Nonlinear optical properties of an exciton in a spherical quantum dot

spherical quantum dot

by

Jefferson Fl´orez Guti´errez

Advisor

Dr.rer.nat. ´ Angela Stella Camacho Beltr´an

Submitted in partial fulfillment of the requirements for the degree of

Master in Sciences - Physics

Universidad de los Andes

Departamento de F´ısica

Grupo de F´ısica de la Materia Condensada

Bogot´a, D. C., Colombia

May 23, 2011

Abstract

We study the nonlinear optical properties of an exciton confined in a quantum dot and placed in an uniform static electric field. The exciton is confined using a parabolic potential. The electric field is parallel to the z-axis of the quantum dot, providing the asymmetry needed to get nonlinear optical phenomena. Within the effective-mass approach, we write the Hamil- tonian of the exciton in terms of the center-of-mass and the relative coordinate, obtaining two separated Hamiltonians. The center-of-mass motion is a harmonic oscillator, while the relative motion includes both a parabolic and a Coulomb potential from the electron-hole electrostatic interaction, so that this Hamiltonian is unsolvable analytically.

In order to solve approximately the relative Hamiltonian, we treat perturbatively the Coulomb potential in the so-called strong-confinement regime, where the matrix elements of this interaction are much smaller than the unperturbed energies. Thus we analyze to first order the energy corrections to the ground state and the triply degenerate first excited states. We obtain that the corrections to the eigenvalues are only reliable in the strong regime, which in terms of the QD characteristic sizes means that they must be much smaller than 18 nm for GaAs/AlGaAs QDs. These sizes are reduced even more when the electric field is introduced. To obtain results for larger quantum dots, from a few to tens of nanometers, we solve nu- merically the relative Hamiltonian in the intermediate regime using a finite elements method. In this regime the Coulomb interaction and the confinement potential are taken exactly. As an interesting result, we obtain that the eigenfunctions are not only shifted in the opposite direction of the electric field, but also they are elongated. Both effects, the shift and the elon- gation, enhance the asymmetry of the QD compared with the results obtained in the strong regime where the eigenfunctions are only shifted.

With the numerical results, we compute the nonlinear optical rectification susceptibility as well as the linear and nonlinear absorption coefficient and refractive index changes as a function of the incident photon energy. The main result is that the nonlinear optical proper- ties of the QD are improved in the intermediate regime as a straightforward consequence of the enhanced asymmetry. We also find that the nonlinear optical coefficients exhibit a wide spectrum of behaviors, such as a sign inversion. This fact offers the possibility of tuning the confinement energy (or the size) of a QD to obtain the desirable optical properties.

Keywords

Quantum Dot, Exciton, Nonlinear Optical Properties. ii

Contents

Abstract ii

1 Introduction 1

2 An exciton in a spherical QD 3

2.1 The effective-mass Hamiltonian . . . 3

2.2 Characteristic quantities of H_rel . . . 4

2.3 Perturbation solution to H_relin the strong-confinement regime . . . 7

2.4 Accuracy of the perturbation solution . . . 9

3 Numerical solution in COMSOL Multiphysics 12 3.1 Statement of the problem . . . 12

3.2 Description of COMSOL Multiphysics . . . 14

3.3 Solving H_rel in COMSOL Multiphysics . . . 14

3.3.1 Confinement energies of the QDs . . . 14

3.3.2 The geometry . . . 14

3.3.3 The mesh . . . 16

3.3.4 Boundary conditions . . . 16

3.3.5 Parameters of the Coefficient Form PDE interface . . . 17

3.4 Numerical results . . . 18

3.4.1 Eigenfunctions . . . 18

3.4.2 Eigenvalues . . . 18

4 Nonlinear optical properties 23 4.1 Theoretical framework . . . 23

4.1.1 Nonlinear optics . . . 23

4.1.2 Density-matrix approach . . . 24

4.1.3 Nonlinear optical coefficients . . . 25

4.2 Results . . . 26

4.2.1 OR . . . 26

4.2.2 Optical absorption . . . 29

iii

CONTENTS iv 4.2.3 Refractive index changes . . . 34

Conclusions 38

A GaAs/AlGaAs QD parameters 39

Bibliography 41

Chapter 1

Introduction

Nonlinear optical properties (NLOP) of quantum dots (QDs) have received much attention in the last two decades due to their relevance in actual and novel applications [1–3], as well as from the fundamental point of view [4–17]. In the nineties of the last century the attention on the NLOP of QDs had centered in nanocrystals, as is summarized in the reviews of A. D. Yoffe [4] and G. Banfi et al. [5]. However, there have been several theoretical publi- cations [10, 12, 16, 17] about optical nonlinearities of self-assembled QDs since the article published by B.-Y. Yu et al. in 2004 [9]. The latter works have used nonlinear optical coeffi- cients obtained by a density-matrix approach and an iterative method. Such coefficients were calculated by Rosencher et al. [18, 19] in the context of quantum wells.

In articles [9, 10, 12, 16, 17] the exciton dynamics has been studied using the effective- mass approximation for GaAs/AlGaAs QDs, and the exciton eigenvalues and eigenstates have been found in the so-called strong-confinement regime. Within this regime the Coulomb interaction between the electron-hole pair is taken approximately or even ignored. The strong regime is reached when the QD sizes are much smaller than the corresponding exciton Bohr radius, which is roughly satisfied by GaAs/AlGaAs QDs due to their actual sizes are around nanometers and the corresponding exciton Bohr radius is 18 nm. Moreover in the case of II- VI semiconductor QDs such as ZnS, CdS, CdSe and CdTe nanocrystals the strong regime is unsuitable because of the experimental QD sizes are comparable with their exciton Bohr radii. We are interested in to obtain the NLOP of QDs when the QD size is smaller or of the same order as the corresponding exciton Bohr radius, defining in the latter case the intermediate- confinement regime. We present our results for GaAs/AlGaAs QDs in the size range from a few to tens of nanometers, but they are applicable to any other direct-gap semiconductor by simply replacing the material parameters.

We consider an exciton confined in a QD by means a parabolic potential. The simplest potential to study theoretically excitons in QDs is a square-well potential that confines both electrons and holes [20]. However Kumar et al. [21] showed that the confinement potential for electrons in a QD is well described by a parabolic potential. Furthermore, this potential is convenient when only the lowest excited states of the carriers are considered [22]. In this

1

INTRODUCTION 2 work we use the model proposed by Que [23] where both electrons and holes are confined in QDs by means a parabolic potential with the same oscillator frequency.

Within the optical nonlinearities of QDs studied in the last ten years there are optical rectification (OR), second harmonic generation (SHG), third harmonic generation (THG), and nonlinear absorption and refractive index changes coefficients. We focus our attention on those nonlinear optical phenomena which involve a two-level system, e.g. the ground and the first excited level of the exciton in the QD: OR and nonlinear absorption and refractive index changes. The purpose of this work is to illustrate the effect of the Coulomb interaction in the intermediate regime by comparing approximate and exact results. In a further work it can be addressed the calculation of optical nonlinearities that imply second or more excited states.

The nonlinear optical phenomena require that the system presents an asymmetric nature to obtain large nonlinear optical coefficients [18]. There are two ways to obtain such asymmetry [10]. One is to use advanced material growing technology to obtain nanostructures with asymmetric confining potentials [18, 24–26]. In this kind of QDs we can include conical, tetrahedral, lens shape, and many other asymmetrical shapes. The other method is through the application of a static electric field to a structurally symmetric system in order to model an asymmetric nanostructure [27, 28]. We choose the latter to get an asymmetrical QD.

The chapters of this work are strongly interconnected, but are written to some extent in a self-contained style. The general context has been established in this introductory chapter, and additional introductory details and concluding remarks are added to each chapter. In Chapter 2 we define the relevant scales of the problem and we reproduce a perturbation solution of the relative Hamiltonian from reference [16] in the strong-confinement regime. We also test the accuracy of this solution in the intermediate and strong regimes. We obtain that the sizes of GaAs/AlGaAs QDs must be equal or smaller than 4 nm to become reliable perturbation results.

The numerical solution is presented in detail in Chapter 3. We use a software to solve by finite elements method the stationary Schr¨odinger equation of the relative motion. We present the eigenfunctions and eigenvalues obtained as a function of different confinement energies and values of the electric field. Finally we compare the numerical solutions with the results from perturbation theory, and we obtain that they differ in the intermediate regime.

In Chapter 4 the nonlinear optical coefficients of an exciton in a QD are studied. We make an introduction to the density-matrix approach in order to obtain the nonlinear optical coefficient expressions. Using the eigenfunctions and eigenvalues computed numerically in chapter 3, we plot such coefficients as a function of the incident photon energy. Our results let us to conclude that the NLOP of QDs are improved in the intermediate regime, leading to wide spectrum of interesting behaviors. The last chapter is dedicated to discussion of our results and conclusions.

Chapter 2

An exciton in a spherical QD

2.1 The effective-mass Hamiltonian

The earliest and simplest model for charge carriers in a QD is based on the effective-mass approximation for the bulk material [20] which can be applied if the QD is larger than the lattice constant of the crystal structure. Under this approximation the carrier properties are the same as those from the bulk crystal, and the quantum confinement of the QD is introduced via a confinement potential [22, 29]. The lattice constant of gallium arsenide is ∼ 6 ˚A, so GaAs/AlGaAs QDs are well described by the effective-mass approximation.

The effective-mass Hamiltonian of an exciton confined in a QD by means of a parabolic potential and in the presence of an uniform static electric field along the z-axis (see figure 2.1) is given by [30]

H= ^p

2e

2m^∗_e ⁺ 1 2^m

∗

e^ω0^2re²⁺

p²_h 2m^∗_h⁺

1 2^m

∗ h^ω

2 0^rh²⁻

e²

ε |r_e− r_h|+ |e|F · (r_e− r_h), (2.1) where m^∗_e(m^∗_h) and r_e(r_h) are the effective-mass and the position vector of the electron (hole), respectively. The oscillator frequency ω₀gives the strength of the confinement, ε denotes the dielectric constant of the medium in which the electron and hole are moving, and F = F ˆk is the uniform static electric field.

In terms of the relative coordinate r = r_e− r_h and the center-of-mass coordinate R = (m^∗_er_e+ m^∗_hr_h)/(m^∗_e+ m^∗_h) the Hamiltonian (2.1) can be written as

H = H_cm+ H_rel, (2.2)

with H_cm the Hamiltonian of the center-of-mass, H_cm= ^P

2

2M⁺ 1 2^Mω

2

0^{R2, (2.3)}

3

CHAPTER 2. AN EXCITON IN A SPHERICAL QD 4

Figure 2.1. Spherical QD placed in a uniform static electric field F parallel to the z-axis.

and H_rel the Hamiltonian of the relative motion, H_rel= ^p

2

2µ ⁺ 1 2^{µ ω}

2

0^r2+ |e|F · r −^e

2

ε r^{. (2.4)}

Here M = m^∗_e+ m^∗_h is the total mass, and µ = m^∗_em^∗_h/M is the electron-hole reduced mass. Notice that the center-of-mass Hamiltonian is independent of the electric field. This is be- cause of the electron and the hole have equal absolute electric charge, so the force exerted by the electric field on both particles has the same magnitude but opposite direction. The net effect is a center-of-mass motion unaffected by the electric field.

The center-of-mass Hamiltonian corresponds to a spherical harmonic oscillator and its eigenvalues and eigenstates are well-known, while the relative Hamiltonian does not have analytical solution. Therefore the Schr¨odinger equation for H_rel must be solved using either approximate or numerical methods.

2.2 Characteristic quantities of H

We shall first use the stationary perturbation theory to solve approximately H_rel. In order to determine the applicability of this method, we study the characteristic lengths and energies of the relative motion problem. Ignoring the electric field term for a while¹, H_relcontains two

1In the next section we will introduce the length β which describes the effect of the electric field on the relative motion.

typical lengths: one is the characteristic size L of the QD set by ω₀as

L= s

¯h µ ω0

, (2.5)

and the other is the exciton Bohr radius,

a^∗₀= ^ε¯h

2

µ e^{2. (2.6)}

There are also two typical energies: one is the energy quanta ¯hω0due to parabolic confine- ment, which is related to L by

¯hω0⁼

¯h²

µ L^{2. (2.7)}

and the other energy quantity is the exciton Rydberg energy, R^{∗= e2}

ε a^∗₀ ⁼ µ e⁴

¯h²ε²

. (2.8)

Both a^∗₀and_R^∗are associated to the Coulomb potential in equation (2.4), so these quantities are independent of ω₀.

The comparison between the two characteristic lengths, or equivalently between the two characteristic energies, defines the strong-confinement regime, where L  a^∗₀, or ¯hω0R^∗, and the weak-confinement regime, where L  a^∗₀, or ¯hω0 R^∗ [23]. The intermediate- confinement regime is established when L and a^∗₀, or ¯hω0and_R^∗, are of the same order.

In figure 2.2 we plot the characteristic lengths and energies of H_relfor a GaAs/AlGaAs QD as a function of the confinement energy ¯hω0. The parameters used for this kind of QDs are summarized in Appendix A. In figure 2.2(a) we can see that lengths L and a^∗₀are of the same order for ¯hω0around 6 meV. In figure 2.2(b) we also observe that ¯hω0^andR^∗exhibit similar values around a confinement energy equal to 6 meV. In terms of ¯hω0, that means the strong regime is achieved for a GaAs/AlGaAs QD when ¯hω0 6 meV, whereas the weak regime occurs when ¯hω0 6 meV. The intermediate-confinement regime happens for confinement energies around 6 meV. In terms of L, this confinement energy defines a characteristic size approximately equal to 18 nm, as can be seen in figure 2.2(a), which means that the strong, weak and intermediate regimes are reached for L  18 nm, L  18 nm and L ∼ 18 nm, respectively.

Within the strong-confinement regime some authors have neglected the Coulomb term in the relative Hamiltonian for one-dimensional [9, 10, 12] and disklike QDs [23], reducing the problem to a pure harmonic oscillator for the relative motion. It is obvious that the Coulomb interaction between the electron and the hole alters the energy eigenvalues of the exciton, so these results cannot be precise. Others authors have treated the Coulomb potential in the strong regime numerically for one-dimensional QDs [17], and considering it as a perturbation

CHAPTER 2. AN EXCITON IN A SPHERICAL QD 6

0.1 1 10 100 1000

2 5 10 20 50 100

ÑΩ₀@meVD

Typicallengths@nmD

a₀^* L

(a)

0.1 1 10 100 1000

0.1 1 10 100 1000

ÑΩ₀@meVD

Typicalenergies@meVD

R^*

ÑΩ₀

(b)

Figure 2.2. Characteristic (a) lengths and (b) energies of the relative Hamiltonian (2.4) for a GaAs/AlGaAs QD as a function of the confinement energy.

for three-dimensional QDs [16]. Although the latter results are more reliable than the formers, their solutions are restricted to the strong regime. In other words, these results are only applicable for QDs with characteristic sizes much smaller than the exciton Bohr radius. We are not only interested in solving the relative Hamiltonian in the strong regime without neglect the Coulomb potential, but also we want to explore its solutions in the intermediate regime, i.e. when the QD size is of the same order as its exciton Bohr radius.

In the next section we use the perturbation theory to solve H_rel in the strong regime. In the following chapter we solve H_relthrough numerical methods in the strong and intermediate regimes, and we compare the latter results with those from the perturbation theory.

2.3 Perturbation solution to H

in the strong-confinement

regime

In this section we treat the Coulomb interaction as a perturbation in the strong regime, repro- ducing the calculations performed by Xie [16]. To calculate the eigenvalues and eigenstates of the relative Hamiltonian, we shall limit ourselves to first-order for the eigenvalues and to zeroeth order for the eigenstates.

We rewrite H_rel as

H_rel= H₀+ H⁰, (2.9)

where

H₀= ^p

2

2µ ⁺ 1 2^{µ ω}

2

0^r2+ |e|F · r, (2.10)

H⁰= −^e

2

ε r^{. (2.11)}

Since F is along the z-axis, the Hamiltonian H₀can be expressed in cartesian-coordinates as H₀= ^p

2x

2µ⁺ 1 2^{µ ω}

2 0^x2+

p²_y 2µ ⁺

1 2^{µ ω}

2 0^y2+

p²_z 2µ ⁺

1 2^{µ ω}

2

0^{z2+ |e|Fz. (2.12)}

Adding and subtracting the term |e|Fβ /2, with β a term with length units defined by β = ^|e|F

µ ω₀^{2, (2.13)}

equation (2.12) transforms into H₀= ^p

2x

2µ⁺ 1 2^{µ ω}

02^x2+

p²_y 2µ ⁺

1 2^{µ ω}

02^y2+

p²_z 2µ ⁺

1 2^{µ ω}

02^{(z + β )2−}

1

2^{|e|Fβ . (2.14)} Thus, the use of cartesian-coordinates allows us to express H₀ as three independent one- dimensional harmonic oscillators along each cartesian axis², but one of them spatially shifted by the lenght β . The shifted oscillator is that which corresponds to the electric field direction (the z-direction). Note that for a fixed ω₀ the magnitude of the spatial shift depends on the strength of the electric field, hence we can describe the effect of F in terms of β .

The eigenvalues and eigenfunctions of the unperturbed Hamiltonian are E_N⁽⁰⁾=

 N+³

2



¯hω0⁻

1

2^{|e|Fβ , N= nx+ ny+ nz, nx, ny, nz}= 0, 1, . . . , (2.15) ψ_n⁽⁰⁾_x,ny,nz(x, y, z) = ϕnx^(x)ϕny^(y)ϕnz^{(z + β ), (2.16)} 2The last term of equation (2.14) is a constant, so it only modifies the eigenvalues of H₀but not its eigen- functions.

CHAPTER 2. AN EXCITON IN A SPHERICAL QD 8 with

ϕ_n(ζ ) = Nn^{exp(−α2ζ2/2)H}n(αζ ), ^Nn⁼

 α

√ π 2ⁿn!

1/2

, (2.17)

and

α =

rµ ω₀

¯h ^{. (2.18)}

Here H_nis the Hermite polynomial of order n, and ϕ_n(ζ ) is the normalized wave function of one-dimensional oscillator. Note that α = L⁻¹, with L given by equation (2.5).

From equation (2.15), the ground level (N = 0) of H₀ is a non-degenerate level. Thus, following the non-degenerate perturbation theory, the first-order correction to E₀⁽⁰⁾ is given by the matrix element

H₀₀⁰ _{= hψ0,0,0}⁽⁰⁾ | H⁰|ψ_0,0,0⁽⁰⁾ i = −^e

2

ε

Erf(αβ )

β ^{, (2.19)}

where Erf(ζ ) is the error function. The next energy level (N = 1) of H₀is triply degenerate, and we label its associated eigenstates |1i, |2i, and |3i, according to |1i = |ψ_0,0,1⁽⁰⁾ i, |2i =

|ψ_0,1,0⁽⁰⁾ i, and |3i = |ψ_1,0,0⁽⁰⁾ i. The first-order correction to E₁⁽⁰⁾ is obtained in the following manner. If i 6= j

H_{i j}⁰ = H⁰_ji= 0, (2.20)

with

H_{i j}⁰ = hψ_i⁽⁰⁾| H⁰|ψ⁽⁰⁾_j i , (2.21) and each subscript i and j denoting one of the degenerate eigenstates |1i, |2i, |3i. If i = j, we get

H₁₁⁰ = −^e

2

ε

1 + α²β²

_√

π Erf(α β ) − 2α β e^−α

2β^2

√

π α²β^{3 , (2.22)}

H₂₂⁰ = H₃₃⁰ = −^e

2

ε

2αβ e^−α2β2+ 2α²β²− 1
^√π Erf(α β )

2^√π α²β^{3 , (2.23)}

Equations (2.20), (2.22) and (2.23) show that the Coulomb potential is diagonal in the sub- space_E⁽⁰⁾₁ associated with the degenerate energy level E₁⁽⁰⁾. Therefore, the diagonal elements H₁₁⁰ , H₂₂⁰ , and H₃₃⁰ , are the energy corrections to the eigenstates |1i, |2i, and |3i, respectively. We also analyse the case when the electric field is zero. In this case, the results from the perturbation theory are

H˜₀₀⁰ = −^e

2

ε

√2α

π^{, (2.24)}

H˜₁₁⁰ = ˜H₂₂⁰ = ˜H₃₃⁰ = −^e

2

ε 4α

3^√π^{, (2.25)}

and ˜H_{i j}⁰ = 0 when i 6= j. The fact that ˜H₁₁⁰ = ˜H₂₂⁰ = ˜H₃₃⁰ is because of the three unperturbed eigenfunctions associated to the degenerate eigenvalue E₁⁽⁰⁾ are the same when β = 0 (see equation (2.16)), i.e. when F = 0. For F 6= 0 the eigenfunction ψ₁⁽⁰⁾ differs from both ψ₂⁽⁰⁾ and ψ₃⁽⁰⁾ giving rise to a privileged spatial direction set by the direction of the electric field. When the perturbation H⁰is introduced, the degeneracy of the level E₁⁽⁰⁾is partially removed leading to the result H₁₁⁰ 6= H₂₂⁰ = H₃₃⁰ .

2.4 Accuracy of the perturbation solution

The perturbation theory is valid only if the matrix elements of the perturbation H⁰ are much smaller than the differences between the energy of the level under study and those of the other levels [31]. Although we expect the perturbation results of section 2.3 fulfill the latter condition in the strong regime, we test their accuracy in that regime and also in the interme- diate one. Thus, we choose for GaAs/AlGaAs QDs a range of confinement energies from 1 meV to 1 eV, and we compare the energy corrections with the energy difference between two consecutive unperturbed levels, i.e. ¯hω0^.

To introduce the effect of the electric field, we take values of β ranging from 0 to 1.5L.³ With this restriction and using equations (2.5) and (2.13) the values of the electric field are

F= γF⁰, 0 ≤ γ ≤ 1.5, (2.26)

with

F⁰= ^ω0

|e| p

µ¯hω0^. (2.27)

In figures 2.3(a) through 2.3(d) we compare the energy corrections with ¯hω0 for different values of β . For β = 0 we observe in figure 2.3(a) that terms | ˜H₀₀⁰ | and | ˜H₁₁⁰ | = | ˜H₂₂⁰ | = | ˜H₃₃⁰ | are at least one order of magnitude smaller than ¯hω0 for confinement energies of 100 meV and above, i.e. in the strong regime. As β increases from 0 to 1.5L the terms |H₀₀⁰ |, |H₁₁⁰ | and

|H₂₂⁰ | = |H₃₃⁰ | become smaller than those for β = 0 as can be seen in figures 2.3(b), 2.3(c) and 2.3(d) for β = 0.5L, 1L and 1.5L respectively, but the confinement energies for which the energy corrections are much smaller than ¯hω0 are still above 100 meV. This analysis let us to conclude that the perturbation results are valid for GaAs/AlGaAs QDs with confinement energies equal or greater than 100 meV, or equivalently with QD sizes smaller than 4 nm (see figure 2.2(a)), but not for ¯hω0< 100 meV, i.e the intermediate regime.

Finally, the behavior of the energy corrections in figures 2.3(a) to 2.3(d) as a function of β can be explained using figure 2.4. In that figure we plot the energy corrections in units of e²/(εL) as a function of β /L ranging from 0 to 1.5. As can be observed in figure 2.4, the term |H₀₀⁰ | is greater than both |H₁₁⁰ | and |H₂₂⁰ | for β /L between 0 and 1, which explain the

3Recall that β is directly proportional to F via equation (2.13).

CHAPTER 2. AN EXCITON IN A SPHERICAL QD 10

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD ^ÈH

Ž

11 '

È ÈH^Ž00 '

È

(a) β = 0.

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD

ÈH22^'È ÈH11^'È ÈH00^'È

(b) β = 0.5L.

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD

ÈH₂₂^'È ÈH₁₁^'È ÈH₀₀^'È

(c) β = L.

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD

ÈH₂₂^'È ÈH₁₁^'È ÈH₀₀^'È

(d) β = 1.5L.

Figure 2.3. Comparison between the energy corrections and ¯hω₀for different values of β = 0.

0.0 0.5 1.0 1.5

0.5 0.6 0.7 0.8 0.9 1.0 1.1

L

Energy@e2 ΕLD ^ÈH22^{' È}

ÈH11^{' È}

ÈH₀₀^' È

Figure 2.4. Energy corrections as a function of the length β .

order of those terms in figures 2.3(a) and 2.3(b). At β /L = 1, |H₁₁⁰ | exhibit its maximum and the same value as |H₀₀⁰ |. For that reason |H₁₁⁰ | overlaps |H₀₀⁰ | in figure 2.3(c). When β /L > 1,

|H₁₁⁰ | is greater than both |H₀₀⁰ | and |H₂₂⁰ |, therefore |H₁₁⁰ | appears above the other two graph in figure 2.3(d). We attribute this particular behavior of |H₁₁⁰ | to the lift of degeneracy of the level E₁⁽⁰⁾ when the electric field is introduced. It seems that the field significantly affects the behavior of |H₁₁⁰ | leading to a growth up to β /L = 1, but when the electric field is greater,

|H₁₁⁰ | starts to decrease like the other ones.

Figure 2.4 is similar to figure 2 from reference [16], but our figure is quite general because it is independent of the value of the electric field.

Chapter 3

Numerical solution in COMSOL

Multiphysics

3.1 Statement of the problem

In this chapter we solve numerically the relative motion of the exciton, i.e. the stationary Schr¨odinger equation for the Hamiltonian (2.4). Before applying the numerical method, we reduce the complexity of the problem solving analytically the azimuthal angular dependence of the relative motion. Thus, we reduce the degrees of freedom of the relative particle from three to two, optimizing computational resources.

Introducing cylindrical-coordinates (ρ, φ , z) and following a similar procedure to that for equation (2.14), H_reltransforms into

H_rel= −^¯h

2

2µ^∇

2 c.c.⁺

1 2^{µ ω}

2 0^ρ2+

1 2^{µ ω}

2

0^{(z + β )2−}

1

2^{|e|Fβ −}

e²

ε (ρ²+ z²)^{1/2, (3.1)} where ∇²_c.c.is the Laplacian operator in cylindrical coordinates,

∇²_c.c.= ¹ ρ

∂

∂ ρ

 ρ ^∂

∂ ρ

 + ¹

ρ²

∂²

∂ φ²⁺

∂²

∂ z^{2. (3.2)}

Therefore, the Schr¨odinger equation takes the form H_relψ = Eψ ,

−^¯h

2

2µ

 1 ρ

∂

∂ ρ

 ρ ^∂

∂ ρ

 + ¹

ρ²

∂²

∂ φ²⁺

∂²

∂ z²



ψ + V (ρ , z)ψ = Eψ , (3.3) with

V_{(ρ, z) =} ¹ 2^{µ ω}

2 0^ρ2+

1 2^{µ ω}

2

0^{(z + β )2−}

1

2^{|e|Fβ −}

e²

ε (ρ²+ z²)^{1/2. (3.4)} 12

Since the potential V (ρ, z) does not have θ -dependence, we consider the wave function ψ (ρ , φ , z) as the product of two functions, one with an angular dependence and the other with a radial and z-coordinate dependence,

ψ (ρ , φ , z) = ψ1(φ )ψ2(ρ, z). (3.5) Substituting the last expression into equation (3.3) gives

− ^¯h

2

2µ

ψ₁ ρ

∂

∂ ρ

 ρ^{∂ ψ2}

∂ ρ

 +^ψ2

ρ²

∂²ψ₁

∂ φ^{2 + ψ1}

∂²ψ₂

∂ z²



+V (ρ, z)ψ1ψ2= Eψ1ψ2^{. (3.6)}

Multiplying by 2µρ²/(¯h²ψ₁ψ₂) and rearranging terms results in

−

 ρ ψ₂

∂

∂ ρ

 ρ^{∂ ψ2}

∂ ρ

 + ^ρ

ψ₂

∂

∂ z

 ρ^{∂ ψ2}

z



+Cρ²[V (ρ, z) − E] = ¹ ψ₁

∂²ψ₁

∂ φ^{2 , (3.7)} where C = 2µ/¯h², or equivalently

− ^ρ ψ2

[∇ · (ρ∇ψ2)] +Cρ²[V (ρ, z) − E] = ¹ ψ1

∂²ψ₁

∂ φ^{2 , (3.8)}

with

∇ = ^∂

∂ ρ^{ρ +ˆ}

∂

∂ z^{ˆz. (3.9)}

The left side of equation (3.8) is independent of φ and the right side is independent of both ρ and z, so the equation is equal to a constant which we call −m². Thus, equation (3.8) can be splited into two parts,

− ^ρ ψ2

[∇ · (ρ∇ψ2)] +Cρ²[V (ρ, z) − E] = −m², (3.10) and

1 ψ₁

∂²ψ₁

∂ φ^{2 = −m}

2_{. (3.11)}

The equation for ψ1gives the solution

ψ₁(φ ) = ^√1

2π^{exp(imφ ), (3.12)}

where the periodicity condition ψ₁(φ + 2π) = ψ1(φ ) implies that m = 0, ±1, ±2, . . . .

Using equation (3.10) with the restriction over m included, and multiplying by ψ₂/ρ, we arrive at the differential equation for ψ₂,

− [∇ · (ρ∇ψ₂)] +



CρV (ρ, z) +^m

2

ρ



ψ₂= CρEψ₂. (3.13)

The equation now exists in a form that we can immediately transfer into COMSOL Mutli- physics.

CHAPTER 3. NUMERICAL SOLUTION IN COMSOL MULTIPHYSICS 14

3.2 Description of COMSOL Multiphysics

COMSOL Multiphysics is a comercial software to solve and simulate several physics and engineering problems by finite element analysis. Its graphical user interface allows to build the corresponding geometry of the physics problem. The geometry can be either 3D, 2D or 1D, and it can reproduce from a carbon nanotube to a photonic crystal, without mention thousands of systems studied in engineering applications. In the field of QDs, it has been used to model a conical QD [32].

After building the geometry, the user should mesh it. In 2D (which will be our case), the mesh can be done with triangular elements, and their maximun size can be adjusted in each subdomain of the geometry.

COMSOL Multiphysics has a special partial differential equation (PDE) interface called Coefficient Form with the following default equation for an eigenvalue problem

∇ · (−c∇u − α u + γ ) + β · ∇u + au = d_aλ u (3.14) Thus, introducing the expressions for the parameters c, α, γ, β , a, and d_a, and the appro- priate boundary conditions, COMSOL Multiphysics will give the desired eigenvalues λ and eigenfunctions u of a particular eigenvalue problem.

3.3 Solving H

in COMSOL Multiphysics

3.3.1 Confinement energies of the QDs

Additionally to the reduced mass and dielectric constant which are listed in Appendix A, equation (3.13) requieres particular values of the confinement energy and the electric field to be solved. Hence we choose ten values of ¯hω0ranging from 1 meV to 1 eV, and for each one we take four values of F in a similar way as in section 2.4: β = 0, 0.5L, L and 1.5L. The chosen ¯hω0values and the corresponding characteristic sizes (equation (2.5)) and F⁰ values (equation (2.27)) are plotted in figure 3.1. Note that L varies from a few to tens of nanometers, while F⁰ranges from 1 to 10⁴kV/cm. We include the confinement energy 6.1 meV for which we have L = a^∗_B.

3.3.2 The geometry

We use a 2D geometry consisting of two concentric semicircular subdomains, being the hor- izontal axis ρ and the vertical z. The radius of the inner semicircle was chosen in such a way that the eigenfunctions were contained entirely within it. The outer semicircle was built with twice the radius of the inner one to ensure that the eigenfunctions vanish completely inside the geometry.

To construct this geometry we take two concentric circles having centers at the origin and with radii 5L and 10L, respectively. Then, we remove the portions to the left of the z-axis

æ æ

æ æ æ

æ æ

æ

æ æ

à à

à à

à à

à à

à à

1 5 10 50 100 500 1000

1 10 100

0.1 1 10 100 1000 10⁴

ÑΩ₀@meVD

L@nmD F' @kVcmD

Figure 3.1. Confinement energies considered in the numerical calculations and their corresponding QD characteristic sizes and F⁰values.

Figure 3.2. Geometry built to solve H_rel. ¯hω₀= 1 meV, i.e. L ≈ 44 nm. The axes units are meters and the horizontal and vertical axes are ρ and z, respectively.

CHAPTER 3. NUMERICAL SOLUTION IN COMSOL MULTIPHYSICS 16 to avoid negative ρ values because the use of cylindrical coordinates makes ρ positive. The resulting geometry is illustrated in figure 3.2 where subdomains 1 and 2 correspond to the semicircles with radii 10L and 5L, respectively.

3.3.3 The mesh

We mesh each subdomain of the geometry using triangular elements with a maximum size of 1% the corresponding semicircle radius. This percentage satisfies that the maximum element size is neither too small to make so long the calculation time, nor too large so that the solutions become unreliable. In this way, we study the convergence of the ground state energy for

¯hω₀= 1 meV and β = 0 as a function of the maximum element size relative to the semicircles radii, and the result is showed in figure 3.3. In that figure the energy of the ground state presents jumps as the relative size decreases, but when it reaches 2.5% the ground energy takes a stable value until 0.5%. Then an optimal relative size is 1%.

æ æ æ æ æ æ _æ

æ æ æ

æ æ

æ æ æ

æ æ

æ æ

æ

2 4 6 8 10

-3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6

Relative maximum element size @%D

Groundstateenergy@meVD

Figure 3.3. Study of convergence of the ground state energy for ¯hω₀= 1 meV and β = 0 as a function of the maximum element size relative to the semicircles radii.

A zoom of the resulting mesh for the geometry of figure 3.2 is shown in figure 3.4.

3.3.4 Boundary conditions

The boundary conditions are as follows. The z-axis is not a boundary in the original problem, but here it becomes one. We give the artificial boundary condition n · (ρ∇ψ₂) = 0. Because ρ = 0 on the z-axis, we should not view this boundary condition as a Neumann condition that imposes a constraint. Rather it indicates the absence of constraints.

On the outer semicircular boundary, the Dirichlet condition ψ₂= 0 applies. The Dirichlet condition corresponds to a zero probability for the relative particle to be outside the specified geometry.

Figure 3.4. Zoom of the resulting mesh of the geometry of figure 3.2. The right (left) side of the figure corresponds to the subdomain 1 (2). The axes units are meters and the horizontal and vertical

axes are ρ and z, respectively.

3.3.5 Parameters of the Coefficient Form PDE interface

Comparing equations (3.13) and (3.14), we set the parameters of the Coefficient Form PDE interface as shown in table 3.1.

Parameter Expression

c ρ

a CρV (ρ, z) + m²/ρ^

d_a Cρ

Table 3.1. Parameter expressions of the Coefficient Form PDE interface. The parameters α, γ and β are set to 0.

CHAPTER 3. NUMERICAL SOLUTION IN COMSOL MULTIPHYSICS 18

3.4 Numerical results

3.4.1 Eigenfunctions

We have obtained the eigenfunctions of the states |0i, |1i and |2i = |3i for the confinement energies shown in figure 3.1 and the values of β mentioned in section 3.3.1. In this section we shall discuss two interesting effects of the electric field over the relative particle eigen- functions. The first one is the shift of the eigenfunctions illustrated in figure 3.5 for ¯hω0^{= 6.1}

meV and two values of β , 0 and 1.5L. In those figures we observe that for β = 0 the eigen- functions are symmetrically distributed along the z-axis. However when the electric field is introduced by means of β = 1.5L the eigenfunctions are shifted toward negative values of z due to the electrostatic force exerted by the electric field over the electron and the hole. The amount of displacement is about 1.5L ≈ 27 nm, as we expected from the definition of β . The results for β = 0.5L and L also exhibit this behavior.

The second interesting effect is the deformation of the eigenfunctions for certain confine- ment energies. To show this effect we set β = 1.5L and we study the ground state eigenfunc- tions for six confinement energies: 2, 4, 6.1, 10, 20 meV and 1 eV. The resulting eigenfunc- tions are shown in figure 3.6. In that set of figures we find that the eigenfunctions are not only shifted by a length β but also they are elongated for ¯hω0= 2, 4, 6.1 and 10 meV, i.e. for confinement energies in the intermediate regime, while as we move towards the strong regime the eigenfunctions are only shifted, as it occurs for ¯hω0= 20 meV and 1 eV. We attribute the elongation to an interplay between the electric field, which displaces the eigenfunctions, and a relevant electron-hole Coulomb interaction (compared with the confinement potential, i.e. the parabolic potential) that binds the relative particle to the origin. In the strong confinement the Coulomb interaction is screened by the confinement potential so that the eigenfunctions are weakly binded to the origin.

Thus, we can conclude that the asymmetry introduced through the electric field is en- hanced in the intermediate regime compared with the strong regime since the eigenfunctions are both shifted and deformed.

3.4.2 Eigenvalues

We present in figure 3.7 the eigenvalue results comparing them with the unperturbed energies of the relative motion (equation 2.15). For each confinement energy of figure 3.1 and each studied state we plot |E_η− E_N⁽⁰⁾| as a function of ¯hω0, where E_η is the obtained numerical energy of the state |ηi, with η = 0, 1, 2, 3. In figure 3.7 we also plot the energy correction from the perturbation solution of Chapter 2 in order to compare the energy modifications when the Coulomb interaction is taken exactly. As for the eigenfunctions we present our results for β = 0 and 1.5L.

For all the states and β values considered the numerical results match very well with the perturbation results of section 2.3 in the strong-confinement regime, i.e. ¯hω0> 100 meV, but

(a) State |0i and β = 0. (b) State |0i and β = 1.5L.

(c) State |1i and β = 0. (d) State |1i and β = 1.5L.

(e) State |2i and β = 0. (f) State |2i and β = 1.5L.

Figure 3.5. Unnormalized eigenfunctions for ¯hω₀= 6.1 meV. The axes units are meters and the horizontal and vertical axes are ρ and z, respectively.

CHAPTER 3. NUMERICAL SOLUTION IN COMSOL MULTIPHYSICS 20

(a) ¯hω0= 2 meV. (b) ¯hω0= 4 meV.

(c) ¯hω0^{= 6.1 meV.} (d) ¯hω0^{= 10 meV.}

(e) ¯hω0= 20 meV. (f) ¯hω0= 1 eV.

Figure 3.6. Unnormalized ground state eigenfunctions for β = 1.5L. The axes units are meters and the horizontal and vertical axes are ρ and z, respectively.

they begin to differ as we approach to the intermediate-confinement regime, i.e. for lower confinement energies. We observe that this mismatch in the intermediate regime is more remarkable for the ground state than for the other ones, as is shown in figures 3.7(a) and 3.7(b). This may be explained in terms of the ∼ 1/r dependence of the Coulomb potential that alters mainly the eigenfunctions that are closest to the origin. As a suggestion, it is possible to employ the results of methods specially designed to solve the ground state like the non-local potential approach [33] and to get higher energy states by perturbation method for approximate calculations in the intermediate regime.

In figure 3.7 we also observe that when the electric field is introduced via the constant length β the mismatch in the intermediate regime between the numerical and perturbation results is enhanced. Therefore, the perturbation results are less reliable when the electric field increases, specially in the case of the ground state.

In conclusion, the figure 3.7 confirms that the perturbation results from section 2.3 can only be taken into account for confinement energies in the strong regime, and also it allows us to compare our numerical results with another method and that we trust them in the inter- mediate regime.

CHAPTER 3. NUMERICAL SOLUTION IN COMSOL MULTIPHYSICS 22

æ ^æ

æ ^æ æ

æ æ

æ

æ æ

à à

à à

à à

à à

à à

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD ^{à ÈH}

Ž

00 '

È æ ÈE0-E₀^H0LÈ

(a) State |0i and β = 0.

æ ^{æ æ}

æ ^æ æ

æ æ

æ æ

à à

à ^à à

à à

à

à à

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD ^{à ÈH00'È} æ ÈE0-E₀^H0LÈ

(b) State |0i and β = 1.5L.

æ æ

æ ^æ æ

æ æ

æ

æ æ

à à

à ^à à

à à

à

à à

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ0@meVD

Energy@meVD ^{à ÈHŽ}11 '

È æ ÈE1-E₁^H0LÈ

(c) State |1i and β = 0.

æ æ

æ ^æ æ

æ æ

æ

æ æ

à à

à ^à à

à à

à

à à

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ0@meVD

Energy@meVD ^{à ÈH}11 'È æ ÈE1-E₁^H0LÈ

(d) State |1i and β = 1.5L.

æ æ

æ ^æ æ

æ æ

æ

æ æ

à à

à à

à à

à à

à à

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD ^à ÈH^Ž₂₂^'È æ _ÈE2-E

1 H0LÈ

(e) State |2i and β = 0.

æ æ

æ ^æ æ

æ æ

æ

æ æ

à à

à ^à à

à à

à

à à

1 5 10 50 100 500 1000

1 2 5 10 20 50 100

ÑΩ₀@meVD

Energy@meVD ^{à ÈH}22^'È

æ _ÈE2-E

1 H0LÈ

(f) State |2i and β = 1.5L. Figure 3.7. Comparison between the numerical and the unperturbed energies.

Chapter 4

Nonlinear optical properties

4.1 Theoretical framework

4.1.1 Nonlinear optics

The nonlinear optical phenomena occur when the response of a material to an incident elec- tromagnetic wave depends nonlinearly on the intensity of the field. In the linear case the response of the material, described by the polarization P(t), depends linearly on the magni- tude E(t) of the electric field, i.e.

P_{(t) = ε0}χ⁽¹⁾E(t), (4.1)

where ε₀ is the vacuum permittivity, and the proportionality constant χ⁽¹⁾ is the linear sus- ceptibility.

In nonlinear optics P(t) depends not only on powers of first order of E(t), but also of higher orders. Thus, P(t) is expressed as a power series of E(t)

P_{(t) = ε0[χ}⁽¹⁾E_{(t) + χ}⁽²⁾E²_{(t) + χ}⁽³⁾E³(t) + · · · ]

= P⁽¹⁾(t) + P⁽²⁾(t) + P⁽³⁾(t) + · · · . (4.2) The quantities χ⁽²⁾ and χ⁽³⁾ are known as the nonlinear optical susceptibilities of second and third order, respectively. For simplicity, we assumed that the applied field is polarized, and that the material is isotropic and non-dispersive. In the general case, χ⁽¹⁾ becomes a second-rank tensor, χ(2) becomes a third-rank tensor, and so on.

In general, the nonlinear susceptibilities depend on the frequencies of the applied fields. As an example [34] consider a material with a nonzero second-order susceptibility, and an electric field

E(t) = Ee^iωt+ c.c.. (4.3)

23

CHAPTER 4. NONLINEAR OPTICAL PROPERTIES 24 The second-order nonlinear polarization that is created in such a material is given according to equation (4.2) as

P⁽²⁾(t) = ε0χ⁽²⁾E²(t) = 2ε0χ₀⁽²⁾EE^∗+ (ε0χ_2ω⁽²⁾E²e^−i2ωt+ c.c.). (4.4) We see that the second-order polarization consists of a contribution at zero frequency (the first term on the right hand side), and a contribution at frequency 2ω (the second term). The first contribution does not lead to the generation of electromagnetic radiation, but it leads to a process known as optical rectification (OR), in which a static electric field is created across the nonlinear material. The associated susceptibility is known as the OR susceptibility χ₀⁽²⁾. The second contribution can lead to the generation of radiation at second-harmonic frequency, a process called second-harmonic generation (SHG), and the corresponding susceptibility χ_2ω⁽²⁾ is the SHG susceptibility.

4.1.2 Density-matrix approach

The interaction of an electromagnetic wave in the far infrared range (in which we will focus our attention) with charge carries confined in a QD can be described in a dipole approxima- tion. For an electron-hole pair, the interaction Hamiltonian is given by [16]

H_dip= −e(r_e− r_h) · E(t), (4.5)

where E(t) is the electric field. In terms of the exciton relative coordinates, H_dip is expressed as

H_dip= −ezE(t), (4.6)

where we have assumed that the electric field is polarized along the z-direction. Since H_dip does not depend on the center-of-mass coordinates, the electromagnetic wave cannot excite the exciton center-of-mass motion, and so affects merely the relative motion.

Consider an exciton excited by an incidence electric field of the form (4.3). The evolution of the density matrix is given by the master equation

∂ ρ_{i j}(t)

∂ t ⁼ 1

i¯h^[Hrel− ezE(t), ρ(t)]_{i j}− Γ_{i j}(ρ_{i j}(t) − ρ_{i j}⁽⁰⁾(t)), (4.7) with ρ⁽⁰⁾(t) the density matrix in absence of any external electromagnetic field, and Γ_{i j} the diagonal and off-diagonal relaxation rates. The last equation can be solved using an iterative method [18],

ρ (t) =

∞ n=0

∑

ρ⁽ⁿ⁾(t). (4.8)

Substituting equation (4.8) into equation (4.7) we get the following recurrence formula

∂ ρ_{i j}⁽ⁿ⁺¹⁾(t)

∂ t ⁼

1 i¯h

n

[H_rel, ρ⁽ⁿ⁺¹⁾]_{i j}− i¯hΓi j^ρ (n+1) i j

o− ¹ i¯h^{[ez, ρ}

(n)_]

i j^{E(t). (4.9)}