††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Non-trivial quantum magnetotransport oscillations
in pure and robust topological $\alpha$-Sn films

Ivan Madarevic  Quantum Solid State Physics, KU Leuven, Celestijnenlaan 200D,
3001 Leuven, Belgium  Niels Claessens  Quantum Solid State Physics, KU Leuven,
Celestijnenlaan 200D, 3001 Leuven, Belgium  IMEC, Kapeldreef 75, 3001 Leuven,
Belgium  Aleksandr Seliverstov  Quantum Solid State Physics, KU Leuven,
Celestijnenlaan 200D, 3001 Leuven, Belgium  Chris Van Haesendonck  Quantum
Solid State Physics, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
Margriet J. Van Bael  Quantum Solid State Physics, KU Leuven, Celestijnenlaan
200D, 3001 Leuven, Belgium

###### Abstract

We report experimental evidence of topological Dirac fermion charge carriers
in pure and robust $\alpha$-Sn films grown on InSb substrates. This evidence
was acquired using standard macroscopic four-point contact resistance
measurements, conducted on uncapped films with a significantly reduced bulk
mobility. We analyzed and compared electrical characteristics of the
constituting components of the $\alpha$-Sn/InSb sample, and propose a three-
band drift velocity model accordingly. A surface band, with low carrier
density and high mobility, is identified as the origin of the observed
Shubnikov – de Haas oscillations. The analysis of these quantum oscillations
results in a non-trivial value of the phase shift $\gamma=0$, characteristic
for topologically protected Dirac fermions. For the same uncapped samples we
estimate the momentum relaxation time $\tau\approx 300\ \mathrm{fs}$, which is
significantly larger in comparison with the previous reports on grown
$\alpha$-Sn films.

††preprint: APS/123-QED

The alpha phase of Sn ($\alpha$-Sn) is a diamond structured crystal which is a
zero band gap semimetal. Due to its suitable band structure, $\alpha$-Sn was
predicted as a material capable of acquiring several topological phases [1,
2]. Crystal strain can bring about the topologically protected states in
$\alpha$-Sn, more specifically: in-plane tensile strain turns it into a 3D
topological insulator (3DTI), while in-plane compressive strain converts it
into a topological Dirac semimetal (DSM). Topologically protected bands of
$\alpha$-Sn crystalline thin films were characterized by experiments during
the last decade. This was particularly done using angle-resolved photoemission
spectroscopy (ARPES), which revealed a Dirac cone near the Fermi level in the
surface electronic structure of Te and/or Bi doped $\alpha$-Sn thin films,
compressively strained on InSb substrates [3, 4, 5, 6, 7]. Interestingly, the
most recent (soft X-ray) ARPES investigations have suggested the existence of
a topologically protected bulk band (DSM phase) in compressively strained
$\alpha$-Sn films [8, 5, 9]. These reports confirm that this material is an
excellent platform to explore several topological phases [1, 2], making it
very interesting for possible spintronics [10, 11] and quantum computing
applications [12, 13, 14]. Moreover, as a very well structurally defined non-
toxic elemental material with a robust topological phase, $\alpha$-Sn could be
more favorable for applications when compared to prototypical DSM binary
compounds Na3Bi [15, 16] and Cd3As2 [17, 18, 19].

There are recent reports indicating topological magnetotransport in
$\alpha$-Sn. In the case of $\alpha$-Sn on InSb there is, until now, one
report [7] suggesting topologically protected carriers based on
magnetotransport measurements. However, these experiments were conducted on a
relatively complicated system (Bi-doped $\alpha$-Sn$\mid$InSb$\mid$GaAs), in
which the Dirac point was positioned below the Fermi level. There is another
report on the $\alpha$-Sn magnetotransport properties in the case of a thin
film grown on CdTe [20]. The use of insulating CdTe should allow more
straightforward detection of the $\alpha$-Sn topological bands. However, both
Cd and Te bond with Sn (by metallic and covalent bonding respectively),
complicating the morphology of the films, disturbing its epitaxial growth and
the introduction of strain [21, 20]. At this moment, no unambiguous
experimental evidence has been presented for topological Dirac charge carriers
in completely pure (no dopants) $\alpha$-Sn films.

Very recently we presented [22] a straightforward method to grow pure,
compressively strained $\alpha$-Sn films on InSb(100) substrates. The
structure and morphology of these films were fully characterized and the
presence of a Dirac cone was confirmed by ARPES. Moreover, these films
demonstrated an excellent surface quality and a notable robustness against
ambient conditions. In this letter we report prominent accessibility of the
topologically protected Dirac charge carriers in pure and uncapped 30 nm thick
$\alpha$-Sn films compressively strained on InSb(100) [22]. This was made
possible due to the decreased mobility of the trivial bulk charge carriers in
the grown films – most likely a consequence of increased grain-boundary
barrier due to the small grain size. We present evidence for the existence of
the Dirac charge carriers, as an absolutely intrinsic property of the
topological $\alpha$-Sn films. This evidence is based on our analysis of the
Shubnikov – de Haas (SdH) oscillations originating from a surface band. The
Dirac charge carriers exhibit a significantly enhanced relaxation time
compared to previous reports [7, 11].

For the unambiguous interpretation of electronic transport experiments on
composite systems (i.e. film/substrate) it is generally required to analyze
the transport properties of the constituting components. Therefore, along with
the transport measurements on the 30 nm $\alpha$-Sn$\mid$InSb(100) system, we
conducted the same measurements on a non-covered InSb(100) substrate, prepared
according to the procedure presented in our previous report [22]. The
magnetotransport experiments were conducted at a temperature of 1.5 K, with
silver paste contacts in a four-point geometry, at constant current (between
50 and 100 $\mu$A), and magnetic field ($B$) up to 8 T.

Figure 1 depicts the results of the conducted Hall effect measurements,
revealing a clear qualitative difference between the off-diagonal
magnetoresistance ($R_{xy}$) of the InSb(100) substrate and the grown
$\alpha$-Sn$\mid$InSb(100) sample. While the substrate’s off-diagonal
magnetoresistance shows a linear behavior ($n$-type), the grown sample clearly
displays a non-linear Hall effect, indicating the presence of multiple types
of carriers. At low fields ($<$ 2 T), the substrate is shunting the
$\alpha$-Sn film, and the behavior of $R_{xy}$ is slightly non-linear. From
the results of our detailed study of the transport properties of the
substrate, there are some indications that the substrate preparation (Ar+
plasma etching and thermal annealing [22]) leaves it electronically
heterogenous in depth, which may cause such non-linear behavior. However, it
is beyond the scope of this letter to discuss the cause of the non-linear
behavior at low fields. Since it can be attributed to the substrate, for which
the linear $n$-type behavior becomes dominant at higher fields, we omit the
low field region from further analysis.

Figure 1:  Measured off-diagonal magnetoresistance tensor element ($R_{xy}$)
of a 30 nm thick film of $\alpha$-Sn(100) on InSb(100), and a non-covered
InSb(100) substrate. Measurements were done using the Van der Pauw technique
[23]. The inset shows the four-point contact (1 – 4) configuration on the
surface of the sample. Figure 2:  Fit based on the three-band drift velocity
model (orange) for the high-field off-diagonal magnetoresistance above 2 T
(black) of a 30 nm thick film of $\alpha$-Sn on InSb(100).

The method to identify different contributions leading to the observed Hall
effect is to fit a multi-band drift velocity model to the measured $R_{xy}$
data. $R_{xy}$ can, in general, be expressed using the following equations:

$R_{xy}=\ \frac{{\sigma}_{xy}}{{{\sigma}_{xy}}^{2}+{{\sigma}_{xx}}^{2}}~{},$
(1)

with the conductance matrix components of the different contributions ($i$) to
$\sigma$ given by:

${\sigma}^{i}_{xx}=\ \frac{n^{i}_{s}\cdot
e\cdot{\mu}^{i}}{1+{\left({\mu}^{i}\cdot B\right)}^{2}}~{},$ (2)
${\sigma}^{i}_{xy}=\ \pm\frac{n^{i}_{s}\cdot
e\cdot{\left({\mu}^{i}\right)}^{2}\cdot B}{1+{\left({\mu}^{i}\cdot
B\right)}^{2}}~{},$ (3)

under the following constraint for the measured sheet resistance at zero
field:

$R_{s}=\ \left(\sum^{k}_{1}{n^{i}_{s}\cdot}e\cdot{\mu}^{i}\right)^{-1}~{}.{}$
(4)

Fitting the measured $R_{xy}$ behavior of the $\alpha$-Sn sample to a two-band
($k=2$) drift velocity model results in one $p$-type band ($i=p$) and one
$n$-type band ($i=n$), with sheet carrier concentrations
$n^{p}_{s}=3.2(1)\cdot{10}^{14}\ \mathrm{cm}^{-2}$,
$n^{n}_{s}=1.65(4)\cdot{10}^{13}\ \mathrm{cm}^{-2}$, and mobilities
${\mu}^{p}=0.184(2)\cdot 10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$,
${\mu}^{n}=1.4(8)\cdot{10}^{4}\ \mathrm{cm^{2}V^{-1}s^{-1}}$. In this case,
the reduced chi-square value (${\chi}^{2}$) for the fit equals 0.78, at a
$R_{xy}$ relative error of 5 %. On the other hand, repeating the procedure for
a three-band ($k=3$) drift velocity model (Fig. 2) results in two $p$-type
bands ($i=p1,~{}p2$) and one $n$-type band ($i=n$), with sheet carrier
concentrations $n^{p1}_{s}=3.2(1)\cdot{10}^{13}\ \mathrm{cm}^{-2}$,
$n^{p2}_{s}=1.3(2)\cdot{10}^{12}\ \mathrm{cm}^{-2}$,
$n^{n}_{s}=1.65(4)\cdot{10}^{13}\ \mathrm{cm}^{-2}$, and mobilities
${\mu}^{p1}=0.177(2)\cdot 10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$,
${\mu}^{p2}=3.47(12)\cdot 10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$,
${\mu}^{n}=1,4(8)\cdot{10}^{4}\ \mathrm{cm^{2}V^{-1}s^{-1}}$, with
${\chi}^{2}=0.53$ at the same relative error. Considering these values, it is
tempting to choose the latter model over the first one. Indeed, knowing the
film/substrate thickness [22], the bands with the large number of carriers
($n$ and $p1$) are typical bulk bands expected for InSb [24] and $\alpha$-Sn
grown on InSb [25], which exhibits a reduced mobility compared to the bulk
single crystal case [26]. At the same time, the band with the high mobility
and low carrier density ($p2$) may be identify as a possible topological
surface band of $\alpha$-Sn [27]. Although the second model fits the data
better, a nicer fit on its own does not prove the correctness of the model.

To exclude one of the two models, we extracted the SdH oscillations from the
diagonal magnetoresistance $R_{xx}$ measurement of a grown $\alpha$-Sn sample
(Fig. 3). The experiments were again conducted at a temperature of 1.5 K and
with magnetic field (up to 8 T) applied perpendicularly to the surface of the
samples ($\vec{B}\perp(100)$), with macroscopic silver paste contacts arranged
in a linear four-point geometry. Figure 4 (inlay) depicts magnetoresistance
oscillations ($\Delta R$) of a grown $\alpha$-Sn sample, extracted after
removing the background which was obtained from a $3^{\mathrm{rd}}$ order
polynomial fit. The maximum of the fast Fourier transform (FFT) of $\Delta R$
vs. $1/B$ provides an estimate of the characteristic frequency $f=14(6)\
\mathrm{T}$ of the oscillations. A more accurate value of 13.5(8) T was
extracted from the Landau index plot (Fig. 4), as shown later.

Figure 3:  Measured diagonal magnetoresistance tensor element ($R_{xx}$) of a
30 nm thick film of $\alpha$-Sn(100) on InSb(100) and a non-covered InSb(100)
substrate, with the magnetic field perpendicular to the film/substrate
surface. The inset shows the four-point contact configuration on the surface
of the sample.

Assuming the oscillations originate from a bulk band, the estimated value for
$f$ would correspond to a carrier density of $n=(4\pi
f/\mathrm{\Phi_{0}})^{{3}/{2}}/{3{\pi}^{2}}=2.8(2)\cdot{10}^{17}\
\mathrm{cm}^{-3}$, where ${\mathrm{\Phi}}_{0}=h/e$ is the elementary flux
quantum, with Planck’s constant $h$ and the electron charge $e$. If we convert
this to the hypothetical sheet carrier densities for such bulk bands of the
$\alpha$-Sn film and InSb substrate, by multiplying with their thicknesses
[22], this would give $8.4(7)\cdot{10}^{11}\ \mathrm{cm}^{-2}$ and
$1.75(16)\cdot~{}{10}^{16}\ \mathrm{cm}^{-2}$ respectively. From these sheet
carrier densities, which differ from the bulk bands extracted using the Hall
effect measurements ($p1$ and $n1$, respectively), it appears that neither the
two- nor three-band model would be compatible with the observed SdH
oscillations if the oscillations would originate from a bulk band.

Figure 4:  Landau index plot of the magnetoresistance oscillations after
removal of the background ($\vec{B}\perp(100)$). The Landau index for the
minima and maxima of the observed SdH oscillations is plotted against the
inverse of the field strength (red dots) at which they are observed. The fit
of Eq. (6) (black line) through the data points is shown together with the low
field limit ($A_{2}=0$) (red dashed line). The inlay shows magnetoresistance
oscillations in the $\vec{B}\perp(100)$ (blue line) and the
$\vec{B}\parallel(100)$ orientation (black dashed line).

If we instead assume that the experimentally observed quantum oscillations
arise from a surface band, the sheet carrier density corresponding to the
observed SdH frequency is given by $n_{s}=2\cdot
n=2\cdot(2f/{\mathrm{\Phi}}_{0})={1.30(7)\cdot{10}^{12}\ \mathrm{cm}^{-2}}$.
This carrier density value is in very good agreement with the second $p$-type
band ($p2$) in the above fitted three-band drift velocity model, but in
contradiction with the two-band model. Note as well that the SdH oscillations
are only observable when their cyclotron frequency is larger than the inverse
of the relaxation time [28] – equivalent to $\mu>{1}/{B}\approx 3.33\cdot
10^{3}\ \mathrm{cm^{2}V^{-1}s^{-1}}$ for oscillations starting at $\sim 3$ T.
This condition is satisfied for the $p2$ band in the three-band drift velocity
model, but it is not met for the two-band model. Moreover, the reduced
mobility of the trivial bulk carriers (increased grain boundary scattering) in
this model (${\mu}^{p1}$) is consistent with the fact that the grown
$\alpha$-Sn films have a granular morphology, with grain size around 10 – 20
nm [22]. Therefore, it can be concluded that the two-band drift velocity model
fails, while the three-band model explains our observations. The second
$p$-type band of the three-band model ($p2$) can thus be identified as a
surface band of the $\alpha$-Sn(100) film. The first $p$-type band ($p1$) we
assign to the bulk $\alpha$-Sn and the $n$-type band to the bulk of the InSb
substrate.

Using Eq. (4) we can now estimate the sheet resistance contribution of the
bands separately: $R^{~{}p1}_{s}=1.10(4)\ \mathrm{k\Omega/\Box}$ ($\alpha$-Sn
bulk band), $R^{~{}p2}_{s}=1.4(2)\ \mathrm{k\Omega/\Box}$ ($\alpha$-Sn surface
band) and $R^{~{}n}_{s}=27(15)\ \mathrm{\Omega/\Box}$ (InSb bulk band).

We will now investigate the possible Dirac nature of the carriers in the
second $p$ band ($i=p2$). The phase offset $\gamma$ of the SdH oscillations
contains the information on the dispersion of the corresponding charge
carriers [29]. The SdH oscillations can be expressed as:

$\Delta
R\propto{\mathrm{cos}\left[2\pi\cdot\left(\frac{f}{B}-\gamma\right)\right]}~{},$
(5)

where $\gamma$ can be related to Berry’s phase acquired by the carriers within
a cyclotron orbit [30]. When the zero-field electronic dispersion would be
followed by the carriers, then $\gamma=0$ (non-trivial Berry’s phase of $\pi$)
for Dirac fermions, while $\gamma=1/2$ (trivial Berry’s phase of 0) for
“normal” fermions [31]. The intercept of a linear fit of the Landau level
index N with respect to $1/B$ then gives the $\gamma$ value. However, for
finite field strengths, as pointed out by Wright and McKenzie, the fermions
can no longer be expected to follow the zero-field dispersion, and a deviation
from a perfect quantization, where $\gamma=\gamma(B)$, is to be expected [31].
For that reason, a more accurate low-field fit procedure was proposed to
extract the zero-field phase offset:

$N=\frac{f}{B}+A_{1}+A_{2}\cdot B~{},{}$ (6)

where $f$ (characteristic frequency), $A_{1}$ and $A_{2}$ are the fit-
parameters. The zero-field limit of the Eq. (6) is equivalent with
$\gamma(0)=A_{1}$. Therefore, the parameter $A_{1}$ is the quantized offset
related to the zero-field Berry’s phase. Figure 4 presents the fit of Eq. (6)
to the Landau indices of the observed SdH oscillations minima and maxima
extracted from the magnetoresistance measurement with the magnetic field
applied perpendicular to the sample surface plane
($\vec{B}\perp\mathrm{(100)}$). Here, due to the dominance of bulk
contributions to the conductance [32], the minima of the SdH oscillations
correspond to integer Landau level indices ($N$), while the maxima correspond
to half-integer Landau level indices ($N+1/2$). The resulting fit-parameters
are $A_{1}=0.02(15)$ and $A_{2}=0.00(6)$, and $f=13.5(8)\ \mathrm{T}$.
Contrary to the near-zero $A_{1}$ value for the $\alpha$-Sn samples, the fit
of the SdH oscillations extracted from the magnetoresistance measurement on a
non-covered InSb substrate gave $A_{1}=0.52(13)$, $A_{2}=-0.04(2)$ and
$f=39.4(8)\ \mathrm{T}$. The characteristic frequencies were determined by a
linear fit of the Landau index plot (Fig. 4) and held fixed as the linear fit
yields more accurate results on the frequency compared to the estimation of
the FFT maximum of $\Delta R$, which is the usual procedure in order to reduce
the number of fit parameters. As the analysis in the case of $\alpha$-Sn gives
$A_{1}\approx 0$, it can be concluded that the second $p$-type band ($i=p2$)
indeed has a topological Dirac fermion nature. Based on this conclusion, we
can now estimate the value of the characteristic Fermi wavevector for this
$p2$ band as $k_{F}=\sqrt{{4\pi}{f/\Phi_{0}}}=0.203(6)\ \mathrm{nm}^{-1}$.
Assuming a linear energy dispersion [33] and using the previously extracted
Fermi velocity value [22], we then estimate the Fermi energy at $E_{F}=\hslash
k_{F}v_{F}=67(2)\ \mathrm{meV}$. These values, estimated from the transport
measurements, are in agreement with the ones extracted from our recent ARPES
and scanning tunneling spectroscopy (STS) study ($k_{F}^{~{}ARPES}\sim 0.18\
\mathrm{nm}^{-1}$, $E_{F}^{~{}ARPES}\sim 60\ \mathrm{meV}$ and
$E_{F}^{~{}STS}\sim 70\ \mathrm{meV}$ below the Dirac point) [22].

From the calculated $R^{~{}p2}_{s}$ value, using the resistance expression for
non-degenerate 2D bands with a linear dispersion, i.e.,
$R^{p2}_{s}=(4\hbar^{2}\pi)/(e^{2}E_{F}\tau)$ [34], we estimate the momentum
relaxation time $\tau=300(40)\ \mathrm{fs}$. This value is about five times
larger than the one estimated for $\alpha$-Sn films with an AlOx capping layer
[7], and almost two orders of magnitude larger compared with $\alpha$-Sn films
covered with Ag [11], indicating that there are fewer parallel momentum
relaxation channels in our samples.

In the case of a 2D gas of Dirac fermions existing on top (and at the bottom)
of the grown $\alpha$-Sn film, one expects the SdH oscillations to vanish for
the $\vec{B}\parallel\mathrm{(100)}$ alignment. However, for our samples this
is not the case. Surprisingly, in the same type of grown $\alpha$-Sn samples,
SdH oscillations were also observed (inlay of Fig. 4), when the magnetic field
is applied parallel to the film surface (equivalent to
$\vec{B}\perp\mathrm{(010)}$). These SdH oscillations have a characteristic
frequency $f=10.1(15)\ \mathrm{T}$, which is quite similar to the
$\vec{B}\perp\mathrm{(100)}$ case, and which cannot be attributed to InSb or
$\alpha$-Sn bulk bands. The phase offset analysis of these oscillations gave a
value $A_{1}=0.1(3)$ close to the non-trivial Berry’s phase of $\pi$. From the
topographic details of our samples [22] we know that the surface of the film
is not atomically flat, but in fact consists of crystalline grains with non-
zero roughness. Therefore, a significant portion of the surface area is never
in parallel alignment with the applied magnetic field.

All of the above presented results do strongly support the existence of
topological Dirac fermion charge carriers, but our discussion so far has not
yet specified which type of Dirac material, 3DTI or DSM, our $\alpha$-Sn films
belong to. In the case of the 3DTI $\alpha$-Sn, a surface phase with a clear
topological Dirac signature should arise in the electrical transport
experiments [1, 2]. The detected $p2$ surface band appears to agree with the
3DTI picture. However, this phase is predicted to emerge for tensile strained
$\alpha$-Sn, while our samples are compressively strained [22]. On the other
hand, the most recent ARPES studies confirmed that for in-plane compressive
strain (which is the case for our samples) $\alpha$-Sn is a DSM [8, 9]. This
portrays $\alpha$-Sn as a material which is topologically non-trivial
regardless of the type of strain. For in-plane compressively strained film,
the $\alpha$-Sn DSM phase should host two Dirac nodes, projected to the (100)
plane as a single Dirac cone [1]. Topological surface states and Fermi arcs
(between multiple Dirac points) can form as a consequence of strain, and/or
the applied magnetic field [2]. However, magnetotransport experiments, aside
of the detected topological Dirac-like 2D surface band, do not show the
existence of the (DSM) Dirac bulk charge carriers in our samples (e.g. through
an additional frequency in the SdH oscillations). For our experiments the lack
of sensitivity to the topology of the bulk is due to the low mobility of the
film bulk charge carriers and the obvious low resistance of the InSb substrate
(resulting in pronounced shunting of the $\alpha$-Sn film). For the above
stated reasons, based on our transport results it is not possible to
unambiguously resolve the type of topological phase (3DTI or DSM) responsible
for the Dirac cone observed in our $\alpha$-Sn films [22].

We conclude that we obtained evidence for the existence of topological Dirac
fermion charge carriers in uncapped pure and robust $\alpha$-Sn(100) films. A
$p$-type surface band with low carrier density and high mobility has been
identified to be the origin of the observed SdH oscillations. To analyze these
quantum oscillations, Landau index plots were fitted, revealing a non-trivial
value of the phase shift, characteristic for topologically protected Dirac
fermions. Such findings strongly support earlier theoretical reports of
topologically protected charge carriers being an intrinsic property of
strained $\alpha$-Sn. A significantly longer momentum relaxation time of the
detected carriers suggests more prominent accessibility of the topological
properties in $\alpha$-Sn(100) grown on InSb(100). The remarkable fact that it
is possible to detect topological Dirac fermion-like transport in our samples,
using standard macroscopic four-point resistance measurements, provides unique
opportunity to further investigate and apply this elemental topological
material.

We thank J. Van de Vondel, B. Raes and G. Lippertz for valuable discussions.
This work was supported by the Research Foundation – Flanders (FWO) and by the
KU Leuven C1 program grants No. C14/18/074 and C12/18/006.

## References

  * Huang and Liu [2017] H. Huang and F. Liu, Tensile strained gray tin: Dirac semimetal for observing negative magnetoresistance with Shubnikov–de Haas oscillations, Phys. Rev. B 95, 201101 (2017).
  * Zhang  _et al._ [2018] D. Zhang, H. Wang, J. Ruan, G. Yao, and H. Zhang, Engineering topological phases in the Luttinger semimetal $\alpha$-Sn, Phys. Rev. B 97, 195139 (2018).
  * Barfuss  _et al._ [2013] A. Barfuss, L. Dudy, M. R. Scholz, H. Roth, P. Höpfner, C. Blumenstein, G. Landolt, J. H. Dil, N. C. Plumb, M. Radovic, A. Bostwick, E. Rotenberg, A. Fleszar, G. Bihlmayer, D. Wortmann, G. Li, W. Hanke, R. Claessen, and J. Schäfer, Elemental topological insulator with tunable Fermi level: Strained $\alpha$-Sn on InSb(001), Phys. Rev. Lett. 111, 157205 (2013).
  * Ohtsubo  _et al._ [2013] Y. Ohtsubo, P. Le Fèvre, F. Bertran, and A. Taleb-Ibrahimi, Dirac cone with helical spin polarization in ultrathin $\alpha$-Sn(001) films, Phys. Rev. Lett. 111, 216401 (2013).
  * Rogalev  _et al._ [2017] V. A. Rogalev, T. Rauch, M. R. Scholz, F. Reis, L. Dudy, A. Fleszar, M.-A. Husanu, V. N. Strocov, J. Henk, I. Mertig, J. Schäfer, and R. Claessen, Double band inversion in $\alpha$-Sn: Appearance of topological surface states and the role of orbital composition, Phys. Rev. B 95, 161117 (2017).
  * Scholz  _et al._ [2018] M. R. Scholz, V. A. Rogalev, L. Dudy, F. Reis, F. Adler, J. Aulbach, L. J. Collins-McIntyre, L. B. Duffy, H. F. Yang, Y. L. Chen, T. Hesjedal, Z. K. Liu, M. Hoesch, S. Muff, J. H. Dil, J. Schäfer, and R. Claessen, Topological surface state of $\alpha$-Sn on InSb(001) as studied by photoemission, Phys. Rev. B 97, 075101 (2018).
  * Barbedienne  _et al._ [2018] Q. Barbedienne, J. Varignon, N. Reyren, A. Marty, C. Vergnaud, M. Jamet, C. Gomez-Carbonell, A. Lemaître, P. Le Fèvre, F. Bertran, A. Taleb-Ibrahimi, H. Jaffrès, J.-M. George, and A. Fert, Angular-resolved photoemission electron spectroscopy and transport studies of the elemental topological insulator $\alpha$-Sn, Phys. Rev. B 98, 195445 (2018).
  * Xu  _et al._ [2017] C.-Z. Xu, Y.-H. Chan, Y. Chen, P. Chen, X. Wang, C. Dejoie, M.-H. Wong, J. A. Hlevyack, H. Ryu, H.-Y. Kee, N. Tamura, M.-Y. Chou, Z. Hussain, S.-K. Mo, and T.-C. Chiang, Elemental topological Dirac semimetal: $\alpha$-Sn on InSb(111), Phys. Rev. Lett. 118, 146402 (2017).
  * Rogalev  _et al._ [2019] V. A. Rogalev, F. Reis, F. Adler, M. Bauernfeind, J. Erhardt, A. Kowalewski, M. R. Scholz, L. Dudy, L. B. Duffy, T. Hesjedal, M. Hoesch, G. Bihlmayer, J. Schäfer, and R. Claessen, Tailoring the topological surface state in ultrathin $\alpha$-Sn(111) films, Phys. Rev. B 100, 245144 (2019).
  * Kondou  _et al._ [2016] K. Kondou, R. Yoshimi, A. Tsukazaki, Y. Fukuma, J. Matsuno, K. S. Takahashi, M. Kawasaki, Y. Tokura, and Y. Otani, Fermi-level-dependent charge-to-spin current conversion by Dirac surface states of topological insulators, Nat. Phys. 12, 1027 (2016).
  * Rojas-Sánchez  _et al._ [2016] J.-C. Rojas-Sánchez, S. Oyarzún, Y. Fu, A. Marty, C. Vergnaud, S. Gambarelli, L. Vila, M. Jamet, Y. Ohtsubo, A. Taleb-Ibrahimi, P. Le Fèvre, F. Bertran, N. Reyren, J.-M. George, and A. Fert, Spin to charge conversion at room temperature by spin pumping into a new type of topological insulator: $\alpha$-Sn films, Phys. Rev. Lett. 116, 096602 (2016).
  * Kitaev [2003] A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. (N. Y.) 303, 2 (2003).
  * Tewari  _et al._ [2007] S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, and P. Zoller, Quantum computation using vortices and Majorana zero modes of a ${p}_{x}+i{p}_{y}$ superfluid of fermionic cold atoms, Phys. Rev. Lett. 98, 010506 (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).
  * Liu  _et al._ [2014a] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain, and Y. L. Chen, Discovery of a three-dimensional topological Dirac semimetal, Na3Bi, Science 343, 864 (2014a).
  * Xu  _et al._ [2014] S.-Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T.-R. Chang, H.-T. Jeng, C.-Y. Huang, W.-F. Tsai, H. Lin, P. P. Shibayev, F.-C. Chou, R. J. Cava, and M. Z. Hasan, Observation of Fermi arc surface states in a topological metal, Science 347, 294 (2014).
  * Borisenko  _et al._ [2014] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, and R. J. Cava, Experimental realization of a three-dimensional Dirac semimetal, Phys. Rev. Lett. 113, 027603 (2014).
  * Liu  _et al._ [2014b] Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain, and Y. L. Chen, A stable three-dimensional topological Dirac semimetal Cd3As2, Nature Materials 13, 677 (2014b).
  * Neupane  _et al._ [2014] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin, _et al._ , Observation of a three-dimensional topological Dirac semimetal phase in high-mobility Cd3As2, Nat. Commun. 5, 1 (2014).
  * Vail  _et al._ [2019] O. Vail, P. Taylor, P. Folkes, B. Nichols, B. Haidet, K. Mukherjee, and G. de Coster, Growth and magnetotransport in thin-film $\alpha$-Sn on CdTe, Phys. Status Solidi B 257, 1800513 (2019).
  * Tu  _et al._ [1989] L. W. Tu, G. K. Wong, S. N. Song, Z. Zhao, and J. B. Ketterson, Shubnikov–de Haas effect in thin epitaxial films of gray tin, Appl. Phys. Lett. 55, 2643 (1989).
  * Madarevic  _et al._ [2020] I. Madarevic, U. Thupakula, G. Lippertz, N. Claessens, P.-C. Lin, H. Bana, S. Gonzalez, G. D. Santo, L. Petaccia, M. N. Nair, L. M. Pereira, C. Van Haesendonck, and M. J. Van Bael, Structural and electronic properties of the pure and stable elemental 3D topological Dirac semimetal $\alpha$-Sn, APL Mater. 8, 031114 (2020).
  * Van der Pauw [1958] L. J. Van der Pauw, A method of measuring specific resistivity and Hall effect of discs of arbitrary shape, Philips Res. Rep 13, 1 (1958).
  * Madelung  _et al._ [2002] O. Madelung, U. Rössler, and M. Schulz, eds.,  _Group IV Elements, IV-IV and III-V Compounds. Part b - Electronic, Transport, Optical and Other Properties_ (Springer-Verlag, 2002).
  * Farrow  _et al._ [1981] R. Farrow, D. Robertson, G. Williams, A. Cullis, G. Jones, I. Young, and P. Dennis, The growth of metastable, heteroepitaxial films of $\alpha$-Sn by metal beam epitaxy, J. Cryst. Growth 54, 507 (1981).
  * Ewald and Tufte [1958] A. W. Ewald and O. N. Tufte, Gray tin single crystals, J. Appl. Phys. 29, 1007 (1958).
  * Franz and Molenkamp [2013] M. Franz and L. Molenkamp,  _Topological insulators_ (Elsevier, 2013).
  * Ashcroft  _et al._ [1976] N. W. Ashcroft, N. D. Mermin, _et al._ , Solid state physics (1976).
  * Shoenberg [2009] D. Shoenberg,  _Magnetic oscillations in metals_ (Cambridge university press, 2009).
  * Fuchs  _et al._ [2010] J. N. Fuchs, F. Piéchon, M. O. Goerbig, and G. Montambaux, Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models, Eur. Phys. J. B 77, 351 (2010).
  * Wright and McKenzie [2013] A. R. Wright and R. H. McKenzie, Quantum oscillations and Berry’s phase in topological insulator surface states with broken particle-hole symmetry, Phys. Rev. B 87, 085411 (2013).
  * Xiong  _et al._ [2012] J. Xiong, Y. Luo, Y. Khoo, S. Jia, R. J. Cava, and N. P. Ong, High-field Shubnikov – de Haas oscillations in the topological insulator Bi2Te2Se, Phys. Rev. B 86, 045314 (2012).
  * Neto  _et al._ [2009] A. C. Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009).
  * Gantmakher and Levinson [2012] V. Gantmakher and Y. Levinson,  _Carrier scattering in metals and semiconductors_ (Elsevier, 2012).