An Equation of Motion Approach to the Calculation of Optical Responses in Semiconductors Including Excitonic Effects

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An Equation of Motion

Approach to the

Calculation of

Optical Responses in

Semiconductors

Including Excitonic

Effects

Carlos Diogo Monteiro Fernandes

Mestrado em Física

Departamento de Física 2018

Orientador

João Manuel Borregana Lopes dos Santos, Professor Catedrático, Faculdade de Ciências

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Todas as correções determinadas pelo júri, e só essas, foram

efetuadas.

O Presidente do Júri,

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Master's Thesis

An Equation of Motion Approach to the Calculation of Optical Responses in Semiconductors Including Excitonic Eects

Author: Carlos Diogo Monteiro Fernandes

21st December 2018

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Acknowledgements

The most important scientic contributor this thesis was was my supervisor Professor João Lopes dos Santos, who suggested this theme and helped to clarify much of the discussion in this thesis. I would also like to thank the rest of our condensed matter physics group for providing helpful support and insight. It is also imperative for me to thank my parents José Carlos de Melo Fernandes and Laura da Conceição Gonçalves Monteiro for raising me and for their sacrices in paying for my studies without which I could have never written this thesis.

The authors acknowledge nancing of Fundação da Ciência e Tecnologia, of COMPETE 2020 program in FEDER component (European Union), through projects POCI-01-0145-FEDER-02888 and UID/FIS/04650/2013.

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Reconhecimentos

A contribuição cientíca mais importante para a elaboração desta tese foi a do Professor Doutor João Lopes dos Santos, que foi o responsável pela sugestão deste tema e ajudou à claricação de vários temas discutidos nesta tese. Também gostaria de agradecer ao resto do grupo de física de matéria condensada do Centro de Física do Porto pelo seu apoio e contribuições. É também imperativo agradecer apos meus pais José Carlos de Melo Fernandes e Laura da Conceição Gonçalves Monteiro por me criarem e sustentarem durante todos os meus estudos e sem os quais eu nunca poderia ter escrito esta tese.

Os autores reconhecem o nanciamento da Fundação da Ciência e Tecnologia, do programa COMPETE 2020 na componente FEDER (União Europeia), através dos projetos POCI-01-0145-FEDER-02888 e UID/FIS/04650/2013.

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Abstract

Ever since Geim and Novoselov rst isolated graphene in 2004, there has been great interest in studying and characterising the properties of 2D materials. One important property is the material's response to an applied external electric eld, which is described by its conductivity.

In the University of Porto our condensed matter physics group has been studying a formalism which allows us to calculate these conductivities in semiconductors with particular interest in the nonlinear response. This formalism is known as the reduced density matrix formalism. This has been the subject of a prior master's thesis [1] in this group and of two published articles [2, 3].

While in our previous work we neglected the electron-electron interaction, the objective of this thesis is treating the problem of including these eects in the calculation of the optical conduct-ivities in the density matrix formalism. We start by an introduction to the physics of excitons in semiconductors and the properties of nonlinear responses. Then we show how to include this term in our model and that it leads to an eective single particle Schrödinger equation which must be solved. We identify the new excitonic resonances predicted by our models and derive an expression for the second order nonlinear conductivity. We also investigate the issue of gauge invariance on this formalism and explicitly demonstrate this equivalency in rst order.

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Resumo

Desde que Geim e Novoselov isolaram pela primeira vez uma camada de grafeno em 2004, tem havido um grande interesse no estudo e na caracterização das properiedades dos materiais bidimensionais. Uma propriedade importante é a resposta do material mediante a aplicação de um campo elétrico externo, que é descrita pela condutividade do material.

O grupo de matéria condensada da Universidade do Porto tem estudado um formalismo que permite o cálculo dessas condutividades em semicondutores com interesse particular na resposta não linear. A este formalismo dá-se o nome de formalismo da matriz de densidade reduzida (reduced density matrix). Este foi o assunto de uma tese de mestrado anterior [1] e de dois artigos já publicados [2, 3].

Enquanto que nos trabalhos anteriores se ignoraram as interações eletrão-eletrão, o objetivo desta tese é tratar o problema da inclusão desses efeitos no cálculo das condutividades óticas no formalismo da matriz de densidade reduzida. Neste trabalho começamos por uma introdução à física dos ex-citões em semicondutores e às propriedades da resposta não linear. Depois mostramos como incluir este termo no nosso modelo e como se chega a uma equação de Schrödinger efetiva para uma única partícula. Conseguimos identicar as novas ressonâncias excitónicas previstas pelos nossos mode-los e derivamos uma expressão para a condutividade não linear de segunda ordem. Investigamos também a questão da invariância de padrão (gauge invariance) e demonstramos explicitamente esta equivalência para primeira ordem.

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1 Introduction to the Physics of Excitons in

Semiconductors

Semiconductors are a class of solid crystalline materials with wide industrial applications. In band theory of solids, they are dened as materials having a Fermi energy situated in the middle an energy gap. At low temperatures, pristine, intrinsic semiconductors are insulators with a set of fully occupied states called the valence states and a set of empty states called the conduction states which are separated from the former [4, 5, 6, 7].

The optical response of these materials is dominated by the excitation of electrons from the valence states to the conduction states. Such a transition can be described by the creation of a pair of eective particles (quasiparticles), one being a conduction electron which carries the same negative charge of an electron, and the other being a hole in the valence band which behaves as a particle with the opposite charge of an electron. They can be described by their eective masses which will be dierent from (and, in semiconductors, generally lower than) the free electron mass. The electron and hole masses are inversely proportional to the curvature of the conduction and valence bands respectively [6, 7].

Semiconductors are transparent insulators at low temperatures for low frequencies as they do not have particles able to carry charge [7]. For frequencies above the band gap it is possible to excite a transition from the valence to the conduction band, meaning that absorption becomes nite. When we measure the absorption of certain materials at low temperatures, we can observe absorption peaks below the band gap. These peaks can be explained by the presence of exciton states in our material [6].

As the electron and hole are oppositely charged particles there will naturally be a Coulomb interaction between them. This lead to the possibility of formation of bound states of these particles in analogy with the hydrogen atom orbitals formed by an electron and a proton. These excitations have large oscillator strengths in their coupling with the electric eld E and narrow linewidths sparking interest in their possible application in semiconductor devices. Also, since excitons are bosonic particles they should be able to form Bose-Einstein Condensates and exhibit superuid behaviour [4, 8]. Unfortunately, the large screening and small mass of these particles means the exciton binding energy in bulk materials is in the meV scale, which is small compared to the energy of thermal uctuations at room temperature, which is about 25 meV. These uctuations drown out excitonic eects unless the material is suciently cooled [4, 9].

Excitons in condensed matter can be categorized depending on their binding energy and extent relative to the unit cell. In molecular crystals and certain ionic solids such as NaCl and LiF, which have small dielectric constants ( ≈ 1), the electron-hole coupling is strong, having binding energy

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1 Introduction to the Physics of Excitons in Semiconductors

in the order of 1 eV, much higher than the thermal energy at room temperature of around 25 meV. Their size also tends to be small having an extent comparable to the size of a unit cell. These excitons are called Frenkel excitons. For semiconductors, which have large dielectric constants ( ≈ 5 − 10), their size is large compared to the unit cell and they are named Wannier-Mott excitons [5]. Our treatment in this text is applicable to the second category.

The beginning of the 21st century witnessed the development of methods allowing the isolation of 2D materials in quantities allowing their study. Semiconductor materials for which screening is reduced, like transition metal dicalchogenides (TMDs) and phosphorene, with exciton binding energies in the order of 100 meV, made excitons more accessible to experimental study and more viable for application in the design of materials, as they are observable even at room temperature, renewing interest in possible engineering applications [9, 10]. For example, in WSe2 (tungsten

diselenide) on a fused quartz substrate, two excitonic peaks at 1.65 and 2.08 eV were observed in the linear absorption (gure 1.1) [11]. Further peaks were identied by analysing the second derivative of the absorption. It was also concluded that the positions of the peaks were inconsistent with simple Coulomb interactions, showing the importance of screening in the electron-electron interaction of those materials (gure 1.2). Similar eects are also observed for WS2 [12]. One commonly used

potential which incorporates screening is the Keldysh potential [13]. In Fourier space, it takes the form v(q) = −e/(2q(r0q + εm)), where εm is the dielectric function of the environment and r0 is a

material dependent parameter. For r0 → 0this potential reduces to the Coulomb potential1.

Figure 1.1: Plot of the linear absorption (in red) and two photon absorption (blue dot) spectrum for monolayer WSe2 with a gap at 2.08 eV, experimental results. [11]

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Figure 1.2: Contrast between the experimentally observed exciton lines (left panel) and the 2D coulomb energy levels for monolayer (right panel) WSe2. [11]

An interesting property of these materials are that under illumination with a intense laser pulse they can generate light at frequencies dierent from the original frequencies. A particularly import-ant and widely studied eect is that of second harmonic generation, where the emitted wave has double the frequency of the incident one. This property is a consequence of the nonlinearity of the material response. In order to observe this eect, it is necessary to either generate large electric elds or to use materials with large second order response; the former is achievable by ultrafast laser pulses, while the latter is a property of the 2D materials we have mentioned [2].

In order to calculate these eects the condensed matter group in our University of Porto studied a theory that permits the calculation of these and higher order quantities, based on the equation of motion of a free electron in a lattice [2]. In such a theory, there are two possible choices on how to describe the interaction of the electron with the external eld, depending on whether the eld is derived from a scalar φ or vector A potential. The results take a markedly dierent analytical form, so that proving their equivalence is a decidedly non trivial task. Furthermore, their equivalence depends on the validity of certain sum rules, which can violated in the employment of various approximation schemes if appropriate care is not taken. Later, with our formalism, we were able to conrm their equivalence of both methods analytically and numerically [3].

If we are to accurately describe the response of materials like the aforementioned TMDs, we must include electron-electron interaction in our model. Papers dealing with the theory of nonlinear reponse including excitonic eects are scarce. In 2015 Pedersen published a paper explaining how to extend previous theory to include excitonic eects [14], and later by May 2018 claried the equivalence between the length and velocity gauges when including excitonic eects, and numerically demonstrated their equivalence [15] (see g. 1.3). These papers are discussed on chapters 4 and 5 of this

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thesis.-1 Introduction to the Physics of Excitons in Semiconductors

Figure 1.3: Plot of computed excitonic second harmonic generation spectrum in a two band model of a hBN monolayer obtained from both gauges. The blue line represents two calculations A and B from the length gauge, while the circles C and crosses D represent results from the velocity gauge. The conductivity spectrum in the independent particle limit is shown as the lled blue curve. The red line C' represents a calculation done in a naïve velocity gauge calculation. The dotted lines are the resonances present in the independent particle limit [15].

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2 Mathematical Description of Optical Linear

and Nonlinear Response

2.1 Introduction

The simplest experiment we can make consists of exerting on a system some external inuence we can quantify, and measure the change in the properties of that system as a result of that inuence. This chapter will introduce the formalism we shall subsequently use in for the rest of this thesis. We will follow Boyd's Nonlinear optics as a reference[16].

Optical or electrical experiments measure how the electric charges and currents develop in a material under an applied electromagnetic eld.

In general we write,

Jind(t, x) = FE(t0, x0), B(t0, x0) , (2.1)

where F is some functional. This can be a very complicated relation, but we know that there are systems which can be described by simpler expressions. One such case is Ohm's law, where the relation Jind(t, x) and E(t0, x0)is a relation of proportionality.

Jind(t, x) = σE(t, x) (2.2)

In this thesis we will consider systems for which the induced current at a point depends only on the local electric eld at that point, but at dierent times t, t0_.

Jind(t, x) = F E(t0, x)

(2.3) If we are working in the long wavelength limit, we can ignore the spatial dependence and consider only the time dependence. This is especially reasonable for 2D materials under normal incidence.

Jind(t) = FE(t0)

(2.4) In most cases E we can be considered a small perturbation an we expand the relation around E(t) = 01. Jind(t) = J(0)ind(t) + Z dt1R(1)(t, t!)E(t1) + Z dt1dt2R(2)(t, t2, t1)E(t2)E(t1) + ... (2.5)

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2 Mathematical Description of Optical Linear and Nonlinear Response

This expression can also be more conveniently written in Fourier space. We adopt the following conventions for the Fourier transform and inverse Fourier transform in time and frequency domains:

f (ω) ≡ Z dtf (t)eiωt, (2.6) f (t) = Z _dω 2πf (ω)e −iωt_. _(2.7)

Using our conventions and the equality R dteiωt _{= 2πδ(ω)}_{, we can write the Fourier Transform of}

(2.5): Jind(ω) = J(0)_ind(ω) + Z dω1 2π R (1)_{(ω, ω} 1) · E(ω1) + Z dω1 2π dω2 2πR (2)_{(ω, ω} 2, ω1) · E(ω2)E(ω1) + ... (2.8)

Notice that both Jind and E are vector quantities this means that R(n)(ω, ω1, ..., ωn) is an n + 1

order tensor. In contracted index notation (2.5) can be written as

J_indα (ω) = J_ind(0)α(ω) + Z _dω 1 2π R (1)αβ (ω; ω1)Eβ(ω1) + Z _dω 1 2π dω2 2π R (2)αβγ_{(ω; ω} 2, ω1)Eβ(ω2)Eγ(ω1) + ... (2.9) .

2.2 Current and Polarization Descriptions

In general there are two equivalent approaches to the calculation of the optical response of a medium. The rst consists on calculating the induced currents as a function of the electric eld. This is our approach and basically requires determining the average equilibrium velocity of the charge carries as a function of the electric eld. The alternative is to calculate the induced electric dipole density in the material and requires the determination of the average displacement from the equilibrium position of charge carriers under an applied electric eld. It is then easy to conclude that these are related simply by a time derivative.

In the polarization approach the expansion in the electric eld is written as:

P(ω) = P(0)(ω) + Z _dω 1 2π χ (1)_{(ω; ω} 1)E(ω1) + Z _dω 1 2π dω2 2π χ (2)_{(ω; ω} 2, ω1)E(ω2)E(ω1) + ... (2.10)

The relation between the two approaches is: dP(t)

dt = Jind(t) (2.11) or in Fourier space:

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2.3 Symmetry Restrictions to the Form of Response Functions This allows us to relate the optical susceptibilities and optical conductivities in all orders:

iωχ(n)(ω; ω1, ..., ωn) = R(n)(ω; ω1, ..., ωn). (2.13)

Then we can easily pass from one to the other formalism.

2.3 Symmetry Restrictions to the Form of Response Functions

Having developed a formalism to describe the optical responses, we will now place restrictions on our quantities arising from the nature of our media and elds. These restriction are important in that they reduce the number of independent quantities we need to calculate.

2.3.1 Reality of the Fields

The rst restriction is due to the fact that the electric eld and current are real quantities. This means:

J(ω) = J(−ω)∗ (2.14)

E(ω) = E(−ω)∗, (2.15) which implies:

R(n)(ω; ωn, ..., ω1) = R(n)(−ω; −ωn, ..., −ω1)∗. (2.16)

2.3.2 Intrinsic Permutation Symmetry

If we look at (2.9) we can see that we can freely rearrange the E factors, that is, if we take X α1...αn Z _dω 1 2π... dωn 2π R ααn...α1_{(ω; ω} n, ..., ω1)Eαn(ωn)...Eα1(ω1) (2.17)

and permute the {1, 2, ..., n} indices in the electric elds around, the result of the integration is the same.

This can be seen by noticing that the αi and ωi are dummy indices, we can rewrite them to get

the same permutation in the conductivity, which means that these indices are permuted around in the expression of the conductivity without changing the resulting current, this is the intrinsic permutation symmetry. This symmetry allow us to write our conductivities such that its form remains the same under permutations of the indices. For example, in the second order conductivity this would look like:

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It should be noted that this is simply a matter of convention. If our function does not have that symmetry then it is the sum of a symmetric part and a non-symmetric part, and the contribution to the integral in 2.17 from the non-symmetric part is zero.

2.3.3 Time Translation Symmetry Considerations

The rst symmetry consideration we take is that for many systems the optical response does not depend in the choice of time origin, meaning

R(n)(t; t1, ..., tn) = R(n)(t − t0; t1− t0, ..., tn− t0) (2.19)

for any t0_{. By choosing t}0 _{= t} _{the expression for the optical conductivity at order n can be written}

as R(n)(ω; ω1, ..., ωn) = Z dtdt1...dtne−i(ω−ω1−...)tei P iωi(ti−t)_...R(n)_{(0; t} 1− t, ..., tn− t). (2.20)

This means that the frequency response will be proportional to a Dirac delta function R(n)(ω; ω1, ..., ωn) ∝ 2πδ(ω −

n

X

i

ωi). (2.21)

We then write the conductivity

R(n)(ω; ω1, ..., ωn) = 2πδ(ω − n

X

i

ωi)σ(n)(ω1, ..., ωn). (2.22)

This, therefore, imposes that the frequency of the current is a sum of each frequency of the eld. This a quite severe restriction on the optical response of the material, eliminating one degree of freedom.

2.3.4 Linear Kramers-Kronig Relations

This is a condition that appears when we require that the response is causal, this is, the response at a time t should depend only on the eld at times before t. Formally this requires:

R(n)(t; t1, ..., tn) = 0 if t < t1or t < t2, ..., or t < tn (2.23)

We will now see the implications of causality for the linear response of time translation invariant systems:

σ(ω) = Z ∞

0

dtR(t, 0)e−iωt. (2.24) From this expression we try to analytically continue our function to complex values of ω, and we see that when ω has a positive imaginary part the integrand decays exponentially in time, therefore

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2.3 Symmetry Restrictions to the Form of Response Functions the integral converges and is analytic in the upper half2_{of the complex plane. Because of analyticity,}

we know that the real and imaginary part of the conductivity are not independent as they must full the Cauchy-Riemann equations of complex analysis. With this in mind, we will now provide a method of relating this two quantities. We can use the method of contour integration of the function σ(ω0_)/(ω0_{− ω)}_{over a semicircle on the upper half plane (see 2.1) to argue that:}

iπσ(ω) = P Z ∞ −∞ dω0 σ(ω 0₎ ω0_{− ω}, (2.25) where P R∞

−∞is the principal value of the integral. For convenience we write separately the real and

imaginary part of the conductivity Re(σ(ω)) = 1 πP Z ∞ −∞ dω0Im (σ(ω 0₎₎ ω0_{− ω} = 2 π Z ∞ 0 dω0ω 0_{Im (σ(ω}0₎₎ ω02_{− ω}2 (2.26) Im(σ(ω)) = −1 πP Z ∞ −∞ dω0Re (σ(ω 0₎₎ ω0_{− ω} = − 2ω π Z ∞ 0 dω0Re (σ(ω 0₎₎ ω02_{− ω}2 (2.27)

These are the Kramers-Kronig relations for the linear conductivity [16].

Figure 2.1: Contour used in proving 2.25.

2.3.5 Nonlinear Kramers-Kronig Relations

Having introduced the Kramers-Kronig relations for the linear case we now present similar results for nonlinear susceptibilities following the lines of reference [17]. For nonlinear response the condition of causality means that we can analytically extend the frequency domain to the upper half complex plane for each frequency. It is then clear that, holding the other frequencies constant, there is a Kramers-Kronig relation for each individual frequency

σ(ω1, ..., ω, ..., ωn) = 1 iπP Z dω0σ(ω1, ..., ω 0_{, ..., ω} n) ω0_{− ω} . (2.28)

It is possible to prove a more general Kramers-Kronig relation for nonlinear response functions. We will start by noticing that:

2_{This depends on our convention for dening Fourier transforms. If we had instead dened them with opposite sign}

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2 Mathematical Description of Optical Linear and Nonlinear Response σ(ω1+ aiΩ, ..., ωn+ anΩ) = Z dt1...dtnσ(t1, ..., tn)ei P i(ωi+aiΩ)ti (2.29) P Z dΩσ(ω1+ aiΩ, ..., ωn+ anΩ) Ω − ω = Z dt1...dtnσ(t1, ..., tn)ei P iωiti_P Z dΩe iP iΩaiti Ω − ω (2.30) Assuming that all ai are positive numbers, we can employ the previous method of contour

integ-ration to arrive at the following identity: P

Z eiax x − x0

= iπeiax0 (2.31)

We can make use of this result to obtain

P Z dΩσ(ω1+ aiΩ, ..., ωn+ anΩ) Ω − ω = iπ Z dt1...dtnσ(t1, ..., tn)ei P iωiti_eiPiωaiti (2.32) = iπσ(ω1+ a1ω, ..., ωn+ anω),

which is true as long as ai are positive numbers.

This is the general Kramers-Kronig relation for nonlinear responses [16, 17].

2.3.6 Symmetry Considerations for Lossless Media

Away from resonances we expect the net transfer of energy from the electric eld to the medium to be small, so there is interest in knowing what are the conditions for the absence of dissipation the material, which will be treated in this section.

When we induce a electric polarization in a medium we cause charges to be displaced in the presence of a electric eld and therefore we do work on that medium. That work per unit volume is given by:

δW = E · dP. (2.33)

This work is in turn responsible for the increase in the density of internal energy of that material. If we adiabatically induce a polarization in a non-magnetic material, then the change in internal energy is given by:

dU = E · dP (2.34)

For a lossless medium, over a cycle of oscillation of the electromagnetic eld, the energy given to the material is the same as the energy taken. This implies that the average of U over many cycles is constant over time, this means that the average of the time derivative of U is zero.

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2.3 Symmetry Restrictions to the Form of Response Functions dU dt = E ·dP dt = 0 (2.35)

Substituting in 2.11 we get the condition for the absence of dissipation: hE · Ji ≡ lim T →∞ 1 2T Z T −T dtE(t) · J(t) = 0. (2.36) Writing this relation in Fourier space we get

Z

dωE(−ω) · J(ω) = 0, (2.37) expanding J and writing explicitly each of its components

Z dω

Z

dωiRα...βi(ω, ..., ωi)...Eα(−ω)...Eβi(ωi) = 0. (2.38)

For the above result to be zero for any E(ω), the fully symmetrized part of the current response function R (and therefore the conductivity) must be zero in all orders.

For the linear case:

σij(ω) + σji(ω)∗ = 0 (2.39) This means that the diagonal components of the conductivity are imaginary.

For the second order conductivity [18].

σijk(ω1, ω2) + σjik(−ω3, ω1) + σkji(ω2, −ω3) = 0. (2.40)

This allows us to identify which quantities are responsible for absorption of energy by the material for any order.

2.3.7 Spatial Symmetry Considerations

Finally, we mention the restrictions placed by spatial symmetry of the medium. Dierent media can be classied according to their symmetry under rotations and reections, and the nonlinear response must reect that symmetry. Each symmetry group has its own set of restrictions. For example, media with inversion symmetry cannot have a nonlinear response at even orders. This can be proved by considering how the nth order contribution changes upon inversion.

We say a medium has inversion symmetry if it remains the same if we invert the system with respect to some inversion point. For example a cubic lattice remains the same after inversion with respect to a point in the lattice site, while a honeycomb lattice like that of graphene lattice remains the same if we invert it with respect to a point in the centre of a hexagonal cell. An example of a lattice without inversion symmetry is that of hBN (hexagonal boron nitride), a honeycomb lattice where adjacent points correspond (alternately) to nitrogen and boron atoms.

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Figure 2.2: Left: Drawing of a honeycomb lattice. This lattice has inversion symmetry around the red point. Right: Depiction of a hBN lattice. Because adjacent atoms are dierent, this lattice does not have inversion symmetry.

For media with inversion symmetry the induced current should be inverted under a spatial inver-sion, which also implies an inversion of the applied eld. This means that the current should be an odd function of the electric eld. By applying this transformation to each term in the various order of the eld: J(n)_ind(ω) = Z _dω 1 2π ... dωn 2π σ (n)_(ω 1, ..., ωn) · E(ω1)...E(ωn), (2.41) → (−1)n Z dω1 2π ... dωn 2π σ (n)_(ω 1, ωn)E(ω1)...E(ωn).

As J must be odd, we conclude that only odd orders can be non zero.

This is just one of the many restrictions placed by spatial symmetry. The complete set of sym-metries should take into account the whole space group of the crystal and can be obtained by means of group theory. Tables listing them are available in the standard references for the lower orders in the 3D case such as Boyd [16].

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2.4 Relation to Other Physical Quantities

Figure 2.3: List of allowed elements of linear response tensors for the many 3D crystal systems, taken from Boyd [16].

2.4 Relation to Other Physical Quantities

Finally we will also be mention how these quantities relate to other physical quantities used to describe the optical response. Common ones are the dielectric function, the refractive index and the absorption constant.

The dielectric function is given by a simple relation to the linear electrical susceptibility:

(ω) = 1 + χ(ω) (2.42) in S.I units, or

(ω) = 1 + 4πχ(ω) (2.43) in gaussian units.

From the dielectric function we can write the (complex) refractive index

n(ω) =p(ω) (2.44) If we explicitly separate the real and imaginary parts [5].

(ω) = 1(ω) + 2(ω) (2.45) Re(n) = √1 2 r 1+ q 2 1+ 22 (2.46)

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2 Mathematical Description of Optical Linear and Nonlinear Response Im(n) = √1 2 r −1+ q 2₁+ 2₂ (2.47) The imaginary part of the refractive index describes the absorption of energy by the material from the electromagnetic eld.

The absorption coecient describing the exponential decay of the intensity of the electromagnetic eld along the direction of propagation in the Beer-Lambert law is given by:

α(ω) = 2ω

c Im(n) (2.48)

If the absorption is small the square root can be expanded and we get:

α(ω) ≈ √ω 1c 2(ω) = √ω 1cIm(χ(ω)) = Re(σ(ω)) cnb (2.49)

where nb is the background refractive index.

2.5 Summary

In this chapter we introduced the usual formalism used in the description of nonlinear optical systems. With those considerations taken to account we understand the meaning of the quantities calculated in the following chapters.

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3 Equation of Motion Method for Calculating

Optical Responses

Having introduced the formalism we will use to describe the response, we now need a way to calculate our optical response functions. A number of simplications need to be introduced in order to make the problem tractable. First we will consider that in our system the electrons (and electron holes) are the sole charge carriers, a very reasonable assumption in solid state physics given that electrons are much more mobile that the lattice ions. This means that our objective is to calculate the mean velocity of the electrons in the material. For that we will use the standard tools of quantum theory. So polarization can be written in therms of the average of the position operator and the current density can be written in terms of the velocity operator [2]. Since we are treating a many particle system and we will later include two particle interactions, we will use the language of second quantization as in [18]. The polarization and current operators are

P = −e hri , (3.1)

J = −e hvi , (3.2)

where e is the elementary charge.

In second quantization the position operator is given as: r ≡

Z

dxx ˆψ†(x) ˆψ(x), (3.3) where ˆψ(x) are the standard second quantization eld operators.

The creation operator for a particle in state i is given by a†_i =

Z

dxφi(x) ˆψ†(x), (3.4)

where φi(x) is the single particle wavefunction of the state i. These operator obey fermionic

anti-commutation relations.

{a†_i, aj} = δij (3.5)

Any single particle operator ˆO can be written as ˆ O =X

ij

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3 Equation of Motion Method for Calculating Optical Responses where Oij is the matrix element

D i Oˆ

jE in the single particle basis.

The velocity operator v is the single particle operator whose expected values equal the derivative of the expected values of the position operator.

hvi = d

dthri (3.7)

We can then use the Heisenberg equation to dene

i~v ≡ [r, H] (3.8)

We dene the density matrix as:

ρji≡a+i aj

(3.9) With these denitions we can write the average of operators as:

P(t) = −e

V tr(ρ(t)r) (3.10) J(t) = −e

V tr(ρ(t)v) (3.11) Where the trace is dened as:

tr(Oρ) ≡X

ij

Oijρji (3.12)

We will also make use of the standard results of the physics of electrons moving in a periodic lattice with a potential U(x) = U(x + R), where R is a vector of the crystal lattice. The fundamental result is known as Bloch's theorem and states that an Hamiltonian periodic in a lattice can be diagonalized by wave functions of the type

ψkλ(x) = eikxukλ(x) (3.13)

where k is called the crystal momentum and is a vector in the Wigner-Seitz cell of the reciprocal lattice, known as the rst Brillouin zone, λ refers to the remaining quantum numbers necessary to fully specify the wavefunction and ukλ(x) = ukλ(x + R) has the same periodicity as the crystal

lattice.

Most of the operators whose expected values we want to calculate, such as the Hamiltonian and the velocity, are diagonal in crystal momentum space, meaning they only have non zero matrix elements between states of equal k. In that case only the density matrix elements between states of equal k contribute to their expected values.

We then dene the reduced density matrix (RDM) as: ρss0(k, t) ≡

D

a†_s0(k)as(k)

E

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The trace in (3.11) then simplies to a sum over the rst Brillouin zone of traces involving the reduced density matrix. For the current, the formula is:

J(t) = −eX

k

tr(ρ(k, t)v(k)) (3.15) Now the problem consists basically on nding an appropriate equation of motion for our system. Since we are treating problem in crystalline semiconductors with both electron-electron interaction and interaction with an exterior electric eld, our Hamiltonian will include three terms; H0 being

the standard Hamiltonian for an electron in a lattice including a lattice-periodical potential; Hee

describing electron-electron interaction; HI describes the interaction with the external eld.

H = H0+ Hee+ HI (3.16)

H0 =

p2

2m+ U (x) (3.17)

For the electron-electron interaction we will consider a two-particle interaction depending only on the distance between the particles. This is not necessarily the Coulomb potential as we may need to include corrections due to screening and other eects. For example, in two dimensional materials between two dielectric substrates, like the aforementioned TMDs and h-BN, a modied form of the potential known as the Keldysh potential is necessary to describe accurately the observed, non-hydrogenic excitonic energy levels instead of the Coulomb potential because of screening [19].

There are many equivalent ways to describe the interaction with the external eld, depending on our choice of gauge. The more conventional is the length gauge where the electric eld is written as the gradient of a scalar potential. The interaction Hamiltonian has the form:

HI = eE · r (3.18)

The use of this form is not without technical diculties. The rst relates to the fact that the potential cannot satisfy the periodic boundary conditions commonly used in the treatment of a nite crystal, therefore these matrix elements can only be well dened in an innite volume crystal [20]. The second is that the potential breaks the lattice symmetry of the Hamiltonian and, as a consequence, it is not diagonal in k space. This second diculty can be surmounted by noticing that the position operator acts like a dierential operator in k space [2].

Having identied our Hamiltonian we use the quantum Liouville equation to determine the evol-ution of the density matrix

i~dρss0 dt = D [a†_s0as, H] E . (3.19)

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3 Equation of Motion Method for Calculating Optical Responses

3.1 Explicit Calculation for the Linear Independent Electron Case

First we will present the calculations for the simple case of a two band quadratic semiconductor neglecting the electron-electron interaction.

We will use the interaction picture to write.

O → Oint= eiH0tOe−iH0t (3.20)

where O is initially written in the Schrödinger picture. We write the eigenstates of H0 as:

H0|ksi = s(k) |ksi (3.21)

The matrix elements of a single particle operator O between eigenstates of H0 are:

Oss0(k)_int= O_ss0(k)ei(s0(k)−s(k))t (3.22)

The equation of motion for the interacting density matrix is given by i~d

dtρint(k, t) = [Hint(t), ρint(k, t)], (3.23) where we have

ρint(t) = eiH0tρs(t)e−iH0t

=X

ss0

|si ei(s(k)−s0(k))t_ρ_s_(t)

ss0s0 (3.24) and

Hint(t) = eiH0tHI(t)e−iH0t

=X

ss0

|sλi ei(s(k)−s0(k))t_d(k)

ss0ks0 (3.25) where we chose to ignore the non-diagonal part in k space of the position operator once in the case of low temperature the intra band terms approach zero. Substituting this in the Liouville equation we have: d dtρint(t)ss0 = X µ E(t) i~ (ρint(t)sµe i(µ−λ0)t_d µs0− ρ_int(t)_µs0ei(s−µ)td_sµ). (3.26)

In our two band case the reduced density matrix has only two independent matrix elements, so Liouville equation can be written as two equations:

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3.1 Explicit Calculation for the Linear Independent Electron Case d dtρint(t)cv= E(t) i~ (ρint(t)cce i(c−v)t_d cv− ρint(t)vvei(c−v)tdcv) = E(t) i~ e i(c−v)t_d cv(ρint(t)cc− ρint(t)vv) (3.27) d dtρint(t)cc = E(t) i~ ρint(t)cv(e i(v−c)t_d vc− ei(c−v)tdcv) (3.28)

The remaining elements are given by ρvc= ρ∗cv e ρcc+ ρvv = 1.

We will now use this knowledge to obtain the linear susceptibility for the case of a semiconductor with charge carrier near to an equilibrium distribution. Using the equation for the polarization we get P (t) = 1 Vtr(P ρ(t)) (3.29) = 1 V X k ρcv(k)dvc(k, k0)ei(v(k)−c(k 0_))t + c.c. (3.30) now we will resort to equations (3.27) e (3.28) in order to determine the evolution of the polarization. In the case of a near-equilibrium distribution the diagonal matrix elements take the form of a Fermi-Dirac distribution ρλλ≡ fλ= 1/ eβ(Eλ−µ)+ 1

and we will ignore their evolution. The time evolution of ρcv will be determined by integration of (3.27); we then have:

ρint(k, t)cv= Z dt0E(t 0₎ i~ (fc(k) − fv(k))e i(k,c−k,v)t0_d cv(k) = Z dt0dω 2π E(ω) i~ (fc(k) − fv(k))e i(ω+c(k)−v(k))t0_d cv(k) = Z dω 2π E(ω) i~(ω + c(k) − v(k) + i0+) ei(ω+c(k)−v(k))t_(f c(k) − fv(k))dcv(k), (3.31)

where i0+ _{term takes into account that the external eld in turned on innitely slowly}1 _.

Replacing (3.31) em (3.30): P (t) = 1 V X k |dcv(k)|2(fc(k) − fv(k)) Z dω 2π E(ω) i~(ω + c(k) − v(k) + i0+) eiωt+ c.c. (3.32) Now we take the Fourier transform of this expression and make use of the notation cv= c− v

to write

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3 Equation of Motion Method for Calculating Optical Responses P (ω) = −1 V X k (fc(k) − fv(k))|dcv(k)|2E(ω)[ 1 i~(ω + cv(k) + i0+) − 1 i~(ω − cv(k) + i0+) ]; (3.33) J (ω) = −iω V X k (fc(k) − fv(k))|dcv(k)|2E(ω)[ 1 i~(ω + cv(k) + i0+) − 1 i~(ω − cv(k) + i0+) ]; (3.34) χ(ω) = −1 V X k (fc(k) − fv(k))|dcv(k)|2[ 1 i~(ω + cv(k) + i0+) − 1 i~(ω − cv(k) + i0+) ]. (3.35) Now we shall calculate the absorption coecient in the case where the imaginary component of the dielectric constant is much smaller than the real component 0

r

00

r (low dissipation). We make

use of the result that

Z dx f (x) x + i0+ = P Z dxf (x) x − iπf (0) (3.36) which can also be expressed as

1

x + i0+ = P (

1

x) − iπδ(x). (3.37) Then the absorption is given by

α(ω) = ω nc 00 r(ω) = ω ncχ 00 (ω) (3.38) = −πω nc[ 1 V X (fc(k) − fv(k)) |dcv(k)|2 ~ (δ(ω + cv(k)) + δ(ω − cv(k)))] (3.39) α(ω) = −πω nc 1 V X (fc(k) − fv(k)) |d_cv(k)|2 ~ δ(ω − cv(k)) , ω > 0, (3.40) where n is the material's refractive index and c is the speed of light in vacuum.

Because the χ00_(ω)_{for negative frequencies can be obtained through the relationχ}00_{(ω) = −χ}00_(−ω)

all the information is contained in the positive part of (3.39), we can forget the Dirac delta which is zero on the positive part of the real axis, whose function is to ensure the reality of α.

We will work near the band edge and use the eective mass approximation for the dispersion relation to get: α(ω) = −πω nc 1 V X (fc(k) − fv(k)) |d_cv(k)|2 ~ δ(ω − ~2k2 2mr − E_G). (3.41) where mr is the electron-hole reduced mass dened by

1 mr = 1 me + 1 mh (3.42)

and the masses of the electron and hole are given by the curvature of the conduction and valence bands, respectively.

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3.1 Explicit Calculation for the Linear Independent Electron Case 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0

Figure 3.1: Plot of the absorption as a function of frequency for a 3D semiconductor at low temperature.

We will now do the sum over k by taking P = V/(2π)D_{R d}D_k_{. To keep the problem analytically}

tractable we focus on transitions near the band edge in the regime where we can approximate |d_cv(k)|2 by its value in the centre of the Brilloiun zone, ignoring dependence in k.

α(ω) = −πω nc |dcv|2 ~ 1 (2π)D Z dDk(fc(k) − fv(k))δ(ω −~ 2_k2 2mr − EG) (3.43) = πω nc |dcv|2 ~ ΩD−1 (2π)D Z dkkD−1(fc(k) − fv(k))δ(ω −~ 2_k2 2mr − EG) = πω nc |dcv|2 ~ ΩD−1 (2π)D( 2mr ~2 ) D 2 Z ∞ 0 dεεD−22 (f_v,ε− f_c,ε)δ(ω − ε − E_G) = πω nc |dcv|2 ~ ΩD−1 (2π)D( 2mr ~2 ) D 2(f_v,ω−E G− fc,ω−EG)(ω − EG) D−2 2 Θ(ω − E_G), (3.44) where fi,x≡ 1 eβ(E0+x mr miµi) + 1 (3.45)

and i runs over the band indices.

This formula has a simple physical interpretation in that the rst term represents the probability of a photon to excite an electron which is proportional to the density of nal states the electron can reach, which is in agreement with Fermi's golden rule. The second term has the interpretation as representing the probability of a photon causing stimulated emission which again is in agreement with Fermi's golden rule, and is also consistent with the known result that the transition probability matrix element is the same for stimulated emission and absorption.

Finally we evaluate the integral in 3D (3.43) at zero temperature fc= 0, fv = 1 the absorption

becomes: α(ω)=m 3 2 rωp2(ω − EG) πnc |dcv|2 ~ 5 2 Θ(ω − EG) (3.46)

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We see that the absorption follows the well documented square root dependence for three dimen-sions on the frequency near the absorption edge.

3.2 General Treatment of Nonlinear Optical Responses

In the previous section we treated the special simple case of two band quadratic semiconductor in the linear regime while ignoring the form of the dipole matrix elements. In this section we will develop a systematic formalism to treat responses for any system of independent electrons.

We go back to the quantum Liouville equation and see that the evolution of the RDM is given by its commutator with the position operator.

Since we are working in the Bloch basis it is fundamental the evaluation of the matrix elements of the position between dierent Bloch waves hks0_{|r| ksi}_{. The usually imposed periodic boundary}

conditions these matrix elements are ill-dened in a system of nite volume [2, 20]. In the thermo-dynamic limit, however, these matrix elements can be expressed as a dierential operator acting in kspace [2]. k0_s0_{|r| ks =}Z _drψ∗ k0_,s0(r)rψ_k,s(r) = Z dre−ik0ru∗_k0_,s0(r) −i∇keikr uk,s(r) = −i∇k Z drψ_k∗0_,s0(r)ψ_k,s(r) + i Z drei(k−k0)ru∗_k0_,s0(r)∇_ku_k,s(r) = −iδss0∇_kδ(k0− k) + δ(k − k0)ξ_k,s0_s, (3.47)

where we use the result

∇_kψk,s(r) = irψk,λ(r) + eikr∇kuk,s(r), (3.48)

where

ξk,s0_s ≡ i

Z

cell

dru∗_k,s0(r)∇_k(r)u_k,s(r)/V_cell (3.49)

is a quantity known as the Berry connection.

Writing the action of the position operator on a single particle state in a basis of Bloch states we get hks |r| ψi =X k0_s ks |r| k0_{s k}0 s0|ψ = −iX s0 δss0∇_k− iξ_k,ss0 ks0|ψ . (3.50)

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3.2 General Treatment of Nonlinear Optical Responses

Dks0_s≡ δ_ss0∇_k− iξ_k,s0_s (3.51)

so that

D = ir (3.52)

when r acts in a basis of Bloch states.

It is possible to make an analogy with gauge theories by noticing that ∇k is not invariant under

a change of the choice of phase for Bloch waves, this is, it is not invariant under the transform-ation ψks(r) → eiθs(k)ψks(r). In order to build a dierential operator which obeys the relation

Deiθs(k)_ψ

ks(r) = eiθs(k)Dψk(r)it is necessary to add additional terms to ∇k to achieve the desired

transformation law. In gauge theories these terms are the gauge connections. In crystalline sys-tems this job is done by the Berry connections. This means that the position operator acts like a covariant derivative in Bloch momentum space.

To evaluate the commutation of the position operator with an observable diagonal in k we make:

[D, O]ss0(k) =ks0|DO(k) − O(k)D| ks

=ks0∇_k(O(k) |ksi) −ks0|O(k)| ∇_kks + i[ξ(k), O(k)]_ss0 (3.53)

=ks0∇_kO(k) |ksi + i[ξ(k), O(k)]_ss0

= ∇kO(k)ss0+ i[ξ(k), O(k)]_ss0

Given that we can express all relevant quantities in a way that the position operator appears only in commutators, equation (3.53) overcomes the diculties associated with the non diagonalizability of the position operator.

We now insert our results in equation (3.19) isolating the interaction term in the right hand side.

i~d

dt − ss0(k)

ρss0(k) = eiE(t) · [D, ρ(k)]_ss0 (3.54)

We will now treat this equation perturbatively, which will then naturally allow us immediately to identify the contributions to the induced current according to their order in the electric eld and therefore extract the nonlinear conductivities from the resulting expression.

We write density matrix as the sum of the contributions of dierent order in perturbation theory: ρ(t) = ρ(0)+ ρ(1)(t) + ρ(2)(t) + ... (3.55) Then the perturbative version of (3.54) is simply:

i~ d

dt − ss0(k)

ρ(n)_ss0(k, t) = eiE(t) · [D, ρ(n−1)(k, t)]ss0 (3.56)

We write this equation in terms of Fourier components, which permits us to write a recurrence relation for the various orders in the perturbation

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3 Equation of Motion Method for Calculating Optical Responses (~ω − ss0(k)) ρ(n) ss0(k, ω) = Z dω0 2πeiE(ω 0_{) · [D, ρ}(n−1)_{(k, ω − ω}0_)] ss0 (3.57)

so we can recursively derive higher order results from lower order ones. If we want our results to take the form in (3.15) it is convenient to separate the current in its contributions from each order in the eld

J(ω) = J(0)(ω) + J(1)(ω) + J(2)(ω) + ... (3.58) Each contribution is given according to (3.15).

J(n)(ω) = −eX

k

tr(ρ(n)_{(k, t)v(k))} _(3.59)

We know are completely able to calculate the nonlinear optical of any independent electron system to any order in perturbation theory. We will calculate the rst and second order conductivity which will be helpful to compare to our results when we include electron-electron interactions. Namely in the limit where the interaction goes to zero these results should be recovered.

To rst order the result is: ρ(1)_ss0(k, ω) =

Z _dω0 2π

eE(ω0) · i∇kρ(0)(k, ω − ω0)ss0 − [ξ, ρ(0)(k, ω − ω0)]_ss0

~ω − ss0(k) (3.60)

Which for the special case of a clean two-band semiconductor at zero temperature we have ρ(0)_cc (k) = ρ(0)_cv(k) = ρ(0)_vc(k) = 0, (3.61)

ρ(0)_vv(k) = 1. (3.62) therefore we can ignore the derivative term in (3.60).

In rst order the only non-zero contributions to the density matrix are in the o-diagonal elements. These are: ρ(1)_cv(k, ω) = − Z dω0 2π eE(ω0) · [ξ, ρ(0)(k, ω − ω0)]cv ~ω − cv(k) = − Z dω0 2π eE(ω0) · ξcv(k) ~ω − cv(k) 2πδ(ω − ω0) = −eE(ω) · ξcv(k) ~ω − cv(k) . (3.63)

Now we make use of equation (3.15) to calculate the nal expression for the electric current. We must rst express the matrix elements of the velocity operator v, which is dened as

v ≡ [r, H]

i~ . (3.64)

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3.2 General Treatment of Nonlinear Optical Responses

~vss0(k) = δ_ss0∇_k_s(k) − i_s0_s(k)ξ_ss0(k). (3.65)

Using the above equation and (3.59) we get an expression for the current

J(1)(ω) = −eX k ρ(1)_cv(k, ω)vvc(k) + c.c.(−ω) =X k −ie2 cv(k)ξvc(k)ξcv(k) ~ω − cv(k) · E(ω) + c.c.(−ω) (3.66) where c.c.(−ω) represents the complex conjugate of the rst term with ω replaced by −ω.2

This can be seen to be equivalent to the result in (3.34) if we use the result: ω − + ω + = ω ω − − ω ω + (3.67)

The second order response is now given by the recurrence relation (3.57).

ρ(2)_ss0(k, ω) = Z dω0 2π ieE(ω0) · [D, ρ(1)(k, ω − ω0)]ss0 ~ω − ss0(k) (3.68)

Now we have contributions in all elements of the density matrix ρ(2)_cv(k, ω) = −ie2 Z _dω0 2π E(ω0)· ~ω − cv(k) (i∇k+ ξcc(k) − ξvv(k)) E(ω − ω0) · ξcv(k) ~(ω − ω0) − cv(k) (3.69) ρ(2)_cc (k, ω) = ie2 Z _dω0 2π E(ω0)· ~ω − cv(k) E∗_{(ω − ω}0_{) · ξ} vc(k) ~(ω − ω0) − cv(k) ξcv(k) + E(ω − ω0) · ξcv(k) ~(ω − ω0) − cv(k) ξvc(k) (3.70) The other elements are given by the relations ρ(2)

cc (k) + ρ(2)vv(k) = 0, and ρ(2)cv(k) = ρ(2)vc(k)∗. We

then obtain the nal expression for the current.

J(2)(ω) = −e2 Z dωX k ivvc(k) E(ω0)· ~ω − cv(k) (i∇k+ ξcc(k) − ξvv(k)) E(ω − ω0) · ξcv(k) ~(ω − ω0) − cv(k) + c.c.(−ω) + (vcc(k) − vvv(k)) −ieE(ω0_)· ~ω − cv(k) eE∗_{(ω − ω}0_{) · ξ} vc(k) ~(ω − ω0) − cv(k) ξcv(k) − eE(ω − ω0) · ξcv(k) ~(ω − ω0) − cv(k) ξvc(k) (3.71) These equations have poles in their denominators, which makes their value undened at resonance frequencies. This issue is solved by adding a i0+_{term to the frequencies in denominator as in (3.31).}

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3.3 Chapter Summary

We now have a theory to calculate optical response to arbitrary order in perturbation theory for non interacting systems. We veried that it reproduces previous methods in the linear regime and wrote explicitly the result for second order. We saw that the absorption spectrum for independent electrons in the eective mass approximation is characterized by zero absorption for frequencies below the gap and absorption proportional to the free electron density of states above the gap as expected.

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4 Optical Response with Electron-Electron

Interactions

Having studied the main features of the optical response in the independent electron regime we will now include electron-electron interaction.

Before, we introduce a method of calculating the linear and nonlinear optical response in a semi-conductor without electron-electron interaction. In this section we will introduce a general method of obtaining expressions for the calculation of excitonic eects in rst and second orders in a two band semiconductor independent of the form of the electron-electron interaction potential. This is important since the general form of the electron-electron interaction is not that of a Coulomb inter-action, due to the fact that electron wavefunctions in a lattice are not plane waves (overlap eects) and because of screening interactions both between the electrons of a material as well as between the electron and the surrounding medium. We conclude that, in a mean eld approximation, the inclusion of electron-electron interaction leads us to a problem of diagonalization of a Hamiltonian representing the interaction between an electron and a hole. Writing the matrix elements of the relevant quantities in the eigenbasis of that Hamiltonian we can get a clear expression for the optical conductivity of the material in question.

4.1 Mathematical Formalism

For that purpose we write the most general two-particle interaction Hamiltonian, H = X indices hlka † lak+ 1 2V lm kna † ka † laman (4.1)

where the single particle term includes both the crystal and external Hamiltonians. For simplicity of notation we will include in the indices of ai all the necessary quantum numbers, including crystal

momentum, band number and spin.

We want to calculate the time evolution of the average of the density matrix, which is given by d dt D a†_iaj E = D [a†_iaj, H] E i~ . (4.2)

We will treat the general problem of the evolution of the density matrix for a general system with a Hamiltonian involving a two-particle interactions. The commutator [a†

iaj, H] is calculated in the

appendix 7.3 on page 62.

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4 Optical Response with Electron-Electron Interactions i~d dtρba = X l {h_blρla− hlaρbl}+ +X lnm V_lnmaDa†_la†_namab E − Vln bm D a†_aa†_lanam E . (4.3)

We now have a problem because the two operator average is coupled to a four operator average. We could then try to nd the equation of motion for the average of four operators, however we would then nd out that that quantity would couple to a six operator average and so on. We therefore must nd some approximation scheme that allows us to truncate this sequence. To do that we assume we can factorize the four operator terms in two operator terms in accordance to Wick's theorem. i~d dtρba≈ X l {hblρla− hlaρbl} + V_bmln(ρmaρnl− ρniρml) −V_maln(ρbmρnl− ρblρnm) i ≈X l {hblρla− hlaρbl}+ +X lnm {Vln bm− Vbnlm}ρnlρma− {Vlnma− Vmnla}ρnlρbm. (4.4)

We will now apply this general result to the special case of our problem. Then each state is described by a band index and a Bloch wavevector in the rst Brillouin zone. We make also use of denition of the reduced density matrix ρab(k)as the components of the density matrix along the

diagonal in k. The correspondence is:

a → a, ka; (4.5)

ρba → ρba(kb, ka); (4.6)

ρba(k) ≡ ρba(k, k); (4.7)

V_adbc→ V_adbc(ka, kd; kb, kc). (4.8)

The form of the interaction matrix element between the states is given in the appendix 7.4 on page 63.

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4.1 Mathematical Formalism i~d dtρba(k) = X l {h_bl(k)ρla(k) − hla(k)ρbl(k)} + W (4.9)

where W is the two-particle term. This term is given by

W =X lnm {V_bmln(k, km; kl, kn) − Vbnlm(k, kn; kl, km)}× ρnl(kn, kl)ρma(km, k)− X lnm {V_lnma(kl, kn; km, k) − Vmnla(km, kn; kl, k)}× ρnl(kn, kl)ρbm(k, km). (4.10)

In the dipole approximation there is only the contribution of matrix elements diagonal in k. We then consider kl= kn e km= k. W =X lnm {V_bmln(k, k; kn, kn) − Vbnlm(k, kn; kn, k)}ρnl(kn)ρma(k)− X lnm {V_lnma(k, k; kn, kn) − Vmnla(k, kn; kn, k)}ρnl(kn)ρbm(k). (4.11) TheVab

cd factor is proportional to overlap integrals of modulating functions of Bloch waves Ia,k;b,k0 =

R

celldre

i(k−k0)r_u∗

a(k, r)ub(k0, r). By orthogonality of Bloch waves we can easily see that Im,k;n,k =

δmn. This suggests an approximation where the Im,k;n,k0 factors are much larger in the case where

m = nrelatively to the case in which m 6= n. This is the Tamm-Danco approximation [14].

W =X l V_bbll(k, k; kn, kn)ρll(kn)ρba(k) − Vbbll(k, kn; kn, k)ρbl(kn)ρla(k)− X l V_llaa(k, k; kn, kn)ρll(kn)ρba(k) + Vllaa(k, kn; kn, k)ρla(kn)ρbl(k). (4.12)

Rearranging terms we have

W =X l V_bbll(k, k; kn, kn) − Vllaa(k, k; kn, kn) ρll(kn)ρba(k) +X l,kn V_llaa(k, kn; kn, k)ρbl(k)ρla(kn) − Vbbll(k, kn; kn, k)ρbl(kn)ρla(k). (4.13)

We notice that there are terms in W which are proportional to ρba(k). These are self energy

energy terms which appear in the Hartree-Fock approximation (Appendix 7.6 on page 66) . We can include them by dening the Hartree-Fock Hamiltonian

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4 Optical Response with Electron-Electron Interactions hhf_ab(k) ≡X l,kn V_abll(k, k, kn, kn) − Valla(k, kn; knk)δabδal ρll(kn). (4.14)

This lead us to dene the self energy

Σba(k) ≡ X l,kn V_bbll(k, k; kn, kn) − Vllaa(k, k; kn, kn) ρll(kn) + V_aaaa(k, kn; kn, k)ρaa(kn) − Vbbbb(k, kn; kn, k)ρbb(kn), (4.15)

which is separated in a direct term Σd_ba(k) ≡X l,kn V_bbll(k, k; kn, kn) − Vllaa(k, k; kn, kn) ρll(kn), (4.16)

and an exchange term Σe_ba(k) ≡X

kn

V_aaaa(k, kn; kn, k)ρaa(kn) − Vbbbb(k, kn; kn, k)ρbb(kn). (4.17)

Having worked out the eects of our two particle interaction we can nally write our equation of motion: i~d dtρba(k) = (ba+ Σba(k)) ρba(k) + X l {hint_bl (k)ρla(k) − hintla (k)ρbl(k)} +X l,kn V_llaa(k, kn; kn, k)ρbl(k)ρla(kn)(1 − δla) − X −l,kn V_bbll(k, kn; kn, k)ρbl(kn)ρla(k)(1 − δlb). (4.18)

We now introduce our two band approximation. Replacing the indices a and b in (4.18) respect-ively by the conduction band index c and valence band index v, and dening

Eba(k) ≡ ba(k) + Σba(k) (4.19)

we get the equation of motion for the o diagonal matrix elements

i~d dtρvc(k) = Evc(k)ρvc(k) + X l {hint_vl (k)ρlc(k) − hintlc (k)ρvl(k)} (4.20) +X kn V_vvcc(kn, k; k, kn)ρvv(k)ρvc(kn) −X kn V_vvcc(k, kn; kn, k)ρvc(kn)ρcc(k).

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4.2 The Excitonic Hamiltonian and the Bethe-Salpeter Equation Knowing the value of ρcv(k), ρvc(k) can be determined by complex conjugation. To complete our

system of equations we now need the equations of motion for the diagonal elements, which are obtained by the same procedure.

i~d dtρcc(k) = X l {hint_cl (k)ρlc(k) − hintlc (k)ρcl(k)} (4.21) + V_vvcc(kn, k; k, kn)ρvc(kn)ρcv(k) − Vccvv(kn, k; k, kn)ρcv(kn)ρvc(k) i~ d dtρvv(k) = X l {hint vl (k)ρlv(k) − hintlv (k)ρvl(k)} (4.22) + V_ccvv(kn, k; k, kn)ρcv(kn)ρvc(k) − Vccvv(kn, k; k, kn)ρvc(kn)ρcv(k)

It can be veried that the derivative of the sum ρcc(k) + ρvv(k), which represents the population

of electrons in the k point of the Brilloiun zone, is zero. Since we start with all k states occupied in the valence band and empty in the conduction band, the total occupation number of each k point remains constant and equal to one.

4.2 The Excitonic Hamiltonian and the Bethe-Salpeter Equation

We now have the general nonlinear equation of motion of our system. As we are interested in calculating the induced current by the application of an external electric eld it is natural to treat the system perturbatively using the amplitude of the electric eld as a expansion parameter. In zero order we consider the zero temperature equilibrium condition ρ(0)

cc (k) = ρ(0)cv(k) = 0e ρ(0)vv(k) = 1.

Then in rst and second order:

i~d dtρ (1) cc = 0 (4.23) ρ(1)_cc = ρ(1)_vv = 0 (4.24) i~ d dtρ (1) vc(k) = Evc(k)ρ(1)vc(k) + X kn V_ccvv(kn, k; k, kn)ρ(1)vc(kn) + ieE · (∂kρ(0)_cv,k− i[ξvl(k)ρ(0)_lc (k) − ξlc(k)ρ(0)_vl (k)]) (4.25) = Evc(k)ρ(1)vc(k) + X kn V_ccvv(kn, k; k, kn)ρ(1)vc(kn) − eE · ξvc(k) (4.26)

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4 Optical Response with Electron-Electron Interactions i~d dtρ (2) vc(k) = Evc(k)ρ(2)vc(k) + X kn V_ccvv(kn, k; k, kn)ρ(2)vc(kn) + ieE(t) · (∂kρ(1)vc(k) − i[ξvl(k)ρ(1)_lc (k) − ξlc(k)ρ(1)_vl (k)]) = Evc(k)ρ(1)vc(k) + X kn V_ccvv(kn, k; k, kn)ρ(1)vc(kn) + ieE(t) · (∂kρ(1)vc(k) − i[ξvv(k) − ξcc(k)]ρ(1)vc(k)) (4.27)

Equation (4.26) and (4.27) have the same homogeneous part, which can be recast as i~ d dtρ (2) vc(k) = X kn Hvc(k, kn)ρ(2)vc(kn), (4.28)

where we dene V (k, kn) ≡ Vccvv(kn, k; k, kn) and

Hvc(k, kn) ≡ Evc(k)δk,kn+ V (k, kn) (4.29)

We call Hvc the electron-hole or excitonic Hamiltonian. This Hamiltonian couples the motion of

electron hole transitions in dierent points of the Brillouin zone.

Equation (4.28) has the same form of a single particle Schrödinger's equation acting on a space of Bloch momentum vectors, so we have reduced our many particle problem to an eective single particle model. Compared to the problem of non interacting electrons we now are required to nd the eigenvalues and eigenvectors of 7.85 on page 74 to solve the equation of motion for ρ(2)

vc(k),

which must be done numerically except for some exactly solvable cases.

After solving this eigenvalue problem, we arrive at eigenfunctions which diagonalize Hvc. We

write them as

X

k0

Hvc(k, k0)ψn(k0) = Enψn(k). (4.30)

These eigenfunctions can be used as an orthogonal basis for our space of vertical transitions. This means we can write ρvc(k)as linear combination of these functions

ρvc(k) =

X

n

Cnψn(k) (4.31)

with coecients Cn. Additionally we can write

X k ψ_n∗(k)ψm(k) = δn,m. (4.32) We dene ρn≡ X k ψ∗_n(k)ρvc(k) (4.33)

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4.2 The Excitonic Hamiltonian and the Bethe-Salpeter Equation so that multiplying (4.31) by ψ∗

n(k), summing over k and substituting (4.32) we get

ρn= Cn. (4.34)

We can use the same procedure to get δk,k0 =

X

n

ψ∗_n(k0)ψn(k). (4.35)

Taking (4.30) multiplying by ψn(k00) and summing over k00 we get

H(k, k0) =X

n

Enψn∗(k0)ψn(k). (4.36)

To make our notation analogous to the bra-ket notation we write

ψn(k) ≡ hk|ni (4.37)

ρvc(k) ≡ hk|ρi (4.38)

Hvc(k, k0) ≡k |Hvc| k0

(4.39) To solve the inhomogeneous equation we will use a Green's function method. We must nd the function G(k0_{, k, t)}_{that solves the equation:}

i~d dtG(k 0 , k, t) + Ecv(k)G(k0, k, t) − X kn V_ccvv(kn, k; k, kn)G(k0, kn, t) = δk,k0δ(t). (4.40)

Taking the Fourier transform we get X

kn

G(ω, k0, kn)[−~ω + Hcv](kn, k) = δk,k0 (4.41)

writing this result in a basis independent form we have

G(ω)[−~ω1 + Hcv] = 1 (4.42)

which implies

G(ω) = 1 Hcv− ~ω

. (4.43)

As G(ω) is an hermitian operator acting in the space of exciton wave functions we can use the spectral theorem to write it as a diagonal operator:

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4 Optical Response with Electron-Electron Interactions G(ω) = 1 Hcv− ~ω =X n 1 En− ~ω |ni hn| . (4.44) Writing the matrix elements of G in k space, we have:

G(ω, k, k0) =X

n

hk|ni hn|k0i En− ~ω

. (4.45)

The o diagonal velocity and position operators can also be written in a basis of eigenfunctions vn≡ X k hn|ki v_vc(k), (4.46) ξn≡ X k hn|ki ξ_vc(k). (4.47)

4.2.1 First Order Result

Then the solution to the rst order equation is ρ(1)_vc(ω, k) = −eX

k0

G(ω, k, k0)E(ω) · ξvc(k0). (4.48)

In the exciton eigenbasis the result is

ρ(1)_n (ω) = −eE(ω) · 1 En− ~ω

ξn (4.49)

We can the use (3.59) to calculate the current and conductivities:

J(1)(ω) = −eX k vcv(k)ρ(1)vc(ω, k) + vvc(k)ρ(1)cv(ω, k) = −eX k v∗_nρ(1)_n (ω) + vnρ(1)∗n (−ω) (4.50) σ(1)(ω) =X k,k0 e2 v ∗ nξn En− ~ω + c.c(−ω). (4.51)

4.2.2 Second Order Results

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4.2 The Excitonic Hamiltonian and the Bethe-Salpeter Equation ρ(2)_vc(ω, k) = ie Z dω0 2π X k0 G(ω, k, k0)E(ω − ω0) · ρ(1)_vc(ω0, k0);k0 = −ie2 Z _dω0 2π X k0_k00 E(ω − ω0)E(ω0) · ·G(ω, k, k0)G(ω0, k0, k00);k0ξ_vc(k00), (4.52)

where the generalized derivative is dened as the intraband part of the covariant derivative

f (k);k ≡ ∇kf (k) − i[ξvv(k) − ξcc(k)]f (k). (4.53)

We can then relate:

[D, (k)]vc= Ovc(k);k− iξcv[Occ(k) − Ovv(k)] . (4.54)

We now have all the necessary elements to calculate the optical current generated in the material:

J(ω) = −eX k tr (v(k)ρ(k, ω)) = −eX kmn vmn(k)ρnm(k, ω) (4.55)

We then expand the current in powers of the electric eld (or vector potential) and remember the denition of the rst and second order conductivities as:

J(1)(ω) = Z _dω 1 2π R (1)_{(ω, ω} 1)E(ω1) (4.56) J(2)(ω) = Z _dω 1 2π dω2 2π R (2)_{(ω, ω} 1, ω2)E(ω1)E(ω2) (4.57)

From the current we obtain an expression of the conductivity which can then be symmetrized. This symmetrization is equivalent to obtaining the conductivity by taking

2πδ(ω − ω1− ω2)σ(2)(ω = ω1+ ω2) ≡ 1 2(2π) 2 δJ(2)(ω) δE(ω1)δE(ω2) E=0 , (4.58) which is analogous to the denition of the Taylor expansion coecients for the nite dimensional case.

For the general second order conductivity tensor: σ(2)ijk(ω1, ω2) = − ie3 2 X k0_k00 v_cvi (k)G(ω, k, k0)G(ω1, k0, k00)j_;k0ξk_vc(k00) + c.c(−ω) (4.59)

A particularly important case is that of second harmonic generation, in that case the previous expression becomes:

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4 Optical Response with Electron-Electron Interactions σ(2)(ω, ω) = −ie31 2 X k0_k00 vi_cv(k)G(2ω, k, k0)G(ω, k0, k00)j_;k0ξk_vc(k00) + c.c(−ω) (4.60)

To express this result in the base of eigenvectors of the exciton Hamiltonian we dene Qmn ≡

X

k

i hm|ki hk|ni_;k. (4.61) This last operator can be proved to be Hermitian by taking:

Q∗_mn= −iX

k

hk|mi [∇_khn|ki + i[ξ_vv(k) − ξcc(k)] hn|ki]

= iX

k

(hn|ki ∇khk|mi − i[ξvv(k) − ξcc(k)] hn|ki hk|mi)

=X

k

i hn|ki hk|mi_;k

= Qnm (4.62)

Our result then takes the form σ(2)(ω1, ω2) = −e3 X nm v∗_mQmnξn (Em− ~(ω1+ ω2)(En− ~ω1) + c.c(−ω). (4.63) For the special case of second harmonic generation

σ(2)(ω, ω) = −e3X

nm

v_m∗Qmnξn

(Em− 2~ω)(En− ~ω)

+ c.c(−ω). (4.64) A full calculation of the conductivity would require choosing an appropriate form of the Vvv

cc (kn, k; k, kn)

potential. Then nding the full solution of (4.30). Then we would have to evaluate of numerical derivatives to get the values of Qnm and nally substitute in (4.63).

4.3 Independent Electron Limit

In order to check the acceptability of our results we will evaluate our results in the independent electron limit where there is no electron-electron interaction v → 0. We will also make a quadratic band approximation as in.

In that case the exciton Hamiltonian is simply Hcv(k, k0) = Ecv(k)δ(k − k0), then we can conclude

that k is an eigenbasis of the Hamiltonian and therefore we can substitute the exciton indicesn and energies En by the Bloch wave indexesk and energies Ecv(k).