Thermoelectric properties of solution-synthesized n

Nano Res.

Electronic Supplementary Material

Thermoelectric properties of solution-synthesized n-type Bi2Te3 nanocomposites modulated by Se: An experimental and theoretical study Haiyu Fang1,†, Je-Hyeong Bahk2,‡, Tianli Feng3, Zhe Cheng4, Amr M. S. Mohammed2, Xinwei Wang4, Xiulin Ruan3, Ali Shakouri2, and Yue Wu5 () 1

School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA 3 School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA 4 Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA 5 Department of Chemical and Biological Engineering, Iowa State University, Ames, IA 50011, USA † Present address: Materials Research Laboratory, UC Santa Barbara, Santa Barbara, CA 93106, USA ‡ Present address: Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221, USA 2

Supporting information to DOI 10.1007/s12274-015-0892-x

S1

Measurement of thermoelectric properties

S1.1 Cross-plane thermal conductivity measurement

The pellets were cut and polished into required dimensions. Seebeck coefficient was measured in home built system by bridging the sample between a heater and heat sink and testing the voltage and temperature difference between the hot and the cold sides in a vacuum chamber. Electrical conductivity was measured with Van der Pauw method in a system where a MMR K-20 temperature stage was used to control sample temperature and an Agilent was connected to provide source current and collect voltage signals. Hall Effect was carried out by applying magnetic field up to 1 Tesla to the electrical conductivity measurement system. The cross-plane thermal conductivity was calculated via the equation κ = αρCp (ρ is the density and Cp is heat capacity) and the thermal diffusivity (α) was measured through the laser flush method. All the measurements were carried out under vacuum in the temperature range from 300 to 500 K. S1.2

In-plane thermal conductivity measurement

The schematic of the experimental setup for characterizing the in-plane thermal conductivity is shown in Fig. S3. The sample was fixed on the aluminum heat sink with silver paste that was also used to connect the Address correspondence to [email protected]

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thermal couple to the suspended end of the sample. To suppress heat convection, the setup was placed in a vacuum chamber with pressure of 2.1–2.3 mTorr. A 1,550 nm laser was used to heat the sample. To enhance laser energy absorption, a black marker was used to paint the sample surface facing the laser. Two lenses were used to enlarge the laser spot before the laser entered the vacuum chamber. Since the laser spot is much larger than the sample size, the laser energy flux hit on the sample surface can be taken as uniform. The heat sink was at room temperature. After laser was applied, the temperature of the suspended sample end rose and it was measured by the thermal couple. The temperature rise is related to the sample’s thermal conductivity as T  Ql (2 wd) . Here, Q is the heating energy, l is the sample length, κ is the thermal conductivity, w and d are the width and thickness respectively. Then the thermal conductivity can be expressed as   Ql (2 Twd) . To measure the heating energy, a piece of hard and thick paper was placed at the position of the sample. The paper was used to block the laser and a hole sharing the same size of the sample was cut in the paper. The light passed through the hole and a laser power meter (POWER MAX500D) was placed underneath the paper to measure the laser energy. The sample length, width and thickness were measured with micrometer and optical microscope. Then the thermal conductivity of the sample was calculated. The calculated thermal conductivity includes the effect of thermal radiation. The effect of radiation can be calculated as  rad  8 T 3 l 2 (d2 ) . Here,  is the emissivity,   is Stefan-Boltzmann constant, T is the surface temperature. For the emissivity, the value for the surface facing the laser is 1 and the value for the other surface is estimated to be 0.5, so the average emissivity of the sample is estimated to be 0.75. The maximum effect of radiation is very small (about 3%) so that it would not bring significant error to the final results. The real thermal conductivity of the sample is  real     rad . To improve the accuracy, one sample was measured for four times with different laser energy and temperature rises. The main error source in this experiment is the laser energy absorption rate, namely the emissivity. The sample surface is coarse and black so the emissivity is taken as 1 in this experiment. The error due to the emissivity is estimated to be 5%. The relative error of the geometrical measurement is estimated to be 1% and the relative error of the thermal couple is also 1%. Therefore, the total relative error of the thermal conductivity is estimated to be 5.4%.

S2

Theoretical modeling

S2.1 Carrier transport modeling

We modeled the carrier transport in the Bi2(Te1–xSex)3 nanocomposites based on the linearized Boltzmann transport equations (BTE) under the relaxation time approximation. The differential conductivity σd(E) is defined as  f0    E 

 d ( E)  e 2 ( E)v 2 ( E)  DOS ( E)  

(S1)

where e is the electron charge, τ is the total relaxation time, ρDOS is the density of states, v is the carrier velocity in one direction, and f0 is the Fermi-Dirac distribution. For the multiple-band transports in Bi2(Te1–xSex)3, the transport properties are calculated by summing all the contributions from each band. The electrical conductivity σ, the Seebeck coefficient S, and the electronic thermal conductivity e are given, respectively, by

     d ( E)dE

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(S2)

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S

1   d ( E)( E  EF )dE qT 

(S3)

 e  L T

(S4)

where Σ is sum over the bands, q is −e for conduction bands, and +e for valence bands, T is the absolute temperature, and EF is the relative position of the Fermi level to each of the band edge. In fact, the Lorenz number is a function of the Fermi level and band structure, but we found that all the samples in this paper were highly degenerate, so that the Lorenz number can be assumed to be the conventional value, 2.44 × 10–8 W·Ω–1·K–2. More information about the electron transport modeling is found elsewhere [S1]. We included two major conduction bands and two valence bands in our transport modeling, all of which are modeled as non-parabolic bands with 6 band degeneracy each. The band gap, effective masses are all dependent on the Se content, x, found from literature, but assumed to be temperature-independent in this paper. The relaxation time is determined by several major scattering mechanisms in Bi2(Te1–xSex)3. Acoustic phonon deformation potential scattering is a major scattering mechanism for electrons in bulk. In the nanocomposites, we added intense ionized defect scatterings at grain boundaries with the defect density as a fitting parameter to fit the mobility. S2.2

Lattice thermal conductivity modeling

The calculation of lattice thermal conductivity as a function of electron mobility consists of two steps. The first step is to figure out the average grain diameter for a given electron mobility by using electron grain boundary scattering. The second step is to estimate the thermal conductivity according to the average grain diameter and the phonon-boundary scattering. S2.2.1 Calculate the average grain diameter The electron mobility  and the grain diameter D in the nanocomposites are related by the electron scattering with the Matthiessen’s rule 1



  

1

imp 1

 bulk 1

 bulk

  

1

def



1

latt



1

 bound

1

(S5)

 bound m* vF eD

where 1 / imp , 1 / def , 1 / latt , etc., represent electron impurity scattering, electron defect scattering, electron phonon (lattice) scattering, etc., in bulk Bi2Te3 with 10% Se. The summation of them gives the reciprocal mobility 1 /  bulk in the bulk Bi2Te3 with 10% Se. The last term 1 /  bound represents the electron boundary scattering

in nanostructured Bi2Te3 with 10% Se, which is determined by  bound 

e e D  * . Here m* and vF are the * m m vF

effective electron mass and the Fermi velocities of electrons, respectively. Based on Eq. S5, the average grain diameter D for a given electron mobility  is

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1 1 D       bulk

1

 m* v   e F 

(S6)

For the three carrier concentrations, i.e. 2 × 1019, 9 × 1019, and 1.6 × 1020 cm–3, the bulk mobilities  bulk are 320, 250, and 220 cm2·V–1·s–1, and the Fermi velocities vF are estimated as 1.5 × 105, 2.2 × 105, and 2.7 × 105 m·s–1, respectively. The effective electron mass is m*  0.7 m0 , with m0 representing the electron static mass. Based on Eq. (S6), the average grain diameter of samples 5 and 6 are estimated around 4.35 nm. Such grain size reduces the electron mean free path  e  vF bulk 

vF  bulk m* (24 nm) in bulk Bi2Te3 with 10% Se and carrier concentration e

of 1.6 × 1020 cm–3 to 3.7 nm in nanostructured Bi2Te3 with 10% Se (e.g. samples 5 and 6). S2.2.2 Calculate the lattice thermal conductivity Similar to electron transport, the effective phonon mean free path p in nanocomposites is determined by phonon scattering in bulk and phonon-boundary scattering in nano-grains 1

p



1

 p,bulk



1 D

Based on kinetic theory, the lattice thermal conductivity is  l 

(S7) 1 max c( )vg ( )p,sp ( )d  p , where  is 3 0

1 max c( )vg ( )d with  , c, vg and  p,sp representing the phonon frequency, specific heat, 3 0 group velocity and spectral mean free path, respectively. Based on our previous calculation [S2] and the literature [S3, S4],  is estimated as 5.767 × 108 W·m–2·K–1. Multiplying Eq. (S7) by  gives the grain diameter dependent lattice thermal conductivity of nanocoposites

defined as  

1

l



1

 l,bulk



1

(S8)

D

Here the unknown  l,bulk represents the lattice thermal conductivity of bulk Bi2Te3 with 10% Se and is obtained in the following approach. Samples 5 and 6 have the electrical conductivities of 907 and 829 S·cm–1, and thus electronic thermal conductivities  e  L T of 0.664 and 0.607 W·m–1·K–1, respectively. Their in-plane thermal conductivities were measured as 1.63 and 1.42 W·m–1·K–1, respectively. Thus their lattice thermal conductivities are estimated as 0.966 and 0.813 W·m–1·K–1, respectively. Sample 5 has a bit larger lattice thermal conductivity is owing to its lower Se concentration and thus less phonon impurity scattering. Since the Se concentration of 10% in our modeling is very close and in between our samples 5 (8.20% ± 0.97% Se) and 6 (11.54% ± 0.68% Se), and the grain diameters of samples 5 and 6 are both around 4.35 nm, thus, the lattice thermal conductivity of our modeling material, i.e. Se concentration of 10%, with grain diameter of 4.35 nm is estimated as  l  0.89 W·m–1·K–1, 1

1 1  the average of 0.966 and 0.813 W·m–1·K–1. Finally  l,bulk is obtained as  l,bulk      1.38 W·m–1·K–1.   D  l  Since both the constants  l,bulk and  have been obtained, the lattice thermal conductivity as a function of

grain diameter D, or electron mobility  , can be calculated based on Eq. (S8).


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S3

Supplementary figures

Figure S1

Scanning electron microscope images at the cross section of hot press pellets. Sample ID is labeled on the images.

Figure S2 Temperature dependent thermal conductivity of samples 1, 3, 5 and 6 along the cross-plane direction.

Figure S3 Schematic of the experimental setup for characterizing the in-plane thermal conductivity. www.theNanoResearch.com∣www.Springer.com/journal/12274 | Nano

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Reference [S1] Bahk, J.-H.; Shakouri, A. Electron transport engineering by nanostructures for efficient thermoelectrics. In Nanoscale Thermoelectrics, Lecture Notes in Nanoscale Science and Technology; Wang, X. D.; Wang, Z. M., Eds; Springer International Publishing: Switzerland, 2014; vol. 16, pp41–92. [S2] Wang, Y. G.; Qiu, B.; McGaughey, A. J. H.; Ruan, X. L.; Xu, X. F. Mode-wise thermal conductivity of bismuth telluride. J. Heat Trans. 2013, 135, 091102. [S3] Mavrokefalos, A.; Moore, A. L.; Pettes, M. T.; Shi, L.; Wang, W.; Li, X. G. Thermoelectric and structural characterizations of individual electrodeposited bismuth telluride nanowires. J. Appl. Phys. 2009, 105, 104318. [S4] Hellman, O.; Broido, D. A. Phonon thermal transport in Bi2Te3 from first principles. Phys. Rev. B 2014, 90, 134309.
