Quantized Transmission in an Asymmetrically Biased Quantum Point Contact

Linköping University | Department of Physics, Chemistry and Biology Master’s thesis, 30 hp | Master’s programme in Physics and Nanoscience Autumn term 2016 | LITH-IFM-A-EX—16/3274--SE

Erik Johansson

Examinator, Magnus Johansson Supervisors, Irina Yakimenko & Karl-Fredrik Berggren

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Theoretical Physics 2016-11-07 Department of Physics, Chemistry and Biology Linköping University, SE-581 83 Linköping, Sweden

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Titel Title

Quantized Transmission in an Asymmetrically Biased Quantum Point Contact

Författare Author

Erik Johansson

Sammanfattning Abstract In this project work we have studied how a two-dimensional electron gas (2DEG) in a GaAs/AlGaAs semiconductor heterostructure can be locally confined down to a narrow bottleneck constriction called a quantum point contact (QPC) and form an artificial quantum wire using a split-gate technique by application of negative bias voltages. The electron transport through the QPC and how asymmetric loading of bias voltages affects the nature of quantized conductance were studied. The basis is Thomas-Fermi simulations that within the Büttiker model give results somewhat similar to experimental work in aspects regarding electron density effects. An extension of the model to include exchange and correlation interaction was investigated, as well as compared to density functional theory.

Nyckelord Keyword

QPC, Conductance, Transport, Thomas-Fermi, Büttiker, Asymmetric, Potentials, AlGaAs, GaAs, Heterostructure

Linköping University

Quantized Transmission in an Asymmetrically Biased Quantum Point Contact

Master’s Thesis LITH-IFM-A-EX—16/3274—SE

Author: Supervisors: Erik Johansson Correction Irina Yakimenko Theoretical Physics, IFM Karl-Fredrik Berggren Theoretical Physics, IFM

Examiner: Magnus Johansson Theoretical Physics, IFM Ausbildung heißt, das zu lernen, von dem du nicht einmal wußtest, daß du es nicht wußtest. Abstract In this project work we have studied how a two-dimensional electron gas (2DEG) in a GaAs/AlGaAs semiconductor heterostructure can be locally confined down to a narrow bottleneck constriction called a quantum point contact (QPC) and form an artificial quantum wire using a split-gate technique by application of negative bias voltages. The electron transport through the QPC and how asymmetric loading of bias voltages affects the nature of quantized conductance were studied. The basis is Thomas-Fermi simulations that within the Büttiker model give results somewhat similar to experimental work in aspects regarding electron density effects. An extension of the model to include exchange and correlation interaction was investigated, as well as compared to density functional theory. Acknowledgements I would like to thank my supervisors Irina Yakimenko and Karl-Fredrik Berggren for all help, advice and discussion, and also for giving me the opportunity to do my project course with you that lead to this diploma work. A huge thanks goes out to my classmates for all support and shared moments throughout my studies, especially Jimmy and Andreas who kept fighting alongside me during the master’s degree. I would also like to thank my family for their everlasting support through thick and thin. Lastly, a special thanks goes out to Yvonne for helping me keep my mind off the physics when needed. Contents

1 Introduction 1

2 GaAs/AlGaAs Heterointerface Ballistic Regime 4

3 Model of split gates QPC 5 3.1 Model GaAs/AlGaAs Heterostructure ...... 5 3.2 Fundamental Theory ...... 7 3.2.1 Derivation: Landauer-Büttiker Formula ...... 7 3.2.2 Quantization of Conductance in Nanostructures ...... 9 3.3 Numerical Schemes ...... 12 3.3.1 Calculation of total potentials within Thomas-Fermi model ...... 12 3.3.2 Finite Diﬀerence Method ...... 15 3.3.3 Calculation Method ...... 16

4 Results of calculations and discussion 17 4.1 Numerical Büttiker Model ...... 18 4.2 Typical Electron Density - n0 ...... 19 4.3 Higher Electron Density - 3/2 n0 ...... 24 4.4 Lower Electron Density - 1/3 n0 ...... 27 4.5 Comments on Eigenvalues ...... 30 4.6 Extension of the model - Exchange and Correlation Interactions ...... 32 4.6.1 Density proﬁles ...... 32 4.6.2 Potentials ...... 34

5 Density functional theory model comparison 41

6 Conclusions 42 6.1 Thomas-Fermi model review ...... 42 6.2 Future Work ...... 42

7 References 43

8 Code 44

Appendix A 45 A.1 Typical Electron Density - n0 ...... 46 A.2 Higher Electron Density - 3/2 n0 ...... 58 A.3 Lower Electron Density - 1/3 n0 ...... 63 A.4 Exchange and Correlation ...... 69 A.4.1 Density proﬁles ...... 69 A.4.2 Potentials ...... 72 1 Introduction

The quantum-well-based high-mobility semiconductor heterostructures are now commonplace in nanoscience and technology. The high-quality nanostructures based on GaAs/AlGaAs semiconductors grown by molecular beam epitaxy have become the genuine quantum laboratories to study a number of fundamental issues in low-dimensional physics [1]. The reason is that the split-gate technique inherent in these systems is free of various imperfections and allows one to change the geometry of the system and the density of the underlying two-dimensional electron gas (2DEG) continuously down to the regime of a very low density electron gas when Wigner crystallization and lattice formation appear.

The physics of low-dimensional GaAs/AlGaAs based semiconductor structures such as quantum dots (QDs), quantum wires (QWs) and quantum point contacts (QPCs) has developed into an important part of nanotechnology, especially in connection with spintronics and quantum information processing. Systems fabricated in this way are also very versatile: QWs can be simply connected to electron reservoirs that serve as source and drain and the conductance measurement can be used as a tool for the identification of different electron configurations.

It is well known that when the wires become narrow they show the quantum properties at low temperature, i.e., the conductance tends to be quantized in units of the fundamental conductance quanta 2e2/h under changing gate voltage. This phenomenon known as the conductance quantization has been discovered experimentally by van Wees et al. [2] and Wharam et al. [3]. In theory, the existence of this phenomenon follows from the general Landauer-Büttiker formula [4], derived within the frame of a simple model of non-interacting electrons. According to this formula, the total conductance is G = N(2e2/h), where N is the number of open subbands (energy levels below the Fermi energy) in the QPC which is deﬁned by the value of the applied gate voltage. Thus we have a step-like behavior of the conductance as a function of the gate voltage, each step being 2e2/h. In the presence of an external magnetic ﬁeld the subbands are split, each level contributes e2/h to the conductance, and thus the conductance step becomes e2/h in magnitude.

In mid 2015 the group of Prof. M. Pepper and colleagues at University of Cambridge (UK) and London Centre for Nanotechnology, UCL has studied the conductance behaviour in asymmetrically biased split gates QPC. If the voltage on one split gate was ﬁxed and another swept it has been shown that in the cases of highly asymmetric voltages the behaviour of conductance follows one predicted by Büttiker model while for intermediate voltage settings the plateaus of the conductance become smeared or disappear. The (unpublished) experimental measurements are presented in Figure 1(a). An initial gate-1 voltage of Vsg1 = −0.22 V was ﬁxed while sweeping gate-2. For the consecutive measurements gate-1 was raised in 20 mV increments. In this measurement series the gate voltages undergo an asymmetric-symmetric-asymmetric transition with respect to their magnitudes. Once again, somewhere along this transition the conductance plateaus become smeared. This occurs for the curves starting close to Vsg2 = −1.50 V.

Some other work within this field includes the published paper of S. Kumar et al. from 2014 [5] where they, in connection to this diploma subject, investigated electron transport in a quasi-one dimensional electron gas as a function of the confinement potential. They found a transition similar to the one in Figure 1(a), represented by the blue curve in Figure 1(b). According to them this effect is owing to an anticrossing of the subbands, leading to the formation of a Wigner lattice. Noteworthy is that their gates have different dimensions than that of M. Pepper and the device structure includes an additional uniform top gate placed between the two gates.

1 (a) Image courtesy of M. Pepper and S. Kumar [6]. Here S stands for source and D for drain.

(b) Image courtesy of S. Kumar [5]. Here TG stands for top gate and SG for split gate.

Figure 1: Experimentally measured conductance.

2 Our motivation is based upon this peculiar experimental feature and it is of our highest interest to gain an understanding of the origin of this transition in conductance. The model device of choice will have the same structure and dimensions as the group of M. Pepper and colleagues. The aim is to study total electron potentials for diﬀerent settings of the split-gates voltage and to use the Büttiker model to calculate the conductance. The potentials will be generated by Thomas-Fermi simulations. A key point in our investigation is to see if it is possible to describe and understand the transition of conductance curves within the Thomas-Fermi model.

The Büttiker model describes one dimensional (1D) conductance and the smooth quantized steps are only achievable in the ballistic regime, wherein quantum effects arise and the charge carrier transport is unscattered and phase-coherent. Electron-electron scattering effects experi- enced in the diffusive regime would spoil and complicate the shape and can not be described by this model. In other words the Büttiker model does not cover Ohmic transport. To ensure that we are within the ballistic regime, a conducting channel length smaller than the mean free path is required. This channel will be created by constricting a 2DEG by gates.

There are some conditions that must be fulfilled in order to observe perfect 1D conductance. First of all, the electron mean free path in the 2DEG must be much greater than the length of the channel. This is because when the channel is formed, the screening effect of the gas is no longer present, which leads to the effective mean free path becoming shorter [7].

A second condition is that the quantum wire transitions are nonmixing adiabatic, meaning the electron eigenstates must be invariant in the process, i.e., ingoing subband equals outgoing subband.

Another condition is that the Fermi wavelength λF = 2π/kF (de Broglie wavelength of electrons near the Fermi energy) should be comparable to or smaller than the length of the channel. This is important since the wave nature of electrons can give rise to macroscopically observable phenomena only when the dimensions are comparable to the wavelength. This is ensured by introducing a suﬃcient carrier density into the channel.

One ﬁnal condition is that the thermal energy is smaller than the subband separation between two adjacent energy levels (kBT En+1 − En). Therefore, the quantum conductance measurements are done at low temperatures (≤ 4.2 K). This is discussed in [8].

3 2 GaAs/AlGaAs Heterointerface Ballistic Regime

The mean free path is the distance travelled before the initial momentum of the electron is destroyed [7] (where also the physics below is discussed, although notations may diﬀer). Let it be denoted by Λe and deﬁne it as the product of the average time between electron scattering events τsc and the Fermi velocity vF (since the collisions only involve electrons near the Fermi surface)

Λe = τscvF . (2.1) For a 2D system the density of states %(E) is given by m∗ %(E) = , (2.2) π~2 where m∗ is the electron eﬀective mass. We can express for such a 2D system the Fermi wave vector kF , the Fermi velocity vF and the Fermi energy EF as 1/2 kF = 2πne (2.3) 1/2 kF ~ 2πne v = ~ = (2.4) F m∗ m∗ 2k2 2πn E = ~ F = ~ e , (2.5) F 2m∗ m∗ where ne is the electron density. According to the Drude-Sommerfeld model we can express the average time between scattering events in terms of the electron mobility µe via the following equation eτ µ = sc . (2.6) e m∗ It is then trivial to deduce m∗µ τ = e . (2.7) sc e Thus a formula for the mean free path can be acquired. 5 2 For the 2DEG in the GaAs layer we have for our device µe = 8.5 · 10 cm /Vs and 11 −2 ne = 2 · 10 cm , meaning that the mean free path is µ Λ = τ v = ~ e 2πn 1/2 (2.8) e sc F e e Js cm2 = (6.5821189 · 10−16 )(8.5 · 105 )2π · 2 · 1011 cm−21/2 C Vs = 6272 nm.

In terms of nanostructure scales this is very large, making it possible to really observe scattering free transport in the ballistic regime and its quantum eﬀects in the GaAs/AlGaAs heterojunction. The Fermi wavelength of this system is 1/2 λF = 2π/kF = 2π/ 2πne = 56 nm. (2.9) In comparison with a typical metal, the 2DEG in the GaAs/AlGaAs heterojunction has a Fermi wavelength which is about a hundred times larger. For metals like sodium it turns out to be a few Ångströms and comparable to the lattice constant. Altogether, as long as the channel length L is within the interval 56 nm < L < 6272 nm, we can expect good conductance.

4 3 Model of split gates QPC

3.1 Model GaAs/AlGaAs Heterostructure As the basic model we use the gated modulation-doped GaAs/AlGaAs heterostructure [9]. By appropriate combination of semiconductor materials, patterned gates, doping and applied gate voltages the electrons can be trapped at the AlGaAs/GaAs heterointerface forming the 2DEG laying 90 nm below the surface and having the density 2 · 1011 cm−2 in our present model.

As for all undoped semiconductors, the band structure near k = 0 is characterized by having a valence band (VB) and a conductance band (CB) separated either directly or indirectly by a bandgap, usually denoted Eg. The regular way to introduce electrons into the conduction band of an undoped semiconductor is to promote them via excitation from the valence band, leaving behind positively charged vacancies called holes. One can also achieve this by doping the semiconductor with impurity atoms that have more (donors) or fewer (acceptors) electrons in the outmost orbitals than the atoms of the lattice, introducing delocalized electrons into the CB and holes into the VB. Doing so, the number of charge carriers is increased while also leaving the semiconductor in its ground state. There can arise a problem with scattering processes related to Coulomb interaction between the carriers and the charged donors or acceptors which are left behind when the carriers are released. This scattering can negatively aﬀect the carrier propagation within the structure and disrupt the coherence of their wavefunctions which is necessary for observing the desired quantum eﬀects in their pure form. One can overcome this problem by using modulation doping, whereby doping is grown up in one region but then the carriers migrate to another region. In the heterostructure in question, the undoped GaAs layer has a lower bandgap than the negatively doped AlGaAs, forming discontinuity in the conduction band. The electrons may therefore penetrate, via the undoped AlGaAs spacer layer, into GaAs to lower their energy. When the donors become ionized, an electrostatic potential arises, trying to drive the electrons back into the AlGaAs layer. This potential warps the discontinuity in the conduction band (owing to the boundary conditions when the Poisson equation is solved), forming a narrow potential well at the heterointerface. The shape of the well is roughly triangular and it extends over 10 nm. In it, the electrons become trapped as they cannot climb the potential well. In the other two dimensions there are no restrictions and thus a 2DEG is formed in the AlGaAs/GaAs heterointerface, as shown in Figure 2 and also 3.

Figure 2: Conduction band structure of the device. Original image courtesy of J. Hakanen [10]. Here eVs is a Schottky barrier appearing at the top interface. The thicknesses of the cap, donor and spacer layers are given by c, d and s respectively. µm is the chemical potential of AlGaAs and µs is the chemical potential of GaAs. ∆Ec is a discontinuity in the conduction band Ec. Lastly, vG denotes an applied gate voltage.

5 The 2DEG may then be properly shaped by means of lithography and application of negative voltage to the metallic split-gates on the top of the structure. The electron transport is studied in the confined QPC with the use of the device consisting of a pair of split-gates that are 200 nm long and deposited on the modulation-doped GaAs/AlGaAs heterostructure. The two gates are separated by a 300 nm wide gap, as depicted in the top view of the device seen in Figure 3. The conducting channel will be formed within the dashed square area between the split-gates. One of these split-gates is held at fixed voltage while the another gate voltage is swept. This allows one to measure the conductance as a function of an asymmetric confinement strength.

Figure 3: Schematic picture of the split gates device used in our modelling. Original device image courtesy of K.-F. Berggren and I. Yakimenko [11]. The red areas in the top view represent depletion regions arising for some arbitrary negative gate biases Vg1 and Vg2 that repel nearby electrons.

6 3.2 Fundamental Theory 3.2.1 Derivation: Landauer-Büttiker Formula As we will show below, in the ballistic regime, the conductance of the 2DEG through the constriction shows quantized behavior with the conductance changing in quantized steps of (2e2/h) when the eﬀective width of the constricting channel is varied by controlling the voltages of the gates above the 2DEG. We ﬁrst give a derivation of the quantization of the conductance [9, 13]. Consider the areas surrounding the QPC (the 1D quantum wire) as a Fermi sea of electrons. The current through the wire coming from left-to-right propagating electrons is given by Z ∞ IL = e fFD((k), µL)n1D(k)v(k)T (k)dk (3.1) 0

Here, the product of the charge e, velocity v(k) and the density of states n1D(k) represents a current density (J = env). The Fermi-Dirac function fFD((k), µL) gives the probability that each state is occupied, governed by the Fermi level of the left sea µL. T (k) is the transmission coeﬃcient that gives the probability that a charge carrier passes through the barrier and contributes to the current. The integral is only over positive k-values as only electrons propagating from the left will contribute to the right-going current.

−1 Using n1D(k) = π and then changing variables by using the following deﬁnition of veloc- −1 ity, v = ~ ∂E/∂k, dk 1 dk = dE = dE, (3.2) dE ~v gives Z ∞ dE 2e Z ∞ IL = e fFD(E, µL)vT (E) = fFD(E, µL)T (E)dE. (3.3) 0 π~v h 0 Note that the velocity cancels in this expression, an important feature that underlies the quantized conductance which we will discuss later. Namely, though the states at higher energy have a higher velocity, and might therefore be expected to carry more current, this is exactly cancelled by the reduction in their density of states.

The expression for the current due to electrons arriving from the right is almost identical. The only diﬀerence is the sign, since the electrons are travelling in the opposite directions, and the Fermi level. Therefore, 2e Z ∞ IR = − fFD(E, µR)T (E)dE. (3.4) h 0 The total current is then given by the sum of the expressions 2e Z ∞ I = IL + IR = [fFD(E, µL) − fFD(E, µR)]T (E)dE. (3.5) h 0

An obvious check of this relation is that I = 0 when µL = µR, i.e. there is no "driving force". When a nonzero current ﬂows, it is driven by a chemical potential gradient, which in turn appears when a bias voltage is applied along the device. This is usually referred to as a source-drain bias, denoted VSD. However, in general, the current is not proportional to the bias; typically, it is a complicated function and Ohm’s law does not hold.

7 To simplify the result, we consider the case of low temperature, where the Fermi occupation function can be approximated by a step function. Then only electrons with energies between µL and µR contribute to the current, and hence we get

2e Z µL I = T (E)dE. (3.6) h µR If we further assume that the applied voltage is small, such that the energy dependence of T(E) is negligible, the relation simpliﬁes to 2e I = T (µ − µ ). (3.7) h L R Finally, by connecting the diﬀerence in the chemical potential to the source-drain bias voltage

eVSD = µL − µR (3.8) the relation takes on the form 2e2 I = TV (3.9) h SD or in terms of the conductance I 2e2 G = = T, (3.10) VSD h which is the single-channel Landauer-Büttiker formula. Its generalization to the case when N subbands are populated at the Fermi energy is

N 2e2 X G = T . (3.11) h i i=1 We can see that the conductance is given by the product of the transmission coeﬃcient and the 2e2 fundamental conductance quanta h . The inverse of this quanta corresponds to a resistance of 12906.4 Ω. One of the characteristics is that this conductance does not decrease inversely with the length as it would classically. The other one is that the mesoscopic systems may be governed by transport through a number of the channels, and the conductance changes precisely with the 2e2 factor h .

8 3.2.2 Quantization of Conductance in Nanostructures Typical result of experimental measurements is shown in Figure 4 below [2, 3]. We can observe that the conductance follows a staircase-like shape with plateaus separated by the fundamental conductance quanta, in accordance to formula (3.11).

Figure 4: Experimentally measured QPC conductance as a function of gate voltage. The gates are symmetrically loaded and the conductance shows a step-like behaviour. Image from [14]

This result can be theoretically derived by calculating the transmission coefficient for the constriction. As thoroughly discussed in [8], an appropriate negative bias on the gates results in the formation of a saddle potential. This does not only confine the electron movement across the channel, but also presents a potential barrier at the saddle minimum in the direction of current flow along the channel (see Figure 5 for how such a confinement usually looks).

Figure 5: Example of a typical conﬁnement potential. Image courtesy of I. Yakimenko.

9 The form of this potential evolves smoothly with change of gate voltage, with the saddle minimum rising in energy when the width of the QPC is made smaller, as the bias voltages get increasingly more negative.

In a close proximity to this saddle bottom, the potential energy can be expanded, in terms of appropriate coordinates x (along the channel) and y (across the channel), and be described by a parabolic form 1 1 V (x, y) = V − m∗ω2x2 + m∗ω2y2. (3.12) 0 2 x 2 y ∗ Here, V0 is the electrostatic potential at the saddle (barrier height), m is the electron effec- ∗ tive mass (m = 0.067me) and the curvatures of the potential are expressed in terms of the characteristic frequencies ωx and ωy. However, these do not physically correspond to oscillatory frequencies in the potential landscape. The energy splitting of the subbands in the QPC is determined by ωy, while ωx dictates how sharply the transmission drops to zero when the saddle minimum becomes greater than the Fermi level. p2 The total energy is given by the potential (3.12) supplemented by a kinetic energy 2m∗ . The Hamiltonian is separable into a transverse wave function associated with energies 1 ω n + , n = 0, 1, 2, ... (3.13) ~ y 2 and a wave function for motion along x in an effective potential 1 1 V + ω n + − m∗ω2x2. (3.14) 0 ~ y 2 2 x This effective potential can be viewed as the band bottom of the nth quantum channel (or subband) in the region of the saddle point. In the absence of quantum tunneling the channels with the threshold energy 1 E = V + ω n + (3.15) n 0 ~ y 2 below the Fermi energy are open, and the channels with the threshold energy En above the Fermi energy are closed. Due to quantum tunneling, the channels are neither completely open nor completely closed but permit transmission with a probability Tmn. Here, the index n refers to the incident channel, and the index m refers to the outgoing channel. The transmission probabilities for this case can be calculated to: 1 Tmn = δmn , (3.16) 1 + e−πn where 2 h 1 i n = E − ~ωy n + − V0 . (3.17) ~ωx 2 Here the convention EF ≡ E is used.

A short motivation for the calculation of Tmn is given below, for full detail see [15].

Consider the transmission probability for a particle passing through a one-dimensional inverted parabolic barrier. The Schrödinger equation can be transformed and written in a compact form d2 + X2 + φ(X) = 0. (3.18) dX2

10 The solutions take on the form of parabolic cylindrical functions. As discussed in the reference, for every value of , there is an even and an odd solution, denoted as φe(X) and φo(X). These may be expressed in terms of conﬂuent hypergeometric functions F (a|b|u) as 2 1 1 1 φ (X) = e−iX /2F + i | | iX2 (3.19) e 4 4 2 2 3 1 3 φ (X) = Xe−iX /2F + i | | iX2 . (3.20) o 4 4 2

For large values of |X| (far away from the potential barrier) the asymptotic forms of the wave function associated with the incoming and outgoing current are denoted φin(u) and φout(u) respectively. The transmission coeﬃcient may then be deﬁned as

2 |φout(X)| T1D = lim 2 . (3.21) X→∞ |φin(−X)| After many steps of manipulation including the Gamma function the transmission coeﬃcient takes on the form seen in (3.16). Thus, the Büttiker model holds if the scattering potential is parabolic.

P The total transmission probability T = n Tnn for diﬀerent ωy/ωx cases as a function of (E − V0)/~ωx is shown in Figure 6. The opening of successive quantum channels leads to the quantization of the conductance.

ωy/ωx 2

2 G/(2e2/h)

0 10 5 0 (E-V0)/ℏωx

Figure 6: Conductance curve as a function of (E − V0)/~ωx for diﬀerent saddle potentials characterized by the ratio ωy/ωx (Recreation from [4]).

11 3.3 Numerical Schemes 3.3.1 Calculation of total potentials within Thomas-Fermi model Before getting into the Thomas-Fermi model, it is worth mentioning that it is assumed that the donors are fully ionized and that the electron gas is homogeneous. It is restricted to slowly varying potentials, has a hard time to model impurity states and cannot accurately treat kinetic energy.

A full description of modelling within Thomas-Fermi approximation is given in [12] and is sum- marized here below. Please note that the point of origin used in the calculations below diﬀers from what was previously shown in Figures 2-3.

To find the electronic configuration of the device one can assume that the electron gas is strictly two-dimensional, the donor layer is fully ionized and the density of electrons is sufficiently high. We also assume that two boundary conditions are satisfied: 1) the heterostucture is electrically neutral, therefore the electric field vanishes at the infinity; 2) there is a Schottky barrier for electrons at the interface with the metallic gate and its value is eVs = 0.8 eV, recall Figure 2. As an initial step to describe the equilibrium electronic properties the semi-classical Thomas-Fermi approximation has been exploited. The corresponding Thomas- Fermi equation for the electron density has the form:

π 2 ~ n(r) + U c(r) + U e(r) = µ. (3.22) m∗ Here, n(r) is the local 2D electron density, m∗ is the electron eﬀective mass in GaAs (in our ∗ calculations m = 0.067me, me being the mass of free electron); µ is the chemical potential being constant everywhere in the 2DEG interface (we put µ = 0 as the reference energy). The conﬁnement potential

c U (r) = eVg(r, z) + eVd + eVs (3.23) comprises contributions from the gates Vg, donors Vd and surface states Vs (eVs = 0.8 eV) [9]. The potential caused by the gates at the point r = (x, y) in the 2DEG plane at depth z0 = 90 nm from the gates is given by the well known expression [9] 1 Z |z | eV (r, z) = dr0eV (r0, 0) 0 , (3.24) g g 0 2 2 3/2 2π (|r − r | + z0) 0 where Vg(r , 0) is the distribution of the potential along the surface of the gates. The contribution from the inﬁnite donor layer: e2 eVd = − ρdd(c + d/2). (3.25) 0 17 −3 where ρd = 6 × 10 cm is the density of donors, and c = 24 nm and d = 36 nm are the thicknesses of the cap and donor layer, correspondingly, and is the dielectric constant which we take equal to 12.9 in GaAs. The resulting bare conﬁnement potential for a QPC has a saddle point shape at the center of a QPC. The direct Coulomb (Hartree) interaction U e(r) is written in the form: " # e2 Z 1 1 U e(r) = dr0n(r0) − . (3.26) 4π |r − r0| p 0 2 2 0 |r − r | + 4z0

12 The second term in the brackets represents the eﬀect of mirror charges introduced to satisfy the boundary conditions. To avoid the diﬃculties with convergence of the iterative scheme, the value of the Hartree potential at each iteration is mixed with the one at the previous step

U e,(i+1) = αU e,(i) + (1 − α)U e,(i−1). (3.27) The mixing parameter α is 0.01 in our calculations in order to damp the Coulomb integral which has a large magnitude in the 2D case. Following this iterative process will yield a 2D total electron potential that I will denote by UTF (x, y). Three examples of such a total electron potential obtained from the Thomas-Fermi equation are shown in Figure 7. These particular ones come from calculations with exchange-correlation interactions not yet discussed and for other device dimensions. However, they are included as the overall shapes of the potentials are basically the same, even with the added interactions. The used notation in this ﬁgure, V (x, y), is meant to represent a general total electron potential.

13 0.8

0.6

0.4

V(x,y) 0.2

400 0

200 −0.2 x (nm) 400 350 300 250 200 150 0 100 50 0 y (nm)

(a) Vg1 = −3.0 V, Vg2 = −4.4 V.

0.7

0.6

0.5

0.4

0.3 V(x,y) 0.2 400 0.1 0 200 x (nm) −0.1 400 350 300 250 0 200 150 100 50 0 y (nm)

(b) Vg1 = −4.0 V, Vg2 = −3.5 V.

1.2

0.8

0.6 V(x,y)

0.4

0.2 400 x (nm) 0 200

−0.2 0 400 350 300 250 200 150 100 50 0 y (nm)

Figure 7: Three examples of the total electron potential V (x, y) in the device. As mentioned, these potentials include exchange-correlation interactions not yet discussed, but still visualize typical potential shapes in a nice manner. Also, the plot origin diﬀers from what is shown in Figures 2-3. Image courtesy of I. Yakimenko.

14 3.3.2 Finite Diﬀerence Method Thomas-Fermi simulations generate two-dimensional potential landscapes of the system that will be analyzed and used for the conductance calculations. Obtaining the energy eigenvalues and eigenstates can be done by solving the Schrödinger equation that governs the dynamics of the 2DEG in the heterostructure. In its full form using the total potential from the Thomas-Fermi model it can be written as 2 ∂2 ∂2 − ~ + ψ(x, y) + U (x, y)ψ(x, y) = Eψ(x, y). (3.28) 2m∗ ∂x2 ∂y2 TF

Solving the equation given above requires intricate integration and can be tedious. However it is not necessary for the Büttiker model as we will calculate the conductance in one dimension. Also, we make the assumption that the potential is separable in x and y which makes the problem easier to handle. The generated two-dimensional Thomas-Fermi landscape comes in discrete points, allowing us to employ a discretization method to simplify the equation. The x − y plane in the GaAs/AlGaAs is discretized into a Nx × Ny grid. Nx denotes the number of elements in the x-direction and Ny the number of elements in the y-direction. The second derivatives can be replaced by the wavefunction ﬁnite diﬀerences

Diψ = ψi−1,j − 2ψi,j + ψi+1,j ; δi = xi+1 − xi (3.29)

Djψ = ψi,j−1 − 2ψi,j + ψi,j+1 ; δj = yj+1 − yj (3.30) where δi is the separation between two points in the x-direction and δj is the separation between two points in the y-direction. In our model the spatial resolution is δi = δj = 10 nm.

Following this schema yields a system of equations that can be solved in an iterative manner using proper boundary conditions. In Thomas-Fermi, as well as in density functional theory (DFT) that will be introduced later, Dirichlet boundary conditions are used in y-direction and periodic boundary conditions in x-direction.

2 ~ Diψ Djψ − ∗ 2 + 2 + UT F,ijψi,j = Eψi,j (3.31) 2m δi δj where ψi,j and UT F,ij are the wavefunction and the total potential at a point (xi,yj) of the 2DEG plane, and E is an energy eigenvalue. We will calculate one-dimensional conductance along x and have to look for one-dimensional subbands for which we only must consider the lateral restriction. Thus only the y cross-sections are to be considered for the eigenvalue problem. The set of equations are then reduced to one dimension

2 ~ − ∗ 2 (ψj−1 − 2ψj + ψj+1) + UT F,i0jψj = Eψj. (3.32) 2m δj

Here, the total potential UT F,i0j is a cross-section running across the system for a ﬁxed xi0 .

The proper x and y cross-sections intersect the minimum of the saddle shaped potential (recall the shapes shown in Figure 7) and how to ﬁnd them goes as follows:

For y: Due to the geometry of the device, the proper y cross-section is always located at the middle with respect to x, i.e., xi0 = 500 nm and does not depend on the applied gate biases.

15 For x: Although this cross-section is not involved in the eigenvalue problem, its importance comes in later in the Büttiker model when locating the correct path along x. For every ﬁxed yj the potential cross-section will approximately have an inverse parabolic shape. The key of ﬁnding the proper one is to locate the yj0 that gives the lowest inverse parabola maximum value. This is dependent on the values of the gate biases as the saddle minimum can move in the y-direction.

In order to facilitate the solving of the system of equations (3.32), it is helpful to represent it in matrix form. Let H be the three-diagonal Hamiltonian matrix

2 ~ Hn,n = ∗ 2 + UT F,i0n m δj 2 ~ Hn,n+1 = Hn+1,n = − ∗ 2 (3.33) 2m δj

Hn,m = 0 otherwise.

The on-site terms lie in the matrix diagonal elements and the oﬀ-diagonal elements are coupling terms to the adjacent nearest-neighbour sites. The energy spectrum {Ek} can be obtained by solving the eigenvalue problem for Hψ = Eψ. However, we are only looking for the levels below and close to the Fermi energy, not the whole spectrum. This is due to the fact that the electron transport only occurs in the occupied subbands below the Fermi energy. Thus, only the grid points in proximity to the centre of the channel must be considered in the calculations. If the desired number of accurate eigenvalues is given by m one should generally speaking choose an M × M sized matrix such that m M is satisﬁed.

3.3.3 Calculation Method All visualization, parametrization, calculations, etc. of the Thomas-Fermi simulation raw data are done in Wolfram Mathematica 10.0. The Thomas-Fermi simulations themselves are done on the NSC cluster.

16 4 Results of calculations and discussion

As previously stated, the objective is to study how the conductance behaves when the split gates are loaded asymmetrically. It is in our interest to see if the Thomas-Fermi model can explain the nature of the transition in conductance seen in the yet unpublished work of Prof. M. Pepper and colleagues. Looking back at the Büttiker model we can deduce that we need to work out the characteristic frequencies ωx and ωy as well as the energy spectrum for every single simulation in order to calculate the conductance. The energy spectrum will give information about the number of occupied subbands below the Fermi energy simply by looking at the sign, as in the simulations EF = 0 as a reference. The characteristic frequencies are calculated from the curvature of the proper cross-sections selected as described in section 3.2.2. The Thomas-Fermi simulations were done at three diﬀerent electron densities. The starting value was at a typical 11 −2 11 −2 density of ne ≡ n0 = 2·10 cm . It was done at a higher density of (3/2)n0 = 3·10 cm and 11 −2 at a lower density of (1/3)n0 ≈ 0.67·10 cm . The reason behind this being to investigate how the electron density aﬀects the potential geometry and the conductance, but also because it was experimentally proven that increasing the electron population, by means of device illumination, leads to a more equispaced distribution of conductance curves along the swept voltage axis [6].

Some more details regarding the calculation of the characteristic frequencies. This is done by parametrization of the cross-sections where we find polynomial functions that fit arbitrarily close to the total potential. It is not important that they accurately fit on a global scale, but in a close proximity to the channel, i.e., where the curve is mostly below the Fermi energy. Reconsider the potential model (3.12), now with shifted origin to where the proper cross-sections meet in the point (xi0 ,yj0 ) ≡ (x0,y0) 1 1 V (x, y) = V − m∗ω2(x − x )2 + m∗ω2(y − y )2. (4.1) 0 2 x 0 2 y 0 As will be shown in a moment, the tunneling barrier along the channel (x cross-section) is smooth and symmetric around its origin. Three interpolation points centred at the barrier maximum and separated by 60 nm where thus chosen for a sufficient fitting, resulting in a second degree polynomial. For the asymmetric y cross-sections a first parametrization attempt of five interpolation points was made, resulting in a fourth degree polynomial. The method of finding ωx and ωy is then simple coefficient identification. Let Vˆx and Vˆy denote the interpolated polynomials of the respective cross-section 1 1 Vˆ (x) = −ax2 + bx − c ≡ − m∗ω2x2 + m∗ω2x x − m∗ω2x2 (4.2) x 2 x x 0 2 x 0 ∂2 ∂2 1 1 Vˆ (y) = (αy4 − βy3 + γy2 − δy + ) ≡ m∗ω2y2 − m∗ω2y y + m∗ω2y2. (4.3) ∂y2 y ∂y2 2 y y 0 2 y 0

The right hand sides are simply expansions of the terms in (4.1). I identified ωy by evaluating the remaining constant terms at y = 0. The fourth degree polynomials gave very accurate graphical fits, however this method gave ωy values about one order of magnitude too low. Post defence note: As pointed out by my examiner, the error originates from not accounting for the shifted origin that requires the evaluation to occur at y = y0. In the second attempt three interpolation points were chosen for the y cross-sections as well, 1 1 Vˆ (y) = αy2 − βy + γ ≡ m∗ω2y2 − m∗ω2y y + m∗ω2y2. (4.4) y 2 y y 0 2 y 0 The separation from the centre point to the edge ones were chosen in an asymmetric manner for the best possible fit. This had to be manually tuned for well over a hundred cases, but proved to give better values for ωy in roughly the same range as ones calculated in [16].

17 4.1 Numerical Büttiker Model The expression for the conductance is

2e2 X G = T , where T is from (3.16). h mn mn m,n

This model (as discussed in [4]) assumes that the gates are symmetrically loaded, that potentials are completely harmonic and that the shape of the lateral constriction is static when the bias voltage is varied. A consequence of this assumption is that the model predicts the eigenenergies to be equispaced and that the spacing is independent of the bias voltage. For some cases this can lead to rough estimates, especially here for asymmetric lateral constrictions that are far from parabolic. 1 I will step away from the harmonic oscillator model with its energies En,HO = V0 + ~ωy(n + 2 ) and instead calculate the energies numerically, letting n → n,Num while keeping in mind that the Fermi energy is set to zero as a reference. 2 n,Num = − En,Num. (4.5) ~ωx This is done with the ﬁnite diﬀerence method as discussed in section 3.3.2. The characteristic frequency ωy will still be calculated as the fraction ωy/ωx can be used as a bridge back to the Büttiker model. As can be seen in Figure 6, this frequency fraction controls how fast the conductance staircase-like shape grows. For higher values the quantized plateaus become wider and further spaced apart. Keeping this in mind, we can calculate the conductance and the characteristic frequencies separately and compare to what degree the curve shape agrees with this prediction.

As previously stated, in the simulation model the grid spacing is 10 nm and for the eigenvalue problem we consider the centre of the wire and 200 nm to each side as our calculation region, resulting in the Schrödinger equation being expressed by a 40 × 40 matrix. This should give a sufficient numerical accuracy. The eigenenergies are easily obtained as nontrivial solutions to the matrix equation Hψ = Eψ, i.e., finding the E spectrum that satisfies det(H − EI) = 0. This is solved in Mathematica.

18 4.2 Typical Electron Density - n0 11 −2 The starting electron density used in our model is as previously mentioned n0 = 2 · 10 cm . In Figure 8 the confinement potential, obtained from (3.23), is plotted for the case Vg1 = −0.5 V where Vg2 is swept from −4.9 V to −1.3 V. This is the bare electrostatic potential where electrons are yet to be introduced into the system. By comparing its figure to that of the total potential, obtained by following the iterative process (3.27), seen in Figure 9, one can note that electron-electron interactions tend to have a flattening effect on the potential, making them less harmonic and shifted upward. Notice in the insets of these figures that the minimum is shifted across the wire as the bias voltage Vg2 is varied (compare this to hanging sheets over chairs with different back heights). These insets are zoomed in at where the channel is located. In Figure 10 the potential tunneling barrier is shown for the same gate bias cases. By looking at the figures we can conclude that by lowering the voltage of the swept gate, the asymmetry is increased leading to an upwards shift on the saddle minimum close to the Fermi energy.

Then in Figure 11 the six first eigenvalues are plotted for each Vg2 case. Once again, each energy spectrum is obtained by solving the discrete Schrödinger equation (3.32) using the total potentials seen in Figure 9. Each level varies in a smooth manner with respect to Vg2. By simply varying the bias voltage the number of open occupied subbands can be controlled, effectively opening and closing the channel. To the left the channel is pinched off as the energy spectrum is strictly positive, above the Fermi energy and thus unoccupied. In this data range up to five subbands can be occupied, meaning that one can expect a conductance up to five times the 2e2 conduction quanta h .

In the nextcoming Figure 12, the result of the characteristic frequency calculations, for several cases, is shown. They are multiplied by ~ to scale down their values to reasonable magnitudes in terms of energy. The total potential cross-sections that we just have seen give rise to the two black curves, the upper one being ~ωy and the lower ~ωx. Notice how the scaled y-frequency greatly increases as Vg2 becomes more negative in accordance to the curvature seen in the inset of Figure 9. The change in the x-frequency is less noticeable.

For the other cases ranging from Vg1 = −1.0 V to −5.0 V, please look at the supplementary material and Figures 34-57 (Plot ranges may vary!). Notice how the x cross-sections are similar for all cases, but the total potential geometry becomes very intricate for the y cross-sections when Vg1 is successively lowered. Especially when it goes below −2.8 V for which we almost can expect the formation of double-well potentials, but they all ﬂatten out. Also interesting, the bottom curves in Figure 52, 54 and 56, i.e., y cross-sections of (Vg1,Vg2) = (−4.0 V,−0.01 V), (−4.5 V,−0.05 V) and (−5.0 V,−0.05 V) are curiously close to the potential in the particle in a box problem.

Back to Figure 12, the eﬀect of lowering the bias voltage of Vg1 down to −5.0 V results in more rapidly varying ω curves as the QPC shape becomes more sensitive to small variations in the swept gate Vg2. Recalling the ωy/ωx magnitude eﬀect on the conductance seen in Figure 6 we expect wider plateaus for high values of the fraction. The ωy and ωx curves seem to converge toward a common point when Vg2 approaches zero, i.e., ωy/ωx → 1 for which the Büttiker model predicts the quantized steps to almost vanish.

19 The conductance is calculated and shown in Figure 13. I did this by letting Mathematica find numerical functions that describe the energy and characteristic frequency curves. Doing this I could interpolate between neighbouring calculated points and fill in the gaps. One can see that when the electrons become less laterally confined (Vg2 less negative), the plateaus become more packed together. The spacing between the conductance curves also decreases. Compare the distance between the first two and the last two curves. For both pairs the difference in Vg1 is 0.5 V. As predicted above the staircase-like shape is also less prominent when Vg1 is more negative and when Vg2 approaches zero. The widest plateaus are toward the left in the plot where the characteristic frequency fraction is at its largest.

To understand the nature of the conductance we must discuss the lateral total potential again. Consider once again the cross-section in the inset of Figure 9 (Vg1 = −0.5 V). In the conductance region of the channel, the total potential is parabolic in its shape. Compare this with Figure 56 (Vg1 = −5.0 V) where the total potential loses its parabolic nature after the second subband becomes occupied (roughly at Vg2 = −0.30 V) and then ﬂattens. These two cases have completely diﬀerent total potential geometry evolutions when Vg2 is varied.

We can conclude that the total potential geometry really matters for the explanation of the conductance behaviour.

20 V V a) g2 = - 1.3 V δ) g2 = - 4.8 V Vg1 = - 0.5 V , X= 500 nm b) Vg2 = - 1.5 V ϵ) Vg2 = - 4.9 V c) V = - 1.6 V 2000 g2 d) Vg2 = - 1.7 V V e) g2 = - 1.8 V 0 f) Vg2 = - 1.9 V g) Vg2 = - 2.0 V -20 h) V = - 2.1 V g2 -40 i) Vg2 = - 2.2 V V 1500 j) g2 = - 2.3 V -60 k) Vg2 = - 2.5 V l) Vg2 = - 2.6 V -80 m) V = - 2.7 V g2 -100 n) Vg2 = - 2.8 V o) Vg2 = - 2.9 V V -120 1000 p) g2 = - 3.0 V q) Vg2 = - 3.2 V 300 400 500 600 700 r) Vg2 = - 3.3 V s) Vg2 = - 3.4 V t) Vg2 = - 3.5 V u) Vg2 = - 3.6 V v) Vg2 = - 3.7 V Confinement potential(meV) 500 w) Vg2 = - 3.8 V x) Vg2 = - 3.9 V y) Vg2 = - 4.0 V z) Vg2 = - 4.1 V å) Vg2 = - 4.2 V ä) Vg2 = - 4.3 V V 0 ö) g2 = - 4.4 V α) Vg2 = - 4.5 V β) Vg2 = - 4.6 V γ) Vg2 = - 4.7 V -200 0 200 400 600 800 1000 Y(nm)

Figure 8: Lateral cross-section of the conﬁnement potential for the case Vg1 = −0.5 V.

V V a) g2 = - 1.3 V δ) g2 = - 4.8 V Vg1 = - 0.5 V , X= 500 nm b) Vg2 = - 1.5 V ϵ) Vg2 = - 4.9 V c) V = - 1.6 V 2000 g2 d) Vg2 = - 1.7 V V e) g2 = - 1.8 V 6 f) Vg2 = - 1.9 V g) Vg2 = - 2.0 V 4 h) V = - 2.1 V g2 2 i) Vg2 = - 2.2 V V 1500 j) g2 = - 2.3 V 0 k) Vg2 = - 2.5 V l) Vg2 = - 2.6 V -2 m) V = - 2.7 V g2 -4 n) Vg2 = - 2.8 V o) Vg2 = - 2.9 V V -6 1000 p) g2 = - 3.0 V q V 3.2 V ) g2 = - 300 400 500 600 700 r) Vg2 = - 3.3 V s) Vg2 = - 3.4 V t) Vg2 = - 3.5 V Total potential(meV) u) Vg2 = - 3.6 V v) Vg2 = - 3.7 V 500 w) Vg2 = - 3.8 V x) Vg2 = - 3.9 V y) Vg2 = - 4.0 V z) Vg2 = - 4.1 V å) Vg2 = - 4.2 V ä) Vg2 = - 4.3 V V 0 ö) g2 = - 4.4 V α) Vg2 = - 4.5 V β) Vg2 = - 4.6 V γ) Vg2 = - 4.7 V -200 0 200 400 600 800 1000 Y(nm)

Figure 9: Lateral cross-section of the total potential for the case Vg1 = −0.5 V.

21 V V 0 a) g2 = - 1.3 V δ) g2 = - 4.8 V Vg1 = - 0.5 V b) Vg2 = - 1.5 V ϵ) Vg2 = - 4.9 V c) Vg2 = - 1.6 V d) Vg2 = - 1.7 V e) Vg2 = - 1.8 V f) Vg2 = - 1.9 V g) Vg2 = - 2.0 V -2 h) Vg2 = - 2.1 V i) Vg2 = - 2.2 V j) Vg2 = - 2.3 V k) Vg2 = - 2.5 V l) Vg2 = - 2.6 V m) Vg2 = - 2.7 V V -4 n) g2 = - 2.8 V o) Vg2 = - 2.9 V p) Vg2 = - 3.0 V q) Vg2 = - 3.2 V r) Vg2 = - 3.3 V s) Vg2 = - 3.4 V t) Vg2 = - 3.5 V Total potential(meV) -6 u) Vg2 = - 3.6 V v) Vg2 = - 3.7 V w) Vg2 = - 3.8 V x) Vg2 = - 3.9 V y) Vg2 = - 4.0 V z) Vg2 = - 4.1 V å) Vg2 = - 4.2 V -8 ä) Vg2 = - 4.3 V ö) Vg2 = - 4.4 V α) Vg2 = - 4.5 V β) Vg2 = - 4.6 V γ) Vg2 = - 4.7 V -200 0 200 400 600 800 1000 X(nm)

Figure 10: Longitudinal cross-section of the total potential for the case Vg1 = −0.5 V.

V = - 0.5 V 20 g1

En,Num (meV)

-5

-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5

Vg2 (V)

Figure 11: Energy eigenvalues as a function of Vg2 for the case Vg1 = −0.5 V.

22 Vg1 = - 0.5 V ℏωy -● Vg1 = - 1.0 V ℏω -▲ Vg1 = - 1.5 V x Vg1 = - 1.9 V Vg1 = - 2.2 V Vg1 = - 2.5 V V = - 2.8 V 6 g1 ● V 3.0 V g1 = - ● ● ● Vg1 = - 3.2 V ● ● ● ● ● ● ● ● ● Vg1 = - 3.5 V ● ● ● ● ● ● ● ● ●● ● V = - 4.0 V ● ● g1 ● ● ● ● ● ● ● ● Vg1 = - 4.5 V ● ● ● ● ● ● ● ● ● ● Vg1 = - 5.0 V ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ● ● ● ℏω(meV) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ▲ ● ● ▲ ● ▲ ● ● ▲ ● ● ▲ ● ● ● ● ● 2 ● ▲ ▲ ▲ ▲ ● ● ▲ ● ● ● ● ▲ ● ● ● ▲ ● ● ▲ ● ● ▲ ● ▲ ▲ ● ▲ ● ● ● ● ▲ ● ● ▲ ● ● ▲ ●●● ▲ ▲ ▲ ▲ ▲ ●● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ●●● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ●● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ●● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲▲ ▲▲▲▲▲ ▲ ▲▲▲▲●▲ 0 -5 -4 -3 -2 -1 0

Vg2 (V)

Figure 12: Characteristic frequencies at typical density, for the cases Vg1 = −0.5, −1.0, −1.5, −1.9, −2.2, −2.5, −2.8, −3.0, −3.2, −3.5, −4.0, −4.5, −5.0 V.

Vg1 = - 0.5 V 12 Vg1 = - 1.0 V

Vg1 = - 1.5 V

Vg1 = - 1.9 V 10 Vg1 = - 2.2 V

Vg1 = - 2.5 V

Vg1 = - 2.8 V

/h) 8 2 Vg1 = - 3.0 V

Vg1 = - 3.2 V

Vg1 = - 3.5 V 6 Vg1 = - 4.0 V

Vg1 = - 4.5 V Conductance(2e Vg1 = - 5.0 V 4

0 -5 -4 -3 -2 -1 0

Vg2 (V)

Figure 13: Conductance at typical density, for the cases Vg1 = −0.5, −1.0, −1.5, −1.9, −2.2, −2.5, −2.8, −3.0, −3.2, −3.5, −4.0, −4.5, −5.0 V.

23 4.3 Higher Electron Density - 3/2 n0 The Figures 14-17 are similar to those in the previous section, but for these simulations the 11 −2 higher electron density of (3/2)n0 = 3·10 cm was used. Have a look at Figures 14-15 as well as Figures 58-67 found in supplementary material and take special notice of the y cross-sections. These are deeper than before as a consequence of having more electrons in the system and they all have the similar parabolic to ﬂat potential evolution. There is no drastic change in the potential barriers other than them being higher and some cases seem to be more cone-shaped rather than parabolic.

In Figure 16 (diﬀerent scale on the Vg2-axis compared to Figure 12) one can notice that the overall magnitude of the ω curves is now increased. The most interesting feature of having an increased electron density is seen in Figure 17, which as predicted [6] shows a more equispaced distribution of the conductance curves compared to Figure 13. This can be understood as more electrons provide higher shielding from the electrostatic potential in the device.

A few of the conductance curves do not successfully interpolate all points towards the right in the ﬁgure. In this region ωy/ωx approaches unity, meaning the plateaus are expected to start vanishing. I believe this interpolation problem boils down to the numerics as a consequence of having too few data points in an area where the subbands are closely packed. If too many subbands are opened in between two data points the interpolation may slightly fail. However the case Vg1 = −5.5 V (red curve) manages ﬁne to do this without spoiling the staircase-like shape.

Post defence note: The interpolation problem related to having too few data points could probably be ﬁxed by running more simulations in this Vg2 regime, but there was no time to do so.

24 V = - 3.0 V , X= 500 nm a) Vg2 = - 0.05 V g1 Higher Density

b) Vg2 = - 0.10 V V 2000 c) g2 = - 0.20 V 20 d) Vg2 = - 0.30 V e) V = - 0.40 V g2 10 f) Vg2 = - 0.50 V

g) Vg2 = - 0.60 V V 0 1500 h) g2 = - 0.80 V i) Vg2 = - 1.00 V

j) Vg2 = - 1.20 V -10

k) Vg2 = - 1.60 V

l) Vg2 = - 1.70 V 300 400 500 600 700 800 900 V 1000 m) g2 = - 1.80 V n) Vg2 = - 1.90 V

o) Vg2 = - 2.00 V Total potential(meV) p) Vg2 = - 2.10 V

q) Vg2 = - 2.30 V V 500 r) g2 = - 2.50 V s) Vg2 = - 2.70 V

-200 0 200 400 600 800 1000 Y(nm)

Figure 14: Lateral cross-section of the total potential for the case Vg1 = −3.0 V.

V = - 3.0 V a) Vg2 = - 0.05 V g1

b) Vg2 = - 0.10 V c) V = - 0.20 V 0 g2 Higher Density d) Vg2 = - 0.30 V

e) Vg2 = - 0.40 V

f) Vg2 = - 0.50 V

g) Vg2 = - 0.60 V

h) Vg2 = - 0.80 V

i) Vg2 = - 1.00 V V -5 j) g2 = - 1.20 V

k) Vg2 = - 1.60 V

l) Vg2 = - 1.70 V

m) Vg2 = - 1.80 V

n) Vg2 = - 1.90 V

o) Vg2 = - 2.00 V Total potential(meV) p) Vg2 = - 2.10 V -10 q) Vg2 = - 2.30 V

r) Vg2 = - 2.50 V

s) Vg2 = - 2.70 V

-15

0 200 400 600 800 1000 X(nm)

Figure 15: Longitudinal cross-section of the total potential for the case Vg1 = −3.0 V.

25 ● V 3.0 V HD ● ℏωy -● g1 = - ( ) 7 ● ● ● ● ● ℏω -▲ Vg1 = - 3.5 V(HD) ● ● ● ● ● x ● ● Vg1 = - 4.0 V(HD) ● ● ● ● ● 6 ● Vg1 = - 4.5 V(HD) ● ● ● Vg1 = - 5.0 V(HD) ● ● ● ● ● Vg1 = - 5.5 V(HD) ● ● 5 ● ● ● ● ● ● ● ● ● ● ● 4 ● ● ● ℏω(meV) ● ●

3 ● ● ● ▲ ● ● ● ▲ ● ▲ ▲ ● ● ● ● ▲ ● ● 2 ● ▲ ● ● ● ● ● ▲ ● ● ● ▲ ▲ ▲ ● ● ● ● ● ▲ ▲ ▲ ● ▲ ● ● ▲ ▲ ▲ ▲ ▲ ▲ ● ● 1 ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

Vg2 (V)

Figure 16: Characteristic frequencies at higher density, for the cases Vg1 = −3.0, −3.5, −4.0, −4.5, −5.0, −5.5 V.

V 15 g1 = - 3.0 V(HD) Vg1 = - 3.5 V(HD)

Vg1 = - 4.0 V(HD)

Vg1 = - 4.5 V(HD)

Vg1 = - 5.0 V(HD)

Vg1 = - 5.5 V(HD)

10 /h) 2 Conductance(2e 5

0 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

Vg2 (V)

Figure 17: Conductance at higher density, for the cases Vg1 = −3.0, −3.5, −4.0, −4.5, −5.0, −5.5 V.

26 4.4 Lower Electron Density - 1/3 n0 11 −2 These simulations were done at a low electron density of (1/3)n0 ≈ 0.67 · 10 cm . In Figures 18-19 and 68-79 the x and y cross-sections are shown for all simulated cases. As for both previous electron densities the potential barriers behave in a similar manner, now having the lowest magnitudes in comparison. The total potentials in the lateral direction are shifted upward, shallow and evolve from parabolic at pinch-oﬀ to wide ﬂat-bottomed potentials at weak Vg2 bias voltages. Since they are shallow one can expect lower conductance as fewer open subbands will be available.

At the lowest of all three used electron densities, we notice how the ω curves now are the most sensitive to variations in Vg2, most notably in the case Vg1 = −1.4 V (red curves) in Figure 20. This is no surprise since now when the system has the least amount of electrons the conﬁnement potential from the device itself has the most inﬂuence, and not outbalanced by electron-electron interactions. The overall ω magnitude is however lowered. Notice in Figure 21 that by lowering the density in the simulations, we once again obtain a pattern with apparent 2e2 curve packing, but having a maximum conductance of six times the quanta h . Compare this to the maximum of 12 quanta for the typical density (Figure 13) and 16 for the higher density (Figure 17).

Post defence note: Figures 75, 77 and 79 might have missing labels in printed versions. This problem is not present in the PDF.

27 V V 700 a) g2 = - 0.05 V g1 = - 0.3 V , X= 500 nm Lower Density V b) g2 = - 0.10 V 20

c) Vg2 = - 0.20 V 600 15 d) Vg2 = - 0.30 V 10 e) Vg2 = - 0.40 V V 5 500 f) g2 = - 0.45 V

g) Vg2 = - 0.50 V 0

h) Vg2 = - 0.55 V -5 400 V i) g2 = - 0.60 V -10

j) Vg2 = - 0.65 V 300 400 500 600 700 800 V 300 k) g2 = - 0.70 V

l) Vg2 = - 0.80 V

Total potential(meV) m) Vg2 = - 1.00 V 200 n) Vg2 = - 1.10 V

o) Vg2 = - 1.20 V V 100 p) g2 = - 1.30 V

-200 0 200 400 600 800 1000 Y(nm)

Figure 18: Lateral cross-section of the total potential for the case Vg1 = −0.3 V.

V V a) g2 = - 0.05 V g1 = - 0.3 V

b) Vg2 = - 0.10 V

1 c) Vg2 = - 0.20 V Lower Density

d) Vg2 = - 0.30 V

e) Vg2 = - 0.40 V

f) Vg2 = - 0.45 V 0 g) Vg2 = - 0.50 V

h) Vg2 = - 0.55 V

i) Vg2 = - 0.60 V V -1 j) g2 = - 0.65 V

k) Vg2 = - 0.70 V

l) Vg2 = - 0.80 V

Total potential(meV) m) Vg2 = - 1.00 V -2 n) Vg2 = - 1.10 V

o) Vg2 = - 1.20 V

p) Vg2 = - 1.30 V

-3

0 200 400 600 800 1000 X(nm)

Figure 19: Longitudinal cross-section of the total potential for the case Vg1 = −0.3 V.

28 7 ℏωy -● Vg1 = - 0.3 V(LD)

ℏωx -▲ Vg1 = - 0.5 V(LD)

Vg1 = - 0.7 V(LD) 6 Vg1 = - 0.9 V(LD)

Vg1 = - 1.0 V(LD) V = - 1.2 V(LD) 5 g1 Vg1 = - 1.4 V(LD)

● 4 ● ● ℏω(meV) ● ● ● ● ● ● ● ● ● ● ● 3 ● ● ● ● ● ● ● ● ● ● 2 ● ● ▲ ▲ ● ● ● ▲ ● ● ▲ ●▲ ● ▲ ● ● ● ▲ ● ▲ 1 ● ● ▲ ● ● ● ● ▲ ● ● ●● ● ● ●● ▲ ▲ ▲ ▲ ● ● ▲ ▲ ▲ ▲ ● ● ▲ ▲ ▲ ▲ ▲ ▲ ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0 -2.0 -1.5 -1.0 -0.5 0.0

Vg2 (V)

Figure 20: Characteristic frequencies at lower density, for the cases Vg1 = −0.3, −0.5, −0.7, −0.9, −1.0, −1.2, −1.4 V.

Vg1 = - 0.3 V(LD)

Vg1 = - 0.5 V(LD) V = - 0.7 V(LD) 5 g1 Vg1 = - 0.9 V(LD)

Vg1 = - 1.0 V(LD)

Vg1 = - 1.2 V(LD) 4 Vg1 = - 1.4 V(LD) /h) 2

Conductance(2e 2

0 -2.0 -1.5 -1.0 -0.5 0.0

Vg2 (V)

Figure 21: Conductance at lower density, for the cases Vg1 = −0.3, −0.5, −0.7, −0.9, −1.0, −1.2, −1.4 V.

29 4.5 Comments on Eigenvalues I have noticed a trend for eigenvalues, especially for somewhat symmetrical gate bias voltages, that are far above the Fermi energy. It isn’t necessarily the truth for all such cases, but at high electron density it is prominent. As the potential geometry transitions from asymmetric into the parabolic region, where equispaced energy levels ought to be expected, the observed effect is the opposite. Look back at Figure 14 and the cases Vg2 = −1.70 V to Vg2 = −2.70 V (green to red curves); these potentials are parabolic even on a global scale up to 500 meV above the Fermi energy. However when plotting the eigenvalues as a function of Vg2 as in Figure 22 below, they are not equispaced. It seems like the higher ones come in pairs. I have the energy levels come in triplets for some other cases not shown here. Nevertheless these eigenvalues are far above the Fermi energy and will not contribute to the conductance. For higher Vg2, in the right region, the eigenvalues look equispaced, even if the finite difference method samples points into the asymmetric anharmonic potential region. I think the reason behind the odd behaviour in the left region arises from having too crude grid resolution for the width of the channel for those potentials. In this picture, the channel width starts to get smaller than 100 nm when Vg2 goes below −1.2 V. Since δj = yj+1 − yj = 10 nm, neighbouring elements in the Hamiltonian will vary too much for the narrow channels, resulting in this presented distribution of energies.

Post defence note: We could have checked for some isolated example and simu- late with a ﬁner grid spacing, but there was no time to do so.

The case Vg1 = −0.5 V at typical electron density (Figure 9) provides one of the most suitable sets of total potentials for possible usage of the harmonic oscillator model out of all simulated cases. For the other cases the harmonic model might only be used for the first couple of energy levels close to pinch-off where the potentials still are parabolic. After that it is challenging to get an arbitrarily close fitting with a second degree polynomial and we can’t expect accurate energies from the harmonic oscillator model. Even so, when comparing the first ten energy levels calculated numerically and with the harmonic oscillator model they do not agree well, as shown in Figure 23 below. Although, the first two levels differ at most with 1 meV, which is acceptable.

Vg1 = - 3.0 V(HD)

120

100

En,Num (meV) 60

-2.5 -2.0 -1.5 -1.0 -0.5 0.0

Vg2 (V)

Figure 22: Eigenvalues as a function of Vg2 for the case Vg1 = −3.0 V at higher electron density.

30 50

Vg1 = - 0.5 V Harmonic Oscillator-● n=0 Numerical-▲ n=1 40 n=2

▲ n=3 ▲ ▲ n=4 30 ▲ n=5 ▲ ▲ ▲ n=6 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ n=7 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ n=8 (meV) ▲ ▲ ▲ 20 ▲ ▲ ▲ ▲ ▲ n ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ E ▲ ▲ ▲ ▲ ▲ n=9 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 10 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0 ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5

Vg2 (V) (a) Eigenenergies calculated using the harmonic oscillator model compared to the numerical method.

Vg1 = - 0.5 V n=0 10 n=1 n=2 n=3 n=4 n=5

5 n=6

(meV) n=7 n=8 n,Num

-E n=9 n,HO E

-5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5

Vg2 (V)

(b) Diﬀerences in the ﬁrst ten energies as functions of Vg2 for the case Vg1 = −0.5 V.

Figure 23: Comparison of the two methods of calculating the eigenenergies.

31 4.6 Extension of the model - Exchange and Correlation Interactions Having failed to reproduce any transition in the conductance as the bias voltage is changed, we chose to extend the model by including terms for exchange and correlation interactions,

δEσ [nσ(x, y)] U σ (x, y) = U σ (x, y) + U σ (x, y) = xc (4.6) xc ex corr δnσ(x, y)

1 where σ = ± 2 is the electron spin, n(x, y) is the 2D electron density and Exc is the exchange- correlation energy. Exc includes all many-body eﬀects, but it has no universal form. These extra interactions lead to an increase in the simulation time and thus a smaller 2D grid of 400 × 400 nm2 was chosen instead of keeping it 1000 × 1000 nm2, but the spatial resolution of 10 × 10 nm2 was maintained and the geometry of the gates was preserved.

A full description of how to model this system is given in [11, 17]. The electrostatic potential and Hartree interaction follows the same schema as discussed in section 3.3.1. For the term including the exchange-correlation potential Uxc a parametrization scheme from [18] is applied. A ficticious Zeeman term associated with a very tiny in-plane magnetic field is used to trigger the spin splitting. It is initially ∼ 10−4 T, but turned off after a few iterations.

The harmonic oscillator model for the eigenenergies is exclusively used here as the channel widths proved to be far too narrow for the numerical method to handle; it only gave one or two open subbands which clearly disagreed with my supervisor Irina Yakimenko’s own calculations. Narrow channels combined with low grid resolution results in bad numerical precision.

With a broken spin-degeneracy the expression for the conductance is modiﬁed. Each electron spin contributes with quantized steps of e2/h [19],

e2 X e2 X G = T ↑ + T ↓ = G↑ + G↓, (4.7) h h where T is from (3.16) and the characteristic frequencies are calculated separately for each spin.

4.6.1 Density profiles With a lifted spin-degeneracy it is important to investigate possible spin-polarization in the channel that can affect the transport properties. A first step is to compare the electron densities for each spin. Electron density profiles at the center of the channel in the lateral and longitudinal directions are presented below in Figures 24 and 25.

Notice how the electrons are shifted across the channel towards the (in this case) weaker loaded Vg2 gate, in agreement with the asymmetrical potentials that are being investigated. Additional cases for Vg1 = −3.0 V, Vg1 = −4.5 V and Vg1 = −5.0 V are presented in supplementary material in Figures 80-85. The electron densities show similar features for all cases. Most interestingly, the x cross-sections seem to have recurring bumps appearing at both sides of the centre as the bias voltage is swept. One can plot the diﬀerence of the densities and conﬁrm that there is small spin-polarization, although not included here.

32 0.0012 a) Vg2 = - 0.01 V å) Vg2 = - 2.20 V b) Vg2 = - 0.03 V ä) Vg2 = - 2.50 V c) Vg2 = - 0.05 V ö) Vg2 = - 2.60 V Vg1 = - 4.0 V d) Vg2 = - 0.08 V α) Vg2 = - 2.80 V e) Vg2 = - 0.10 V β) Vg2 = - 2.90 V V V 0.0010 f) g2 = - 0.20 V γ) g2 = - 3.00 V g) Vg2 = - 0.30 V δ) Vg2 = - 3.20 V h) Vg2 = - 0.40 V ϵ) Vg2 = - 3.50 V i) Vg2 = - 0.50 V ζ) Vg2 = - 3.80 V j) Vg2 = - 0.60 V k) Vg2 = - 0.80 V ) 0.0008 V -2 l) g2 = - 1.00 V m) Vg2 = - 1.20 V n) Vg2 = - 1.40 V o) Vg2 = - 1.42 V p) Vg2 = - 1.43 V 0.0006 q) Vg2 = - 1.44 V r) Vg2 = - 1.45 V s) Vg2 = - 1.46 V Spin Up ______t) Vg2 = - 1.47 V u) V = - 1.48 V Electron density(nm g2 Spin Down ____ 0.0004 v) Vg2 = - 1.50 V w) Vg2 = - 1.60 V x) Vg2 = - 1.70 V y) Vg2 = - 1.80 V z) Vg2 = - 2.00 V 0.0002

0.0000 0 100 200 300 400 Y(nm) Figure 24: Electron density across the wire, y cross-section at x = 200 nm (middle of the tunneling barrier), for the case Vg1 = −4.0 V. Due to some numerical discrepancies with Mathematica, some of the curves (red to blue) are not fully plotted.

a) Vg2 = - 0.01 V 0.0014 Vg1 = - 4.0 V b) Vg2 = - 0.03 V c) Vg2 = - 0.05 V d) Vg2 = - 0.08 V e) V = - 0.10 V 0.0012 g2 f) Vg2 = - 0.20 V g) Vg2 = - 0.30 V h) Vg2 = - 0.40 V i) Vg2 = - 0.50 V 0.0010 j) Vg2 = - 0.60 V k) Vg2 = - 0.80 V ) V -2 l) g2 = - 1.00 V m) Vg2 = - 1.20 V V 0.0008 n) g2 = - 1.40 V o) Vg2 = - 1.42 V p) Vg2 = - 1.43 V q) Vg2 = - 1.44 V r) V = - 1.45 V 0.0006 g2 s) Vg2 = - 1.46 V t) Vg2 = - 1.47 V ä) Vg2 = - 2.50 V u) V = - 1.48 V ö) V = - 2.60 V Electron density(nm g2 g2 v) Vg2 = - 1.50 V α) Vg2 = - 2.80 V 0.0004 w) Vg2 = - 1.60 V β) Vg2 = - 2.90 V x) Vg2 = - 1.70 V γ) Vg2 = - 3.00 V V V y) g2 = - 1.80 V δ) g2 = - 3.20 V ______Spin Up z) Vg2 = - 2.00 V ϵ) Vg2 = - 3.50 V ____ 0.0002 å) Vg2 = - 2.20 V ζ) Vg2 = - 3.80 V Spin Down

0.0000

0 100 200 300 400 X(nm)

Figure 25: Electron density along the wire, x cross-section, for the case Vg1 = −4.0 V. Spin up and spin down curves are very closely overlapping for most cases.

33 4.6.2 Potentials Potential cross-sections of the channel with exchange and correlation eﬀects included are presented below. Having such a small spin-polarization we do not expect strongly spin-split energy levels. In Figure 26, 86, 88 and 90 we can see that all y cross-section potentials have parabolic shapes ﬁt for the usage of the harmonic oscillator model. With added exchange and correlation interactions it appears that the intricate potential geometry evolution now is shifted to the x cross-sections (as opposed to the Thomas-Fermi simulations). These are presented in Figures 27 as well as 87, 89 and 91 in supplementary material.

V V 10 a) g2 = - 0.01 V å) g2 = - 2.20 V b) Vg2 = - 0.03 V ä) Vg2 = - 2.50 V c) Vg2 = - 0.05 V ö) Vg2 = - 2.60 V d) Vg2 = - 0.08 V α) Vg2 = - 2.80 V e) Vg2 = - 0.10 V β) Vg2 = - 2.90 V f) Vg2 = - 0.20 V γ) Vg2 = - 3.00 V g) Vg2 = - 0.30 V δ) Vg2 = - 3.20 V h) Vg2 = - 0.40 V ϵ) Vg2 = - 3.50 V 5 i) Vg2 = - 0.50 V ζ) Vg2 = - 3.80 V j) Vg2 = - 0.60 V k) Vg2 = - 0.80 V l) Vg2 = - 1.00 V m) Vg2 = - 1.20 V n) Vg2 = - 1.40 V o) Vg2 = - 1.42 V V 0 p) g2 = - 1.43 V q) Vg2 = - 1.44 V r) Vg2 = - 1.45 V s) Vg2 = - 1.46 V t) Vg2 = - 1.47 V Total potential(meV) u) Vg2 = - 1.48 V v) Vg2 = - 1.50 V w) Vg2 = - 1.60 V -5 x) Vg2 = - 1.70 V y) Vg2 = - 1.80 V V z) g2 = - 2.00 V ______Spin Up ____ Spin Down

Vg1 = - 4.0 V -10

0 100 200 300 400 Y(nm) Figure 26: Total potential across the wire, y cross-section, at x = 200 nm (middle of the tunneling barrier) for the case Vg1 = −4.0 V.

34 The process of the opening of a subband seemingly gives rise to a bump on both sides of the tunneling barrier, as can be seen in Figure 27, and must be related to the bumps also seen in the electron density. However, these features are not caught in the Büttiker model as only the close proximity to the top is considered when calculating the characteristic frequency ωx. By in- vestigating the functions for the eigenenergies, I found numerically that the first subband opens at Vg2 ≈ −2.975 V, the second at Vg2 ≈ −1.525 V, the third at Vg2 ≈ −0.737 V, the fourth at Vg2 ≈ −0.428 V and the fifth at Vg2 ≈ −0.051 V. When comparing these values to the ones of the curves in Figure 27 it appears that the bumps appear slightly before the subband is open. For instance, at Vg2 = −3.5 (-curve), the ground state is 1.93 meV above the Fermi energy and there are noticable bumps in the tunneling barrier. Meanwhile at Vg2 = −3.2 (δ-curve), it is only 0.47 meV above the Fermi energy and now the bumps seem to have disappeared. Having such a low difference the electrons are likely to tunnel through, even if the channel isn’t fully opened.

As a matter of fact, and this goes for all conductance curves, one can easily make the es- timation and ﬁnd out that the bottom-half of a conductance step has its contribution from weakly positive energy levels, 1 1 Tmn = δmn < 1 + e−πn 2 ⇔ e−πn > 1

⇔ −πn > 0 2π ⇔ En,HO > 0 ~ωx ⇔ En,HO > 0.

a) Vg2 = - 0.01 V å) Vg2 = - 2.20 V Vg1 = - 4.0 V 2 b) Vg2 = - 0.03 V ä) Vg2 = - 2.50 V ______c) Vg2 = - 0.05 V ö) Vg2 = - 2.60 V Spin Up d) Vg2 = - 0.08 V α) Vg2 = - 2.80 V ____ Spin Down e) Vg2 = - 0.10 V β) Vg2 = - 2.90 V f) Vg2 = - 0.20 V γ) Vg2 = - 3.00 V g) Vg2 = - 0.30 V δ) Vg2 = - 3.20 V 0 h) Vg2 = - 0.40 V ϵ) Vg2 = - 3.50 V i) Vg2 = - 0.50 V ζ) Vg2 = - 3.80 V j) Vg2 = - 0.60 V k) Vg2 = - 0.80 V l) Vg2 = - 1.00 V m) V = - 1.20 V -2 g2 n) Vg2 = - 1.40 V o) Vg2 = - 1.42 V p) Vg2 = - 1.43 V q) Vg2 = - 1.44 V r) Vg2 = - 1.45 V V -4 s) g2 = - 1.46 V t) Vg2 = - 1.47 V Total potential(meV) u) Vg2 = - 1.48 V v) Vg2 = - 1.50 V w) Vg2 = - 1.60 V x) Vg2 = - 1.70 V -6 y) Vg2 = - 1.80 V z) Vg2 = - 2.00 V

-8

0 100 200 300 400 X(nm) Figure 27: Total potential along the wire, x cross-section, for the case Vg1 = −4.0 V.

35 Looking back to Figures 12, 16 and 20 we can make the fair approximation ~ωx ≤ 2 meV which 2π neatly gives us En,HO ≥ πEn,HO and we can now roughly estimate the contribution to the ~ωx conductance. A subband at 1/π meV translates to Tmn having a magnitude less than 0.27 times a step quanta. At a mere 1.47 meV the contribution is already less than a percent of the step and at 1.93 meV as mentioned above, the contribution is 0.0023 times a step quanta. As noticed for the δ-curve in Figure 27 the bump is gone and at that point the ground state contributes with roughly 18.6% of a whole step meaning that the channel isn’t fully pinched oﬀ. One could thus argue that the bumps in the tunneling barrier come from the accumulation of the electrons at the barrier edge, eﬀectively bending it, as they are stuck and cannot pass through. Then as the channel becomes partially open, transport is available and the bumps disappear. This process is repeated for the opening of each subsequent subband. For a closer look at this the spin polarization of the system is calculated as

p(x, y) = ρ↓(x, y) − ρ↑(x, y), (4.8) i.e., the diﬀerence between the densities of ↓-spin and ↑-spin electrons. The densities themselves are calculated by the method described in [11,17]

σ X σ 2 ρ (x, y) = φk (x, y) , (4.9) σ Ek ≤µ

σ σ where Ek are the eigenenergies and φk are the eigenstates that are calculated at each iterative 1 step for each spin σ = ± 2 .

In Figure 28 the spin polarization is plotted and the aforementioned bumps are outlined by the white circles to the left. Going from left to right the gate voltage is increased. The channel is ﬁrst pinched oﬀ and then the potential at the QPC is lowered such that the ground state is approaching the Fermi energy. This allows the electron density to drift inwards, enabling transport when the bridge is made. Related results regarding the accumulation and release of electrons in this manner is discussed in [20].

Figure 28: Spin-polarization of the system for the ε (left) and δ (right) cases.

36 The characteristic frequencies are calculated in the same way as in the Thomas-Fermi simulations and are shown in Figure 29 for all Vg1 cases. Worth noting is that the bump features are not caught in the Büttiker model as only the close proximity of the barrier maximum is considered when choosing the interpolation points. These do not extend to the bumps. The frequency curves have an almost linear appearance and this lowest order approximation is made when calculating the conductance. Although not shown here, similar approximations are made for the energy eigenvalues as they too can be accurately approximated by linear curves. This makes it possible to then extrapolate the conductance beyond the simulated data points. However this approximation can lead to shifts in the opening of subbands, i.e., an eigenvalue curve may not cross the Fermi energy at the exact same Vg2 value as its linear approximation.

The calculated conductance is presented in Figure 30. I have chosen to plot G↑ and G↓ separately instead of their sum. Going from left to right, the order in which the ﬁrst step should come is Vg1 = −3.0, −4.0, −4.5, −5.0 V. If we look closely the order is instead Vg1 = −3.0, −4.5, −4.0, −5.0 V by looking at the colours of the curves. We can notice that the ground state of Vg1 = −4.0 V has been shifted and opens later than expected, at a higher Vg2 value. This is a consequence of a lowest order approximation of the energy eigenvalue curve. Proper position can likely be achieved by being more careful when numerically calculating ωy for the energies. One could also consider estimating the energy curves by piecewise linear functions or using splines, although then losing the ability to extrapolate beyond the data points.

Any evidence whatsoever of possible transitions in the conductance with smeared plateaus is yet to be found. It is very likely that the Büttiker model is oversimpliﬁed. It merely cannot explain this experimentally found feature.

37 8

Vg1 = - 3.0 V ↑ ↓ V = - 4.0 V ↓↑ g1 ↑ ↓ ↓↑ ↓↑ ↓↑ ↓↑ Vg1 = - 4.5 V ↑ ↓ ↓↑ ↓↑ ↓↑ Vg1 = - 5.0 V ↑ ↓ ↓↑ ↓↑ ↓↑ ℏωy 6 ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↑ ↓↑ ↑ ↓ ↓↑ ↓ ↓ ↓↑ ↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ 4 ↓↑ ↓↑ ↓↑ ↓↑↓↑↓↑ ℏω(meV) ↑ ↓↑↓↑↓↑ ↓↑ ↓↑ ↓↑ ↑ ↓↑ ↓↑ ↓ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑↓↑ 2 ℏωx ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ↓↑↓↑ ↑ ↓↑↓↑↓↑ ↓↑ ↓ ↓↑ ↑↓ ↑↓ ↓↑ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↓↑ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↓↑ ↓↑ ↓↑ ↑↓ ↑↓ ↑↓ ↓↑ ↑↓ ↓↑ ↓↑ ↑↓ ↑↓ ↓↑↓↑↑↓↑ ↓↑ ↓↑ ↓↑ ↑↓ ↑↓ ↓↑ ↓↑ ↑ ↓↑ ↑↓ ↓↓↑↓↑↓↑ ↓↑ ↑↓ ↓↑ ↓↑ ↑↓ ↑↓ ↓↑ ↓↑ ↓↑↑↓↑↓↓↑↑↓

0 -4 -3 -2 -1 0

Vg2 (V)

Figure 29: Energy-scaled characteristic frequencies, ωx and ωy, for four Vg1 cases. The dashed lines here are linear approximations that are used when interpolating the conductance.

V g1 = - 3.0 V ↑ ↓

Vg1 = - 4.0 V 4 ↑ ↓ V g1 = - 4.5 V ↑ ↓ V g1 = - 5.0 V ↑ ↓

3 /h) 2

2 Conductance(e

-4 -3 -2 -1 0

Vg2 (V)

Figure 30: Calculated conductance for four Vg1 cases.

38 Another check with experimental data is to look at the so called transconductance, deﬁned as ∂G/∂Vg2. As the subbands successively become occupied and adding to the conductance, each opening will give rise to a peak in the transconductance, and it will be zero on the plateaus. Experimental transconductance measured on the device from the group of M. Pepper is presented below in Figure 31. I have roughly outlined where the transitions occur and connected it to the transconductance plot. There seems to be some sort of peak crossings where I have drawn the thin lines (highlighted with arrows to the right). It would therefore be interesting to see if this can be reproduced.

Figure 31: Experimental conductance and related transconductance. Image courtesy of M. Pepper and S. Kumar [6]. Left one is the same as the top picture in Figure 1. In the right one the derivatives of the curves seen in the left picture are plotted. Their magnitude is represented by the black-to-white colour gradient, black being zero. The blue lines are my own modiﬁcation to highlight interesting regions.

Let Φˆ denote the normalized transconductance (I normalize for easier visualization). I have calculated it using the conductance curves seen in Figure 30. The contribution from both electron spins are added up, 1 ∂ Φˆ = G↑ + G↓, (4.10) N ∂Vg2 where N is a normalization constant that I find numerically in Mathematica. In Figure 32 the resulting transconductance is shown for the four Vg1 cases. Even if this is weeks worth of data it is far from enough to draw any valuable conclusion. It would require a lot of work to fill out this plot with more points. Perhaps a completely different model must be considered in order to find an explanation for the transition in the conductance curves.

39 Transconductance 1.0

0.8

0.6

0.4

0.2

Figure 32: Normalized transconductance. Each peak corresponds to a sublevel crossing the Fermi energy.

40 5 Density functional theory model comparison

For a description of the modelling see [11, 17, 21]. DFT provides a more rigorous model that makes less assumptions. The conductance is calculated diﬀerently here. The current through the QPC is calculated using the quantum mechanical deﬁnition for current density

Eσ≤µ+eV/2 k σ σ∗ σ ~ X h σ∗ ∂ϕk ∂ϕk σi J = ϕk − ϕk (5.1) 2mi σ ∂x ∂x Ek ≤µ−eV/2 and integrating it over the middle y cross-section of the QPC. Here, σ = ±1/2 for spin up/down and −eV is the drop of bias voltage applied along the QPC between the source and drain. The total current is obtained by summing the contribution for each spin. Assuming that the applied bias is small makes it possible to then calculate the conductance by dividing the total current by the applied source-drain voltage. Compare this to equations (3.6)-(3.11). Examples of such a conductance is presented in Figure 33 below.

Figure 33: Conductance using DFT. Image courtesy of I. Yakimenko

Going from left to right in this ﬁgure, the total potential undergoes an asymmetric-symmetric- asymmetric transition similar to the tipping of a balance scale, see Figure 7 at the beginning where the plots actually correspond to points on three of the curves here. Although no clear staircase-like shape is visible here, one can almost suspect that some transition has happened when going from the black to the blue curve, especially since the green curve is the only one 2e2 that has no clear plateau at 1.25 · h . However this could simply be numerical noise. It is not entirely understood why DFT gives such a diﬀerent result compared to the conductance curves shown in Figure 30.

41 6 Conclusions

We have shown that within the Büttiker model the conductance behaviour is dependent on the strength in asymmetry of the split gates voltages. This was done using the simple Thomas-Fermi model for our total potentials. Our results from varying the electron density agrees with the unpublished work of Prof. M. Pepper and colleagues, in the aspect that an increased density gives a more equispaced distribution of the conductance curves. An extended model with exchange and correlation interactions was investigated. Interesting features in form of bumps in the total potentials were noticed and explained by the accumulation and release of electrons.

An explanation for the experimentally observed transition in the conductance was not found.

A quick comparison of the Büttiker model was made with DFT, but the reason behind the diﬀering results is not explained, other than Büttiker is an oversimpliﬁed model that does not catch every interesting feature in the potential landscape. Even though DFT is a more rigorous model that ought to take the features into account, it does not provide the entire truth.

As mentioned in the introduction and discussed in [5], it may well be that the conductance transition is owing to Wigner crystallization. The explanation being that an anticrossing of the ground state and the ﬁrst excited state leads to a hybridization on the form

2 2 |ψ1s + ψ2s| + |ψ1s − ψ2s| that gives two independent and separated edge states. However, no such electron density proﬁle was observed in our model for any of the simulations. This boils down to a geometrically dependent problem. Is our channel too long, short, wide, narrow, deep, shallow? Is our electron density not within the right range? Will the transition in conductance be explained by something completely diﬀerent? Not even the group of M. Pepper have come up with their own explanation with certainty and one year has passed.

6.1 Thomas-Fermi model review Albeit relatively simple, the Thomas-Fermi model is suﬃcient at explaining the packing of conductance curves. It also manages to explain experimental results regarding what eﬀect the electron density has on the curve spacing. When a sample is illuminated to increase the electron density, the conductance measurements show more equispaced curves, which was also observed numerically in our simulations. For all investigated cases, we did not observe any level crossing nor anticrossing that has been mentioned on the experimental side of the problem. As it is not included in the model, Thomas-Fermi cannot say anything about possible spin-polarization in the device. Overall it is a powerful model.

6.2 Future Work

Investigate the possibility of combining Vg2 interpolating functions with Vg1 for some complete function Γ(Vg1,Vg2) in order to generate, for instance, a full transconductance spectrum.

Leave the Büttiker model and perhaps modify the exchange and correlation interactions.

Last remark: 805 simulations and 14 782 400 data points were analyzed.

42 7 References

[1] K.-F. Berggren and M. Pepper. In: Phil. Trans. Roy. Soc. A 368, 1141-1162 (2010).

[2] B. J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988).

[3] D. A. Wharam et al., J. Phys. C 21, 209 (1988).

[4] M. Büttiker, Phys. Rev. B 41, 7906 (1990).

[5] S. Kumar et al., Phys. Rev. B 90 201304(R) (2014).

[6] Private communication between my supervisor K.-F. Berggren and M. Pepper as well as S. Kumar (Unpublished).

[7] S. Datta, Electronic Transport in Mesoscopic Systems, (Cambridge University Press, 1995).

[8] D. K. Ferry, S. M. Goodnick and J. P. Bird, "Transport in Nanostructures", 2nd ed. (Cambridge, 2010).

[9] J. H. Davies, The Physics of Low-Dimensional Semiconductors: An Introduction (Cambridge University Press, Cambridge, 1998).

[10] J. Hakanen, Diploma work conducted at IFM, LiU-IFM-Ex-1121 (2004).

[11] K.-F. Berggren and I. Yakimenko, J. Phys.: Condens. Matter 20 164203 (2008).

[12] I. Yakimenko, I. V. Zozoulenko and K.-F. Berggren, Semicond. Sci. Technol. 14 949-957 (1999).

[13] I. Yakimenko, "Lecture Notes in Quantum Dynamics", (Linköping University, 2014) (with permission).

[14] K.J. Thomas, J.T. Nicholls, M. Y. Simmons, et al., Phys. Rev. Lett. 77, 135 (1996).

[15] H. A. Fertig and B. I. Halperin, Phys. Rev. B, 36, 7969 (1987).

[16] K.-F. Berggren and I. Yakimenko, Phys. Rev. B 66, 085323 (2002).

[17] I. Yakimenko, V. S. Tsykunov and K.-F Berggren, J. Phys.: Condens. Matter 25 072201 (2013).

[18] B. Tanatar and D. M. Ceperley, Phys. Rev. B 39 5005 (1989).

[19] R. Landauer, IBM J. Res. Dev. 1, 223 (1957).

[20] T. Rejec and Y. Meir, Nature Lett. Vol 442: 900-903 (2006).

[21] I. Yakimenko and K.-F. Berggren, J. Supercond. Nov Magn 22: 449-454 (2009).

43 8 Code

Example how Mathematica was used to obtain ωy and the energy spectrum for one y cross- section. All code used in this project would in its entirety cover somewhere between 50 and 100 pages here and can thus not be included. dataa = Import["C:\\Users\\Erik\\Desktop\\TFYA17 \Projekt\\Set1\\ Y_V1-0.5_V2-1.dat", "Table"]; Tablea = ConstantArray[1, 100]; For[i = 1, i < 101, i++, Tablea[[i]] = {dataa[[i, 2]], dataa[[i, 4]]*10^3}]; Pointsa = ConstantArray[1, 5]; For[i = 1, i < 101, i++, If[dataa[[i, 4]]*10^3 == Min[Tablea], {Pointsa[[1]] = {dataa[[i - 12, 2]], dataa[[i - 12, 4]]*10^3}, Pointsa[[2]] = {dataa[[i - 6, 2]], dataa[[i - 6, 4]]*10^3}, Pointsa[[3]] = {dataa[[i, 2]], dataa[[i, 4]]*10^3}, Pointsa[[4]] = {dataa[[i + 6, 2]], dataa[[i + 6, 4]]*10^3}, Pointsa[[5]] = {dataa[[i + 12, 2]], dataa[[i + 12, 4]]*10^3}}, 0]]; Polya = InterpolatingPolynomial[Pointsa, y]; Aint = Plot[Polya, {y, 300, 700}, PlotStyle -> {Thick, Dashed, Darker[Färga]}, Frame -> True, Axes -> False, ImageSize -> Full]; PlotPointsa = ListPlot[Pointsa, PlotStyle -> {Dashed, Darker[Färga], PointSize[0.0125]}];

Omegaya = Sqrt[(2*Abs[D[Polya, {y, 2}] /. y -> 0]*1.60217646*10^-22)/( 0.067*9.1093819*10^-31)]*1/(Pointsa[[3, 1]]*10^-9); hbarOmegaya = 6.5821189*10^-16*Omegaya*10^3;

Subscript[N, 0] = 40; (*\[HBar]=6.5821189*10^-16 ; \[HBar] = 1.05457160*10^-34; SuperStar[m] = 0.067*9.1093819*10^-31; \[Delta] = 10^-8; q = 1.60217646*10^-19; \[HBar]^2/(SuperStar[m] \[Delta]^2); \[Kappa] = \[HBar]^2/(SuperStar[m] \[Delta]^2)*10^3/q;

UtotA = ConstantArray[1, 100]; For[i = 1, i < 101, i++, UtotA[[i]] = dataa[[i, 4]]*10^3]; HelementsA[i_, j_] := If[i == j, \[Kappa] + UtotA[[i]], If[i == j + 1, -\[Kappa]/2, If[i == j - 1, -\[Kappa]/2, 0]]]; HA = Table[HelementsA[i, j], {i, Pointsa[[3, 1]]/10 - Subscript[N, 0]/2 + 1, Pointsa[[3, 1]]/10 + Subscript[N, 0]/2}, {j, Pointsa[[3, 1]]/10 - Subscript[N, 0]/2 + 1, Pointsa[[3, 1]]/10 + Subscript[N, 0]/2}]; EigenvaluesA = Roots[Factor[Det[HA - \[CapitalEpsilon] IdentityMatrix[Subscript[N, 0]]]] == 0, \[CapitalEpsilon]];

44 Appendix A Supplementary Material

Here I include some plots of cases that did not fit into the bulk of the thesis, but still are of great value for understanding how the sweeping of a gate bias affects the potential geometry. Plots of the electron density for different spins are included at the end.

45 A.1 Typical Electron Density - n0

a) Vg2 = - 1.0 V

b) Vg2 = - 1.5 V 20 c) Vg2 = - 1.6 V

d) Vg2 = - 1.8 V V 15 e) g2 = - 2.0 V

f) Vg2 = - 2.2 V

g) Vg2 = - 2.3 V 10 h) Vg2 = - 2.4 V

i) Vg2 = - 2.5 V j) V = - 2.6 V 5 g2 k) Vg2 = - 2.7 V Total potential(meV) l) Vg2 = - 2.8 V

0 m) Vg2 = - 2.9 V

n) Vg2 = - 3.0 V

o) Vg2 = - 3.1 V -5 p) Vg2 = - 3.2 V

-10 Vg1 = - 1.0 V , X= 500 nm

300 400 500 600 700 Y(nm)

Figure 34: Lateral cross-section of the total potential for the case Vg1 = −1.0 V.

2 Vg1 = - 1.0 V

a)Y= 500 nm Vg2 = - 1.0 V b)Y= 490 nm Vg2 = - 1.5 V c)Y= 490 nm Vg2 = - 1.6 V 0 d)Y= 490 nm Vg2 = - 1.8 V e)Y= 480 nm Vg2 = - 2.0 V f)Y= 480 nm Vg2 = - 2.2 V g)Y= 480 nm Vg2 = - 2.3 V h)Y= 480 nm Vg2 = - 2.4 V -2 i)Y= 470 nm Vg2 = - 2.5 V j)Y= 470 nm Vg2 = - 2.6 V k)Y= 470 nm Vg2 = - 2.7 V l)Y= 460 nm Vg2 = - 2.8 V m)Y= 460 nm Vg2 = - 2.9 V -4 n)Y= 460 nm Vg2 = - 3.0 V o)Y= 460 nm Vg2 = - 3.1 V p)Y= 460 nm Vg2 = - 3.2 V Total potential(meV) -6

-8

-10

0 200 400 600 800 1000 X(nm)

Figure 35: Longitudinal cross-section of the total potential for the case Vg1 = −1.0 V.

46 25

a) Vg2 = - 1.0 V

b) Vg2 = - 1.2 V 20 c) Vg2 = - 1.4 V

d) Vg2 = - 1.6 V V 15 e) g2 = - 1.7 V

f) Vg2 = - 1.8 V

g) Vg2 = - 2.0 V 10 h) Vg2 = - 2.1 V

i) Vg2 = - 2.2 V j) V = - 2.3 V 5 g2 Total potential(meV)

-5

-10 Vg1 = - 1.5 V , X= 500 nm

300 400 500 600 700 Y(nm)

Figure 36: Lateral cross-section of the total potential for the case Vg1 = −1.5 V.

5 Vg1 = - 1.5 V

Y= 510 nm a) Vg2 = - 1.0 V

Y= 510 nm b) Vg2 = - 1.2 V

Y= 500 nm c) Vg2 = - 1.4 V

Y= 500 nm d) Vg2 = - 1.6 V

Y= 500 nm e) Vg2 = - 1.7 V

0 Y= 490 nm f) Vg2 = - 1.8 V

Y= 490 nm g) Vg2 = - 2.0 V

Y= 490 nm h) Vg2 = - 2.1 V

Y= 490 nm i) Vg2 = - 2.2 V

Y= 490 nm j) Vg2 = - 2.3 V Total potential(meV) -5

-10

0 200 400 600 800 1000 X(nm)

Figure 37: Longitudinal cross-section of the total potential for the case Vg1 = −1.5 V.

47 25

a) Vg2 = - 0.3 V

b) Vg2 = - 0.5 V 20 c) Vg2 = - 0.7 V

d) Vg2 = - 1.0 V V 15 e) g2 = - 1.2 V

f) Vg2 = - 1.5 V

g) Vg2 = - 1.9 V 10

5 Total potential(meV)

-5

-10 Vg1 = - 1.9 V , X= 500 nm

300 400 500 600 700 Y(nm)

Figure 38: Lateral cross-section of the total potential for the case Vg1 = −1.9 V.

5 Vg1 = - 1.9 V

Y= 550 nm a) Vg2 = - 0.1 V

Y= 550 nm b) Vg2 = - 0.5 V

Y= 550 nm c) Vg2 = - 0.7 V

Y= 530 nm d) Vg2 = - 1.0 V

Y= 520 nm e) Vg2 = - 1.2 V

0 Y= 510 nm f) Vg2 = - 1.5 V

Y= 500 nm g) Vg2 = - 1.9 V Total potential(meV) -5

-10

0 200 400 600 800 1000 X(nm)

Figure 39: Longitudinal cross-section of the total potential for the case Vg1 = −1.9 V.

48 25

a) Vg2 = - 0.1 V

b) Vg2 = - 0.2 V 20 c) Vg2 = - 0.4 V

d) Vg2 = - 0.6 V V 15 e) g2 = - 0.8 V

f) Vg2 = - 1.0 V

g) Vg2 = - 1.2 V 10 h) Vg2 = - 1.3 V

i) Vg2 = - 1.4 V j) V = - 1.5 V 5 g2 k) Vg2 = - 1.6 V Total potential(meV) l) Vg2 = - 1.8 V 0

-5

-10 Vg1 = - 2.2 V , X= 500 nm

300 400 500 600 700 Y(nm)

Figure 40: Lateral cross-section of the total potential for the case Vg1 = −2.2 V.

10 Vg1 = - 2.2 V

a)Y= 580 nm Vg2 = - 0.1 V

b)Y= 570 nm Vg2 = - 0.2 V

c)Y= 560 nm Vg2 = - 0.4 V

d)Y= 540 nm Vg2 = - 0.6 V

5 e)Y= 530 nm Vg2 = - 0.8 V

f)Y= 530 nm Vg2 = - 1.0 V

g)Y= 520 nm Vg2 = - 1.2 V

h)Y= 520 nm Vg2 = - 1.3 V

i)Y= 520 nm Vg2 = - 1.4 V 0 j)Y= 510 nm Vg2 = - 1.5 V

k)Y= 510 nm Vg2 = - 1.6 V Total potential(meV)

l)Y= 510 nm Vg2 = - 1.8 V

-5

0 200 400 600 800 1000 X(nm)

Figure 41: Longitudinal cross-section of the total potential for the case Vg1 = −2.2 V.

49 1500 Vg1 = - 2.5 V , X= 500 nm

25 20 15 10 1000 5 0

a) Vg2 = - 0.3 V -5

b) Vg2 = - 0.5 V -10 300 400 500 600 700 c) Vg2 = - 0.7 V

d) Vg2 = - 0.8 V 500 Total potential(meV) e) Vg2 = - 0.9 V

f) Vg2 = - 1.0 V

g) Vg2 = - 1.1 V

h) Vg2 = - 1.2 V

i) Vg2 = - 1.3 V

j) Vg2 = - 1.5 V 0

0 200 400 600 800 1000 Y(nm)

Figure 42: Lateral cross-section of the total potential for the case Vg1 = −2.5 V.

25 Vg1 = - 2.5 V

a)Y= 550 nm Vg2 = - 0.3 V

b)Y= 550 nm Vg2 = - 0.5 V 20 c)Y= 550 nm Vg2 = - 0.7 V

d)Y= 550 nm Vg2 = - 0.8 V e)Y= 550 nm V = - 0.9 V 15 g2 f)Y= 550 nm Vg2 = - 1.0 V

g)Y= 550 nm Vg2 = - 1.1 V

10 h)Y= 550 nm Vg2 = - 1.2 V

i)Y= 550 nm Vg2 = - 1.3 V

j)Y= 550 nm Vg2 = - 1.5 V 5 Total potential(meV)

-5

-10 0 200 400 600 800 1000 X(nm)

Figure 43: Longitudinal cross-section of the total potential for the case Vg1 = −2.5 V.

50 10 a) Vg2 = - 0.01 V

b) Vg2 = - 0.03 V

c) Vg2 = - 0.05 V

d) Vg2 = - 0.07 V

e) Vg2 = - 0.09 V 5 f) Vg2 = - 0.1 V

g) Vg2 = - 0.2 V

h) Vg2 = - 0.3 V

i) Vg2 = - 0.4 V

0 j) Vg2 = - 0.5 V

k) Vg2 = - 0.6 V

l) Vg2 = - 0.7 V Total potential(meV) m) Vg2 = - 0.8 V

n) Vg2 = - 0.9 V -5 o) Vg2 = - 1.0 V

p) Vg2 = - 1.1 V

q) Vg2 = - 1.2 V

r) Vg2 = - 1.3 V

-10 Vg1 = - 2.8 V

400 500 600 700 800 900 Y(nm)

Figure 44: Lateral cross-section of the total potential for the case Vg1 = −2.8 V.

a)Y= 650 nm Vg2 = - 0.01 V Vg1 = - 2.8 V

b)Y= 620 nm Vg2 = - 0.03 V

c)Y= 610 nm Vg2 = - 0.05 V d)Y= 600 nm V = - 0.07 V 5 g2

e)Y= 600 nm Vg2 = - 0.09 V

f)Y= 590 nm Vg2 = - 0.1 V

g)Y= 580 nm Vg2 = - 0.2 V

h)Y= 570 nm Vg2 = - 0.3 V

i)Y= 560 nm Vg2 = - 0.4 V 0 j)Y= 560 nm Vg2 = - 0.5 V

k)Y= 550 nm Vg2 = - 0.6 V

l)Y= 550 nm Vg2 = - 0.7 V Total potential(meV)

m)Y= 540 nm Vg2 = - 0.8 V

n)Y= 540 nm Vg2 = - 0.9 V V -5 o)Y= 540 nm g2 = - 1.0 V

p)Y= 530 nm Vg2 = - 1.1 V

q)Y= 530 nm Vg2 = - 1.3 V

0 200 400 600 800 1000 X(nm)

Figure 45: Longitudinal cross-section of the total potential for the case Vg1 = −2.8 V.

51 a) V = - 0.01 V 4 g2 b) Vg2 = - 0.03 V

c) Vg2 = - 0.05 V

2 d) Vg2 = - 0.07 V

e) Vg2 = - 0.09 V

f) Vg2 = - 0.1 V 0 g) Vg2 = - 0.15 V

h) Vg2 = - 0.17 V

-2 i) Vg2 = - 0.2 V

j) Vg2 = - 0.3 V

k) Vg2 = - 0.4 V

Total potential(meV) -4 l) Vg2 = - 0.5 V

m) Vg2 = - 0.7 V V -6 n) g2 = - 0.9 V

o) Vg2 = - 1.0 V

-8

Vg1 = - 3.0 V

400 500 600 700 800 900 Y(nm)

Figure 46: Lateral cross-section of the total potential for the case Vg1 = −3.0 V.

Vg1 = - 3.0 V a)Y= 660 nm V = - 0.01 V 4 g2

b)Y= 620 nm Vg2 = - 0.03 V

c)Y= 610 nm Vg2 = - 0.05 V

2 d)Y= 600 nm Vg2 = - 0.07 V

e)Y= 600 nm Vg2 = - 0.09 V

f)Y= 600 nm Vg2 = - 0.1 V 0 g)Y= 590 nm Vg2 = - 0.15 V

h)Y= 590 nm Vg2 = - 0.17 V

-2 i)Y= 580 nm Vg2 = - 0.2 V

j)Y= 570 nm Vg2 = - 0.3 V

k)Y= 570 nm Vg2 = - 0.4 V Total potential(meV) -4 l)Y= 550 nm Vg2 = - 0.5 V

m)Y= 550 nm Vg2 = - 0.7 V V -6 n)Y= 540 nm g2 = - 0.9 V

o)Y= 540 nm Vg2 = - 1.0 V

-8

0 200 400 600 800 1000 X(nm)

Figure 47: Longitudinal cross-section of the total potential for the case Vg1 = −3.0 V.

52 a) Vg2 = - 0.01 V

b) Vg2 = - 0.03 V 4 c) Vg2 = - 0.05 V

d) Vg2 = - 0.07 V V 2 e) g2 = - 0.08 V

f) Vg2 = - 0.09 V

g) Vg2 = - 0.1 V

0 h) Vg2 = - 0.2 V

i) Vg2 = - 0.3 V

j) Vg2 = - 0.4 V -2 k) Vg2 = - 0.5 V

l) Vg2 = - 0.6 V Total potential(meV) m) Vg2 = - 0.7 V -4 n) Vg2 = - 0.8 V

o) Vg2 = - 0.9 V p) V 1.0 V -6 g2 = -

-8 Vg1 = - 3.2 V

400 500 600 700 800 900 Y(nm)

Figure 48: Lateral cross-section of the total potential for the case Vg1 = −3.2 V.

a)Y= 670 nm Vg2 = - 0.01 V Vg1 = - 3.2 V

b)Y= 630 nm Vg2 = - 0.03 V 4 c)Y= 610 nm Vg2 = - 0.05 V

d)Y= 610 nm Vg2 = - 0.07 V V 2 e)Y= 600 nm g2 = - 0.08 V

f)Y= 600 nm Vg2 = - 0.09 V

g)Y= 600 nm Vg2 = - 0.1 V

0 h)Y= 580 nm Vg2 = - 0.2 V

i)Y= 580 nm Vg2 = - 0.3 V

j)Y= 570 nm Vg2 = - 0.4 V -2 k)Y= 560 nm Vg2 = - 0.5 V

l)Y= 560 nm Vg2 = - 0.6 V Total potential(meV)

m)Y= 550 nm Vg2 = - 0.7 V -4 n)Y= 550 nm Vg2 = - 0.8 V

o)Y= 540 nm Vg2 = - 0.9 V p)Y= 540 nm V = - 1.0 V -6 g2

-8

0 200 400 600 800 1000 X(nm)

Figure 49: Longitudinal cross-section of the total potential for the case Vg1 = −3.2 V.

53 a) Vg2 = - 0.01 V

b) Vg2 = - 0.05 V

c) Vg2 = - 0.09 V 5 d) Vg2 = - 0.1 V

e) Vg2 = - 0.2 V

f) Vg2 = - 0.3 V

g) Vg2 = - 0.5 V

h) Vg2 = - 0.7 V

0 i) Vg2 = - 0.9 V

j) Vg2 = - 1.0 V Total potential(meV)

-5

Vg1 = - 3.5 V

400 500 600 700 800 900 Y(nm)

Figure 50: Lateral cross-section of the total potential for the case Vg1 = −3.5 V.

a)Y= 680 nm Vg2 = - 0.01 V Vg1 = - 3.5 V

b)Y= 620 nm Vg2 = - 0.05 V

c)Y= 610 nm Vg2 = - 0.09 V

d)Y= 600 nm Vg2 = - 0.1 V 5 e)Y= 590 nm Vg2 = - 0.2 V

f)Y= 580 nm Vg2 = - 0.3 V

g)Y= 570 nm Vg2 = - 0.5 V

h)Y= 560 nm Vg2 = - 0.7 V

i)Y= 550 nm Vg2 = - 0.9 V

0 j)Y= 540 nm Vg2 = - 1.0 V Total potential(meV)

-5

0 200 400 600 800 1000 X(nm)

Figure 51: Longitudinal cross-section of the total potential for the case Vg1 = −3.5 V.

54 25 a) Vg2 = - 0.01 V

b) Vg2 = - 0.03 V

c) Vg2 = - 0.05 V 20 d) Vg2 = - 0.07 V

e) Vg2 = - 0.09 V f) V = - 0.10 V 15 g2 g) Vg2 = - 0.20 V

h) Vg2 = - 0.40 V V 10 i) g2 = - 0.45 V

j) Vg2 = - 0.50 V

k) Vg2 = - 0.55 V 5 l) Vg2 = - 0.60 V Total potential(meV) m) Vg2 = - 0.65 V

n) Vg2 = - 0.70 V 0 o) Vg2 = - 0.75 V

p) Vg2 = - 0.80 V V -5 q) g2 = - 1.00 V

Vg1 = - 4.0 V -10 400 500 600 700 800 900 Y(nm)

Figure 52: Lateral cross-section of the total potential for the case Vg1 = −4.0 V.

25 a)Y= 700 nm Vg2 = - 0.01 V Vg1 = - 4.0 V

b)Y= 650 nm Vg2 = - 0.03 V

c)Y= 630 nm Vg2 = - 0.05 V 20 d)Y= 620 nm Vg2 = - 0.07 V

e)Y= 610 nm Vg2 = - 0.09 V

f)Y= 610 nm Vg2 = - 0.10 V 15 g)Y= 590 nm Vg2 = - 0.20 V

h)Y= 580 nm Vg2 = - 0.40 V V 10 i)Y= 580 nm g2 = - 0.45 V

j)Y= 570 nm Vg2 = - 0.50 V

k)Y= 570 nm Vg2 = - 0.55 V 5 l)Y= 570 nm Vg2 = - 0.60 V Total potential(meV)

m)Y= 560 nm Vg2 = - 0.65 V

n)Y= 560 nm Vg2 = - 0.70 V 0 o)Y= 560 nm Vg2 = - 0.75 V

p)Y= 560 nm Vg2 = - 0.80 V V -5 q)Y= 550 nm g2 = - 1.00 V

-10 0 200 400 600 800 1000 X(nm)

Figure 53: Longitudinal cross-section of the total potential for the case Vg1 = −4.0 V.

55 30 a) Vg2 = - 0.05 V

b) Vg2 = - 0.10 V

c) Vg2 = - 0.25 V

d) Vg2 = - 0.30 V

20 e) Vg2 = - 0.40 V

f) Vg2 = - 0.45 V

g) Vg2 = - 0.50 V

h) Vg2 = - 0.60 V

10 i) Vg2 = - 0.70 V

j) Vg2 = - 0.80 V

k) Vg2 = - 1.00 V Total potential(meV) 0

-10

Vg1 = - 4.5 V

400 500 600 700 800 900 Y(nm)

Figure 54: Lateral cross-section of the total potential for the case Vg1 = −4.5 V.

30 a)Y= 640 nm Vg2 = - 0.05 V Vg1 = - 4.5 V

b)Y= 620 nm Vg2 = - 0.10 V

c)Y= 600 nm Vg2 = - 0.20 V

d)Y= 600 nm Vg2 = - 0.25 V

20 e)Y= 590 nm Vg2 = - 0.30 V

f)Y= 580 nm Vg2 = - 0.40 V

g)Y= 580 nm Vg2 = - 0.45 V

h)Y= 580 nm Vg2 = - 0.50 V

10 i)Y= 570 nm Vg2 = - 0.60 V

j)Y= 570 nm Vg2 = - 0.70 V

k)Y= 560 nm Vg2 = - 0.80 V

l)Y= 550 nm Vg2 = - 1.00 V Total potential(meV) 0

-10

0 200 400 600 800 1000 X(nm)

Figure 55: Longitudinal cross-section of the total potential for the case Vg1 = −4.5 V.

56 15 a) Vg2 = - 0.05 V

b) Vg2 = - 0.10 V

c) Vg2 = - 0.20 V

d) Vg2 = - 0.25 V 10 e) Vg2 = - 0.30 V

f) Vg2 = - 0.40 V

g) Vg2 = - 0.45 V V 5 h) g2 = - 0.50 V

i) Vg2 = - 0.60 V

0 Total potential(meV)

-5

Vg1 = - 5.0 V -10

400 500 600 700 800 900 Y(nm)

Figure 56: Lateral cross-section of the total potential for the case Vg1 = −5.0 V.

15 a)Y= 650 nm Vg2 = - 0.05 V Vg1 = - 5.0 V

b)Y= 620 nm Vg2 = - 0.10 V

c)Y= 610 nm Vg2 = - 0.20 V

d)Y= 600 nm Vg2 = - 0.25 V 10 e)Y= 600 nm Vg2 = - 0.30 V

f)Y= 590 nm Vg2 = - 0.40 V

g)Y= 580 nm Vg2 = - 0.45 V V 5 h)Y= 580 nm g2 = - 0.50 V

i)Y= 570 nm Vg2 = - 0.60 V

0 Total potential(meV)

-5

-10

0 200 400 600 800 1000 X(nm)

Figure 57: Longitudinal cross-section of the total potential for the case Vg1 = −5.0 V.

57 A.2 Higher Electron Density - 3/2 n0

a) Vg2 = - 0.10 V

b) Vg2 = - 0.20 V

c) Vg2 = - 0.30 V 20 d) Vg2 = - 0.40 V

e) Vg2 = - 0.50 V

f) Vg2 = - 0.60 V

g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V V 10 i) g2 = - 1.20 V j) Vg2 = - 1.40 V

k) Vg2 = - 1.50 V

l) Vg2 = - 1.60 V

m) Vg2 = - 1.70 V

n) Vg2 = - 1.80 V

0 o) Vg2 = - 1.90 V Total potential(meV) p) Vg2 = - 2.00 V

q) Vg2 = - 2.10 V

r) Vg2 = - 2.20 V

s) Vg2 = - 2.30 V t) V = - 2.40 V -10 g2 u) Vg2 = - 2.50 V

Vg1 = - 3.5 V Higher Density

300 400 500 600 700 800 900 Y(nm)

Figure 58: Lateral cross-section of the total potential for the case Vg1 = −3.5 V.

a) Vg2 = - 0.10 V V = - 3.5 V b) Vg2 = - 0.20 V g1

c) Vg2 = - 0.30 V 5 V d) g2 = - 0.40 V Higher Density e) Vg2 = - 0.50 V

f) Vg2 = - 0.60 V

g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V

0 i) Vg2 = - 1.20 V

j) Vg2 = - 1.40 V

k) Vg2 = - 1.50 V

l) Vg2 = - 1.60 V

m) Vg2 = - 1.70 V V -5 n) g2 = - 1.80 V

o) Vg2 = - 1.90 V

p) Vg2 = - 2.00 V Total potential(meV) q) Vg2 = - 2.10 V

r) Vg2 = - 2.20 V V -10 s) g2 = - 2.30 V t) Vg2 = - 2.40 V

u) Vg2 = - 2.50 V

-15

0 200 400 600 800 1000 X(nm)

Figure 59: Longitudinal cross-section of the total potential for the case Vg1 = −3.5 V.

58 a) Vg2 = - 0.10 V b V 20 ) g2 = - 0.20 V c) Vg2 = - 0.40 V

d) Vg2 = - 0.60 V

e) Vg2 = - 1.20 V

f) Vg2 = - 1.40 V 10 g) Vg2 = - 1.60 V

h) Vg2 = - 1.80 V

i) Vg2 = - 2.00 V

0 Total potential(meV)

-10

Vg1 = - 4.0 V Higher Density

300 400 500 600 700 800 900 Y(nm)

Figure 60: Lateral cross-section of the total potential for the case Vg1 = −4.0 V.

5 Vg1 = - 4.0 V

a) Vg2 = - 0.10 V

b) Vg2 = - 0.20 V Higher Density

c) Vg2 = - 0.40 V

d) Vg2 = - 0.60 V 0 e) Vg2 = - 1.20 V

f) Vg2 = - 1.40 V

g) Vg2 = - 1.60 V

h) Vg2 = - 1.80 V V -5 i) g2 = - 2.00 V Total potential(meV)

-10

-15

0 200 400 600 800 1000 X(nm)

Figure 61: Longitudinal cross-section of the total potential for the case Vg1 = −4.0 V.

59 a) Vg2 = - 0.10 V b V 20 ) g2 = - 0.20 V c) Vg2 = - 0.30 V

d) Vg2 = - 0.40 V

e) Vg2 = - 0.50 V

f) Vg2 = - 0.60 V 10 g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V

i) Vg2 = - 1.20 V

j) Vg2 = - 1.40 V k V 0 ) g2 = - 1.60 V

Total potential(meV) l) Vg2 = - 1.80 V

-10

Vg1 = - 4.5 V Higher Density

300 400 500 600 700 800 900 Y(nm)

Figure 62: Lateral cross-section of the total potential for the case Vg1 = −4.5 V.

5 Vg1 = - 4.5 V

a) Vg2 = - 0.10 V

b) Vg2 = - 0.20 V Higher Density

c) Vg2 = - 0.30 V

d) Vg2 = - 0.40 V 0 e) Vg2 = - 0.50 V

f) Vg2 = - 0.60 V

g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V V -5 i) g2 = - 1.20 V

j) Vg2 = - 1.40 V

k) Vg2 = - 1.60 V

Total potential(meV) l) Vg2 = - 1.80 V

-10

-15

0 200 400 600 800 1000 X(nm)

Figure 63: Longitudinal cross-section of the total potential for the case Vg1 = −4.5 V.

60 a) Vg2 = - 0.10 V b V 20 ) g2 = - 0.20 V c) Vg2 = - 0.30 V

d) Vg2 = - 0.40 V

e) Vg2 = - 0.50 V

f) Vg2 = - 0.60 V 10 g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V

i) Vg2 = - 1.20 V

j) Vg2 = - 1.40 V k V 0 ) g2 = - 1.60 V

Total potential(meV) l) Vg2 = - 1.80 V

-10

Vg1 = - 5.0 V Higher Density

300 400 500 600 700 800 900 Y(nm)

Figure 64: Lateral cross-section of the total potential for the case Vg1 = −5.0 V.

15 Vg1 = - 5.0 V

a) Vg2 = - 0.10 V

b) Vg2 = - 0.20 V Higher Density V 10 c) g2 = - 0.30 V

d) Vg2 = - 0.40 V

e) Vg2 = - 0.50 V V 5 f) g2 = - 0.60 V

g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V V 0 i) g2 = - 1.20 V

j) Vg2 = - 1.40 V

k) Vg2 = - 1.60 V

Total potential(meV) V -5 l) g2 = - 1.80 V

-10

-15

0 200 400 600 800 1000 X(nm)

Figure 65: Longitudinal cross-section of the total potential for the case Vg1 = −5.0 V.

61 a) Vg2 = - 0.10 V b V 20 ) g2 = - 0.20 V c) Vg2 = - 0.30 V

d) Vg2 = - 0.40 V

e) Vg2 = - 0.50 V

f) Vg2 = - 0.60 V 10 g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V

i) Vg2 = - 1.20 V

j) Vg2 = - 1.40 V k V 0 ) g2 = - 1.50 V

Total potential(meV) l) Vg2 = - 1.60 V

m) Vg2 = - 1.70 V

n) Vg2 = - 1.80 V

-10

Vg1 = - 5.5 V Higher Density

300 400 500 600 700 800 900 Y(nm)

Figure 66: Lateral cross-section of the total potential for the case Vg1 = −5.5 V.

Vg1 = - 5.5 V

a) Vg2 = - 0.10 V Higher Density b V 20 ) g2 = - 0.20 V c) Vg2 = - 0.30 V

d) Vg2 = - 0.40 V

e) Vg2 = - 0.50 V

f) Vg2 = - 0.60 V 10 g) Vg2 = - 0.80 V

h) Vg2 = - 1.00 V

i) Vg2 = - 1.20 V

j) Vg2 = - 1.40 V k V 0 ) g2 = - 1.50 V

Total potential(meV) l) Vg2 = - 1.60 V

m) Vg2 = - 1.70 V

n) Vg2 = - 1.80 V

-10

0 200 400 600 800 1000 X(nm)

Figure 67: Longitudinal cross-section of the total potential for the case Vg1 = −5.5 V.

62 A.3 Lower Electron Density - 1/3 n0

a) Vg2 = - 0.05 V

b) Vg2 = - 0.10 V

30 c) Vg2 = - 0.20 V

d) Vg2 = - 0.40 V

e) Vg2 = - 0.60 V

f) Vg2 = - 0.80 V 20 g) Vg2 = - 1.00 V

h) Vg2 = - 1.30 V

i) Vg2 = - 1.50 V

j) Vg2 = - 2.00 V 10 Total potential(meV)

Vg1 = - 0.5 V -10 Lower Density

300 400 500 600 700 800 Y(nm)

Figure 68: Lateral cross-section of the total potential for the case Vg1 = −0.5 V.

Vg1 = - 0.5 V

a) Vg2 = - 0.05 V

b) Vg2 = - 0.10 V Lower Density

c) Vg2 = - 0.20 V 30 d) Vg2 = - 0.40 V

e) Vg2 = - 0.60 V

f) Vg2 = - 0.80 V

g) Vg2 = - 1.00 V 20 h) Vg2 = - 1.30 V

i) Vg2 = - 1.50 V

j) Vg2 = - 2.00 V Total potential(meV) 10

0 200 400 600 800 1000 X(nm)

Figure 69: Longitudinal cross-section of the total potential for the case Vg1 = −0.5 V.

63 20

a) Vg2 = - 0.05 V

b) Vg2 = - 0.10 V

15 c) Vg2 = - 0.20 V

d) Vg2 = - 0.30 V

e) Vg2 = - 0.40 V

10 f) Vg2 = - 0.50 V

g) Vg2 = - 0.60 V

h) Vg2 = - 0.70 V

5 i) Vg2 = - 0.80 V Total potential(meV) 0

-5

Vg1 = - 0.7 V -10 Lower Density

300 400 500 600 700 800 Y(nm)

Figure 70: Lateral cross-section of the total potential for the case Vg1 = −0.7 V.

Vg1 = - 0.7 V

a) Vg2 = - 0.05 V 6 b) Vg2 = - 0.10 V Lower Density

c) Vg2 = - 0.20 V

d) Vg2 = - 0.30 V V 4 e) g2 = - 0.40 V

f) Vg2 = - 0.50 V

g) Vg2 = - 0.60 V

h) Vg2 = - 0.70 V 2 i) Vg2 = - 0.80 V Total potential(meV) 0

-2

-4 0 200 400 600 800 1000 X(nm)

Figure 71: Longitudinal cross-section of the total potential for the case Vg1 = −0.7 V.

64 20

a) Vg2 = - 0.05 V

b) Vg2 = - 0.10 V

15 c) Vg2 = - 0.20 V

d) Vg2 = - 0.30 V

e) Vg2 = - 0.40 V

10 f) Vg2 = - 0.50 V

g) Vg2 = - 0.60 V

5 Total potential(meV) 0

-5

Vg1 = - 0.9 V -10 Lower Density

300 400 500 600 700 800 Y(nm)

Figure 72: Lateral cross-section of the total potential for the case Vg1 = −0.9 V.

Vg1 = - 0.9 V

a) Vg2 = - 0.05 V 6 b) Vg2 = - 0.10 V Lower Density

c) Vg2 = - 0.20 V

d) Vg2 = - 0.30 V V 4 e) g2 = - 0.40 V

f) Vg2 = - 0.50 V

g) Vg2 = - 0.60 V

2 Total potential(meV) 0

-2

-4 0 200 400 600 800 1000 X(nm)

Figure 73: Longitudinal cross-section of the total potential for the case Vg1 = −0.9 V.

65 20

a) Vg2 = - 0.01 V

b) Vg2 = - 0.03 V

15 c) Vg2 = - 0.05 V

d) Vg2 = - 0.10 V

e) Vg2 = - 0.20 V

10 f) Vg2 = - 0.30 V

g) Vg2 = - 0.40 V

5 Total potential(meV) 0

-5

Vg1 = - 1.0 V -10 Lower Density

300 400 500 600 700 800 Y(nm)

Figure 74: Lateral cross-section of the total potential for the case Vg1 = −1.0 V.

Vg1 = - 1.0 V

a) Vg2 = - 0.01 V b) V = - 0.03 V 0 g2 Lower Density

c) Vg2 = - 0.05 V

d) Vg2 = - 0.10 V

e) Vg2 = - 0.20 V

f) Vg2 = - 0.30 V -1 g) Vg2 = - 0.40 V

Total potential(meV) -

-3

0 200 400 600 800 1000 X(nm)

Figure 75: Longitudinal cross-section of the total potential for the case Vg1 = −1.0 V.

66 20

a) Vg2 = - 0.01 V

b) Vg2 = - 0.03 V

15 c) Vg2 = - 0.05 V

d) Vg2 = - 0.10 V

e) Vg2 = - 0.20 V

10 f) Vg2 = - 0.30 V

5 Total potential(meV) 0

-5

Vg1 = - 1.2 V -10 Lower Density

300 400 500 600 700 800 Y(nm)

Figure 76: Lateral cross-section of the total potential for the case Vg1 = −1.2 V.

Vg1 = - 1.2 V

a) Vg2 = - 0.01 V b) V = - 0.03 V 0 g2 Lower Density

c) Vg2 = - 0.05 V

d) Vg2 = - 0.10 V

e) Vg2 = - 0.20 V

f) Vg2 = - 0.30 V -1

Total potential(meV) -

-3

0 200 400 600 800 1000 X(nm)

Figure 77: Longitudinal cross-section of the total potential for the case Vg1 = −1.2 V.

67 20

a) Vg2 = - 0.01 V

b) Vg2 = - 0.03 V

15 c) Vg2 = - 0.05 V

d) Vg2 = - 0.10 V

e) Vg2 = - 0.20 V

10 f) Vg2 = - 0.30 V

5 Total potential(meV) 0

-5

Vg1 = - 1.4 V -10 Lower Density

300 400 500 600 700 800 Y(nm)

Figure 78: Lateral cross-section of the total potential for the case Vg1 = −1.4 V.

3 Vg1 = - 1.4 V

a) Vg2 = - 0.01 V

b) Vg2 = - 0.03 V Lower Density 2 c) Vg2 = - 0.05 V

d) Vg2 = - 0.10 V

e) Vg2 = - 0.20 V 1 f) Vg2 = - 0.30 V

-1 Total potential(meV)

-2

-3

0 200 400 600 800 1000 X(nm)

Figure 79: Longitudinal cross-section of the total potential for the case Vg1 = −1.4 V.

68 A.4 Exchange and Correlation A.4.1 Density proﬁles

a) Vg2 = - 0.1 V 0.0012 b) Vg2 = - 0.2 V c) Vg2 = - 0.4 V Vg1 = - 3.0 V d) Vg2 = - 0.8 V e) Vg2 = - 1.0 V f) Vg2 = - 1.4 V 0.0010 g) Vg2 = - 1.8 V h) Vg2 = - 2.2 V i) Vg2 = - 2.6 V j) Vg2 = - 2.8 V k) Vg2 = - 3.0 V ) V -2 0.0008 l) g2 = - 3.1 V m) Vg2 = - 3.4 V n) Vg2 = - 3.6 V o) Vg2 = - 3.8 V p) Vg2 = - 4.0 V V 0.0006 q) g2 = - 4.2 V r) Vg2 = - 4.4 V s) Vg2 = - 4.6 V t) Vg2 = - 4.8 V Electron density(nm 0.0004