Tunneling spectroscopy of disordered two-dimensional electron gas in the quantum Hall regime

The tunneling density of states (TDOS) into a disordered two-dimensional electron gas is calculated in the quantum Hall regime, including electron-electron interactions. Combining general arguments and a detailed Hartree-Fock calculation, it is demonstrated that the quenching of the kinetic energy by the applied magnetic field leads to several universal features, nearly independent of the disorder potential, corresponding to the addition and removal spectrum in the valleys and tips of the disorder potential. These features are manifested as “sashes” in the TDOS spectra, and are in quantitative agreement with recent measurements. It is predicted that more such features will become observable with decreasing potential fluctuations.

Figure 1

(a) Schematic of the TDOS dependence on the filling factor. For ν $=$ 1, the TDOS comprises a disorder-broadened band (I). Energies are defined with respect to the chemical potential. For small filling factors, populated states (II) comprise electrons sitting at the ground state of disorder wells. Under strong magnetic field, the wave functions of the lowest-energy states in a disorder potential well are only weakly dependent on details of the disorder potential. A second Hubbard band is created (H1), resulting from the strong Coulomb interaction of adding an opposite-spin electron to an occupied state. Arrows indicate spin orientation. (b) Schematic of the TDOS dependence on the filling factor, including secondary bands arising from interactions with higher angular momentum eigenstates. In addition to the high-energy band H1, a lower band (H2) arises from the injection of an electron into the more extended, higher-energy eigenstate. A higher-energy band (H3) is expected for larger filling factors, corresponding to the addition of an electron to a disorder potential valley already occupied by two particles.

Figure 2

(a) The Hartree-Fock model employed to study the influence of Coulomb interactions on the tunneling density of states. (b) The background disorder potential (black line) gives rise to a broad tunneling density of states. When an electron occupies a potential valley, the resulting Coulomb interaction creates high-energy bands for additional occupation of the same site with an opposite spin electron, as well as occupation of nearby sites. (c) The resulting tunneling density of states is comprised of the low-energy disorder-broadened band, and several high-energy bands corresponding to the discrete energies to occupy neighboring states of an occupied site.

Figure 3

(a) A Hartree-Fock calculation of the TDOS, when the screening length is comparable to the magnetic length ($l_{\mathrm{sc}}$ ≈ $l_B$). White features correspond to high TDOS. The strong feature marked by the dashed blue line is a “second Hubbard band” created due to the added energy required to inject a particle into a disorder well already occupied by an opposite spin particle. (b) Calculated TDOS when the screening length is larger than the magnetic length ($l_{\mathrm{sc}}$ ≈ 4$l_B$). Additional high-energy features arise due to interactions between neighboring states (see the text).

Figure 4

(a), (b) Simplified model for the 2DEG density of states, presenting the nontrivial influence of temperature. (a) For $T$ $=$ 0, all states below the Fermi level (set here to $E$ $=$ 0) are occupied, giving rise to a duplicate high-energy band H1. (b) For $T$0, occupation of the low-energy states is broadened. The high-energy second Hubbard band duplicates this broadening. (c) The calculated TDOS at $T$ $=$ 0 (blue) for low filling factor of ν $=$ 0.03. A temperature increase to 4 K smears the TDOS structure, but the characteristic structure is still observable (red). Considering, in addition, the influence of the heated charge distribution in the tunneling electrode renders this structure undistinguishable (green).