Basics of the Einstein Relation

It is well known that the Einstein relation for the diffusivity-mobility ratio (DMR) of the carriers in semiconductors occupies a central position in the whole field of semiconductor science and technology [1] since the diffusion constant (a quantity very useful for device analysis where exact experimental determination is rather difficult) can be obtained from this ratio by knowing the experimental values of the mobility. The classical value of the DMR is equal to (kBT ǀ ǀeǀ) , (kB, T, and ǀeǀ are Boltzmann's constant, temperature and the magnitude of the carrier charge, respectively). This relation in this form was first introduced to study the diffusion of gas particles and is known as the Einstein relation [2,3]. Therefore, it appears that the DMR increases linearly with increasing T and is independent of electron concentration. This relation holds for both types of charge carriers only under non-degenerate carrier concentration although its validity has been suggested erroneously for degenerate materials [4]. Landsberg first pointed out that the DMR for semiconductors having degenerate electron concentration are essentially determined by their energy band structures [5, 6]. This relation is useful for semiconductor homostructures [7, 8], semiconductor—semiconductor heterostructures [9, 10], metals—semiconductor heterostructures [11–19] and insulator-semiconductor heterostructures [20–23]. The nature of the variations of the DMR under different physical conditions has been studied in the literature [1–3, 5, 6, 24–50] and some of the significant features, which have emerged from these studies, are:

• (a) The ratio increases monotonically with increasing electron concentration in bulk materials and the nature of these variations are significantly in- fluenced by the energy band structures of different materials;

• (b) The ratio increases with the increasing quantizing electric field as in inversion layers;

• (c) The ratio oscillates with the inverse quantizing magnetic field under magnetic quantization due to the Shubnikov de Hass effect;

• (d) The ratio shows composite oscillations with the various controlled quantities of semiconductor superlattices.

• (e) In ultrathin films, quantum wires and field assisted systems, the value of the DMR changes appreciably with the external variables depending on the nature of quantum confinements of different materials.