Strong electron-electron interactions of a Tomonaga-Luttinger liquid observed in InAs quantum wires

We report strong electron-electron interactions in quantum wires etched from an InAs quantum well, a material generally expected to have strong spin-orbit interactions. We find that the current through the wires as a function of the bias voltage and temperature follows the universal scaling behavior of a Tomonaga-Luttinger liquid. Using a universal scaling formula, we extract the interaction parameter and find strong electron-electron interactions, increasing as the wires become more depleted. We establish theoretically that the spin-orbit interaction cause only minor modifications of the interaction parameter in this regime, indicating that genuinely strong electron-electron interactions are indeed achieved in the device. Our results suggest that etched InAs wires provide a platform with both strong electron-electron interactions and the strong spin-orbit interaction.

Figure 1

(a) Microscope photographs of the parallel-wire device before (left) and after (right) depositing a top gate above a cross-linked PMMA layer. The wires are along [010]. (b) Schematic structure of the wafer, grown along [001]. (c) Gate voltage dependence of the current through the wires measured at a constant source-drain voltage as given in the figure caption. This measurement is performed at 2.9 K.

Figure 2

Current ($I$) flowing through the wires as a function of the bias voltage ($V$) for the top gate voltage $V_g=-0.6\mathrm{V}$.

Figure 3

A rescaled current $I/T^{1+\alpha }$ as a function of $eV/k_{\text{B}}T$ from the data points in Fig. 2. The black-solid curve is drawn using Eq. (2) with the parameters $(\alpha ,\gamma ,I_0)=(1.3,0.38,3.6{\times }^{-10}\mathrm{A}/{\mathrm{K}}^{2.3})$, which were extracted by fitting the data in Fig. 2 to Eq. (2). We note that the unit of $I_0$ scales with $\alpha$.

Figure 4

Fitted values of $\alpha$ as a function of the top gate voltage $V_g$ (red circle, left axis). The error bars of the fitting are smaller than the markers. Fitted values of $\gamma$ as a function of $V_g$ are also plotted (blue squares, right axis). For $V_g>-0.25\mathrm{V}$, the fitted values of $\gamma$ are not very reliable, as reflected by the larger error bars.

Figure 5

(a) A schematic illustration of scenario A: there are bulk barriers (crosses), each acting as a TLL-TLL junction, and many weak impurities (not shown). (b) A schematic illustration of scenario B: there are boundary barriers, each acting as a TLL-Fermi liquid (FL) junction, and many weak impurities. In (a) and (b), for illustration, we plot two barriers, motivated by the observed $1/\gamma \lesssim 2$. (c) Extracted values of the interaction parameter $g_{\text{c}}$ as a function of the top gate voltage $V_g$ for the two scenarios. The red-solid and green-dashed curves are the fits to Eq. (6), with the fitting parameter $w$ (the wire transverse size) being 87 and 47 nm, respectively.

Figure 6

SEM images of quantum wires chemically etched out from the wafer grown along [001] direction. The wires are along (a) $\left[1\overline{1}0\right]$, (b) [110], and (c) [010] direction, respectively. The scale bar in each figure indicates the length of 500 nm. From the brightness of the edges compared to dots surrounding the wires, each wire's cross-section is inferred as a trapezium, a reverse trapezium and a rectangle, respectively.

Figure 7

Current as a function of bias voltage with various $V_g$. The first row is the raw data and the second is fitted and scaled data. We evaluate $(\alpha ,\gamma ,I_0)=(0.42,1.0,2.9\times {10}^{-9}{\mathrm{A}/\mathrm{K}}^{1.42}), (0.51,0.61,2.2\times {10}^{-9}{\mathrm{A}/\mathrm{K}}^{1.51})$, and $(2.5,0.38,7.7\times 10-13{\mathrm{A}/\mathrm{K}}^{3.5})$ for $V_g=-0.2, -0.4$ and $-0.8\mathrm{V}$, respectively.

Figure 8

Effective interaction parameter $g_{\mathrm{c},\mathrm{eff}}$ as a function of the actual interaction parameter $g_{\text{c}}$ for various values of the ratio $\delta v/v_F$. (Top) In the case of tunnel barriers, $g_{\mathrm{c},\mathrm{eff}}^{\mathrm{bulk}}$ is defined in Eq. (B6). (Bottom) In the case of weak impurities, $g_{\mathrm{c},\mathrm{eff}}^{\mathrm{imp}}$ is defined in Eq. (B9).