization is maximal

for photon energies corresponding to the gap, and it decreases for larger photon energies. The

injection coefficient of an arbitrary quantity for circularly polarized light, ηh (ω), is given in


terms of its Cartesian components as

ηh (ω) =12

[ηxx (ω) + ηyy (ω)

]+ ih

12

[ηyx (ω) − ηxy (ω)

]= ηxx (ω) + i h ηyx (ω) , (3.22)

where the relations ηxx (ω) = ηyy (ω) and ηyx (ω) = −ηxy (ω) due to the symmetries of a buck-

led honeycomb lattice were used. For carrier and spin densities in stanene, we also have

ξxx (ω) = Re ξxx (ω) and ξyx (ω) = 0, as well as ζzxx (ω) = 0 and ζzyx (ω) = i Im ζzyx (ω). So the

coefficients for circular polarizations are simply ξh (ω) = Re ξxx (ω) and ζh (ω) = −h Im ζzyx (ω).

We present plots of the spin density injection coefficient ζzxy (ω) computed with our effective

model in Fig. 3.9 a), which shows the same frequency regimes discussed for ξxx (ω). In Fig. 3.9

b) we show the spin polarization of injected carriers for circularly polarized light. Even for ex-

citations at the Γ valley there is still a helicity-spin coupling, although the net spin polarization

is partially canceled by the excitations at the K and K′ valleys.

We note that helicity-spin coupling is due to the sign of the mass term ∆K in each Dirac

cone [102–104], which also explains why stanene shows the spin Hall effect. We also point

out that the helicity-spin coupling in stanene is analogous to the helicity-valley coupling in

TMDs [105].

As an example of application of our model to study non-linear optics, now we present

results for the two-photon absorption (2PA) rate, defined by

ddt

n2 (2ω) =ξabcd (2ω) Ea (ω) Eb (ω) Ec (−ω) Ed (−ω) , (3.23)

where ξabcd (2ω) is the two-photon absorption coefficient, expressed as

ξabdg (2ω) =2πe4

~4ω4

∑cv

∫d2k

(2π)2

∑c′

vacc′v

bc′v

ω − ωc′v−

∑v′

vbcv′v

av′v

ω − ωcv′

∗ × ∑c′′

vdcc′′v

gc′′v

ω − ωc′′v−

∑v′′

vgcv′′v

dv′′v

ω − ωcv′′

× δ (2ω − ωcv) , (3.24)


0.0 0.2 0.4 0.6 0.8 1.0 1.2

2hω (eV)

−50

0

50

100

150

200

250

300

350

ξabcd

2(2ω

)(m

2/V

4s)

a)

K + K′ + Γregions

xxxx

xxyy

xyxy

0.4 0.6 0.8 1.0 1.2−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.2 0.4 0.6 0.8 1.0 1.2

2hω (eV)

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

ξabcd

2(2ω

)(m

2/V

4s)

only Γregion

b)

xxxx

xxyy

xyxy

Figure 3.10: Two-photon absorptionof stanene. On panel a) we showthe contribution from k-points in theneighbourhood of the three specialpoints Γ, K and K′. The inset is anamplification around the two-photonenergy range where the Γ point con-tributes to this responce. On panelb) we present solely the contributionsfrom k-points around Γ.

where all quantities are as defined below Eq. (3.19). From symmetry considerations, we find

that the only nonzero components of this coefficient are ξxxxx, ξxxyy and ξxyxy. On Fig. 3.10

we present the evaluation of ξabcd (2ω). As expected, the onset of these coefficients start when

the two-photon energy 2~ω equals the fundamental bandgap Egap, which occurs at K. The

contributions from k-points in the neighbourhood of K dominate the 2PA signal, whereas the

contributions arising form the Γ point are negligible as compared to those from K. On the

inset of Fig. 3.10-a) we present a zoom-in over the photon energy range over which the two

band edges at Γ occur (cf. Fig. 3.4). To clarify further the contributions from the Γ point, on

Fig. 3.10-b) we evaluate the 2PA coefficients considering solely k-points near Γ. From both

panels a) and b) we clearly identify two onsets at 0.472 eV and at 0.808 eV, arising from the


two band-edges at Γ (i.e., transitions from the two top valence bands to the first conduction

band). Although the relative magnitude of the 2PA from Γ is much smaller than that from

K(K′), the relevance of the former in an experimental or applied setting would depend on the

photon energy of the applied light field.

3.5 Tight binding model

Tight-binding models (TBM) have successfully been used to describe electronic states in the

full Brillouin zone (BZ) in different monolayer materials, such as silicene, germanene and

stanene [106, 107]; the description of the full BZ usually requires the inclusion of basis sets

with s, px, py and pz orbitals. In this section we discuss a TBM that includes only pz orbitals

on a buckled honeycomb lattice, which is enough to describe the states around the K and K′

points in the BZ, and in that region it agrees with the models of Liu [106] and Ezawa [107].

This TBM is unable to describe the states at the Γ point because their orbital character are not

purely pz. For instance, from a DFT calculation, we find that at the Γ point the orbital character

of the first conduction band is 73% s, 24% pz and 3% d, while that of the top valence band is

96% a mix of px and py, and about 4% d character.

This basis is not in Bloch’s form5, and it allows us to write all the hopping parameters in

terms of the nearest neighbor vectors δn instead of the lattice vectors an. Using the notation

employed in the main text, the Hamiltonian is written in terms of the matrices si ⊗ σ j. With

these conventions and employing a usual tight-binding framework [55], the nearest-neighbour

(NN) hopping term in the Hamiltonian is

HNNk = −t

3∑n=1

s0 ⊗

0 e−ik·δn

eik·δn 0

= −t

3∑n=1

s0 ⊗[cos (k · δn)σx + sin (k · δn)σy

],

(3.25)

5When the periodic functions of a basis satisfy the condition u`k+G (r) = e−iG·ru`k (r) , where G is a reciprocallattice vector and ` is a band index, the Bloch wavefunctions are periodic over the Brillouin zone, φ`k+G (r) =

φ`k (r) , and the basis is said to be in Bloch’s form.


and the next-nearest-neighbor (NNN) term is

HNNNk = −t′

∑n,m,n

[eik·(δm−δn) + e−ik·(δm−δn)

]s0 ⊗ σ0

= −2t′∑

n,m,ncos (k · (δm − δn)) s0 ⊗ σ0,

(3.26)

without the spin-orbit coupling. The spin-orbit coupling changes the next-nearest-neighbor

hopping matrices according to

t′s0 ⊗ σ0 → t′s0 ⊗ σ0 + iλS O

(δm × δn

)· s ⊗ σz, (3.27)

where λS O is the spin-orbit coupling parameter. The last term in the above equation can be

further separated in two parts by decomposing the δm × δn vector as

λS Oδm × δn = λzz + λbz × (δm − δn) , (3.28)

where

λz =a

√a2 + 4b2

λS O, λb =2b

√a2 + 4b2

λS O, (3.29)

according to the lattice buckling; the lattice parameters a and b are depicted in Fig. 3.1.

In order to compare the tight-binding model with the one described in Sec. 3.3, we now

perform an expansion in powers of κ around the K and K′ points in the BZ, to which we re-

spectively associate τ = +1 and τ = −1. Applying a further change of basis to the B sublattice,

uBk (r)→ ieiK·δ3uBk (r), the linear term is

H(1)τκ = −τ 9

2λza2sz ⊗ σz + 32 tas0 ⊗

(κxσx + τκyσy

)+3t′s0 ⊗ σ0 −

92λba3

(κysx − κxsy

)⊗ σz,

(3.30)


where the term 3t′s0 ⊗ σ0 is simply an energy shift and can be removed. The quadratic term is

H(2)τκ = −3

4 ta2s0 ⊗[τκxκyσx + 1

2

(κ2

x − κ2y

)σy

]−9

4 t′a2κ2s0 ⊗ σ0 +(

32

)3λza4τκ2sz ⊗ σz

+(

32

)3λba4τ

[(κ2

x − κ2y

)sx − 2κxκysy

]⊗ σz.

(3.31)

Now we compare this tight-binding model described by Eqs. (3.30)-(3.31) to our effective

model around the K and K′ points in the BZ described by Eqs. (3.12)-(3.13). The relations

between the respective first order parameters are

∆K =92λza2, ζ(1)

K =32 t, λ(1)

K =92λba2, (3.32)

and for the second order ones, we have

ζ(2)K =3

4 t, v(2)K =9

4 t′, ϑ(2)K =

(32

)3λza2, η(2)

K =(

32

)3λba2, (3.33)

Since λb = 2bλz/a, we can take t, t′ and λz to be the only independent parameters of the

tight-binding model; numerical values for them can be obtained from Table 3.1. Consequently,

λ(1)K =

2ba

∆K , ζ(2)K =

12ζ(1)

K , ϑ(2)K =

34

∆K , η(2)K =

3b2a

∆K . (3.34)

This tells us that ∆K , ζ(1)K and v(2)

K can be taken as the only independent parameters in Table 3.1,

just as the 3 independent parameters for the tight-binding. The parameters t′ and v(2)K can be

neglected, though, so the relevant parameters are only two: t and λz for tight-binding, and ∆K

and ζ(1)K in our effective model. The relations above are satisfied by the parameters shown in

Table 3.1.



We have presented an effective model that accurately describes the electronic and optical prop-

erties of stanene for low photon energies. We started from an ab initio calculation of the

bandstructure of stanene, which allowed us to identify the parameters in the model. Our model

includes a minimum set of energy states: 6 bands around the Γ point in the BZ, and 4 bands

around the K and K′ points. We provided measures for the accuracy of the approximations for

states and for energies, so we can identify the range of validity of the model.

We found that a quadratic model with respect to the lattice momentum is the best suited for

calculations based on the bandstructure. Even the band warping from DFT calculations is better

reproduced by the quadratic rather than a cubic model. We also found that the lattice buckling

can be neglected. This is confirmed by verifying that a separation of the states according to

spin-z subsectors is a good approximation for the band energies. In Section 3.5, we discuss the

physical significance of some parameters in our model by comparing it to a pz-orbital tight-

binding model expanded around the K and K′ regions of the BZ. Finally, we illustrated the

applicability of the model by computing linear optical absorption rates of stanene. We high-

lighted the coupling of circularly polarized light with the electronic spin, which underscores

the potential of stanene for optical-spintronic applications.

The model proposed here can accurately describe optical properties of stanene up to pho-

ton energies of 1.1 eV, which is suitable for a wide range of optical experiments. Compared

with a usual k · p method, our model requires fewer parameters to describe the bandstructure;

we also provide a figure of merit to determine the portion of the Brillouin Zone where the ap-

proximation is sensible. We expect that this simple model will be useful in understanding and

suggesting experiments on this promising material, and that the procedure described here will

be used to extract effective models from ab initio calculations for other 2D materials.

Chapter 4

Coherent Control of Two- and

Three-photon Absorption in AlGaAs

Abstract

As described in Chapter 2, optical absorption in a semiconductor crystal can lead to carrier

and spin injection into specific regions of the bandstructure of a crystal, and the interference

of different pathways of absorption can lead to current injection of charge and spin carriers.

The magnitude and direction of such injected currents can be manipulated by controlling the

phases and polarizations of the optical fields that lead to their respective carrier injections. This

technique is to referred as Quantum Interference Control (QuIC), and can also been exploited

to measure the parameters of the incident fields.

In this Chapter I study QuIC technique involving two- and three-photon (“2+3”) absorp-

tion processes. As compared to QuIC of one- and two-photon (“1+2”) absorption processes

(Chapter 2), the 2+3 QuIC has different requirements for the laser frequencies and the opti-

cal processes occur in smaller regions of the Brillouin zone. Experiments demonstrating 2+3

QuIC current injection in AlGaAs are being finalized at by Prof. Steven Cundiff’s experimental

group at the University of Michigan. A modified version of this chapter is being prepared for

submission to the Physical Review B.

83

Chapter 4. Coherent Control of Two- and Three-photon Absorption in AlGaAs 84

4.1 Introduction

Quantum interference between different optical processes arises when two optical beams of

different frequencies can lead to the same transition. In a crystal, amplitudes for different opti-

cal processes leading to electron-hole excitations can interfere constructively in some regions

of the Brillouin zone (BZ), and destructively in others. By controlling the polarizations and

phases of the incident fields, it is possible to excite carriers in selected localized regions of the

BZ. Such Quantum Interference Control (QuIC), using 1- and 2-photon absorption processes,

has been used for current injection in semiconductors [4, 108, 109], graphene [3, 9, 110], topo-

logical insulators [6, 111, 112], and transition metal dichalcogenides [7, 113], as well as spin

current injection in semiconductors [114–118]. It has also been theoretically investigated for

current injection in graphene nanoribbons [119], spin currents in topological insulators [6], and

spin and valley currents in transition metal dichalcogenides [7]. The currents injected via QuIC

have also been exploited to determine parameters of the optical fields responsible for their in-

jection [120–122]. This method has found application in the measurement [123, 124] and

stabilization of the carrier-envelope phase of a train of octave-spanning laser pulses [125–127].

In crystalline materials, every instance of QuIC studied to date has involved 1- and 2-photon

absorption processes.

In this chapter I present a theoretical study of QuIC with 2- and 3-photon processes in Al-

GaAs. The derivation of the expressions for the optical injection coefficients is presented, and

then I evaluate them for different stoichiometries of AlGaAs using a 30-band k·pmodel. I com-

pute all the symmetry-allowed injection coefficients corresponding to different polarizations of

the incident fields, and analyze their frequency dependence over a range where the injection of

carriers that do not contribute to the current is avoided. That is, considering 3-photon absorp-

tion of photons at energy ~ω and 2-photon absorption of photons at energy 3~ω/2, we require

2~ω to be less than the band gap. The alloy AlGaAs is an ideal material for 2+3 QuIC, as

its stoichiometry can be chosen to yield a band gap appropriate for the available laser wave-

lengths. The 2+3 QuIC process is of interest both because the use of all-nonlinear optical


processes leads to excitations even more localized in the BZ than does 1+2 QuIC, and because

2+3 QuIC could be used to stabilize the carrier-envelope phase of a train of laser pulses that do

not span an octave. Experiments demonstrating 2+3 QuIC of injection current in AlGaAs are

in progress [128].

The outline of this chapter is the following: In Sec. 4.2 I present the equations necessary

to compute the optical injection rates for a generic material. In Sec. 4.3 I describe the model

used for AlGaAs. In Sec. 4.4 I present our results for carrier and current injection from 2- and

3-photon absorption (2PA and 3PA) processes in AlGaAs. I also discuss the efficiency of the

current injection by analyzing the swarm velocity, and computing the optimal laser intensities.

In Sec. 4.5 I discuss the implications of our results and present our conclusions.

4.2 Optical injection rates

Assuming the independent particle approximation, I consider a system in the presence of the

external perturbationVext (t) described by a Hamiltonian

H (t) = H0 +Vext (t) , (4.1)

whereH0 is the Hamiltonian in the absence of any external perturbation. In the basis of eigen-

states ofH0,

H0 =∑nk

~ωnk a†nk ank, (4.2)

Vext (t) =∑mnk

a†mk (t) Vmnk (t) ank (t) . (4.3)

In the interaction picture, the creation and annihilation fermion operators are

a†nk (t) = a†nkeiωnt and ank (t) = anke−iωnt, (4.4a)


and the time-evolution operator can be expanded as

U (t) = 1 +

∞∑N=1

∫ t

t0

dtN

i~Vext (tN)· · ·

∫ t2

t0

dt1

i~Vext (t1) . (4.5)

The terms of each order inVext can be obtained from the previous one by

UN (t) =

∫ t

t0

dtN

i~

∑mnk

a†mkVmnk (tN) ankeiωmntNUN−1 (tN) , (4.6)

whereU0 (t) = 1, ωmnk = ωmk − ωnk, and Emk = ~ωmk is the bandstructure. We are interested

in the excitation of an electron from a valence band v to a conduction band c due to the external

field. This excited state is |cvk〉 = a†ckavk|gs〉, where |gs〉 is the ground state of H0 with filled

valence bands. The state of the system is described by

|ψ〉 = U (t) |gs〉 = γ0|gs〉 +∑cvk

γcvk (t) |cvk〉 + . . . , (4.7)

where the coefficients

γcvk (t) = 〈cvk|U (t) |gs〉 (4.8)

contain the information we want. The expectation value of the density 〈M〉 of a quantity

associated with an operator

M(t) =∑mnk

a†mk(t) Mmnk ank(t), (4.9)

due only to the excitation of states |cvk〉 is

〈M〉 =1

LD

∑cvc′v′k

γ∗c′v′kγcvk 〈c′v′k|M |cvk〉

=1

LD

∑cvc′v′k

(Mc′ckδv′v − Mv′vkδc′c) γ∗c′v′kγcvke−iωc′v′kteiωcvkt, (4.10)


where L is a normalization length, and D is the spatial dimension of the system. For a full

HamiltonianH (t) that follows from a Hamiltonian for a single particle of the form

H (x,p; t) =1

2m

[p − eA (t)

]2+ HSO

(x,p − eA (t)

)+ Vlat (x) , (4.11)

where x and p are position and momentum operators, HSO is the spin-orbit term, and Vlat (x)

is the lattice potential energy. Here we neglect a contribution to the interaction that is solely

a function of time (∼ [A (t)]2), for it will not lead to any transitions, and we work in a gauge

where the electric fieldE (t), assumed independent of position, is fully described by the vector

potentialA (t). The interaction term in the Hamiltonian and the velocity operator take the form

Vext (t) = −ev ·A (t) and v = −1e∂H

∂A, (4.12)

respectively, where e = − |e| is the charge of the electron. Indeed, the interaction is of the form

we consider for any unperturbed Hamiltonian for a single particle that is at most quadratic in

the momentum. Taking the vector potential as

A (t) =∑α

Aαe−i(ωα+iε)t = −∑α

iωα

Eαe−i(ωα+iε)t, (4.13)

with ωα = ±ω, ±3ω/2; here ε → 0+ describes turning on the field from t = −∞. The γcvk (t)

coefficients can be expanded as γ(N)cvk (t) = 〈cvk| UN (t) |gs〉 following the expansion ofU (t) as

given by Eq. (4.6) for an incident optical field, so we can write the coefficients γ(N)cvk (t) as

γ(N)cvk (t) = R(N)

cvk

e−i(ΩN−ωcv+iε)t

ΩN − ωcvk + iε, (4.14)

where ΩN = ω1 + . . . + ωN . The coefficients R(N)cvk involve the electric field amplitudes Eα

R(N)cvk = R(N)a...b

cvk

(ωα, . . . , ωβ

)Eaα . . . E

bβ (4.15)


where repeated indices are summed; here superscripts refer to Cartesian indices and subscripts

to incident frequency components. For the lower orders we have

R(1)acvk (ωα) =

∑α

ie~ωα

vacvk, (4.16)

R(2)abcvk

(ωα, ωβ

)=

∑αβ

−e2

~2ωαωβ

∑c′

vacc′kvb

c′vk

ωβ − ωc′vk−

∑v′

vbcv′kva

v′vk

ωβ − ωcv′k

, (4.17)

R(3)abdcvk

(ωα, ωβ, ωδ

)=

∑αβγ

ie3

~3ωαωβωδ

[X −Y −Z

], (4.18)

where

X =∑

c′

vacc′k

ωα − ωcc′k

∑c′′

vbc′c′′kvd

c′′vk

ωδ − ωc′′vk−

∑v′

vdc′v′kvb

v′vk

ωδ − ωc′v′k

(4.19)

Y =∑

v′

vav′vk

ωα − ωv′vk

∑c′

vbcc′kvd

c′v′k

ωδ − ωc′v′k−

∑v′′

vdcv′′kvb

v′′v′k

ωδ − ωcv′′k

(4.20)

Z =∑c′v′

vbcv′kva

v′c′kvdc′vk

(ωα − ωv′c′k) (ωδ − ωc′vk)+

vdcv′kva

v′c′kvbc′vk

(ωδ − ωcv′k) (ωα − ωv′c′k)

(4.21)

We are interested in the non-oscillatory response of the system, so we focus on the ΩN = ΩN′

contributions to Eq. (4.10). In order to compute the injection rate d 〈M〉 /dt associated with

that equation, it is important to realize that

ddt

(γ∗c′v′kγcvke−iωc′v′kteiωcvkt

)t=0,ε→0

=∑

ΩN=ΩN′

2π δ (ΩN − ωcv) R(N′)∗c′v′k R

(N)cvk

∣∣∣∣∣∣ωcv=ωc′v′

. (4.22)

The fact that the R(N)cvk coefficients are always accompanied by δ (ΩN − ωcv) in the expression

for the response allows for substitutions 3~ω − ωcv = 0; this is used to simplify R(3)abdcvk in

Eq. (4.18). The expression for the injection rate of 〈M〉 due to the interference of an N′ photon

process with an N photon process is

ddt〈M〉 = µabd...,pq... (Ω) Ea

−αEb−βE

d−δ . . . E

pρEq

σ . . . + c.c., (4.23)


3ω/2

ω

ω

ω

3ω/2

3ω/2

ω

ω

ω

3ω/2Figure 4.1: Depiction of 2+3 QuICshowing the destructive (left) andconstructive (right) interference indifferent regions of the Brillouinzone.

where there are N′ frequency labels (α, β, δ, ...) and N frequency labels (ρ, σ, ...). The injection

rate coefficient µabd...,pq... (Ω) is assembled from extracting the terms multiplying the electric

field amplitudes in R(N′)∗c′v′kR(N)

cvk of Eq. (4.22) together with the appropriate matrix elements

appearing in Eq. (4.10) for 〈M〉. I give examples below.

Quantum interference of 2- and 3-photon processes

The processes of 3PA with frequency ω and 2PA with frequency 3ω/2 can interfere since the

total frequency for each of them is Ω = 3ω. For such processes the frequencies are all equal

in the equations (4.17) and (4.18) for the coefficients R(2)cvk and R(3)

cvk, and symmetrizing their

components leads to some simplifications. Using ωβ = ωcvk − ωα and ωα = ωβ = 3ω/2 in

Eq. (4.17), the second order coefficient can be written as

R(2)abcvk

(3ω2,

3ω2

)=−4e2

9~2ω2

∑m

vacmkvb

mvk(3ω2 − ωmvk

) , (4.24)

and using ωβ +ωγ = ωcvk −ωα and ωα = ωβ = ωγ = ω in Eq. (4.18), the third order coefficient

can be written as

R(3)abdcvk (ω,ω, ω) =

ie3

~3ω3

∑mn

vacmkvb

mnkvdnvk

(ω − ωcmk) (ω − ωnvk). (4.25)


Notice that the denominators in Eqs. (4.24) and (4.25) are minimal for m, n = c, v, so the domi-

nant contributions to R(2)cvk always involve intraband velocity matrix elements, but R(3)

cvk also has

contributions from interband velocity matrix elements 1. Intraband velocity matrix elements

are associated with the corresponding band dispersion, vannk = ∂a

kωnk, which vanishes at the k

point corresponding to the bandgap. Thus R(2)cvk is zero for total photon energies corresponding

to the band gap, and increases for larger excess photon energies. The dependence of R(3)cvk on

the total photon energy is different, as it depends on both interband and intraband velocity ma-

trix elements. For total photon energies just above the gap, R(3)cvk is determined mainly by the

interband matrix elements, but as the photon excess energy increases R(3)cvk becomes dominated

by the intraband matrix elements, since the electronic transitions occur at k points with larger

band dispersion.

The injection rate coefficients corresponding to the interference of 2- and 3-photon pro-

cesses can then be computed as

µabd, f g2+3 (Ω) = 2π

∫dk

(2π)D

∑cvc′v′

(Mc′ckδv′v − Mv′vkδc′c

)δωcv=ωc′v′ R(3)abd∗

c′v′k R(2) f gcvk δ (Ω − ωcv) ,

(4.26)

following Eqs. (4.10) and (4.23), where we have taken the continuous momentum limit. For

the plots in the next sections we use a frequency broadening ∆ corresponding to ~∆ = 13meV.

The factor R(3)abd∗c′v′k R(2) f g

cvk changes sign under a transformation k → −k, and this is the mech-

anism for constructive versus destructive interference in opposite points of the Brillouin zone.

In Fig. 4.1 we illustrate constructive versus destructive interference of 2- and 3-photon pro-

cesses at opposite points in the Brillouin zone.

1The denominators in Eqs. (4.24) and (4.25) do not lead to any divergences because of the assumption that2~ω is below the gap.


4.3 Electronic model of AlGaAs

I use a 30-band k ·pmodel for computing the electronic bands. The model has free parameters

associated with energies and momentum matrix elements at the Γ point, and the parameters

are adjusted to match the experimental results for band energies from −5 eV to 4 eV, such that

computations of optical absorption coefficient are expected to be reliable for photon energies

up to 6 eV.

L Γ X U,K3

2

1

0

1

2

3

4

5

ω (eV

)

Al0. 2Ga0. 8As

L Γ X U,K3

2

1

0

1

2

3

4

5

ω (eV

)

Al0. 35Ga0. 65As

Figure 4.2: Electronic bandstructure of AlαGa1−αAs for two different stoichiometries.

Using the Γ point as the expansion point for a k · p model, the effective Hamiltonian that

acts only on the periodic part of an energy eigenfunction of crystal momentum k is

Heff = H +~

mk · p +

~2k2

2m, (4.27)

where H is the Hamiltonian (4.11) with the vector potential set equal to zero; in this model [88]

we neglect the k dependence of the effective spin-orbit term. The second term on the right-

hand-side is the usual k·p contribution, and the last term is the contribution to the kinetic energy

only due to the lattice momentum. The basis of states has 8 sets [88], 4 of them corresponding

to the Γ1 representation of the point group Td (or 43m), 3 corresponding to the Γ4 representation,

and 1 to the Γ3 representation. The Γ1 representation has only 1 state, Γ4 has 3 states, and Γ3

has 2 states, so in total we have 4 × 1 + 3 × 3 + 1 × 2 = 15 states before considering spin; we

denote these states as |A〉, |B〉, etc. Tensor products of these are taken with spin states to get 30


states in all. Terms 〈A|Heff |B〉 are then 2 × 2 matrices, and take the form

〈A|Heff |B〉 = EAδABσ0 +i3∆AB · σ + iPAB · kσ0 +

~2k2

2mδABσ0, (4.28)

where σ0 is the unit 2 × 2 matrix and the components of σ are the usual Pauli matrices. The

free parameters of the model are the energies EA, the matrix elements of the spin-orbit term

∆AB, and the matrix elements of the momentum operator PAB. Since the basis for the states is

the same at every k point [129], the corresponding 2×2 matrices corresponding to the velocity

operator 〈A|v|B〉 are diagonal in the spin sector,

〈A| va |B〉 =1~

∂

∂ka 〈A|Heff |B〉 =

(i~

PaAB +

~ka

mδAB

)σ0, (4.29)

from which the matrix elements of the velocity operator between the energy eigenstates can be

determined.

For GaAs [88] and AlAs [130] I use reported parameters adjusted for room temperature,

while the parameters for AlαGa1−αAs are obtained from a linear interpolation according to the

stoichiometry. This approximation is accurate within an energy tolerance corresponding to

room temperature 2. The chosen parameters lead to effective masses and g-factors that are in

good agreement with experimental data. More important for the problems I consider, the band

structures and linear optical absorption spectra are also in good agreement with experimental

data. In Fig. 4.2 I show the relevant electronic bands for two different stoichiometries. As a

validation of the bandstructures obtained with the 30-band model, I computed bandstructures

for α = 0 (GaAs) and for α = 1 (AlAs) and found good agreement with first-principles (DFT)

calculations (not shown here).

In Fig. 4.3 I show the imaginary parts of the corresponding dielectric functions, which are

related to the 1-photon absorption rates (or carrier injection) by Im ε (Ω) = ~ξxx (Ω) /2ε0.

2The corrections that are quadratic on the stoichiometry parameter α are small, and do not lead to significantchanges in the band energies within a tolerance given by room temperature.


1.8 2.0 2.2 2.4 2.6 2.8

Ω(eV)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Im ε

(Ω)/ε 0

broadening = 13 meV

α= 0. 20

α= 0. 35

.

Figure 4.3: Imaginary part of the dielec-tric function for different stoichiometries α.The vertical lines indicate the correspondingbandgaps.

4.4 Quantum interference control using two- and three-photon

absorption in AlGaAs

I consider two incident fields of different frequencies with amplitudes

Eω = Eω eiφω eω and E3ω/2 = E3ω/2 eiφ3ω/2 e3ω/2, (4.30)

where Eω > 0 and E3ω/2 > 0 are the field magnitudes, the unit vectors eω and e3ω/2 indicate

their polarizations, and φω and φ3ω/2 indicate their phases. We also define the phase parameter

∆φ = 2φ3ω/2 − 3φω, which will be useful later. We assume that the field at 3ω/2 has a weaker

intensity than the field at ω, and we demand that the frequencies satisfy 2ω < ∆g < 3ω, where

Eg = ~∆g is the electronic gap. Therefore only 3PA processes are important for the lower

frequency fieldEω, while only 2PA processes are relevant for the higher frequency fieldE3ω/2;

the 3PA associated with E3ω/2 is weaker due to the lower intensity of the field, and we neglect

it.

I focus on Al concentrations α such that 0.18 . α . 0.38, since AlαGa1−αAs with α too

small has a band gap smaller than 2~ω for telecommunication wavelengths (~ω ≈ 0.8 eV), and

AlαGa1−αAs with α too large is too reactive.


1.8 2.0 2.2 2.4

hΩ (eV)

0.0

0.2

0.4

0.6

0.8

1.0ξ 2

(Ω)

(m/V

4s)

×105 Al0.2Ga0.8As

xxxxxyxyxxyy

2.0 2.2 2.4 2.6 2.8

hΩ (eV)

0

1

2

3

4

5

6

7

ξ 2(Ω

)(m/V

4s)

×104 Al0.35Ga0.65As

xxxxxyxyxxyy

Figure 4.4: Two-photon carrier injection coefficients for two different stoichiometries.

1.8 2.0 2.2 2.4

hΩ (eV)

0

1

2

3

4

5

6

7

ξ 3(Ω

)(m

3/V

6s)

×10−13 Al0.2Ga0.8As

xxxxxxxxxxyyxxyxxyxyzxyzxxyyzz

2.0 2.2 2.4 2.6 2.8

hΩ (eV)

0.0

0.5

1.0

1.5

2.0

2.5

ξ 3(Ω

)(m

3/V

6s)

×10−13 Al0.35Ga0.65As

xxxxxxxxxxyyxxyxxyxyzxyzxxyyzz

Figure 4.5: Interference of 2- and 3-photon carrier injection coefficients for two different stoi-chiometries.

4.4.1 Carrier injection

I track the number of injected carriers by calculating the number of electrons in the conduction

bands, which corresponds to the operator

N =∑ck

a†ckack, (4.31)


so we use ncc′ = δcc′ and nvv′ = 0 for the carrier density matrix elements in Eq. (4.26). The

optical injection of carriers due to 2PA and 3PA processes, as well as their interference, is

characterized by the tensors ξ, according to

ddt〈n〉2 = ξabcd

2 (3ω) Ea−3ω/2Eb

−3ω/2Ec3ω/2Ed

3ω/2, (4.32)

ddt〈n〉3 = ξ

abcde f3 (3ω) Ea

−ωEb−ωEc

−ωEdωEe

ωE fω, (4.33)

ddt〈n〉2+3 = ξabcde

2+3 (3ω) Ea−ωEb

−ωEc−ωEd

3ω/2Ee3ω/2 + c.c., (4.34)

where ~Ω = 3~ω is the total transition energy. If the band gap of the material is smaller than

5ω/2 there would be an additional contribution to n2 proportional to E−ωE−3ω/2EωE3ω/2, but

these carriers do not contribute to the interference between 2PA and 3PA, which is our main

interest. The coefficients are calculated as

ξabde2 (3ω) = 2π

∫dk

(2π)D

∑cv

Rab∗cvkRde

cvkδ (3ω − ωcv) , [three IC] (4.35)

ξabde f g3 (3ω) = 2π

∫dk

(2π)D

∑cv

Rabd∗cvk Re f g

cvkδ (3ω − ωcv) , [three IC] (4.36)

ξabde f2+3 (3ω) = 2π

∫dk

(2π)D

∑cv

Rabd∗cvk Re f

cvkδ (3ω − ωcv) [five IC]. (4.37)

The number of independent non-zero components of the tensors ξ2, ξ2+3, and ξ3 is deter-

mined by the symmetries of the zincblende lattice. The text within square brackets in Equa-

tions (4.35)–(4.37) indicates the number of nonzero, independent components (“IC”). I present

the complete list in the Appendix A.

In Figs. 4.4, 4.6 and 4.5, we show the frequency dependence of the independent compo-

nents of the coefficients ξabcd2 (3ω), ξabcde f

3 (3ω), and ξabcde2+3 (3ω), respectively. Notice that the

3PA coefficient is large for frequencies right above the band gap, while the coefficient for 2PA

nearly vanishes for similar frequencies. As discussed following Eqs. (4.24) and (4.25), the

dominant contribution to 2PA always involves intraband velocity matrix elements, which cor-

respond to the band dispersion, so they vanish at the Γ point of the Brillouin zone. The 3PA


1.8 2.0 2.2 2.4

hΩ (eV)

0.0

0.2

0.4

0.6

0.8

1.0

−ξ 2

+3(Ω

)(m

2/V

5s)

×10−5 Al0.2Ga0.8As

xxxyzxyyyzxyzxx

2.0 2.2 2.4 2.6 2.8

hΩ (eV)

0

1

2

3

4

5

6

7

−ξ 2

+3(Ω

)(m

2/V

5s)

×10−6 Al0.35Ga0.65As

xxxyzxyyyzxyzxx

Figure 4.6: Interference of two- and three-photon carrier injection coefficients for differentstoichiometries.

has contributions from interband velocity matrix elements, which in general do not vanish at

Γ.

1.8 2.0 2.2 2.4

hΩ (eV)

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

η 2+

3(Ω

)(A

m3/V

5s)

×10−17 Al0.2Ga0.8As

xxxxxxxxxxyyyxxxxyyxxyxxyxxyyyxxxyxyyyzzxxzxxyyzxxyzyz

2.0 2.2 2.4 2.6 2.8

hΩ (eV)

0.0

0.5

1.0

1.5

η 2+

3(Ω

)(A

m3/V

5s)

×10−17 Al0.35Ga0.65As

xxxxxxxxxxyyyxxxxyyxxyxxyxxyyyxxxyxyyyzzxxzxxyyzxxyzyz

Figure 4.7: Two- and three-photon interference current injection coefficients for differentstoichiometries.


4.4.2 Current injection

The injected current density 〈J〉 due to the quantum interference between two- and three-

photon absorption processes is given by

ddt〈Ja〉2+3 = η

abcde f2+3 (3ω) Eb

−ωEc−ωEd

−ωEe3ω/2E f

3ω/2 + c.c., (4.38)

where ~Ω = 3~ω is the total photon energy. I list the independent components of the injection

tensor coefficient ηabcde f2+3 (3ω) in Appendix A. In Fig. 4.7 I show the frequency dependence of

the independent components of the coefficient ηabcde f2+3 (3ω) for different stoichiometries. The

plots show that some components change sign as the frequency increases. This is due to the

competing contributions due to intraband and interband velocity matrix elements to the R(3)cvk

coefficients. For low excess photon energies, the excited carriers are close to the Γ point in the

BZ, and the interband contribution is the most important, as the band dispersion is small. For

larger photon excess energies, the excited carriers are located further from the Γ point in the

BZ, so the band dispersion is large and the intraband contributions are more important.

0.0 0.5 1.0 1.5 2.0

θ/π

−3

−2

−1

0

1

2

3

η 2+

3(Ω

)(A

m3/V

5s)

×10−17 Al0.2Ga0.8As

e3ω/2 = θ

eω = θ

0.0 0.5 1.0 1.5 2.0

θ/π

−3

−2

−1

0

1

2

3

η 2+

3(Ω

)(A

m3/V

5s)

×10−17 Al0.2Ga0.8As

e3ω/2 = eω

e3ω/2 = z × eω

Figure 4.8: Coefficient for injection current along the x direction as the polarizations of theincident fields are rotated in the x-y plane. (Top) Either eω or e3ω/2 is rotated while the otheris fixed along the x direction. (Bottom) In both cases eω = θ is rotated, and e3ω/2 is eitherparallel or perpendicular to it. The total photon energy is ~Ω = 2.4 eV (≈ 520 nm) in bothcases.

In order to illustrate some aspects of the different tensor components, in Fig. 4.8 I plot


the injection current for different polarizations of the incident fields in a typical experimental

scenario. We assume that the sample has electrodes mounted such that they always measure

the current along the [100] crystal direction, which we denote by x. In the first case we keep

either eω or e3ω/2 fixed along the x direction, while the other field is rotated in the x-y plane

and points along the direction θ = x cos θ + y sin θ, where y corresponds to the [010] crystal

direction. In the second scenario, the polarizations of both incident fields are rotated in the

x-y plane and they are kept either parallel or perpendicular to each other. In Fig. 4.8, we show

that the current is largely along the eω direction regardless of the e3ω/2 direction. However,

the magnitude of the current depends significantly on the e3ω/2 direction, and it is maximal for

e3ω/2 = eω.

1.8 2.0 2.2 2.4

hΩ (eV)

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Vsw

arm

(m/s

)

×106 Al0.2Ga0.8As

e3ω/2 = eω

e3ω/2 = z × eω

2.0 2.2 2.4 2.6 2.8

hΩ (eV)

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

Vsw

arm

(m/s

)

×106 Al0.35Ga0.65As

e3ω/2 = eω

e3ω/2 = z × eω

Figure 4.9: Swarm velocity assuming optimum interference between 2PA and 3PA. Thecurrent is measured along the x direction, and so is the polarization of the lower frequencyfield eω = x, while we consider two cases for the polarization of the higher frequency field:e3ω/2 = x and e3ω/2 = y.

Swarm velocity

Since the excited carriers respond to the induced voltage due to the injected current, and usually

screen it at least partially, a good measure of the efficiency of the current injection is the swarm


velocity, defined as

vswarm =

ddt 〈J〉

e ddt 〈n〉

, (4.39)

which represents the average contribution to the injection current due to one excited electron.

Since 〈n〉2+3 〈n〉2 + 〈n〉3, the total density of carriers is 〈n〉 ' 〈n〉2 + 〈n〉3, and for both light

beams polarized along the x direction we have a swarm velocity of magnitude

vswarm =2∣∣∣ηxxxxxx

2+3 (3ω)∣∣∣ E3

ωE23ω/2

|e|(ξxxxxxx

3 (3ω) E6ω + ξxxxx

2 (3ω) E43ω/2

) , (4.40)

where we have chosen ∆φ = π/2 to optimize the magnitude of the numerator. The whole ex-

pression is optimized by choosing the intensities of the two beams appropriately; the condition

to be satisfied is ξxxxx2 (3ω) E4

3ω/2 = ξxxxxxx3 (3ω) E6

ω , which corresponds to an equal number of

carriers injected by 2-photon absorption and 3-photon absorption. If this holds,

vswarm =

∣∣∣ηxxxxxx2+3 (3ω)

∣∣∣|e|

√ξxxxxxx

3 (3ω) ξxxxx2 (3ω)

. (4.41)

In Fig. 4.9 I plot this expression, together with the expression that would result if the beam

of frequency ω were polarized in the x direction while the one of frequency 3ω/2 in the y

direction, which is the same as Eq. (4.41) but with ηxxxxxx2+3 (3ω) replaced by η

xxxxyy2+3 (3ω); as

well, ξxxxx2 (3ω) should also be replaced by ξyyyy

2 (3ω), but they are equal. We see that different

stoichiometries give similar values for the swarm velocity if the frequency is adjusted according

to the electronic band gap of the system. The fact that higher Ga concentrations lead to larger

injected currents (see Fig. 4.7) is only due to a higher carrier injection. Yet with appropriate

laser intensities it is possible to reach the same levels of injected current densities with any Al

concentration, although the laser frequencies and intensities at which the maximum is achieved

depend on the Al concentration.

Notice that the 2+3 QuIC swarm velocity is about twice its equivalent for 1+2 QuIC. This

is an indication that the distribution of carriers injected in the BZ is sharper for 2+3 QuIC com-


pared to 1+2 QuIC. We further confirm that by computing the variance of the lattice momentum

k of the electrons injected in the conduction band for both 1+2 QuIC and 2+3 QuIC,

σa1+2 =

⟨(ka)2

⟩1+2− 〈ka〉

21+2 , σa

2+3 =⟨(ka)2

⟩2+3− 〈ka〉

22+3 (4.42)

respectively. For the incident fields polarized along the x direction, we find

〈k〉1+2 = (4.9, 0, 0) × 10−2Å−1, 〈k〉2+3 = (5.7, 0, 0) × 10−2Å

−1, (4.43a,b)

σ1+2 = (3.4, 4.3, 4.3) × 10−3Å−2, σ2+3 = (2.8, 2.2, 2.2) × 10−3Å

−2, (4.43c,d)

which indicates that the distribution of injected electrons in the BZ is sharper for 2+3 QuIC,

especially in the directions transverse to the polarization of the field (y and z).

Laser intensities

Our calculations are performed in the perturbative regime, the validity of which requires that

the fraction of the injected carrier population density relative to the total density of states nmax

in the range of energies covered by the laser pulse be small. We thus consider these calculations

to be valid when

〈n〉2 + 〈n〉3 < 0.1 nmax, (4.44)

where the fraction 0.1 is chosen arbitrarily, as an estimate of unsaturated regime. The carrier

injection due to the 2- and 3-photon interference 〈n〉2+3 mostly has the effect of concentrating

the carrier injection in some region of the BZ, but it does not contribute significantly to the total

number of injected carriers compared to 〈n〉2 and 〈n〉3. For the estimates of laser intensities we

consider the incident fields to be both polarized along the x direction, so for a laser pulse of

duration T we require

[ddt〈n〉2 +

ddt〈n〉3

]T < 0.1nmax, (4.45)


[ξxxxx

2 (3ω) E43ω/2 + ξxxxxxx

3 (3ω) E6ω

]T < 0.1nmax. (4.46)

The maximum density of states nmax that can be injected is determined by analyzing the vol-

ume V corresponding to the excited states in the BZ. We denote by kΩ the momentum cor-

responding to the energy difference ~ωcvk = ~Ω between the conduction and valence bands,

so V = 4πk2Ω∆k, where ∆k = dk

dωcvk∆ωcvk is related to the frequency broadening ∆ω = 2π/T

associated with the time duration of the pulse. The derivative of the band energy corresponds

to the velocities of electrons in the conduction and valence bands, vΩ = dωcvdk = dωc

dk −dωvdk , so

V = 8π2k2Ω/(vΩT ). The volume in the BZ associated with one quantum state is V1 = (2π/L)3,

where L is the normalization length of the sample. The number of states that can be excited is

then V/V1, and their spatial density is

nmax =V

V1L3 =k2

Ω

πvΩT. (4.47)

For optimal interference, there should be equal densities of carriers injected by 2- and 3-photon

absorption, 〈n〉2 = 〈n〉3, which according to Eq. (4.46) gives

ξxxxx2 (3ω) E4

3ω/2 = ξxxxxxx3 (3ω) E6

ω < 0.05k2

Ω

πvΩT2 . (4.48)

The maximal amplitudes Eω and E3ω/2 of the incident fields can then be estimated from the

extreme of the inequality in the above equation. For the stoichiometry of α = 0.2, pulses with

duration T = 150 fs, and total photon energy ~Ω = 2.4 eV, we have

Eω = 1.24 × 108 Vm, and E3ω/2 = 6.05 × 107 V

m. (4.49)

The intensities in the material medium with these field amplitudes are given by Iϕ = (2ε0c) nϕ E2ϕ,

with ϕ =ω, 3

2ω,

Iω = 26.5GWcm2 , I3ω/2 = 6.54

GWcm2 . (4.50)


For these values, the injected current density is

〈Jx〉 = 2ηxxxxxx2+3 (3ω) E3

ωE23ω/2T = 6.25

MAcm2 . (4.51)

It should be noted that these are just estimates, as the limit of carrier density is set arbitrarily

in Eq. (4.44). Note that we are ignoring scattering of the injected carriers. This means that

the true maximal intensities would be larger than our estimates here, since there is room for

more photon absorption as scattering depletes some of the excited states. Also note that in

this treatment the electron-electron interaction has been neglected; were it included, the phase

parameter would be shifted. However this shift is usually very small for zincblende semicon-

ductors, except for frequencies very close to the band gap [131].


QuIC using 2- and 3-photon processes leads to carrier excitation even more localized in the

BZ than 1+2 QuIC. One consequence of this is the fact that the swarm velocity is higher in

2+3 QuIC than in 1+2 QuIC. Another interesting difference is that in 2+3 QuIC the current

injection coefficients tend to change sign as the total photon energy is increased, while in 1+2

QuIC they typically do not. This happens because interband velocity matrix elements are

responsible for the largest contribution to the 3PA coefficient at low photon energies, but at

higher photon energies the intraband velocity matrix elements dominate. As well, for a given

material 2+3 QuIC can result from a larger set of laser frequencies than 1+2 QuIC. Since

only nonlinear optical processes are involved in 2+3 QuIC, the laser intensities required for

maximal effect are higher than for 1+2 QuIC, but still moderate. Also, the fields have a power

law attenuation as they propagate through the absorbing material, instead of the exponential

attenuation of linear absorption. Thus a waveguide geometry is desirable. And while QuIC in

waveguides presents some challenges, as it raises issues of phase and mode matching, it also

presents opportunities as it would allow for easy integration with devices on-chip.

Chapter 5

Conclusions

Effective models to compute electronic and optical properties of semiconductor materials are

of fundamental and practical interest; these simple models enable researchers to study physical

properties at low computational cost, and also provide the simplicity that allows for a focus on

particular phenomena. The study of reduced dimensionality materials is crucial not just from

a fundamental point if view, but also from a practical perspective; as progress on the minia-

turization of technology continues, and as more complex on-chip devices become feasible, the

study of quantum confinement effects becomes more relevant. An important area of research

in this field is the study of optical processes in semiconductor materials. The relevance of this

topic is only expected to grow as opto-electronic devices continue to become ubiquitous; as

these gadgets tend towards nanometric dimensions, the study of the optoelectronic properties

becomes crucial, since quantum confinement strongly modifies the (joint) density of electronic

states accessible to a given photon energy. In this thesis I have presented a study of reduced-

dimensionality semiconductor materials employing effective Hamiltonian models; I presented

a study of narrow strips of monolayer graphene (1D), of monolayer stanene (2D) and of the

alloy AlαGa1−αAs (3D). I presented calculations of linear and nonlinear optical response, in-

cluding charge-carrier injection by one- and two-photon absorption processes, spin-injection,

and the coherent control of injected charge current. Some of these results were contrasted

against first-principles calculations based on density-functional theory. In the following lines

103

Chapter 5. Conclusions 104

I present the relevant findings of these studies and conclude this chapter with an outline of

possible routes for future research.

In Chapter 2 I presented a study of narrow strips of graphene, commonly known as graphene

nanorribons. Electrons are confined along two dimensions, i.e., periodicity is unidimensional.

I confirmed that the electronic dispersions strongly depend on the boundaries of the graphene

strip. I focused on zigzag boundaries, because they led to strongly confined edge states along

the zigzag borders; these states are found to be precisely at the Fermi level for an undoped

material, and as such, the large number of occupied states per unit length allocated in the edge

states is easily controlled by doping. Another important characteristic of this system is that

—within the k · p framework— the bandstructure is entirely composed of parabolic-like en-

ergy bands, aside from the edge-states. These parabolic bands lead to a rich structure due to

singularities in the joint-density-of-states (JDOS). This was followed by a detailed study of

the velocity matrix elements, in which their selection rules were identified. Then I presented

the one- and two-photon carrier injection coefficients, together with their respective selection

rules. Similarly to graphene, the zigzag ribbons support stimulated electronic Raman scatter-

ing (ERS), which in general is present when the fundamental photon energy is larger than the

optical bandgap; this scattering process is characterized by virtual absorption at 2~ω followed

by emission at ~ω. In general, due to the boundary conditions, and the uni-dimensional peri-

odicity of the ribbons, different nonzero optical coefficients have different selection rules, in

contrast with a monolayer of graphene.

One of the key findings in this chapter is that, aside from the JDOS divergencies present

in all the optical coefficients studied here, the second order absorption processes (2PA and

ERS) also possess a class of divergences associated with resonant conditions, when the optical

process in consideration connects real states. Once these absorption processes were analyzed,

I proceeded to study interference processes among them. This interference process leads to

current injection of charge carriers. First I studied current injection arising from the interfer-

ence of sequential absorption of two single photons with two-photon absorption, and then that

from the interference of ERS absorption with 1PA. Although all these absorption and current


injection coefficients inherit the rich structure of the JDOS, I found they follow a general trend

similar to that of 2D graphene.

Since the outstanding feature of zigzag nanoribbons is the existence of localized edge states

along their length and confined states across their width, in this work I identified the contri-

bution to the optical responses that involves edge states (edge ↔ edge and edge ↔ confined

transitions). This is referred here as the “only-edge” contribution, and it is thoroughly com-

pared with the “full” contribution that originates from any combination of optical transitions

(“only-edge” plus confined↔ confined). From this comparison I found that the edge states are

always involved in the onset of the absorption and the current injection processes, and that at

low-photon energies the edge states have the largest contributions. Since the energy of these

edge states is near zero, this suggested a study of the effects of positive doping. I found that, at

low photon energies, all the optical coefficients of ZGNR are extremely sensitive to changes in

the chemical potential. Due to the relative ease with which the chemical potential of graphene

can be altered, zigzag nanoribbons offer an excellent opportunity to generate and control charge

currents by optical fields and by doping techniques. Moreover, due to the strong dependence of

the electronic bandstructure with respect to the width and boundary conditions (edge shape),

the study of graphene ribbons proves relevant from both fundamental and practical perspec-

tives.

In Chapter 3 I studied another graphene-like semiconductor material, stanene, a monolayer

of tin atoms arranged in a buckled honeycomb lattice. Stanene is a promising material for

fundamental studies and practical applications, given the topological properties and the signif-

icant spin-orbit interaction present in its electronic structure; moreover, monolayer stanene on

a substrate and free-standing few-layered stanene have recently been obtained experimentally.

My contribution in this chapter was the development of an effective Hamiltonian model that

accurately describes the bandstructure of stanene at low photon energies employing a minimal

basis set. This simple model is based on a singular value decomposition of overlap matrices

obtained in a preliminary first-principles calculation. This effective model resembles a tradi-

tional k · p method but it differs from the latter in that the electronic energies described by


the simple model presented here uses a minimal basis set and it is not restricted to a second

order dependence on the crystal momentum. I also presented a detailed analysis of the approx-

imations made, and I provided figures of merit for the eigenenergies and eigenstates obtained

with this simple model. During this analysis I found that a second order expansion of the

bandstructure energies is more accurate than higher order expansions on k for stanene; indeed,

the two-dimensional band warping of the bandstructure is well described by the second order

model, as compared with the band warping predicted by a first-principles calculation. Another

significant finding in this work was that the lattice buckling in stanene can be neglected for the

calculation of the optical responses presented in this third chapter; this simplification led to a

separation of the effective Hamiltonian matrix into spin subsectors. I also found that the model

predicts the coupling of circularly polarized light with the spin of the electron; this points to

a potential application of stanene for spintronic applications. Since the procedure described

in this chapter depends solely on a preliminary first-principles calculation, that is, since this

effective model is free of phenomenological input, I expect this method can be applied to study

other bidimensional materials.

After having studied mono- and bi-dimensional novel materials in Chapter 2 and Chapter

3, respectively, in Chapter 4 I focused my attention on a “traditional” tridimensional alloy

material, AlαGa1−αAs . Here I presented results for charge injection based on two- and three-

photon (2+3) absorption processes, and results of optical coherent control of charge current

based on the interference of these two color optical phenomena. Although a vast amount

of literature exist on this material, this class of properties has not yet been reported in the

literature. As in previous chapters, I performed this study using a k · p model, but here I used

a 30-band basis set and I also included the description of spin-orbit interaction, as I also did

on Chapter 3. In general terms, this chapter presents an example of bandstructure engineering

in which the objective was to study optical coherent control of charge currents for suitable

values of the stoichometric parameter α in AlαGa1−αAs . As compared with optical coherent

control based on one- and two-photon (1+2) absorption processes (reported elsewhere), the

study presented here showed that coherent control based on 2+3 processess occurs on a smaller


Brillouin Zone volume, and consequently the swarm velocities reported here are higher in this

latter case. The laser intensities required to maximize the injection of current based on 2+3

processes are higher as compared with 1+2 processes, since in the former case only non-linear

processes are involved.

Future Work

In this thesis I have presented studies of electronic structure and optical properties of some

semiconductors in the framework of simple model Hamiltonians. Although some of these

properties were contrasted with first-principles results, an obvious extension of this work could

include the calculation of all the optical coefficients from first principles. This extension

could still be restricted to a single-particle (non-interacting) picture (as was done here with

simple Hamiltonians) in the framework of density-functional theory and the local-density ap-

proximation (DFT/LDA). Yet a further level of sophistication could include the description of

many-body effects to describe the electron-electron interaction beyond the scope of exchange-

correlation pseudopotentials, as it is commonly done in the DFT/LDA. The best description

of these many-body effects to date is given by the GW approximation, which is commonly

performed with a preliminary DFT calculation of the ground-state wavefunctions. These DFT

wavefunctions are then used as a basis set to express the Green functions in the GW method.

Further sophistications can also include the description of electron-hole (neutral) excitations by

solving the Bethe-Salpeter equations. Another way to extend the work presented here is to em-

ploy time-dependent first-principles methods to describe time- and frequency-dependent quan-

tities; to this day time-dependent DFT is one of the most widely used schemes for this kind of

description. As time-dependent-DFT is a single-particle framework, it has a much lower com-

putational cost than the (many-body) GW method, however it faces important challenges, such

as the accurate description of time-dependent correlation effects and time-dependent structural

forces.

Extensions of the work done in this thesis for specific materials or specific material geome-

tries can include the study of nanoribbons made of monolayered materials other than graphene,


e.g., silicene, germananene, stanene, h-BN, and SiC, among others. Several of these monolayer

materials have recently been produced. The effective model presented in Chapter 3 could be

extended to study the optical coherent control of charge currents and the effects of external

DC fields. Finally, the study presented in Chapter 4 can be extended to study alloys other

than AlGaAs and by studying waveguide heterostructures to for potential applications of the

optically-injected charge currents.

Appendix A

Nonzero injection coefficient components

of zincblende lattices

AlGaAs in the virtual crystal approximation forms a zincblende lattice, which has the sym-

metry of point group Td (or 43m). The optical responses we consider in this work involve

tensors of rank 4 up to 6. With Td symmetries [132], generic rank-4 tensors have 21 non-zero

components of which 4 are independent, rank-5 tensors have 60 non-zero and 10 independent

components, and rank-6 tensors have 183 non-zero and 31 independent components.

However, the tensors representing the optical processes have a few more specific restric-

tions due to their relation to the optical fields, as the indices associated with the same incident

field are symmetrized. With these considerations the number of nonzero independent compo-

nents are

Number of

Tensor independent components

ξ2 three

ξ3 five

ξ2+3 three

η2+3 nine

109

Appendix A. Nonzero injection coefficient components of zincblende lattices 110

The complete list is as follows. ξ2 has three independent components

ξxxxx2 = P (x, y, z) , ξ

xyxy2 = ξ

xyyx2 = P (x, y, z) , ξ

xxyy2 = P (x, y, z) , (A.1a,b,c)

where P (x, y, z) indicates all the possible permutations of (x, y, z) in the indices. The tensor

ξ2+3 has three independent components

ξxxxyz2+3 = P (x, y, z) , (A.2a)

ξxxyxz2+3 = ξ

xxyzx2+3 = ξ

xyxxz2+3 ξ

xyxzx2+3 = ξ

yxxxz2+3 = ξ

yxxzx2+3 = P (x, y, z) , (A.2b)

ξxyzxx2+3 = ξ

yxzxx2+3 = ξ

yzxxx2+3 = P (x, y, z) , (A.2c)

while ξ3 has five independent components

ξxxxxxx3 = P (x, y, z) , (A.3a)

ξxxxxyy3 = ξ

xxxyxy3 = ξ

xxxyyx3 ξ

yyxxxx3 = ξ

yxyxxx3 = ξ

xyyxxx3 = P (x, y, z) , (A.3b)

ξxxyxxy3 = ξ

xxyxyx3 = ξ

xxyyxx3 = ξ

xyxxxy3 = ξ

xyxxyx3 = ξ

xyxyxx3

= ξyxxxxy3 = ξ

yxxxyx3 = ξ

yxxyxx3 = P (x, y, z) , (A.3c)

ξxxyyzz3 = ξ

xxyzyz3 = ξ

xxyzzy3 = ξ

xyxyzz3 = ξ

xyxyzyz3 = ξ

xyxzzy3

= ξyxxyzz3 = ξ

yxxyzyz3 = ξ

yxxzzy3 = P (x, y, z) , (A.3d)

ξxyzxyz3 = ξ

xyzzxy3 = ξ

xyzyzx3 = ξ

xyzzyx3 ξ

xyzxzy3 = ξ

xyzyxz3 = P (x, y, z) . (A.3e)

Appendix A. Nonzero injection coefficient components of zincblende lattices 111

Finally, the tensor η2+3 has nine independent components

ηxxxxxx2+3 = P (x, y, z) , (A.4a)

ηxxxxyy2+3 = P (x, y, z) , (A.4b)

ηxxyyxx2+3 = η

xyxyxx2+3 = η

xyyxxx2+3 = P (x, y, z) , (A.4c)

ηxxxyxy2+3 = η

xxxyyx2+3 = η

xxyxxy2+3 = η

xxyxyx2+3 = η

xyxxxy2+3 = η

xyxxyx2+3 = P (x, y, z) , (A.4d)

ηyxxxxy2+3 = η

yxxxyx2+3 = P (x, y, z) , (A.4e)

ηyxxyxx2+3 = η

yxyxxx2+3 = η

yyxxxx2+3 = P (x, y, z) , (A.4f)

ηxxyyzz2+3 = η

xyxyzz2+3 = η

xyyxzz2+3 = P (x, y, z) , (A.4g)

ηxxyzyz2+3 = η

xxzyyz2+3 = η

xyxzyz2+3 = η

xyzxyz2+3 = η

xzxyyz2+3 = η

xzyxyz2+3 = P (x, y, z) , (A.4h)

ηxyyzxz2+3 = η

xyzyxz2+3 = η

xzyyxz2+3 = η

xyyzzx2+3 = η

xyzyzx2+3 = η

xzyyzx2+3 = P (x, y, z) . (A.4i)

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