Quantum transport properties of two-dimensional electron gases under high magnetic fields

We study quantum transport properties of two-dimensional electron gases under high perpendicular magnetic fields. For this purpose, we reformulate the high-field expansion, usually done in the operatorial language of the guiding-center coordinates, in terms of vortex states within the framework of real-time Green functions. These vortex states arise naturally from the consideration that the Landau levels quantization can follow directly from the existence of a topological winding number. The microscopic computation of the current can then be performed within the Keldysh formalism in a systematic way at finite magnetic fields $B$ (i.e., beyond the semiclassical limit $B=\infty$). The formalism allows us to define a general vortex current density as long as the gradient expansion theory is applicable. As a result, the total current is expressed in terms of edge contributions only. We obtain the first and third lowest order contributions to the current due to Landau-levels mixing processes, and derive in a transparent way the quantization of the Hall conductance. Finally, we point out qualitatively the importance of inhomogeneities of the vortex density to capture the dissipative longitudinal transport.

Figure 1

(Color online) Density of states at ${\epsilon }_F$ as a function of the inverse magnetic field for various disorder broadening parameters $\Gamma$. The solid curve is obtained for $\Gamma =0.1{\epsilon }_F$, the dashed curve for $\Gamma =0.03{\epsilon }_F$, and the dotted curve for $\Gamma =0.01{\epsilon }_F$.

Figure 2

(Color online) Oscillations of the chemical potential as a function of the inverse magnetic field (through the ratio ${\epsilon }_F∕\hslash {\omega }_c$) for various disorder broadening parameters $\Gamma$ (same values as in Fig. 1; we have used the same convention for the correspondence between the curves and the parameters). The plot is made for the temperature $T={\epsilon }_F∕200$. For large broadening, the oscillations are damped.

Figure 3

(Color online) Hall conductance as a function of the inverse magnetic field for various disorder broadenings. Here, the temperature is $T={\epsilon }_F∕200$. The three disorder parameters $\Gamma$ considered are the same as in Figs. 1, 2 (we have followed the same convention for the curves). The nicest plateaus are obtained for the largest broadening parameter.

Figure 4

(Color online) Hall conductance as a function of the inverse magnetic field for various temperatures. Here, the disorder broadening parameter $\Gamma ={\epsilon }_F∕10$, corresponding to a relatively flat density of states (see Fig. 1). The widest Hall plateau is obtained for the lowest temperature, here $T={\epsilon }_F∕80$. The width of the plateau shrinks with increasing temperature. The other curves correspond to the higher temperatures $T={\epsilon }_F∕40$, $T={\epsilon }_F∕20$, and $T={\epsilon }_F∕10$.

Figure 5

(Color online) The chemical potential $\mu$ over $\hslash {\omega }_c$ as a function of the magnetic field. For a small broadening of the Landau levels (dotted curve), the ratio $\mu ∕\hslash {\omega }_c$ remains close to the value $n+1∕2$ for a wide range in magnetic fields. For a large broadening (solid curve), the chemical potential approaches the energy $(n+1∕2)\hslash {\omega }_c$ only for the fields corresponding to ${\epsilon }_F=(n+1∕2)\hslash {\omega }_c$. For the correspondence between the different curves and the chosen parameters, see Figs. 1, 2, 3.