Analytic Solution for the Density of States of the Disordered Quasi-One-Dimensional Electron Gas in a Quantum WireMathematics and Statistics
Analytic Solution for the Density of States of the Disordered
Quasi-One-Dimensional Electron Gas in a Quantum Wire
There has been a great deal of recent interest1) in semi-conductor quantum wire (QWR) structures, where elec-tron dynamics is essentially restricted to be one dimen-sional. The quasi-one dimensional electron gas, hereafter called for short a one-dimensional electron gas (1DEG), has been intensively studied both experimentally and theoretically. The QWR structures have opened up the potential for various device applications.1-3)
In practice, the 1DEG is generally strongly affected by disorder caused by some random field present in the wire. The field is of different origins, e.g. impurity dop-ing,4-8) surface roughness4-10) and alloying.4, 6, 11) The disorder has been shown to lead to considerable mod-ifications in the energy spectra of the 1DEG for both one-electron states12-15) and elementary collective exci-tations.16) These in turn result in remarkable changes in many phenomena occurring in the wire, e.g. optical ab-sorption. Since the density of states (DOS) is the main ingredient of the integral expressions for many observ-able properties of 1DEG’s (optical absorption coefficient, partition function, etc.), having a useful, especially ana-lytic, solution for the DOS is of fundamental importance in explaining the phenomena in QWR’s as well as in ana-lyzing the performance of modern semiconductor devices based on them.
So far, only a few theoretical investigations12-15) have been made in order to understand the one-particle energy spectrum of disordered 1DEG’s in QWR’s, and merely numerical results have been available in the literature. It should be noted that disorder sources other than impu-rity doping have recently been confirmed experimentallyto be of importance in very thin QWR’s, e.g. surfaceroughness in wires made from GaAs/AlxGa1−xAs17-19) and alloying in wires from In1−xGaxAs/InP.6, 20) Never-theless, it is surprising to note that the existing theories of disorder effect on the 1D DOS have been focused only on the disorder arising from impurity doping, ignoring the other disorders. The basic idea has then been to extend the relevant approaches already established for the 3D case of bulk semiconductors,21) taking into ac-count the main features of quasi-1D electrons in the wire. Takeshima12) proposed a 1D version of the Kane’s semiclassical model for calculating the DOS in heavily doped square wires, while Ghazali et al.13) developed a 1D ver-sion of the Klauder’s best multiple-scattering approach to the DOS in doped cylindrical wires. There, it is the potential created by an individual center of force, e.g. a single ionized impurity in doped QWR’s, that has been chosen to be the input function for disorder interaction. The single-center potential is useful for describing the disorder interaction due to impurity doping, but, it is obviously inadequate for describing other interactions, e.g. due to surface roughness. Moreover, in the case of doping-caused disorder, the single-impurity potential is screened by interacting electrons and is generally seen very complicated for realistic 1DEG’s.4-8) The why is probably that the symmetry of the screened impurity potential in QWR’s is drastically reduced because the screening 1D electrons may be redistributed merely along one dimension (the wire axis). Therefore, the current theories have to invoke severe approximations, e.g. the use of a 3D (bulk-like) screened Coulomb impurity potential12) and the separable-potential approximation.13)Despite these the theories are, however, computationallystill very complicated for realistic 1DEG’s.12, 13)
Recently, Quang and Tung22) have proposed another 1D version of the Kane’s semiclassical model which em-ploys the autocorrelation function of the random field as the input function for disorder interaction. This enables one to analytically evaluate on equal footing the influence on the 1D DOS from disorder of arbitrary origins without any simplifying assumption on the potential shape. Nev-ertheless, the approach has assumed the random field to be smooth, involving only long-range fluctuations in the disorder potential. As a result, this cannot be applied to the case of impurity doping inside a very thin wire and/or at a rather low doping level nor to alloy disorder with a δ-potential. Furthermore, the theory may describe prop-erly the high-energy region (above and near the subband edge) and is inapplicable to the low-energy region (deep tail), where the short-range potential fluctuations make clearly the key contribution. Electron energy states far under the subband edge surely need a quantum descrip-tion.
Thus, the aim of the present paper is to derive an ana-lytic solution for the electronic DOS of actual disordered 1DEG’s in semiconductor QWR’s which is applicable to disorder not only arising from impurity doping but also of any origin, e.g. surface roughness and alloying, and, in addition, which must describe the entire energy spec-trum. For this purpose, we will modify the path-integral method suggested by Quang and Tung23) for disordered electron systems in two dimensions.
In §2 below, we start with a collection of the formulae to be used for calculating the DOS of 1DEG’s subjected to a Gaussian random field. In §3, the 1D DOS is approx-imately evaluated in dependence on the curvature of a non-local harmonic trial well. For finding a best value of this curvature, different variational equations are derived in §4, where a scheme for obtaining the 1D DOS over all energies is proposed. In §5, the asymptotic behavior of the 1D DOS is given in a very deep tail. The theory is specified in §6 for a modulation-doped cylindrical QWR from GaAs/AlxGa1−xAs. Finally, some advantages of the theory are listed in §7.
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