Supplemental material for the manuscript
Thermoelectric properties of topological insulator Bi2Te3, Sb2Te3 and Bi2Se3 thin film
quantum wells
By Hermann Osterhage, Johannes Gooth, Bacel Hamdou, Paul Gwozdz, Robert Zierold and
Kornelius Nielsch
Thermoelectric performance of the bulk channel
In our calculations, we treat the bulk channel as a three-dimensional semiconductor with a
parabolic dispersion relation:
2 2
 2 k x2  k y  2 k z2
E k x ,k y ,k z  


2m x
2m y
2m z
(1)
Table S1 shows the energy gaps ΔEb between the valence and the conduction band for all
three materials. The effective masses for holes (VB) and electrons (CB) are displayed in
Table S2. The mobility along the direction of transport µx and the phonon thermal
conductivity κph are depicted in Table S3. µx is presumed to have equal values for electrons
and holes. Both, µx and κph are considered to be constant coefficients with respect to the film
thickness. We chose the direction of transport parallel to the x-axis in the hexagonal unit cell
to obtain maximum ZT values and hence maximum thermoelectric efficiency. The
thermoelectric coefficients Sb, σb and κb of the bulk channel are shown in Figure S1.
Thermoelectric performance of the surface channel
We calculated the thermoelectric transport coefficients Ss, σs and κs of the surface states
assuming a symmetrical splitting of the Dirac cone around the Dirac point into two separate
bands, induced by hybridization of the two surfaces, in thin films.1 The thickness-dependent
development of the hybridization gaps ΔEHyb is shown in Table S4. Bi2Se3, Bi2Te3 and Sb2Te3
have a layered hexagonal crystal structure. Atomic layers of the two specific compounds are
alternately stacked upon each other perpendicular to the c-axis. Five of those layers form a
quintuple layer (QL) with a stoichiometric composition.1-3 Furthermore, we presume the mean
free path4, 5 λ = 200 nm and the Fermi velocity6 vF = 4×105 ms-1 to be constant and equal for
carriers in the hybridized and in the gapless system. The relaxation time τ therefore yields
τ = λ/vF = 5×10-13 ms. Further important assumptions are, that the surface states do not
penetrate the material, which is not entirely true for real systems, and that there is no
phononic contribution to the thermal conductivity of the surface channel. Figure S2 shows the
thermoelectric transport coefficients of the surface channel for several film thicknesses d. The
thermoelectric performance of the surface channel only depends on the width of the energy
gap ΔEHyb. The maximum Seebeck coefficient S increases with increasing ΔEHyb, while the
electric conductivity σ as well as the electronic thermal conductivity κ is decreased.
Consequently, ZT = σS²/κ increases for big hybridization gaps ΔEHyb and hence for thin films.
Bi2Se3
Bi2Te3
Sb2Te3
ΔDP [meV]
145
-265
-38
ΔEbulk [meV]
300
105
90
Table S1: Gap between Dirac point and VBM (ΔDP)7 and bulk band gap (ΔEb).
Bi2Se3
Bi2Te3
Sb2Te3
VB
CB
VB
CB
VB
CB
mx [m0]
-0.125
0.109
-0.024
0.178
-0.054
0.045
my [m0]
-0.125
0.122
-0.134
0.178
-0.054
0.045
mz [m0]
-0.125
0.240
-1.921
0.835
-0.102
0.114
Table S2: Effective masses for valence band (VB) and conduction band (CB) of Bi 2Se38,
Bi2Te3 and Sb2Te3.9
Bi2Se3
Bi2Te3
Sb2Te3
μx [cm²V-1s-1]
600
1200
3500
κph [Wm-1K-1]
4.0
1.5
2.4
Table S3: Carrier mobility10,11 μx along the transport direction and phonon thermal
conductivity8,12 κph for Bi2Se3, Bi2Te3 and Sb2Te3, respectively. The carrier mobility is
presumed to be equal for electrons and holes.
1QL
2QL
3QL
4QL
5QL
6QL
Bi2Se3
0.71 eV
0.098 eV
0.04 eV
0.012 eV
0.004 eV
0 eV
Bi2Te3
0.14 eV
0 eV
0 eV
0 eV
0 eV
0 eV
Sb2Te3
not avail.
0.255 eV
0.06 eV
0 eV
0 eV
0 eV
Table S4: Energy gaps ΔEHyb in the surface states‘ dispersion relation induced by
hybridization effects as measured via angle resolved photon emission spectroscopy (ARPES)
on thin films of several thicknesses for Bi2Se313, Bi2Te32 and Sb2Te33. The declaration QL is
short for quintuple layer, which is a composition of five atomic layers (e. g. 2xBi and 3xSe).
A quintuple layer of each material is about 1 nm thick. For a 1 nm thick film of Sb2Te3 is no
data available.
Figure S1. Thickness-dependent thermoelectric key figures of the bulk channel in thin films
for Bi2Se3, Bi2Te3 and Sb2Te3. (a), (b), (c) Seebeck coefficient S; (d), (e), (f) electrical
conductivity σ; and (g), (h), (i) thermal conductivity κ = κel+κph as a function of the Fermi
level EF.
Figure S2. Thickness-dependent thermoelectric transport properties of the surface channel in
thin TI films for Bi2Se3, Bi2Te3 and Sb2Te3. (a), (b), (c) Seebeck coefficient S; (d), (e), (f)
electrical conductivity σ; and (g), (h), (i) electron thermal conductivity κ as a function of the
Fermi level EF and for several film thicknesses d. The blue graphs correspond to surface states
with a gapless Dirac cone in the dispersion relation (no hybridization, see Table S4).
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