Figure of Merit of one-dimensional resonant transmission systems in the quantum regime

The figure of merit, ZT, for a one-dimensional conductor displaying a Lorentzian resonant transmission probability is calculated. The optimum working conditions for largest ZT values are determined. It is found that, the resonance energy has to be adjusted to be several resonance widths away from the Fermi level. Similarly it is better for the temperature to be equal to several resonance widths. The approximate relationships, which can be a fairly good guide for designing devices, between different parameters under optimum working conditions are given.


I. INTRODUCTION
Recent advances in fabrication and material growth technologies have made possible the production of devices whose dimensions are order of a few nanometers. The thermoelectric power, thermal and electrical conductivities of these mesoscopic scale materials are of interest for thermoelectric device applications 1 , such as heat pumps 2 and power generators. The performance of a thermoelectric device is usually quantified by a dimensionless number called as figure of merit, ZT , which measures the efficiency of thermoelectric energy conversion.
Higher values of ZT correspond to higher thermoelectric energy conversion efficiency so that, for example, if ZT tends to infinity, the efficiency approaches to that of an ideal Carnot engine. Recent work on superlattice semiconducting devices demonstrated ZT ≈ 2.5 at room temperature 3 and ZT ≈ 1.4 at high temperatures 4 , breaking the long-standing limit of ZT ≈ 1 for most of best known thermoelectric materials.
In 1D nanoscale systems, the increase of ZT might be further enhanced by the quantum confining of electrons and phonons in low dimensions 5 . In the recent experimental study, for instance, ZT could be increased by embedding nanoparticles in a crystalline semiconductors 6 . High values of ZT can also be obtained for one-dimensional structures displaying resonant transmission. In this case, the transmission probabilities are very sensitive to the electron energies leading to large values of the thermopower and hence of the figure of merit. In this contribution, relationships between different thermoelectric coefficients are calculated, and probably the first, ZT values of a resonant tunnelling device is computed and its dependence on device parameters is investigated. We are expecting that our calculations will be a good guide to researchers studying on thermoelectric devices.

II. TRANSPORT COEFFICIENTS
In here a one-dimensional mesoscopic device is considered. It is assumed that there is only one transverse mode that is occupied by the electrons. The thermoelectric currents in such a device under linear regime can be expressed in terms of the energy dependent transmission probability T (E) of the electrons 7 . The electric current, I, and the heat current,Q, under a potential difference ∆V and a temperature difference ∆T can be expressed as 8 where g n are µ is the chemical potential and f (x) = 1/(1 + e x ) is the Fermi-Dirac distribution function.
Frequently measured transport coefficients, the electrical conductance G el , the thermal conductance G th , the Seebeck coefficient S, and the dimensionless thermoelectric figure of merit ZT can be expressed as It can be seen that, to obtain large values of ZT , the factor g 1 , which also appears in the Seebeck coefficient, has to be large. Equation (3) implies that at low temperatures, g 1 is basically proportional to the first derivative of T (E). For this reason, large values of g 1 can be achieved when T (E) is strongly energy dependent, and the fastest change in it occurs around the Fermi level. In this contribution we investigate a resonant tunnelling device where the transmission probability is assumed to have a Lorentzian form where E o is the resonance energy and Γ is the half width. The factor g 1 and the Seebeck The maximum attainable value of figure of merit, ZT max and the optimum temperature, θ max are shown in Fig. (4) as a function of ǫ. When the Fermi level is almost coincident with the resonance energy (ǫ < ∼ 1), the temperature at which maximum attained is around θ ≈ 0.6. However, in this region, ZT values are low. In the opposite case, for ǫ > ∼ 1, the optimum temperature and the best value of ZT appear to be linear in ǫ with the following  The value of θ at maxima is plotted as a function of ǫ.
6 numerical relationships The approximate relation θ max ≈ 0.9ZT max + 0.6 seems to hold over the whole range investigated in this study.
It is seen that for one-dimensional systems displaying resonant transmission, it is possible to obtain large values of the figure of merit ZT by fine-tuning the device parameters, the resonance width and the place of the resonance energy relative to the Fermi level. Lorentzian form can be used as a fairly good guide for designing devices.
An important problem, connected with being far away from the resonance, is the smallness of the electrical and thermal conductances. For this reason, efficient devices can only work with very small power. Moreover, precautions should be taken to make the phonon contribution to the heat conductance to be very small as this will always decrease the ZT value.

IV. CONCLUSIONS
The figure of merit has been computed for a system whose transmission probability displays a Lorentzian peak. By adjusting the resonance width to be small and carefully placing the Fermi level for the working temperatures of interest, significantly high values of ZT can be obtained. Approximate numerical relationships, which can be a guide for