Content uploaded by Alexander Fedotov

Author content

All content in this area was uploaded by Alexander Fedotov on Jan 13, 2021

Content may be subject to copyright.

SIMULATION OF POLYCRYSTALLINE BISMUTH FILMS SEEBECK

COEFFICIENT BASED ON EXPERIMENTAL TEXTURE IDENTIFICATION

Alexander S. Fedotov1, Vasiliy Shepelevich1, Sergey Poznyak2, Lyudmila Tsybulskaya2,

Alexander Mazanik1, Ivan Svito1, Sofia Gusakova1, Pawel Zukowski3,

Tomasz N. Koltunowicz3,

1 Belarusian State University, Minsk, Belarus

2 Research Institute for Physical Chemical Problems, Belarusian State University, Minsk,

Belarus

3 Lublin University of Technology, Lublin, Poland

Abstract. Seebeck coefficient values measured for textured polycrystalline Bi films have

been compared with numerically computed ones. The proposed computation procedure

requires crystallographic texture mapping and values of single-crystalline transport

coefficients as input data. The comparison of computed and measured Seebeck coefficients

allowed determining the factors affecting films’ transport properties except of crystalline

texture. Two particular cases are considered: when the film properties are affected only by

texture and when they depend on texture and grain size simultaneously.

Keywords: Bismuth; Polycrystalline films; Crystal structure; Transport properties.

Introduction

It is common knowledge that properties of polycrystalline materials are determined by

such factors as interface electronic states, mean free path limitations, and, in the case of

materials with a pronounced anisotropy of single-crystalline properties, by texture type [1-3].

It is a challenge usually to distinguish these physical mechanisms, especially when crystallites

are highly anisotropic and film shows a clearly defined texture.

Bismuth is known as semimetal possessing a significant anisotropy of transport

properties in the single-crystalline form. It is a basic material for many of thermoelectrical

applications [4, 5] that justifies a search for the ways of its Seebeck coefficient improvement

[6].

In the present report Seebeck coefficient of polycrystalline bismuth films with different

grain sizes and crystallographic texture has been studied. We propose a technique for

computation of polycrystalline film’s Seebeck coefficient on the basis of solution of a partial

differential equation system, considering each grain as a continuous medium with transport

coefficients corresponding to the texture. Additional factors, except of crystalline texture,

affecting the transport properties of polycrystalline Bi films are also considered.

Preparation of samples and texture identification

Two types of polycrystalline bismuth films were studied: (a) films prepared by the melt

spinning (MS) and (b) ones obtained by the electrochemical deposition (ECD). Both synthesis

techniques were well-developed and tested in our previous works [7,8].

In the MS technique, melted Bi was spilled on the room-temperature surface of a

rotating copper cylinder. Cooling rate of Bi reached appr. 106 K/s [9]. Thickness of the

fabricated MS films was about 40 µm.

ECD synthesis was performed in the galvanostatic mode from an aqueous electrolyte,

containing 0.174 M bismuth perchloride dissolved in 3 M perchloric acid. The

Corresponding author e-mail:t.koltunowicz@pollub.pl, fax: +48-81-5384575, tel: +48-81-5384713

electrodeposited Bi films were rinsed with distilled water and then picked off from the

substrate. Thickness of the electrodeposited films was about 70 µm.

Several series of samples were fabricated by both methods under the same conditions.

Five samples obtained by the MS and more than ten ECD films were investigated by scanning

electron microscopy and demonstrated good reproducibility in their structure.

For experimental determination of Seebeck coefficient, a heater element was powered

by an Agilent U3606B DC power source creating a temperature gradient about 1.5 K on the

sample investigated. Temperatures on cold and hot ends of the sample were determined using

platinum resistance thermometers connected to Agilent 34410A digital multimeters and

providing an accuracy of temperature determination about 0.05 K. Thermal EMF was

measured by an Agilent 34411A multimeter with an accuracy of about 3.5 μV.

Samples for electrophysical measurements were cut from the center of each as-

synthesized film to avoid an influence of boundaries on the structure. Seebeck coefficients

determined for different samples of each type fluctuate around the corresponding average

values within 1%.

A HZG-4M X-ray diffractometer (Cu Kα radiation) and a LEO 1455VP scanning

electron microscope with an add-on for electron backscatter diffraction (EBSD) analysis were

used for identification of the texture.

According to the SEM data, Bi grain sizes range from 5 to 15 µm for the films

fabricated by the MS and from 0.5 to 1.5 µm for the ECD ones.

(a) (b)

(c)

Fig. 1. EBSD texture mapping for particular grains (in-film plane) of the MS (a) and ECD

films (b) and the color legend (c).

As follows from Fig. 1, two types of in-plane growth texture are observed: (001) and

(012). XRD analysis (Fig. 2a) demonstrated only the presence of reflex corresponding to

(012), but this divergence is well-known and related to the relaxation of the (001) line in

bismuth [10]. Quantitative EBSD analysis showed that the films fabricated by the MS possess

(001) and (012) textures for 39% and 54% of grains, respectively. The ECD films have (001)

texture for 36% and (012) texture for 57% of the grains. Relevant crystallographic planes are

shown in Fig. 2b.

20 40 60 80 100

0.0

0.5

1.0

0.0

0.5

1.0

2deg.

012

Intensity, a.u.

Electrochemical deposition

024

024

012

Melt spinning

(a) (b)

Fig. 2. XRD patterns (a) and the corresponding crystallographic planes (b) in Bi lattice.

Mathematical model and Seebeck coefficient computation

We can use the following equation system [11] to describe the coupled processes of

heat and charge transfer:

VJJSTTk

t

T

C

TSVJ

)(ρ

)(σ

p

, (1)

where J

is the current density, T – the temperature, V – the electric potential, S – the Seebeck

coefficient, σ – the electrical conductivity, k – the thermal conductivity, Cp – the heat

capacity, ρ – the density, and t – the time.

We have set the computational domain consisted of 48 subdomains (according to the

SEM data, the MS films possess a column-like structure). Each subdomain is 15×15×40 µm3

parallelepiped (Fig. 3), corresponding to the measured foil thickness (40 µm) and average

lateral grain size (15 µm). The whole domain size is 120×90×40 µm3.

For computation of Seebeck coefficient, boundary conditions were set to correspond to

the open circuit regime at a small temperature gradient between two opposite faces, while the

rest boundary faces were considered as thermally insulated:

294 0

293 120

0,

( ) 0, 0 120, ( , )

T , x , (y, z)

Γ

T , x , (y, z)

Γ

n J x

n k T x y z

, (2)

where Г is the boundary of the domain and n

is the unit vector normal to it.

For one of the faces, between which the temperature gradient was set, the potential was

considered as equal to zero. The value of thermo-EMF was obtained by averaging the voltage

through entire face with non-zero-voltage condition.

Fig. 3. Boundary conditions over the computational domain.

For quantitative predictions of the electrical conductivity, additional initial-boundary

value problems were solved on the same computational domain:

),( ,1200 ,0

),( ,120 ,101

),( ,0 ,101

),,( ,293

5

5

zyxJn

zyxJn

zyxJn

zyxT

. (3)

The voltage drop across the rectangular sample was obtained by averaging potential

distributions on the both ends of the sample (for x=0 and x=120 boundaries) and subtracting

one from another. The conductivity of the sample was determined as a ratio of the current

density to the voltage drop, multiplied by the sample length.

All the simulations were performed for stationary cases using the finite element method.

Mesh consisted of 5∙105 tetrahedral elements. Conventional PARDISO 5.0 solver was used

with pivoting perturbation order of 10-8 [12, 13].

Transport coefficients of single crystal, such as conductivity, Seebeck coefficient and

others, are tensors [14]. To take into account the crystalline texture, it is necessary to

transform the tensors using the rotation matrix [15]. However, some geometrical relations

could simplify the procedure of transport properties assignment to the simulation domain.

Bismuth lattice cell is suitable to be described in the hexagonal coordinate system [10, 14].

Therefore it is common to distinguish two types of crystallographic directions with their own

transport coefficients. The first one coincides with C3 symmetry axis of the crystal cell and is

called c direction, whereas the directions lying in the plane normal to the C3 axis are called a

directions (Fig. 2b). If the (001) plane is parallel to the film surface, it leads to a direction

coefficients for film in-plane transport (Fig. 1b). In the case of the (012) plane parallel to the

surface, transport processes occur with the values of transport coefficients ranging between a

lower limit (values for a direction ) and an upper limit Cup, which can be approximated as

( )sin

up a c a

C C C C

, (4)

where θ is the angle between the (001) and (012) planes, Ca and Cc – the transport coefficients

for corresponding directions taken from [14].

For the MS films, we excluded from the consideration 7% of the grains, which do not

show (001) or (012) dominating textures. Then, we assumed that every subdomain has the

probability to possess transport coefficients of a direction equal to 71%. The rest 29% of

grains oriented with the (012) plane parallel to the sample surface have equal probabilities to

possess transport coefficients corresponding to a direction or coefficients as given in (4)

(bimodal distribution with equal probabilities). Similar analysis was also performed for the

ECD films taking into account peculiarities of their texture.

Fig. 4 shows the dependence of Seebeck coefficient and electrical conductivity on the

volume percentage, x, of the grains oriented with c direction parallel to the temperature

gradient, whereas the rest of the grains are oriented with a direction parallel to the

temperature gradient. A linear dependence of Seebeck coefficient on the fraction of grains

oriented with c direction is obtained as a result of the calculations.

0 25 50 75 100

7x105

8x105

9x105

, Sm/m

x, %

Conductivity

Seebeck coefficient

60

80

100

|S|, V/K

Fig. 4. Calculated dependences of Seebeck coefficient and electrical conductivity for ideal bi-

textured film on the volume percentage of the grains oriented with c direction parallel to the

temperature gradient.

The performed calculation for the MS films returned the value of Seebeck coefficient

equal to 62 µV/K, whereas the experimentally measured value is 63 µV/K. For the ECD films

the divergence is rather large: 66 µV/K according to the calculations and 75 µV/K in the

experiment.

As mentioned above, the grain size for the ECD films ranges from 0.5 to 1.5 µm. The

mean free path of charge carriers in bismuth at 300 K is about 250 nm [16]. It is reasonable to

suggest that the limitation of charge carrier mean free path by the scattering on grain

boundaries results in modification of the transport properties. Therefore, it is improperly to

use experimental data related to large-sized single-crystals for defining the transport

coefficients of the fine-grained ECD films.

Mobilities of electrons μn and holes μp can be calculated using Seebeck coefficient and

electrical conductivity:

)μμ(σ pn pne , (5)

pn

ppnn

μμ

μμ

pn

pSnS

S

. (6)

Here the concentrations of electrons, n, and holes, p, are independent on texture and can be

taken from [8]. Partial Seebeck coefficients Sn and Sp can be calculated using the Lax model

[16].

Results of the computation demonstrate (Fig. 5) that the mobility of electrons is almost

independent on the texture, while the mobility of holes decreases appreciably with x. This fact

can be explained taking into account that T-holes pocket is prolate in k-space along the c

direction, while three pockets of L-electrons are distributed symmetrically [17]. Such form of

the T-holes pocket leads to a higher effective mass of holes in the direction of prolongation as

compared with that in the other two directions and, consequently, to the reduction of holes

mobility in c direction.

0 20 40 60 80 100

0.1

0.2

0.3

0.4

0.5

0.6

, m

2

/(VS)

x, %

n

p

Fig. 5. Calculated mobilities of electrons (squares) and holes (circles) for bi-textured film

depending on the volume percentage of grains oriented with c crystallographic direction

parallel to the temperature gradient.

Conclusions

Two types of polycrystalline bismuth films were examined to highlight factors affecting

their transport properties depending on crystalline texture.

The proposed numerical technique predicts successfully Seebeck coefficient for the

samples synthesized by the melt spinning and possessing a clearly defined texture. The results

of calculation coincide with the experimental data with a high accuracy (about 1 %). For the

samples synthesized by the electrochemical deposition, discrepancy between the calculated

and experimental values of Seebeck coefficient reaches 15 %, indicating the presence of

additional factors affecting Seebeck coefficient in such films. The last is in agreement with

SEM analysis data showing that grain size for the ECD samples is of the order of charge

carriers’ mean free path that produces an additional impact on the transport properties.

The computation technique described can be used for determining Seebeck coefficient of

anisotropic polycrystalline (or just composite-like) materials and has to be applicable to the

films with grain sizes significantly larger than mean free path of charge carriers.

References

[1] U.F. Kocks, C.N. Tomé, R. Wenk, Preferred Orientations in Polycrystals and their

Effect on Materials Properties, Cambridge University Press, (2000).

[2] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press,

(1995). doi:http://dx.doi.org/10.1017/CBO9780511805776.

[3] M. Lundstrom, C. Jeong, Near-Equilibrium Transport. Fundamentals and Applications,

World Scientific Publishing, (2013). doi:http://dx.doi.org/10.1142/7975.

[4] D.M. Rowe, Thermoelectrics Handbook: Macro to Nano, CRC Press, (2005).

doi:http://dx.doi.org/10.1201/9781420038903.

[5] H.J. Goldsmid, Bismuth Telluride and Its Alloys as Materials for Thermoelectric

Generation, Materials 7 (2014) 2577–2592. doi:http://dx.doi.org/10.3390/ma7042577

[6] H.J. Goldsmid, Introduction to Thermoelectricity, Springer, Berlin, (2010).

doi:http://dx.doi.org/10.1007/978-3-642-00716-3.

[7] A.S. Fedotov, S.K. Poznyak, L.S. Tsybulskaya, V.G. Shepelevich, I.A. Svito, A. Saad,

A. Mazanik, A.K. Fedotov, Polycrystalline bismuth films: correlation between grain

structure and electron transport, Phys. Status Solidi B, 252 (9) (2015) 2000-2005.

doi:http://dx.doi.org/10.1002/pssb.201552051.

[8] A.S. Fedotov, S.K. Poznyak, L.S. Tsybulskaya, I.A. Svito, V.G. Shepelevich, A. V.

Mazanik, A.K. Fedotov, T. V. Gaevskaya, Synthesis, structure and electrophysical

properties of polycrystalline bismuth films, Proceedings of International Conference

Sviridov Readings 2015, Minsk. http://elib.bsu.by/handle/123456789/118049.

[9] W.J. Boettinger, J H. Perepezko, J.H. Liebermann, Rapidly Solidified Alloys:

Processes, Structures, Properties,Applications, Marcel Dekker, Inc, New York, (1993).

[10] P. Cucka, C.S. Barrett, The crystal structure of Bi and of solid solutions of Pb, Sn, Sb

and Te in Bi, Acta Crystallogr, 15 (9) (1962) 865–872.

doi:http://dx.doi.org/10.1107/s0365110x62002297.

[11] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media Edition,

Butterworth Heinemann, Oxford, (1984). doi:http://dx.doi.org/10.1016/B978-0-08-

030275-1.50024-2.

[12] O. Schenk, M. Bollhöfer, and R. Römer, On large-scale diagonalization techniques for

the Anderson model of localization, SIAM Review 50 (2008) 91-112.

doi:http://dx.doi.org/10.1137/070707002.

[13] O. Schenk, A. Wächter, M. Hagemann, Matching-based Preprocessing Algorithms to

the Solution of Saddle-Point Problems in Large-Scale Nonconvex Interior-Point

Optimization, Comput Optim Appl 36 (2-3) (2007) 321-341.

doi:http://dx.doi.org/10.1007/s10589-006-9003-y.

[14] B.S. Chandrasekhar, The seebeck coefficient of bismuth single crystals, J Phys Chem

Solids 11 (1959) 268–273. doi:http://dx.doi.org/10.1016/0022-3697(59)90225-2.

[15] O. Engler, V. Randle, Introduction to Texture Analysis: Macrotexture, Microtexture,

and Orientation Mapping, CRC Press, (2009).

doi:http://dx.doi.org/10.1201/9781420063660.

[16] D. Nakamura, M. Murata, H. Yamamoto, Y. Hasegawa, T. Komine, Thermoelectric

properties for single crystal bismuth nanowires using a mean free path limitation model,

J Appl Phys 110 (2011) 053702. doi:http://dx.doi.org/10.1063/1.3630014.

[17] Z. Zhu, B. Fauque, Y. Fuseya, K. Behnia, Angle-resolved Landau spectrum of electrons

and holes in bismuth, Phys Rev B 84 (2011) 115137.

doi:http://dx.doi.org/10.1103/physrevb.84.115137.