Thermoelectric properties of topological insulator Bi2Te3, Sb2Te3, and Bi2Se3 thin filmquantum wellsHermann Osterhage, Johannes Gooth, Bacel Hamdou, Paul Gwozdz, Robert Zierold, and Kornelius Nielsch Citation: Applied Physics Letters 105, 123117 (2014); doi: 10.1063/1.4896680 View online: http://dx.doi.org/10.1063/1.4896680 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of carrier recombination on ultrafast carrier dynamics in thin films of the topological insulator Bi2Se3 Appl. Phys. Lett. 105, 171905 (2014); 10.1063/1.4901052 Intrinsic Rashba-like splitting in asymmetric Bi2Te3/Sb2Te3 heterogeneous topological insulator films Appl. Phys. Lett. 105, 082401 (2014); 10.1063/1.4893987 Enhanced surface mobility and quantum oscillations in topological insulator Bi1.5Sb0.5Te1.7Se1.3 nanoflakes Appl. Phys. Lett. 103, 163111 (2013); 10.1063/1.4826092 Topological insulator Bi2Te3 films synthesized by metal organic chemical vapor deposition Appl. Phys. Lett. 101, 162104 (2012); 10.1063/1.4760226 Epitaxial growth of Bi2Se3 topological insulator thin films on Si (111) J. Appl. Phys. 109, 103702 (2011); 10.1063/1.3585673

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Thermoelectric properties of topological insulator Bi2Te3, Sb2Te3,and Bi2Se3 thin film quantum wells

Hermann Osterhage,a) Johannes Gooth, Bacel Hamdou, Paul Gwozdz, Robert Zierold,and Kornelius Nielschb)

Institute of Applied Physics, Universit€at Hamburg, Jungiusstrasse 11, 20355 Hamburg, Germany

(Received 21 August 2014; accepted 16 September 2014; published online 26 September 2014)

The thermoelectric (TE) figure of merit ZT of topological insulator Bi2Te3, Sb2Te3, and Bi2Se3 thin

film quantum wells is calculated for thicknesses below 10 nm, for which hybridization of the

surface states as well as quantum confinement in the bulk are individually predicted to enhance ZT.

Here, the question is addressed what ZT can be expected from coexisting surface and bulk states in

such quantum wells. It is demonstrated that the parallel contributing bulk and surface channels tend

to cancel each other out. This is because the surface-to-volume ratios of the thin films prevent the

domination of transport through a single channel and because the individual bulk and surface ZTs

are optimized at different Fermi levels. VC 2014 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4896680]

In thin Bi2Te3 films (thickness d< 10 nm), quantum con-

finement is predicted to significantly enhance the thermoelec-

tric (TE) figure of merit ZT¼ S2rT/(jelþjph) at optimal

Fermi level position EF, far beyond the best thermoelectric

bulk efficiencies at room temperature of ZT ffi 1.1 Here, Sdenotes the Seebeck coefficient, r is the electrical conductiv-

ity, T is the temperature, jel is the electron, and jph is the pho-

non thermal conductivity. However, these predictions were

made before the topological insulator (TI) nature of Bi2Te3

was discovered.2 TIs exhibit a semiconducting bulk channel

and topological surface states, protected by time-reversal sym-

metry, which carry massless Dirac fermions.3,4 Hybridization

of the TI surface states leads to a gap-opening at the Dirac

point and has been individually predicted to enhance ZT.5–8

In TI thin films, both transport channels—bulk and sur-

face—contribute parallel to the charge transport in the film

plane. However, in any system of multiple parallel transport

channels, i.e., surface and bulk, the total ZT suffers from

mutually “parasitic” operation of the individual channels.9

This is particularly evident for systems with strongly diver-

gent single channel efficiencies. In fact, for classical topolog-

ical insulator thin films (d> 10 nm), it has been recently

shown that the interplay between surface and bulk channel

causes a remarkable modification of the thin films’ total ZTcompared to the individual efficiencies of the surface states

ZTs and the bulk ZTb,10 where the subscripts b and s denote

the bulk and surface channel, respectively.

Here, we demonstrate that the promising high maximum

efficiency predictions based on the single surface or single

bulk channel model cannot be held for TI thin films. We apply

a multiple-channel model for calculating the thermoelectric

transport properties of such thin film quantum wells with

thicknesses between 1 and 10 nm, where confinement effects

in the bulk and at the surface play a significant role. We find

that the total ZT of a TI thin film is rather the result of

competing ZTs and ZTb. ZT enhancement compared to classi-

cal TI films with thicknesses in the sub-10 nm size range is,

however, still possible but requires accurate control of the

thickness d and the position of the Fermi energy EF.

Specifically, we have investigated the ZT of an ideal Bi2Te3,

Sb2Te3, and Bi2Se3 TI thin film (as sketched in Fig. 1(a)) at

FIG. 1. (a) Sketch of a topological insulator film with two parallel transport

channels: bulk (green) and surface (blue). (b) Schematic diagram of the thick-

ness dependent energy landscape in thin films. Thick TI films show solid

bands (green) separated by an energy gap DEb and a gapless Dirac cone (blue

solid). In the surface channel of thin films, a gap DEHyb (blue dashed) opens

symmetrically around the Dirac point, induced by hybridization of TI states

of the two opposed faces. Quantum confinement leads to a splitting of va-

lence and conduction band into multiple subbands (green dashed). (c) and (d)

DEb and DEHyb as a function of film thickness d for Bi2Se3, Bi2Te3, and

Sb2Te3. Both gaps increase with decreasing thickness. A gap in the surface

states opens at d< 6 nm for Bi2Se3, d< 2 nm for Bi2Te3 and d< 4 nm for

Sb2Te3. The hybridization gaps were taken from literature.18–20

a)hosterha@physnet.uni-hamburg.deb)knielsch@physnet.uni-hamburg.de

0003-6951/2014/105(12)/123117/5/$30.00 VC 2014 AIP Publishing LLC105, 123117-1

APPLIED PHYSICS LETTERS 105, 123117 (2014)

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room temperature as a function of Fermi energy EF and film

thickness d, considering a semiconducting bulk and a topolog-

ical surface channel, which contribute parallel to the total TE

transport in the film plane. A two-channel model9,11,12 for cal-

culating the electronic transport parameters S, r, and jel has

been applied

r ¼ rb þ rs

2

d; (1)

S ¼Sbrb þ Ssrs

2

d

� �

rb þ rs

2

d

� � ; (2)

jel ¼ jel;b þ jel;s2

dþ

rbrs

2

d

rb þ rs

2

d

� � Sb � Ssð Þ2: (3)

The factor of 2 considers the top and the bottom surfaces of

the film. As in all figures, EF¼ 0 refers to the valence band

edge (VBE) of the film’s bulk without confinement effects.

The distance between Dirac point and VBE as well as all

other model parameters are obtained from literature. A com-

pilation of these parameters is given in the supplementary

material.13 We performed all calculations along the crystal

orientation of highest bulk mobility, parallel to the a0-axis,

similar to the previous calculations for the single bulk chan-

nel of Bi2Te3 nanostructures.1,14 The phonon contributions

to the thermal conductivity jph of the bulk channel were

taken from literature and correspond to bulk materials. A

possible reduction of jph in the film due to enhanced surface

scattering or impurities is not considered in our calculations.

The electronic transport properties of each individual chan-

nel are described by standard semi-classical Boltzmann

equations, considering a quantum well under the assumption

of a constant relaxation time s1,14,15

rb=s ¼ L0;b=s; (4)

Sb=s ¼ �1

eT

L1;b=s

L0;s=b; (5)

jel;b=s ¼1

T

L0;b=sL2;b=s � L21;b=s

L0;b=s

: (6)

The semiconducting bulk bands are calculated using the many

subband model introduced by Cornett and Rabin,15 with

Li;b ¼elx

p�h2d

Xn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimx;CBmy;CBp

ð1DEb

2þEn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiE� En

p� @f

@E

� �E� EFð ÞidEþ

Xm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimx;VBmy;VBp

ð�DEb2�Em

�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE� Em

p� @f

@E

� �E� EFð ÞidE

2664

3775

(7)

for the conduction band (CB) and VB, respectively. Here,

f ðEÞ ¼ ½exp ððE� lÞ=kBTÞ þ 1��1is the Fermi-Dirac distri-

bution and DEb denotes the bulk band gap without quan-

tum confinement. The charge mobility lx in the direction

of transport is considered to be equal for electrons and

holes. mi,CB/VB represents the effective mass of the elec-

trons/holes in the specific transport direction i. En/m are the

subbands induced by quantum confinement, where n and mdenote the quantum number of the energy level in the con-

fined dimension for electrons and holes, respectively.14,15

Considering the film bulk as a potential well of infinite

depth, Schr€odinger’s equation leads to a quantization in

energy of

Ej ¼ �h2p2 j2

2mz;CB=VBd2; with j ¼ n;m: (8)

The confinement in the z-direction results in splitting of the

conduction and valence bands into multiple subbands as well

as an increase of the bulk band gap of DEb¼DEb,0þEn¼ 0

þEm¼0 (Figs. 1(b) and 1(c)) with decreasing film thickness.

A correct simulation of the bulk transport properties is

achieved in Eq. (7) by summation over several subbands

(n,m> 20) following the example of Cornett and Rabin.15,16

For the surface states, we use5

Li;s ¼s

2h2

ð1

DEHyb

E2 � DE2Hyb

E

� E� EFð Þi

cosh2 E� EF

2

� �þ �E� EFð Þi

cosh2 Eþ EF

2

� �264

375dE; (9)

where DEHyb is the energy gap of the surface states induced

by hybridization of the opposed top and bottom surfaces of

the film.4,17 DEHyb as a function of film thickness (determined

by spin-orbiting coupling calculations,18 angle-resolved pho-

toemission spectroscopy (ARPES),2,19 and scanning tunnel

microscopy (STM))20 is shown in Figs. 1(b) and 1(d).

In Figs. 2(a)–2(f), the calculated thermoelectric efficien-

cies of both individual transport channels are depicted for the

three analyzed material systems Bi2Se3, Bi2Te3, and Sb2Te3.

As expected from literature, for thicker films (d> 10 nm),

ZTb(EF) exhibits the characteristic double-peak structure of a

two-band semiconductor,15 which transforms into peaks as a

result of the two-dimensional subbands induced by the quan-

tum confinement in the z-direction, when the film thickness is

reduced below 10 nm.16 The maximum ZT of the bulk channel

123117-2 Osterhage et al. Appl. Phys. Lett. 105, 123117 (2014)

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is located a few kBT below the first subband (n,m¼ 0)14,21 and

monotonically increases with decreasing d as formerly stated

by Hicks and Dresselhaus1 for Bi2Te3 thin films. In the surface

states of the thin films, the increasing hybridization gap leads

to a broadening of the electron and hole-cone ZTs peaks as

well as to a monotonic increase of their maximum TE effi-

ciency with decreasing d.

Applying the two channel model (Eqs. (1)–(3)), we

observe that the total ZT of the TI thin films is a function of

competing surface and bulk channel for all thicknesses

investigated. Although the surface channel increasingly

gains influence as d is decreased, ZT of the total films cannot

be solely described by one of the single channel models.

Crucially, when comparing Figs. 2(a)–2(f) with Figs.

2(g)–2(i), the corresponding higher efficiency peak of the

two individual transport channels is strongly suppressed in

the total ZT of the film as a result of the parallel contributing

transport channel with lower efficiency. In fact, in Sb2Te3

and Bi2Se3, the promising high ZTs values for films thinner

than 5 nm are up to two orders of magnitude suppressed by

the “parasitic” bulk channel, respectively. This is related to

the differing Fermi level dependencies of the thermoelectric

transport parameters of each individual channel13 and the

non-negligible weight of the bulk channel in the total trans-

port of the films, determined by their surface to volume ratio.

The question arises what maximum thermoelectric effi-

ciencies can be achieved at all in TI thin films. Therefore, we

have extracted the peak values at optimal Fermi level posi-

tion ZTmax for each film thickness as shown in Fig. 3. Due to

the enhanced surface-to-volume ratio and the comparably

high ZTs, Bi2Se3 thin films show higher ZTmax efficiencies

than the single bulk channel for d< 15 nm. Furthermore, we

observe that ZTmax of Bi2Te3 and Sb2Te3 TI thin films are

only improved for d< 2 nm and d< 4 nm, respectively.

Consequently, the best bulk thermoelectric efficiency of ZTffi 1 of Bi2Te3 known to date is annihilated for nanometer

thick films with d� 2 nm, even though ZTs is comparably

large. The TE efficiency of Bi2Te3 thin films suffer from the

different Fermi level positions of the ZTb and ZTs maxima.

While the individual maximum of ZTb and ZTs monotoni-

cally increase with decreasing film thickness, we observe

that ZTmax of the Bi2Te3 total film shows a local minimum

and ZTmax of Sb2Te3 shows a local maximum as a function

of d, as exemplarily highlighted in the inset in Fig. 3(c).

Such local extrema are the result of: first, interferences

between shifting bulk ZTb-peaks, arising from two-

dimensional subbands, and the ZTs maxima of the surface

states in EF. In Sb2Te3, the local maximum occurs if a bulk

subband ZTb-peak shifts at the Fermi level position of a ZTs-

peak of the surface states with equal polarity; second, a grad-

ual transition of the dominating channel in ZTmax of the total

film due to varying surface-to-volume ratio causing the local

minimum in Bi2Te3. A comparison between our calculations

and experiments would require thermoelectric measurements

on topological insulator Bi2Te3, Sb2Te3, or Bi2Se3 thin films

with thicknesses below 10 nm. Unfortunately, such measure-

ment data are not available at present. However, a reduction

of the total thermoelectric performance has been found in

various experiments on thin film topological insulators with

thicknesses above 20 nm, i.e., on Bi2Te3, Sb2Te3, and

Bi(2�x)SbxTe3.22–24 Although the position of the Fermi level

has neither been varied nor given in these studies, our calcu-

lations reflect this overall experimental observation very

well.

Given these findings, we conclude that the TE perform-

ance of Bi2Te3, Sb2Te3, and Bi2Se3 TI thin film quantum

FIG. 2. Thermoelectric figure of merit

ZT of the bulk channel (top), of the sur-

face channel (middle, logarithmic

scale), and of the combined system of

thin (d< 10 nm) (a), (d), and (g)

Bi2Se3; (b), (e), and (h) Bi2Te3; and (c),

(f), and (i) Sb2Te3 TI films as a function

of Fermi level EF for various film thick-

nesses d at 300 K. In general, the

hybridization of the surface states is

stronger, the smaller d, but its absolute

strength depends on the material of

choice. The blue curves correspond to

non-hybridized surface states and thus

to a gapless Dirac cone. Down to

d¼ 1 nm, the ZT of the total film of all

investigated systems is the result of

competing surface and bulk channels

and is not solely shaped by one of the

single channels. With decreasing film

thickness, the surface states increasingly

govern influence in ZT of the total film

as a simple consequence of enhancing

surface-to-volume ratio. This leads to

equivalent thermoelectric efficiencies

for electrons and holes in thin Bi2Se3

and Sb2Te3 thin films and a change of

polarity in the maximum ZT for Bi2Te3

thin films compared to bulk, from n- to

p-type.

123117-3 Osterhage et al. Appl. Phys. Lett. 105, 123117 (2014)

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wells is strongly influenced by both, the surface states and

the bulk for 1 nm� d� 20 nm. The details depend on the

film thickness, the height of the ZTs and ZTb maxima, their

Fermi level position, as well as their polarity. As a conse-

quence, the promising predictions based on the single TI sur-

face state cannot be held for TI thin films wherein both

channels contribute to the TE transport. Applying a two-

channel model to calculate the thermoelectric properties of

TI thin films, ZT can be actually enhanced compared to the

predictions based on a single bulk-model. However, values

beyond ZT¼ 2 cannot be achieved in these structures. In

order to observe a significant enhancement of the TE effi-

ciency in TI thin film quantum wells far beyond ZT¼ 1 in

the experiments, care must be taken in the ongoing develop-

ment of thermoelectric TI nanostructures. The total ZT of a

TI thin film can be remarkably suppressed even if the maxi-

mum efficiencies of the surface and the bulk channel individ-

ually remain large because ZTb and ZTs are optimized at

different Fermi levels or have different polarities. Thus, indi-

vidual tuning of the carrier concentration in the two channels

by, e.g., a gate voltage25–27 or gradual doping28–30 might

pave the way to overcome these difficulties. Strong bulk con-

finement and strong surface state hybridization are benefi-

cial. Since the thermoelectric properties of a TI thin film will

always suffer from the transport channel with lower effi-

ciency, a complete suppression of the low-efficiency channel

could also be a future route to create highly efficient TI ther-

moelectric materials.

This work was supported by the Deutsche

Forschungsgemeinschaft (DFG) via Graduiertenkolleg 1286

“Functional Metal-Semiconductor Hybrid Systems,” SPP

1386 “Nanostructured Thermoelectrics,” SPP 1666

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FIG. 3. Maximum ZT of the single bulk (green) and the single surface (red)

channel as well as of the total film (blue) as a function of thickness d for a

(a) Bi2Se3; (b) Bi2Te3; and (c) Sb2Te3 film. In all investigated systems, mu-

tual shortening of the parallel contributing surface and bulk channels causes

a suppression of the maximal ZT of the total film compared to the highest ZTof the single channels. Due to increasing hybridization and confinement

strength with decreasing film thickness, the ZTmax of each individual trans-

port channel monotonically increases with decreasing d for all systems

investigated, while local extrema occur in the thickness-dependent optimum

TE performance of Bi2Te3 and Sb2Te3 films.

123117-4 Osterhage et al. Appl. Phys. Lett. 105, 123117 (2014)

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