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Linköping University | Department of Physics, Chemistry and Biology Master’s Thesis, 30 hp | Educational Program: Applied Physics and Electrical Engineering - international Autumn 2016 | LITH-IFM-A-EX—16/3273—SE

Interaction of sublevels in gated

biased semiconductor nanowires

Henrik Karlsson

Examinator: Magnus Johansson Adviser: Irina Yakimenko, Karl-Fredrik Berggren

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Master’s Thesis

LITH-IFM-A-EX—16/3273—SE

Interaction of sublevels in gated biased

semiconductor nanowires

Henrik Karlsson

Adviser: Irina Yakimenko

IFM

Karl-Fredrik Berggren IFM

Examiner: Magnus Johansson

IFM

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Datum Date 2016-10-21

Avdelning, institution

Division, Department

Department of Physics, Chemistry and Biology Linköping University

URL för elektronisk version http://urn.kb.se/resolve?urn=urn:nbn:

se:liu:diva-132380

ISBN

ISRN: LITH-IFM-A-EX—16/3273—SE

_________________________________________________________________

Serietitel och serienummer ISSN

Title of series, numbering ______________________________ Språk Language Svenska/Swedish Engelska/English ________________ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport _____________

Titel

Interaktion av subband i nanotrådar med pålagd drivspänning

Title

Interaction of sublevels in gated biased semiconductor nanowires

Författare Henrik Karlsson Author

Nyckelord Nanowire, spintronics, lateral spin-orbit coupling, asymmetrical confinement potential Keyword

Sammanfattning Abstract

Mesoscopic devices, such as nano-wires, are of interest for the next step in creating spintronic devices. With the ability to manipulate electrons and their spin, spintronic devices may be realised. To that end the different effects found in low-dimensional devices must be studied and understood. In this thesis the influence that lateral spin-orbit coupling (LSOC) has on a nanowire, with asymmetrical confinement potential, is studied. The nanowire is studied through a numerical approach, using the Hartree-Fock method with Dirac interactions to solve the eigenvalue problem of an idealised infinite nanowire. The nanowire has a split-gate that generates the electrostatic asymmetrical confinement potential. It is found that the lateral spin-orbit coupling has little to no effect without any longitudinal effects in the wire, such as source-drain bias. The electrons will spontaneously create spin-rows in the device due to spin polarization. The spin polarization is triggered by using LSOC, numerical noise or from a weak magnetic field.

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Abstract

Mesoscopic devices, such as nano-wires, are of interest for the next step in cre-ating spintronic devices. With the ability to manipulate electrons and their spin, spintronic devices may be realised. To that end the different effects found in low-dimensional devices must be studied and understood. In this thesis the influence that lateral spin-orbit coupling (LSOC) has on a nanowire, with asymmetrical confinement potential, is studied. The nanowire is studied through a numeri-cal approach, using the Hartree-Fock method with Dirac interactions to solve the eigenvalue problem of an idealised infinite nanowire. The nanowire has a split-gate that generates the electrostatic asymmetrical confinement potential. It is found that the lateral spin-orbit coupling has little to no effect without any longitudinal effects in the wire, such as source-drain bias. The electrons will spontaneously create spin-rows in the device due to spin polarization. The spin polarization is triggered by using LSOC, numerical noise or from a weak magnetic field.

Sammanfattning

Mesoskopiska anordningar, som nano-tr˚adar, tros vara ett viktigt steg f¨or att skapa spinnelektronik. F¨or att kunna skapa spinnelektronik beh¨ovs kunskap om hur elek-troner kan manipuleras. Generellt m˚aste d¨arf¨or existerande fenomen i nanoelek-tronik studeras. I denna avhandling studeras hur ”lateral spin-orbit koppling” (LSOC) influerar en nanotr˚ad som har en asymmetrisk potentialbarri¨ar. Hartree-Fock metoden, med Dirac potential f¨or elektron-elektron interaktioner, anv¨andes f¨or att ber¨akna energiniv˚aerna f¨or en idealisk, o¨andligt l˚ang nanotr˚ad. Nanotr˚aden har en split-gate som alstrar den elektrostatiska, asymmetriska potentialbarri¨aren. ”Lateral spin-orbit koppling” visar sig ha minimal effekt d˚a longitudinella effek-ter, exempelvis sp¨anning, saknas. Elektronerna placerar sig spontant i spinn-rader

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i tr˚aden vid spontan spinn polarisation. Spinn polarisationen s¨atts ig˚ang av LSOC, numeriska st¨orningar eller fr˚an svagt p˚alagt magnetf¨alt.

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Acknowledgements

First and foremost, I would like to thank my supervisors. Without the insights and fruitful discussions from my supervisors, Irina Yakymenko and Karl-Fredrik Berggren, this thesis would not have been possible. I would also like to thank my examiner Magnus Johansson for all support and practical help.

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Contents

1 Introduction 3

2 Model 5

3 Theoretical Background 7

3.1 Schr¨odinger Equation . . . 7

3.2 Effective Mass Approximation . . . 8

3.3 Hartree-Fock Method . . . 8

3.4 Confinement Potential . . . 10

3.5 Magnetic Field . . . 11

3.6 Spin-Orbit Interaction . . . 11

3.7 Electron-density . . . 12

3.8 Finite Difference Method . . . 14

3.9 Theoretical Model Summary . . . 15

4 Results and Discussion 17 4.1 Spin Polarization Trigger . . . 17

4.2 Biased Potentials . . . 23

4.2.1 Gamma Parameter . . . 24

4.2.2 GaAs . . . 25

4.2.3 InAs . . . 34

4.3 Case without LSOC . . . 39

4.4 Electron Localization . . . 40

4.5 Varying beta . . . 43

4.5.1 Higher Beta and Positive k-values . . . 49

4.6 Artificial LSOC . . . 54

4.7 Computational Aspects . . . 59

5 Conclusions 61 5.1 Future work . . . 62

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Notations

Some abbreviations and words used within this work are explained here.

Dictionary

Biased Used to describe the potential when the split-gate has different gate volt-ages. Interchangeable with asymmetrical and uneven.

Subband Energy eigenvalue for a system, derived from the Schr¨odinger equation. In this project subband refers specifically to lateral energy.

Sublevels See subband.

Spin-orbit interaction Interaction between motion of electrons and their spin. Spin-orbit coupling See spin-orbit interaction.

Spin polarization Used to describe an uneven amount of spin-up and spin-down electrons.

Abbreviations

Abbreviation Meaning

LSOC Lateral Spin-Orbit Coupling

FDM Finite Difference Method

GaAs Gallium-Arsenide

AlGaAs Aluminium-Gallium-Arsenide

InAs Indium-Arsenide

InAlAs Indium-Aluminium-Arsenide

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Chapter 1

Introduction

Our personal computers’ processing units are getting smaller and faster every year, pushing the limits of semiconductor physics. Smaller and faster devices allow for more advanced computer simulations on personal computers, which has made this diploma work possible. These developments together with the limitations on conventional processing units also promote the creation of even smaller devices. With the technological advancements made throughout the years it is possible to fabricate devices that can confine electrons in space, and limit their movements to fewer dimensions. These low-dimensional devices, such as quantum dots, quantum wires and quantum point contacts, allow us to study the electrons behaviour, and are important for bringing us closer to quantum computations.[1] In these low-dimensional structures there are several interesting effects that can be studied such as spontaneous spin polarization [2, 3], conductance anomalies [4], tuning of g-factor [5], effects of spin-orbit coupling [6–8] etc. One of the goals is to create transistors (spintronics), using the electrons’ spin instead of the electrons’ charge.

The different low-dimensional structures are studied actively, both experimentally [9–11] and theoretically [3–5, 12]. The theoretical studies include modelling the structures using computational simulations with some kind of numerical approach. Different approaches are being used in order to study different effects found in these structures. Some common computational methods used in other works are Green’s function formalism [13], density functional theory [12] and Hartree-Fock method [14]. In this project the Hartree-Fock method is used. Different methods mean different approaches and assumptions, where different aspects of the structures become clearer. In this work the focus is on nanowires, and how an asymmetrical potential will influence the electrons.

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Chapter 2

Model

In this project a gated biased semiconductor nanowire is modelled. The model used is based upon a model developed by K.F Berggren, P. Jaksch and I. Yakimenko.[15] The modelled wire is assumed to be of infinite length in x-direction and confined in z- and y-direction. In z-direction it is confined by a layered heterostructure, and in y-direction by an electrostatic confinement potential. See figure 2.1 for a schematic overview.

Figure 2.1: Schematic figure of a split gate quantum wire. Source [15], text is edited.

The strong confinement in the z-direction makes the electrons strongly constricted in this direction and this confinement is much stronger than for the y-direction. In the scope of this project it is assumed that only the lowest energy level in z-direction is occupied. It is also assumed that the energy contributed from the z-direction is constant, therefore it will only effect the system with an energy-offset. This energy-offset will not affect the qualitative results and is therefore

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not taken into consideration in any calculations done in this project. With all these assumptions the model is effectively a quasi one-dimensional system, with electrons forming an electron gas confined in the lateral direction. Furthermore, it is assumed that the electrons that form the electron gas are travelling ballistically, i.e. the electrons do not scatter within the device.[14–17].

The model wire can be realised by having a layered heterostructure with gallium-arsenide and aluminium-gallium-gallium-arsenide layers (GaAs/AlGaAs) with a split gate to deplete the electron gas at the interface which will create a nanowire. See figure 2.1 for a typical schematic device. Another possibility for creating the wire is to use an indium-arsenide and indium-aluminium-arsenide heterostructure (InAs/InAlAs).[6, 16].

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Chapter 3

Theoretical Background

This chapter will present a brief overview of the physics used in this diploma project. The focus is to briefly introduce the physics and equations needed to describe a split gate quantum wire, which was introduced in chapter 2. For a more in-depth explanation, basic textbooks in quantum physics and/or semiconductor physics are recommended, such as [16, 18, 19].

3.1

Schr¨

odinger Equation

When the behaviour of electrons and atoms in a material is studied, the partial differential equation known as the Schr¨odinger equation is commonly used. The Schr¨odinger’s equation determines allowed energy levels and the behaviour of the particles in a given system:

ˆ

HΨ(r, t) = i~∂t∂Ψ(r, t). (3.1)

Here ˆH is a quantum mechanical operator known as the Hamiltonian, r is the positional vector, t is the time and Ψ is the system’s wave function. The wave function contains all information about the state of the system. Using different quantum mechanical operators different measurable variables can be calculated. The Hamiltonian is the operator for calculating a system’s total energy.

The Schr¨odinger equation can be solved for a stationary state when the system is not time dependent, this is the case considered for the model used. The time-independent Schr¨odinger equation for one particle (1P), influenced by an external potential V, has the following form:

ˆ H1PΨ(r) = −~ 2 2m∇ 2 + V (r)  Ψ(r) = EΨ(r) (3.2) 7

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where E is the energy and is also referred to as subband energy, or simply subband in this thesis. A differential equation, such as the Schr¨odinger equation, is said to be separable if the solution can be written as a product of functions, where each function depends on a single variable.[20] This is an assumption made for the differential equations considered within this project. The wave function will be written in the following form:

Ψ(x, y, z) = ψ(y)φ(x)χ(z) (3.3)

where the three different functions on the right hand side are the wave functions for the different axes.

3.2

Effective Mass Approximation

Since the electrons for different materials are subject to this study, it is important to have correct parameters for the materials. Correct parameters means parameter values that give results relatable to experimental results. It is known that in materials the mass of electrons has to be treated differently than that of free electrons (moving in vacuum). Therefore, the effective mass (m∗) approximation is used.[19, 21, 22] The effective mass in GaAs is 0.067 me and 0.023 me in InAs

[23].

3.3

Hartree-Fock Method

Equation 3.2 is limited to the case of one particle; in order to consider many-particle systems there are different approaches. One approach is the so called Hartree-Fock method [24, 25]. The focus in this project is how an electron gas behaves, and thus the interactions between the electrons and atomic nuclei are not discussed here but it can be included in the Hartree-Fock model. All elec-trons have the same charge and will repel each other following Coulomb’s law. From Coulomb’s law the potential between two electrons can be derived. For two electrons at position x1 and x2the Coulomb potential is:

VC = e2 4π0 1 |x1− x2| (3.4)

where e is the electron charge, 0 is the permittivity in vacuum and  is the

rela-tive permittivity. The relarela-tive permittivity is material dependent, and scales the Coulomb force to the correct strength for different materials. Using the Coulomb potential between the electrons the Coulomb operator ˆj will take the form:

ˆ j(x1)f (x1) = e2 4π0 N X i=1 Z Ψ∗i(x2)Ψi(x2) 1 |x1− x2| f (x1) dx2 (3.5)

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3.3 Hartree-Fock Method 9 The operator is a sum over all N particles in the system. Since two identical fermions (here electrons) cannot occupy the same quantum state, commonly known as the Pauli-exclusion principle, another operator is needed, the ˆk operator. The ˆ

k operator is also known as the exchange operator: ˆ k(x1)f (x1) = e2 4π0 N X i=1 Z Ψ∗i(x2)f (x2) 1 |x1− x2| Ψi(x1) dx2. (3.6)

Putting these two operators together we get the Coulomb-exchange operator [24]: ˆ

g = ˆj − ˆk. (3.7)

Adding this operator together with the one-particle Hamiltonian, equation 3.2, the full time-independent Hamiltonian for a many-body system in the Hartree-Fock approximation is: ˆ H = −~ 2 2m∇ 2+ V (r)  + ˆg = ˆH1P+ ˆg (3.8)

The Hartree-Fock Hamiltonian includes the Coulomb potential between particles. The Coulomb potential is computationally heavy, but a less computationally heavy interaction potential can be used, the so called Dirac potential, as proposed in [13]. This interaction potential will simplify the calculation and speed up simula-tions,

Vint= γδ(x1− x2). (3.9)

The γ determines the interaction strength and its value can be approximated from a screened Coulomb potential, which will be in the order of γ = π~2/m.

Depending on the screening length (range of interactions) in a material and the effective width of the channel this is a good approximation. For GaAs the screening length is ≈ 5 nm and for InAs it is ≈ 17 nm.1 To validate the Dirac potential the

screening length can be compared to the width of the wire. The effective width for a typical unbiased parabolic potential can be calculated from:

W∗=r 2

πρ (3.10)

where ρ is the (two-dimensional) electron density. From experiments a typical density value is 1.8 × 10−11cm−2 for GaAs; this will give an effective width of 20 nm.[13] Increased densities will decrease the effective width, as seen in equation 3.10. For the symmetrical case of GaAs wire with densities around this value the screening length is smaller than the effective width of this channel, showing that the Dirac potential is a very rough approximation.[13] For a strongly biased confinement potential the effective width will increase and be larger than the

1The screening length is calculated for GaAs in [13]. For InAs it can be calculated using the

effective mass, 0.023 me, and relative permittivity as 15.1.[23] The relative permittivity is for

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estimated value of 20 nm for GaAs. On the other hand, InAs has a screening length that is larger than the effective width of a parabolic potential, assuming the same density. Thus this approach disregards the possible influence of charges that come from outside the channel. This approach can also let some electrons that are inside the channel but outside the screening length of each other interact. The trade off for using the Dirac potential and eventually missing out on some effects is the significantly faster computational speed.

With the use of the Dirac delta interaction potential and limiting ourselves to a one-dimensional channel the Coulomb-exchange operator simplifies to

ˆ

HHFσ = γn−σ(y). (3.11)

Here n−σ is the density for all electrons with opposite spin, i.e. the Hartree-Fock

operator only depends on the electron density for the electrons with the opposite spin. Since the Hartree-Fock operator depends on the current state of a system the equations have to be solved until self-consistency is reached. One way of solving them are by using an iterative method, i.e. using the previous results as input and solve until self-consistency is reached within some decided limit.

Potential Mixing

When calculating the eigensolutions to the Hartree-Fock Hamiltonian with an iterative method, there is a risk that the potential will oscillate between solutions without settling down to a value (i.e. reaching a stable solution). A solution to this problem is to introduce a mixing parameter and to mix the previous potential with the currently calculated one. The mix is then used as the current Hartree-Fock potential. This mixing will hopefully steer the eigensolutions to stable and correct values, and damp any undesirable oscillations. The value of the mixing parameter α is normally set to 0.2 in the course of this project:[12, 25]

ˆ

HHF,new= (1 − α) ˆHHF,previous+ α ˆHHF,current. (3.12)

3.4

Confinement Potential

In order to confine the particles in the system a confinement potential is used. In this project a confinement potential derived from electrostatics for a split-gate wire is used [26]. The surface of the simulated device is taken to be ”perfect” in the sense that surface contamination is not taken into account in the potential. Furthermore the surface of the device is assumed to be an equipotential with a constant charge density that creates a potential contribution, Φ(x, y, 0) = Vs.

The charges from the donor layer are not in the potential explicitly, the potential from this layer is assumed to be equipotential and can be considered as a part of Vs, Φconf(y, d) = Vs− 1 π  Vlarctan  a + y d  + Vrarctan  a − y d  . (3.13)

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3.5 Magnetic Field 11

Here Vrand Vlare the gate voltages from the right and left gate respectively, 2a is

the lateral wire width of the device and d is the depth from the surface to the plane of the electron gas.[26] The possible influence on the potential from the randomly distributed donors are not taken into consideration.[27, 28]

3.5

Magnetic Field

Adding a magnetic field will have different effects depending on how it is aligned. Because of the implementation of LSOC and the aim to use the magnetic field to separate the spin-directions, a field along the growth direction (z-direction) is chosen. Since a magnetic field also influences the magnetic moment of the electrons it is required to take the spin into account. The spin dependent Zeeman term gµBB ˆσ will be added to the Hamiltonian. The operator ˆσ is the spin operator and

has eigenvalues ±1/2, µB is the Bohr magneton, B is the magnetic field running

parallel to the spin direction and the g-factor (g) depends on which material the electrons reside in.[19]. In this diploma project only weak fields are considered. Therefore the only effect the magnetic field is assumed to have is the effect from the Zeeman term.

3.6

Spin-Orbit Interaction

When using the Schr¨odinger equation the relativistic effects on electrons are not taken into consideration. To take these effects into consideration the Dirac equa-tion which includes the relativistic effects can be used. One effect that is derived from the Dirac equation is the spin-orbit interaction, how the particles’ spin cou-ples with their motion. This project is focused on the lateral spin-orbit interaction, also called lateral spin-orbit coupling (LSOC). This spin-orbit effect can be added to the Schr¨odinger equation with the term [22, 29]:

ˆ

HLSOC = βσ · ˆp × (∇Vtot) (3.14)

Here σ = (σx, σy, σz) are the Pauli matrices, and ˆp = −i~∇ is the momentum

operator. β is a constant which determines the strength of the coupling and is dependent on the material. Vtot is the total potential that the electrons interact

with. The value of β differs for different materials and their composition. In vacuum β is in the order of −3.7 × 10−6˚A2 and in GaAs and InAs it is in the order of 5 ˚A2 and 200 ˚A2 respectively.[6, 8, 30]

For the infinite wire model used in this project the electrons are unconfined in x-direction with a wave-function on the form as φl(x) = √1Lexp{iklx}, and total

wave function in equation 3.3. This together with the fact that the potential only affects electrons in the y-direction implies that the LSOC-term simplifies to:

ˆ HLSOC= β  kl dVtot(y) dy σz  (3.15)

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where σz is the Pauli matrix for z, and in the end there will be two contribu-tions: ˆ HLSOC↑ = β  kl dVtot(y) dy  (3.16) ˆ HLSOC↓ = −β  kl dVtot(y) dy  . (3.17)

It should be noted that the difference between these two Hamiltonians is the sign. In the model the only other effect that influences the electrons’ spin directly is the choice of the magnetic field. As such, choice of negative or positive β will not influence the qualitative results2 when a magnetic field is not taken into consideration together with the spin-orbit interaction. This is important due to a modelling choice where the β was set to negative for some simulations. Therefore, for some simulations the spin orbit interaction has switched sign for spin-up and down.

Lateral spin-orbit coupling (LSOC) is named from the spin interaction with an electric field that is induced by the lateral potential in the model. Other types of spin-orbit coupling exist, such as Rashba spin-orbit coupling (RSOC) and Dres-selhaus spin-orbit coupling. Rashba spin-orbit coupling is related to the fields the electrons feel from the growth-direction of the wire (z-direction) and influencing the energy levels in the transverse direction (x-direction). The effects of the RSOC can be controlled by adding a top gate. Since a top gate is not included in the project and without a top gate the value of an eventual effect of a Rashba term is expected to be minimal, it is thus not included in this project. The Dresselhaus spin-orbit coupling is an effect due to the material properties and with a ”correct” material choice the effect can be minimized and is not included in this model. [6, 29]

3.7

Electron-density

To get the density for free electrons travelling ballistically in an infinite one-dimensional long channel, one can divide the wire in x-direction into stripes of length L and add some boundary conditions. A common choice for the periodic boundary conditions is:

φ(0) = φ(L), ∂φ ∂x 0 = ∂φ ∂x L . (3.18)

Solving the one-dimensional Hamiltonian (equation 3.2) with these boundary con-ditions will give us the wave-functions as travelling waves in the form:

φl(x) = 1 √ Lexp{iklx}, kl= 2πl L , l = 0, ±1 ± 2, . . . (3.19)

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3.7 Electron-density 13

Now, to get the density of states per unit length, one only has to count the avail-able states, for example, in a region of k to k + δk there are ρ(k)δk = δk/2π available states per unit length. For calculating how many of the available states are occupied with electrons in the system the Fermi-Dirac occupation function can be used: fF D(E, EF, T ) =  exp E − EF kBT  + 1 −1 (3.20) where E is the energy, kB is Boltzmann’s constant, EF is the Fermi energy in the

system at temperature T. Putting the density of state per unit length together with the Fermi-Dirac function and integrating over all space will give the amount of electrons in our system at the given energy, Fermi energy and temperature. The energy in the model used is the lateral energy3, σn, together with the energy from the free electrons in x-direction, ~2kl2/2m. The total energy for subband n will be on the form

Eσn = σn+ ~2kl2/2m. (3.21)

The total number of electrons per unit length, for the infinite wire, can be calcu-lated with an integral when L goes to infinity.[19] For each subband n and spin σ the number of electrons per unit length, the density, will now be:

ρσn = ∞ Z −∞ 1 2π  exp E σ n− EF kBT  + 1 −1 dk (3.22)

When adding LSOC to the model the lateral energy will have dependency on the k-values:

n,l= σn,l+ ~2kl2/2m. (3.23) This k-dependency means that the density has to be calculated for every l value separately: ρσn =X l 1 2π  exp Eσ n,l− EF kBT  + 1 −1 . (3.24)

Letting L reach infinity would now mean that an infinite amount of differential equations would have to be solved to get the lateral energy. Computational lim-itations hinder L to be infinite, instead L is set to a finite value. This allows dividing the k-values into chunks with corresponding lateral energy which then can be integrated: ρσn =X l kl+kl+1 2 Z kl−kl−1 2 1 2π " exp ( σn,l+ ~2k2/2m − EF kBT ) + 1 #−1 dk . (3.25)

The summation here is over all l, however not all kl values will be occupied. In

figure 3.1 two subbands are plotted against k-values. The darkblue line represents a kl chunk.

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k-values (2π L) E (meV)

E

F kl kl-1

E

1 kl+1

E

2

Figure 3.1: Two subbands plotted against k-values in units of 2π/L. The dotted blue line represents the Fermi energy. The darkblue region represents a ”klchunk”.

The number of electrons per unit length is the same for every length unit along the wire, but in the lateral direction amount of electrons will be localized since there is a split gate creating a confinement potential, which is influencing the electron gas. In order to take this localization into account local density of states is used. The density is weighted by the wave-function for every subband and then summed together, over all existing subbands for spin-up and down separately [19, 22]: nσ(y) = X n |ψσ n(y)| 2 ρσn (3.26)

3.8

Finite Difference Method

To solve a partial differential equation such as the Schr¨odinger equation in an efficient way there are several numerical methods. In this project the Finite Dif-ference Method (FDM) is used. The definition of the derivative in one dimension, equation 3.27, together with limiting to a finite space which is broken into a set of discrete points with a spacing of h, will create a good start for a numerical method:

ψ0(y) = lim

h→0

ψ(y + h) − ψ(y)

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3.9 Theoretical Model Summary 15

In this now discrete space where the function is split into M parts, at the m-th point the first derivative will be:

dψ dy =

ψm+1− ψm

h (3.28)

Using the central difference approximation for the second derivative: d2ψ

dy2 =

ψm+1− 2ψm+ ψm−1

h2 (3.29)

For the boundary of the finite space where we will solve the Schr¨odinger equa-tion (y-direcequa-tion), boundary condiequa-tions have to be used. In this project Dirichlet boundary conditions are used in transverse direction, ψ(W ) = ψ(0) = 0, where W is the width of the wire. These boundary conditions correspond to an infinite potential well. Now when solving the Hamiltonian (equation 3.2) the FDM matrix will have the form:

          V1+ Ch22 −C 2 h2 0 . . . 0 −C 2 h2 V2+ Ch22 −C 2 h2 0 . . . 0 0 −C 2 h2 V3+ Ch22 −C 2 h2 . .. 0 .. . . .. . .. . .. . .. 0 0 −C 2 h2 0 0 0 0 −C 2 h2 VM + Ch22                      ψ1 .. . ψM            = E            ψ1 .. . ψM            (3.30) Here C is −~2/2m, and V

mis the potential at point m, Hartree-Fock, LSOC, and

confinement potential.[25] Calculating the eigenvalues and corresponding eigen-vectors of the matrix representing the Hamiltonian will give the system’s energy levels and corresponding wave-functions. Solving this eigenvalue problem can be done with different approaches. In this project it was solved using Matlab’s eigs function.

3.9

Theoretical Model Summary

Using the Dirac Hartree-Fock potential together with the confinement potential for the infinite wire, the magnetic field and spin-orbit interaction, the complete mathematical model for the system is achieved. It is a simplified model for the system but should give qualitative results. The full Hamiltonian for spin-up and down respectively: ˆ Hf ull,↑= −~ 2 2m∗ d2

dy2 + γn↓(y) + eΦconf(y, d) + β

 kl dVtot(y) dy  + gµBB (3.31) ˆ Hf ull,↓= −~ 2 2m∗ d2

dy2 + γn↑(y) + eΦconf(y, d) − β

 kl dVtot(y) dy  − gµBB (3.32)

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The final differential equations for spin-up and down respectively will be: ˆ

Hf ull,↑ψ↑(y) = E↑ψ↑(y)

 −~2

2m∗

d2

dy2 + γn↓(y) + eΦconf(y, d) + β

 kl dVtot(y) dy  + gµBB  ψ↑(y) = E↑ψ↑(y) (3.33) ˆ

Hf ull,↓ψ↓(y) = E↓ψ↓(y)

 −~2

2m∗

d2

dy2 + γn↑(y) + eΦconf(y, d) − β

 kl dVtot(y) dy  − gµBB  ψ↓(y) = E↓ψ↓(y) (3.34)

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Chapter 4

Results and Discussion

In this chapter all results gained from simulations done with the model described in previous sections are presented. For all simulations the wire width is set to 400 nm, 2a in equation 3.13. The depth from the substrate surface to the plane of electron gas, d in equation 3.13, is chosen as 100 nm. These values are close to values used in constructed devices [6, 31]. Tests have been done with shorter wires but only energy differences have been found. The temperature is set to 15 mK, since low temperatures are needed for ballistic transport (≤4.2 K).[16]

With increased temperature the amount of electrons in the system will increase, see equation 3.22. Testing has been done for 15 mK and 50 mK. However, no difference has been found except for a small increase in density for higher temper-ature. Temperature of 15 mK is therefore used for all simulations presented in this project.

4.1

Spin Polarization Trigger

For models like the one described in this project, it is proven that there is an effect that causes the occupation of the subbands for spin-up and down to differ, so called spin polarization. The cause of this effect is coming from the electron-electron interaction in the system. A subband will be populated according to the Fermi-Dirac distribution, equation 3.20. The lower the subband energy is compared to the Fermi energy, the more the subband will be occupied. Some occupation will start to happen before the subband energy reaches the Fermi energy, depending on the temperature and the subbands energy. When a subband starts to become heavily populated it will trigger a spin splitting, as can be seen in figure 4.1. The energies here are for a GaAs with an effective mass set to 0.067 me.

Neither LSOC nor magnetic field is used for this simulation. LSOC is not included in the model for any of the results showed in this section. The electron-electron interaction strength controlled by γ is set to 0.45 ∗ 2π ~2

2m∗, which is the same γ

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0 1 2 3 4 5 6 7 E F (meV) 0 1 2 3 4 5 6 7 8 9 E (meV)

Figure 4.1: The 10 lowest subbands for spin-up (red) and spin-down (blue) plotted against Fermi energy, the black line shows the Fermi energy. Neither LSOC nor magnetic field is included. The spin polarization is triggered by numerical noise.

used for all results in this section.1 The parameters can be seen in equations 3.31-3.32. The voltage at the left gate (Vl) is 22.3 mV, the right gate voltage (Vr) is

33.5 mV and Vsis 15 mV, see equation 3.13. This is close to a quantum harmonic

oscillator with ~ω = 1 meV, but with a bias. The stepping between the Fermi energies is 1 × 10−5eV. In the figure the 10 lowest subbands are plotted against the Fermi energy, the black line shows the Fermi energy. When a new subband starts becoming populated there will be a splitting of the degenerated subbands for spin-up and down. This will polarize the one-dimensional channel. When increasing the Fermi energy, after a splitting occurs, one of the spin direction’s subband will not start to populate but instead the system will favour one spin direction and only occupy that subband. When the Fermi energy is raised further the spin splitting will collapse into degenerate subbands again.

One can compare this work to the work of Hans Lind [14, 32], where it has been found that the spin splitting will occur when the Fermi energy reaches the degen-erate subbands and return to the degendegen-erate state when the Fermi energy reaches the elevated band. Looking at figure 4.1 we can see that the splitting occurs after the degenerate band has started to occupy, and the spin splitting stops before the Fermi energy reaches the elevated band. One explanation for this difference is

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4.1 Spin Polarization Trigger 19 0 1 2 3 4 5 6 7 E F (meV) 0.02 0.04 0.06 0.08 0.1 0.12 ρ (nm -1)

Figure 4.2: Separated density ρ for spin-up (red) and spin-down (blue) plotted against Fermi energy (EF). Simulation done without any magnetic field or LSOC,

but with numerical noise from eigenvalue solver.

how the density is calculated, in this work there is no cutoff length used for the wire length and the infinity is used as integral limits in equation 3.22. Another explanation would be that another numerical solution is used in [32] to solve the Hartree-Fock equations and one can expect that different methods will give dif-ferent results. Also this simulation is at difdif-ferent temperature and not the exact same potential, which can be influencing factors.

One way to trigger the spin polarization is to add a small magnetic field to simulate a disturbance that might come from impurities in the material, experimentation setup or failure to shield the system from other disturbances.[28, 31] A suggested trigger for the polarization is the addition of spin-orbit coupling to the model. Spin-orbit coupling together with an asymmetrical potential will create a small initial polarization that then will be enhanced by the electron-electron interaction and polarize the system, thus no other disturbance might be needed.[6]

The trigger for spin polarization in this model is the small disturbance that comes from solving the eigenvalue problem numerically. The function eigs in Matlab uses a random vector as a starting point to solve the eigenvalue problem by iter-ation, and depending on this vector the eigenvalues will differ slightly (order of 1 × 10−16eV). This small energy difference will cause enough difference in the

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density (order of 1 × 10−22nm−1) and allow for spin polarization to happen in the model.

In figure 4.2 the densities, that corresponds to the energies shown in figure 4.1, are plotted against the Fermi energy. The spin polarization is triggered by the afore-mentioned numerical noise. The spin polarization occurs without any particular bias for the polarization. Several simulations with the same settings together with different random seed for the random vector generator have been done to make sure this is the case.

When using eigs in Matlab it is possible to remove the random effect from the random vector by manually setting it to a fixed value and use the same random vector for spin-up and down. This will effectively remove the numerical noise that previously allowed the system to find a polarized solution. Doing this without adding a magnetic field creates a situation where the iteration method used to solve the Hartree-Fock equations does not converge to any value at the points where it should supposedly become spin split. In order to get spin polarization and convergence, a small magnetic field can be added. This field will create an unbalance due to the Zeeman term. In figure 4.3 the density is plotted against the Fermi energy when the numerical noise between spin-up and down is removed for different magnetic fields. The lowest possible field that did not give any conver-gence issues was in the order of 1 × 10−14T, smaller fields seems not be able to create an initial unbalance big enough to neither create a spin-splitting or allow convergence in the system. This field is several orders of magnitude smaller than earth’s magnetic field which is in the order of 1 × 10−5T.[23] The strength of the magnetic field needed for spin splitting and convergence is believed to be depen-dent on the numerical precision for the numerical method and eigenvalue solver, with better precision a weaker field might give convergence.

Adding a strong magnetic field (orders of 1 × 10−1T) would allow control of the spin splitting direction.[14] For the fields used here, the interaction energy between the electrons will become stronger than the Zeeman energy term. The Zeeman energy term (gµBBkσ) for a field at 1 × 10ˆ −3T is in the order of 1 × 10−8eV, and

lower for weaker fields. As the Fermi energy in the system is raised the effective interaction potential, from the Coulomb-exchange operator, that is acting on the system will dominate and make the effect of the magnetic field negligible. The dominating interaction potential is in the order of 1 × 10−4eV, for higher Fermi energy. In figure 4.3 this can be observed. For the field at 1 × 10−3T (figure 4.3a) the first three occurring spin splittings all favour the spin splitting direction induced by the Zeeman term, increased density for spin down. At higher Fermi energy the interaction potential will dominate the system, see the fourth spin splitting. When the interaction potential dominates the system it will make the polarization not favour the polarization direction that the Zeeman term would induce. For weaker fields the interaction potential will dominate over the Zeeman term at lower Fermi energies, compare the different cases in figure 4.3. For the weak field at 1 × 10−14T, figure 4.3d, the interaction potential dominates over the weak Zeeman term before the first spin splitting.

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4.1 Spin Polarization Trigger 21

Running repeated simulations with different but fixed ”random vectors” and with a small magnetic field, they will yield the same polarization behaviour for the spin-up and spin-down electrons for every simulation. This is believed to be caused by the Zeeman energy term effectively acting as ”fixated” numerical noise, compa-rable to the difference between the random vectors used by eigs. When higher densities are reached so that the interaction potential influences the system more than the magnetic field, the Zeeman energy can be seen as a small disturbance. This disturbance is however several orders of magnitude larger than the difference between the fixed ”random vectors” for the two spin-directions that are set for eigs to use. The Zeeman energy term effectively creates a semi-random effect for the polarization directions. The polarization directions will be hard to predict but will be the same for every simulation.

Instead of adding a magnetic field to solve the convergence issue connected with fixed ”random vector” for spin-up and down, adding LSOC to the model will cre-ate enough of an energy difference between the spin-directions for convergence. The use of LSOC to trigger spin polarization, which is then enhanced by the electron-electron interaction creating the polarization, is confirmed by others.[6] Using LSOC, a small magnetic field or the numerical noise to trigger the spin-polarization will not affect the subbands final energy. All tested spin-polarization trig-gers have given the same final qualitative result.

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0 2 4 6 EF (meV) 0.02 0.04 0.06 0.08 0.1 0.12 ρ (nm -1 ) (a) 1 × 10−3T 0 2 4 6 EF (meV) 0.02 0.04 0.06 0.08 0.1 0.12 ρ (nm -1 ) (b) 1 × 10−5T 0 2 4 6 EF (meV) 0.02 0.04 0.06 0.08 0.1 0.12 ρ (nm -1 ) (c) 1 × 10−10T 0 2 4 6 EF (meV) 0.02 0.04 0.06 0.08 0.1 0.12 ρ (nm -1 ) (d) 1 × 10−14T

Figure 4.3: Separated density (ρ) for spin-up (red) and spin-down (blue) plot-ted against the Fermi energy (EF) for four different magnetic fields (1 × 10−3,

1 × 10−5, 1 × 10−10 and 1 × 10−14T) and without numerical noise from the eigen-value solver.

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4.2 Biased Potentials 23

4.2

Biased Potentials

The confinement potential defines the channel where the electrons will travel through. Lower potential strength will increase the density and create a wider channel, while a higher potential will decrease the density and limit the electrons into a more narrow channel. In experiments, typical values for the gates for a split gate potential is around 1 V.[6, 11] When making the potential asymmetrical the LSOC is starting to affect the wire and will need to be accounted for. The expected outcome is spin polarization. Also expected is spin accumulation at the edges of the channel, spin-up on one side and spin-down on the opposite side. The spin accumulation is coming from the effective magnetic field the electrons feel from the confinement potential.[6–8, 33]

-200 -150 -100 -50 0 50 100 150 200 y (nm) 0 1 2 3 4 5 6 7 V(y) (eV)

Figure 4.4: Potential when the right gate voltage is kept constant at 1 V, and left gate voltage 1, 1.5, 2, 4, 5, 5.5, 6 and 8 V respectively, and Vs= 0. The depth from

surface (d) is at 100 nm and the wire is 400 nm wide. The minimum is marked with a cross.

In figure 4.4 the confinement potential defined at equation 3.13 is plotted for some increasingly biased cases. The right gate voltage is kept constant at 1 V while the left gate voltage is increased. With increased bias the minimum of the potential, where the electrons will localize themselves, will move. When the confinement potential is 6 V and higher while the other gate voltage is kept at 1 V, the minimum of the potential will be at the substrate edge. For a highly asymmetric potential

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it will be considerably flatter around the minimum and the electron density will become higher.

In order to add spin-orbit interaction into the infinite wire model in theory the continuous k-space has to be divided into discrete parts in order to solve the differential equations. The L in equation 3.19 is chosen to a constant value at ≈200 nm which has proven to give a good trade-off between computational speed and numerical accuracy. In theory L would be chosen to be infinite, creating a continuous space of k-values, this is however hard to deal with due to compu-tational limitations and is instead chosen to a constant value. With a finite L, density calculations can be split up into kl chunks, as described in chapter 3.7,

making it easier to calculate. When solving the final Hamiltonians, equations 3.31 and 3.32, it has to be done for every occupied klvalue. Compared with the case of

absence of LSOC, it turns the model into a more complex system which requires a lot more computational time to solve.

4.2.1

Gamma Parameter

The strength of the Dirac potential is controlled by the γ parameter. Lassl et al.[13] estimates the strength of γ analytically to 2π ~2

2m∗. For these simulations different γ

values have been considered. In GaAs with an asymmetrical confinement potential with 4 V and 1 V at left and right gate respectively and γ at zero, no spin splitting occurs for the subbands. This can be viewed in figure 4.5a. In the figure the respective subband’s energy without the energy from the longitudinal direction, equation 4.1, are plotted for the subbands that are occupied, and will become occupied, against the Fermi energy together with the different klvalues:

n,l= En,lσ −~

2k2 l

2m∗. (4.1)

The LSOC is added in these simulations with β at 5 ˚A2, which is the only ef-fect that differs for different spin-directions. It should be noted that LSOC does not induce any spin-polarizations, in the total density, by itself.2 With γ set to

the strength of the theoretical value from Lassl et al., figure 4.5b, after the first subband for spin-up (or down) gets occupied it will increase the energy for the opposite spin with such strength that the Fermi energy does not catch up. For this strength the channel will only be occupied with one spin-direction, always creating a fully polarized wire. A more realistic choice of γ is 0.45 ∗ 2π ~2

2m∗, 45 %

of the analytical value. This value gives the polarizations seen in section 4.1 and agrees with experiments. This γ value is used in all further work. It is comparable to the value used in other works, π ~2

2m∗ in [14], 3.7 ~ 2

2m∗ in [13], and will give results

comparable to experiments.[13, 14, 33]

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4.2 Biased Potentials 25

(a) γ = 0 (b) γ = 2π ~2

2m∗

Figure 4.5: The subbands plotted against Fermi energy and k-values for two dif-ferent γ cases. Spin up is red and spin down is blue, the black plane represents the Fermi energy. The left gate voltage is 1 V and the right gate voltage is 4 V. LSOC is included in the model, with β at 5 ˚A2.

4.2.2

GaAs

GaAs has an effective mass of 0.067 me and a β estimated at 5 ˚A 2

. β is however chosen as −5 ˚A2. These parameters are kept constant through all the following simulations for GaAs if nothing else is stated. The steps between the Fermi energy is set to 5 × 10−5eV, which is also used for all further presented simulations. In figure 4.6 the electron densities are plotted against the k-values for an unbiased gate voltage at 1 V. Unbiased gate voltage means Vl is 1 V and Vr is 1 V, see

equation 3.13. The density of electrons for different k-values are shown to be fully symmetrical around k = 0. This is to be expected since there is nothing included in the model that is affecting the balance of the electrons positive and negative momentum in the wire. In the figure it is also visible that before a polarized region collapses, a higher klvalue will start to fill before collapsing again. This is because

non-raised energy levels from the spin splitting will increase slower than the Fermi energy, thus more kl will end up being occupied. These extra occupied kl values

will drop down again after the polarized region returns to the degenerate state, and returns to the same amount of occupation for spin-up and down. For the different asymmetrical potentials discussed in this and next section the electron densities and k-values will not behave differently than the unbiased case. Only difference will be the amount of occupied kl values due to the difference in subband energy.

In figure 4.7 the electron distribution (nσ(y)) are plotted against the Fermi energies

for the unbiased case. The effects of the interaction potential at the spin splitting become clear. Looking at the region 0.01 to 0.012 eV, at the second polarization region the electrons order themselves into rows for spin-up and down. At a spin split region one subband more of down (or up) will become occupied for the

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non-Spin up 0 5 10 15 20 EF (meV) -6 -4 -2 0 2 4 6 k -v al u es ( 2 π ) L Spin down 0 5 10 15 20 EF(meV) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 ρ (k l ) (nm -1)

Figure 4.6: The density plotted at different Fermi energies (x-axis) for the different k-values (y-axis), For spin-up (left) and spin-down (right).

raised subband. The wave function in the y-direction closely resembles that of an harmonic oscillator, and it is the wave function together with the electron density that influence the interaction potential. The extra occupied subband for spin-down (or up) will create valleys in the interaction potential which the raised subband will localize into, creating a row-formation for the spin-up and down at the spin polarized regions. The more subbands that are occupied the more ”humps” will the wave function have, which in return creates more rows at the polarized regions. In the non polarized regions the electrons wave functions and eigenenergies are the same for spin-up and down, creating a degenerate state.

When making the split-gate biased various effects will be seen. Eight different cases for increased biased confinement potential are presented in figure 4.9, the potentials can be seen in figure 4.4. The subband’s energy, as discussed in section 4.2.1, equation 4.1, is plotted against the Fermi energy together with the different kl values. As before the spin splitting will occur when the Fermi energy reaches

a set of unoccupied degenerate spin-up and down subbands, and collapses into a degenerate state when the Fermi energy is increased. The system is solved for a Fermi energy at 0 eV to 20 meV with an increment of 0.05 meV. The Vs, from

equation 3.13, is set so that the lowest subbands will be at 0 eV. This is done for ease of comparison for different cases without having to search for which Fermi energy is relevant for different cases. The main focus is qualitative results so a more realistic value of Vs is not deemed necessary (Vs is only an energy offset).

The setting of Vsis also why the system will always start at a polarized state, the

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4.2 Biased Potentials 27 Spin up 0 5 10 15 20 E F (meV) -200 -150 -100 -50 0 50 100 150 200 y-position (nm) Spin down 0 5 10 15 20 E F (meV) 0 1 2 3 4 5 6 7 n(y) (nm -2) ×10-3

Figure 4.7: The density plotted at different Fermi energies (x axis) for an unbiased confinement potential at 1 V.

For the asymmetrical potential biased at 1.5 V and 2 V on the left gate and 1 V on the other gate the difference between a fully symmetrical potential is found to be negligible. The spin splitting occurs as expected and no new feature is visible. Compare figures 4.9b and 4.9c for the asymmetrical cases and figure 4.9a for the symmetrical case. The difference is limited to the difference in density and separation of the subbands which will differ with the confinement potential’s strength. At a stronger bias (4 V), figure 4.9d and 4.10, where only the subbands for kl = 0 is plotted, a new feature occurs. When the Fermi energy reaches the

second set of degenerate subbands, mark S1, there is a jump in subband energy. The second set of subbands has a small increase in energy, mark S1, while the lower set of subbands, mark S2, has a decrease in energy, different amount of energy increase/decrease for spin-up and down. This state only lasts for a few iteration steps, 4 steps i.e. 0.2 eV, at the first occurrence, and even shorter before the next spin splitting when this occurs again, mark S3. This state seems to be an intermediate state that occurs immediately before the new subbands become occupied. When increasing the left gate voltage to 5 V, figure 4.9e, the same intermediate steps happen before the spin splitting. The intermediate state occurs in the same fashion as seen in figure 4.10, except different polarization directions and for one more set of subbands.

When the asymmetrical potential is raised to 5.5 V, the confinement potential has a minimum close to the wire edge and is mostly flat around the minimum. For high densities the system is effectively a broad channel limited by the

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confine-ment potential and the infinite potential at the edge. When the system with this confinement potential gets to higher densities the subbands no longer return to a degenerate state after spin splitting, instead the system stays polarized. Further increasing the potential at the right gate to 6 V the intermediate state before spin splitting does not occur since the system goes into a polarized state. Further in-creasing the voltage to 8 V the energy difference between the subbands is big, so that for the calculation limit at 20 meV only the first spin splitted subband will get occupied. The reason that the system goes into this polarized state for the highly asymmetrical wires is believed to be caused by the broad channel.

The stronger the asymmetry is, the closer the potential minimum will be to the edge of the device, and the channel width will increase due to flattening around the potential minimum. For the extreme cases of 5.5 V (or stronger) at the left gate, 1 V at the right gate the confinement potential is very flat and as such the densities (figure 4.11f) will get higher compared to the other cases shown. Comparing the maximum density reached for all the different potentials, figure 4.8, it can be seen that for the symmetrical case and increasing the bias the maximal total density will drop until the 4 V at the left gate is reached. Increasing left gate voltage from 4 V to 5.5 V, the maximum density is increased. For left gate voltage at 6 V and 8 V the energy decreases again. How the maximum density behaves can more easily be understood when looking at the confinement potential, figure 4.4. For the low biased cases the potential still resembles a harmonic potential at the minimum, where the electron channel is situated. When increasing one of the gate voltages to 4 V and above the potential is flatter around the minimum, giving a wider channel for the electrons, thus increasing the densities. For the 6 V and 8 V the channel is cut in half because of the boundary of the material. For the 8 V case the total density and the channel width is smaller than the unbiased 1 V case. The energy difference between the subbands is also directly related to the channel width, the broader the channel, the smaller energy gap between subbands, i.e. the broader the channel the faster it will populate.

Simulations have been done with decreased and increased L with symmetrical potential, which have not given any other result than the ones viewed in figure 4.9a. Simulations have been done without LSOC for the biased 5.5 V case, and the same results that are seen in figure 4.9f were reached. Simulations have also been done for the 5.5 V case with lower γ, 30% of the analytical value, and without LSOC. They showed that spin splittings occurred on the same form as 5 V case, figure 4.9e, except smaller in magnitude and without intermediate states.

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4.2 Biased Potentials 29 1 2 3 4 5 6 7 8 V l (V) 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 ρ (nm -1 )

Figure 4.8: The maximum total density reached for different potentials when the left gate voltage is increased (x-axis) and right gate voltage is kept constant at 1 V. The circles correspond to data-points.

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(a) Vl= Vr= 1V (b) Vl= 1.5V , Vr= 1V

(c) Vl= 2V , Vr= 1V

(d) Vl = 4V , Vr = 1V . The zoomed in

insets shows how the density behaves at the intermediate states.

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4.2 Biased Potentials 31

(g) Vl= 6V , Vr= 1V (h) Vl= 8V , Vr= 1V

Figure 4.9: The energy plotted for the occupied subbands against the Fermi energy and the kl values, for GaAs. Red for spin-up and blue for spin-down, the Fermi

energy is represented by the black plane.

0 2 4 6 8 10 12 14 16 18 20 EF (meV) 0 2 4 6 8 10 12 14 16 18 20 E (meV) 17.7 17.9 18 19 9 9.2 9.4 9 10 9 9.2 9.4 4 5

S2

S1

S3

Figure 4.10: The energy, red for spin-up and blue for spin-down, plotted for the occupied subbands with kl= 0 against the Fermi energy, the black line. The figure

corresponds to figure 4.9d, potential at Vl= 4V , Vr= 1V . The intermediate state

behaviour shown is the typical behaviour for all intermediate states, for both GaAs and InAs.

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0 5 10 15 20 E F (meV) 0 0.02 0.04 0.06 0.08 0.1 ρ (nm -1 ) (a) Vl= Vg= 1V 0 5 10 15 20 E F (meV) 0 0.02 0.04 0.06 0.08 ρ (nm -1 ) (b) Vl= 1.5V , Vr= 1V 0 5 10 15 20 E F (meV) 0 0.02 0.04 0.06 0.08 ρ (nm -1 ) (c) Vl= 2V , Vr= 1V 0 5 10 15 20 EF (meV) 0 0.02 0.04 0.06 0.08 0.1 ρ (nm -1 ) (d) Vl= 4V , Vr= 1V 0 5 10 15 20 EF (meV) 0 0.02 0.04 0.06 0.08 0.1 0.12 ρ (nm -1 ) (e) Vl= 5V , Vr= 1V 0 5 10 15 20 EF (meV) 0.05 0.1 0.15 ρ (nm -1 ) (f) Vl= 5.5V , Vr= 1V

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4.2 Biased Potentials 33 0 5 10 15 20 EF (meV) 0 0.05 0.1 ρ (nm -1 ) (g) Vl= 6V , Vr= 1V 0 5 10 15 20 EF (meV) 0 0.01 0.02 0.03 0.04 0.05 ρ (nm -1 ) (h) Vl= 8V , Vr= 1V

Figure 4.11: The density plotted for the different spin-directions against the Fermi energy, for GaAs. Red for spin-up and blue for spin-down.

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4.2.3

InAs

InAs has an effective mass of 0.023 meand β at 200 ˚A 2

, but it is chosen as −200 ˚A2 for the following simulations. The effect of spin-orbit interaction is expected to have a greater influence for this system than for GaAs. This is because InAs has smaller effective mass and a small band gap which makes the effect of spin-orbit interaction more prominent.[6]

For the case of a non-biased confinement potential, figures 4.12a and 4.13a, and a lowly biased potential at 1.5 V, figures 4.12b and 4.13b, no new effects was found. These cases only differ in energy levels compared to GaAs. The energies differ because of the effective mass for the different materials. At 2 V, figure 4.12c, the same intermediate state that happens before a spin splitting at 4 V (and higher) for GaAs occurs. Further increase of the potential still creates these intermediate states, as they do for GaAs.

In the case of GaAs the energy was not considerably changed for different k-values, but for InAs the energy levels are not perfectly horizontal for different k-values. Looking at the spin interaction term, equation 3.15, the higher klvalue, the larger

will the spin-orbit interaction term be, with different signs for spin-up and down. Looking at the subbands in figure 4.12 it can be seen that with larger k-values for the spin-up the energy increases and the energy decreases with smaller k, symmetrically around k = 0. The effect is easiest to see in figure 4.12h which only contains one set of subbands. While for spin-down it is the opposite, energy decreases with increase of k, energy increases with decreased k. It can also be seen that, the higher the subband, the larger is the energy difference between k-values, figure 4.12g shows this clearly. In figure 4.12 it is also clear that at increasing bias of the confinement potential this effect increases.

At 6 V at the left gate and above the system stays in a fully polarized state, comparable to 5.5 V and above for GaAs. The reason that the constant polarized state occurs at a higher left gate voltage for InAs, than for GaAs, is because of different subband-energies. At 5.5 V for InAs the electron channel is not limited by the infinite potential at the wire-edge, as it is for the GaAs case.

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4.2 Biased Potentials 35

(a) Vl= Vr= 1V (b) Vl= 1.5V , Vr= 1V

(c) Vl= 2V , Vr= 1V (d) Vl= 4V , Vr= 1V

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(g) Vl= 6V , Vr= 1V (h) Vl= 8V , Vr= 1V

Figure 4.12: The energy plotted for the occupied subbands against the Fermi energy and the kl values, for InAs. Red for spin-up, blue for spin-down, the black

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4.2 Biased Potentials 37 0 5 10 15 20 E F (meV) 0 0.01 0.02 0.03 0.04 ρ (nm -1 ) (a) Vl= Vg= 1V 0 5 10 15 20 E F (meV) 0 0.01 0.02 0.03 0.04 ρ (nm -1 ) (b) Vl= 1.5V , Vr= 1V 0 5 10 15 20 E F (meV) 0 0.01 0.02 0.03 0.04 ρ (nm -1 ) (c) Vl= 2V , Vr= 1V 0 5 10 15 20 E F (meV) 0 0.01 0.02 0.03 0.04 ρ (nm -1 ) (d) Vl= 4V , Vr= 1V 0 5 10 15 20 EF (meV) 0 0.01 0.02 0.03 0.04 ρ (nm -1 ) (e) Vl= 5V , Vr= 1V 0 5 10 15 20 EF (meV) 0 0.01 0.02 0.03 0.04 0.05 0.06 ρ (nm -1 ) (f) Vl= 5.5V , Vr= 1V

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0 5 10 15 20 E F (meV) 0 0.02 0.04 0.06 ρ (nm -1 ) (g) Vl= 6V , Vr= 1V 0 5 10 15 20 E F (meV) 0 0.005 0.01 0.015 0.02 0.025 0.03 ρ (nm -1 ) (h) Vl= 8V , Vr= 1V

Figure 4.13: The density plotted for the different spin-directions against the Fermi energy, for InAs. Red for spin-up and blue for spin-down.

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4.3 Case without LSOC 39

4.3

Case without LSOC

Setting the spin-orbit interaction term β to 0 will show if the inclusion of this interaction will have any effect in this model. A case of an asymmetrical potential with the confinement potential set to 5 V at the left gate and 1 V at the right gate is done. In figure 4.14 the subband energies are plotted against Fermi energy and k-values. Without the spin-orbit interaction term influencing the system it is clear that the subbands should have no k dependency, and thus the subbands should not differ between different kl. This expected effect can be seen in the

figure 4.14. Furthermore, the same spin splitting occurs as before, and at the same Fermi energy for GaAs and InAs as it did with a non-zero β. Compare with figures 4.9e and 4.12e for GaAs and InAs respectively. The same intermediate state that occurs before the spin splittings is also occurring as it did with spin-orbit interaction. This shows that LSOC is not triggering spin-polarization, and is not the cause of the intermediate state.

The densities for zero β is shown in figure 4.15. Compared to nonzero case, figures 4.11e and 4.13e for GaAs and InAs respectively, there is no visible difference between the cases. The spin splitting that occurs for the no LSOC cases shows no preferred way of splitting. Calculating the total density for the zero and nonzero β case and comparing them numerically shows a mean difference in the order of 1 × 10−8nm−1 for GaAs and 1 × 10−6nm−1 for InAs, which can be considered negligible in these numerical simulations.

(a) GaAs (b) InAs

Figure 4.14: The subbands energy plotted against Fermi energy and k-values when Vl = 5V , Vr = 1V and β = 0, for GaAs and InAs respectively. Red for spin-up

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0 5 10 15 20 EF (meV) 0 0.02 0.04 0.06 0.08 0.1 0.12 ρ (nm -1 ) (a) GaAs 0 5 10 15 20 EF (meV) 0 0.01 0.02 0.03 0.04 ρ (nm -1 ) (b) InAs

Figure 4.15: The density for the separate spin-directions plotted against Fermi energy when Vl= 5V , Vr= 1V and β = 0, for GaAs and InAs respectively. Red

and blue represents spin-up and spin-down, respectively.

4.4

Electron Localization

When doing the simulations the system is solved for the lowest possible state, which means that the electrons will strive to minimize their energy. To minimize the energy, the electrons will have to localize and fill up from the minimum of the confinement potential. As more electrons added to the wire, the more will the channel be filled out. Looking at how the electrons localize for some of the different cases it is found that at the degenerate states for spin-up and down the electrons slowly spread out more and more with increased Fermi energy. The electron concentration is higher where the confinement potential is the lowest. The electrons will for the symmetrical case spread out symmetrically around the potential minimum, and for the non-symmetrical cases the electrons will spread out to fill out where the confinement potential is the lowest, as expected. In figure 4.16 the total density is plotted against the lateral direction for increasing Fermi energy for a symmetrical and asymmetrical confinement potential. The cases are both for GaAs with LSOC, and β at −5 ˚A2. The densities and subbands can be seen in figures 4.9a and 4.11a for the symmetrical case, and figures 4.9e and 4.11e for the asymmetrical case.

When increasing the energy for the system and the Fermi energy reaches a not yet occupied subband the spin polarization occurs as discussed in section 4.1. At this point the electrons will create stripes of spin-up and down electrons. The spin polarization is plotted for one symmetrical and one asymmetrical case in figure 4.17. The polarization p is calculated from:

p(y) = n↑(y) − n↓(y). (4.2)

The symmetrical case is found in figure 4.17a. First the system starts in a fully polarized state (both the symmetrical and asymmetrical case). When increasing

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4.4 Electron Localization 41 0 5 10 15 20 EF (meV) -200 -150 -100 -50 0 50 100 150 200 y-position (nm) 0 0.002 0.004 0.006 0.008 0.01 0.012 n(y) (nm -2) (a) Vl= 1V , Vr= 1V 0 5 10 15 20 EF (meV) -200 -150 -100 -50 0 50 100 150 200 y-position (nm) 0 0.002 0.004 0.006 0.008 0.01 0.012 n(y) (nm -2) (b) Vl= 5V , Vr= 1V

Figure 4.16: The total electron density plotted against the y-position and Fermi energy for an asymmetrical case and one symmetrical, both cases are for GaAs.

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0 2 4 6 8 10 12 14 16 18 20 EF (meV) -200 -150 -100 -50 0 50 100 150 200 y-position (nm) -4 -3 -2 -1 0 1 2 3 4 p(y) (nm -2) ×10-3 (a) Vl= 1V, Vr= 1V 0 2 4 6 8 10 12 14 16 18 20 EF (meV) -200 -150 -100 -50 0 50 100 150 200 y-position (nm) -3 -2 -1 0 1 2 3 p(y) (nm -2 ) ×10-3

S3

S1

S2

(b) Vl= 5V, Vr= 1V

Figure 4.17: The electron polarization plotted against the y-position and Fermi energy for an asymmetrical case and one symmetrical, both cases are for GaAs. Dark red represents strong spin-up polarization and dark blue represents strong spin-down polarization, no polarization is green.

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4.5 Varying beta 43

the energy and the subbands become degenerate, at this state the system is not polarized, and the electrons do not localize in any particular texture. When the second subband starts to populate, the electrons localize themselves into stripes. These stripes could be interpreted as signs of a Wigner lattice. Wigner lattice is an electron lattice structure where electrons minimize their energy by positioning themselves in either a row structure or a zig-zag structure.[34, 35] Since this model is limited to one dimension the two different cases can not be told apart here. When further increasing the Fermi energy and reaching higher spin polarized states more stripes will occur. The amount of stripes is related to the number of subbands that are occupied, every occupied subband is represented by one stripe. When the wire is polarized there will be one less subband occupied for spin-up or spin-down, which will be one less stripe. The reason of the stripe formation is because of the electron-electron interaction as mentioned before.

For highly asymmetrical wire, figure 4.17b, the stripe formation discussed in pre-vious paragraph is occuring at the strongly polarized states. Before the spin po-larization occurs there is a state where the second set of subbands makes a jump in energy and the first subbands get a small difference between up and spin-down (intermediate state), marked as S1. At this state the spins align in something called spin Hall configuration.[3, 36] Spin Hall configuration is when the spin-up and down electrons localize themselves at the edges of the lateral channel, one spin-direction at each edge. This state only occurs for strongly biased confine-ment potentials. For GaAs the spin Hall effect occurs for gate voltages Vr= 1 V

and Vl = 4 V and higher. InAs has this configuration at Vr = 1 V and Vl = 2

V and higher. For higher Fermi energy there are states where the non-occupied subbands have a small energy increase, and the occupied subbands have a small energy decrease, which differ for spin-up and spin-down which leads to a polar-ized state, marked as S2 and S3 in the figure (intermediate states). At this state the electrons create row structures, but the same amount of rows for spin-up and down. Both the Hall configuration and the row structure are something observed in both simulations and experiments.[6]

Other works have shown that for highly asymmetrical confinement potentials a net polarization can occur, together with spin-up and down localizing to the wire edges. Also switching the asymmetrical gate voltages switches the spin polarization, i.e. determines which spin-direction that is occupied and which is raised above the Fermi energy.[6] However a switch between left and right gate voltage for the asymmetrical case has shown no bias for the spin polarization, the polarization direction is random.

4.5

Varying beta

Simulations have been done to see if a stronger β will have any effects on the system. The testing has been done for GaAs for which β is estimated at 5 ˚A2, and results are here presented for the cases of 50 and 500 ˚A2. Starting with difference

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0 2 4 6 8 10 12 14 16 18 20 E F (eV) 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 ρ (nm -1 ) β = 5 β = 50 β = 500

Figure 4.18: The total density at different Fermi energies (x axis) for a confinement potential at 4 and 1 V at right and left gate respectively. Plotted for three different β values, 5, 50 and 500 ˚A2.

in total density for the three different cases, seen in figure 4.18. The total densities for β at 5 and 50 ˚A2are not differing any significant amount, difference on the scale 1 × 10−6nm−1. For the 500 ˚A2there is a small offset in the densities compared to the other two cases. Higher densities mean that the subbands energies are lower i.e. filled with more electrons. The energy shift for the subbands also shift at which Fermi energies the polarization will occur. This is since the Fermi energy reaches the unoccupied subbands at slightly different energies with stronger LSOC. The polarization can be seen as the jumps in the total densities.

In figure 4.19 the electron polarization is plotted against Fermi energy, for dif-ferent β cases. Looking at the figure and how the electrons localize themselves inside the wire, it is seen that there are no major difference between the different β values. The polarization direction is still random, and the spin-up and down electrons do not favour any particular part of the channel. The electrons also localize themselves into stripes and Hall texture as before. The reason that the increased strength of the LSOC has no effect on the wire can be understood when considering how the kl values populate. In the model the only influence the

References

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