Magnetoresistance oscillations in MBE-grown
Sb2Te3 thin films
Supplementary information
I.
Details on MBE-growth
Standard parameters for MBE growth of Sb2Te3-films in our system are
TSi = 300 °C and TTe = 330 °C as stated in Ref. [1]. With these we usually
obtain mobilities of 200 − 500 cm2 ⁄Vs and carrier concentrations around
1 × 1014 cm-2 . To improve these parameters, we therefore tried to increase the
temperatures of the Si(111) substrate TSi and the Te-effusion cell TTe during
growth, which were raised to TSi = 350 °C and TSi = 380 °C. The latter already
being at the upper limit of our Te-Knudsen cell. This resulted in a Sb2Te3-film
with more than twice the mobility. Atomic force microscopy (AFM) was then
used to get an insight about the origin of this increase.
S1. a) AFM picture of a 19 nm thick film, grown using standard parameters. b) Same for the
film grown with increased temperatures TSi and TTe.
From these measurements, shown in S1, it is obvious that the elevated
temperatures during growth lead to a significant reduction in mount formation,
which directly causes a reduction of defects in the film as there are less starting
islands which have to coalesce to form a closed film. These borders between
islands are likely to be a main source of scattering for the charge carriers, which
could explain the general increase in mobility that enables the observation of
quantum oscillations in the present film. We were also able to reduce the carrier
concentration to about a half of that obtained with standard conditions. Thus, the
increased temperatures may have reduced the amount of antisite defects, which
are generally believed to be responsible for the intrinsic doping of the crystal.[2]
II.
Determination of the background
As can be seen in S2, the measured sheet resistance resembles a decaying
exponential function when the raw data is plotted against the reciprocal field 1⁄𝐵.
In the derivative, the oscillations are directly visible as modulations at smaller
values.
S2 Sheet resistance 𝑅□ (green) plotted against the reciprocal field and the fit using four
𝑒-functions (red). The insert shows the derivatives of the same 𝑅□ (blue) and fit curves (red).
Here, the oscillations of 𝑅□ are more pronounced, while the fit is clearly smooth over the entire
range.
Although there is no physical motivation to use an exponential function, this
resemblance signifies that the points can be traced by such a function. Moreover,
𝑒-functions are smooth and strictly decreasing, so there is no risk that the fitted
curve itself produces modulations that could alter the oscillations after
subtraction. Therefore, the following method was used to extract the oscillations
from the measurements.
A sum of exponential factors of the form
0
𝐹(𝑥) = 𝑅𝑥𝑥
+ ∑ 𝐴𝑖 𝑒
−(𝑥−𝑥0 )⁄
𝛾𝑖
𝑖
0
was used to fit the data with 𝑅𝑥𝑥
, 𝑥0 , 𝐴𝑖 and 𝛾𝑖 being fit parameters and 𝑥 = 1⁄𝐵.
The cut-off at low fields was set to 1 T. However, when calculating the reciprocal
fields from roughly equidistant values in 𝐵, the density of data points is no longer
constant since most of the points will be located at small values of 1⁄𝐵 (high
fields) while they become increasingly separated at smaller fields. Fitting the
curve like this would mean that the high field values completely dominate the fit
and the resulting curve will not trace the lower field ranges correctly. To prevent
this, the curve was interpolated linearly at equidistant points in 1⁄𝐵 over the entire
fitting range. To ensure, that no information is lost, the interpolation was done
using at least 30.000 data points (about 10 times the original amount) for the
whole curve to maintain the high density at higher fields. Also, padding the data
with additional points at smaller fields does not lead to a significant alteration
since the slope in this region is already strongly reduced. The actual fitting was
done with 𝑖 = 4 as using more terms did not change the resulting fit. The initial
parameters were chosen to be 𝐴𝑖 = 𝐴0 ⁄𝑖 and 𝛾𝑖 = 𝛾0 ⁄𝑖 respectively. The
parameters obtained from the fits were then used to calculate the final background
at the field values of the original data.
III.
The mixed fermion system
To see if the intersection from the fan diagram can be determined more
sophistically, we applied a model which takes into account a mixed system formed
by regular and Dirac fermions and also includes Zeeman splitting. Such a system
has already been discussed in various works [3 - 6] and can be described by the
sum of the Hamiltonians of the individual systems
Π2
1
⃗)=
𝐻(𝑘
+
𝑣
𝜎
+
Π
𝜎
−
𝑔 𝜇 𝐵𝜎 .
(Π
)
𝐹
𝑥
𝑦
𝑦
𝑥
2𝑚∗
2 𝑠 𝐵 𝑧
The first term describes the quadratic dispersion of the regular fermions with the
⃗ + 𝑒𝐴, 𝐴 being the vector potential. The second
effective mass 𝑚∗ and Π = ℏ𝑘
term describes the linear dispersion of the Dirac fermions with the Fermi velocity
𝑣𝐹 and the Pauli matrices 𝜎𝑖 . The third term is responsible for the Zeeman splitting
with the surface 𝑔-factor 𝑔𝑠 and the Bohr magneton 𝜇𝐵 . When solved separately,
the first two terms yield their well known energy eigenvalues
𝑝𝑎𝑟
𝐸𝑛
1
= ℏ𝜔𝑐 (𝑛 + ) and 𝐸𝑛𝑙𝑖𝑛 = 𝑣𝐹 √2𝑒ℏ𝐵𝑛
2
respectively, with the Landau level index 𝑛, the cyclotron frequency 𝜔𝑐 = 𝑒𝐵⁄𝑚𝑐
and the cyclotron mass 𝑚𝑐 . While 𝑛 can only be integer, we can define an
analogous index 𝑛̃ which can also assume half-integral values. Thus, we can set
𝑛̃ = 𝑛 for the maxima in the resistance and 𝑛̃ = 𝑛 + 1⁄2 for the minima
accordingly. With this, extrapolating the fan diagram to 1⁄𝐵 → 0 should give an
intercept at 𝑁 = 𝑛̃ + 1⁄2 = 0 for the parabolic dispersion and 𝑁 = 1⁄2 for the
linear one, when integer 𝑁 are assigned to the minima of the oscillations to count
the number of fully occupied Landau levels. This is used to determine if
oscillations originate from topological surface states or not. For the mixed system
however, the intercept can deviate from these two values, rendering the
identification ambiguous, since the dispersion of the surface states is known to
become increasingly nonlinear at higher energies. The full Hamiltonian can also
be solved analytically and gives
1
𝐸𝑛𝑚𝑖𝑥 = 𝐸0 𝑛 ± √𝑛𝜅 + (𝐸0 − 𝜆)2 .
4
Here, the parameters are 𝐸0 = 𝑒ℏ𝐵⁄𝑚∗ , 𝜅 = ℏ𝑣𝐹2 𝑒𝐵 and 𝜆 = 𝑔𝑠 𝜇𝐵 𝐵. In the
quantum limit (1⁄𝐵 → 0) this gives 𝑁 = 1⁄2 ∗ (1 ± |1 − 𝑔𝑠 𝜇𝐵 𝑚∗ ⁄𝑒ℏ|), so the
intercept now depends on the 𝑔-factor and the effective mass of the surface states.
All of these systems (parabolic, linear and mixed) lead to oscillations in the
magnetoresistance which are periodic in 1⁄𝐵 as the magnetic field is swept. This
follows immediately from the Onsager relation 𝑓 = ℏ⁄2𝜋𝑒 ∙ 𝐴(𝐸𝐹 ) [7], which
connects the oscillation frequency 𝑓 with the area 𝐴(𝐸𝐹 ) of the Fermi surface
perpendicular to the magnetic field. For all three systems the area of the cross
section is a circle, thus 𝐴(𝐸𝐹 ) = 𝜋𝑘𝐹2 . The Fermi-wave vector 𝑘𝐹 is defined as
𝑘𝐹 = √4𝜋𝑛2𝐷 ⁄𝑠 for a two-dimensional system, independently of the dispersion
relation. 𝑠 is the spin degeneracy of each 𝑘-state and should be equal to 1 for the
surface states, due to the spin-momentum locking.
The SdH-oscillations can be described by Δ𝑅𝑥𝑥 ∝ cos[2𝜋(𝑓 ⁄𝐵 − 𝛾)] where 𝑓 is
given by the Onsager relation and the phase 𝛾 depends on the system. For the
𝑝𝑎𝑟
parabolic dispersion, 𝑘𝐹 = √2𝑚∗ 𝐸𝐹 ⁄ℏ2 and 𝐸𝐹 = 𝐸𝑛 every time Δ𝑅𝑥𝑥 has a
maximum at a field 𝐵=𝐵𝑛 . Therefore
1
𝑝𝑎𝑟
𝑓(𝐸𝑛 )⁄𝐵𝑛 − 𝛾 = 𝑛 ⇒ (𝑛 + ) − 𝛾 = 𝑛
2
1
⇒𝛾= .
2
An analogous calculation for the linear dispersion leads to 𝛾 = 0. This phase is
directly related to the Berry phase Φ𝐵 of the system [8], according to
𝛾=
1 Φ𝐵
−
.
2 2𝜋
For normal fermions this yields Φ𝐵 = 0 and for the Dirac fermions of the
topological surface states Φ𝐵 = 𝜋.
Since the intercept from fitting a straight line to our fan diagram does not give a
conclusive answer to the dispersion relation in our system, we apply the method
proposed in Ref. [5]. By solving the equation 𝐸𝑛𝑚𝑖𝑥 =𝐸𝐹𝑚𝑖𝑥 for the field values 𝐵𝑛
we obtain a rather complicated formula that can be fitted to the measured values
of our extrema. To reduce the number of free parameters we can use the value
𝑣𝐹 = 4.36 x 105 m⁄s,
obtained
from
ARPES
measurements
by
Plucinski et al.[9] Furthermore, using the general definition of the cyclotron mass
𝑚𝑐 = ℏ2 𝑘𝐹 (𝜕𝐸 ⁄𝜕𝑘)−1 [10] we find 𝑚∗ = 𝑚𝑐 ⁄(1 − 𝑚𝑐 𝑣𝐹 ⁄ℏ𝑘𝐹 ) using 𝐸 of the
mixed system. With 𝑚𝑐 = 0.08 𝑚𝑒 and 𝑣𝐹 from above we can estimate
𝑚∗ ≈ 0.56 𝑚𝑒 . This leaves 𝑔𝑠 as the only free parameter. Fitting 𝐵𝑛 (𝑛, 𝑔𝑠 ) to our
values of 𝐵𝑛 , we obtain 𝑔𝑠 = 91.5 or 𝑔𝑠 = −84.3 which cannot be distinguished
due to the squaring of the bracket in 𝐸𝑛𝑚𝑖𝑥 . Although these values appear to be
very large, Taskin and Ando find similarly big values for several topological
insulator materials and 𝑔𝑠 = 0 only for graphene.[5]
The fit to the 𝐵𝑛 of the mixed system is shown in S3. The strong deviation from a
purely linear dependence at higher fields (smaller N) can be attributed to the
pronounced field dependence of the phase 𝛾, which itself is due to the high surface
𝑔-factor obtained from the fit. This is can be seen in the insert where 𝛾 is plotted
against the index 𝑁. The Zeeman term shifts the phase so much, that it leaves the
range between 𝛾 = 0 − 0.5, which is associated with the pure linear and parabolic
dispersions. With the parameters used for the fit, we can also calculate the Fermienergy for the mixed system and obtain
S3 Fan diagram showing the common linear fit (blue) which gives an interception at 𝑁 ≈ 0.15
and the fit to the mixed system (red). The insert shows the evolution of the phase 𝛾 with
decreasing index 𝑁.
ℏ2 𝑘 𝐹 2
𝐸𝐹 =
+ℏ𝑣𝐹 𝑘𝐹 ≈ 8.4 meV + 100.9 meV
2𝑚∗
for the individual contributions. This means that the parabolic contribution is
rather small (less than 10 %) compared to the linear one, which is due to the quite
large effective mass 𝑚∗ .
Another analysis of the mixed system was done by Wright and McKenzie [6] who
derive a simple model of the form
𝑁=
𝐵0⁄
𝐵 +𝐴1 +𝐴2 B,
with 𝐵0 , 𝐴1 and 𝐴2 being numerical constants. They also give a recipe to extract
the relevant phase shift by fitting the whole formula to the fan diagram and then
extrapolating only the linear term to 1⁄𝐵 → 0. However, applying their model to
our data does not change the intersection of the extrapolated line fit to any
significant degree from the simple line fit. We attribute this to the lack of data
points at lower fields, since we only observe five extrema starting at 𝐵 = 8.5 T.
IV.
Dingle factor
No Dingle analysis was performed because of the pronounced increase in sheet
resistance of the background channel as the magnetic field is swept. Earlier works
on 2DEG-systems with parasitic parallel channels have shown that those tend to
alter the evolution of the oscillations with the magnetic field in a nontrivial
manner.[10 - 12] This cannot be accounted for by a simple fit to the Dingle
damping term 𝑅𝐷 .
[1] M. Lanius et al., Journal of Crystal Growth, (2016)
[2] P. Lošťák et al., physica status solidi (a) 115, 87 (1989)
[3] B. Seradjeh et al., Phys. Rev. Lett. 103, 136803 (2009)
[4] Z. Wang et al., Phys. Rev. B 82, 085429 (2010)
[5] A. A. Taskin and Y. Ando, Phys. Rev. B 84, 035301 (2011)
[6] A. R. Wright and R. H. McKenzie, Phys. Rev. B 87, 085411 (2013)
[7] L. Onsager, The London, Edinburgh, and Dublin Philosophical Magazine
and Journal of Science 43, 1006 (1952)
[8] G. P. Mikitik and Yu. V. Sharlai, Phys. Rev. Lett. 82, 2147 (1999)
[9] L. Plucinski et al., Journal of Applied Physics 113, 053706 (2013)
[10] V. Ariel and A. Natan, arXiv:1206.6100 (2012)
[10] E. F. Schubert et al., Applied Physics A 33, 63 (1984)
[11] S. J. Battersby et al., Solid-State Electronics 31, 1083 (1988)
[12] S. Contreras et al., Journal of Applied Physics 89, 1251 (2001)