Statistical study of conductance properties in one-dimensional quantum wires focusing on the 0.7 anomaly

The properties of conductance in one-dimensional (1D) quantum wires are statistically investigated using an array of 256 lithographically identical split gates, fabricated on a GaAs/AlGaAs heterostructure. All the split gates are measured during a single cooldown under the same conditions. Electron many-body effects give rise to an anomalous feature in the conductance of a one-dimensional quantum wire, known as the “0.7 structure” (or “0.7 anomaly”). To handle the large data set, a method of automatically estimating the conductance value of the 0.7 structure is developed. Large differences are observed in the strength and value of the 0.7 structure [from $0.63$ to $0.84\times (2e^2/h)$], despite the constant temperature and identical device design. Variations in the 1D potential profile are quantified by estimating the curvature of the barrier in the direction of electron transport, following a saddle-point model. The 0.7 structure appears to be highly sensitive to the specific confining potential within individual devices.

Figure 1

(a) Typical trace of conductance $G$ as a function of the voltage applied to the split gate $V_{\mathrm{sg}}$, from $V_{\mathrm{sg}}$ = 0 to $V_p$. The vertical (horizontal) dashed line indicates $V_d$ $\left(G_d\right)$. The inset shows a schematic diagram of a split gate, where S and D correspond to the source and drain contacts, respectively. (b) Scatter plot of $G_d$ against $V_p$ for which $r=-0.81$, indicating a strong negative correlation. (c) Histogram of $V_d$ from 240 split gates, for a bin size of 1 mV. (d) Color scale of $V_d$ as a function of spatial location, where each box represents a split gate in the array. Clear boxes indicate devices which did not define a 1D channel. The boxes are equally spaced in the $x$ and $y$ directions for convenience, although in reality the split gates have horizontal and vertical pitch lengths of 100 and 130 $\mu$m, respectively.

Figure 2

(a)–(c) Conductance against $V_{\mathrm{sg}}$ for three nominally identical split gates. Well-developed conductance anomalies occur below $G_0$ in panels (a) and (b), indicated by the arrows. A much weaker shoulder-like feature occurs near $0.7G_0$ in panel (c), marked by the arrow. (d) Conductance as a function of $V_{\mathrm{sg}}$ for a fourth example split gate (solid line). First, second, and third derivatives $dG/d{V}_{\mathrm{sg}}$, $d^{2}G/d{V}_{\mathrm{sg}}^2$, and $d^{3}G/d{V}_{\mathrm{sg}}^3$ are shown by the dashed, dotted, and dash-dotted curves, respectively. The $dG/d{V}_{\mathrm{sg}}$ trace is offset vertically for clarity, and all derivative data are scaled vertically in order to be shown on the plot. The dashed horizontal line shows $G_{0.7}$, corresponding to the local minimum in $dG/d{V}_{\mathrm{sg}}$. The upper and lower error bounds of $G_{0.7}$ are shown by the horizontal dotted lines, given by the nearest maxima in $d^{2}G/d{V}_{\mathrm{sg}}^2$ and $d^{3}G/d{V}_{\mathrm{sg}}^3$, respectively. (e), (f) Conductance against $V_{\mathrm{sg}}$ for two example split gates showing evidence of disorder. In panel (e), the conductance is no longer quantized in units of $G_0$, while in panel (f) the second plateau is unusually weak and both the second and third plateaus occur below the expected values (the arrow indicates a 0.7 structure).

Figure 3

(a) Histogram of $G_{0.7}$, for a bin size of $0.005G_0$. Data are obtained from 36 devices, where $G_{0.7}$ is defined as the local minimum in $dG/d{V}_{\mathrm{sg}}$. (b)–(f) Scatter plots of $G_{0.7}$ against $\Delta R_{0.5}$, $W_1$, $V_p$, $G_d$ and $V_d$, respectively, where $\Delta R_{0.5}=\Delta V_{\mathrm{sg}}$ from $G=0$ to $0.5G_0$, and $W_1=\Delta V_{\mathrm{sg}}$ between $G=0.5$ and $1.5G_0$. No strong correlations are apparent; $r=0.07$, $0.07$, $-0.11$, $0.10$, and $0.33$ for panels (b), (c), (d), (e), and (f), respectively. Error bounds on the estimate of $G_{0.7}$ are given by the width of the 0.7 anomaly [28].

Figure 4

(a) The solid line shows the measured $G$ as a function of $V_{\mathrm{sg}}$ from an example device [same as Fig. 2]. Dot-dashed lines show a fit to the data $\left(G_n\right)$ for individual subbands $n=1$, 2, and 3, using a modified saddle-point model [33]. The points at which $G_n=0.5$ are aligned with the corresponding $G=(n-0.5)G_0$ values on the experimental data, and the dashed line shows ${\sum }_n{G}_n$. (b) Lever arm $\alpha$ as a function of 1D subband index $n$. The inset shows a grayscale diagram of the transconductance $dG/d{V}_{\mathrm{sg}}$ as a function of $V_{\mathrm{sg}}$ and $V_{sd}$. The dark (white) regions correspond to high (low) transconductance. The conductance values of low transconductance regions are labeled in units of $2e^2/h$, and $\Delta E_{1,2}$ is given by the point at which the two dashed lines cross.

Figure 5

(a) Scatter plot of $\hslash {\omega }_{x,1}$ against $\Delta R_{0.5}$, where $\Delta R_{0.5}=\Delta V_{\mathrm{sg}}$ from $G=0$ to $0.5G_0$. Error bounds are obtained by finding $\hslash {\omega }_{x,1}$ for the upper and lower estimates of ${\alpha }_1$. The strong correlation illustrates the accuracy of the fit to the experimental data. (b), (c) Scatter plots of $\Delta R_{0.5}$ against $G_d$, and $\hslash {\omega }_{x,1}$ against $G_d$, respectively. Panel (b) shows a reasonably distinct diagonal cutoff such that there are no data points in the upper-left section of the plot. This is weakly reflected in panel (c). (d)–(f) Scatter plots of $\hslash {\omega }_{x,1}$ against $V_d$, $V_p$ and $W_1$, respectively, where $W_1=\Delta V_{\mathrm{sg}}$ between $G=0.5$ and $1.5G_0$.

Figure 6

(a) Scatter plot of $G_{0.7}$ against $\hslash {\omega }_{x,1}$ $\left(r=0.12\right)$. Error bounds in $\hslash {\omega }_{x,1}$ are obtained from the upper and lower estimates of lever arm ${\alpha }_1$. The bounds on $G_{0.7}$ are related to the width of the conductance anomaly.

Figure 7

Conductance as a function of $V_{\mathrm{sg}}$ for an example split gate, where an anomalous feature appears near $0.5G_0$, indicated by the arrow.