The Einstein Relation in Compound Semiconductors Under Magnetic Quantization

It is well known that the band structure of electronic materials can be dramatically changed by applying external fields [1]. The effects of the quan¬tizing magnetic field on the band structure of compound semiconductors are more striking and can be observed easily in experiments. Under magnetic quantization, the motion of the electron parallel to the magnetic field remains unaltered while the area of the wave vector space perpendicular to the direc¬tion of the magnetic field gets quantized in accordance with Landau's rule of area quantization in the wave-vector space [2]. The energy levels of the carri¬ers in a magnetic field (with the component of the wave-vector parallel to the direction of magnetic field equated with zero) are termed as the Landau levels and the quantized energies are known as the Landau sub-bands. It is important to note that the same conclusion may be arrived either by solving the single-particle time independent Schrodinger differential equation in the presence of a quantizing magnetic field or by using the operator method. The quantizing magnetic field tends to remove the degeneracy and increases the band gap. A semiconductor, placed in a magnetic field B,can absorb radiated energy with the frequency ω₀ (= (|e| B/m*)).This phenomenon is known as cyclotron or diamagnetic resonance. The effect of energy quantization is experimentally noticeable when the separation between any two consecutive Landau levels is greater than k _B T.A number of interesting transport phenomena originate from the change in the basic band structure of the semiconductor in the pres¬ence of a quantizing magnetic field. These have been widely been investigated and have also served as diagnostic tools for characterizing the different ma¬terials having various band structures. The discreteness in the Landau levels leads to a whole crop of magneto-oscillatory phenomena, important among which are (a) Shubnikov-de Haas oscillations in magneto-resistance; (b) de Haas-Van Alphen oscillations in magnetic susceptibility; (c) magneto-phonon oscillations in thermoelectric power, etc.

In Sect. 3.2.1, of the theoretical background, the Einstein relation has been investigated in tetragonal materials in the presence of an arbitrarily ori¬ented quantizing magnetic field by formulating the density-of-states function. Section 3.2.2 contains the results of III–V, ternary and quaternary compounds in accordance with the three and the two band models of Kane and forms the special case of Sect. 3.2.1. In the same section the well known result of DMR in relatively wide gap materials has been presented. Section 3.2.3 contains the study of the Einstein relation for the II–VI semiconductors under mag¬netic quantization. In Sect. 3.2.4, the magneto-DMR for Bismuth has been investigated in accordance with the models of McClure and Choi, Cohen, Lax non-parabolic ellipsoidal and the parabolic ellipsoidal respectively. In Sect. 3.2.5, the Einstein relation in IV–VI materials has been discussed. In Sect. 3.2.6, the magneto-DMR for the stressed Kane type semiconduc¬tors has been investigated. Section 3.3 contains the result and discussions in this context.