The Einstein Relation in Compound Semiconductors Under Magnetic Quantization

It is well known that the band tails are being formed in the forbidden zone of heavily doped semiconductors and can be explained by the overlapping of the impurity band with the conduction and valence bands [1]. Kane [2] and Bonch Bruevich [3] have independently derived the theory of band tailing for semiconductors having unperturbed parabolic energy bands. Kane's model [2] was used to explain the experimental results on tunneling [4] and the optical absorption edges [5, 6] in this context. Halperin and Lax [7] developed a model for band tailing applicable only to the deep tailing states. Although Kane's concept is often used in the literature for the investigation of band tailing [8, 9], it may be noted that this model [2, 10] suffers from serious assumptions in the sense that the local impurity potential is assumed to be small and slowly varying in space coordinates [9]. In this respect, the local impurity potential may be assumed to be a constant. In order to avoid these approximations, we have developed in this chapter the electron energy spectra for heavily doped semiconductors for studying the DMR based on the concept of the variation of the kinetic energy [1,9] of the electron with the local point in space coordinates. This kinetic energy is then averaged over the entire region of variation using a Gaussian type potential energy. On the basis of the E–k dispersion relation, we have obtained the electron statistics for different heavily doped materials for the purpose of numerical computation of the respective DMRs. It may be noted that a more general treatment of many-body theory for the density-of-states of heavily doped semiconductor merges with one-electron theory under macroscopic conditions [1]. Also, the experimental results for the Fermi energy and others are the average effect of this macroscopic case. So, the present treatment of the one-electron system is more applicable to the experimental point of view and it is also easy to understand the overall effect in such a case [11]. In a heavily doped semiconductors, each impurity atom is surrounded by the electrons, assuming a regular distribution of atoms, and it is screened independently [8, 10, 12]. The interaction energy between electrons and impurities is known as the impurity screening potential. This energy is determined by the inter-impurity distance and the screening radius, which is known as the screening length. The screening radius changes with the electron concentration and the effective mass. Furthermore, these entities are important for heavily doped materials in characterizing the semiconductor properties [13,14] and the devices [8,15]. The works on Fermi energy and the screening length in an n-type GaAs have already been initiated in the literature [16–18], based on Kane's model. Incidentally, the limitations of Kane's model [9], as mentioned above, are also present in their studies.

At this point, it may be noted that many band tail models are proposed using the Gaussian distribution of the impurity potential variation [2, 9]. In this chapter, we have used the Gaussian band tails to obtain the exact E—k dispersion relations for heavily doped tetragonal, III–V, II–VI, IV–VI and stressed Kane type compounds. Our method is not at all related to the density-of-states (DOS) technique as used in the aforementioned works. From the electron energy spectrum, one can obtain the DOS but the DOS technique, as used in the literature cannot provide the E–k dispersion relation. Therefore, our study is more fundamental than those in the existing literature, because the Boltzmann transport equation, which controls the study of the charge transport properties of the semiconductor devices, can be solved if and only if the E–k dispersion relation is known. We wish to note that the Gaussian function for the impurity potential distribution has been used by many authors. It has been widely used since 1963 when Kane first proposed it. We will also use the Gaussian distribution for the present study.