The Einstein Relation in Compound Semiconductors Under Size Quantization

In recent years, with the advent of fine lithographical methods [1], molecular beam epitaxy [2], organometallic vapor-phase epitaxy [3], and other experimental techniques, the restriction of the motion of the carriers of bulk materials in one (ultrathin films, quantum wells, nipi structures, inversion layers, accumulation layers), two (quantum wires), and three (quantum dots, magnetosize quantized systems, magneto inversion layers, magneto accumulation layers, quantum dot superlattices, magneto quantum well superlattices and magneto NIPI structures) dimensions has in the last few years, attracted much attention not only for its potential in uncovering new phenomena in nanoscience but also for its interesting quantum device applications [4–6]. In ultrathin films, the restriction of the motion of the carriers in the direction normal to the film (say, the z direction) may be viewed as carrier confinement in an infinitely deep 1D rectangular potential well, leading to quantization [known as quantum size effect (QSE)] of the wave vector of the carrier along the direction of the potential well, allowing 2D carrier transport parallel to the surface of the film representing new physical features not exhibited in bulk semiconductors [7]. The low-dimensional heterostructures based on various materials are widely investigated because of the enhancement of carrier mobility [8]. These properties make such structures suitable for applications in quantum well lasers [9], heterojunction FETs [10], high-speed digital networks [11], high-frequency microwave circuits [12], optical modulators [13], optical switching systems [14], and other devices. The constant energy 3D wave-vector space of bulk semiconductors becomes 2D wave-vector surface in ultrathin films or quantum wells due to dimensional quantization. Thus, the concept of reduction of symmetry of the wave-vector space and its consequence can unlock the physics of low dimensional structures.

In Sect. 5.2.1 of this chapter, the expressions for the surface electron concentration per unit area and the 2D DMR for ultrathin films of tetragonal materials have been formulated on the basis of the generalized dispersion relation, as given by (2.2). In Sect. 5.2.2, it has been shown that the corresponding results of the 2D DMR in ultrathin films of III–V, ternary and quaternary compounds form special cases of our generalized analysis as given in Sect. 5.2.1. In Sect. 5.2.3, we have studied the same for ultrathin films of II–VI semiconductors. In Sect. 5.2.4, the 2D DMR has been derived for ultra-thin films of bismuth in accordance with the McClure and Choi, the Cohen, the Lax nonparabolic ellipsoidal, and the parabolic ellipsoidal models respectively. In Sects. 5.2.5 and 5.2.6, the formulations of the 2D DMR in ultrathin films of IV–VI and stressed Kane type materials has been presented. The last Sect. 5.2.7 contains the result and discussions for this chapter.