Probing Electron-Electron Interaction in Quantum Hall Systems with Scanning Tunneling Spectroscopy

Using low-temperature scanning tunneling spectroscopy applied to the Cs-induced two-dimensional electron system (2DES) on $p$-type InSb(110), we probe electron-electron interaction effects in the quantum Hall regime. The 2DES is decoupled from bulk states and exhibits spreading resistance within the insulating quantum Hall phases. In quantitative agreement with calculations we find an exchange enhancement of the spin splitting. Moreover, we observe that both the spatially averaged as well as the local density of states feature a characteristic Coulomb gap at the Fermi level. These results show that electron-electron interaction can be probed down to a resolution below all relevant length scales.

Figure 1

(a) Calculated band bending using 1D Poisson equation; first two 2D subband energies $E_0=-118\mathrm{meV}$, $E_1=-65\mathrm{meV}$ as calculated using the triangular well approximation [17, 35] are marked; corresponding electron distribution is shown in 17. (b) Spatially averaged $dI/dV$ curves across an area $A_{\mathrm{Av}}$: (I): without contacted 2DES, (II), (III): with contacted 2DES at $B=0\mathrm{T}$, 7 T as marked; gray area: bulk band gap of $p$-InSb [36]; $E_0$, $E_1$: subband energies from experiment (solid) and calculation (broken lines); arrows in (III) mark spin-split LLs. (I) $V_{\mathrm{stab}}=400\mathrm{mV}$, $I_{\mathrm{stab}}=100\mathrm{pA}$, $V_{\mathrm{mod}}=1{\mathrm{mV}}_{\mathrm{rms}}$, $A_{\mathrm{Av}}=200\times 160{\mathrm{nm}}^2$; (II) $V_{\mathrm{stab}}=300\mathrm{mV}$, $I_{\mathrm{stab}}=50\mathrm{pA}$, $V_{\mathrm{mod}}=3{\mathrm{mV}}_{\mathrm{rms}}$, $A_{\mathrm{Av}}=100\times 100{\mathrm{nm}}^2$; (III) $V_{\mathrm{stab}}=300\mathrm{mV}$, $I_{\mathrm{stab}}=200\mathrm{pA}$, $V_{\mathrm{mod}}=0.4{\mathrm{mV}}_{\mathrm{rms}}$, $A_{\mathrm{Av}}=300\times 300{\mathrm{nm}}^2$.

Figure 2

(a) $dI/dV$ spectra (gray scale) recorded at the same lateral position at different $B$ fields as indicated. $I_{\mathrm{stab}}$ is increased for each $B$ from 100 pA to 2000 pA (left to right). $V_{\mathrm{stab}}=300\mathrm{mV}$, $V_{\mathrm{mod}}=1.6{\mathrm{mV}}_{\mathrm{rms}}$. (b) Measured lowest LL positions (spin-up $\uparrow$; spin-down $\downarrow$) at $B=7\mathrm{T}$ (symbols) in comparison with calculated LL positions (line). Note that the current $I$ at the peak position and not $I_{\mathrm{stab}}$ is used as the $x$ axis.

Figure 3

(a) Landau fan showing $dI/dV$ as grayscale; $B$ field ramped downwards, $V_{\mathrm{stab}}=300\mathrm{mV}$, $I_{\mathrm{stab}}=400\mathrm{pA}$, $V_{\mathrm{mod}}=1.6{\mathrm{mV}}_{\mathrm{rms}}$. (b) Spin splitting energy $E_{\mathrm{SS}}$ of lowest LL extracted by Gaussian fits as shown in the inset. Straight line marks $E_{\mathrm{SS}}=|g^{*}|{\mu }_{B}B$. (c) Deviation $\Delta$ from the linear fit in (b). The inner bright line is smoothed. Vertical bars show the calculated values of EE between neighboring odd and even filling factors. Local filling factors $\nu \propto 1/B$ are matched by counting spin levels below $E_F$ [compare Fig. 4b] and verifying their $B$ dependence.

Figure 4

(a) Zoom of Fig. 1b (III) (thick line), additional Gaussian fits for the LL at $E_F$ (thin lines), and a plot of the difference (medium, blue line) in comparison with the expected bare Coulomb gap of Eq. (1) ($V$-shaped line) and the Coulomb gap taking finite temperature, screening and energy resolution into account (dotted line); absolute scale of DOS is deduced by fitting the lowest spin-split LL to a two-peak Gaussian and matching the integral of one Gaussian to the known level degeneracy $eB/(2\pi \hslash )$. (b) $dI/dV$ spectra taken at the same position at $B$ as marked. Arrows of same color follow the same peak or minimum across $E_F$; a movie of the data is available in 17 (c) $dI/dV$ value of the marked peak and minimum in (b) as a function of energy (voltage).