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Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3



A.A. Javansiah Bigdilu and Hasan Hossein Zadeh
 
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ABSTRACT

In this study, mathematical calculation of thermoelectric figure of merit (ZT) has been derived in low-dimensional semiconductor materials on the based on density of states for quantum systems. The quantum mechanical behavior, temperature and size dependent of these nanostructures have been described more completely.

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  How to cite this article:

A.A. Javansiah Bigdilu and Hasan Hossein Zadeh, 2012. Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3. Journal of Applied Sciences, 12: 863-869.

DOI: 10.3923/jas.2012.863.869

URL: https://scialert.net/abstract/?doi=jas.2012.863.869
 
Received: January 08, 2012; Accepted: April 27, 2012; Published: June 27, 2012



INTRODUCTION

In this study, mathematical calculation of thermoelectric figure of merit (ZT) has been derived in low-dimensional semiconductor materials on the based on density of states for quantum systems. The quantum mechanical behavior, temperature and size dependent of these nanostructures have been described more completely. The study of thermoelectric phenomena in semiconductor materials is not new; it does back to Seebeck's works in the years 1821-1823 (Treatise, 2001a). However, within recent years has been renewed and intensive study of thermoelectric effect (Carotenuto et al., 2009; Heremans and Thrush, 1999) due to realization of the need both experimentally and theoretically to study more behavior of thermoelectric materials (Glatz and Beloborodov, 2009; Radwan, 2001). Generation of nanostructure materials for thermoelectric applications has attracted great interest in recent years because of their high efficiencies and unique physical properties which are different from those of either the bulk materials (Goldsmid, 1964; Treatise, 2001a; Zheng, 2008). The efficiency of thermoelectric power for a material is usually made in terms of dimensionless figure of merit (ZT) defined by Carotenuto et al. (2009), Heremans and Thrush (1999), Miller (2000), Treatise (2001a), Senthilkumaar and Thamiz Selvi (2008), Zheng (2008) and Zhang et al. (2009):

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(1)

where, S is the see beck coefficient, σ is the electrical conductivity, T is the temperature (Kelvin), κph is the phonon thermal conductivity and κe is the electronic thermal conductivity. In order to achieve high thermoelectric performance (or high ZT), one requires a high thermoelectric power S, where S denotes the voltage generated by thermal gradient, a high electrical conductivity (σ) and low thermal conductivity (κeph). Since an increase in S for a typical material leads to a decrease in σ, due to carrier density consideration and an increase in σ leads to an increase in the electronic contribution to κeph, so, it is very difficult to increase Z in typical thermoelectric materials (Malyarevich et al., 2007; Heremans and Thrush, 1999; Zhang et al., 2009).

Confinement of charge carriers in a nanometer scale thermoelectric materials increases the local carrier density of states per unit volume near the Fermi energy increasing the see beck coefficient (Treatise, 2001a, b; Bhushan, 2007), while the thermal conductivity can be decreased due to phonon confinement (Bhandari, 1995; Muller, 2005; Liu et al., 2008; Zhang and Xu, 2008; Pendyala and Rao, 2009) and phonon scattering at the material interfaces in nanostructures. In the present work, we describe the temperature and size dependence of thermoelectric efficiency (ZT) in Bi2Te3 nanostructures (Treatise, 2001c; Naves et al., 2006; Boberl et al., 2008).

THEORY

At first we need to calculate electrical conductivity (σ), electronic thermal conductivity (κe) and Seebeck coefficient (S) for 1D, 2D and 3D nanostructures.

Calculation of electrical conductivity for 1D, 2D and 3D: Whit definition of distribution function of Fermi-Dirac f0 and integral of Fermi and density of state g (E), we calculate electrical conductivity for 1D, 2D, 3D as follows (Muller, 2005):

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(2)

Where:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

Fermi integral is:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

and density of state for 3D:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

With substitute in relation 2 for 3D we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

where, m = (mxmymz)3/2, therefore:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(3)

and by definition of Fn we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

Then:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(4)

Now, for achieve electrical conductivity 1D and 2D, we use from Wiedemann-Franz's law σ = niDeμ, where i = 1, 2, 3:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(5)

And:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(6)

With substitute f0 and density of state 2D at Eq. 6 we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(7)

where, Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3, therefore:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

Therefore, from definition Fn, we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(8)

Finally:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(9)

Similarly, for achieve electrical conductivity 1D with substitute f0 and:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(10)

and:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(11)

where, m = mx, therefore:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(12)

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(13)

by relation Eq. 5 we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(14)

Calculation of thermal conductivity (κe) for 1D, 2D and 3D: For calculate thermal conductivity 3D we substitute f0, τ and density of state 3D in relation:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(15)

We have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(16)

where, Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3 Therefore, thermal conductivity for 3D is:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(17)

Now for calculation thermal conductivity 1D, 2D from Wiedemann-Franz's law we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(18)

Hence, for thermal conductivity 2D with notation to g2D (E) and other supposition we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(19)

where, Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3 Therefore, thermal conductivity 2D is:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(20)

Now, for 1D, we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(21)

Therefore:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(22)

That m = mx. Then finally:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(23)

Calculation of Seebeck coefficient (S) for 1D, 2D and 3D: For calculate Seebeck coefficient we use from following relation:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(24)

for calculate Seebeck coefficient for 3D with notation to Fn and f0 and g3D we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(25)

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(26)

Where:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

Now for achieve Seebeck coefficient 2D with iteration up stages and relation Eq. 25 we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(27)

Similarly, for 1D, we can achieve as follows:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(28)

Calculation of figure of merit (ZT) for 1D, 2D and 3D: By substitution of Eq. 14, 23 and 28 in relation (Eq. 1) we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(29)

By simplification we can write:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(30)

Where:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(31)

But since in calculation ZT in 1D, six charge carrier supposed, therefore, B1D becomes as follows:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(32)

For calculate ZT in 2D similarly to 1D, by substitution of relations Eq. 9, 20 and 27 in Eq. 1 we have:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(33)

Where:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

Then:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(34)

For calculate ZT in 3D similarly to two considered cases, we substitute relations Eq. 4, 17 and 26 in Eq. 1 and so:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(35)

Where:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3

Then:

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
(36)

DISCUSSION

By using the above mentioned theoretical expression of thermoelectric figure of merit and Seebeck constant the size and temperature dependent of these factors has been investigation at low-dimensional quantum systems. The results presented in Fig. 1, in the case of a 1-dimensional Bi2Te3 semiconductor crystal.

Due to differences of effective masses in a rectangular directions in Bi2Te3 crystals, the temperature and size dependent properties will be different in each of this direction, this is also clear in the figure.

The behavior of thermoelectric figure of merit (ZT) with different quantum sizes and temperatures in all directions of effective masses (mx, my, mz) are same but the quantities are different. The amounts of (ZT) for mz are smaller than my and also the cases of my are smaller than mx. The amounts of ZT in the case of mx are large the temperature dependent of ZT are also identified for all cases. The results shows an increase in (ZT) with increase in temperature in directions of mx, my and mz it is appear from figures and observation, with increase in temperature, it will be occur an overlap between, mx, my and mz, under special temperature higher than 1000 K actually in this case we assume a fix temperature for my and a variable temperature for mz. This claim presented in Fig. 2. Obviously the overlap between mx and mz will occur in temperature 4550 K. In the case of 1-dimensional Bi2Te3 nanowiers.

Calculation of 2-dimensional quantum systems were presented in, using Eq. 33. Figure 3 shows the size and temperature dependent of thermoelectric. Figure of merit (ZT) in 2-dimensional Bi2Te3 semiconductor crystal. Due to existence of 2-dimensional in well like system or thin films, the number of cases to be consider for this systems are in the form of mymz, mxmz and mxmy. The behavior of thermoelectric figure of merit (ZT) by varying of temperatures and sizes, are similar to 1-dimensional cases. In these cases also we come control and optimized the amount of ZT by size and temperature.

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
Fig. 1: Merit Z1D T for Bi2Te3 in T = 300,…, 1000

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
Fig. 2: Merit Z2D T for Bi2Te3 in T = 300,…, 1000

It is clear from Fig. 3, the maximum thermoelectric figure of merit (ZT) obtained in the case of mxmy direction, the smallest figure of merit (ZT) obtained in the case of mymz directions.

Comparative studies between Fig. 1 and 2 shows the variation of thermoelectric figure of merit with different quantum sizes and temperature are similar and parabolic in nature but the amount of sharpness in curves (bending) are different each other and this phenomena confirms that the quantum size effect in 1-dimensional systems (quantum wires) are effective and sharp at very small sizes (sizes less than 5 nm). However in the case of 2-dimensional systems, this effect is weak. It means that the quantum size effect occurs in such size and dimensions. The quantum size effect in quantum wires more intensive to size of system and this is refer to the quantum nature of 1-dimensional systems.

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
Fig. 3: Merit Z3DT for Bi2Te3 in T = 300,…, 1000 K

Image for - Investigation of Temperature Dependence Thermoelectric Figure of Merit (ZT) in Low-dimensional Bi2Te3
Fig. 4: Merit Z4DT for Bi2Te3 in T = 3000,…, 4550, T = 300 K for mx and T = 300,…, 4550 K for mz

For bulk systems, the behavior of thermoelectric materials is linear. As shown in Fig. 4 the behavior of thermoelectric figure of merit (ZT) are completely classical nature, also the influence of temperature in this case are classical but the thermoelectric figure of merit (ZT) increase by temperature (Fig. 4).

CONCLUSION

In this research, the theory of size and temperature dependent of thermoelectric figure of merit (ZT) in low-dimensional Bi2Te3 semiconductor crystals have been investigated. Analysis of observation shows that, the general behaviors in (ZT) at different low-dimensional are same and shows quantum mechanical nature and completely differs from its bulk behavior. Size and temperature dependent of (ZT) identifies that the quantum mechanical behavior of in the case of 2 and 1-dimensional depends on the size of systems. In the case of 1-dimensional systems, the quantum matters observed in comparison of 2-dimensional systems.

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