Simulation of polycrystalline bismuth films Seebeck coefficient based on experimental texture identification

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.

Full text

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
2deg.
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
/(VS)
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.
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