Quasiparticle tunneling in the lowest Landau level

We measure quasiparticle tunneling across a constriction in the first Landau level. In the limit of weak backscattering, the dependence of the tunneling conductance on temperature and dc-bias is in qualitative disagreement with existing theories. For stronger backscattering, data obtained in the $\nu =1/3$ state can be fitted to weak backscattering theory with the predicted effective fractional charge of $e^{*}=e/3$. The scaling parameter $g$ is however not universal and depends strongly on the gate voltage applied to the constriction. At $\nu =4/3$, a more complex picture emerges. We propose an interpretation in terms of selective tunneling between the multiple modes present at the edge.

Figure 1

(a) Schematic of the sample. Crosses represent Ohmic contacts to the Hall bar, voltage probes are indicated by dotted lines. (b) SEM image showing typical QPC geometries. The QPC on the left side has been used to acquire the data shown in Figs. 2, 3, and 4. The QPC on the right side has a similar design as the one used to acquire the data shown in Figs. 5 and 6. The tunneling conductance $g_{\text{tun}}$ between two counterpropagating edge modes in the QPC is represented schematically.

Figure 2

(a) Diagonal resistance measured across QPC 1 (solid colored lines) for three different values of gate voltage. The black dotted line shows $R_{\text{xy}}$ in the bulk of the sample, which is in the $\nu =1/3$ state at $B=12.35$ T (value marked by the vertical dashed orange line). Labels indicate gate voltages $V_{\text{G}}$ and the transmission $t$ of the $\nu =1/3$ edge channel at $B=12.35$ inferred from $R_{\text{Diag}}$. No dc bias was applied in this measurement. (b) dc-bias dependence of the tunneling conductance $g_{\text{tun}}$ across QPC 1 at $B=12.35$ for the same three values of gate voltage as in (a). All data shown in this figure have been acquired at an electronic temperature of $71\mathrm{mK}$.

Figure 3

(a) to (c) Temperature and dc-bias dependence of the tunneling conductance across QPC 1. The gate voltage and the electronic density extracted from low-field data are indicated on top of each graph. The data acquired at $71\mathrm{mK}$ are the same as shown in Fig. 2. Note that the seven $y$ axes in each column are independent, and that their scales are adapted for an optimal readability of the data. In (c), the black crosses follow the best fit to weak QP tunneling theory. Electronic temperatures are indicated on the right-hand side and are color coded for the convenience of the reader.

Figure 4

Best-fit parameters $e^{*}$ and $g$ obtained in a least-square fit of Eq. (2) to the temperature and dc-bias dependence of the tunneling conductance across QPC 1. (a) Gate voltage dependence of the best-fit parameters for a fixed value of magnetic field $B=12.35\mathrm{T}$ [vertical dashed line in (b)]. The top axis shows the transmission $t$ of the $\nu =1/3$ mode at zero dc bias. The underlying data have been acquired in the same measurement as those displayed in Figs. 2 and 3. Data acquired for values of gate voltage less negative than $-1.35\mathrm{V}$ cannot be fit to QP tunneling theory. (b) Magnetic field dependence of the best fit parameters for a fixed value of gate voltage $V_{\text{G}}=1.52\mathrm{V}$ [vertical dashed line in (a)]. The underlying data have been acquired under the same experimental conditions as for the results shown in (a), in a temperature range of $96\mathrm{mK}$ to $411\mathrm{mK}$.

Figure 5

(a) Diagonal resistance measured across QPC 2 (solid colored lines) for three different values of gate voltage. The black dotted line shows $R_{\text{xy}}$ in the bulk of the sample, which is in the $\nu =4/3$ state at $B=3.4\mathrm{T}$ (value marked by the dashed orange line). Labels indicate gate voltages $V_{\text{G}}$ and the transmission $t$ of the $\nu =4/3$ edge channels at $B=3.4\mathrm{T}$ inferred from $R_{\text{Diag}}$. (b) dc-bias dependence of the tunneling conductance $g_{\text{tun}}$ across QPC 2 for the same three values of gate voltage as in (a). All data shown in this figure have been acquired at an electronic temperature of $9.6\mathrm{mK}$.

Figure 6

Temperature and dc-bias dependence of the tunneling conductance across QPC 2 with a gate voltage of $-1.35\mathrm{mV}$. (a) Tunneling conductance calculated from raw data following Eq. (1). The data obtained at $9.6\mathrm{mK}$ (uppermost) are the same as shown in Fig. 5. Dashed lines in the two uppermost graphs mark the value $g_{\text{tun}}=4/9\approx 0.44$. (b) Tunneling conductance of the outermost $\nu =1/3$ mode calculated according to Eq. (3) on the basis of the same data as in (a). The edge structure model corresponding to the method chosen for calculating $g_{\text{tun}}$ is symbolically represented on top of the figure. Note that the eight $y$ axes in each column are independent, and that their scales are adapted for an optimal readability of the data. In (a) and (b), the black crosses follow QP tunneling theory with $g=1/3,e^{*}=e/3$, and manually adjusted amplitude and offset. Electronic temperatures are indicated on the left-hand side and are color coded for the convenience of the reader.