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17Physics of Complex Systems, 2022, vol. 3, no. 3 _www.physcomsys.ru
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Physics of Semiconductors. Semiconductors
UDC 538.9 + 535.3
EDN RVKOKJ
https://www.doi.org/10.33910/2687-153X-2022-3-3-144-148
Nernst mobility of holes in Bi2Te3
S. A. Nemov™, G. V. Demyanov1, A. V. Povolotskiy2
1 Peter the Great Saint Petersburg Polytechnic University, 29 Polytechnicheskaya Street, Saint Petersburg 195251, Russia 2 Saint Petersburg State University, 7/9 Universitetskaya Emb., Saint Petersburg 199034, Russia
Authors
Sergey A. Nemov, ORCID: 0000-0001-7673-6899, e-mail: nemov s@mail.ru Grigorij V. Demyanov, e-mail: demyanov.g@bk.ru Alexey V. Povolotskiy, e-mail: apov@inbox.ru
For citation: Nemov, S. A., Demyanov, G. V., Povolotskiy, A. V. (2022) Nernst mobility of holes in Bi2Te3.
Physics of Complex Systems, 3 (3), 144-148. https://www.doi.org/10.33910/2687-153X-2022-3-3-144-148 EDN RVKOKJ
Received 1 June 2022; reviewed 6 June 2022; accepted 6 June 2022.
Funding: The study did not receive any external funding.
Copyright: © S. A. Nemov, G. V. Demyanov, A. V. Povolotskiy (2022). Published by Herzen State Pedagogical University of Russia. Open access under CC BY-NC License 4.0.
Abstract. This paper presents the results of a study of the electrical conductivity coefficient, the Hall coefficient, the Seebeck coefficient, the transverse Nernst-Ettingshausen coefficient and their anisotropy in a Bi2Te3 single crystal with a hole concentration p =1 x 1010-19 cm-3 at temperatures 77-350 K. It is established that hole scattering occurs mainly on long-wave acoustic phonons. Despite the fact that the chemical potential level is located near the top of the additional extreme of the valence band, no interband scattering was detected. The complex structure of the valence band is confirmed.
Keywords: Bismuth telluride, hole conductivity, temperature dependence, specific electrical conductivity, thermopower coefficient, Hall coefficient, Nernst-Ettingshausen coefficient, hole scattering mechanism, Nernst mobility, acoustic phonons
Introduction
Currently, there is an unusual situation with the interpretation of experimental data on the transport phenomena in hole chalcogenides of elements V of group V of the Mendeleev table Bi1-xSbxTe1-ySey, including in Bi2Te3. Single-band and double-band models are used to describe experimental data on transport phenomena in the temperature range of 100-600 degrees Celsius. For example, at temperatures above room temperature, minor carriers are also taken into account.
On the one hand, temperature dependences of the coefficients of specific electrical conductivity (a) and thermopower (a) do not have any features and have the form typical for strongly doped semiconductors. The electrical conductivity decreases according to the law a ~ T-n, n ~ 1.7, and the thermopower is proportional to the temperature a ~ T.
Such dependences a (T) and a (T) are well described by a single-band model at temperatures of 77-300 K with the participation of one class of current carriers in the transport phenomena. This model is convenient for estimating the fundamental parameters of the band structure, determining the dominant scattering mechanisms, and calculating the thermoelectric properties of material (Golts-man et al. 1972; Lukyanova et al. 2005).
On the other hand, there is a significant increase in the Hall coefficient R(T) with temperature, which is traditionally associated in semiconductor physics with the redistribution of holes between two non-equivalent extremes of the valence band. Thus, such a dependence of R(T) indicates a complex structure
of the valence band and the participation of several types of current carriers in the transport phenomena. In this case, the calculations usually use a two-band model with two types of holes with different effective masses m^ and m*2, respectively.
It should also be noted that most of the previous studies of transport phenomena were not comprehensive. They were carried out on poly-crystal samples and had a practical orientation.
Considering the practical significance of solid solutions Bi1_xSbxTe1_ySey, which are the main components of materials used in thermoelectric energy converters, operating in the temperature range of 200-600 K it is is advisable to continue studying the electrophysical properties of crystals of these materials and, first of all, the main component Bi2Te3.
Experiment
In the present work, studies of the kinetic coefficients were carried out on a single crystal grown by the Czochralski method with a Hall hole concentration p = 1 x 1010-19 cm-3. The choice of the crystal is due to the fact, that, according to quantum oscillation data, the chemical potential level p at 4.2 K at such a hole concentration is located near the additional extreme of the valence band, in this case |i~AEv « 0.03 эВ, where AEv is the energy gap between the peaks of the nonequivalent bands (Sologub et al. 1975). In this case, if the two-band model of the valence band is valid, we can expect the manifestation of features in the kinetic coefficients and, first of all, in the transverse Nernst-Ettingshausen effect, related to the interband scattering of "light holes" during their transition to additional extrema at low temperatures (T > 77 K).
In this paper, the temperature dependences of the main kinetic Hall (R), electrical conductivity (o), thermopower (a), and the transverse Nernst-Ettingsgazuen (Q) coefficients and their anisotropy in the temperature range 77-350 K were measured. The obtained values of the coefficients R, o, and a and their temperature dependences almost coincided with the literature data for samples with close Hall hole concentrations. Their values used in calculations are shown in Table 1.
Table 1. Basic kinetic coefficients of the p-Bi2Te3 crystal
T, K ffn, Om 1cm 1 ^ cm3/C a11, mcV/K Qm, cm 2 k0/e ' V-xS x < 0 r
1 100 4800 0.38 50 -450 -0.427 0.051
2 150 2600 0.45 90 -451 -0.370 0.094
3 200 1600 0.52 140 -400 -0.297 0.157
4 250 1100 0.56 180 -280 -0.218 0.230
5 300 800 0.54 205 -140 -0.137 0.319
6 350 500 0.38 202 -10 -0.225 0.323
Temperature dependences of two components of the Nernst-Ettingsgazuen tensor Q123 and Q132 are shown in Figure 1.
200-
tfl 0)
c
O
100 ■ 0
-100 -200 -300-400-500-
■
■ ^123
A Q 132 A' -
A. A A' ■
A k ■
■ ■ ■ ■ ■
\ ...
50 100 150 200 250 300 350 400 450
T, K
Fig. 1. Temperature dependence of the Nernst-Ettingsgazuen coefficient for a Bi2Te3 crystal2Te3 with a hole
concentration of 1 x 1010-19 era-3.
Nernst mobility of holes in Bi2Te3
Note that the thermopower coefficient is isotropic and increases linearly with the temperature in the range of 85-250 K. The differences in the values of the Seebeck coefficients in the cleavage plane (a11) and along the inversion-rotation axis 3 (a33) are several MV/K, which does not exceed the measurement error. This result indicates that there is no noticeable anisotropy in the energy dependence of the hole relaxation time t (e) in Bi2Te3.
Discussion of experimental data
First of all, we note that measuring the four main kinetic coefficients R, a, a, Q and their temperature dependences allows us to determine the main parameters of the band and structure of semiconductors, as well as the dominant scattering mechanisms in a single-band model (Zhitinskaya et al. 1966), and to conduct a correct analysis of experimental data for several types of current carriers.
Let us take a closer look at the obtained data. Let us start with the Hall effect. Both components of the Hall coefficient tensor R123 and R132 in hole chalcogenide materials Bi1 _ xSbxTe1 _ Se (including in Bi2Te3) (Goltsman et al. 1972; Sologub et al. 1975) grow with increasing temperature. Therefore, in the materials of group A it is customary to determine the concentration of holes p from the larger component of the Hall tensor R77 at a temperature of 77 K. In p-Bi2Te3 there is a large component of the Hall tensor R132. The concentration of holes is calculated by the formula
p = {eRliy (1)
The concentration determined in this way is called the Hall concentration of holes and is used in all calculations of the parameters of the zone structure.
Note that Formula (1) does not take into account the hall factor associated with the anisotropy of the effective mass (it is approximately 0.7-0.8 for the coefficient R132 in p-Bi2Te3 (Goltsman et al. 1972). Thus, the real concentration of holes is about 20-30% more.
The Bi2Te3 single crystals studied by us had a Hall concentration of holes p ~ 1 x 1019 cm-3, the maximum was observed in unalloyed bismuth telluride crystals obtained by deviation from the stoichiomet-ric composition (Goltsman et al. 1972).
Our data on the Hall effect are consistent with the literature data (Goltsman et al. 1972). Both components of the Hall tensor R123 and R132 grow with increasing temperature from the value of R77 to R300 (at room temperature) by about 1.4 times.
Let us discuss in more detail the data on the Nernst-Ettingshausen transverse effect, which are presented in Figure 1 as a temperature dependence of the Nernst-Ettingshausen coefficient Q divided by a multiplier k0/e (where k0 is the Boltzmann constant, e is the electron charge modulus).
In this form, experimental data on the Nernst-Ettingshausen transverse effect are usually presented. The fact is that the coefficient Q/(k0/e) has a dimension of cm2/V-s, and its module is called the Nernst mobility. In such units of measurement, it is convenient to compare the Nernst and Hall mobility (Ra).
As can be seen from Figure 1, both components of the Nernst-Ettingsgazuen tensor Q123 and Q132 are negative at T < 350 K. This means that in the studied temperature range according to the formula for Q, valid for degenerate statistics
Q = !^-R(Tn2 VSInT
e 3 n Sine
• (2)
So, the energy dependence of the relaxation time of holes T(e) is a function decreasing with energy e. This dependence has only two mechanisms of current carriers scattering: on acoustic phonons and interband scattering.
Note that the idea of interband scattering was introduced by N. V. Kolomoets (Zhitinskaya et al. 1966). The inclusion of this mechanism of hole scattering in samples with a chemical potential located near the top of the additional extremum of the valence band, that is, under the condition of |i « AEv in the region
of nitrogen temperatures, can lead to features in the temperature dependence of the Nernst mobi
5 It! X""
The fact that the interband scattering is characterized by the large negative value of the derivative
Sine
ity.
e=n
as the relaxation time of "light" holes from the main extremum, in this case it is described by the expression according to (Kolomoets 1966):
T"1'
y¡£ by £< A Ey
+ W12\I£ ~AEv fy>e >AEv '
where t0 is a constant coefficient in expression for the relaxation time of light holes when scattering by acoustic phonons = r0e"0'5; w12 = (m*2 / m[) (S12 / S11 )2is a parameter characterizing the probability of an interband transition of holes from the main to the additional extremum; m1 and m2 are the effective masses of the density of the states of holes in the first and second zones; Hn and H12 are the
constants of the deformation potential for intraband and interband scattering of holes. Using the data on the four kinetic coefficients R, a, a and Q, we will estimate the scattering parameter r, which is included in the expressed t = t0 er - 0,5 as part of a single-band model. Since the coefficients a and Q are equal, respectively
Kn2k£(r +1), (4)
a =
e 3 ¡u
e 3 fi V 2
(5)
we get an expression for their relationship
Qm _ r-1/2
«11*123 °il r + l
(6)
where index 3 means the direction along the trigonal axis 3. Denoting the experimental values of the relation by x
(7)
k/pkp
A-q / L- 0
we obtain an equation for determining the scattering parameter r
r = (8)
1-JC
The value of r obtained from experimental data is shown in Table 1, from which it can be seen that in the low temperature region the parameter r is close to zero. This means that in bismuth telluride at temperatures of 77 K and above, hole scattering on long-wave acoustic phonons dominates (parameter r = 0) and there are no signs of the inclusion of a new interband hole scattering mechanism.
As can be seen from Table 1, the parameter r does not decrease, rather, it shows a slight increase. So, there are no large modulo negative values of the scattering parameter characteristic of the interband scattering of holes. Thus, interband hole scattering is inefficient in p-Bi2Te3.
Perhaps, this is due to the fact that the transfer phenomena involve holes with slightly different effective masses of the density of states m* ~ (0,5 - 0,6) m0and m*2 ~ (0,9 -1,2) mff where m0 is the mass of a free electron. Estimates of effective masses are made for the two-band model (Goltsman et al. 1972; Sologub et al. 1975). In addition, the thermal blurring of the Fermi distribution function k0T is comparable to the energy gap AEv in the two-band model and the chemical potential |i, which eliminates the features in the energy dependence of the relaxation time of holes. Theoretical calculations of Bi2Te3 performed from the first principles confirm the complex structure of the valence band (Scheidemantel et al. 2003).
Nernst mobility of holes in BiTe
In conclusion, we note that at temperatures T > 250 K, non-basic carriers (electrons) appear and, in accordance with the theory of kinetic phenomena, the Hall coefficients and thermal EMF decrease, and the electrical conductivity increases. The Nernst-Ettingshausen coefficient at the same time sharply decreases in modulus and at temperatures T = 350 K it changes its sign to positive.
Conclusion
Thus, as a result of studies of the specific electrical conductivity, Hall coefficients, Seebeck coefficients and the Nernst-Ettingshausen transverse effect on a single crystal p-Bi2Te3 at temperatures of 77-350 K, it was found that the dominant mechanism of hole scattering is the scattering by long-wave acoustic phonons.
Due to the location of the chemical potential level ц in the studied crystal with a hole concentration of 1 x 1019 sm-3 near the top of the additional extremum (ц = AEv) interband scattering was expected, however, it was not detected. Perhaps this is due to the fact that the holes involved in the transfer phenomena have slightly different effective masses, and the amount of thermal blurring of the Fermi distribution function k0T is comparable to the values of ц and AEv.
The observed deviations of the experimental data on kinetic coefficients from the calculation results in the single-band approximation are associated with the complex structure of the valence band and the participation in the transfer phenomena of several groups of current carriers with different effective masses.
Conflict of Interest
The authors declare that there is no conflict of interest, either existing or potential.
Author Contributions
The authors have made an equal contribution to the preparation of the text.
References
Goltsman, B. M., Kudinov, V. A., Smirnov, I. A. (1972) Poluprovodnikovye termoelektricheskie materialy na osnove Bi2Te3 [Semiconductor thermoelectric materials based on bismuth telluride Bi2TeJ. Moscow: Nauka Publ., 320 p. (In Russian)
Kolomoets, N. V. (1966) Vliyanie mezhzonnykh perekhodov na termoelektricheskie svojstva veshchestva [Influence of interband transitions on thermoelectric properties of matter]. Fizika tverdogo tela, 8 (4), 997-1003. (In Russian)
Lukyanova, L. N., Kutasov, V. A., Konstantinov, P. P. (2005) Effektivnaya massa i podvizhnost' v tverdykh rastvorakh p-Bi2-x Sbx Te3-y Sey dlya temperatur < 300 K [Effective mass and mobility in solid solutions of p-Bi2-xSbxTe3-ySey for temperatures < 300 K]. Fizika tverdogo tela, 47 (2), 224-228. (In Russian) Scheidemantel, T. J., Ambrosch-Draxl, C., Thonhauser, T. et al. (2003) Transport coefficients from first-principles calculations. Physical Review B, 68 (12), article 125210. https://doi.org/10.1103/PhysRevB.68.125210 (In English) Sologub, V. V., Parfenyev, R. V., Goletskaya, A. D. (1975) Struktura valentnoj zony tellurida vismuta [Structure of the valence band of bismuth telluride]. Pis'ma v Zhurnal eksperimental'noj i teoreticheskoj fiziki — JETP Letters, 21 (12), 711-715. (In Russian) Zhitinskaya, M. K., Kaidanov, V. I., Chernik, I. A. (1966) O neparabolichnosti zony provodimosti tellurida svintsa [On the nonparabolicity of the conduction band of lead telluride]. Fizika tverdogo tela, 8 (1), 295-297. (In Russian)
