Hope, Stephen Alexander
(1979)
Einstein diffusion-mobility relations in semi-conductors.
*University of Southampton, Doctoral Thesis*.

## Abstract

Transport equations for semiconductors can be written with the diffusion term eD1Vn or eV(D2n), where D1 and D2 are diffusion coefficients. In Chapter 1 the issues arising from this ambiguity are clarified and convenient macroscopic transformation formulae between the two formulations, for application in semiconductor devices at low electric fields, are developed. A number of new macroscopic formulae for the Soret coefficient, the thermoelectric power and the thermal conductivity are derived. A consistent macroscopic scheme of equations is developed, which also include the Einstein diffusion - mobility ratio eD1u 2nnlrJ (1) T,V and a corresponding D2-Einstein relation. Transformation formulae between the D1 D2 formulations for application at high fields in the Gunn effect are given in Appendix H. In Chapter 2, (1) and the D2-Einstein relation are obtained microscopically from the Boltzmann transport equation for both zero and non-zero magnetic fields and for an arbitrary density of states. This is a steady state nonequilibrium linear argument valid, for low fields. In Chapter 3 generalised macroscopic theory is given and kept general enough to incorporate the effects of two types of non-linearity: (a) The ensemble average of the particle density operator is represented by a simple power series expansion, valid for equilibrium conditions, in the electrostatic potential ¢(r) up to terms in ~2(r). (b) For strong fields (E) and carriergradients (Vn) terms in E2, (Vn)2, V2n and EVn are considered in the total current. Thus one obtains new macroscopic non-linear Einstein-type equations valid for equilibrium situations, in terms of the coefficients occuring in the expansion (a), in addition to (1). The amendment (a) is of importance for weak non-uniformly-doped semiconductors even when the effect of (b) is negligible. Both (a) and (b) may be neglected in uniformly-doped semiconductors. In Chapter 4, using an equilibrium density matrix, a Kubo-formalism i.e. response theory is employed to obtain microscopic expressions for the coefficients occuring in the expansion in (a). The Einstein relation (1) and the non-linear relations obtained macroscopically in Chapter 3 are now given microscopic interpretations, in the sense that the coefficients in the expansion (a) are now known. In Section 4.5, which deals with heavily non-uniformly-doped crystals, it is suggested that (1) may no longer be valid. In Section 5.1 response theory is again used and integral expressione are obtained for the conductivity a, the diffusivity Dl and the coefficients of proportionality of the driving forces in amendment (b) which are seen to satisfy the Einstein relations given in Chapters 3 and 4. In Chapter 6 the main conclusions of the thesis are given. and various avenues for future research are discussed.

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