Eindhoven University of Technology MASTER Towards showing the topological phase transition in Pb1-xSnxTe nanowires Mientjes, M.G.C.

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Eindhoven University of Technology

MASTER

Towards showing the topological phase transition in Pb1-xSnxTe nanowires

Mientjes, M.G.C.

Award date:

2021

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E ^INDHOVEN U NIVERSITY OF T ^ECHNOLOGY

M ASTER THESIS

Towards showing the topological phase transition in Pb _1-x Sn _x Te

nanowires

M.G.C. Mientjes

Supervisors

M.S.M. Hoskam M.Sc.

Dr. Y. Uzun

Prof. dr. E.P.A.M. Bakkers

January, 2021

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1 Abstract

We have worked on optimizing the back gate design of Pb_1-xSnxTe nanowire devices. Due to the high p-type carrier density in Pb1-xSnxTe nanowires, achieving pinch-off, which is a requirement to reliably measure topological transport properties, is difficult. Therefore, optimizing the back gate, is a first step towards being able to measure these properties for a large range of Sn contents.

We report on some experimental methods to measure the topological properties of Topological crystalline insulators(TCI). Additionally, the viability and effect of using HfOxas a gate oxide was investigated. We determined the breakdown field of the HfOx and the behaviour of the displacable charge of the gate. Also, the influence of adding an Al2Oxcapping layer on top of the HfOxon the breakdown voltage was investigated. An electrical breakdown field of 9.95 MV/cm was found for the HfOx. It was found that the use of HfOxas a gate oxide leads to an improved gating effect and the capping of the HfOx with Al2Ox increases the breakdown voltage of the stack by a factor of 2.5. Finally, we report on achieving pinch-off in Pb1-xSnxTe nanowires with a growth x-value of 0.2 and a IV/VI ratio of 0.49. Mobility’s were found ranging from 0.1-0.14 cm²/Vs and the carrier density was determined to be p = 3.9∗10²² cm^-2. Since Pb_1-xSnxTe is a TCI, being able to achieve pinch-off in these devices is the first step towards using TCI’s for quantum computing purposes.

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2 Introduction

Since the invention of the first programmable digital electronic computer, the development of this technology has been very rapid. The number of transistors on integrated circuit chips, which is a measure of the computing power, has doubled approximately every 14 months for the last 40 years. This empirical trend is known as Moore’s law and is depicted in figure 1. However, as the size of the transistors reach the same length scales as atoms, quantum effects will start to play a larger role. One of the major limitations for down scaling the size of transistors. is the size of it’s building blocks. Besides this, at these very small length scales, quantum effects will start to play a larger role and losses due to tunneling currents will limit the efficiency of the transistors [1].

Figure 1: A representation of Moore’s law. The number of transistors per die doubles approximately every 14 months. This trend has been persistent through multiple technological innovations in the industry [2].

As we cannot increase our transistor density as efficiently as before, we need larger circuits in order to increase our computation power. This will costs more money and energy. Therefore, we cannot realistically increase our computation power at the same rate as before. This means that there are types of simulations which cannot conceivably be solved in the near future using classical computation methods. A shared characteristic is the fact that they are complex multi-body systems with high degrees of freedom. This includes, amongst others simulations of the effects that a new drug has on cancer cells. Being able to simulate the effect of many types of drugs could greatly increase the speed with which new drugs are discovered. Another example could be simulations in which not much is known about the environment, like the early universe. In these situations we do not know enough about the environment to make meaningful simplifica- tions in our models and we are again limited by computation power. These are very interesting subjects which could potentially have a large impact on society if we succeed in computing such a type of simulation [3].

A promising solution to this problem is the use of a quantum computer instead of classical computation methods. Classical computation uses classical bits, which can be either 0 or 1, while a quantum computer uses quantum bits (qubits), which can be 0, 1 and a superposition of these states. This property itself is not necessarily useful for faster computation. However, this allows for the design of algorithms which can solve problems in a parallel manner instead of linearly.

This could greatly decrease computation time for certain complex problems, like the simulation problems described above [4].

Currently, a lot of research is focused on theoretically improving these quantum algorithms and experimentally realizing these quantum computers. At this moment in time, the largest quantum

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computer holds 65 qubits [5]. Although this is a big milestone on itself, much work is still to be done before a quantum computer can be used reliably by the industry. One of the main experimental challenges researchers face when building a many qubits quantum computer, is the inherent introduction of errors in the system. These errors originate from interactions of the qubits with the environment and limited control over the qubits due to technical limitations. It is possible to correct for these errors by implementing so called quantum error correction(QEC) algorithms [4] [6]. However, in order to achieve a fault-tolerant quantum computing system, the number of errors introduced in the system needs to be lower than the number of errors that is corrected by QEC algorithms [7]. Therefore, there is an ongoing search for qubit systems which produce less errors.

One very promising candidate for such a system is one comprised of bound Majorana fermions.

Majorana fermions can appear as excitation’s in solid state systems. These quasi particles have many interesting properties as has been discussed elsewhere [8]. For now it is sufficient to know that these Majorana fermions can bind to a topological defect. It is then possible to store infor- mation in the topologically protected degrees of freedom of the combined system of Majorana fermion with defect. Since these degrees of freedom are topologically protected, they are robust against small perturbations of the environment and are therefore also less susceptible to errors caused by the environment [9] [10].

One of the systems that is predicted to be able to harbor Majorana fermions is a topological superconductor [11]. Topological superconductors can be formed by coupling a nanowire with a strong spin-orbit coupling to an s-wave superconductor through the proximity effect. An example of a system which meets these requirements can be seen in figure 2[12]. In order to produce this topological superconductor state, the chemical potential of the nanowire must be fine tuned so that the electron degeneracy is lifted [11]. This can for example be realized by applying an external magnetic field to open up a topological gap due to the Zeeman splitting. The chemical potential can then be tuned to lie within this gap by external electric fields [12].

Figure 2: A schematic image of topological insulator nanowire (blue) with a proximity coupled s-type superconductor (brown) and a magnetic field along the nanowire direction (red). At the interface between the nanowire and the superconductor, the topological super- conducting phase will occur [12].

The size of this gap is relatively small, making the fine tuning of the chemical potential experimentally challenging. A typical size of the Zeeman gap in such a device is about 1 meV at a field of 1 T [12]. This experimental challenge can in principle be circumvented by using a different material system to form a topological superconductor. Instead of using a semiconductor nanowire, a topological insulator (TI) can be used. In section 3.1, we will go deeper into the properties of TI’s, but for now it suffices to know that the TI bulk gap is a few orders of magnitude larger than the Zeeman gap in semiconductors, making the fine tuning of the chemical potential much easier. Additionally, the TI bulk gap is topologically protected by time reversal symmetry. So as long as the time reversal symmetry is not broken, the gap is very stable against material disorder [12].

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A new subclass of topological materials consists of Topological Crystalline Insulators (TCI). In TCI’s the bulk gap is not be protected by time reversal symmetry but by crystal symmetries.

Not only do these TCI’s open up a whole new class of materials which can potentially harbor Majorana fermions, they are also interesting from a fundamental point of view [13].

It has been known for some time now that TI’s adhere to the bulk-boundary correspondence.

For ”conventional” three-dimensional TI’s this means that the bulk is insulating while it has conducting surface states. So the dimensionality of the transport shifts one down. Recently, it was found that it is also possible to achieve systems where the dimension shift can be more than one, leading to Higher Order Topological Insulators(HOTI) [14]. This generalized form of the bulk-boundary correspondence is expected to hold for both TI’s and TCI’s and has been shown experimentally for the metallic TI bismuth using Scanning Tunneling Microscopy(STM) measurements [15]. Ideally, this would be shown using a transport experiment. However, it is difficult to eliminate all background conductance in the bulk due to the metallic nature of bismuth. That is where tin telluride (SnTe) comes in.

SnTe is a semiconductor which is predicted to be a TCI [16]. Since SnTe is a semiconductor, it should be possible to tune the SnTe into the intrinsic regime. In this regime, the bulk conductance is gapped while the surface conductance is topologically protected by the crystal symmetry. This should make it possible to probe only the topological surface states and study it’s properties.

Furthermore, SnTe is predicted to also be an HOTI [14]. It is predicted that by breaking the crystal symmetry of SnTe the surface states get gapped but the edge channels remain open.

This opens up a whole new field of both theoretical and experimental research. On the theoretical side, it is a quest to find ways to break the crystal symmetry which can be experimentally validated, like exerting a mechanical strain on the nanowire or applying an electric field in specific directions [14]. Additionally, HOTI’s are predicted to be able to give rise to an exciting new type of quasi-particle: parafermions [17]. These fractionalized Majorana particles are not only very interesting from a fundamental point of view, they may also pave the way to new types of quantum operations [18]. Finally, since the discovery of the TCI phase happened very recently, there are still many interesting phenomena which are not yet fully understood, like Spin-momentum locking and topologically protected transport. We will come back to these concepts in section 3, but a more complete overview can be found elsewhere [13].

The research into new TCI materials and properties is not only interesting from a fundamental point of view, but also holds great potential for the future of quantum computing. However, before we can dream of creating industrial quantum computing devices with Majorana fermions or even parafermions, there are still many experimental challenges to be overcome. Firstly, although SnTe is a semiconductor, it is very highly p-doped and is found to have a carrier concentration in the order of p= 10²¹cm⁻³[19]. In order to measure the topological properties of SnTe and eliminate the background conductance, the material needs to be tuned to its intrinsic regime. However, this is very difficult with such a high carrier concentration. This high carrier concentration can be decreased by alloying the SnTe with Pb forming Pb1-xSnxTe [20].

PbTe is a trivial insulator and by tuning the Sn content in Pb1-xSnxTe it is possible to tune the material from the trivial state to the topological state [21]. This phase transition is depicted in figure 3. From this figure, we also observe that the phase transition is not only dependent on the Sn content but also on the measurement temperature. This allows for observations of the topological phase transition in Pb1-xSnxTe nanowires with a constant x-value by tuning the temperature.

This could be measured by performing I-V sweeps at different temperatures and observing the closing and opening of the band gap. To our knowledge, this would be the first time that such a

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topological phases transition would be observed in a TCI using transport measurements.

Figure 3: The topological phase transition for Pb1-xSnxTe for increasing Sn content. The size of the band gap was determined by multiple authors using different types of both optical as well as electrical measurements and is summarized by ref [21]. As the Sn content increases, the bandgap closes, gets inverted at 46% at a temperature of 150 K and opens again entering the topological state. This band inversion is schematically depicted in the top graph .

The goal of this research is to pave the way for showing the topological phase transition in Pb_1-xSnxTe using a transport experiment. In this research, we will mainly focus on optimizing the back gate geometry. Previously, a global back gate geometry has been used with SiO2

oxide layers with a thickness of 100 nm. Here we will look into the possibility of using differ- ent high-κ gate oxides like HfOxand AlOxwith varying thickness. Furthermore, some electrical transport experiments are proposed which could be used to show the topological phase transition in Pb1-xSnxTe nanowire devices.

First, the necessary theoretical background for the topics described in the introduction is pro- vided in section 3. The growth of the nanowires, the fabrication of the devices and the measurement techniques are discussed in section 4. Then, the results will be presented and discussed in section 5. Finally, our conclusions are presented in section 6 after which we give an outlook on future research in section 7.

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3 Theory

In this section we will discuss the relevant concepts related to this project. First we will discuss some of the general properties of TCI’s. We will then focus on the relevant material parameters of Pb1-xSnxTe. Next, we will discuss the concept regarding gates and their effect on semiconductors. Finally, we will focus on some of the transport signatures of TCI nanowires which are expected to be present in Pb1-xSnxTe and some thoughts on how to experimentally observe these signatures.

3.1 General properties of TCI nanowires

Below a brief description is given of the general concepts concerning TI’s and TCI’s. A more concise summary of the history and theory on TI’s and TCI can be found in references [22] and [23], respectively.

An insulator can be described as a material in which all the electronic interactions are local [24].

This means that the individual wave functions are not overlapping and there is no interaction.

This is called an atomic insulator. Therefore, materials where all electrons are closely bound to the atoms like ionic and covalent materials fall into this class and can therefore be classified as an insulator. These materials have in common that their Hamiltonian can be adiabatically transformed into that of an atomic insulator without going through any phase transition [25].

We can classify these materials as topologically trivial materials.

In contrast, the Hamiltonian of a topological insulator cannot be adiabatically transformed back into an atomic insulator [26]. This topological state arises from symmetries in the Hamiltonian.

It is impossible to go from a non-trivial state to a trivial state without breaking this symmetry or by closing the band gap [22]. The topological state is therefore very robust against disorder in the material and other external influences. In conventional topological insulators, the topological state is protected by time-reversal symmetry and can thus be broken by the presence of an external magnetic field.

As mentioned before, it is not possible for a topological phase transition to occur without the closing of this band gap. However, at the boundary of a TI, the TI phase must terminate and a trivial phase must begin. Therefore, at the boundary of a 3D TI with a non trivial insulator or a vacuum for example, the band gap must close. This creates an insulating state in the bulk of the material, while we have conducting surface states. Since these surface states are topologically protected, they are very robust against perturbations which do not break time-reversal symmetry.

This is schematically depicted in figure 4a. In general, a N-dimensional topological insulator will have N-1 dimensional gapless states. This principle is called the bulk-boundary correspondence.

As mentioned in section 2, there are also systems for which this step in dimension can even be more than one as we will discuss in more detail in section 3.4.6

As can be seen from figure 4a, these gapless surface states have a linear dispersion. This is due to the fact that these surface states are solutions to the relativistic Dirac equations for massless particles. These surface states form a cone in 3 dimensions which is conventionally known as the Dirac cone. The electrons in these surface states also have very high scattering lengths since there are no bulk states to scatter into. This leads to a principle called spin-momentum locking in TI’s as predicted in one of the first theoretical models for TI’s [26]. The spin gets paired to one momentum direction in the surface state and because of the high scattering length, the spin and momentum direction stays locked. This is visualized in figure 4b.

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(a) (b)

Figure 4: (a) A schematic representation of a band structure of a topological insulator. Within the bulk band gap lie topologically protected spin polarized Dirac surface states. The yellow and blue filling represents the inverted band structure present in most topological insulators. The spin direction is indicated by the red arrows. (b) A schematic overview of spin-momentum locking in real space. The red and blue line indicate the momentum direction, while the spin direction is indicated by the grey arrows [22].

TI’s and TCI’s have an inverted band structure in their topological phase. For example, in the trivial material PbTe, the behavior of the valence band is characterized by p-type orbitals of the anion (Te), while the behaviour of the conduction band is dominated by p-type orbitals of the cation (Pb). For the non-trivial SnTe, we observe an inversion of the band structure, where now the behaviour of the valence band is dominated by p-type orbitals of the cation (Sn) and the behaviour of the conduction band is dominated by p-type orbitals of the anion (Te) [27] [28].

Whether a material is trivial or non-trivial can be characterized by the so-called topological invariants. There are many different types of topological invariants, depending on the symmetry that protects the topological phase. In topological insulators, where the topological phase is protected by time-reversal symmetry, the relevant topological invariant is the Z2invariant as was predicted by Kane and Mele in 2005 [26]. Examples of topological insulators are HgTe/CdTe quantum wells, Bi2Se3and Sb2Te3[29] [30] [31].

As mentioned before, in topological crystalline insulators, the topological state arises not from time-reversal symmetry but from crystal point symmetries [23]. Therefore, TCI’s have a trivial Z2invariant, but instead have a non-trivial mirror Chern number, nM, which is the topological invariant associated with crystal symmetries [32]. Although there is still much to be experimentally validated, many of the concepts discussed for TI’s, such as Dirac cones, protected surface states and spin-momentum locking, are expected to also hold for TCI’s.

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3.2 Material properties of Pb

_1-x

Sn

_x

Te

The material of interest of this research is Pb_1-xSnxTe . In this subsection we will discuss some of the properties of this material relevant to this research. However, for clarity, we will discuss the transport signatures of TCI’s in section 3.4.

SnTe is a TCI for which the topological phase is protected by crystal symmetries [16]. At room temperature, SnTe has a rocksalt crystal structure, with an even number of Dirac cones on the (001), (110) and (111) surfaces (or the surfaces equivalent to these surfaces due to symmetry) which are all mirror symmetric to the (110) mirror plane [16]. This is visualized in figure 5a.

Additionally, SnTe is a semiconductor, which allows for easier electrical characterization of the topological surface transport with respect to metallic TI’s or TCI’s, due to a lower amount of parasitic bulk transport.

(a) (b)

Figure 5: (a) The crystal structure of SnTe. The (001), (111) and the (110) planes are expected to have an even number of Dirac cones and are symmetric to the (110) mirror plane. (Image adapted from [28]) (b) The hall carrier concentration of Pb1-xSnxTe . We see a transition from n-type to p-type around a Sn-concentration of 15%. (Image adapted from [21])

Although SnTe is a a semiconductor, it is very heavily p-doped (p = ₁₀²¹), meaning that the Fermi energy lies deep withing the valence band. This extremely high carrier concentration is caused by the presence of negatively charged Sn vacancies [20] and, as mentioned in section 2, can be partially negated by alloying SnTe with Pb. By adding more lead (decreasing the x- value) the material gets closer to the intrinsic regime as can be seen figure 5b. Up until a critical, temperature-dependent Pb-Sn ratio, Pb_1-xSnxTe is expected to hold it’s topological phase.

As was discussed in section 3.1, SnTe has an inverted band structure whereas PbTe has a trivial band structure. To go from the topological to the trivial state by alloying these materials, the band gap must close. The point where the band gap closes, the phase transition occurs. The x-value for which this closing of the band gap occurs has been measured for different temperatures using multiple measurements techniques as is described by Dimmock et al. [33] and summarised by Volobuev et al. in figure 3 [21].

Many of the electronic properties of semiconductor heterostructures are determined and can be understood by the alignments of the band structure and work functions. Since this is a whole field of research on it’s own and falls outside the scope of this research, the reader is referred elsewhere [34]. However, the case of metal-semiconductor contacting will be discussed, as that will play a role in interpreting the results obtained.

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Depending on the relative size of the work functions of the metal contact and the nanowire, φ_mand φnw respectively, we expect either ohmic contacts (for φm > φ_nw), or Schottky contacts (for φm < φnw). As can be seen in figure 6a, for ohmic contacts the electrons can move easily from the nanowire to the metal in the forward bias regime. In the reverse bias regime, there is a small barrier which vanishes quickly with an increasing voltage. Therefore, we expect a linear IV-characteristic. For Schottky contacts we have a non vanishing barrier in both the forward bias and reverse bias regimes, as can be seen in figure 6b. For very small voltages, the electrons can’t cross the barrier resulting in a non-linear I-V characteristic with a very high resistance. For larger voltages, the electrons can cross the barrier which will again result in a linear I-V characteristic [34].

In our research, the nanowires are in contact with 10 nm of Ti. This layer is assumed to be thick enough that the work function of Ti contributes most to the work function of the metal contact.

Based on the work function of Ti (φm = 4.3eV) and that of PbTe and SnTe (φnw = 4.6eV and φ_nw =5.1eV respectively), we thus expect Schottky barriers to form for a Ti-Pb1-xSnxTe contact [35] [36] [37].

Ec Ei EF Ev

Metal contact Nanowire (p-type SC)

(a)

Ec

Ei

EF Ev

Metal contact Nanowire (p-type SC)

(b)

Figure 6: A schematic of the band bending which occurs when a metal and a p-type semiconductor (SC) are brought together. Based on the work functions of both the metal and the semiconductor either an ohmic contact (a) or a Schottky contact (b) is formed.

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3.3 Gating physics

As mentioned in section 2, in order to isolate the topological surface conduction from the trivial bulk conduction, we need to tune the chemical potential to within the band gap and, as mentioned in section 3.2, the chemical potential lies deep within the valence band. Intuitively, one can see that there is an excess of holes, and by pushing these holes away we can tune the chemical potential to lie within the band gap. Therefore, by applying positive gate voltage to the nanowire we can push away a certain amount of charge, Q. The amount of charge that can be displaced is related to the capacitance of the system and the voltage applied to the gate as

Q=C∗V, (1)

where C is the capacitance and V is the voltage. Since the carrier density in SnTe is very high, a lot of charge has to be displaced and therefore we need to maximize the amount of charge we can displace. To that end, we will first discuss the capacitance of the system and its relevant parameters. Then, we will discuss the limitations of the voltage given in equation 1. Finally we will discuss the effect of the charge on a semiconductor nanowire.

3.3.1 Capacitance

In a sense, the capacitance is a purely geometrical quantity, as it relates the size and direction of electric fields to a certain geometry. Accordingly, there is no simple equation for the capacitance.

Instead, the capacitance is heavily dependent on geometry. In this research, the focus lies on optimizing the back gate geometry as is depicted in figure 8a. Therefore we will limit our discussion to that geometry, while a full derivation and a summary of related concepts can be found in ref[38].

First, the case of a parallel plate capacitor is discussed. We consider two conducting plates with a homogeneous electric field between them. Following the derivation of ref [38], we find

C= ^Ae⁰

d , (2)

where A is the area of the plates, e₀ the vacuum permittivity and d the distance between the plates. However, in order to obtain the maximum amount of gating in such a system, instead of having a vacuum between the two conductors, we can use a dielectric material. A dielectric material can be thought of to contain many microscopic electric dipoles, which can align with an applied electric field, effectively enhancing the electric field. This effect is characterized by the dielectric constant er, which is defined as

e_r = ^e

e0, (3)

where e is the permittivity of the material. Now for the case of two parallel plates with a dielectric material in between we find

C= ^Ae⁰^e^r

d . (4)

ercan take a large range of values and therefore adding a dielectric layer to a system is an effective way to increase the capacitance in a system. We will discuss some prerequisites of these materials for our system in section 3.3.2.

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As mentioned in section 2, the focus of the research is on rectangular nanowire systems. How- ever, to the best of our knowledge, to this date no analytical expression has been found for such a system. Therefore we will first determine the capacitance for a cylindrical nanowire and then try to extend this theory for rectangular nanowires.

Having a cylindrical nanowire instead of a second conducting plate makes the case a lot more complicated as we do not have the nice symmetrical case of two parallel plates with homogeneous electric fields. Instead we have an asymmetrical case with inhomogeneous electric fields.

The biggest challenge is determining the form of the electric fields in the space around the nanowire. If the electric field is known, we can use Gauss’ law to determine the capacitance of the system. To determine the form of the electric field, we follow the derivation in ref [39]

using the mathematical method of conformal transformations. In this derivation we will consider a conducting cylinder instead of a semiconductor and thus assume that all the charge in the semiconductor resides at the surface. This assumption is valid for sufficiently high carrier densities as was shown in triangular GaN nanowires [40]. We expect this assumption to also be valid for our nanowires with an initial carrier density of p=10²¹. It is good to keep in mind that this assumption becomes increasingly less valid as the nanowire gets more depleted.

In a conformal transformation, one maps the field equations from one complex plane(e.g. the Z-plane) to another complex plane (e.g. the W-plane). In this mapping, the specific geometry can change, but the relative angle between the field equations is preserved [41]. It can then be shown that this condition implies that this transformation must also satisfy the Laplace equations. Therefore, we can use a conformal mapping to transform our geometry in the Z-plane to a solvable geometry in the W-plane and determine our field equations. We can then transform these equations back to the Z-plane and from that, determine the capacitance.

Let’s assume an infinite conducting plate at the origin and an infinite conducting cylinder at a distance d parallel to that plate in a vacuum as is visualized in figure 7a. We apply a voltage V0

to the cylinder, while keeping the conducting plate at zero potential. First, we use the method of images to change our problem to that of two parallel infinite conducting cylinders with opposite charge at a distance d of the origin as can be seen in figure 7b.

(a) (b)

Figure 7: (a) An infinite conducting cylinder placed at a distance d of an infinite conducting plate. A voltage of V = V0is applied to the conducting cylinder. (b) The method of images is used on the case of (a) to form a more symmetrical case of two parallel infinite conducting cylinders with opposite charge. (Images adapted from [39])

Now, define

Z=x+yi (5)

and

W=u+vi, (6)

where Z and W are the complex planes which are spanned by x, y and u, v respectively and i is the imaginary unit. The conformal transformation

W=K₁ln Z−d Z+d

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transforms this case to the complex W-plane. By combining equations 5, 6 and 7 and assuming K1to be real we find

u= ^K¹

2 ln (x−a)²+y² (x+a)²+y²

!

(8) and

v=K1

arctan

y x−d

−arctan

y x+d

. (9)

Since, W is an harmonic function, if we take u to be the potential function, v must be related to the flux function viaΨ= −e₀v [41]. The constant K₁can be determined by implementing the boundary conditions of the conducting cylinder. We then find the following potential function

Φ=u= ^V⁰

2 arcosh

d R

ln

(x−a)²+y² (x+a)²+y²

!

, (10)

where R is the radius of the conducting cylinder. For the flux function we find

Ψ= −e₀v= −e₀V0

2 arcosh

d R

arctan

y

x−d

−arctan

y

x+d

. (11)

Gauss’s law can now be used to determine the charge on the conducting cylinder by integrating the flux lines that end on it, from which we find that the capacitance per unit length has the surprisingly simple form of

C

L = ^2πe⁰ arcosh

d R

F m

. (12)

So for a nanowire of finite length completely enclosed by a dielectric layer, we can again modify the capacitance to be approximately

C= ^2πe⁰^e^r arcosh

d R

L, (13)

where L is the length of the nanowire. In reality, the nanowire is not completely surrounded by the dielectric material. For dielectric materials with a low dielectric constant (e.g.SiO2), this effect can be approximated by simple mathematical averaging of the dielectric constant giving an effective dielectric constant e_{e f f}. This method can also be used to determine the effective dielectric constant of a stack of different gate oxides [42].

An attempt was made to find an analytical expression for the capacitance of an infinite rectangular nanowire parallel to a conducting plate. However, no relevant results were obtained due to the even more complex geometry. This difficulty arises from the fact that it is difficult to describe rectangles in an elegant mathematical way, preventing us to find a suitable transformation to the complex W-plane.

One way to solve this problem, is to create a custom conformal transformation for this geometry using a Schwarz-Christoffel transformation [41]. However, as this lies outside the field of our expertise, this remains an open question. In the rest of the work we will use equation 13 to determine the capacitance of this system. This approximation is reasonable, as the capacitance for rectangular nanowires follows the same trend as the capacitance for cylindrical nanowires [42].

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3.3.2 Breakdown voltage of dielectric oxides

Besides the capacitance, the other factor of relevance in equation 1 is the voltage. This is the voltage that is applied to the gate and the charge that we can displace scales proportional with this voltage. In principle, we can increase the voltage until the electric field is high enough that the current can leak through the vacuum separating the conductors. However, the situation changes when a dielectric material is present between the two conductors, as this must be a worse insulator then vacuum. We thus need a dielectric material which not only has a high dielectric constant, but is a good insulator as well. It turns out that the class of oxides meets these requirements. In figure 8b we see multiple gate oxides and their dielectric constants and band gap.

(a) (b)

Figure 8: (a) A schematic of the gate design used for nanowire devices (not on scale). A global back gate is used and a gate oxide with variable thickness is deposited on top of highly doped silicon. (b) The dielectric constant and band gap of multiple dielectric materials [43].

When applying a voltage to a capacitor with a gate oxide, the electric field can damage the oxide in many ways and eventually the oxide will break down and a leakage channel will open within the dielectric material. This principle is conventionally known as the gate breakdown. A lot of research has been devoted to describing the mechanism of this breakdown and although there is still debate on the specifics of the correct model, all models agree on the fact that the breakdown is time-dependent [44].

For our research purposes, the gate does not need to work for extended amounts of times as we are more interested in applying a high gate voltage for a short time period. Therefore, these models are not useful for our research. From these models we learn that because of the high voltages and short timescales, the voltage for which we observe breakdown is highly dependent on the crystal structure in the oxides and the presence of crystal defects. Therefore, the breakdown voltage can be seen as a stochastic process. Nonetheless, we can define a breakdown field, E_bd (MV/cm)) which relates to the voltage as E = V/d, for which on average the breakdown will occur within very short time scales.

As the breakdown process is very dependent on the exact growth conditions, no all encompass- ing model has been developed. Therefore, this must be determined for each specific oxide with specific growth conditions. For nanoscale SiO2HfOxand Al2Ox layers, the breakdown field is typically in the order of 1−10 MV/cm [45] [46] [47].

For this research, three different gate oxides are considered. Thermally grown SiO2, Plasma assisted Atomic Layer Deposition (ALD) of HfOxand thermal ALD of Al2Ox. The SiO2is used as control sample, as previous devices were all fabricated using a SiO2gate. As can be seen in figure 8b, although SiO2 has a large band gap, the dielectric constant is quite low. HfOx and Al2Oxare both commonly used as gate oxides which were both available to be grown using ALD and both have a higher dielectric constant than SiO2.

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Besides the breakdown voltage and dielectric constant, it is important to consider charging effects in gate oxides. Due to their chemical composition, gate oxides can trap either positive or negative charges at the interfaces. These charges might cause a screening effect which reduces the effect of an applied gate voltage, essentially shifting the gate curve to either higher or lower gate voltages, depending on the sign of the trapped charges. This effect is especially pronounced in Al₂Ox. Here it is known that thin layers can get negatively charged which thus reduces the effectiveness of a positive gate voltage [48].

3.3.3 Gating in a semiconductor nanowire

In order to understand the effect of a gate on a semiconductor, we need to understand the band diagram of such a gate. This is visualized in the case of zero gate voltage and a non zero gate voltage in figure 9a and figure 9b, respectively. By applying a positive voltage, Vgto the gate, the band of the oxide and nanowire are pulled downwards. This creates a depletion layer at the interface between the oxide and the nanowire, since the holes are being ”pushed” away from this interface. As we increase Vg, the depletion region gets wider. The width of such a depletion region is given by

W = s

2enwΨs

eN_A , (14)

where enwis the dielectric constant of the nanowire,Ψsis the electrostatic surface potential, and N_A is the acceptor density within the nanowire. As we increase Vg even further, the intrinsic Fermi energy, Ei, gets pulled below the Fermi energy of the system, creating a region of n-type conductance. This is called the inversion layer and this thus limits the maximum width of the depletion region. Therefore, the maximum depletion width is given forΨs = 2ΨF, whereΨF is the Fermi potential.

Ec

Ei

EF Ev

Metal contact Nanowire

(p-type SC)

Oxide

qΨF qΨs

(a)

Ec

Ei

EF Ev

Metal contact Nanowire

(p-type SC)

Oxide

qΨF qΨs

qV0

(b)

Figure 9: A schematic of the band diagrams for a metal-oxide-semiconductor(p-type) contact with no applied voltage (a) and in the depletion regime (b).

If this depletion region becomes larger than the nanowire, pinch-off is achieved. The full thickness of the nanowire is then depleted of carriers and becomes insulating. In terms of the chemical potential, this means that the chemical potential of the nanowire is tuned to lie within the band gap. For nanowires in the pinch-off regime, the mobility and carrier density can be extracted from gate traces, where a large range of gate voltage is swept [49].

The mobility of a semiconductor nanowire is given by

µ= ^g^m^L

2

CV_bias, (15)

where gmis the trans conductance of the device, L is the channel length, C is the capacitance of the gate and V_biasis the bias voltage applied to the device. The transconductance is defined as gm=

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dI/dVg, which is thus the slope of the gate trace. The carrier density of a p-type semiconductor nanowire is defined as

p= ^CV^th

eAnwL = ^CV^th

e(2r)²L, (16)

where V_th is the treshold voltage, Anw the area of the nanowire, which for cubic nanowires is given by(2r)². V_ths defined as the gate voltage necessary to achieve pinch-off in the nanowire, which is thus the point where the gate trace reaches zero drain voltage.

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3.4 Transport signatures of TCI’s

Now that we have gained familiarity with some of the general concepts concerning TI’s and TCI’s, we will dive deeper in some of the transport properties which can be used as an indication that a certain material is indeed a TCI. Furthermore, we will discuss some of the experimental challenges one would face when measuring these properties.

3.4.1 Opening and closing of the bulk band gap

As mentioned in section 3.1, if a material goes from a trivial phase to a topological phase, the band gap must close during this phase transition. During this phase transition, the band structure gets inverted. Although it is very difficult to measure this band inversion using transport measurements, it is possible to measure this closing and opening of the band gap. The size of the band gap can for example be determined by performing IV-sweeps on an ambipolar device.

This should also be possible in Pb1-xSnxTe . As can be seen in figure 3, it is possible to tune the material from a trivial phase to the topological phase by decreasing the temperature and keeping the x-value constant. If one would for example take Pb_1-xSnxTe with an x-value of around 0.46, we see that at room temperature the band gap is positive, and for decreasing temperatures the band gap first decreases and then increases depicting the closing and opening of the band gap.

A successful observation of the closing and opening of the band gap would, to our knowledge, be the first observation of a topological phase transition in a TCI using an electrical transport experiment.

3.4.2 Measuring of Dirac surface states

Another key signature of the topological phase is the observation of topologically protected surface conduction. In principle, this can be done by just measuring the current through the material since the bulk is insulating and the only conduction possible is through the surface. However, to experimentally validate this, we need to not only show that the measured current indeed comes from the surface, we should also show the Dirac nature of the the electron transport.

To show that the measured current is indeed a surface current, a Hall measurement can be performed in a geometry as visualized for a nanowire system in figure 10a with a magnetic field applied in the z-direction. The hall resistivity, ρxy, is then defined as the resistivity in the y- direction (between leads 1-3 or 2-4) if a voltage is applied between the source and the drain in the x-direction. If all the transport would be through surface channels, we would expect the Hall resistivity to change linearly for a varying magnetic field. If the transport would go through a combination of bulk and surface channels, we would expect ρxy to vary non-linearly with the magnetic field due to parallel contributions of the bulk and the surface to the conductance[50]

[51].

When it is indeed verified that the measured current comes through surface states, it is necessary to verify that the surface transport actually has a topological nature and is not just trivial surface transport. If the transport would be topological in nature, we would expect a Dirac dispersion for these electrons, as can be seen from figure 4b. Ideal Dirac fermions are expected to have a Berry phase of π [52]. We can probe this berry phase by measuring Shubnikov de Haas (SdH) oscillations. These oscillations in the conductivity in the high magnetic field regime hold infor- mation about the cyclotronic motion of the electrons in the material [53]. During this cyclotronic motion, Dirac fermions pick up a Berry phase of1

2.

The oscillations in conductivity can be probed in a similar setup as depicted in figure 10a [54]. If

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(a) (b)

Figure 10: (a) An example of a configuration able to measure the hall receptivity’s on a nanowire (green). (b) A schematic of the influence of the magnetic field on the density of states and the location of the Landau levels. By increasing the magnetic field, the spacing between the Landau levels increases [22].

the magnetic field is applied orthogonal to the surface and increased to high magnetic fields (≈ 9T), the conductivity in the longitudinal direction of the nanowire, ρxx, will start to oscillate as the Fermi energy moves through Landau levels. This can be understood by relating the Density Of States (DOS) in the material to the conductivity. Due to the magnetic field, Landau levels appear in the DOS as is visualized in figure 10b. If EF lies at the center of a Landau level, the DOS is maximum and the conductivity will thus also be at a maximum. If E_F lies just outside of a Landau level, the DOS will have a minimum and thus the conductivity will also have a minimum. This minimum then corresponds to an integer number of filled Landau levels, N [53].

This oscillation follows

∆σxx∼cos

2π F

B+¹ 2− ^γ

2π

, (17)

where F is the oscillation frequency, B the magnetic field and γ is the berry phase around a closed electron orbit [53] [55].

From equation 18, we see that we the Nth minimum occurs if

2π F B +¹

2− ^γ 2π

= (2N−1)π. (18)

We can now think of an experimental setup where we measure the longitudinal conductivity, σxx, while varying the magnetic field. We then assign integers N to the observed minima. If we then plot 1

BN against N we see a straight line of which the slope equals the frequency F. This is known as a Landau fan diagram. We can perform a linear fit through the data points in which we can fix the slope to F, which can be obtained by taking the Fourier transform of the SdH oscillations.

The point at which this line intersects with the line 1 BN

=0 gives us the factor γ 2π [22].

In reality, there are a few more subtleties involved with this method. For example, the dispersion within a TI is often not perfectly linear but instead has some nonlinear components for which we have to correct [53]. Also, the choice to use σxxinstead of ρxxis not trivial, since it is carrier dependent whether we expect the maxima and minima of σxx to ρxx coincide or not. A nice overview of some of these subtleties is given by ref [22].

In conclusion, observing a linear ρxy, for a varying magnetic field and additionally establishing that electrons carry a berry phase of π, is a clear signature of topological surface transport. The above discussion is based on TI experiments. However, the discussed properties of the surface

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states of TI’s are also expected to hold for TCI’s [22]. Therefore, we expect these experiments to be valid for TCI’s.

3.4.3 Aharonov-Bohm effect

Another transport effect that can be measured in TI/TCI nanowires is the Aharonov-Bohm (AB) effect [56]. If a magnetic field is applied along the longitudinal direction of a nanowire, electrons that move around the circumference of the nanowire will pick up a phase factor of 2πΦ

Φ0 which will lead to measurable interference effects. Here,Φ is the total flux going through the nanowire andΦ0 = ^h

e is the flux quantum. This effect in itself is not topological of nature. However, by examining the effect of the magnetic field on the band structure of the surface states in a TI, one can make predictions on the transport properties which are specific to TI’s and TCI’s. First we will have a look at an ideal ballistic TI/TCI nanowire to get a grasp of the underlying physics.

Then we will make the system more realistic by adding disorder to the system and observe its effects.

Figure 11: Calculations of the surface state excitation spectra in a TI nanowire for a magnetic field applied in the longitudinal direction by Cook and Franz [12]. By increasing the magnetic field and thus the magnetic flux, the energy bands go from doubly degenerate (solid lines) to non degenerate (dashed line). ForΦ = 0.5Φ0we observe gapless surface states (dashed red lines).

When applying the magnetic field in the longitudinal direction of a TI nanowire, there is only one (periodic) value ofΦ= ^Phi⁰

2 for which we observe gapless surface states, as can also be seen in figure 11 [57]. So, if the chemical potential is tuned exactly at the Dirac point, we expect only one channel to be open. This channel has to be a perfectly transmitting mode [58]. From this, we expect that the magnetoconductance of an ideal ballistic TI/TCI nanowire oscillates with a period ofΦ0 and has maxima at multiples ofΦ = ^Φ⁰

2 . This is in contrast to trivial materials, for which we expect an oscillation period of Φ0

2 [12]. This discrepancy is caused by the fact that electrons in a TI/TCI pick up a berry phase of π, while electrons in trivial materials do not.

However, for disordered systems, with a Fermi energy away from the Dirac point, the physics becomes more complicated. In 2010 Peng et al. measured the AB effect in the TI Bi2Se3nanoribbons [59]. Instead of the above explained result for TI’s, a period ofΦ0and a maximum at multiples of Φ= ^Φ⁰

2 , they indeed observed a period ofΦ0, but they found a maximum atΦ=0. Badarson et al. explained this result by expanding on the simple band picture and including time-reversal preserving disorder [57]. They find that the presence of disorder in the system can shift the maximum of the conductance. However, the oscillation period remainsΦ0. Therefore, the observation of AB oscillations with a period ofΦ0is a clear signature that the measured surface conductance is indeed topological in nature.

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3.4.4 DOS of the surface states and Van Hove singularities

As mentioned in section 3.1, a characteristic of TI’s and TCI’s is the linear energy dispersion of the gapless surface states, while the bulk states have a parabolic dispersion. This combination should lead to a unique DOS of the material. To get a first indication on how this DOS looks in a TI/TCI, we can do a simple calculation to get an estimate of the DOS.

The DOS in energy space can be determined by knowing the energy dispersion and the density of states in momentum-space. By using fixed end boundary conditions as described in ref [60], we can determine the DOS in momentum space. This yields

g_2D(k) =2 L²

(2π)²^2πk= ^L

2k

π (19)

and

g3D(k) =2 L³ (2π)³^4π

2k²= ^L

3k²

π² , (20)

for the two-dimensional and three-dimensional case.

Let’s first discuss the DOS of the surface states within the bulk gap. Here we expect a linear energy dispersion of the form

E=¯hvFk, (21)

where ¯h is the reduced Planck constant and vF is the Fermi velocity. By combining equation 19 and equation 21, we can determine the DOS in energy space, which yields

D2D(E) = ^L

2

π(¯hv_F)²^E. ⁽²²⁾

Outside the gap, we expect a parabolic dispersion of the form

E= ^¯h

2k²

2m^∗, (23)

where m∗is the effective mass of the electrons. We also expect that the transport in this regime is governed by bulk transport, and we can thus determine the DOS in energy space by combining equations 20 and 23, which yields

D3D(E) = ^L

3

2π²

2m^∗

¯h²

³₂√

E. (24)

Therefore, within the gap we expect a DOS which scales linearly with E and outside the gap a DOS which scales with√

E. For completeness, the DOS in all dimensions for both a linear and parabolic dispersion is given in table 1

Another interesting feature of the DOS for certain TCI’s is the presence of van Hove singularities (VHS). VHS are singularities in the DOS, or in our case a discontinuity in the DOS. These singularities are not a general feature of TI’s or even TCI’s but band structure calculations have predicted their presence in SnTe for the (001) surface [28]. Due to symmetry, this crystal surface has two distinct Dirac cones, which are both a combination of two different L points which are

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D(E) for linear dispersion D(E) for parabolic dispersion

1D L

π¯hvF

L π

2m^∗

¯h²

1/2

√1 E

2D L²

π(¯hvF)²^E

L² π

2m^∗

¯h²

3D L³

π²(¯hvF)³^E

2 L³

2π²

2m^∗

¯h²

3/2√ E

Table 1: The density of states in energy space in multiple dimensions for both a linear dispersion and a parabolic dispersion.

projected onto the same symmetry point on the surface. The two Dirac cones are located at~_X₁ and~_X₂_points.

Since two L points are projected onto a~X point, these L points are allowed to interact with each other. Ab initio band calculations were performed by Liu et al., as can be seen in figure 12a [28].

We see two low energy Dirac cones located symmetrically around the~X point. Interestingly, Liu et al. find that the two low-energy bands are forbidden to interact along the~_X₁~Γ line, while they are allowed to interact along the~_X₁M direction. The result of this is depicted in figure 12b. Close~ to the Dirac point, we observe two disconnected Dirac pockets, as we move away from the Dirac point, the pockets will connect with each other at the saddle points~_S₁_and~_S₂, which leads to the VHS in the DOS as is observed in figure 12b. Therefore, measuring a VHS in SnTe would be a clear signature of it’s topological nature.

(a) (b)

Figure 12: (a) The band structure of the (001) crystal surface of SnTe at one of the ~X. The two low energy Dirac cones connect along the

~X1M at the saddle points ~S~ 1and ~S2. (b) The band structure of the (001) crystal surface of SnTe and the DOS associated with this band structure. VHS occur at the saddle points ~S1and ~S2. (Image adapted from [28])

The most common method for measuring the DOS in a material, is Scanning Tunneling spec- troscopy (STS), which is a subclass of Scanning Tunneling Microscopy(STM). One would then measure the dI/dV, for varying bias voltage while keeping the tip position constant. An example of this set-up and data interpretation can be found in ref [61]

3.4.5 Long scattering length in the surface states

As mentioned in section 3.1, the electrons in the surface states have a very long scattering length, which is in part the cause of spin-momentum locking, and is typical for topological surface states.

One easy way to measure this long scattering length, is by showing that the resistance in a topological nanowire is not dependent on the channel length, since that implies that the electrons do not scatter within the nanowire. The difficulty herein lies that one needs to either fabricate

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a device with a variable channel length, of which no device geometry is known to the Author, or one needs to compare the resistance of multiple devices with different channel lengths. Al- though the nanowires used for these devices are grown in the same growth run, there can still be variations in the compound of the nanowire which influences the resistivity of such a wire. This complicates a direct comparison between different devices.

Another way to observe the long scattering length is to measure the spin-momentum locking di- rectly. As is visualized in figure 4b, the spin-momentum locking leads to spin polarized transport channels within the bulk band gap where the spin direction is locked perpendicular to the momentum direction. This can be measured via optical experiments or by transport experiments.

The optical experiments are based on spin resolved ARPES measurements, where shining circu- larly polarized light on the material selectively excites electrons in one of the surface states based on their spin polarization [62] [63].

This can also be observed via transport experiments based on the spintronics field of research.

By applying an unpolarized charge current along the longitudinal direction of a nanowire, the spin-momentum locking induces a spin-current perpendicular to this [64] [65]. It is possible to measure this spin-current using Ferromagnetic contacts as is proposed and measured by Li et al.

in Bi2Se3films [66].

A concept schematic of a possible measurement configuration based on the measurement setup of Li et al. can be found in figure 13. An unpolarized charge current between the source and the drain causes all the spins to align with the y-direction. Since the magnetic orientation of FM1 is orthogonal to the spin direction, we expect to measure no voltage drop caused by the spin current. However, if the magnetic direction of FM2 is anti-parallel or parallel, we do expect a spin-related signal in the voltage measurements. A convincing measurement to observe the spin current can now be done by rotating the magnetic orientation by applying a small magnetic field to the ferromagnetic contact. When going from an anti-parallel alignment of spin and magnetic orientation to a parallel alignment, we expect a voltage drop to occur [67]. Therefore, this method gives a direct measurement of the spin-momentum locking induced spin polarized surface conduction in TI’s and TCI’s.

Figure 13: A concept schematic of a possible way to measure the spin current induced by spin momentum locking. The nanowire( green) is contacted by two gold leads, which provide an unpolarized charge current in the x-direction. Two Ferromagnetic contacts (brown) are contacted to the nanowire through a magnet separated by an oxide (grey) which limits the magnetic exchange interaction between the nanowire and the contact. The arrow on FM1 and FM2 indicates the direction of the magnetic orientation. A voltage drop can be measured between the source and FM1 and FM2 respectively.

3.4.6 Effects of symmetry breaking

A final transport signature that we will discuss here, is one specific for TCI’s. As mentioned in section 2, the topological phase in a TCI is protected by the band gap and by the crystal symmetry.

Therefore, if we would be able to break this symmetry, this should destroy the topological phase, which should give a measurable effect.

Although the mathematics behind these principles is very complex, the resulting physics is quite intuitive. As long as a facet is symmetric with respect to the (110)-plane, that facet should remain

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topological and it should have a bulk band gap with gapless surface states connecting the conduction and valence band. Since this is valid for each facet individually, it is possible to break only certain facets while keeping the symmetry intact for other facets. This breaking of the symmetry is a gradual process and as we start breaking the symmetry, the Dirac cones start gaining curvature and the surface states get gapped [61].

Intuitively, we can think of a few ways to break the symmetry of some of these facets. For example we could apply a strain to nanowire so that we break the symmetry of the nanowire.

However, it is quite difficult to tune this symmetry breaking in a controlled manner.

Another way to break the symmetry is by applying electric fields in specific directions. This method is expected to be much easier since the fabrication needed is very similar to the fabrication process used for this research. Therefore, we will now focus on measuring the symmetry breaking by electric fields.

Let’s first consider the simple case of a nanowire on a back gate as can be seen in figure 14a. A global back gate can generate an electric field which can be considered homogeneous and points in the perpendicular direction of the gate. The constant potential lines therefore lie parallel to the substrate. From this we see that the top and the bottom facets of the nanowire lie along these lines and therefore the symmetry of these facets remains conserved. However, this is not the case for the two vertical side facets. The electric potential is not constant along these facets and therefore the symmetry will be broken.

(a) (b)

Figure 14: (a) The schematic of a back gated nanowire device. Applying a voltage to the back gate is expected to break the symmetry of the vertical side facets while keeping the symmetry of the top and bottom facets conserved. (b) The schematic of a nanowire device with both a back gate and a top gate. In this case all the side facets are expected to conserve their symmetry since they all lie along equipotential lines.

In contrast, for a nanowire with both a back gate and a top gate which also covers the side facets, the electric field is approximately perpendicular to all the side facets as can be seen in figure 14b.

Therefore, we expect that all the side facets lie approximately along the equipotential lines. This thus conserves the symmetry of all facets.

We can now contrive an experiment where we try to measure the AB oscillations discussed in section 3.4.3 for both a system with only a back gate and for a system with both a back gate and a top gate. For the first system, we do not expect to measure the oscillations since two of the surface facets are fully gapped. For the second system we do expect to measure AB oscillations since the surface transport should still be topologically protected in all side facets.

After these measurements, there are many variations to this experiment possible with more difficult configurations. For example, an electric field applied in the diagonal direction is expected to break the symmetry of all the side facets, bringing us fully in HOTI territory, where all bulk and surface conduction is gapped and we instead have gapless 1D edge states. [14]. This could open up a whole new field of research.

Eindhoven University of Technology MASTER Towards showing the topological phase transition in Pb1-xSnxTe nanowires Mientjes, M.G.C.

Eindhoven University of Technology

MASTER

Towards showing the topological phase transition in Pb1-xSnxTe nanowires

Mientjes, M.G.C.

E INDHOVEN U NIVERSITY OF T ECHNOLOGY

M ASTER THESIS

Towards showing the topological phase transition in Pb 1-x Sn x Te

nanowires

M.G.C. Mientjes

Supervisors

M.S.M. Hoskam M.Sc.

Dr. Y. Uzun

Prof. dr. E.P.A.M. Bakkers

January, 2021

Contents

1 Abstract

2 Introduction

3 Theory

3.1 General properties of TCI nanowires

3.2 Material properties of Pb

Sn

Te

Metal contact Nanowire (p-type SC)

Metal contact Nanowire (p-type SC)

3.3 Gating physics

3.4 Transport signatures of TCI’s

E ^INDHOVEN U NIVERSITY OF T ^ECHNOLOGY

Towards showing the topological phase transition in Pb _1-x Sn _x Te