Coulomb interactions and effective quantum inertia of charge carriers in a macroscopic conductor

We study the low-frequency admittance of a quantum Hall bar of a size much larger than the electronic coherence length. We find that this macroscopic conductor behaves as an ideal quantum conductor with vanishing longitudinal resistance and purely inductive behavior up to $f\lesssim 1\mathrm{M}\mathrm{Hz}$. Using several measurement configurations, we study the dependence of this inductance on the length of the edge channel and on the integer quantum Hall filling factor. The experimental data are well described by a scattering model for edge magnetoplasmons taking into account the effective long range Coulomb interactions within the sample. We find that the inductance's dependence on the filling factor arises predominantly from the effective quantum inertia of charge carriers induced by Coulomb interactions.

Figure 1

(a) The device under test (DUT) is measured using four coaxial cables and the impedance-meter Agilent HP4294A, which measures the current $I$ at $L_{\text{cur}}$, the potential $V$ at $H_{\text{pot}}$, and gives $G=I/V$ (for details, see Refs. [36, 37]). Note that the potentials of $H_{\text{pot}}, L_{\text{pot}}, H_{\text{cur}}$, and $L_{\text{cur}}$ are the potentials of the four connectors of the impedance meter. (b) Scheme of the multiterminal Hall bar with only three ohmic contacts wire-bonded onto the sample holder. In this geometry, the impedance is $Z_{23}^{\left(\text{expt}\right)}\left(\omega \right)=-(\partial V_3/\partial I_2)\left(\omega \right)$ for $V_2=0$.

Figure 2

The reactance $X$ as a function of the frequency $f$ for different $\nu$ and $B_{<0}$, in the measurement configuration shown here. Inset: The longitudinal resistance $R\left(f\right)$ vanishes quadratically for integer filling factors at low frequency.

Figure 3

For measurement configurations with $B<0$ corresponding to $l=600$, 1000, and $1600\mu \mathrm{m}$, the inductance increases with $1/\nu$. Dashed lines correspond to model $v_{\text{d}}\left(\nu \right)=v_{\text{d}}/\nu$ with $v_{\text{d}}=15$, 5, and 40 (in units of ${10}^5\mathrm{m}{\mathrm{s}}^{-1}$) from top to bottom. The $v_{\text{d}}\left(\nu \right)=v_{\text{d}}/\sqrt{\nu }$ model leads to (solid lines) $v_{\text{d}}=6$, 3, and 17 in units of ${10}^5\mathrm{m}{\mathrm{s}}^{-1}$ from top to bottom. The blue points corresponds to the experimental data displayed in Fig. 2.

Figure 4

$v_{\text{eff}}$ as a function of $\nu$. Dashed curves correspond to $v_{\text{d}}\left(\nu \right)=v_{\text{d}}/\nu$ and solid lines to $v_{\text{d}}\left(\nu \right)=v_{\text{d}}/\sqrt{\nu }$. Colored lines correspond to the expressions (1) taking into account interedge Coulomb interactions. Thin gray curves correspond to plots of the bare velocity $v_{\text{d}}\left(\nu \right)$. The solid lines have been obtained with (starting from the top curve) $v_{\text{d}}=17, v_{\text{d}}=6$ (fit of the data from Fig. 2), $v_{\text{d}}=3, v_{\text{d}}=1.2$, and $v_{\text{d}}=0.66$ in units of ${10}^5\mathrm{m}{\mathrm{s}}^{-1}$. For the dashed lines, we have starting from the top curve: $v_{\text{d}}=40, v_{\text{d}}=15, v_{\text{d}}=5, v_{\text{d}}=2$, and $v_{\text{d}}=1$ in units of ${10}^5\mathrm{m}{\mathrm{s}}^{-1}$.