Determination of a Mathematical Discrete Model for the Study of Thermoelectric Materials with the Use of the Microprobe

The understanding of the temperature profile in a semiconductor structure in each of its directions is essential for determining its intrinsic characteristics. By means of the use of tools like the microprobe, it is possible to find some of these characteristics [Plazek, D, et. al., 2003]. A problem with the microprobe is to know with precision the transient evolution of the initial temperature when it comes in contact with the sample. With the help of suitable mathematical models, it is possible to know the evolving temperature flux as a function of time and space. In the linear space and with the use of Laplacian solutions, a useful solution can be obtained. In this paper, a temperature fluctuation evolution model by discrete state-space has been developed. The most direct application is to develop valid algorithms to apply to support software. The impact in a thermoelectric system is basically to give improved knowledge of the temperature fluctuations throughout a semiconductor, taking into account more than one dimension. The verification of results using semiconductor compound samples used in thermoelectrics (BiTe, Skutterudite, etc.), allows us to determine graphically the thermoelectric properties that characterize them. By means of the use of thermal scanning and the use of thermocouples, it is possible to build up a function of the Seebeck coefficient in the measurement direction (microprobe). It is quite interesting to know the initial transitory measurements to characterize in an optimal way the results obtained from the thermal system (????). The diffusivity (beta) of the material is a parameter that modulates the exponential response of the system. By means of the use of a discrete-time system [Dominguez, S, et. al., 2001] it is possible to simplify the analysis and the development of the algorithms. There are few ways to do the sampling: (1) Conventional periodical sampling or uniform sampling. In this case the interval of sampling is a constant: t k = T (k = 0,1,2,...). (2) Sampling of multiple order: in this case t_k will repeat periodically, where t_k+r-t_k is a constant for all k. (3) Sampling of multiple order: in this case there is a simultaneous agreement of two sample operations in t_k = pT₁ and qT₂ where T₁ and T₂ are constant and p and q are integer values. (4) Random sampling: in this case the instant moments of the sampling t_k is a random variable. In this work a periodic sampling has been considered, with sampling time-collection: t(k) = kT (k = 0,1,2,... )