Eindhoven University of Technology MASTER Strain-induced g-factor tuning in individual InGaAs quantum dots Tholen, H.M.G.A.

(1)

Eindhoven University of Technology

MASTER

Strain-induced g-factor tuning in individual InGaAs quantum dots

Tholen, H.M.G.A.

Award date:

2015

Link to publication

Disclaimer

This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration.

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

(2)

Strain-induced g-factor tuning in individual InGaAs quantum dots

H.M.G.A. Tholen

Under supervision of:

Prof. dr. P.M. Koenraad

dr. A.Y. Silov

(3)

Abstract

Self-assembled quantum dots are semiconductors nanostructures whose charge carriers are confined in all three spatial dimensions. Quantum dots have often been suggested as fundamental building blocks for future quantum computing and spintronic devices. When exposed to a magnetic field, the energy levels of a quantum dot experience a splitting which is char- acterized by the g-factor. For the envisioned spintronic applications it is highly desirable to gain full control over this g-factor, as tuning this around zero is at the essence of quantum computing. This thesis presents the possibility to manipulate the exciton g-factor by making use of external strain fields.

A numerical model based on k · p-theory was used to predict the effect of strain on the electronic properties of the quantum dots. The electron and hole g-factors were determined by calculating the ground states of the system as a function of magnetic field. The electron and hole g-factors both showed significant changes, with the electron g-factor increasing and the hole g-factor decreasing as a function of strain. The behaviour of the electron g- factor was explained in the framework of orbital momentum quenching. The hole g-factor has proven to be a more complicated matter and more research is required to gain a full physical picture of this.

Magnetoluminescence experiments were performed on individual InGaAs quantum dots.

Strain was incorporated by mounting quantum dot layers onto piezo-electric material. The exciton energies showed a linear increase with applied strain, agreeing with previous experiments and our calculations, both qualitatively and quantitatively. We have shown for the first time that the exciton g-factor indeed depends on strain. We found a linear dependence, with the g-factor decreasing several percent over the investigated range of strain.

This is a very promising effect as state of the art samples are capable of achieving much higher strains (at least a factor of ten) than what has been achieved in our sample.

Furthermore, by playing around with the size and composition of the quantum dots, g- factors close to zero can be engineered. We therefore anticipate that strain is indeed a viable way to tune g-factors and can be used to reach the envisioned goals.

(4)

A brief introduction to quantum dots

The invention of the technology behind computers several decades ago has lead to significant changes in our everyday lives and continues to do so. This seemingly everlasting revolution is only possible due to the constant strive for more powerful computers. Famous is Moore’s law, which states that the density of transistors on a chip, the fundamental building blocks of computers, doubles approximately every two years. To achieve this, the working elements need to be decreased in size, down to the point where their physics is dominated by quantum mechanics [1]. This leads to new physical phenomena which have been subject to much interest from the scientific community.

Quantum wells, wires and dots, which can be seen as respectively 2D, 1D and 0D structures, have been of particular interest. Not only due to their importance in current- day technology, but also for envisioned new technologies. It is generally believed that the current electronic technology will one day reach its limit. Therefore new devices will have to be developed. A promising technology is quantum information processing, for which quantum dots have been suggested as excellent fundamental building blocks [2].

1.1 Fabrication

The most common way to obtain quantum dots is to epitaxially grow one semiconductor material on top of a different one. Here it is essential that these materials have different lattice constants. When doing this, three different growth modes can occur, depending on the growth conditions. In Frank van der Merwe growth the interaction between the atom and the substrate is stronger than the interaction between individual atoms. This causes layer-by-layer growth. Volmer-Weber growth is the opposite case, in the sense that the interaction between individual atoms is stronger. This leads to the formation of islands. If the growth conditions are chosen correctly, a combination of the two will occur, called Stranski- Krastonow growth. First the layer-by-layer growth is energetically favourable. Due to strain, after the growth of a few layers the island-formation becomes energetically favourable. The growth modes are schematically shown in figure 1.1.

These islands are the quantum dots, typically with a lateral size of 20 − 30nm and a height of 6 − 8nm. The layer that is deposited before the formation of quantum dots is

(7)

Figure 1.1: The different growth modes that can occur when growing a semiconductor on top of a different one. Left: Frank van der Merwe growth, where the atom-substrate interaction is larger than the atom-atom interaction, causing layer-by-layer growth. Right:

Volmer-Weber growth, where the atom-atom interaction is the largest, leading to the formation of islands. Middle: Stranski-Krastonow growth, where first layer-by-layer growth is energetically favourable, until island-formation takes over due to strain. This is the most common way of creating quantum dots.

called the wetting layer. Due to their small sizes, the dots exhibit quantum mechanical properties, hence the name quantum dots. The composition of quantum dots can be chosen as desired. By controlling, amongst others, the temperature and time of the growth process, the average quantum dot size can be controlled.

There are several ways to grow semiconductor materials, for instance molecular beam epitaxy (MBE), metalorganic vapour phase epitaxy (MOVPE) or chemical beam epitaxy (CBE). The former was used to create the samples studied during this project. In MBE, the desired semiconductor materials (for instance gallium or arsenic) are, in pure form, heated in separate cells until slow sublimation takes place. The gaseous atoms are released into the main chamber, where they will eventually condense onto the wafer. On the wafer, they are able to react with each other and create the targeted material. For instance, using both gallium and arsenic, a gallium arsenide crystal is grown. Because of the low density (and thus a long mean free path), the evaporated atoms do not interact with each other until they have been deposited onto the wafer. This takes place in ultra-high vacuum and requires no carrier gasses, resulting in the highest achievable purity.

1.2 Controlling electronic properties

The main point of interest of quantum dots comes from their analogy to atoms; due to their nanoscale sizes, particles inside a quantum dot are confined in all three directions and are only allowed to occupy some discrete energy levels. Quantum dots are therefore often called artificial atoms. The energy levels are determined by a constant battle between several physical phenomena among which are quantum confinement, Coulomb interactions and exchange effects. These quantum mechanical interactions have proven to be incredibly sensitive to the exact structure of the quantum dot; its size, shape and composition are the first things that come to mind, but also less straight-forward effects like the strain profile inside the quantum dot play an important role.

(8)

The growth process of self-assembled quantum dots is semi-random. Therefore no two quantum dots are identical and each quantum dot will have unique properties, or fingerprint so to speak. This causes major problems for the envisioned applications of quantum dots.

As it is impossible to produce quantum dots on a large scale while being able to exactly control their electronic properties, post-growth techniques are required to achieve the desired properties. Exposing quantum dots to external magnetic, electric and strain fields has already proven to have large effects on their electronic structure. Therefore these can be used as external knobs to make quantum dots electronically identical, at least to some extent. Magnetic fields are known to split and shift the energy levels of the quantum dot.

This has been thoroughly investigated in the past and is well understood [3]. The effects of strain and external electric fields have been shown more recently [4]. One way to induce strain in quantum dots is to incorporate them in a bowed airbridge structure [5] which has proved to be able to shift the exciton energy by ∼ 6meV . Another approach was recently also tested, where the quantum dots are mounted on top of piezo-electric material, allowing for biaxial strains. This lead to energy shifts up to 15meV . They even showed that the exciton and biexciton states of a quantum dot can be independently controlled by making use of a combination of strain and external electric fields [6]. This is necessary to achieve energy coincidence between the two states, which is relevant for the generation of entangled photon pairs using the time-reordering scheme [7]. This approach to inducing strain will also be used in this thesis.

1.3 Spin

Current-day technology makes use of the charge of electrons. Besides charge, electrons also carry an intrinsic momentum referred to as spin. This spin provides another degree of freedom which can be used to encode and process information. Using this in applications requires a full control over these spins. Manipulating spins is already possible by making use of local magnetic fields. However, making use of the internal electronic structure of the quantum dots provides a more powerful to gain control over this. One important quantity in this is the g-factor, which is a measure of the magnetic moment of a particle and therefore related to its spin. External magnetic fields can couple to this spin, resulting in a splitting of the energy levels proportional to the g-factor. This makes finding a way to manipulate g-factors a necessity. Both electric fields and strain can be used to achieve this. They both lead to a change in confinement that causes a change in orbital momentum of the charge carriers in the quantum dot, which is an important component of the g-factor. It was recently shown that electric fields indeed affect the exciton g-factor [8]. The effects of strain on the g-factor of individual dots have never been investigated. This will therefore be the goal of this thesis.

1.4 Scope of this thesis

In this thesis we will investigate the effects of strain on the electronic structure, emission energies and g-factors of InGaAs quantum dots. In the first chapter the harmonic oscillator potential model will be introduced to discuss some fundamental properties of quantum dots.

This is necessary to understand well the working principle of quantum dots and what interactions play an important role. The second chapter will make use of the more sophisticated k · p-theory, which allows us to incorporate the effects of strain. The third chapter will give

(9)

an overview of the experimental techniques that are used during this research. This mainly includes the confocal microscope and its working principle. A numerical model based on k · p-theory will be presented in the fourth chapter. Using the results of these calculations, we attempt to gain a better understanding of how external strain will affect the quantum dots in our experiment. Finally, experiments performed on individual InGaAs quantum dots will be discussed in the fifth chapter. Its results will be compared to the numerical calculations.

(10)

Chapter 2

Quantum dots in the harmonic oscillator model

The following chapter will introduce the harmonic oscillator potential model. This model can, in good approximation, be applied to quantum dots and will prove to be able to describe some of the quantum dots most fundamental properties. We will elaborate on what interactions play important roles in quantum dots and how an external magnetic field affects their electronic properties.

2.1 Electronic structure of a quantum dot

To understand the physics behind the experiments, a theoretical model describing the electronic structure of the quantum dots is required. In general, the wave function Φi of a particle in a quantum mechanical system can be obtained by solving the Schr¨odinger equation, given by

HΦi= EiΦi, (2.1)

where H is the Hamiltonian describing the system and E_i is the eigenenergy corresponding to wavefunction Φ_i. The exact Hamiltonian is complicated, as many things have to be taken into account (i.e. the composition profile and strain.) Therefore describing the electronic structure of a quantum dot exactly would require an advanced and complicated model. However a simple approximation that will prove to be able to describe the basic physics behind quantum dots quite accurately can be made by using the harmonic oscillator potential (VQD) [9], given by

V_QD= 1

2m^∗(ω²_xx²+ ω_y²y²+ ω²_zz²). (2.2) In 2.2 m^∗ is the effective mass of a particle (electron or hole) and ωx,y,zare the eigenfre- quencies in the various directions. In quantum dots, x and y are usually chosen to be the lateral directions and z to be the growth direction. The system inside a magnetic field [10]

is described by a Hamiltonian given by H = 1

2m^∗(p − qA)²+ VQD, (2.3)

(11)

where q is the electric charge of the particle, p = −i~∇ the momentum operator and A the vector potential describing the magnetic field. The latter is related to the magnetic field B by

B = ∇ × A (2.4)

If we first consider the case that there is no magnetic field (B = 0), the Schr¨odinger equation can be solved [11] [12], resulting in

E_i= (n_x+1

2)~ωx+ (n_y+1

2)~ωy+ (n_z+1

2)~ωz (2.5)

Φi(r) = ψⁿ_x^x(x)ψ_yⁿ^y(y)ψ_zⁿ^z(z) (2.6) where i is a label containing quantum numbers nx, ny and nz and r is the three dimensional vector (x, y, z). The subwavefunctions ψⁿ_α^α are expressed by

ψⁿ_α^α(α) = (mω_α

π~ )¹⁴ 1 2^nα² √

nα!e⁻^2~¹^mω^α^α²H_n_α(αr mω

~

) (2.7)

H_n_α(ξ) = (−1)ⁿ^αe^ξ² dⁿ^α

dξⁿ^αe^−ξ² (2.8)

where α indicates either of the three spatial dimension coordinates x, y or z and H_n_α(ξ) are Hermite polynomials. A one-dimensional harmonic potential and its corresponding eigenenergies and wavefunctions are shown in figure 2.1. The energy of the ground state is

1

2~ω. Energies of excited states are equidistant with a splitting of ~ω.

Figure 2.1: The energy levels of the harmonic oscillator model (left) and the wavefunctions inside a quantum dot (right). The smaller the quantum dot, the larger its energy, where its height plays a more important role than its lateral size.

The lateral size of a quantum dot is usually in the order of ≈ 30nm, whereas the height is in the order of ≈ 5nm. Therefore the frequency in the growth direction ω_z is larger than in the lateral direction, making the confinement energy mostly dependent on the quantum dot height. The wavefunction has a Gaussian shape in the ground state, which decays exponentially into the host material. One can now define a typical lateral extension length lα as

(12)

lα= r 2~

m^∗ω_α (2.9)

where m^∗ denotes the effective mass of the particle. The effective mass of a particle can be seen as the mass it seems to have when responding to other physical quantities.

This lateral extension length will have to match the size of the quantum dot in order to fit the wavefunction inside. If l_α decreases, meaning the dot is smaller, ω_α, and with it the confinement energy, increases.

2.2 Magnetic field

In a classical picture, a charged particle gives rise to a magnetic moment µ when travelling.

When exposed to a magnetic field B, this moment will couple to it. The magnetic energy is then given by Emagnetic = −µ · B. In an attempt to minimize this energy, the magnetic moment will always try to align parallel to the magnetic field. In the case of semiconductors, these charged particles are the electrons and holes that orbit in the semiconductor crystal.

The coupling of this orbital momentum to the magnetic fields results in a so-called diamagnetic shift, which will be explained in the next section. Besides this orbital momentum, the charge carriers also possess an intrinsic magnetic moment referred to as spin, which itself will also couple to the magnetic field. This gives rise to a Zeeman splitting, which will be explained in the section after.

When taking the magnetic field in the growth direction, the energy levels for the harmonic oscillator are described by the Fock-Darwin energies [13];

El,m_z,n= (2l + 1 + |mz|)~

r (ωc

2 )²+ ω²_x,y+mz

2 ~ωc+ (n +1

2)~ω^z (2.10) In 2.10 ωc = _m^qB∗ is the cyclotron frequency, l is the angular momentum, mz is its projection along the z-axis and n is the quantization in the harmonic oscillator potential in the z-direction. In the ground state (l = m = n = 0) the Fock-Darwin equation simplifies to

E0,0,0= ~ r

(ωc

2 )²+ ω²_x,y+1

2~ωz (2.11)

The equation in 2.11 provides us with two limiting cases. If the magnetic field is very large, meaning ωc>> ωx,y, the energy will be linear in magnetic field. However, this effect is not observed for magnetic fields < 10T . Therefore we are only interested in the limit of small magnetic field, ωx,y >> ωc. One can perform a Taylor expansion on the ground state of this system, which results in

E0,0,0= ~ω^x,y+ ~ωc²

4ωx,y

+1

2~ωz, (2.12)

which, using 2.9, can be written as

E0,0,0= E0+e²B²l²_x,y 8m^∗

= E0+ αdB²

(2.13)

(13)

where E0is the ground state energy in absence of a magnetic field and αdis the diamagnetic coefficient. Due to the quadratic dependency of the diamagnetic shift on the lateral extension length, larger dots will have a larger increase in energy by magnetic field as opposed to smaller dots. In essence, the diamagnetic coefficient is a measure for the average size of the wavefunction. This will prove to be a crucial concept for understanding the behaviour of the diamagnetic shift.

2.2.1 Zeeman splitting

As stated at the beginning of this chapter, the charge carriers in the quantum dots also posses a spin s, which will also interact with the magnetic field, called the Zeeman interaction. The magnetic moment µsof a charge carrier is related to its spin by [14]

µ_s= g₀q~

2ms (2.14)

where g₀is the gyromagnetic factor, also called the g-factor. For a free electron its value is about 2. To simplify things, we introduce the Bohr magneton µ_B = _2m^~e. The Zeeman Hamiltonian for a free electron is then given by

H_Zeeman= µ_s· B

= g0µBs · B (2.15)

When considering non-free electrons, for instance the electrons in a semiconductor crystal, the gyromagnetic factor g0 is replaced by an effective electron g-factor ge, which takes into account the band structure of the crystal. The magnetic moment of a particle will then have two major contributions; one from its spin and the other from its orbital motion. For semiconductor nanostructures, the effective g-factor can no longer be determined analytically and numerical calculations are required to find values for it. The coupling of the carrier spin to the magnetic field results in a splitting in the energy, called the Zeeman splitting, which is given by

∆E_Zeeman= g_eµ_BB. (2.16)

For the holes (electron vacancies) in a semiconductor crystal, matters are a little more complicated. The hole states are located in the valence band and experience a spin-orbit coupling. This means that the orbital motion of a hole induces an effective magnetic field which will couple to its own spin. Therefore the spin can no longer be regarded as a proper quantum number. It has been proven that the total angular momentum J = L + s is still a proper quantum number. In the valence band L = 1, which means that for holes J is either ³₂ or ¹₂. The latter corresponds to split-off states which will be neglected here.

J = ³₂ can be split into heavy holes and light holes according to the z-projection of the total angular momentum Jz. For the heavy holes Jz = ±³₂ while for the light holes Jz = ±¹₂. It can be shown that the heavy hole character dominates in the confined state, so the hole pseudospin can be considered to be J_z = ±³₂. Then the hole states can formally be treated the same as the electron states. Going from a bulk semiconductor to a nanostructure complicates matters even more. However, the Zeeman Hamiltonian remains valid and every quantum dot state, both electron and hole, will experience a Zeeman shift, each with its own characteristic g-factor. Due to mixing between the heavy hole and light hole bands the

(14)

g-factor is in general not perfectly linear, but also contains a quadratic term. However, in the quantum dots studied in this report this quadratic dependence is not observed and the heavy hole-light hole mixing is negligible.

2.3 Excitonic states

When doing a photoluminescence experiment, an electron from the valence band is excited (by a laser) to the conduction band, leaving a hole in the valence band. There will be a Coulomb interaction between this electron and hole, creating an electron-hole pair called an exciton. When the electron and hole recombine, and thus the exciton annihilates, a photon is created and emitted, which can then be detected to provide information about the quantum dots band structure. In the neutral exciton X₀, consisting of one electron-hole pair, the electron has spin s_z = ±¹₂, while the hole has pseudospin J_z = ±³₂. This results in four possible spin configurations, denoted as |X, Xz, where X = |sz+ Jz| and Xz its projection along the z-axis, which are degenerate when the spin-Hamiltonian is neglected.

In a quantum dot strain causes a splitting between the light hole and heavy hole states of several tens of meV. As the fine-structure interaction energies are in the order of several meV, only the heavy hole has to be taken into account and the light hole-heavy hole mixing can be neglected. When we choose | ↑ and | ↓ to represent the electron spin and | ⇑ and | ⇓ to represent the heavy hole pseudospin, these four spin configurations are

| ↓| ⇑ = |1, 1

| ↑| ⇓ = |1, −1

| ↑| ⇑ = |2, 2

| ↓| ⇓ = |2, −2

(2.17)

Within the dipole approximation only the X = 1 are optically active. As photons are bosons with spin ±1, the momentum of the X = 1 states can be transferred to a photon.

Therefore these states are referred to as bright states, while the X = 2 states are referred to as dark states. It can also be derived that the photons emitted by bright excitons are either right handed (σ⁺) or left handed (σ⁻) circularly polarized. Using this description, it is possible to demonstrate how the exchange interaction affects the exciton states. The exchange Hamiltonian can be written in terms of spin. When not assuming rotational invariance, which will be the case for most quantum dots, and neglecting heavy hole-light hole mixing the exchange Hamiltonian becomes [16]

H_exchange= − X

α=x,y,z

(a_αs_αJ_h,α+ b_αs_αJ_h,α³ ) ' −azszJh,z− X

α=x,y,z

bαsαJ_h,α³

(2.18)

Here Jiis the hole spin, sithe electron spin and ai, bimaterial parameters. Representing this in a matrix with the possible spin configurations as basis we obtain [17]

(15)

Hexchange= 1 2







δ0 δ1 0 0

δ1 δ0 0 0

0 0 −δ0 δ₂

0 0 δ₂ −δ0







(2.19)

where δ0= ³₂(az+⁹₄bz), δ1 = ³₄(bx− by) and δ2= ³₄(bx+ by). As the exchange matrix has a block diagonal form, the bright and dark excitons do not mix with each other. The two dark states however do mix with each other, while the two bright states will only mix in the absence of rotational symmetry (bx 6= by). The mixing between the excitons will cause an energy splitting, which is generally referred to as the finestructure splitting, but which we will neglect here as we generally do not observe it in our experiments. This is true when measuring in the Faraday geometry, where the growth direction of the quantum dots is the same as the direction of the magnetic field. In the Voigt geometry the magnetic field is perpendicular to the growth direction, which results in a mixing between the bright and dark states. This allows for detection of both the bright and dark states. Applying a magnetic field will result in a diamagnetic shift α_ex and a Zeeman splitting g_X. When we assume the diamagnetic shift to be the same for dark and bright states, the excitonic energies are related to them by

E(H) = E_X=2±1

2g_X=2µ_BH + α_exH² (2.20) E(H) = E_X=1±1

2g_X=1µ_BH + α_exH² (2.21) The Coulomb and exchange interactions will result in small corrections on 2.20 and 2.21, but these will be neglected here. Simple expressions for the diamagnetic shift and g-factors can be then obtained;

E_X=1≈ E_X=2

= |Ee− Eh| (2.22)

αex = αe− αh (2.23)

gX=1= gh− ge

≡ gex

g_X=2= g_h+ g_e

(2.24)

In our experiment these quantities will be determined by measuring the exciton energy as a function of magnetic field. Using the expressions in 2.24, the electron and hole g- factors can be obtained from measuring the exciton g-factors. The experiments discussed in this thesis are performed in the Faraday geometry. Because of this only the exciton g- factor can be determined. To determine the hole and electron g-factors separately, the Voigt geometry would have to be used. The main results of the theory discussed in this section are summarized in figure 2.2.

(16)

Figure 2.2: Excitons at zero magnetic field (top) as opposed to non-zero magnetic field (bottom) for the single particle picture (left) and the many particle picture (right). In the single particle picture both the hole and electron are confined to the quantum dot, where recombination of the two results in a photon with either left-handed or right-handed circular polarization. When applying a magnetic field, the electron and hole each experience a different diamagnetic shift and Zeeman splitting. In the many particle picture, the electron and hole form an exciton through their Coulomb interaction. At zero magnetic field the fourfold degenerate ground state already experiences mixing. As a function of magnetic field, these mixed states shift and split further. Only the excitons with X = 1 (the bright excitons) are luminescent and are able to annihilate upon emission of a photon.

(17)

Chapter 3

The effects of strain on quantum dot properties

In the previous chapter we introduced a simple model based on the harmonic oscillator potential, which was already capable of describing some fundamental properties of quantum dots. However, investigating the effects of strain on these properties requires a more sophisticated model. A very popular approach to do so is the so-called k · p-theory, which will be introduced in this chapter. To get a better understanding of the physics behind strained quantum dots, first the concepts of strain and band structures have to be introduced.

3.1 Strain

This section will discuss the origin of strain and how it can be defined. To explain what strain is, first consider a one dimensional rod with length L. When elongating this rod to length l, the strain is given by

=l − L

L . (3.1)

When considering a three dimensional body, and dividing it into small elements, each element may experience both a normal strain and a shear strain. Normal strain acts perpendicular to the face of the element, while shear strain acts along the face. This builds up a three-dimensional strain tensor, where the diagonal components describe the normal strain and the off-diagonal components the shear strain. Each element can be described by a displacement vector u = u_x+ u_y+ u_z[18]. When choosing infinitesimally small elements, the components of the strain tensor consist of derivatives of this displacement vector via

_ij =1

2(u_i,j+ u_j,i) (3.2)

where u_i,j represents the derivative of displacement u_i with respect to j. The complete strain tensor is then given by

(18)

=





xx xy xz

yx yy yz

zx zy zz



=







∂u_x

∂x

1

2(^∂u_∂y^x+^∂u_∂x^y) ¹₂(^∂u_∂z^x +^∂u_∂x^z)

1

2(^∂u_∂x^y +^∂u_∂y^x) ^∂u_∂y^y ¹₂(^∂u_∂z^y +^∂u_∂y^z)

1

2(^∂u_∂x^z +^∂u_∂z^x) ¹₂(^∂u_∂y^z +^∂u_∂z^y) ^∂u_∂z^z





 (3.3)

For symmetry reasons xy = yx will be true. In a semiconductor crystal, the elements are generally chosen as Bravais cells containing one atom each. The strain tensor is then discretized by using the next-neighbours for each atom. In this thesis only the normal strain is treated and a few terms regarding strain are used. First there is hydrostatic strain, where the normal strain components all equal to each other (_xx = _yy = _zz ≡ ). The strain component is then equal to the relative change in volume of the system, ^∆V_V . In uniaxial strain, one of the diagonal components will be non-zero, while in biaxial strain there will be two non-zero components on the diagonal. As we choose z to be the growth direction of our quantum dots, xxand yy are the in-plane strain components _k, while zz represents the out-of-plane strain component _⊥. In our experiments, an in-plane biaxial strain will be added to the quantum dots. This strain will be assumed to be isotropic, meaning xx= yy. Let us now consider a thin epilayer of one semiconductor material grown on a substrate of another. If the layer is sufficiently thin, all of the strain will be incorporated in the layer, and not in the substrate. The difference in lattice constants of the epilayer and substrate, denoted by respectively ae and as, will result in an in-plane strain in the epilayer given by [19]

_k= xx= yy= as− ae

ae

. (3.4)

The in-plane strain leads to a strain component in the growth direction, which is of opposite sign and given by [20] [21]

_⊥ = − 2σ

1 − σ_k. (3.5)

Here σ represents the Poisson ratio, which for the type of semiconductor studied in this report is roughly ¹₃. Using this value yields ⊥ ≈ −_k. In general the radius of a quantum dot is much larger that its height. Because of this, the analysis for the epilayer is also to some extent applicable to the inner region of a quantum dot. The edges of the dot will show some transitional behaviour.

3.2 Band structure

Before we can investigate the effects of the strain described by the previous section on quantum dots, we must first elaborate on the bandstructure of a semiconductor. Each atom in a crystal has a number of discrete energy levels which its electrons can occupy. The coupling between these atomic states will create energy bands. Electrons are best described by a wavevector k, which represents the direction of motion of the electron. For each wavevector, certain energy levels are accessible and others are not. Altogether this creates a bandstructure given by n(k), where nrepresents the energy of band n. For semiconductors there will be an area in energy that contains no states at all, which is called the band gap.

The bands above and below this gap are referred to as the conduction and valence band respectively. The first state above (below) the bandgap is therefore called the conduction

(19)

(valence) band edge. In most semiconductors the band structure around these bandedges can locally be approximated by [18]

E(k) = E₀+~²k²

2m^∗ (3.6)

where E0 is the energy of the bandedge and m^∗ denotes the effective mass. Knowing this, the effective mass of a particle in band n can be calculated by

1

m^∗_n = ∂²En

∂k² . (3.7)

It is important to note that the valence band has a p-type character, since it originates from p-type atomic orbitals which have an orbital momentum of L = 1. The conduction band comes from atomic orbitals having no orbital momentum (L = 0) and therefore has an s-type character. Electrons fill up the bands up to the Fermi level, which for semiconductors is located inside the band gap. For indirect semiconductors, the conduction and valence bandedges are located at different wavevectors k. An electron in the conduction band will therefore not be able to recombine with a hole (electron vacancy) in the valence band. For direct semiconductors, the wavevectors are the same and will therefore be optically active.

An electron can be excited from the valence band to the conduction band. By generating phonons, the electron (hole) relaxes to the bottom (top) of the valence (conduction) band, after which it is able to recombine with the hole. A photon will be created during this process. In a photoluminescence experiment, these photons can be detected, providing valuable information about the properties of the semiconductor. This recombination will take place around k = 0. We are therefore only interested in the band structure around this point and it is helpful to be able to approximately describe this. This can be done using k · p-theory.

3.3 k · p-theory

In k · p-theory, the band structure in the entire Brioullin zone can be extrapolated from just the zone center energy gaps and optical matrix elements, which can both be determined by experiments. The theory is derived by combining the time-independent one-electron Schr¨odinger equation (3.8) with the Bloch theorem [22]:

Hφ(r) = p²

2m+ V (r)

φ(r) = Eφ(r). (3.8)

The Bloch theorem states that the energy eigenfunction in a periodically-repeating en- vironment (a crystal) can be written as the product of a slowly varying plane wave envelope function and a fast oscillating periodic function with the same periodicity as the crystal lattice. This gives

φnk(r) = e^ık·runk(r), (3.9)

where k is the wave vector and n is the band index. Using the definition of the momentum-operator p = ı~∇ and substituting 3.9 into 3.8 leads to an equation in unk:

p²

2m+~k · p

m +~²k² 2m + V

unk= Enkunk. (3.10)

(20)

At zone center (k = (0, 0, 0)) 3.10 reduces to

p² 2m+ V

u_n0= E_n0u_n0. (3.11)

We treat the complete Hamiltonian as a sum of an unperturbed term H0and a perturbed term H_k⁰;

H = H0+ H_k⁰ (3.12)

where H0 = _2m^p² + V and H_k⁰ = ^~_2m²^k² +^~k·p_m . H0 is independent of k and thus equal to the complete Hamiltonian at zone center. 3.11 now simplifies to

H₀u_n0= E_n0u_n0. (3.13)

The solutions of 3.13 will form a complete and orthonormal set of basis functions. Once these are known, H_k⁰ can be treated as a perturbation in H0, which will lead to terms proportional to k. Therefore, this method will work best for small values of k, but will work also for higher values of k if more terms are included. What results are expressions for Enk

and unk in terms of the energies and wavefunctions at k = 0. The band dispersion near an arbitrary point k₀ can be calculated by expanding around this point instead of k = 0.

However, this approach demands that the wave functions and energies at k₀ are known.

Instead, using a sufficiently large number of u_n0, a complete set of basis functions can be approximated, which can then be diagonalized to obtain the band structure in the entire Brillouin zone. This approach requires the use of computers, as this can no longer be done analytically.

The matrix elements of the Hamiltonian are Hmn = hum|H|uni. The matrix element Hmn represents the mixing between Bloch functions umand un. It is convenient to trans- form the basis Bloch functions uj in such a way that they form a basis for the irreducible representations of the symmetry group of the crystal. This way the off-diagonal matrix element Hmn will represent the mixing of band m and band n, while the diagonal matrix element Hmm will represent the zone center energy of band m. Due to the presence of the k·p-term in the Hamiltonian (3.12), matrix elements will contain terms of Hmn= hum|p|uni, the so-called optical matrix elements, most of which are zero. These optical matrix elements can be measured experimentally, and are together with the zone center energies En0(which can also easily be determined in experiments), the only parameters that are required to do k · p-calculations. This is what makes k · p-theory such a useful and commonly used tool.

In the case of non-degenerate bands, k · p-calculations are very straightforward by using non-degenerate perturbation theories. k ·p-models that treat degenerate bands by the use of degenerate perturbation theory are, among others, the Kane [23] model and the Luttinger- Kohn [24] model.

3.3.1 Including spin-orbit interaction

To include the spin-orbit interaction in the model, an extra term has to be added to the Hamiltonian. This now becomes

H = p²

2m+ V +~k · p

m +~²k²

2m + 1

4m²c²(~σ × ∇V ) · (~k + p), (3.14) where ~σ is a vector consisting of the three Pauli matrices (σx, σy, σz).

(21)

3.4 Effects of strain on the band structure

In this section we will attempt to describe the effects on strain on the band structure of a semiconductor. Doing this for the conduction band is a relatively straightforward approach, whereas the valence band requires some more sophisticated techniques due to its degeneracy. To investigate the conduction band of a bulk semiconductor we can make use of the deformation potential theory.

3.4.1 Conduction band and deformation potential theory

The deformation potential theory was developed in the fifties by Bardeen and Shockley [25], originally to describe the interactions of electrons and phonons. Pikus and Bir applied the theory to strained semiconductors [26]. The main results of their analysis will be provided in this section and give a very straightforward way of describing the behaviour of the conduction band with strain. According to deformation potential theory, the energy shift of the bandedge of band l can be written as a linear combination of the components of the strain tensor. Doing this yields

∆E^l=X

ij

Ξ^l_ijij, (3.15)

where Ξij are elements of the so-called deformation potential tensor, for which the re- lation Ξij = Ξji will also be valid due to the symmetry of the strain tensor. For a cubic lattice this deformation potential tensor generally only contains two independent elements.

These are called the uniaxial deformation potential constant Ξuand the dilatation deformation potential constant Ξd. Those constants have been determined both experimentally and theoretically. For an arbitrary strain tensor, the shift of a local minimum in the conduction band is given by

∆E_C= Ξ_d(_xx+ _yy+ _zz) + Ξ_ua^T · · a, (3.16) where a is the unit vector in the direction of the k-vector at the conduction band edge.

As here we are only interested in the bandedge minimum at k = 0, the second term in 3.16 can be disregarded and we are left with

∆EC= Ξd(xx+ yy+ zz). (3.17) This shows us that the conduction band edge experiences a linear shift with strain.

3.4.2 Valence band and the eight band k · p-model

As always, matters for the valence band are a bit more complicated due to its degeneracy.

Bir and Pikus have constructed Hamiltonians to treat also the valence band [26]. They found three deformation potential constants describing the valence band. However, their description did not yet include the coupling between the valence and the conduction band.

To get a complete picture of the changes of the bandedges in bulk, a different approach is required. In the early nineties Bahder constructed an eight band model based on k ·p-theory that included coupling between the conduction and valence band [27]. This model will be discussed in this section and will later in this thesis also form the basis for our numerical calculations.

(22)

Figure 3.1: Schematic view of the bands and their interactions in the eight-band model.

As the conduction band is twofold degenerate due to spin and the valence band is six-fold degenerate due to spin and an orbital momentum of L = 1, this will lead to an eight by eight Hamiltonian. This eight-band Hamiltonian consists of an interaction matrix H(k), which includes the p-term of the spin-orbit interaction (equation 3.14) and a k-dependent spin-orbit matrix Hso. Because the contribution of the latter is very small compared to the contribution of the first, the latter is neglected. A strain dependent term must also be added. The basis Bloch functions u_i are constructed in such a way that they form a basis for the irreducible representations of the T_d double group [28]. The bands are denoted by Γ^J_n, where J is the total angular momentum of the band and Γ_n the representation of the symmetry group the band belongs to.

In the eight-band model the bands are two conduction bands (Γ^±₆¹²) and six valence bands, split up into two split-off bands (Γ^±₇¹²), two light hole bands (Γ^±₈¹²) and two heavy hole bands (Γ^±₈³²). This is schematically shown in figure 3.1. The conduction and valence band are coupled to each other through the parameter P₀. This is related to the energy E_P by

E_P = 2m₀

~²

P₀² (3.18)

Bahder arrived at two Hamiltonians describing the system; one strain-independent part and one strain-dependent part. These are given in appendix A. They contain material dependent parameters, such as effective masses and interband couplings. These parameters are generally known from experiments. The dispersion relations can be analytically determined by diagonalizing the sum of the two Hamiltonians. For the complete dispersion relations the reader is referred to Ref [29]. At zone center (k = 0) they simplify to

(23)

M aterial Eg(eV ) ∆(eV ) Ep(eV ) a0(eV ) a(eV ) b(eV )

GaAs 1.519 0.341 25 −7.17 −1.16 −2

InAs 0.417 0.39 21.5 −5.08 −1.00 −1.8

In40Ga60As 1.145 0.325 24.0 −5.71 −1.01 −1.88

Table 3.1: Values for the energies and deformation potential constants as taken from Vur- gaftmann.

ECB = Eg+ a0(xx+ yy+ zz) EHH = −a(xx+ yy+ zz) +

rb²

2((xx− yy)²+ (xx− zz)²+ (yy− zz)²) + d²(²_xy+ ²_xz+ ²_yz) E_LH = −a(_xx+ _yy+ _zz) −

rb²

2((_xx− _yy)²+ (_xx− _zz)²+ (_yy− _zz)²) + d²(²_xy+ ²_xz+ ²_yz) ESO= −∆ − a(xx+ yy+ zz)

(3.19) where the level of zero energy is chosen to be at the valence band edge. The equations in 3.19 show that both the conduction band minimum shifts linearly with strain. This agrees with the deformation potential theory discussed in section 3.4.1. The average of the light hole and heavy hole also shifts linearly, however a splitting between the two is induced by an asymmetric strain. In other words, strain induces a symmetry breaking in the crystal. If the strain is purely hydrostatic, meaning xx= yy = zz and xy= xz = yz= 0 this splitting will not be present. If we now consider the case of biaxial strain, where = xx= yy = −zz

and the shear components are 0, we are left with

ECB= Eg+ a0

E_HH = −a + 2b

ELH= −a − 2b

ESO= −∆ − a

(3.20)

The asymmetric strain couples to the heavy hole and light hole bands, inducing a linear splitting between the two. Reliable values for the deformation potential constants have been given by Vurgaftmann [30] and are shown in table 3.1.

Using a strain of = −2.8% for In40Ga60As yields for the energy shifts of the band edges ∆ECB = 160meV , ∆EHH = 134meV , ∆ELH = −77meV and ∆ESO= −28meV .

3.5 g-factors and the Roth formula

In the previous chapter the origin of the g-factor was explained; an external magnetic field couples to the spin of a particle (hole or electron) which results in a splitting of the energy levels called the Zeeman splitting. This splitting is proportional to the g-factor of the particle. For a free electron this value is 2. For electrons in a semiconductor crystal the effective g-factor will deviate from the free electron value. Using k · p-theory an analytical expression for electron g-factors in bulk can be derived. The deviations from the free electron

(24)

g-factor arise from the coupling of the conduction bands to other bands in the system. When taking into account only the influence of the valence band on the conduction band it can be shown that

g_e= 2 − 2EP∆

3E_g(E_g+ ∆) (3.21)

where EP denotes a coupling between the conduction and valence band. The equation in 3.21 was derived by Roth [31] [32] and is therefore called the Roth formula. This simple formula shows the necessity of the spin-orbit coupling; without spin-orbit coupling, ∆ = 0, the effective electron g-factor is equal to the free electron g-factor. Also, without a coupling between the conduction and valence band, the deviations from the free electron g-factor vanish. This is due to the absence of an orbital momentum (L = 0) in the conduction band;

only a coupling to the valence band will induce an effective orbital momentum. The Roth formula is capable of reproducing bulk electron g-factors quite well. For example, for InAs the electron g-factor can be calculated to be −14.4, while experiments show −14.9 [33]. For the hole g-factor it is not possible to derive such an analytical formula. This is due to its degeneracy and the fact that the band itself carries an orbital momentum. For free holes the g-factor is given by the Land´e-factor [11]

gJ,L,S= 1 +J (J + 1) − L(L + 1) + S(S + 1)

2J (J + 1) (3.22)

Applying this formula to the J = ³₂ and J +¹₂ valence bands yields respectively g_h=⁴₃ and g_h = ²₃. Again, in a semiconductor crystal coupling to the other bands will cause deviations from these values.

It was shown by Pryor and Flatt´e that for confined electrons the Roth formula no longer holds [34] [35]. To do so, they performed numerical calculations on spherical InAs quantum dots in vacuum. They used high energy barriers to prevent the wave function from exceeding the quantum dot radius. Due to their approach, effects like strain were not present in their calculations. They found for very large quantum dots values of the effective electron g-factor as predicted by the Roth formula. By decreasing the quantum dot size, thus increasing the confinement and the band gap, both their calculations and the Roth formula start to approach the free electron g-factor value of 2. However, their calculations reached this value much quicker. As the only thing that is taken into account by the calculations and not by the Roth formula is the confinement, this must be the cause of this. They introduced the concept of orbital momentum quenching in order to explain their observation. We already mentioned that usually particles in the conduction band carry no angular momentum, but that the coupling to the valence band induces an effective orbital momentum. This effective orbital momentum causes the effective electron g-factor to deviate from the free electron value. In the framework of the eight-band model, due to coupling between the bands, the ground state is always a linear combination of the eight band states. The conduction band ground state will logically be dominated by the conduction band states. Increasing the confinement leads to an increased band gap and therefore a decreased coupling between the conduction and valence band. Therefore the conduction band ground state will be more heavily dominated by the conduction band states, leading to a decrease of the contribution of the valence band states. This will decrease the effective orbital momentum induced by the valence band; the orbital momentum is said to be quenched. This will prove to be an important concept for qualitatively explaining the results of both our numerical model and our experiments later in this report. For the valence band the concept of orbital momentum

(25)

quenching is not easily applicable. It is likely that the confinement does have an effect on the effective orbital momentum in the valence band, however to our best knowledge it has not yet been investigated how exactly this is happening. Analyzing this problem would be far from straightforward, as the valence band itself already carries a non-zero orbital momentum. We can therefore not intuitively predict what will happen to the effective hole g-factor under the application of strain. In other research it has been shown both numerically and experimentally that the hole and exciton g-factor decrease with increasing confinement energy [34] [36] [37]. As compressive strain leads to an increase of the confinement energy, we hypothesize that this will have the same effect on the g-factor. Whether this hypothesis is correct will be discussed in the following chapters.

(26)

Chapter 4

Photoluminescence and confocal microscopy

In this chapter the experimental techniques used in this research are discussed. First the working principle of our main investigation method, the confocal microscope, is discussed, after which the used setup will be described. To finish the chapter, a description of the sample structure is given.

4.1 Photoluminescence

When focussing a laser on a quantum dot, an electron can be excited from the valence band to the conduction band, creating an electron-hole pair (exciton). When the electron and hole recombine, a photon is emitted. The energy of this photon depends on the characteristics of the quantum dot. In a photoluminescence experiment, these photons are collected, providing valuable information about the properties of the quantum dot. One possibility is to do photoluminescence experiments on an ensemble of quantum dots (macro-PL). This will yield information of the basic characteristics of the sample, among which are the quantum dot density and the average size. Using sophisticated optical techniques, it is also possible to measure PL of single quantum dots (micro-PL). This way valuable information of each individual quantum dot can be determined, like the emission energy and the g-factor, which is of great importance in this report. Only micro-PL measurements are performed in the work discussed in this thesis.

4.2 Micro-photoluminescence

Typical quantum dot densities of ∼ 10⁹QDs/cm² lead to inter quantum dot distances of

∼ 300nm. Therefore in order to measure individual quantum dots a high spatial resolution is required. This resolution can be reached with a so-called confocal microscope. The following sections will discuss its working principle and the setup used for the experiments reported in this thesis.

(27)

Figure 4.1: Schematic representation of the confocal microscope setup. The objective focusses the light from the excitation source on the sample. The PL is collected by the same objective. The pinhole blocks any light from the sample that is out of focus in any way.

4.2.1 Diffraction limit and confocal microscope

To achieve a sufficiently high resolution, diffraction limited optics is necessary. Let us assume that the quantum dot can be considered as a point light source. When illuminating an objective, the spherical waves from the quantum dot converge via the objective to the focal point. Due to the finite aperture of the objective, an Airy diffraction pattern is formed.

The radius of the central Airy disk of this pattern is given by [38]

r_Airy= 0.61 λ

N A (4.1)

where N A is the numerical aperture of the objective and λ the wavelength of the emitted light. The numerical aperture is a measure for how much light can be collected by the objective; the larger the numerical aperture, the larger the imaginary cone that has its light focussed by the objective. When using the Rayleigh criterion, which assumes two points sources to be resolvable if they are separated by a distance of at least the radius of the central Airy disk, the resolution of the objective used in our setup can be determined to be

∼ 500nm [39]. A schematic representation of the confocal microscope is shown in figure 4.1.

The laser beam is focused onto the sample by the objective, after which the photoluminescence is collected by the same objective. Because of this, any light which is out of focus, both vertically and laterally will be blocked by the pinhole. This means that only PL-signal emitted by the part of the sample exactly in focus of the objective is left after the pinhole.

As the diffraction limit of ∼ 500nm is still larger than the average inter dot distance of

∼ 300nm, in general the excitation spot will illuminate more than one quantum dot. This is not an issue as the number of quantum dots will still be small, and due to the spectrally

(28)

Figure 4.2: Overview of the used micro-photoluminescence setup. The magnet is located in the outer chamber of the cryostat and is cooled using liquid helium. At the center of the magnet, inside the insert, the sample is placed on top of a piezo stack. This allows the sample to be positioned. On top of the microscope stick, the optical head is located. The paths of the excitation light (red) and the collection light (orange) are visible.

different properties of each quantum dot they can still be distinguished from one another.

Moreover, the sample studied in this thesis was found to have an unusually low quantum dot density. Because of this, the excitation spot always contained at most one quantum dot.

Now that the working principle of the confocal microscope has been explained, an overview of the used setup will be given. The setup is shown in figure 4.2 and consists of four main components; the cryostat containing the superconducting magnet, the optical head which is in essence our confocal microscope, the microscope stick which contains our sample and a monochromator which is able to detect and analyze the photoluminescence signal.

4.2.2 The cryostat and superconducting magnet

The cryostat consists out of four compartments. The outer layer is a vacuum shield containing super-isolation and is pumped down to typically 10⁻⁵mbar. This thermally isolates the inner chambers from the outside world. Next is the magnet bath, which contains a cooled superconducting magnet, capable of producing magnetic fields up to 10T . This bath is cooled with liquid helium to reach a temperature of 4K. The magnet bath is separated

(29)

from the inside of the system using another vacuum layer. This layer is typically filled with some contact gas, to cool down the sample as well as the magnet. Our microscope stick, which is a closed vacuum chamber containing the sample, is inserted into the inner chamber.

4.2.3 The microscope stick

The microscope stick is a metal tube, the length of which matches the cryostat in such a way that the sample is located at the center of the superconducting magnet. At the bottom of this metal tube, the sample is mounted on top of a piezostack consisting of three independent piezos, allowing us to translate the sample in any of the three directions with a precision of ∼ 10nm. Right above the sample (several millimeters) the objective is placed.

On top of the stick, outside of the chamber, the optical head is mounted (see next section).

After flushing with helium gas to prevent condensation while cooling, the stick is filled with

≈ 10mbar of gaseous helium, to ensure the sample is also cooled down to the liquid helium temperature, 4K.

4.2.4 The optical head

The optical head consists of three arms; an excitation arm, a collection arm and an imaging arm. The excitation source used is a laser diode emitting at 780nm. Through a fiber the laser light is coupled to a lens, which it is collimated by. The light passes through the first beam splitter, which reflects half towards the microscope objective and the other half to a power meter for monitoring purposes. The objective focusses the laser onto the sample and then collects the PL signal. As the objective is chromatic, its focal point depends heavily on the wavelength of the light. Therefore when aligning the optical head for the laser wavelength, the head will have to be slightly realigned when moving to the PL wavelength.

When the PL returns to the optical head, half of the signal is lost by the first beamsplitter.

The second beamsplitter reflects half the signal towards another lens, which images the signal on the CCD camera, again for monitoring purposes, while the other half is reflected towards the collection fiber. Before reaching the fiber, the laser wavelength is filtered out by a high pass filter and the signal is focussed onto the collection fiber via the third lens.

This fiber, which has a diameter of ∼ 5µm acts as the pinhole described in the previous section. From here, the light will be directed to the monochromator.

4.2.5 Monochromator

The monochromator is used to analyse the PL signal. The monochromator consists of three stages, each consecutive one allowing for a better resolving ability. In our experiments, only the first stage is used, which has a focal length of 750mm. The first stage has three different gratings which can be used with 750, 1100 and 1800grooves/mm. Using a mirror, the light from the fiber is directed towards a slit, which reduces the spot diameter to 5µ. As the original laser spot has the same size, this should get rid of any redundant light. The slit is placed in the focus of a curved mirror, which will collimate the light. The collimated light is then diffracted by the grating. Another curved mirror than refocusses the now dispersed light onto the exit slit. An extra mirror is used to direct the light to either an InGaAs detector or a Si CCD camera. As the detection efficiency and noise-signal ratio of the Si CCD camera is a lot better as opposed to the InGaAs detector, the Si detector will be used in this project. While scanning the sample in the search of quantum dots, the 750-grating

Eindhoven University of Technology MASTER Strain-induced g-factor tuning in individual InGaAs quantum dots Tholen, H.M.G.A.