Thermo-electric figure of merit of quasi-two-dimensional Bi2Te3 nano-structures

Quasi-two dimensional nano-structures are characterized by radical changes in electron and phonon properties. Electron behavior in Q-2D nano-structures is modified by the presence of multiple sub-bands. For well widths smaller than 50 Å simplified model calculations based on the assumption of electrons occupying the lowest sub-band appears to be quite reasonable. For larger width others, parabolic sub-bands need to be considered. The present paper investigates the thermoelectric figure-of-merit of bismuth telluride which is five layered (Te-Bi-Te-Bi-Te) crystal with hexagonal symmetry with six equivalent valleys, in the range 40–300 Å and observe that for 100 Å well width three sub-band model and for 200 Å well width six sub-band model are good approximation at room temperature. Further it is observed that optimum carrier density for 200 Å well width is 3.2 × 10 24 m − 3 and maximum of ZT shifts towards lower concentration; and discusses the acceptability of single band approximation.


Introduction
Semiconductor thermoelectric materials are extensively used in power generation and heat pumps. The performance of thermoelectric devices is quantified by dimensionless figure-of-merit ZT, is defined as where α, σ, and λ refer to the Seebeck coefficient, electrical conductivity, and thermal conductivity of the material, respectively [1][2][3][4][5]. Thermoelectric behavior of bulk material [6][7][8][9] is usually understood in terms of electronic and phonon properties, which have bearing on transport coefficients such as electrical and thermal conductivity and the Seebeck coefficient. Both electron and phonon properties show significant changes when the sample size is the order of deBroglie wavelength of the carrier and quantum effects become increasingly evident [10][11][12][13][14][15]. Search for efficient thermoelectric materials have led to intense theoretical and experimental study of conventional as well as engineered materials. Recent decades have witnessed keen interest on lowdimensional systems [16][17][18][19][20][21]. Several physical properties such as electron and phonon density of states undergo radical changes, which are quite unlike that in the bulk. For quasi two-dimensional system density of state at any particular energy is the sum over all parabolic sub-band below that point which is step like function and written as ) where Q is unit step function, m* is effective mass, m is number of parabolic sub-band below particular energy E . m The total number of carrier in Quasi-two-dimensional system may be written as carrier concentration per unit area where E F quasi fermi energy for Q2D parabolic particular sub-band [22]. Moreover, particle interactions may also change significantly such as electron-alloy-disorder, electron-acoustic phonon and all this has an influence on thermoelectric properties in a complex manner [23][24][25].
In Q2-D system restriction on electronic motion along one-direction results is a typical sub-bands structure. Majority of theoretical work has used the simplified model (quantum size limit SQL) in which electron occupy only lowest sub-band. Except at very small width (<50 Å) SQL can be only a crude approximation. However, to obtain an accurate estimate of the influence of sub-band structure, one has to go beyond the confines of SQL. This paper deals with the effect of inclusion of multiple sub-bands on transport properties and thermoelectric figure-of-merit of bismuth telluride.

Outline of theoretical model
Let us consider Quasi-two-dimensional quantum well structure assuming parabolic multiple energy sub-band and current flow in x-direction and quantum confinement in z-direction. The electron dispersion relation is given by [26][27][28][29][30][31] where m m m , , x y z are the effective mass along x-direction, y-direction and confined z-direction, a is the well width, k , x k y are components of the wave vector. This dispersion relation shows free motion of electron in x-y plane and bound states in z-direction. Details of theoretical formulation for obtaining transport coefficient of a Q-2D structures have been presented in earlier work [32]. The electrical conductivity, electronic thermal conductivity, Lorentz factor and Seebeck coefficient can be expressed as where m is effective number of quantized conduction parabolic sub-bands below particular energy E . X is deformation potential, and r is material density [33,34]. From the expression of relaxation time for Q2D one may observe that relaxation time is independent of energy and depends upon the width of the layer.
Thermoelectric figure-of-merit Q-2D structures with their somewhat unusual transport behavior in special situations, provide useful thermoselement materials. The primary aim of this paper is to exploit these properties to obtain a high thermoelectric performance exemplified the figure-of-merit ZT defined in equation (1). The thermal conductivity λ is expressed as a sum of electronic and lattice contributions since confinement direction Z-direction and current flow along X-direction one can defined mobility .
The thermal conductivity λ is given as The figure-of-merit of a quasi-two-dimensional structure with the help of the equations (1), (3), (6) and (15) can be expressed as where the expression B is sensitive upon materials properties of quantum well and B is given by as [26][27][28][29] The figure-of-merit may be optimized with respect to variations of B which depends upon intrinsic properties of the material. B is inversely proportional to width of layer 'a' and proportional to mobility of the carriers along x-direction. The large values of carrier mobility and small width may enhance the figure-of-merit. The figure-of-merit also depends upon doping of the material. In quasi two-dimensional structures due to quantization of conduction sub-bands doping of the material depends itself on layer width. Thus, for optimization of the figure-of-merit multiple parabolic sub-band model must be taken into account. The bismuth telluride has six equivalent valleys along slightly different orientations on the Brillouin zone. For simple numerical estimation of ZT it may be assumed that all six valleys have same orientation. Thus, expression for B equation (17) should be modified and is given by where N v is number of equivalent valleys. The lattice thermal conductivity can be modeled within the framework of the Boltzmann equation approach under the relaxation time approximation. One can express lattice thermal conductivity as where d S j w w ( ) is the contribution to the specific heat form modes of polarization j at frequency ω, v j is phonon group velocity, and t¢ is relaxation time depends upon temperature [23]. Reduction of λ L is one of the primary concerns in any thermoelectric performance enhancement effort. In two-dimensional structures of interest to us the lattice thermal conductivity can be effectively reduced through boundary scattering and through the alteration of phonon spectrum in low-dimensional structures [35][36][37][38][39]. The phonon transport in bulk materials is almost unaffected by the sample size as the phonon mean free path is already orders of magnitude lower than the sample size due phonon-phonon umklapp scattering. The sample size becomes effective when it is almost same or lower than the mean free path due to other processes. In this situation quantum effects become significant with radical changes in electronic and phonon properties. This leads to further reduction of the thermal conductivity due to changes in group velocity, density of states and scattering mechanism. The modification of the phonon group velocities and dispersion due to spatial confinement leads to a significant increase of the phonon relaxation rates, and as a result, there is a significant drop in the lattice thermal conductivity. These results have been used in the estimation of the thermoelectric figure-of-merit in bismuth telluride quantum wells [38,39].

Results and discussions
The theoretical model described here has been applied to polycrystalline bismuth telluride. Various relevant parameters used in calculations are shown in table 1. In figure 1 we presented the power factor w mK    [26]. However, we observe that with increase in well-width more and more sub-bands drop below the Fermi level and contribute to transport; and as well-width increases sub-bands come to nearer to each other and carrier have sufficient thermal energy to hop from one sub-band to other sub-band. From figures 3(b) & 4 one can observe that for 100 Å well width three sub-band model and for 200 Å well width six sub-band model are good approximation at room temperature. Further it is observed that optimum carrier density for 200 Å well width is 3.2 10 m 24 3 and maximum of ZT shifts towards lower concentration. In each sub-band carriers may acquire different velocities. Below 100 Å one observes a rapid rise in ZT. The curves for various sub-bands start coming close below about 60 Å indicating the validity of SQL. Above this SQL is no longer a valid approximation. Figure 6 displays percentage change in ZT over the single Deviation from SQL appears to be more pronounced for larger well widths.
For three sub-bands the percentage change in ZT goes from about 35 to 58 as width increases from 105 to 200 Å.
In conclusions, the main results of Bi Te 2 3 quantum well in which thermal gradient present along x-axes and confinement along z-axes (1) as number of parabolic quantized sub-bands increases optimum power factor increased and maxima shifted towards larger quantum well width. (2) The lattice thermal conductivity is reduced with a reduction in well width. (3) The electronic thermal conductivity shows slow increase and attains a