# Quantum Transport and Potential of Topological States for Thermoelectricity
in Bi2Te3 Thin Films

Prosper Ngabonziza<EMAIL_ADDRESS>Max Planck Institute for Solid
State Research,
Heisenbergstraße 1, 70569 Stuttgart, Germany  Department of Physics,
University of Johannesburg,
P.O. Box 524, Auckland Park 2006, Johannesburg, South Africa

###### Abstract

This paper reviews recent developments in quantum transport and it presents
recent efforts to explore the contribution of topological insulator boundary
states to thermoelectricity in Bi2Te3 thin films. Although Bi2Te3 has been
used as a thermoelectric material for many years, it is only recently that
thin films of this material have been synthesized as 3D topological insulators
with interesting physics and potential applications related to topologically
protected surface states. A major bottleneck in Bi2Te3 thin films has been
eliminating its bulk conductivity while increasing its crystal quality. The
ability to grow epitaxial films with high crystal quality and to fabricate
sophisticated Bi2Te3-based devices is attractive for implementing a variety of
topological quantum devices and exploring the potential of topological states
to improve thermoelectric properties. Special emphasis is laid on preparing
low-defect-density Bi2Te3 epitaxial films, gate-tuning of normal-state
transport and Josephson supercurrent in topological insulator/superconductor
hybrid devices. Prospective quantum transport experiments on Bi2Te3 thin-film
devices are discussed as well. Finally, an overview of current progress on the
contribution of topological insulator boundary states to thermoelectricity is
presented. Future explorations to reveal the potential of topological states
for improving thermoelectric properties of Bi2Te3 films and realizing high-
performance thermoelectric devices are discussed. Keywords: topological
insulators, thermoelectric materials, bismuth telluride, quantum transport,
topological states for thermoelectricity

## 1\. Introduction

A topological insulator (TI) is a material that has an insulating interior
(bulk) and conducting boundaries. The bulk of a TI is characterized by an
energy gap in the electronic band structure, whereas the edges or surfaces
host non-gapped states crossing the bulk band gap, thus enabling conduction on
the boundaries of the material [1]. The surface states of a TI mimic
relativistic Dirac electrons because of their linear energy-momentum relation
[2]. The appearance of metallic surface/edge states is due to the strong spin-
orbit coupling in the material, which causes a band inversion. TIs have
generated much interest, not only as fundamental new electronic states of
matter but also for enabling the creation of devices that exploit the unusual
properties of topological surface/edge states. This allows future
technological applications in such fields as spintronics, topological quantum
computation and others [3, 4, 5, 6, 7, 8].

TIs were first predicted theoretically in 2D quantum-spin Hall systems
(HgTe/CdTe) [9, 10, 1]. Shortly thereafter, the first 3D TI was experimentally
realized in BixSb1-x alloys [11, 12]. Soon afterwards, the materials Sb2Te3,
Bi2Te3, and Bi2Se3 were predicted and experimentally confirmed to be 3D TIs.
These latter compounds (Sb2Te3, Bi2Te3, and Bi2Se3) have the advantage of
exhibiting surface states with a single Dirac cone [13, 14, 15], which makes
them candidate materials for the exploration of exotic topological states such
as Majorana bound states and magnetic monopoles in condensed-matter systems
[3, 16]. The confirmation of the existence of 3D TIs has led to an explosion
of research in this field, both theoretically and experimentally. Since then,
many more materials have been added to the list of 3D TIs, and their diverse
potential technological applications have been explored [17, 18, 8].

This review focuses on Bi2Te3 epitaxial films. The material Bi2Te3 has a
rhombohedral crystal structure consisting of quintuple layers (QLs) of bismuth
and tellurium atoms bound to each other (stacking sequence: Te[1]-Bi-Te[2]-Bi-
Te[1]) through weak Van der Waals (VdW) forces, see Figs. 1(a) and (b). Bi2Te3
is a prime member of 3D TIs that is most extensively studied due not only to
its fascinating TI properties [15, 19, 20, 21, 22, 23], but also its excellent
room-temperature thermoelectric (TE) properties [24, 25]. By exploiting these
properties, Bi2Te3 has been used to fabricate flexible TE devices with a wide
range of technological applications, for example in self-powered wearable
electronics, electrical power-generation devices and precision temperature
control of microchips [26, 27, 28, 29, 30, 31, 32]. Topological surface states
and TE properties of Bi2Te3 are intimately related to the position of the
Fermi level $(E_{\textrm{F}})$ in the electronic band structure [33, 20, 34,
35] and the concentration of intrinsic defects [36, 37, 38, 39, 40, 41].
Furthermore, interest in Bi2Te3 has been heightened recently by proposals for
topological surface-state-enhanced TE devices [42, 43, 44, 45], which
underscore the need for systematic experimental explorations of the
contribution of TI boundary states to thermoelectricity.

However, as bulk conduction in Bi2Te3 complicates the direct exploitation of
TI surface effects, great care must be taken to improve the crystal quality
and obtain samples with fewer intrinsic defects. The most promising approach
to improve the crystal quality and obtain samples with minimal intrinsic
defects and high surface mobilities is to prepare high-quality intrinsically
insulating Bi2Te3 thin films [20, 34, 35, 21]. This paper reviews the latest
progress in studying the quantum transport properties of Bi2Te3 epitaxial
films and the contribution of TI boundary states to thermoelectricity. Recent
advances in thin-film growth, magnetotransport and induced superconducting
proximity effect in Bi2Te3 thin films are discussed. Finally, an overview of
current efforts to explore the contribution of topological boundary states to
thermoelectricity in epitaxial Bi2Te3 thin films is presented.

###

## 2\. Optimizing the Growth Parameters of Bi2Te3 thin films

###

To study the salient topological surface states and TE properties of Bi2Te3,
thin films of this material have been synthesized over the past few years
using different experimental methods. The most widely used techniques include,
but are not limited to: molecular beam epitaxy (MBE) [46, 20, 33, 34, 39],
pulsed laser deposition (PLD) [47, 48, 49], metal-organic chemical vapor
deposition (MOCVD) [50, 51, 52], hot-wall epitaxy (HWE) [53, 54, 55],
electrodeposition [56, 57, 58], ion beam sputtering deposition [59, 60, 61],
and magnetron sputtering deposition [62, 63, 64]. Amongst these techniques,
MBE has been especially well established for growing high-quality crystalline
thin films with precise control of film thickness down to atomic layers [46].
In addition, MBE makes it easy to combine different TI compounds, for example
to form $p$-$n$ junctions [65, 66, 67] or TIs with other materials such as TI-
ferromagnetic insulator heterostructures [68, 69, 22]. In particular,
intrinsically insulating Bi2Te3 samples with low bulk-defect density have been
prepared by carefully controlling the MBE growth conditions [20, 34, 35, 70].
Such samples allow a fine-tuning of $E_{\textrm{F}}$ in the band gap, which is
of key importance for harnessing special properties of topological surface
states in order to optimize the TE performance of TI nanostructures [45]. The
high flexibility in combining different elements, the good control over film
thickness and the possibility of uniform deposition over a large area make MBE
the most versatile method for synthesizing multifunctional TI/TE Bi2Te3
nanomaterials. This section focuses on recent experimental endeavors to
optimize MBE growth conditions for achieving high-quality Bi2Te3 with low
defect density.

Figure 1:  (a) Crystalline structure of Bi2Te3. The unit cell is formed by
stacking three QLs in [001] direction. VdW forces between adjacent QLs through
Te atoms is much weaker than that between Te and Bi inside the QL. (b) Side
view of one QL slab of Bi2Te3. (c) Te-rich growth environment in the MBE
growth process, which is the ideal condition for obtaining high-quality Bi2Te3
TI thin films. (c) Reproduced and adapted with permission from [71], © 2012
Elsevier Science, all rights reserved.

During MBE growth, elemental or compound materials of interest are generally
evaporated from sources contained in electrically heated furnaces or crucibles
[72, 73]. The evaporated species pass through an ultra-high vacuum (UHV)
environment and are deposited on a substrate that is hundreds of millimeters
away from the evaporated materials. Under such UHV conditions, the
incorporation of contaminants from the background gas is minimized. The
temperature of the sources is set according to the type of material used,
because different materials have different melting points and vapor pressures.
The MBE growth process of Bi2Te3, and for other 3D TIs in general, is mainly
affected by the atomic flux ratio of the elements, the UHV conditions in the
growth environment and the substrate temperature [46, 73].

To maximize the surface diffusion for confining nucleation and growth at the
proper atomic sites, the substrate temperature is often tuned to an optimum
value. Compared to bulk crystals (typically synthesized at temperatures above
600°C) [74, 36], high-quality thin films of Bi2Te3 are grown by MBE at much
lower temperatures in the range of 200–350°C [46, 34, 75, 35]. The reduced
kinetics at low growth temperatures helps minimize the formation of Te
vacancies in MBE-grown thin films. Furthermore, Bi2Te3 bulk crystals can be
grown in both $n$\- and $p$-types [76, 17]. This is because the Te anti-site
defects Te${}^{+}_{\text{Bi}}$ (Te ion sitting on a Bi lattice site) or Bi
anti-site defects Bi${}^{-}_{\text{Te}}$ (Bi ion sitting on a Te lattice site)
are formed in Te-rich or Bi-rich conditions, respectively [17].
Te${}^{+}_{\text{Bi}}$ is a donor, whereas Bi${}^{-}_{\text{Te}}$ is an
acceptor, thus suggesting that, by carefully controlling the MBE growth
conditions, it would be possible to prepare high-quality bulk insulating
Bi2Te3 thin films.

Figure 1(c) illustrates the Te-rich growth dynamics required for preparing
high-quality Bi2Te3 films using MBE [46, 77]. To obtain high-quality films, it
is necessary to control the atomic flux ratio of the Bi and Te elements [46,
78, 20]. In practice, to prepare Bi2Te3, the growth should be performed in the
tellurium-rich side of the phase diagram in order to avoid Te vacancies. This
Te-rich condition helps achieve the correct stoichiometry (a ratio of Te:Bi =
3:2) to ensure the best film growth. To obtain stoichiometric Bi2Te3 thin
films, it is often suggested that the Te/Bi atomic ratio should be in the
range of 8 to 20 [46, 20]. Furthermore, when using thermal evaporation, the
condition $T_{\text{Bi}}>T_{\text{sub}}>T_{\text{Te}}$ (where $T_{\text{Bi}}$,
$T_{\text{sub}}$ and $T_{\text{Te}}$ are the Bi-cell, substrate and Te-cell
temperatures, respectively) should be satisfied.

2.1 Surface Passivation and Surface Morphology

###

The lattice constant mismatch between a substrate and a film is a major
challenge for achieving epitaxial growth of thin films. A large lattice
mismatch between the constituent materials leads to defects and worsens the
crystal quality of the epitaxially grown film. However, for growing Bi2Te3
films (also for other 3D TI systems), the lattice-match criteria are
drastically relaxed because the unit cells are coupled by VdW interactions,
see Figs. 1(a) and (b). This growth preference is known as VdW epitaxy [79,
80]. Therefore, a variety of substrates can be chosen, and relatively good
Bi2Te3 films have been obtained for each. Substrates reported so far include,
but are not limited to: Si [111] [81, 82], Al2O3 [0001] [75, 34], SrTiO3 [111]
[34, 21], GaAs [111]B [83, 84], GaAs [001] [85, 86], BaF2 [111] [87, 88] and
Mica [89, 90]. For purposes of avoiding the contribution of conduction from
the substrate in electronic transport measurements, insulating substrates are
preferred. In particular, insulating oxide substrates with a high relative
dielectric constant at low temperatures offer the opportunity of gate-tuning
the film’s transport properties using the substrates as a back gate [21, 91].
Although the atomically well-defined single-termination of these substrates is
not a requirement, proper surface preparation is desirable to obtain good
templates for epitaxial growth [92, 93, 94, 95, 96].

To passivate the substrate surface and achieve atomically sharp interfaces
between the TI film and the substrate, the two-step growth scheme for VdW
epitaxy of Bi2Te3 is often used [75, 34, 46, 22]. After initial treatments of
the substrate, a few nanometers of Bi2Te3 are first deposited at a slightly
lower substrate temperature. This initial low-temperature layer, referred to
as the nucleation layer, effectively fulfills the objective of passivating the
substrate’s dangling bonds for the subsequent high-temperature VdW epitaxy.
The lower-temperature layer is then annealed as the substrate temperature is
raised to higher growth temperatures, and the second layer is epitaxially
deposited on top. This two-step growth procedure often results in Bi2Te3 thin
films of high-quality, terrace-step surface morphology with fewer defects, and
also with larger and fewer triangular mounds, see Fig. 2. Furthermore,
adopting the two-step growth method has been demonstrated to facilitate the
growth of bulk insulating Bi2Te3 films on a wide variety of substrates [34,
35, 97, 22].

Figure 2: Growth morphology of Bi2Te3 films of different thicknesses grown on
distinct substrates at the same optimal substrate temperature of the
nucleation layer (170°C), but with different temperatures for the second layer
(170, 210 and 250°C). Surface morphology of 15-nm Bi2Te3 films grown on
(111)-oriented SrTiO3 (STO) at different substrate temperatures (a) 170°C, (b)
210°C, and (c)–(d) 250°C. The optimized growth temperature (for the second
layer) was found to be around 250°C. The sample in (c) was annealed for 5 min,
whereas the sample in (d) was annealed for 30 min immediately after growth.
The three arrows in (b) point to small and larger 3D defects and quasi-1D
nanostructures present at the surface. The inset in (c) shows the RHEED
pattern of the surface of the sample taken at room temperature after growth.
(e) Morphology of a 70-nm-thick Bi2Te3 film grown on a (0001) Al2O3 substrate
at 230°C and of a (f) 20-nm-thick Bi2Te3 film grown on an STO substrate at
250°C. From the line profile obtained across a series of wide steps (blue
line), the step height was determined to be 1 QL (right-hand panels) for
samples grown on both substrates. Reproduced and adapted from [98].

For practical purposes, large terraces are preferred, the size of which
strongly depends on the substrate temperature during growth, which must be
chosen properly. To establish the correct substrate temperature
($T_{\text{sub}}$) for MBE growth of the Bi2Te3 films, we have performed a set
of depositions at different $T_{\text{sub}}$ for samples grown on different
substrates. Representative samples grown on both sapphire and (111)-oriented
(STO) substrates are presented in Fig. 2. Using the two-step growth method, we
found that the optimal growth temperature for Bi2Te3 prepared on sapphire is
170°C for the nucleation layer and 230°C for the subsequent layer. For samples
grown on STO with the two-step method as well, the optimal temperature for the
nucleation layer is also 170°C, whereas for the second layer it is 250°C.
Figures 2a–d demonstrates the improvement of the surface morphology as
$T_{\text{sub}}$ is increased for four different samples of 15-nm Bi2Te3 grown
on STO [111].

In practice, if $T_{\text{sub}}$ is too low, an amorphous or a polycrystalline
film will be formed, as observed in the film grown at 170°C, see Fig. 2(a).
This is because the adatoms will not have enough energy to diffuse and find
the lowest potential energy site. As the substrate temperature is slightly
increased from 170 to 210°C, an improved morphology with the appearance of
triangular mounds is observed, see Fig. 2(b). However, at this temperature,
there are still notable small and large 3D or 1D defects, indicated by arrows
in Fig. 2(b) [75]. The same features were also reported for MBE growth of
Bi2Se3 films, a prototypical material of Bi2Te3 [99, 97]. For the sample grown
at 250°C and annealed for 5 min in Te flux (after growth of the second layer),
the 3D and 1D defects disappear and the surface is dominated by triangular
mounds as depicted in Fig. 2(c). The corresponding reflection high-energy
electron diffraction (RHEED) image, see inset in Fig. 2(c), taken after
cooling to room temperature, signals that the film is highly oriented.
Annealing samples longer ($\sim$30 min) helps produce a smooth surface, Fig.
2(d). However, it should be noted that longer annealing times often cause Te
to re-evaporate, resulting in Te-deficient films or non-stoichiometric
samples, depending on the sticking coefficient of Te on different substrates
at high temperatures [100, 101].

The morphology of the resulting films at optimal substrate temperatures
demonstrated characteristic triangular-shaped terraces and steps, see Figs.
2(e)–(f). The surface morphology of pure Bi2Te3 films typically exhibits this
type of spiral-like triangular domains [77, 102, 103, 87, 75]. Such triangular
islands highlight the fcc stacking for the rhombohedral crystal structure of
Bi2Te3 [99, 46]. The lateral size of these triangular mounds is $\sim$1.5 µm
for films grown on both sapphire and STO. The adjacent terraces were separated
by an average step height of $\sim$1 nm, consistent with the thickness of a
single QL in the [001] direction of Bi2Te3, see right-hand panels, extracted
line profiles in Figs. 2(e) and (f) [13]. From a surface morphology point of
view on a small scale, namely the lateral size of triangular mounds and the
step height of terraces, the difference in growth on the two substrates is
insignificant, in agreement with previous reports [46, 103, 87, 75].

The formation of spiral-like structures is observed at the top of the
triangular island, see Figs. 2(e)and (f). Previous studies on binary (Bi2Te3
and Bi2Se3) and (off-) stoichiometric ternary Bi-based TI thin films have
shown the formation of spirals for samples grown on a variety of substrates
[104, 46, 97, 75]. At first glance, the spirals’ growth behavior in Bi-based
TI systems might be considered to come from misfit dislocations at the
interface during epitaxial growth, as in conventional semiconductor systems
[105, 106, 107], where the spirals are often considered to come from misfit
dislocations at the interface during heteroepitaxial growth. When the
dislocations have a screw component, they can emerge at the surface to form
additional steps to facilitate the growth of spirals. However, spiral growth
without dislocations in Bi-based TI systems has been demonstrated [108, 109],
where the adjacent QLs are only weakly bonded by VdW forces, and the epilayer
is not expected to bond strongly to any substrate, i.e., no formation of
dislocations is anticipated to seed the spiral growth. Furthermore, Wang et
al. [89] have reported the absence of spirals on the terraces of Bi2Te3 films.
Thus, contrary to dislocation-mediated spiral growth, the observed spirals in
Bi2Te3 thin films most likely form as a result of the pinning of the 2D growth
fronts at the substrate steps or domain boundaries during the initial and
intermediate stages of growth, respectively [108].

2.2 Correlation between Surface Morphology and Structural Properties

###

Different studies have investigated the correlation between surface morphology
and structural defects of epitaxial Bi2Te3 films grown on different
substrates, which demonstrated that such defects depend strongly on the
substrate material and growth conditions [110, 111, 112, 113].

By performing a combined bulk- and surface-sensitive study of the epitaxial
Bi2Te3 films grown on (111)-oriented BaF2, Kriegner et al. [111] resolved
buried TDs not visible on the surface of the samples. Depending on the growth
temperature of Bi2Te3 films, the orientation of the triangle mounds varied.
Triangular pyramidal features with edges in two different orientations rotated
in the surface plane by 60° were observed, see Figs. 3(a) and (b). The
dissimilarities were attributed to the different growth temperatures, which
also resulted in clear differences to the structural properties of samples
grown at different $T_{\text{sub}}$, see Fig. 3(c). For the sample grown at
$T_{\text{sub}}=280$°C, a predominantly threefold symmetry was observed,
highlighting an almost complete absence of any twinning, in agreement with the
surface morphology with only one orientation of the surface pyramids. Whereas
for samples grown at $T_{\text{sub}}=320$°C, a strong twinning with almost
equal abundance of the ordinary domain (OD) and twin domain (TD) regions was
observed in the film. This observation was also in agreement with combined
scanning electron microscopy (SEM) and electron backscatter diffraction
crystal orientation data, see Figs. 3(d) and (f), which further confirmed that
the surface topography is indeed correlated with the local crystal orientation
of the thin film.

Figure 3:  Surface morphology and structural properties correlations for
200-nm-thick Bi2Te3 films grown on (111)-oriented BaF2 substrates. Surface
morphology of two Bi2Te3 samples grown at different substrate temperatures:
(a) $T_{\text{sub}}=280$°C and (b) $T_{\text{sub}}=320$°C. The lattice plane
indices of the preferred surface step orientation are indicated in (a). Both
samples show characteristic triangular-shaped pyramids on the surface, but the
sample grown at $T_{\text{sub}}=320$°C demonstrates pyramids with two
orientations rotated by 60/180°. Dashed lines in (b) mark the boundary between
areas with the two distinct orientations. The side steps of the pyramids are
oriented along $(1\bar{1}0.0)$ lattice planes as indicated in (a). (c) X-ray
diffraction azimuth scans highlighting a threefold symmetry according to the
crystal structure for the sample grown at $T_{\text{sub}}=280$°C (red),
whereas the sample grown at $T_{\text{sub}}=320$°C (black) shows a sixfold
symmetry. The origin of the Bragg peaks from the ordinary domains (OD) and
twin domains (TD) is indicated. (d) SEM image of the sample in (b) overlaid
with a semi-transparent representation of the electron backscatter diffraction
crystal orientation data. In agreement with XRD data shown in (c), only two
types of orientations, i.e. the two twin orientations, are observed. The TD
boundaries are correlated with the surface structure as shown in enlarged SEM
images (e) and (f). Arrows indicate the orientation of the triangular-shaped
pyramids. All areas shown in yellow show the triangular pyramids pointing to
the left, whereas the areas shown in violet point to the right. Reprinted from
[111], © 2017 International Union of Crystallography.

On the other hand, it is also worth mentioning some beneficial effects of the
formation of special defects in epitaxial Bi2Te3 films. As recently indicated
for Bi-rich Bi2Te3 films grown on sapphire substrates, it was observed that,
when the Bi excess is sufficiently high, $n$-type Bi2 planar defects form in
the VdW gap, which were clearly identified in the cross-sectional structural
characteristics of such samples, see Figs. 4(a)–(c). As these Bi2 defects are
excellent electron donors, enhancement of the electron density by over one
order of magnitude was observed, accompanied by the suppression of carrier-
intrinsic excitations. It was also established that such Bi-rich Bi2Te3 films
with Bi2 planar defects possess a much improved TE power compared to that of
the similar Bi2Te3 films without Bi2 intercalations [114].

Figure 4:  Scanning transmission electron microscope (STEM) characterizations
of Bi2Te3 films grown on different substrates under distinct growth
conditions. (a)–(c) STEM images showing the cross-sectional structures for
Bi2Te3 films grown at $T_{\text{sub}}=270$°C on an Al2O3 $(00l)$ substrate:
(a) Low-magnification TEM image of a Bi2Te3 film, the white dotted boxes
indicate the local distorted regions; (b) and (c) show the atomic-resolution
STEM images in the marked areas. The inset in (c) shows the corresponding fast
Fourier transformation image from the sequential stack of Bi2-Bi2Te3. Overview
STEM images of Bi2Te3 films grown using a two-step temperature growth on (d)
Al3O3 [0001] and (g) (111)-oriented STO substrates. Magnified STEM images of
the samples showing: (e,h) QLs of the films and (f,i) a smooth interface
between the films and the substrates. The bottom right-hand inset in (e)
displays a schematic atomic structural model showing the sequence of Te and Bi
atoms in a quintuple layer of Bi2Te3. (a)–(c) Reprinted from [114] with
permission of AIP Publishing. (d)–(f) Reprinted and partly adapted with
permission from [40] © 2018, American Physical Society.

By using an extremely low growth rate of 2.7 nm h-1, Kampmeier et al. [112]
managed to suppress previously reported TDs in epitaxial Bi2Te3 films grown on
(111)-oriented Si substrates [115]. The deposited layer followed the threefold
symmetry of the substrate surface, and high-quality single crystalline and TD-
free Bi2Te3 films were prepared [112]. This finding of achieving TD-free films
corroborates our microstructural analysis using scanning transmission electron
microscopy (STEM) for Bi2Te3 films grown on sapphire [40] and STO substrates.
No noticeable domain boundaries, propagating through the complete films, were
resolved, see Figs. 4(d)–(i). For both samples, no misfit dislocations
generating edge-type defects or threading dislocations extending vertically
through the film were observed, see Figs. 4(d) and (g). Instead, highly
parallel QLs are clearly visible, which are separated by VdW gaps, see Figs.
4(e) and (h). A closer view of the interfacial region between the film and the
substrates reveals highly parallel layers in the film close to the interface,
despite the large lattice mismatch between Bi2Te3 [001] and these substrates
($\sim$8.7% for sapphire and $-10.8\%$ for STO [111]) [46]. This suggests a
high crystal quality at the film–substrate interface, see Figs. 4(f) and (i).
Moreover, contrary to a previous report on Bi2Te3 films grown on Si substrates
[115], no dislocations and distortions of the atomic column were observed
immediately above the substrate, confirming a rapid relaxation of strain at
the interface.

Finally, it is also necessary to note that, in the practice of preparing
stoichiometric Bi2Te3 films with better transport characteristics, it is
important to ascertain that the correct phase has been grown. There are
reports on Bi2Te3 films grown on BaF2 substrates indicating that using small
Te/Bi beam equivalent pressure ratios during growth led to the formation of
bismuth bilayers between QLs [103, 88, 81]. This resulted in the formation of
either BiTe or Bi4Te3 phases in addition to or in place of the desired Bi2Te3
phase, and neither of these phases show evidence of the TI surface state.
Furthermore, samples with such extra phases have considerably lower electron
mobilities and higher carrier concentrations than pure Bi2Te3 films. Thus, the
ability to control the Bi2Te3 film growth is essential for obtaining high-
quality films with low defect density and high electron mobility in order to
explore their transport properties for the electrons in the surface states, as
discussed in the next section.

## 3\. Quantum Transport in Bi2Te3 thin films

###

As most of the fascinating applications of TI materials are for electronic
devices, a fundamental understanding of their electronic transport properties
is needed before they can be implemented in novel electronic devices. For
electronic transport characterizations of most 3D TIs, especially for Bi2Te3
samples, bulk conduction complicates the direct exploitation of the surface
effect. A very promising approach is to improve the crystal quality by
carefully controlling the growth conditions of the films. This allows us to
obtain samples with fewer intrinsic defects and high surface mobilities [20,
34, 35, 70]. Moreover, thin films allow the sample thickness to be controlled
accurately in order to investigate the thickness-dependent transport
properties, thus achieving the thickness range for observing a crossover from
a 3D TI to a 2D TI, in which the films are in the quantum-spin Hall (QSH)
transport regime [116, 117]. Furthermore, (i) the possibility of preparing in
situ films and capping them immediately after growth to avoid extrinsic
defects, (ii) the ability of interfacing films with other materials to make TI
heterostructures, and (iii) the flexibility of preparing several electronic
devices and multilayers on the same film render thin films much more
advantageous than bulk single crystals for quantum transport studies. MBE-
grown thin films are also especially advantageous compared to exfoliated films
because they offer the ability to dope the materials in a controllable manner,
which led, for example, to the observations of the quantum anomalous Hall
effect in magnetic TIs [118, 119, 120].

From a quantum-transport perspective, studying transport properties in an
externally applied magnetic field is a powerful tool for investigating the
topological nature of the surface state. From magnetotransport experiments,
transport characteristics such as the carrier density and mobility in the film
can be extracted. Moreover, interesting phenomena such as nonlinear ordinary
Hall effect [40, 121], planar Hall effect [122], weak antilocalization [123,
124] and Shubnikov–de Haas (SdH) oscillations [125, 84] have been reported in
magnetotransport experiments on Bi2Te3 thin films. In addition, micro- and
nano-structured Bi2Te3 thin films demonstrated gate-tunable changes in
transport properties, the full depletion of bulk carriers, as well as the
successful realization of combined gate-tunable Josephson supercurrents and
normal-state transport of Josephson junctions and Hall bar devices fabricated
side-by-side [21, 126].

3.1 Ordinary Hall Effect and Planar Hall Effect

###

The ordinary Hall effect is the development of a transverse electric field in
a solid material when it carries an electric current. This requires a magnetic
field to be perpendicular to both the electric current direction and the
sample plane. Experimentally, the nonlinear ordinary Hall effect is commonly
measured in transport experiments on 3D TI films at low temperatures [127,
128, 40, 129]. In particular, the nonlinearity in the ordinary Hall signal
$R_{xy}(B)$ has been measured in Bi2Te3 TI films, Bi2Te3-based
heterostructures, thin films of its prototypical material Bi2Se3, as well as
in hybrid Bi2Se3/Bi2Te3 heterostructures [40, 130, 121, 131]. This
nonlinearity in the $R_{xy}(B)$ signal has been suggested to originate from
the coexistence of bulk and surface transport channels. If all carriers
participating in transport had the same mobility, $R_{xy}(B)$ would show
linear behavior with the slope determined by $1/(eR_{\textrm{H}})$, where
$R_{\textrm{H}}$ is the Hall coefficient. However, when there are multiple
types of carriers with different but comparable electronic mobilities,
nonlinearity is observed in the $R_{xy}(B)$ signal, see Fig. 5(a). For
electronic characterizations of TIs, these ordinary Hall measurements also
provide an opportunity to get a much clearer picture of the carrier density
and mobility in the films than those determined, for example, from ARPES
analysis. Table 1 provides a representative overview of reported electrical
properties from transport measurements on Bi2Te3 films grown on different
substrates [81, 21, 40, 112, 123, 126, 89, 132, 131].

To determine an accurate contribution of the surface states to the overall
electronic transport properties, it is crucial to understand the behavior of
the bulk of the sample. This was emphasized recently in a detailed analysis of
the carrier densities and electronic mobilities extracted from the $R_{xy}(B)$
signal of low-defect-density Bi2Te3 films: High-mobility bulk electrons were
identified to contribute significantly to the overall magnetotransport
properties [40]. Ideally, thinner samples minimize the contribution of bulk
modes in electronic transport by decreasing the total number of dopant
charges, thus indicating that the surface transport through the topological
surface states will play an increasingly predominant role when the film
thickness is decreased. However, structural defects such as intrinsic point
defects could have strong negative effects on the overall measured transport
properties of the film. M. Zhang et al. [132] observed a monotonous decrease
of $n_{3\textrm{D}}$ extracted from the ordinary Hall signal when increasing
the thickness of Bi2Te3 films, leading to an obvious increase in resistivity
and mobility, see Figs. 5(b) and (c). These behaviors were attributed to in
situ transformation of point defects, which highlights the particular role
played by point defects for modulating and controlling the electronic
transport properties of Bi2Te3 films.

Figure 5:  Ordinary and planar Hall effects in Bi2Te3 thin films. (a)
Nonlinear ordinary Hall effect in a 70-nm Bi2Te3 film grown on sapphire
$(000l)$ substrates using a two-step temperature growth: The Hall resistance
versus magnetic field at 2 K plotted together with a one-carrier model fitting
to highlight the presence of multiple bands carrier transport. (b)
Temperature-dependent electron concentration extracted from the ordinary Hall
signal and (c) electrical resistivity for Bi2Te3 films grown at 245°C with
different thicknesses. The inset in (c) depicts the temperature-dependent
carrier mobility of the films. (d) Planar Hall transverse resistivity
($\rho_{xy}$) versus azimuthal angle ($\phi$) for 14-nm-thick films of Bi2Te3
grown on Si and sapphire substrates plotted together with a $\sin 2\phi$ fit.
The inset depicts the schematic illustration of the measurement geometry. (e)
$\rho_{xy}$ versus $\phi$ at different temperatures of a 30-nm-thick film of
Bi2Te3 grown on Si. (f) The PHE amplitude versus temperature for 30-nm-thick
films of Bi2Te3 grown on Si and sapphire substrates. (a) Reprinted and partly
adapted with permission from [40], © 2018 American Physical Society. (b)–(c)
Reprinted from [132]; and (d)–(f) from [122] with permission of AIP
Publishing.

Next, the planar Hall effect (PHE) in 3D TI materials is discussed. In the
past few years, the PHE in topological materials has evoked extensive
interest. Contrary to the ordinary Hall effect, the PHE manifests itself as a
detectable transverse voltage in response to a longitudinal current, under an
external in-plane applied magnetic field, in a configuration where the
transverse voltage due to the Lorentz force is zero. Interest in this effect
in topological materials has been stimulated by its role in probing
topological characteristics (PHE has been proposed to originate from surface-
state conduction). Taskin et al. [133] demonstrated that the PHE amplitude
measured in Bi2-xSbxTe3 thin films across the Fermi level exhibited local
maxima on either side of the Dirac point in the surface-state regime. Somewhat
similar observations were also reported in bulk crystals of Sn-doped
Bi1.1Sb0.9Te2S, which showed oscillating PHE characteristics only in the
topologically protected surface-state regime [134]. Although both the linear
[134, 133] and nonlinear [135, 136] PHE in TIs have been confirmed
experimentally to originate from surface states, little is known about their
microscopic mechanisms. Various microscopic mechanisms have been proposed to
explain this phenomenon theoretically [137, 138].

Table 1: Electrical transport properties at low temperatures ($<$50 K) for Bi2Te3 thin films grown on different substrates. The electron mobility $(\mu)$ as well as the sheet density $(n_{\textrm{2D}})$ and bulk density $(n_{\textrm{3D}})$ are shown wherever data were available. Details of material and substrates |  $\mu$
[cm2/Vs] |  $n_{\textrm{2D}}$
[cm-2] |  $n_{\textrm{3D}}$
[cm-3] |  Refs.
---|---|---|---|---
|  |  |  |
Bi2Te3 on Si $[111]$ |  1030 |  – |  $4.7\times 10^{19}$ |  [81]
|  |  |  |
Bi2Te3 (capped with Al2O3) on SrTiO${}_{3}\,[111]$ |  1594 |  $3.8\times 10^{12}$ |  – |  [21]
|  |  |  |
Bi2Te3 (capped with Al2O3) on Al2O${}_{3}\,[0001]$ |  1206 |  $9.5\times 10^{12}$ |  – |  [21]
|  |  |  |
Bi2Te3 on Al2O${}_{3}\,[0001]$ |  2800 |  $5.8\times 10^{13}$ |  $8.2\times 10^{18}$ |  [40]
|  |  |  |
Bi2Te3 on pre-structured Si $[111]$ |  110 |  $1.6\times 10^{14}$ |  – |  [112]
|  |  |  |
Bi2Te3 on $n$-type Si $[111]$ |  35 |  $1.2\times 10^{14}$ |  – |  [123]
|  |  |  |
Bi2Te3 on SrTiO${}_{3}\,[111]$ |  700 to 1900 |  $2.8\times 10^{13}$ |  – |  [126]
|  |  |  |
Bi2Te3 on mica |  5800 |  $2.45\times 10^{12}$ |  – |  [89]
|  |  |  |
Bi2Te3 on Al2O${}_{3}\,[0001]$ |  $\sim$1000 |  – |  $0.5\text{ to }10\times 10^{19}$ |  [132]
|  |  |  |
Bi2Te3 on Al2O${}_{3}\,[0001]$ |  $912$ |  – |  $4.17\times 10^{19}$ |  [131]
|  |  |  |

A. Bhardwaj et al. [122] recently reported linear PHE in Bi2Te3 thin films
grown on Si and sapphire substrates. As expected for the linear PHE, the
planar Hall transverse resistivity $\rho_{xy}$ varied as $\sin 2\phi$ with a
period of $\pi$, see Fig. 5(d). Contrary to the reported experimental linear
PHE in Bi2-xSbxTe3 thin films [133], the PHE amplitude was observed to
increase with increasing temperature for thicker (30 nm) films, see Figs. 5(e)
and (f). Unlike the extensive observation of the ordinary Hall effect in 3D TI
thin films, only a few publications have very recently reported the PHE in
nonmagnetic Bi-based 3D TI thin films [133, 135, 122]. In particular, the
linear PHE has not yet been reported experimentally in Bi2Se3; only nonlinear
unconventional PHE with a periodicity of $2\pi$ has been reported in Bi2Se3
films [135]. This further emphasizes the need for in-depth experimental
exploration and theoretical understanding of the PHE in thin films and
heterostructures of Bi-based 3D TIs. For a detailed comprehensive discussion
of the linear and nonlinear PHE in TIs, see [137, 138, 139].

3.2 Shubnikov–de Haas Oscillations

###

Quantum oscillations are determined by Landau quantization of the energy
levels of crystalline solids in magnetic fields due to the density of states
being periodically modulated as a function of magnetic field [140]. In
particular, the oscillations occurring in conductivity are called Shubnikov–de
Haas (SdH) oscillations. Conductivity oscillates periodically as a function of
the inverse of the magnetic field $(1/B)$. The oscillatory part of the
longitudinal conductivity is given by
$\Delta\sigma_{xx}\sim\cos\left[2\pi\left(\frac{F}{B}-\frac{1}{2}+\phi\right]\right)$,
where $F$ is the oscillation frequency and $\phi$ is the phase factor
$(0<\phi<1)$. The phase factor $\phi$ of the SdH oscillations directly
reflects the Berry phase, thus providing an opportunity to check whether the
electrons leading to these oscillations are Dirac fermions. Experimentally,
$\phi$ can be determined by carefully analyzing the so-called Landau level
(LL) fan diagram. For Dirac systems with a linear energy dispersion
$(\phi=1/2)$, a topologically non-zero Berry phase of $\pi$ is expected [141,
142], whereas for spinless fermions with a parabolic energy dispersion
$(\phi=0)$, the Berry phase is zero [140, 143]. Therefore, SdH oscillations
have been used for the quantitative characterization and careful
disentanglement of the 2D TI surface states that coexist with 3D bulk states,
particularly also because their angular dependence in tilted magnetic fields
provides information about the size, shape and dimensionality of the Fermi
surface (FS) [144, 145].

SdH oscillations have been used as a powerful probe of the nature of the
metallic state in Bi2Te3 TI samples [146, 147, 89]. Notably, the electron
mobility $\mu$ of Bi2Te3 thin films has been reported to be sufficiently high
enough to fulfill the condition $\mu B\gg 1$ for allowing SdH oscillations in
the resistance at high magnetic fields to be detected [40, 125, 89, 148, 149].
For Bi2Te3 films, reported SdH oscillations have been interpreted to originate
from surface states, from high-mobility bulk electrons, or from additional
topologically trivial states due to Rashba splitting in the transport
characteristics of Bi2Te3 FS [40, 89, 125].

Initial quantum transport experiments on Bi2Te3 single crystals and thin films
detected SdH oscillations, which were often attributed to originate from
topological surface states based on a cosine magnetic field-angle dependence
of the oscillations and the magnetic field dependence of the LLs [146, 89, 76,
150]. For example, in an early analysis of SdH oscillations for Bi2Te3 thin
films, K. Wang et al. [89] reported two-frequency SdH oscillations, which they
attributed to originate from the top and bottom surface states with different
carrier densities, see Fig. 6(a). From the LLs analysis, however, deviations
from the expected $\pi$-Berry phase were observed, which were attributed to
arise from the interference between the two oscillation frequencies.
Nevertheless, such a non-zero Berry phase could also originate from a non-
topological trivial band that crosses the Fermi energy due to band bending at
the surface, as observed for Bi1.5Sb0.5Te1.7Se1.3 samples [151]. More
specifically, this could also be due to Rashba splitting states that break the
spin degeneracy and give rise to the observed two frequencies [125].

Figure 6:  Shubnikov–de Haas (SdH) oscillations in Bi2Te3 films grown on
different substrates. (a) SdH oscillations from a 4-nm-thick film grown on a
mica substrate: The Hall resistance $R_{yx}$ at 2 K shows quantum
oscillations. The lower and upper insets depict the first derivative of
$R_{yx}$ against $1/B$ and the fast Fourier transform (FFT) of $dR_{yx}/B$,
respectively. (b) SdH oscillations for a 156-nm-thick film grown on a BaF2
(111) substrate. The oscillations were extracted from the second derivative of
$R_{xx}$ with respect to the applied magnetic field and (c) FFT curves from
the SdH oscillations at various temperatures. SdH oscillations in a 70-nm-
thick film grown on a sapphire $(000l)$ substrate: (d) The oscillatory part of
the longitudinal resistance $\Delta R_{xx}/dB$ as a function of $1/B$. (e) SdH
oscillations for different tilt angles $\theta$ plotted as a function of
$1/B_{\perp}$ with $B_{\perp}=B\cos(\theta)$. The inset shows the direction of
the applied magnetic field. (f) Temperature dependence of the normalized
longitudinal resistance, where the solid red line is the best fit to the
function $\lambda(T)/\sinh(\lambda(T))$. (a) Reprinted from [89]; (b)–(c)
reprinted from [125] with permission of AIP Publishing. (d)–(f) Reprinted from
[40], © 2018 American Physical Society.

Indeed, Holgado et al. [125] recently reported SdH oscillations in Bi2Te3
epitaxial films, and their systematic study demonstrated that such SdH
oscillations did not originate from the topological surface states, but rather
emerge from the Rashba splitting of the FS of Bi2Te3. Figure 6(b) depicts SdH
oscillations from the second derivative of longitudinal resistance as a
function of $1/B$. Two well-developed peaks with frequencies of $f_{1}=59$ T
and $f_{2}=75$ T were extracted from the corresponding fast Fourier transform
(FFT) analysis, see Fig. 6(c). There are several possible interpretations of
these two frequencies. The first is to consider that the electronic transport
through the Bi2Te3 film is composed of normal fermions from the bulk of the
sample and of Dirac fermions from topological surface states. The second is
that these two frequencies originate from the top and bottom surface states,
as concluded from early quantum transport experiments on Bi-based 3D TI
samples [146, 89, 152, 153, 154]. However, the angular dependence of the
measured oscillations in a tilted magnetic field on the same sample did not
follow the expected 2D FS state behavior. Thus, it was concluded that the
measured SdH oscillations originated from 3D bulk states. In particular, it
was indicated in this study that the Rashba effect is responsible for the two
frequencies observed in the FFT, a proposal that is in agreement with previous
reports on several systems (e.g., bulk BiTeCl, SnTe epilayer and HgTe quantum
wells) [155, 156, 157]. The origin of such Rashba splitting in Bi2Te3
epitaxial films was attributed to the intrinsic disorder present in the sample
or due to surface oxidation of the Bi2Te3 films. The latter is supported by
the fact that Bi2Te3 samples are known to experience surface chemical
modifications upon exposure to ambient conditions [40, 158]. There is also
theoretical support indicating that adsorbed atoms from the atmosphere alter
the surface states of Bi2Te3 by inducing Rashba splitting due to the charge
transfer from the adsorbed atoms to the sample surface [159].

Regarding the 2D nature of the SdH oscillations, which is often taken as
evidence of surface-state transport in 3D TIs, it should be emphasized in the
case of Bi2Te3 that SdH oscillations should be analyzed meticulously. Owing to
the cylindrical shape of the Bi2Te3 bulk FS, the angle dependence of the bulk
magnetoresistance oscillations is two-dimensional in nature, thus highlighting
the possibility of detecting 2D quantum oscillations from high-mobility bulk
electrons from such a cylindrical FS. Indeed, our recent magnetotransport
study on low-defect-density Bi2Te3 films showed that SdH oscillations arise
from the high-mobility bulk electrons, see Fig. 6(d). However, the maxima and
minima of the measured SdH oscillations were aligned with each other for
different angles when oscillations are plotted as a function of the
perpendicular component of the magnetic field ($B_{\perp}=B\cos(\theta)$), see
Fig. 6(e) [40]. A careful comparison of the Hall carrier density with the
spherical and cylindrical FS carrier densities from SdH oscillations
demonstrated that the FS has a shape between that of a spherical FS and an
open cylindrical FS, but more elongated towards a cylindrical FS. This
indicates that the measured quantum SdH oscillations originate from bulk
carriers. Figure 6(f) depicts the temperature dependence of the normalized
longitudinal resistance. By analyzing SdH oscillations at different
temperatures using the Lifshitz–Kosevich expression and the Dingle plot [140,
17, 160], a surface mobility was extracted from SdH oscillations of
$\mu_{s}=3600$ cm2 V-1s-1, which is consistent with the mobility value
extracted from the Hall data. This agreement between bulk and surface
mobilities, as determined from Hall and SdH data, respectively, further
indicates that bulk states significantly contribute to the measured SdH
oscillations in magnetotransport [40]. As discussed above for SdH oscillations
originating from the Rashba splitting effect, which are due most probably to
surface oxidation [125], and because our Bi2Te3 films were exposed to air for
days during device fabrication processes, it is reasonable to assume that
there are chemical surface modification effects that effectively dope
electrons into the bulk. This implies that electronic conductivity of the film
is no longer determined by the surface states alone. These observations
highlight that, for further ex situ electronic device fabrication steps and
subsequent quantum transport studies, Bi2Te3 films should first be capped in
situ immediately after growth, before exposure to air and other ex situ
contaminations. This will protect the surface states from degradation and
unintentional doping.

3.3 Protective Capping and Electrostatic Tuning of the Fermi Level

###

Usually, device-structuring processes and quantum-transport measurements are
not performed in vacuum systems connected to the UHV chambers used for
combined in situ TI thin-film deposition and characterizations. Therefore, it
is important to develop methods for in situ capping of TI films to avoid any
possible contamination and protect the TI surface states prior to ex situ
device fabrication steps and subsequent quantum transport experiments. An
advanced multicluster UHV system architecture with multiple deposition and
analysis chambers for in situ synthesis, capping and surface characterization
is ideal for detailed investigations before and after in situ capping of the
films. Preferably, such a multicluster UHV system should also be connectable
to a vacuum suitcase for long-distance transport of samples in UHV conditions
for further in situ analysis to other facilities such as synchrotron radiation
facilities. Hoefer et al. [70, 22] and Ngabonziza et al. [21, 23] have
independently employed variants of such multicluster UHV systems for film
growth, in situ capping and characterizations of high-quality, bulk-insulating
Bi2Te3 epitaxial thin films.

Figure 7:  Protective capping and decapping of epitaxial Bi2Te3 films. (a)
Cross section field-emission gun microscopy image of a 25-nm-thick Bi2Te3 film
grown on $(111)$-oriented BaF2 substrates and capped with a 100-nm Te layer.
(b) Illustration of the mechanical capping layer removal technique that uses
adhesive tape. Atomic force microscopy images of (c) decapped and (d) as-grown
25-nm-thick Bi2Te3 films. (e) X-ray reflectivity of a Bi2Te3 film before and
after decapping procedure. Recovering the pristine surface of 30-nm-thick
Bi2Te3 films grown on sapphire substrates after long-time storage: (f)
Illustrative image of the Te capping and decapping process, highlighting the
need for doing argon-ion etching before decapping. (g) STM image showing the
atomically resolved structure of the tellurium-terminated surface layer. The
pristine surface is recovered after argon-ion sputtering of the long-time
stored Bi2Te3 film. (h) Scanning tunneling spectroscopy spectra from the same
surface (solid line) and after decapping (dots). Electronic transport
characteristics of in situ-capped Bi2Te3 films: (i) In situ sheet resistance
measurements of the pristine and Te-capped 20-nm-thick Bi2Te3 films grown on
$(111)$-oriented BaF2 substrates; (j) Temperature dependence of the sheet
resistance for ex situ-structured Hall bar devices on three different 15-nm-
thick Bi2Te3 films grown on $(111)$-oriented STO substrates (black: capped
with Al2O3, blue: capped with Te and red: uncapped film). (a)–(e) Reprinted
from [103]. (f)–(h) Reprinted from [161]. (i) Reprinted from from [70]. (j)
Reprinted with permission from [21], © 2016 John Wiley and Sons.

For in situ capping of Bi2Te3 films, the first approach has been to identify a
suitable capping layer that would preserve the surface states and avoid
unintentional doping induced by ambient conditions. Systematic
characterizations of the surface properties of intrinsically insulating Bi2Te3
films before and after in situ capping with an epitaxially grown Te layer have
been reported [21, 70]. This epitaxially grown Te layer could be an
appropriate capping material because the surface of Bi2Te3 is naturally Te-
terminated. Moreover, it is a removable capping layer within the temperature
range of Bi2Te3 growth. An amorphous layer of Te [162, 103], Se [163], or an
evaporated layer of Al [84] have also been used as a protective capping layer
for Bi2Te3 films. However, there have been questions related to whether the
intrinsic topological properties of Bi2Te3 films are influenced by such an
amorphous capping overlayer. For example, alterations of the stoichiometry
after removal of a Se/Te capping layer have been reported [162, 163]. In
addition, an Al2O3 capping layer grown by atomic layer deposition was reported
to potentially cause damage to the Bi2Te3 surface states [164].

Nevertheless, the successful removal of Te capping layers on Bi2Te3 films has
been reported [103, 161, 70]. Fornari et al.  [103] demonstrated the
possibility to deposit a protective Te capping layer immediately after Bi2Te3
epitaxial growth and a mechanical decapping process to expose fresh surfaces
by using the adhesive tape technique, as commonly used for cleaving bulk TI
single crystals, see Figs. 7(a) and (b). Systematic surface and structural
characterizations after applying this Te decapping strategy to Bi2Te3
epitaxial films demonstrated that pristine surface and crystal structure are
totally preserved, see Figs. 7(c)–(e). Recently, Liang et al. [161] reported
the successful use of a protective Te capping layer for long-time storage and
transfer of an epitaxial Bi2Te3 film as well as a temperature-controlled
method for removing the Te capping layer, which involved argon-ion etching of
the film surface prior to annealing for decapping, see Fig. 7(f). After argon-
ion sputtering, the Bi2Te3 pristine surface is recovered at a decapping
temperature of $\sim$550 K, Figs. 7(g) and (h), suggesting that nearly perfect
TI surfaces can be reliably attained even after long-time storage through a
combination of an initial argon-ion sputtering process, followed by heating
for decapping.

Hoefer et al. [70] and Ngabonziza et al. [21] conducted systematic studies of
the influence of a Te capping layer on the electrical transport properties of
intrinsically bulk-insulating Bi2Te3 films. Comparative in situ investigations
of the sheet resistance were reported for pristine Bi2Te3 films and for the
film fully capped with a Te layer. After the Te capping step, the sheet
resistance decreased gradually, indicating the contribution of the Te capping
layer to the measured electronic conductivity, see Fig. 7(i) [70].
Temperature-dependent sheet resistance measurements on in situ Te-capped
Bi2Te3 films, which were structured ex situ into Hall bar devices, also
indicated a small contribution of the thin Te protective cap to the transport
properties of the Bi2Te3 [21]. Therefore, for systematic and in-depth quantum
transport investigations, an insulating capping layer would be preferable as
it would help avoid a possible contribution of conduction from the capping
material in measured TI electronic transport characteristics. For example, the
Al2O3-capped Bi2Te3 film was reported to show the highest sheet resistance
compared to uncapped and in situ Te-capped Bi2Te3 films, Fig. 7(j). This
suggests that the insulating Al2O3 capping layer would be more appropriate for
the direct exploitation of the surface states in quantum transport experiments
[21]. Furthermore, such an insulating protective capping layer offers the
opportunity to achieve a dual-gating configuration in thin-film devices
fabricated from in situ-capped Bi2Te3 samples for efficient electrostatic
tuning of the Fermi level.

Figure 8:  Electronic transport characteristics of in situ-capped 15-nm-thick
Bi2Te3 films grown on a (111)-oriented STO substrate. (a) Top panel: Optical
micrograph image of the structured Hall bar device. Bottom panel: Schematic
representation of a typical in situ-capped Bi2Te3 sample, where the total
thickness of the A2O${}_{\text{x}}$ capping layer is $\sim$6 nm. (b) Back-gate
voltage dependence of the Hall resistance, $R_{xy}$ at 100 mK. (c) 2D carrier
density extracted from the Hall resistance zero-field slope of $R_{xy}$ in (b)
and the sheet resistance as a function of the applied back-gate voltage
$V_{\text{bg}}$. (d) Schematic band diagram illustration in which BCB is the
bulk conduction band, BVB is the bulk valence band, and $E_{\text{F}}$ is the
position of the Fermi level. The transport regimes I, II, and III are accessed
in (c) as the Fermi level moves through the respective shaded regions of the
band structure. (e) Back-gate dependence of the weak antilocalization effect,
$\Delta G_{xx}(B)$, for in situ Al2O3-capped Bi2Te3 sample. (f) Extracted pre-
factor $\alpha$ and dephasing length $\l_{\phi}$ at different back-gate
voltages. Reprinted and partly adapted with permission from [21], © 2016 John
Wiley and Sons.

We have demonstrated the effectiveness of combining in situ capping using an
insulating protective cap layer and electrostatic tuning of the Fermi level in
Bi2Te3 thin-film devices [21]. This combined capping and electrostatic gating
strategy provides a technological platform to investigate the topological
properties of intrinsically insulating Bi2Te3 surface states in quantum
transport. This paves the way towards realizing a variety of Bi2Te3-based
topological quantum devices. Figure 8(a) depicts a representative structured
Bi2Te3 Hall bar device and a schematic illustration of a typical in situ-
capped Bi2Te3 film grown on a (111)-oriented STO substrate, where the
substrate is used as the back-gate dielectric. Magnetotransport properties of
Al2O3-capped Bi2Te3 films were modulated by applying back-gate voltages in the
range of $\pm 200$ V. For example, the slope of the Hall resistance ($R_{xy}$)
at different back-gate voltages ($V_{\text{bg}}$) was observed to increase
with decreasing $V_{\text{bg}}$. Distinct nonlinearity in $R_{xy}(B)$ was
observed for $V_{\text{bg}}\leq 50$ V, see Fig. 8(b). Qualitatively similar
gate-tunable magnetotransport characteristics were reported in different Bi-
based TI films grown on (111)-oriented STO substrates [91, 165, 166, 167]. In
the analysis of the response of sheet carrier density and sheet resistance
with respect to $V_{\text{bg}}$, three different gate voltage regions were
identified with distinct characteristics, see Fig. 8(c). At high positive gate
voltages (region I), the Fermi level is located in the bulk conduction band.
Observations of a maximum in the sheet resistance at $V_{\text{bg}}\sim 10$ V,
which is followed by a downturn and saturation at negative gate voltages, were
interpreted as an increase in interband scattering when the Fermi level
touches the bulk valence band [21]. This is because the Dirac point of Bi2Te3
is buried below the bulk valence band maximum. Thus the resistance maximum is
not correlated with the depletion of the surface states at the Dirac point. In
the gate voltage range of $\pm 200$ V, the 2D carrier density was modulated by
a factor of $\sim$7, and the observed low carrier concentration in the
intermediate region was considered to be a strong indication that the Fermi
level is located in the bulk band gap, crossing only the topologically
nontrivial metallic surface states, see Fig. 8(d). At high negative gate
voltages, the carrier density starts to increase slowly again as the bulk
valence band contributes mobile carriers. These results demonstrate the
possibility of achieving full depletion of bulk carriers in the Al2O3-capped
Bi2Te3 films, thus allowing access to the topological transport regime
dominated by surface-state conduction.

Weak antilocalization (WAL) measurements at different gate voltages further
support the observed carrier distribution in Al2O3-capped Bi2Te3 films [21].
WAL is a quantum effect that is commonly observed in TI systems due to the
spin-momentum locking resulting from strong spin-orbit coupling [168]. The
$\pi$-Berry phase of the topological surface states leads to destructive
interference between time-reversed charge carrier paths, which results in an
increase of measured conductivity at zero magnetic field. The quantum
correction to the 2D conductivity due to the WAL effect was calculated by
Hikami, Larkin, and Nagaoka (HLN) [169]: $\Delta G(B)=\alpha\frac{e^{2}}{\pi
h}\left[\Psi\left(\frac{B_{\varphi}}{B}+\frac{1}{2}\right)-\ln\left(\frac{B_{\varphi}}{B}\right)\right].$
The pre-factor $\alpha$ accounts for the WAL contributions from different
conducting channels. From the HLN equation, the pre-factor $\alpha$ is
expected to be $-1/2$ for a single 2D conducting surface with strong spin-
orbit coupling. If there are two independent 2D conducting channels, the pre-
factors add up. Contributions from either a conducting bulk or disorder-
induced electron–electron interactions in the system will cause the value of
$\alpha$ to deviate from $-1/2$ towards more positive values [170, 171]. An
effective dephasing length $l_{\varphi}$ will replace the phase relaxation
length of the individual channel [172, 173]. For elaborated theoretical and
experimental discussions of the WAL effect in TI systems, the reader is
referred to [174, 175, 173, 167, 170].

In capped Bi2Te3 and other Bi-based TI thin films, the WAL effect has been
investigated at different gate voltages [21, 176, 177]. Figure 8(e) depicts
representative sheet conductance $\Delta G_{xx}(B)$ as a function of magnetic
field at various $V_{\text{bg}}$ for Al2O3-capped Bi2Te3 films [21]. These
$\Delta G_{xx}$ data show WAL behavior. The extracted pre-factor $\alpha$ and
dephasing length $l_{\varphi}$ show modulations for a wide range of applied
$V_{\text{bg}}$, see Fig. 8(f). At high positive $V_{\text{bg}}$ (region I),
the value of $\alpha$ is $\sim-0.5$, and it drops to $\sim-1$ in the
intermediate regime (region II). It then saturates at a value slightly lower
than $-1$ for negative back-gate voltages (region III). The same trend was
observed for the $V_{\text{bg}}$ dependence of the dephasing length. These WAL
data on Al2O3-capped Bi2Te3 thin-film devices further demonstrate the
possibility of using a back-gate to tune the Fermi level all the way from the
bulk conduction band to the topological surface states, and then to the bulk
valence band. The ability to prepare Bi2Te3 thin films with low intrinsic
doping, paired with protective capping of the topological surface and
effective depletion of bulk carriers by back gating, is an important step
toward implementing a variety of topological Bi2Te3-based quantum devices.

3.4 Proximity-Induced Superconductivity in Bi2Te3 Thin-Film Devices

###

Next, recent progress in quantum transport studies on hybrid structures,
consisting of Bi2Te3 thin films interfaced to an $s$-wave superconductor in
topological Josephson junction devices, is discussed. The field of topological
superconductivity has drawn considerable interest in the condensed-matter
research community. It offers the unique prospect of combining properties of
band-structure topology with superconducting pairing to generate exotic
topological states, for example Majorana bound states (MBS) [3, 178, 179]. MBS
in condensed-matter systems could potentially be used as topological qubits to
perform fault-tolerant computations because they obey non-abelian statistics
[180, 181]. Recently, signatures of MBS have been experimentally observed in
various materials [182, 183, 184, 185, 186, 187]. Common to all these
experiments is that superconducting correlations are induced by bringing the
material in contact with an ordinary $s$-wave superconductor. For an
elaborated introduction to topological superconductivity and Majorana fermions
in condensed-matter physics, the interested reader can consult available
references on this topic [188, 189, 190, 191].

Figure 9:  Electronic transport characteristics in Nb$/$Bi2Te${}_{3}/$Nb
Josephson junction devices. (a) SEM image of a typical Josephson junction
device. The device dimensions are clearly visible by residues of cross-linked
photoresist from the etch mask. (b) The I–V characteristic of a Josephson
device with a 6-nm-thick weak link, and (c) the measured voltage plateaus
appear in the V–I curves under microwave irradiation with a frequency of 6.02
GHz, as well as (d) the corresponding color-scale plot of the differential
resistance $(dV/dI)$ as a function of the DC and rf current drives $I$ and
$I_{\textrm{rf}}$, respectively (at zero back-gate voltage). The black region
in the color-scale maps corresponds to the voltage plateaus in the V–I curves,
see panel (c). (e) Back-gate voltage dependence of the normal state resistance
and the Josephson critical current in the same Josephson device with a 6-nm-
thick weak link. (f) Experimental carrier density and modeling of the carrier
distribution in a 15-nm-thick Bi2Te3 film: total carrier density as a function
of gate voltage from Hall measurements (black squares) and from band structure
modeling (red filled circles). (a)–(c) and (e)–(f) Reprinted from [126].

A promising approach in the search for MBS is to use 3D TI-based Josephson
junctions. Early on, theoretical proposals were put forward to use a network
of Josephson junctions on the surface states of a 3D TI in order to realize an
artificial topological superconductor that resembles a spinless chiral
$p_{x}+ip_{y}$ superconductor without breaking time-reversal symmetry [3].
Subsequent theoretical work demonstrated that this proximity-induced
superconductivity has an orbital term of the $p+s$-wave type, in which the
$p$-wave part is composed of conventional chirals $p_{x}+ip_{y}$ and
$p_{x}-ip_{y}$ that add to the $s$-wave term to form the total superconducting
order parameter [192]. The first observations of Josephson supercurrent in 3D
TI were reported in devices fabricated from Bi-based single crystals [193,
146]. In these single crystals, most studies demonstrated in the Josephson
transport that a large fraction of the supercurrent is carried by ballistic
Andreev bound states, which was attributed to the topological surface state
[194, 146]. However, no other signatures were observed, and, up to now, the
helical metallic surface states of Bi-based 3D TIs have not unambiguously
demonstrated evidence of unconventional Josephson effects related to Majorana
physics in topological Josephson junction devices.

Proximity-induced superconductivity in Josephson junction devices fabricated
on thin films of Bi2Te3 and other Bi-based TIs has also been reported recently
[195, 196, 126, 197, 198, 199, 200, 201]. Advantages of using thin films over
single crystals include their design flexibility that enable the fabrication
of several Josephson junction devices alongside numerous Hall bar structures
on the same sample. This provides the opportunity to obtain a coherent
quantum-transport picture of the sample that includes the Josephson
supercurrent transport and the transport carrier densities from
magnetotransport, all of which are extracted from the same thin film. Here,
recent experimentally observed induced superconductivity in topological
Josephson junctions fabricated on Bi2Te3 thin films are reviewed. Lastly,
potential future quantum-transport experimental exploration of Bi2Te3-based
film devices is proposed.

To study the proximity effect in the bulk and the contribution of the bottom
surface to the supercurrent, we recently reported proximity-induced
superconductivity in Josephson Nb/Bi2Te3/Nb devices fabricated from thin (6
and 15 nm) Bi2Te3 films with small, but finite residual doping, see Fig. 9(a).
The films were deposited on (111)-oriented STO substrates, which were used as
the back-gate dielectric for combined gate-tunable studies of the Josephson
supercurrent and normal state transport [126]. Hallmarks of the Josephson
effect were observed in these junction devices: namely, a current–voltage
characteristic exhibiting a clear Josephson supercurrent, see Fig. 9(b), and a
Fraunhofer-like magnetic field modulation of the critical current [126]. In
the presence of microwave radiation, current plateaus appear in the I–V
characteristics of the junction with zero differential resistivity, see Fig.
9(c). These plateaus, called Shapiro steps [202], are due to the ac Josephson
effect. A color map of the differential resistance dependence on the microwave
power and critical current is shown in Fig. 9(d), demonstrating the evolution
of the Shapiro steps. Unlike the Fraunhofer-like magnetic diffraction
patterns, Shapiro steps do not depend on the geometry of the junction.
Instead, they depend on the current-phase relation. For a pure
$\sin(\varphi/2)$ current-phase relationship contributing a $4\pi$-periodic
supercurrent [188, 203, 204], an unconventional sequence of even steps, with
missing odd steps, is expected. Conversely, conventional $2\pi$-periodic
Andreev bound states will result in Shapiro steps at voltages $V=nhf/2e$,
where $n$, $h$, $f$, and $e$ are the integer multiples of the step index, the
Planck constant, the microwave frequency, and the elementary charge,
respectively. This characteristic voltage is half the step size of the
unconventional $4\pi$-periodic bound state. From the analysis of the I–V
curves extracted from the differential resistance color-scale maps, see Fig.
9(d), the spacing between Shapiro steps show that both $\textrm{even}-n$ and
$\textrm{odd}-n$ are present at the employed microwave frequency, see Fig.
9(c).

Figure 10:  (a) Back-gate voltage dependence of junction resistance for
Nb$/$Bi2Te${}_{3}/$Nb Josephson junction devices with an 8-nm weak link of
Bi2Te3 grown on an STO substrate. The maximum resistance is located at 68 V,
considered to be the charge neutrality point. (b) $dV/dI$ versus $I$ for
similar Nb$/$Bi2Te${}_{3}/$Nb Josephson junction devices with a junction
length of 134 nm measured at different gate voltages. (c) Corresponding
temperature dependence of the critical current at gate voltages of 120, $-80$
and $-200$ V. (d) Surface morphology of 80-nm-thick film of Bi2Te3 grown on a
GaAs substrate showing a typical growth with aligned pyramidal domains. (e)
Magnetic field pattern of a typical Al/Bi2Te3/Al junction device at 20 mK,
before (cyan) and after (blue points) the thermal cycle. At the first cool
down, a conventional Fraunhofer-like Josephson diffraction pattern is
observed, but after the thermal cycle, the critical current is dramatically
reduced, and a dip at zero magnetic field appears. (f) I–V characteristics as
a function of magnetic field and temperature for Al/Bi2Te3/Al junction
devices. The dark points yield the magnetic pattern of the junction device
with a pronounced dip at zero magnetic field. (g) Colored SEM image of a
typical Al$/$Bi2Te${}_{3}/$Al junction fabricated on a similar sample as in
(d) after the second cool down (white scale bar is 200 nm). (h) Schematic
illustration of a junction device at the surface of a Bi2Te3 film to probe-
induced mixed-parity superconductivity, which illustrates the induced order
parameter and the occurrence of $\pi$-paths across the junction. The $s$
electrodes (Al) induce a mixed $s+p$-wave superconductivity at the surface of
Bi2Te3; only the $p_{x}+ip_{y}$-component is shown. (a)–(c) Reprinted from
[195], © 2021 Chinese Physical Society and IOP Publishing Ltd. (d)–(h)
Reprinted from [196].

In the same study, we also reported a comparative analysis of the gate
dependence of both the supercurrent in Josephson junctions and the carrier
density from Hall bar structures fabricated side-by-side on the same film.
This demonstrated strong correlations between the supercurrent transport and
the charge distribution in Bi2Te3 samples [126]. By using the substrate as a
gate dielectric, the carrier density was tuned electrostatically, and the
Josephson supercurrent and the normal-state transport properties were measured
in Bi2Te3 thin-film devices. Three gate-voltage ranges with distinct behaviors
were identified: one region of intermediate gate bias, where the measured
quantities change rapidly with the applied electric field, and two saturation
regions for large bias of each polarity, Fig. 9(e). Similar gate-dependence
characteristics were also observed in the carrier density obtained by Hall
effect measurements on the same film. Furthermore, excellent agreement was
demonstrated between experimental carrier densities and the calculated carrier
distribution in the thin film using a tight-binding model based on the
solutions of the self-consistent Schrödinger–Poisson equations, Fig. 9(f).
This combined study of the Josephson effect and magnetotransport clearly
demonstrates that the Josephson supercurrent in Nb/Bi2Te3/Nb thin-film devices
maps band structure properties of the thin films, thus providing a coherent
picture that incorporates band bending in the discussion of proximity effects
for 3D TI devices.

Recently, using the STO substrate as a bottom gate dielectric, Wu et al. [195]
also reported qualitatively similar gate-tunable transport measurements on
Josephson junctions of Nb/Bi2Te3/Nb devices with an 8-nm-thick Bi2Te3 film.
Both the normal-state resistance and the supercurrent of the thin-film devices
were tuned by applying a back-gate voltage in the range of $-200$ to $120$ V,
Figs. 10(a) and (b). In the $p$-type regime of the TI film, the Josephson
current at different applied gate voltages was well described by a short
ballistic junction model, Fig. 10(c), whereas in the $n$-type regime,
diffusive bulk modes emerge for $T<0.7$ K. The lack of diffusive bulk modes in
the $p$-type regime was attributed to the formation of $p$–$n$ junctions
[195].

Charpentier et al. [196] reported unconventional magnetic field patterns in
Al/Bi2Te3/Al Josephson junction devices made using flakes that were exfoliated
from Bi2Te3 films grown on (100)-oriented GaAs substrates. The nanostructured
morphology of the flakes exhibited the commonly observed, characteristic
triangular-shaped terraces of Bi2Te3 films, see Fig. 10(d). The Josephson
junction devices were realized by transferring the flakes onto SiO2/Si
substrates [196]. The peculiar result of this work is the observation of a
dramatic change in the magnetic field dependence of the Josephson effect of
devices after thermal cycling (from base to room temperature and back to base
temperature). The critical current of the junctions was reduced by more than
two orders of magnitude, and the magnetic field pattern of the junction
devices was found to have a dip at zero applied magnetic field. An inverted
Josephson magnetic field pattern was observed after thermal cycling, see Figs.
10(e) and (f). Strain-related phenomenology that tune the interquintuple layer
interactions was invoked to explain this observation. As the exfoliated Bi2Te3
flakes were transferred to SiO2/Si substrates, there is strain during thermal
cycling related to the huge difference between the thermal expansion
coefficient of Bi2Te3 and that of the SiO2/Si substrate. Indeed, plastic
deformation induced by compressive strain, resulting in buckling of the Bi2Te3
weak link in the nanogap, was observed after a second cool down, Fig. 10(g).
The observed dip at zero applied magnetic field in the unconventional magnetic
field pattern of the Josephson current was presumably attributed to the
simultaneous existence of the $0$ and $\pi$ couplings across the junction
provided by a mixed $s+p$-wave order parameter, where the $\pi$ coupling is
due to the combined effect of a sign-changing $p$-component of the order
parameter and scattering in the junction devices, Fig. 10(h).

3.5 Prospective Quantum Transport Experiments on Bi2Te3 Thin-Film Devices

###

Lastly, for this section on quantum transport, potential future experiments
for Josephson junction and Hall bar devices fabricated on Bi2Te3 thin films
are discussed. For future investigations, a promising approach to achieving
zero-energy Majorana modes in 3D TIs, particularly in Bi2Te3 TI thin films, is
to use the 1D edge states originating from thin films that are in the QSH
regime. A crossover regime from a 3D to a 2D TI phase is anticipated in the
band structure of Bi2Te3 due to hybridization effects when the sample
thickness is reduced [116, 117]. In such a 2D TI phase, the sample would be in
the 2D QSH regime with the current carried solely by the helical edge states,
Fig. 11(a). In this QSH regime, trivial non-perpendicular channels in the
junctions, which contribute gapped $2\pi$-periodic mode Andreev bound states,
would be significantly reduced, and the Majorana fermion fingerprints would
then appear strongly in ultra-thin Bi2Te3-based devices, Fig. 11(b) [205].

Figure 11:  Outlook in quantum transport for Bi2Te3-based TI nanostructures.
(a) Schematic illustration of a 2D TI with helical edge states between two
superconductors for out-of-plane applied magnetic field. Corresponding
schematic illustration of edge supercurrents in thin Bi2Te3 TI Josephson
device with reduced bulk carriers. (c) Theoretical predicted behavior of
supercurrent versus magnetic flux in an edge-dominated supercurrent regime.
(d) Preliminary signature of quantum interference of edge supercurrents in a
6-nm-thick Bi2Te3 TI Josephson device for a film grown on (111)-oriented STO
substrate. The device dimensions (width and length) are 860 and 250 nm,
respectively. (e)–(f) Dual-gating configuration in Bi2Te3-based TI
nanostructures: (e) schematics of the envisaged ultra-thin in situ-capped
Bi2Te3 samples. Using both top and bottom gates, the sample could potentially
be gated towards the QSH regime. (f) Illustrative image of a TI Josephson
device and a Hall bar structure fabricated side-by-side on the same Bi2Te3
thin film together with the measurement configuration. (a) and (c) Reprinted
with permission from [206], © 2015 American Physical Society.

As discussed in theoretical proposals [206, 207], TI Josephson junction
devices (TIJDs) fabricated on such 2D TI samples act as nanoscale
superconducting quantum interference devices (SQUIDs) in which a magnetic flux
enclosed in the interior of the 2D TI controls the interference of the
Josephson currents flowing at the opposite edges of the sample, Fig. 11(c).
Figure 11(d) depicts preliminary experimental indications of supercurrent
originating from states probably located on the edges of the Josephson
junction fabricated on a 6-nm-thick Bi2Te3 film grown on an STO substrate. For
all back-gate voltages $V_{\textrm{bg}}$, the device exhibits unconventional
Fraunhofer pattern behavior. As $V_{\textrm{bg}}$ is decreased, the critical
current decreases without changing the overall behavior. The measured shift
towards a SQUID-like interference pattern in the 6-nm TIJD suggests a
development of peaks in supercurrent density at both edges, coexisting with
residual two-dimensional current flow in the junction, Fig. 11(b). These
preliminary peculiar $I_{\textrm{c}}(B)$ characteristics provide a promising
sign towards the formation of edge-mode proximity-induced superconductivity in
thin Bi2Te3 TIJDs in the vicinity of the 2D QSH regime.

Additional systematic characterization and a complete set of measurements of
the Josephson properties of ultra-thin Bi2Te3 TIJDs are needed. The first
experimental challenge is to identify and control precisely the thickness
range under which the Bi2Te3 thin films are in the 2D QSH regime. The second
challenge is to ensure that the homogeneity of the desired ultra-thin films is
very high, so that all the regions of the sample are in the QSH regime, thus
avoiding an asymmetry in supercurrent distribution of the subsequently
fabricated TIJDs. As discussed in the next section, interest in controlling
the growth of Bi2Te3 films in the ultra-thin regime is also motivated by
recent theoretical proposals that demonstrate the feasibility of maximizing
the TE figure of merit of Bi2Te3 TI films in the few-quintuple layer regime
[208, 209, 45]. A third challenge would be to prepare such ultra-thin films on
insulating substrates with a high relative dielectric constant, which will be
used as a bottom gate in transport experiments. This challenge includes adding
an efficient top gate on the top film surface and capping the film in situ
before exposing it to ambient conditions to avoid degradation, Fig. 11(e).
Hall bar devices and Josephson junction devices should ideally be patterned
side-by-side on the same ultra-thin Bi2Te3 sample, Fig. 11(f). The dual-gate
configuration will be useful for pushing the sample towards the QSH regime,
and then performing systematic studies of both the gate-dependent carrier
distributions and Josephson properties in the devices.

## 4\. Topological states for improving the Thermoelectric properties of
ultra-thin Bi2Te3 nanostructures

Bi-based 3D TIs (e.g., Bi2Te3, Bi2Se3 and Bi1.5Sb0.5Te1.7Se1.3) and other TI
materials such as Sb2Te3 and SnTe are known to be good TE materials [45, 210,
211, 32, 212]. TE materials directly convert thermal energy into electrical
energy and vice versa, and they are considered clean energy converters [213].
For practical applications, the conversion efficiency of TE materials is often
characterized according to a dimensionless parameter, the TE figure of merit
$ZT$ with the expression $ZT=\frac{S^{2}\sigma}{\kappa}T$ [213, 32], where
$S$, $\sigma$, $\kappa$ and $T$ are the Seebeck coefficient, electrical
conductivity, thermal conductivity and absolute temperature, respectively. For
detailed reviews of the TE properties of Bi2Te3 on compound classes with both
good TE and TI properties, and also discussions of typical TE properties of
TIs with gapless boundary states, the interested reader is referred to [32,
210, 211].

In recent years, there have been several theoretical and experimental attempts
to understand the contribution of Bi2Te3 TI boundary states to
thermoelectricity [45, 211, 209, 214, 215]. This is because Bi2Te3 is one of
the second-generation 3D TIs investigated extensively, and is also the best TE
material around room temperature [32]. In particular, the focus has been on
exploring the possibility of achieving enhanced TE performance in Bi2Te3 TI
films that are in the ultra-thin regime (within the thickness range of one to
six QLs) [45, 208]. Recent progress in exploring the contribution of TI
boundary states to thermoelectricity in epitaxial Bi2Te3 TI thin films is
presented briefly in order to understand the relation between TE properties
and topological nature. Special emphasis will be placed on maximizing the TE
figure of merit $ZT$ of Bi2Te3 TI nanostructures with topologically protected
surface states.

Figure 12:  Potential of topological states for improving TE properties in
ultra-thin Bi2Te3 films. Calculated (a) electrical conductivity $\sigma$, (b)
Seebeck coefficient S, and (c) figure of merit $ZT$ of the $p$-type three-QL
Bi2Te3 film as a function of the relaxation time ratio $r_{\tau}$, where
contributions from the surface and bulk states are also determined. (d) and
(e) Calculated thickness-dependent figure of merit $ZT$ values of Bi2Te3 films
(blue and green regions indicate films with trivial and non-trivial surface
states, respectively), where in the topologically non-trivial region, the
relaxation time ratio is assumed to be (d) 100 and (e) 1000. Dashed lines
indicate the $ZT$ value of the bulk Bi2Te3 sample in its pristine form. (f)
Maximum possible figure of merit $ZT$ of 2D TIs as a function of the
scattering-time ratio $r_{\tau}$. (a)–(e) Reprinted with permission from
[208], © 2016 The Royal Society of Chemistry. (f) Reprinted with permission
from [209], © 2014 American Physical Society.

On the theoretical side, using first-principles calculations and the Boltzmann
transport theory, Liang et al. [208] reported a systematic exploration of the
effects of topologically protected surface states on the TE performance of
ultra-thin Bi2Te3 films. By tuning the relaxation time ratio $r_{\tau}$
between the bulk states and the topological surface states, the maximum $ZT$
value of Bi2Te3 TI films was optimized at a critical thickness of three QLs.
Figures 12(a), (b) and (c) depict room-temperature transport data
(conductivity $\sigma$, Seebeck coefficient $S$) and the $ZT$ value of the
$p$-type three-QL Bi2Te3 film as a function of the scattering-time ratio
$r_{\tau}$. The surface states dominate the electronic transport, Fig. 12(a).
The total $ZT$ value is shown to be dominated by the surface states with only
minor bulk contributions and increases with increasing $r_{\tau}$, Fig. 12(c).
Similar behaviors were also obtained for the $n$-type three-QL Bi2Te3 film.
Notably, the $ZT$ values of the $n$-type system could be enhanced to be as
large as those of the $p$-type system [208]. The $ZT$ values of ultra-thin
$p$-type and $n$-type Bi2Te3 films were also demonstrated to show distinct
non-monotonous dependence on the film thickness with a maximum value at three
QL when the system enters the non-trivial topological region from the trivial
regime, Figs. 12(d) and (e). Independently, Xu et al. [209] also explored the
influence of $r_{\tau}$ on TE performance for a 2D TI by neglecting the
lattice thermal conductance. The upper limit of the TE figure of merit
$ZT_{\text{max}}$ was found to increase monotonically with increasing
$r_{\tau}$, Fig. 12(f). In the same work, the contribution of lattice thermal
conductance was also explored by considering a finite mean-free path of
phonons, and the mean-free path ratio between phonons and boundary electrons
was minimized by introducing defects or disorders into the system, thus
improving TE performance. A recent theoretical work also proposed that tuning
the Dirac point, for example by introducing defects or changing surface
chemistry, is probably an efficient way to retain the enhancement of TE
properties from quantum confinement in TI thin films [216].

These theoretical works clearly demonstrate the possibility of achieving
enhanced TE performance by utilizing TI boundary states in TI films under
certain conditions, taking into account the hybridization effects in boundary
states. In particular, they demonstrate the feasibility of maximizing the TE
figure of merit for Bi2Te3 films with topological surface states in the few-QL
regime by increasing the relaxation time ratio between the surface and bulk
states and by introducing disorders and impurities [208, 209, 45]. These
theoretically predicted, enhanced $ZT$ values for films that are in the few-QL
regime have rarely been reported experimentally for epitaxial Bi2Te3 TI thin
films. This is mainly because it is challenging for MBE-grown films to become
homogeneous ultra-thin ($\leq 3$ QLs) and high-quality epitaxial TI films, in
which all regions of the film are in the same few-quintuple layer regime.
Further experimental efforts are needed to assess the underlying relationship
between TE performance and the topological nature of Bi2Te3 thin films in the
few-QL regime and to explore the advantages of TI boundary states for
improving Bi2Te3 outstanding TE properties.

In summary, recent progress in epitaxial film growth, magnetotransport, and
proximity-induced superconductivity in Bi2Te3 TI films was presented as well
as the potential impact of TI boundary states for improving TE properties of
ultra-thin Bi2Te3 films. Using the MBE technique, we succeeded in preparing
high-quality, bulk-insulating Bi2Te3 TI films. These films offer the
flexibility to control the sample thickness down to a few QLs in the vicinity
of the 2D QSH regime, in addition to the possibility of preparing films in
situ and capping them immediately after growth to avoid extrinsic defects.
Recent quantum transport experiments reported interesting phenomena such as
the planar Hall effect and quantum oscillations arising from high-mobility
bulk electrons. In addition, successful realization of combined gate-tunable
Josephson supercurrent and normal-state transport in Bi2Te3 thin-film devices
have been achieved. For future quantum transport experiments, it is imperative
to use ultra-thin Bi2Te3 films in the 2D QSH regime in order to reach the
single-channel regime, which is the ideal experimental configuration to
explore Majorana physics this system. Several theoretical works have discussed
the inherent connections between thermoelectric properties and topological
states in Bi2Te3 films. However, to explore more the underlying relationship
between TE performance and the topological nature, further experimental
efforts are needed. In conclusion, clearly tremendous opportunities exist at
the intersection of the rapidly evolving fields of TE materials and the new
quantum materials of topological insulators. Progress in controlling the
epitaxial growth of Bi2Te3 films down to few QLs is expected to reveal the
potential of topological states for improving the TE properties of Bi2Te3,
which will help define a path towards realizing a variety of high-performance
TE devices.

##  Acknowledgments

The author thanks Alexander Brinkman and colleagues for valuable comments and
beneficial discussions, in particular Martin P. Stehno and Hiroaki Myoren,
with whom the author collaborated on the TI project. The author also
gratefully acknowledges helpful discussions with Jochen Mannhart and Thomas
Whittles, technical support by Yi Wang and editorial support by Lilli Pavka.

##  References

## References

  * Kane and Mele [2005] C. L. Kane and E. J. Mele, ${Z}_{2}$ topological order and the quantum spin hall effect, Phys. Rev. Lett. 95, 146802 (2005).
  * Fu  _et al._ [2007] L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett. 98, 106803 (2007).
  * Fu and Kane [2008] L. Fu and C. L. Kane, Superconducting proximity effect and majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100, 096407 (2008).
  * Akhmerov  _et al._ [2009] A. R. Akhmerov, J. Nilsson, and C. W. J. Beenakker, Electrically detected interferometry of majorana fermions in a topological insulator, Phys. Rev. Lett. 102, 216404 (2009).
  * Hsieh  _et al._ [2009a] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A tunable topological insulator in the spin helical dirac transport regime, Nature 460, 1101 (2009a).
  * Jozwiak  _et al._ [2013] C. Jozwiak, C.-H. Park, K. Gotlieb, C. Hwang, D.-H. Lee, S. G. Louie, J. D. Denlinger, C. R. Rotundu, R. J. Birgeneau, Z. Hussain, and A. Lanzara, Photoelectron spin-flipping and texture manipulation in a topological insulator, Nature Phys. 9, 293 (2013).
  * Appelbaum  _et al._ [2011] I. Appelbaum, H. D. Drew, and M. S. Fuhrer, Proposal for a topological plasmon spin rectifier, Appl. Phys. Lett. 98, 023103 (2011).
  * He  _et al._ [2019a] M. He, H. Sun, and Q. L. He, Topological insulator: Spintronics and quantum computations, Front. Phys. 14, 43401 (2019a).
  * Bernevig and Zhang [2006] B. A. Bernevig and S.-C. Zhang, Quantum spin hall effect, Phys. Rev. Lett. 96, 106802 (2006).
  * Bernevig  _et al._ [2006] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin hall effect and topological phase transition in hgte quantum wells, Science 314, 1757 (2006).
  * Hsieh  _et al._ [2008] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, A topological dirac insulator in a quantum spin hall phase, Nature 452, 970 (2008).
  * Hsieh  _et al._ [2009b] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of unconventional quantum spin textures in topological insulators, Science 323, 919 (2009b).
  * Zhang  _et al._ [2009] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in bi2se3, bi2te3 and sb2te3 with a single dirac cone on the surface, Nature Phys. 5, 438 (2009).
  * Xia  _et al._ [2009] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of a large-gap topological-insulator class with a single dirac cone on the surface, Nature Phys. 5, 398 (2009).
  * Chen  _et al._ [2009] Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Experimental realization of a three-dimensional topological insulator, bi2te3, Science 325, 178 (2009).
  * Qi  _et al._ [2009] X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Inducing a magnetic monopole with topological surface states, Science 323, 1184 (2009).
  * Ando [2013] Y. Ando, Topological insulator materials, J. Phys. Soc. Jpn. 82, 102001 (2013), and references therein .
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010), and references therein .
  * Alpichshev  _et al._ [2010] Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, Stm imaging of electronic waves on the surface of ${\mathrm{bi}}_{2}{\mathrm{te}}_{3}$: Topologically protected surface states and hexagonal warping effects, Phys. Rev. Lett. 104, 016401 (2010).
  * Li  _et al._ [2010a] Y.-Y. Li, G. Wang, X.-G. Zhu, M.-H. Liu, C. Ye, X. Chen, Y.-Y. Wang, K. He, L.-L. Wang, X.-C. Ma, H.-J. Zhang, X. Dai, Z. Fang, X.-C. Xie, Y. Liu, X.-L. Qi, J.-F. Jia, S.-C. Zhang, and Q.-K. Xue, Intrinsic topological insulator bi2te3 thin films on si and their thickness limit, Adv. Mater. 22, 4002 (2010a).
  * Ngabonziza  _et al._ [2016] P. Ngabonziza, M. P. Stehno, H. Myoren, V. A. Neumann, G. Koster, and A. Brinkman, Gate-tunable transport properties of in situ capped bi2te3 topological insulator thin films, Adv. Electron. Mater. 2, 1600157 (2016).
  * Pereira  _et al._ [2021] V. M. Pereira, C.-N. Wu, K. Höfer, A. Choa, C.-A. Knight, J. Swanson, C. Becker, A. C. Komarek, A. D. Rata, S. Rößler, S. Wirth, M. Guo, M. Hong, J. Kwo, L. H. Tjeng, and S. G. Altendorf, Challenges of topological insulator research: Bi2te3 thin films and magnetic heterostructures, Phys. Status Solidi (b) 258, 2000346 (2021), and references therein .
  * Ngabonziza  _et al._ [2018a] P. Ngabonziza, M. P. Stehno, G. Koster, and A. Brinkman, In Situ Characterization Tools for Bi2Te3 Topological Insulator Nanomaterials, in  _In-situ Characterization Techniques for Nanomaterials_, edited by C. S. Kumar (Springer Berlin Heidelberg, 2018) pp. 223–250.
  * Snyder and Toberer [2008] G. J. Snyder and E. S. Toberer, Complex thermoelectric materials, Nature Mater. 7, 105 (2008).
  * Uher [2016] C. E. Uher,  _Materials Aspect of Thermoelectricity (1 st ed.)_ (Taylor & Francis Group, 2016).
  * Venkatasubramanian  _et al._ [2001] R. Venkatasubramanian, E. Siivola, T. Colpitts, and B. O’Quinn, Thin-film thermoelectric devices with high room-temperature figures of merit, Nature 413, 597 (2001).
  * Chowdhury  _et al._ [2009] I. Chowdhury, R. Prasher, K. Lofgreen, G. Chrysler, S. Narasimhan, R. Mahajan, D. Koester, R. Alley, and R. Venkatasubramanian, On-chip cooling by superlattice-based thin-film thermoelectrics, Nature Nanotechnol. 4, 235 (2009).
  * Wang  _et al._ [2018] L. Wang, Z. Zhang, Y. Liu, B. Wang, L. Fang, J. Qiu, K. Zhang, and S. Wang, Exceptional thermoelectric properties of flexible organic-inorganic hybrids with monodispersed and periodic nanophase, Nature Commun. 9, 3817 (2018).
  * Jin  _et al._ [2018] Q. Jin, S. Jiang, Y. Zhao, D. Wang, J. Qiu, D. M. Tang, J. Tan, D. M. Sun, P. X. Hou, X. Q. Chen, K. Tai, N. Gao, C. Liu, H. M. Cheng, and X. Jiang, Flexible layer-structured bi2te3 thermoelectric on a carbon nanotube scaffold, Nature Mater. 18, 62 (2018).
  * Petsagkourakis  _et al._ [2018] I. Petsagkourakis, K. Tybrandt, X. Crispin, I. Ohkubo, N. Satoh, and T. Mori, Thermoelectric materials and applications for energy harvesting power generation, Sci. Technol. Adv. Mater. 19, 836 (2018).
  * Dagdeviren  _et al._ [2017] C. Dagdeviren, Z. Li, and Z. L. Wang, Energy harvesting from the animal/human body for self-powered electronics, Annu. Rev. Biomed. Eng. 19, 85 (2017).
  * Witting  _et al._ [2019] I. T. Witting, T. C. Chasapis, F. Ricci, M. Peters, N. A. Heinz, G. Hautier, and G. J. Snyder, The thermoelectric properties of bismuth telluride, Adv. Electron. Mater. 5, 1800904 (2019), and references therein .
  * Wang  _et al._ [2011a] G. Wang, X.-G. Zhu, Y.-Y. Sun, Y.-Y. Li, T. Zhang, J. Wen, X. Chen, K. He, L.-L. Wang, X.-C. Ma, J.-F. Jia, S. B. Zhang, and Q.-K. Xue, Topological insulator thin films of bi2te3 with controlled electronic structure, Adv. Mater. 23, 2929 (2011a).
  * Ngabonziza  _et al._ [2015] P. Ngabonziza, R. Heimbuch, N. de Jong, R. A. Klaassen, M. P. Stehno, M. Snelder, A. Solmaz, S. V. Ramankutty, E. Frantzeskakis, E. van Heumen, G. Koster, M. S. Golden, H. J. W. Zandvliet, and A. Brinkman, In situ spectroscopy of intrinsic ${\mathrm{bi}}_{2}{\mathrm{te}}_{3}$ topological insulator thin films and impact of extrinsic defects, Phys. Rev. B 92, 035405 (2015).
  * Hoefer  _et al._ [2014] K. Hoefer, C. Becker, D. Rata, J. Swanson, P. Thalmeier, and L. H. Tjeng, Intrinsic conduction through topological surface states of insulating bi2te3 epitaxial thin films, Proc. Natl. Acad. Sci. 111, 14979 (2014).
  * Tang  _et al._ [2019] C. Tang, Z. Huang, J. Pei, B.-P. Zhang, P.-P. Shang, Z. Shan, Z. Zhang, H. Gu, and K. Wen, Bi2te3 single crystals with high room-temperature thermoelectric performance enhanced by manipulating point defects based on first-principles calculation, RSC Adv. 9, 14422 (2019).
  * Zhu  _et al._ [2016] T. Zhu, L. Hu, X. Zhao, and J. He, New insights into intrinsic point defects in v2vi3 thermoelectric materials, Adv. Sci. 3, 1600004 (2016).
  * Hu  _et al._ [2014] L. Hu, T. Zhu, X. Liu, and X. Zhao, Point defect engineering of high-performance bismuth-telluride-based thermoelectric materials, Adv. Funct. Mater 24, 5211 (2014).
  * Peranio  _et al._ [2013] N. Peranio, M. Winkler, M. Dürrschnabel, J. König, and O. Eibl, Assessing antisite defect and impurity concentrations in bi2te3 based thin films by high-accuracy chemical analysis, Adv. Funct. Mater 23, 4969 (2013).
  * Ngabonziza  _et al._ [2018b] P. Ngabonziza, Y. Wang, and A. Brinkman, Bulk contribution to magnetotransport properties of low-defect-density ${\mathrm{bi}}_{2}{\mathrm{te}}_{3}$ topological insulator thin films, Phys. Rev. Mater. 2, 044204 (2018b).
  * Shrestha  _et al._ [2017] K. Shrestha, M. Chou, D. Graf, H. D. Yang, B. Lorenz, and C. W. Chu, Extremely large nonsaturating magnetoresistance and ultrahigh mobility due to topological surface states in the metallic ${\mathrm{bi}}_{2}{\mathrm{te}}_{3}$ topological insulator, Phys. Rev. B 95, 195113 (2017).
  * Ghaemi  _et al._ [2010] P. Ghaemi, R. S. K. Mong, and J. E. Moore, In-plane transport and enhanced thermoelectric performance in thin films of the topological insulators ${\mathrm{bi}}_{2}{\mathrm{te}}_{3}$ and ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$, Phys. Rev. Lett. 105, 166603 (2010).
  * Tretiakov  _et al._ [2011] O. A. Tretiakov, A. Abanov, and J. Sinova, Holey topological thermoelectrics, Appl. Phys. Lett. 99, 113110 (2011).
  * Fan  _et al._ [2012] Z. Fan, J. Zheng, H.-Q. Wang, and J.-C. Zheng, Enhanced thermoelectric performance in three-dimensional superlattice of topological insulator thin films, Nanoscale Res. Lett. 7, 570 (2012).
  * Xu  _et al._ [2017] N. Xu, Y. Xu, and J. Zhu, Topological insulators for thermoelectrics, npj Quant. Mater. 2, 51 (2017), and references therein .
  * He  _et al._ [2013] L. He, X. Kou, and K. L. Wang, Review of 3d topological insulator thin-film growth by molecular beam epitaxy and potential applications, Phys. Status Solidi RRL 7, 50 (2013), and reference therein. .
  * Obara  _et al._ [2009] H. Obara, S. Higomo, M. Ohta, A. Yamamoto, K. Ueno, and T. Iida, Thermoelectric properties of bi2te3-based thin films with fine grains fabricated by pulsed laser deposition, Jpn. J. Appl. Phys. 48, 085506 (2009).
  * Zhang  _et al._ [2012a] S. Zhang, L. Yan, J. Qi, M. Zhuo, Y.-Q. Wang, R. Prasankumar, Q. Jia, and S. Picraux, Epitaxial thin films of topological insulator bi2te3 with two-dimensional weak anti-localization effect grown by pulsed laser deposition, Thin Solid Films 520, 6459 (2012a).
  * Liao  _et al._ [2019] Z. Liao, M. Brahlek, J. M. Ok, L. Nuckols, Y. Sharma, Q. Lu, Y. Zhang, and H. N. Lee, Pulsed-laser epitaxy of topological insulator bi2te3 thin films, APL Mater. 7, 041101 (2019).
  * You  _et al._ [2012] H. You, S. Hyub Baek, K.-C. Kim, O.-J. Kwon, J.-S. Kim, and C. Park, Growth and thermoelectric properties of bi2te3 films deposited by modified mocvd, J. Cryst. Growth 346, 17 (2012).
  * Cao  _et al._ [2012] H. Cao, R. Venkatasubramanian, C. Liu, J. Pierce, H. Yang, M. Zahid Hasan, Y. Wu, and Y. P. Chen, Topological insulator bi2te3 films synthesized by metal organic chemical vapor deposition, Appl. Phys. Lett. 101, 162104 (2012).
  * Kim  _et al._ [2020] K.-C. Kim, S. K. Kim, J.-S. Kim, and S.-H. Baek, Domain engineering of epitaxial (001) bi2te3 thin films by miscut gaas substrate, Acta Mater. 197, 309 (2020).
  * Ferhat  _et al._ [2000] M. Ferhat, J. C. Tedenac, and J. Nagao, Mechanisms of spiral growth in bi2te3 thin films grown by the hot-wall-epitaxy technique, J. Cryst. Growth 218, 250 (2000).
  * Takagaki  _et al._ [2012] Y. Takagaki, U. Jahn, and M. Ramsteiner, Incorporation of transition metals in the hot-wall-epitaxy growth of bi2te3and sb2te3, Semicond. Sci. Technol. 27, 085006 (2012).
  * Charar  _et al._ [2000] S. Charar, J. Tedenac, V. Potin, R. Viennois, O. Laire, C. Fau, and B. Liautard, Preparation of bi2te3 films by hot-wall epitaxy and characterization of p–n junction, Phys. Status Solidi (a) 182, 669 (2000).
  * Zimmer  _et al._ [2008] A. Zimmer, N. Stein, L. Johann, H. Terryn, and C. Boulanger, Characterizations of bismuth telluride films from mott-schottky plot and spectroscopic ellipsometry, Surf. Interface Anal. 40, 593 (2008).
  * Li  _et al._ [2008] S. Li, H. M. A. Soliman, J. Zhou, M. S. Toprak, M. Muhammed, D. Platzek, P. Ziolkowski, and E. Müller, Effects of annealing and doping on nanostructured bismuth telluride thick films, Chem. Mater. 20, 4403 (2008).
  * Zimmer  _et al._ [2007] A. Zimmer, N. Stein, L. Johann, R. Beck, and C. Boulanger, In situ analysis of bismuth telluride electrodeposition using combined spectroscopic ellipsometry and electrochemical quartz crystal microbalance, Electrochim. Acta 52, 4760 (2007).
  * Zheng  _et al._ [2012] Z. Zheng, P. Fan, T. Chen, Z. Cai, P. Liu, G. Liang, D. Zhang, and X. Cai, Optimization in fabricating bismuth telluride thin films by ion beam sputtering deposition, Thin Solid Films 520, 5245 (2012).
  * hao Zheng  _et al._ [2010] Z. hao Zheng, P. Fan, G. Liang, D. Zhang, X.-M. Cai, and C. Tian-bao, Annealing temperature influence on electrical properties of ion beam sputtered bi2te3 thin films, J. Phys. Chem. Solids 71, 1713 (2010).
  * hao Zheng  _et al._ [2014] Z. hao Zheng, P. Fan, J. Luo, X.-M. Cai, G. Liang, D. Zhang, and F. Ye, Thermoelectric properties of bismuth antimony tellurium thin films through bilayer annealing prepared by ion beam sputtering deposition, Thin Solid Films 562, 181 (2014).
  * Goto  _et al._ [2017] M. Goto, M. Sasaki, Y. Xu, T. Zhan, Y. Isoda, and Y. Shinohara, Control of p-type and n-type thermoelectric properties of bismuth telluride thin films by combinatorial sputter coating technology, Appl. Surf. Sci. 407, 405 (2017).
  * Sasaki  _et al._ [2014] Y. Sasaki, K. Takayama, and M. Takashiri, Film deposition and annealing treatment of sputtered bismuth telluride based thin films, J. Adv. Sci. 26, 23 (2014).
  * kun Cai  _et al._ [2013] Z. kun Cai, P. Fan, Z. hao Zheng, P. juan Liu, T. bao Chen, X. min Cai, J. ting Luo, G. xing Liang, and D. ping Zhang, Thermoelectric properties and micro-structure characteristics of annealed n-type bismuth telluride thin film, Appl. Surf. Sci. 280, 225 (2013).
  * Mayer  _et al._ [2021] T. Mayer, H. Werner, F. Schmid, R. Diaz-Pardo, J. Fujii, I. Vobornik, C. H. Back, M. Kronseder, and D. Bougeard, Transport properties of band engineered $p\text{$-$}n$ heterostructures of epitaxial ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}/{({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_{x})}_{2}{({\mathrm{Te}}_{1-y}{\mathrm{Se}}_{y})}_{3}$ topological insulators, Phys. Rev. Mater. 5, 014202 (2021).
  * Lanius  _et al._ [2016] M. Lanius, J. Kampmeier, C. Weyrich, S. Kölling, M. Schall, P. Schüffelgen, E. Neumann, M. Luysberg, G. Mussler, P. M. Koenraad, T. Schäpers, and D. Grützmacher, P–n junctions in ultrathin topological insulator sb2te3/bi2te3 heterostructures grown by molecular beam epitaxy, Cryst. Growth Des. 16, 2057 (2016).
  * Eschbach  _et al._ [2015] M. Eschbach, E. Młyńczak, J. Kellner, J. Kampmeier, M. Lanius, E. Neumann, C. Weyrich, M. Gehlmann, P. Gospodarič, S. Döring, G. Mussler, N. Demarina, M. Luysberg, G. Bihlmayer, T. Schäpers, L. Plucinski, S. Blügel, M. Morgenstern, C. M. Schneider, and D. Grützmacher, Realization of a vertical topological p–n junction in epitaxial sb2te3/bi2te3 heterostructures, Nature Commun. 6, 8816 (2015).
  * Chang  _et al._ [2018] S.-J. Chang, P.-Y. Chuang, C.-W. Chong, Y.-J. Chen, J.-C. Andrew Huang, P.-W. Chen, and Y.-C. Tseng, Heterostructured ferromagnet–topological insulator with dual-phase magnetic properties, RSC Adv. 8, 7785 (2018).
  * Pereira  _et al._ [2020] V. M. Pereira, S. G. Altendorf, C. E. Liu, S. C. Liao, A. C. Komarek, M. Guo, H.-J. Lin, C. T. Chen, M. Hong, J. Kwo, L. H. Tjeng, and C. N. Wu, Topological insulator interfaced with ferromagnetic insulators: $\mathrm{B}{\mathrm{i}}_{2}\mathrm{T}{\mathrm{e}}_{3}$ thin films on magnetite and iron garnets, Phys. Rev. Mater. 4, 064202 (2020).
  * Hoefer  _et al._ [2015] K. Hoefer, C. Becker, S. Wirth, and L. Hao Tjeng, Protective capping of topological surface states of intrinsically insulating bi2te3, AIP Adv. 5, 097139 (2015).
  * Wang and Xue [2013] G. Wang and Q.-K. Xue,  _Molecular beam epitaxial growth and exotic electronic structure of topological insulators,[Chap. 25, -p.587] in Molecular beam epitaxy: From Research to Mass Production, (1st ed.)_ (Elsevier, Waltham, 2013).
  * Joyce [1985] B. A. Joyce, Molecular beam epitaxy, Rep. Prog. Phys. 48, 1637 (1985).
  * Hanini [2013] M. E. Hanini,  _Molecular beam epitaxy: From Research to Mass Production, (2 nd ed.)_ (Elsevier, Waltham, 2013).
  * Sultana  _et al._ [2018] R. Sultana, G. Gurjar, P. Neha, S. Patnaik, and V. P. S. Awana, Treatment of the magneto-conductivity of bi2te3 topological insulator, J. Supercond. Nov. Magn. 31, 2287 (2018).
  * Harrison  _et al._ [2013] S. E. Harrison, S. Li, Y. Huo, B. Zhou, Y. L. Chen, and J. S. Harris, Two-step growth of high quality bi2te3 thin films on al2o3 (0001) by molecular beam epitaxy, Appl. Phys. Lett. 102, 171906 (2013).
  * Qu  _et al._ [2010] D.-X. Qu, Y. S. Hor, J. Xiong, R. J. Cava, and N. P. Ong, Quantum oscillations and hall anomaly of surface states in the topological insulator bi2te3, Science 329, 821 (2010).
  * Krumrain  _et al._ [2011] J. Krumrain, G. Mussler, S. Borisova, T. Stoica, L. Plucinski, C. Schneider, and D. Grützmacher, Mbe growth optimization of topological insulator bi2te3 films, Journal of Crystal Growth 324, 115 (2011).
  * Liu  _et al._ [2013] Y. Liu, H.-H. Wang, G. Bian, Z. Zhang, S. S. Lee, P. A. Fenter, J. Z. Tischler, H. Hong, and T.-C. Chiang, Interfacial bonding and structure of ${\mathrm{bi}}_{2}{\mathrm{te}}_{3}$ topological insulator films on si(111) determined by surface x-ray scattering, Phys. Rev. Lett. 110, 226103 (2013).
  * Koma [1992] A. Koma, New epitaxial growth method for modulated structures using van der waals interactions, Surf. Sci. 267, 29 (1992).
  * Koma [1999] A. Koma, Van der waals epitaxy for highly lattice-mismatched systems, J. Cryst. Growth 201-202, 236 (1999).
  * Fülöp  _et al._ [2014] A. Fülöp, Y. Song, S. Charpentier, P. Shi, M. Ekström, L. Galletti, R. Arpaia, T. Bauch, F. Lombardi, and S. Wang, Phase transition of bismuth telluride thin films grown by MBE, Appl. Phys. Express 7, 045503 (2014).
  * Liu  _et al._ [2010a] H. W. Liu, H. T. Yuan, N. Fukui, L. Zhang, J. F. Jia, Y. Iwasa, M. W. Chen, T. Hashizume, T. Sakurai, and Q. K. Xue, Growth of topological insulator bi2te3 ultrathin films on si(111) investigated by low-energy electron microscopy, Cryst. Growth Des. 10, 4491 (2010a).
  * Zeng  _et al._ [2013] Z. Zeng, T. A. Morgan, D. Fan, C. Li, Y. Hirono, X. Hu, Y. Zhao, J. S. Lee, J. Wang, Z. M. Wang, S. Yu, M. E. Hawkridge, M. Benamara, and G. J. Salamo, Molecular beam epitaxial growth of bi2te3 and sb2te3 topological insulators on gaas (111) substrates: a potential route to fabricate topological insulator p-n junction, AIP Adv. 3, 072112 (2013).
  * Yu  _et al._ [2012] X. Yu, L. He, M. Lang, W. Jiang, F. Xiu, Z. Liao, Y. Wang, X. Kou, P. Zhang, J. Tang, G. Huang, J. Zou, and K. L. Wang, Separation of top and bottom surface conduction in bi2te3thin films, Nanotechnology 24, 015705 (2012).
  * Liu  _et al._ [2011] X. Liu, D. J. Smith, J. Fan, Y.-H. Zhang, H. Cao, Y. P. Chen, J. Leiner, B. J. Kirby, M. Dobrowolska, and J. K. Furdyna, Structural properties of bi2te3 and bi2se3 topological insulators grown by molecular beam epitaxy on gaas(001) substrates, Appl. Phys. Lett. 99, 171903 (2011).
  * Liu  _et al._ [2012a] X. Liu, D. J. Smith, H. Cao, Y. P. Chen, J. Fan, Y.-H. Zhang, R. E. Pimpinella, M. Dobrowolska, and J. K. Furdyna, Characterization of bi2te3 and bi2se3 topological insulators grown by mbe on (001) gaas substrates, J. Vac. Sci. Technol. 30, 10.1116/1.3668082 (2012a).
  * Steiner  _et al._ [2014] H. Steiner, V. Volobuev, O. Caha, G. Bauer, G. Springholz, and V. Holý, Structure and composition of bismuth telluride topological insulators grown by molecular beam epitaxy, J. Appl. Crystallogr. 47, 1889 (2014).
  * Caha  _et al._ [2013] O. Caha, A. Dubroka, J. Humlíček, V. Holý, H. Steiner, M. Ul-Hassan, J. Sánchez-Barriga, O. Rader, T. N. Stanislavchuk, A. A. Sirenko, G. Bauer, and G. Springholz, Growth, structure, and electronic properties of epitaxial bismuth telluride topological insulator films on baf2 (111) substrates, Cryst. Growth Des. 13, 3365 (2013).
  * Wang  _et al._ [2013] K. Wang, Y. Liu, W. Wang, N. Meyer, L. H. Bao, L. He, M. R. Lang, Z. G. Chen, X. Y. Che, K. Post, J. Zou, D. N. Basov, K. L. Wang, and F. Xiu, High-quality bi2te3 thin films grown on mica substrates for potential optoelectronic applications, Appl. Phys. Lett. 103, 031605 (2013).
  * Rapacz  _et al._ [2015] R. Rapacz, K. Balin, M. Wojtyniak, and J. Szade, Morphology and local conductance of single crystalline bi2te3 thin films on mica, Nanoscale 7, 16034 (2015).
  * He  _et al._ [2012] X. He, T. Guan, X. Wang, B. Feng, P. Cheng, L. Chen, Y. Li, and K. Wu, Highly tunable electron transport in epitaxial topological insulator (bi1-xsbx)2te3 thin films, Appl. Phys. Lett. 101, 123111 (2012).
  * Biswas  _et al._ [2017] A. Biswas, C.-H. Yang, R. Ramesh, and Y. H. Jeong, Atomically flat single terminated oxide substrate surfaces, Prog. Surf. Sci. 92, 117 (2017).
  * Nono Tchiomo  _et al._ [2018] A. P. Nono Tchiomo, G. Babu-Geetha, E. Carleschi, P. Ngabonziza, and B. P. Doyle, Surface characterization of clean srtio3(100) substrates by x-ray photoelectron spectroscopy, Surf. Sci. Spectra 25, 024001 (2018).
  * Braun  _et al._ [2020] W. Braun, M. Jäger, G. Laskin, P. Ngabonziza, W. Voesch, P. Wittlich, and J. Mannhart, In situ thermal preparation of oxide surfaces, APL Mater. 8, 071112 (2020).
  * Chang  _et al._ [2008] J. Chang, Y.-S. Park, and S.-K. Kim, Atomically flat single-terminated srtio3 (111) surface, Appl. Phys. Lett. 92, 152910 (2008).
  * Sánchez  _et al._ [2014] F. Sánchez, C. Ocal, and J. Fontcuberta, Tailored surfaces of perovskite oxide substrates for conducted growth of thin films, Chem. Soc. Rev. 43, 2272 (2014).
  * Ginley  _et al._ [2016] T. P. Ginley, Y. Wang, and S. Law, Topological insulator film growth by molecular beam epitaxy: A review, Crystals 6, 154 (2016), and reference therein. .
  * Ngabonziza [2016] P. Ngabonziza,  _In-situ spectroscopy and nanoscale electronics in superconductor-topological insulator hybrid devices: a combined thin film growth and quantum transport study_ , Ph.D. thesis, University of Twente (2016).
  * Wang  _et al._ [2011b] Z. Wang, H. Li, X. Guo, W. Ho, and M. Xie, Growth characteristics of topological insulator bi2se3 films on different substrates, J. Cryst. Growth 334, 96 (2011b).
  * Mzerd  _et al._ [1995] A. Mzerd, D. Sayah, G. Brun, J. C. Tedenac, and A. Boyer, Crystal growth and sticking coefficient of bi2te3 thin films on si(1 1 1) substrate, J. Mater. Sci. Lett. 14, 194 (1995).
  * Mzerd  _et al._ [1994] A. Mzerd, D. Sayah, J. C. Tedenac, and A. Boyer, Effect of substrate temperature on crystal growth of bi2te3 on single crystal sb2te3, J. Mater. Sci. Lett. 13, 301 (1994).
  * Morelhão  _et al._ [2017] S. L. Morelhão, C. I. Fornari, P. H. O. Rappl, and E. Abramof, Nanoscale characterization of bismuth telluride epitaxial layers by advanced x-ray analysis, J. Appl. Crystallogr. 50, 399 (2017).
  * Fornari  _et al._ [2016] C. I. Fornari, P. H. O. Rappl, S. L. Morelhão, T. R. F. Peixoto, H. Bentmann, F. Reinert, and E. Abramof, Preservation of pristine bi2te3 thin film topological insulator surface after ex situ mechanical removal of te capping layer, APL Mater. 4, 106107 (2016).
  * Li  _et al._ [2010b] H. D. Li, Z. Y. Wang, X. Kan, X. Guo, H. T. He, Z. Wang, J. N. Wang, T. L. Wong, N. Wang, and M. H. Xie, The van der waals epitaxy of bi2se3on the vicinal si(111) surface: an approach for preparing high-quality thin films of a topological insulator, New J. Phys. 12, 103038 (2010b).
  * Karma and Plapp [1998] A. Karma and M. Plapp, Spiral surface growth without desorption, Phys. Rev. Lett. 81, 4444 (1998).
  * Cui and Li [2002] Y. Cui and L. Li, Evolution of spirals during molecular beam epitaxy of gan on 6h-sic(0001), Phys. Rev. B 66, 155330 (2002).
  * Dong  _et al._ [2016] X. Dong, C. Yan, D. Tomer, C. H. Li, and L. Li, Spiral growth of few-layer mos2 by chemical vapor deposition, Appl. Phys. Lett. 109, 051604 (2016).
  * Liu  _et al._ [2012b] Y. Liu, M. Weinert, and L. Li, Spiral growth without dislocations: Molecular beam epitaxy of the topological insulator ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$ on epitaxial graphene/sic(0001), Phys. Rev. Lett. 108, 115501 (2012b).
  * Hao  _et al._ [2013] G. Hao, X. Qi, Y. Fan, L. Xue, X. Peng, X. Wei, and J. Zhong, Spiral growth of topological insulator sb2te3 nanoplates, Appl. Phys. Lett. 102, 013105 (2013).
  * Fornari  _et al._ [2018] C. I. Fornari, P. H. O. Rappl, S. L. Morelhão, G. Fornari, J. S. Travelho, S. de Castro, M. J. P. Pirralho, F. S. Pena, M. L. Peres, and E. Abramof, Structural defects and electronic phase diagram of topological insulator bismuth telluride epitaxial films, Mater. Res. Express 5, 116410 (2018).
  * Kriegner  _et al._ [2017] D. Kriegner, P. Harcuba, J. Veselý, A. Lesnik, G. Bauer, G. Springholz, and V. Holý, Twin domain imaging in topological insulator bi2te3 and bi2se3 epitaxial thin films by scanning x-ray nanobeam microscopy and electron backscatter diffraction, J. Appl. Cryst. 50, 369 (2017).
  * Kampmeier  _et al._ [2015] J. Kampmeier, S. Borisova, L. Plucinski, M. Luysberg, G. Mussler, and D. Grützmacher, Suppressing twin domains in molecular beam epitaxy grown bi2te3 topological insulator thin films, Cryst. Growth Des. 15, 390 (2015).
  * Fornari  _et al._ [2020] C. I. Fornari, E. Abramof, P. H. O. Rappl, S. W. Kycia, and S. L. Morelhão, Morphology control in van der waals epitaxy of bismuth telluride topological insulators, MRS Advances 5, 1891 (2020).
  * Zhang  _et al._ [2021] M. Zhang, W. Liu, C. Zhang, S. Xie, F. Hua, F. Yan, R. Cheng, J. Luo, W. Wang, H. Sang, H. Ge, Z. Wang, Q. Tao, H. Bai, H. Luo, J. Wu, and X. Tang, Evolution of atomic structure and electronic transport properties in n-type bi2te3 films via bi2 planar defects, Appl. Phys. Lett. 118, 103901 (2021).
  * Borisova  _et al._ [2013] S. Borisova, J. Kampmeier, M. Luysberg, G. Mussler, and D. Grützmacher, Domain formation due to surface steps in topological insulator bi2te3 thin films grown on si (111) by molecular beam epitaxy, Appl. Phys. Lett. 103, 081902 (2013).
  * Liu  _et al._ [2010b] C.-X. Liu, H. Zhang, B. Yan, X.-L. Qi, T. Frauenheim, X. Dai, Z. Fang, and S.-C. Zhang, Oscillatory crossover from two-dimensional to three-dimensional topological insulators, Phys. Rev. B 81, 041307 (2010b).
  * Zhang  _et al._ [2010] Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, S.-Q. Shen, Q. Niu, X.-L. Qi, S.-C. Zhang, X.-C. Ma, and Q.-K. Xue, Crossover of the three-dimensional topological insulator bi2se3 to the two-dimensional limit, Nature Phys. 6, 584 (2010).
  * Chang  _et al._ [2013a] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator, Science 340, 167 (2013a).
  * Chang  _et al._ [2015] C.-Z. Chang, W. Zhao, D. Y. Kim, H. Zhang, B. A. Assaf, D. Heiman, S.-C. Zhang, C. Liu, M. H. W. Chan, and J. S. Moodera, High-precision realization of robust quantum anomalous hall state in a hard ferromagnetic topological insulator, Nature Mater. 14, 473 (2015).
  * Chang  _et al._ [2013b] C.-Z. Chang, J. Zhang, M. Liu, Z. Zhang, X. Feng, K. Li, L.-L. Wang, X. Chen, X. Dai, Z. Fang, X.-L. Qi, S.-C. Zhang, Y. Wang, K. He, X.-C. Ma, and Q.-K. Xue, Thin films of magnetically doped topological insulator with carrier-independent long-range ferromagnetic order, Adv. Mater. 25, 1065 (2013b).
  * Zhang  _et al._ [2012b] Z. Zhang, X. Feng, M. Guo, Y. Ou, J. Zhang, K. Li, L. Wang, X. Chen, Q. Xue, X. Ma, K. He, and Y. Wang, Transport properties of sb2te3/bi2te3 topological insulator heterostructures, Phys. Status Solidi RRL 7, 142 (2012b).
  * Bhardwaj  _et al._ [2021] A. Bhardwaj, S. Prasad P., K. V. Raman, and D. Suri, Observation of planar hall effect in topological insulator –bi2te3, Appl. Phys. Lett. 118, 241901 (2021).
  * Roy  _et al._ [2013] A. Roy, S. Guchhait, S. Sonde, R. Dey, T. Pramanik, A. Rai, H. C. P. Movva, L. Colombo, and S. K. Banerjee, Two-dimensional weak anti-localization in bi2te3 thin film grown on si(111)-(7-×-7) surface by molecular beam epitaxy, Appl. Phys. Lett. 102, 163118 (2013).
  * He  _et al._ [2011] H.-T. He, G. Wang, T. Zhang, I.-K. Sou, G. K. L. Wong, J.-N. Wang, H.-Z. Lu, S.-Q. Shen, and F.-C. Zhang, Impurity effect on weak antilocalization in the topological insulator ${\mathrm{bi}}_{2}{\mathrm{te}}_{3}$, Phys. Rev. Lett. 106, 166805 (2011).
  * Holgado  _et al._ [2020] D. P. A. Holgado, K. Bolaños, S. de Castro, H. S. A. Monteiro, F. S. Pena, A. K. Okazaki, C. I. Fornari, P. H. O. Rappl, E. Abramof, D. A. W. Soares, and M. L. Peres, Shubnikov–de haas oscillations and rashba splitting in bi2te3 epitaxial film, Appl. Phys. Lett. 117, 102108 (2020).
  * Stehno  _et al._ [2020] M. P. Stehno, P. Ngabonziza, H. Myoren, and A. Brinkman, Josephson effect and charge distribution in thin bi2te3 topological insulators, Adv. Mater. 32, 1908351 (2020).
  * Steinberg  _et al._ [2011] H. Steinberg, J.-B. Laloë, V. Fatemi, J. S. Moodera, and P. Jarillo-Herrero, Electrically tunable surface-to-bulk coherent coupling in topological insulator thin films, Phys. Rev. B 84, 233101 (2011).
  * Taskin  _et al._ [2012] A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Manifestation of topological protection in transport properties of epitaxial ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$ thin films, Phys. Rev. Lett. 109, 066803 (2012).
  * Ren  _et al._ [2010] Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Large bulk resistivity and surface quantum oscillations in the topological insulator ${\text{bi}}_{2}{\text{te}}_{2}\text{Se}$, Phys. Rev. B 82, 241306 (2010).
  * Bansal  _et al._ [2012] N. Bansal, Y. S. Kim, M. Brahlek, E. Edrey, and S. Oh, Thickness-independent transport channels in topological insulator ${\mathrm{bi}}_{2}{\mathrm{se}}_{3}$ thin films, Phys. Rev. Lett. 109, 116804 (2012).
  * Zhao  _et al._ [2013] Y. Zhao, C.-Z. Chang, Y. Jiang, A. DaSilva, Y. Sun, H. Wang, Y. Xing, Y. Wang, K. He, X. Ma, Q.-K. Xue, and J. Wang, Demonstration of surface transport in a hybrid bi2se3/bi2te3 heterostructure, Sci. Rep. 3, 3060 (2013), (For nonlinear Hall data and extracted transport properties, see supporting info).
  * Zhang  _et al._ [2020] M. Zhang, W. Liu, C. Zhang, J. Qiu, S. Xie, F. Hua, Y. Cao, Z. Li, H. Xie, C. Uher, and X. Tang, Thickness-dependent electronic transport induced by in situ transformation of point defects in mbe-grown bi2te3 thin films, Appl. Phys. Lett. 117, 153902 (2020).
  * Taskin  _et al._ [2017] A. A. Taskin, H. F. Legg, F. Yang, S. Sasaki, Y. Kanai, K. Matsumoto, A. Rosch, and Y. Ando, Planar hall effect from the surface of topological insulators, Nat. Commun. 8, 1340 (2017).
  * Wu  _et al._ [2018] B. Wu, X.-C. Pan, W. Wu, F. Fei, B. Chen, Q. Liu, H. Bu, L. Cao, F. Song, and B. Wang, Oscillating planar hall response in bulk crystal of topological insulator sn doped bi1.1sb0.9te2s, Appl. Phys. Lett. 113, 011902 (2018).
  * He  _et al._ [2019b] P. He, S. S.-L. Zhang, D. Zhu, S. Shi, O. G. Heinonen, G. Vignale, and H. Yang, Nonlinear planar hall effect, Phys. Rev. Lett. 123, 016801 (2019b).
  * Yasuda  _et al._ [2017] K. Yasuda, A. Tsukazaki, R. Yoshimi, K. Kondou, K. S. Takahashi, Y. Otani, M. Kawasaki, and Y. Tokura, Current-nonlinear hall effect and spin-orbit torque magnetization switching in a magnetic topological insulator, Phys. Rev. Lett. 119, 137204 (2017).
  * Rao  _et al._ [2021] W. Rao, Y.-L. Zhou, Y.-j. Wu, H.-J. Duan, M.-X. Deng, and R.-Q. Wang, Theory for linear and nonlinear planar hall effect in topological insulator thin films, Phys. Rev. B 103, 155415 (2021).
  * Zheng  _et al._ [2020] S.-H. Zheng, H.-J. Duan, J.-K. Wang, J.-Y. Li, M.-X. Deng, and R.-Q. Wang, Origin of planar hall effect on the surface of topological insulators: Tilt of dirac cone by an in-plane magnetic field, Phys. Rev. B 101, 041408 (2020).
  * Nandy  _et al._ [2018] S. Nandy, A. Taraphder, and S. Tewari, Berry phase theory of planar hall effect in topological insulators, Sci. Rep. 8, 14983 (2018).
  * Shoenberg [1984] D. Shoenberg,  _Magnetic Oscillations in Metals_ (Cambridge University Press, 1984).
  * Taskin and Ando [2011] A. A. Taskin and Y. Ando, Berry phase of nonideal dirac fermions in topological insulators, Phys. Rev. B 84, 035301 (2011).
  * Mikitik and Sharlai [2012] G. P. Mikitik and Y. V. Sharlai, Berry phase and the phase of the shubnikov–de haas oscillations in three-dimensional topological insulators, Phys. Rev. B 85, 033301 (2012).
  * Mikitik and Sharlai [1999] G. P. Mikitik and Y. V. Sharlai, Manifestation of berry’s phase in metal physics, Phys. Rev. Lett. 82, 2147 (1999).
  * Lahoud  _et al._ [2013] E. Lahoud, E. Maniv, M. S. Petrushevsky, M. Naamneh, A. Ribak, S. Wiedmann, L. Petaccia, Z. Salman, K. B. Chashka, Y. Dagan, and A. Kanigel, Evolution of the fermi surface of a doped topological insulator with carrier concentration, Phys. Rev. B 88, 195107 (2013).
  * Bao  _et al._ [2012] L. Bao, L. He, N. Meyer, X. Kou, P. Zhang, Z.-g. Chen, A. V. Fedorov, J. Zou, T. M. Riedemann, T. A. Lograsso, K. L. Wang, G. Tuttle, and F. Xiu, Weak anti-localization and quantum oscillations of surface states in topological insulator bi2se2te, Sci. Rep. 2, 726 (2012).
  * Veldhorst  _et al._ [2012] M. Veldhorst, M. Snelder, M. Hoek, T. Gang, V. K. Guduru, X. L. Wang, U. Zeitler, W. G. van der Wiel, A. A. Golubov, H. Hilgenkamp, and A. Brinkman, Josephson supercurrent through a topological insulator surface state, Nature Mater. 11, 417 (2012).
  * Veldhorst  _et al._ [2013] M. Veldhorst, M. Snelder, M. Hoek, C. G. Molenaar, D. P. Leusink, A. A. Golubov, H. Hilgenkamp, and A. Brinkman, Magnetotransport and induced superconductivity in bi based three-dimensional topological insulators, Phys. Status Solidi RRL 7, 26 (2013), and references therein .
  * Rosenbach  _et al._ [2020] D. Rosenbach, N. Oellers, A. R. Jalil, M. Mikulics, J. Kölzer, E. Zimmermann, G. Mussler, S. Bunte, D. Grützmacher, H. Lüth, and T. Schäpers, Quantum transport in topological surface states of selectively grown bi2te3 nanoribbons, Adv. Electron. Mater. 6, 2000205 (2020).
  * Jin  _et al._ [2015] H. Jin, K.-C. Kim, J. Seo, S. K. Kim, B.-k. Cheong, J.-S. Kim, and S. Lee, High mobility, large linear magnetoresistance, and quantum transport phenomena in bi2te3 films grown by metallo-organic chemical vapor deposition (mocvd), Nanoscale 7, 17359 (2015).
  * Barua  _et al._ [2014] S. Barua, K. P. Rajeev, and A. K. Gupta, Evidence for topological surface states in metallic single crystals of bi2te3, J. Phys.: Condens. Matter 27, 015601 (2014).
  * Taskin  _et al._ [2011] A. A. Taskin, Z. Ren, S. Sasaki, K. Segawa, and Y. Ando, Observation of dirac holes and electrons in a topological insulator, Phys. Rev. Lett. 107, 016801 (2011).
  * Petrushevsky  _et al._ [2012] M. Petrushevsky, E. Lahoud, A. Ron, E. Maniv, I. Diamant, I. Neder, S. Wiedmann, V. K. Guduru, F. Chiappini, U. Zeitler, J. C. Maan, K. Chashka, A. Kanigel, and Y. Dagan, Probing the surface states in bi2se3 using the shubnikov–de haas effect, Phys. Rev. B 86, 045131 (2012).
  * Devidas  _et al._ [2014] T. R. Devidas, E. P. Amaladass, S. Sharma, R. Rajaraman, D. Sornadurai, N. Subramanian, A. Mani, C. S. Sundar, and A. Bharathi, Role of se vacancies on shubnikov-de haas oscillations in bi2 se3: A combined magneto-resistance and positron annihilation study, Europhys. Lett. 108, 67008 (2014).
  * Zhang  _et al._ [2014] C. Zhang, X. Yuan, K. Wang, Z.-G. Chen, B. Cao, W. Wang, Y. Liu, J. Zou, and F. Xiu, Observations of a metal–insulator transition and strong surface states in bi2–xsbxse3 thin films, Adv. Mater. 26, 7110 (2014).
  * Xiang  _et al._ [2015] F.-X. Xiang, X.-L. Wang, M. Veldhorst, S.-X. Dou, and M. S. Fuhrer, Observation of topological transition of fermi surface from a spindle torus to a torus in bulk rashba spin-split bitecl, Phys. Rev. B 92, 035123 (2015).
  * Okazaki  _et al._ [2018] A. K. Okazaki, S. Wiedmann, S. Pezzini, M. L. Peres, P. H. O. Rappl, and E. Abramof, Shubnikov–de haas oscillations in topological crystalline insulator snte(111) epitaxial films, Phys. Rev. B 98, 195136 (2018).
  * Becker  _et al._ [2003] C. R. Becker, X. C. Zhang, A. Pfeuffer-Jeschke, K. Ortner, V. Hock, G. Landwehr, and L. W. Molenkamp, Very large rashba spin–orbit splitting in hgte quantum wells, J. supercond. 16, 625 (2003).
  * Bando  _et al._ [2000] H. Bando, K. Koizumi, Y. Oikawa, K. Daikohara, V. A. Kulbachinskii, and H. Ozaki, The time-dependent process of oxidation of the surface of bi2te3studied by x-ray photoelectron spectroscopy, J. Phys.: Condens. Matter 12, 5607 (2000).
  * Jin and Jhi [2012] K.-H. Jin and S.-H. Jhi, Effect of atomic impurities on the helical surface states of the topological insulator bi2te3, J. Phys.: Condens. Matter 24, 175001 (2012).
  * Dietl [1978] T. Dietl, Dingle temperature in hgse, J. Phys. Colloques 39, 1081 (1978).
  * Liang  _et al._ [2021] L.-U. Liang, Y.-H. Yen, C.-W. Chou, K.-H. M. Chen, H.-Y. Lin, S.-W. Huang, M. Hong, J. Kwo, and G. Hoffmann, Protected long-time storage of a topological insulator, AIP Adv. 11, 025245 (2021).
  * Harrison  _et al._ [2014] S. E. Harrison, B. Zhou, Y. Huo, A. Pushp, A. J. Kellock, S. S. P. Parkin, J. S. Harris, Y. Chen, and T. Hesjedal, Preparation of layered thin film samples for angle-resolved photoemission spectroscopy, Appl. Phys. Lett. 105, 121608 (2014).