Density of states of a two-dimensional electron gas in a perpendicular magnetic field and a random field of arbitrary correlationMathematics and Statistics
A theory is given of the density of states (DOS) of a two-dimensional electron gas subjected to a uniform perpendicular magnetic field and any random field, adequately taking into account the realistic correlation function of the latter. For a random field of any long-range correlation, a semiclassical non-perturbative path-integral approach is developed and provides an analytic solution for the Landau level DOS. For a random field of any arbitrary correlation, a computational approach is developed. In the case when the random field is smooth enough, the analytic solution is found to be in very good agreement with the computational solution. It is proved that there is not necessarily a universal form for the Landau level DOS. The classical DOS exhibits a symmetric Gaussian form whose width depends merely on the rms potential of the random field. The quantum correction results in an asymmetric non-Gaussian DOS whose width depends not only on the rms potential and correlation length of the random field, but the applied magnetic field as well. The deviation of the DOS from the Gaussian form is increased when reducing the correlation length and/or weakening the magnetic field. Applied to a modulation-doped quantum well, the theory turns out to be able to give a quantitative explanation of experimental data with no fitting parameters.
Since the discovery of the quantum Hall effect  the properties of a disordered two-dimensional electron gas (2DEG) subjected to a perpendicular magnetic field have been extensively studied. The nature of the density of states (DOS) of the 2DEG is a problem of vital importance for the understanding of many quantum phenomena observed in the system, e.g., cyclotron resonance, specific heat, magnetization, magnetocapacitance and magnetotransport.
The problem has attracted a lot of attention from both experimentalists and theorists . Nevertheless, over the years of research the common conclusion has seemed to be only that the DOS in question is composed of disorder-broadened Landau levels with a significant number of electronic states in between. To date, there has been little or no consensus regarding the exact form of the Landau level DOS of the 2DEG and the magnetic-field dependence of the linewidth. This conclusion belongs not only to the calculated DOS but the measured DOS as well. Indeed, some experimentalists have reported a Gaussian lineshape [3-9], while others claim that it is Lorentzian [10, 11]. Some report a broadening which is independent of the applied magnetic field [3, 6, 10],while others report a broadening which is a square root [4, 5, 8] or oscillating [7, 12, 13] function of the magnetic field. In addition, in order to explain their experimental results several authors [3, 5, 8] have invoked a constant background DOS whose origin is unclear.
The first calculation of the Landau level DOS was performed within a self-consistent Born approximation (SCBA) [14-16]. However, based on a single-site picture, the SCBA theory underestimates the disorder effect. Within the many-site picture, the random field is characterized by a correlation function. This is shown [2, 17] to capture microscopic details of the electron system, e.g. the actual origin of the disorder as well as the actual geometry of the sample. Moreover, for true 2DEGs this is found to exhibit a very complicated dependence on its variables in real space. Therefore, in the existing literature one has to adopt a severe approximation, replacing the true spatial correlation of a random field by a simple one with some fitting parameters. The simplest form is a δ-function, which enables an exact solution to be found for the lowest Landau level DOS . However, this model describes a zero-range electron-impurity interaction and none of the theories with this white-noise limit [19-22] predict a remarkable DOS lying between Landau levels.
Thus, the key problem is to keep the correlation length of the disorder finite from the outset [22-32] by assuming a Gaussian or exponential correlation function. Unfortunately, this simplification cannot allow an exact DOS solution and various approximation schemes have to be proposed. The Landau level DOS was evaluated by means of a cumulant expansion [22, 24, 25] for the Green function, or an expansion in the inverse correlation length for different quantities of interest, e.g. the Green function , the partition function  and the correlation function [30-32]. The validity of the approximations was often insufficiently discussed and, instead, the disorder was in practice considered to be a perturbation, as in the SCBA [22, 24, 30]. In addition, the theoretical prediction was found not to be in quantitative agreement with experimental findings .
The fact that so many different experiments seemingly yield so many different results for the Landau level DOS suggests that its exact form is very likely fixed by the very realistic nature of the 2DEG under consideration, i.e. by the experimental conditions in which the 2D electrons have been created and move. This means that, in order to quantitatively describe the measured Landau level DOS and its width, one has to work with the true correlation function. Recently, we have developed semiclassical approaches to a 1D and 2D electron in the presence of a random field of any long-range correlation [33, 34]. These turn out to meet that demand and to result in an analytic DOS solution. In the present paper, we will extend our methods to incorporate also a high perpendicular magnetic field into the theory.
It should be noted that semiclassical calculations of the Landau level DOS of a 2DEG have recently been carried out for a smooth disorder of arbitrary correlation, based on a diagrammatic  or a path-integral technique . Nevertheless, these theories were developed essentially for the case of weak disorder and, hence, turn out to be accurate, especially for high Landau levels. It is clear that, under strong magnetic fields and low temperatures, the low-energy region is likely to be of more physical interest in comparison to the high-energy one. Thus, the aim of this paper is to find another version of the semiclassical approach, which may get rid of the assumption of weak disorder and must then be useful for low Landau levels.
In section 2, the calculation of the Landau level DOS of a disordered 2DEG proceeds within a semiclassical non-perturbative path-integral approach to a smooth random field,taking explicitly into account the realistic correlation function of the random field via its average potential and force. To control the validity of the approximation used, a computational method is exactly derived in section 3, which is applicable to any random field. The theory is applied in section 4 to a quantum well where the disorder is caused by modulation doping. Finally, section 5 is devoted to conclusions.
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