INTRODUCTION
In this study, mathematical calculation of thermoelectric figure of merit (ZT)
has been derived in lowdimensional 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 18211823 (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):
where, S is the see beck coefficient, σ is the electrical conductivity,
T is the temperature (Kelvin), κ_{ph} is the phonon thermal conductivity
and κ_{e} 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 (κ_{e}+κ_{ph}).
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 κ_{e}+κ_{ph}, 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 Bi_{2}Te_{3} nanostructures (Treatise,
2001c; Naves et al., 2006; Boberl
et al., 2008).
THEORY
At first we need to calculate electrical conductivity (σ), electronic thermal conductivity (κ_{e}) 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 FermiDirac f_{0} and integral of Fermi
and density of state g (E), we calculate electrical conductivity for 1D, 2D,
3D as follows (Muller, 2005):
Where:
Fermi integral is:
and density of state for 3D:
With substitute in relation 2 for 3D we have:
where, m = (m_{x}m_{y}m_{z})^{3/2}, therefore:
and by definition of F_{n} we have:
Then:
Now, for achieve electrical conductivity 1D and 2D, we use from WiedemannFranz's law σ = n_{iD}eμ, where i = 1, 2, 3:
And:
With substitute f_{0} and density of state 2D at Eq. 6 we have:
where, ,
therefore:
Therefore, from definition F_{n}, we have:
Finally:
Similarly, for achieve electrical conductivity 1D with substitute f_{0} and:
and:
where, m = m_{x}, therefore:
by relation Eq. 5 we have:
Calculation of thermal conductivity (κ_{e}) for 1D, 2D and 3D: For calculate thermal conductivity 3D we substitute f_{0}, τ and density of state 3D in relation:
We have:
where,
Therefore, thermal conductivity for 3D is:
Now for calculation thermal conductivity 1D, 2D from WiedemannFranz's law we have:
Hence, for thermal conductivity 2D with notation to g_{2D} (E) and other supposition we have:
where,
Therefore, thermal conductivity 2D is:
Now, for 1D, we have:
Therefore:
That m = m_{x}. Then finally:
Calculation of Seebeck coefficient (S) for 1D, 2D and 3D: For calculate Seebeck coefficient we use from following relation:
for calculate Seebeck coefficient for 3D with notation to F_{n} and f_{0} and g_{3D} we have:
Where:
Now for achieve Seebeck coefficient 2D with iteration up stages and relation
Eq. 25 we have:
Similarly, for 1D, we can achieve as follows:
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:
By simplification we can write:
Where:
But since in calculation ZT in 1D, six charge carrier supposed, therefore, B_{1D} becomes as follows:
For calculate ZT in 2D similarly to 1D, by substitution of relations Eq.
9, 20 and 27 in Eq.
1 we have:
Where:
Then:
For calculate ZT in 3D similarly to two considered cases, we substitute relations
Eq. 4, 17 and 26 in
Eq. 1 and so:
Where:
Then:
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 lowdimensional quantum systems. The results presented in Fig. 1, in the case of a 1dimensional Bi_{2}Te_{3} semiconductor crystal.
Due to differences of effective masses in a rectangular directions in Bi_{2}Te_{3} 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 (m_{x}, m_{y}, m_{z}) are same but the quantities are different. The amounts of (ZT) for m_{z} are smaller than m_{y} and also the cases of m_{y} are smaller than m_{x}. The amounts of ZT in the case of m_{x} 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 m_{x}, m_{y} and m_{z} it is appear from figures and observation, with increase in temperature, it will be occur an overlap between, m_{x}, m_{y} and m_{z}, under special temperature higher than 1000 K actually in this case we assume a fix temperature for m_{y} and a variable temperature for m_{z}. This claim presented in Fig. 2. Obviously the overlap between m_{x} and m_{z} will occur in temperature 4550 K. In the case of 1dimensional Bi_{2}Te_{3} nanowiers.
Calculation of 2dimensional quantum systems were presented in, using Eq.
33. Figure 3 shows the size and temperature dependent
of thermoelectric. Figure of merit (ZT) in 2dimensional Bi_{2}Te_{3}
semiconductor crystal. Due to existence of 2dimensional in well like system
or thin films, the number of cases to be consider for this systems are in the
form of m_{y}m_{z}, m_{x}m_{z} and m_{x}m_{y}.
The behavior of thermoelectric figure of merit (ZT) by varying of temperatures
and sizes, are similar to 1dimensional cases. In these cases also we come control
and optimized the amount of ZT by size and temperature.

Fig. 1: 
Merit Z_{1D} T for Bi_{2}Te_{3} in
T = 300,…, 1000 

Fig. 2: 
Merit Z_{2D} T for Bi_{2}Te_{3} in
T = 300,…, 1000 
It is clear from Fig. 3, the maximum thermoelectric figure
of merit (ZT) obtained in the case of m_{x}m_{y} direction,
the smallest figure of merit (ZT) obtained in the case of m_{y}m_{z}
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 1dimensional systems (quantum wires) are effective
and sharp at very small sizes (sizes less than 5 nm). However in the case of
2dimensional 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
1dimensional systems.

Fig. 3: 
Merit Z_{3D}T for Bi_{2}Te_{3} in
T = 300,…, 1000 K 

Fig. 4: 
Merit Z_{4D}T for Bi_{2}Te_{3} in
T = 3000,…, 4550, T = 300 K for m_{x} and T = 300,…, 4550
K for m_{z} 
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 lowdimensional Bi_{2}Te_{3} semiconductor
crystals have been investigated. Analysis of observation shows that, the general
behaviors in (ZT) at different lowdimensional 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 1dimensional depends on the size of systems. In the case of 1dimensional
systems, the quantum matters observed in comparison of 2dimensional systems.