Electrostatic potential shape of gate-defined quantum point contacts

Quantum point contacts (QPCs) are fundamental building blocks of nanoelectronic circuits. For their emission dynamics as well as for interaction effects such as the 0.7 anomaly the details of the electrostatic potential are important, but the precise potential shapes are usually unknown. Here, we measure the one-dimensional subband spacings of various QPCs as a function of their conductance and compare our findings with models of lateral parabolic versus hard-wall confinement. We find that a gate-defined QPC near pinch-off is compatible with the parabolic saddle-point scenario. However, as the number of populated subbands is increased, Coulomb screening flattens the potential bottom and a description in terms of a finite hard-wall potential becomes more realistic.

Figure 1

(a) Scanning electron microscope (SEM) picture of Ti/Au gates (light-gray regions) on the wafer surface (dark region) of ${\mathrm{QPC}}_1$ and sketch of the electric field effect device. Negative voltage $V_{\text{g}}$ applied to the gates (yellow regions) is used to locally deplete the 2DES (blue where conducting, gray where depleted) 107 nm beneath the surface. (b) Pinch-off curve $G\left(V_{\text{g}}\right)/G_{\text{Q}}$ of ${\mathrm{QPC}}_1$ using the source-drain voltage $V=-0.5\mathrm{mV}$. Solid line, raw data; dotted line, corrected for lead resistance $R_{\text{lead}}=4.62\mathrm{k}\mathrm{\Omega }$, which includes $4.4\mathrm{k}\mathrm{\Omega }$ resistance of external RC filters. Inset: Simplified circuit diagram of the measurement. (c) Finite-bias spectroscopy of ${\mathrm{QPC}}_1$, $\text{d}g/\text{d}{V}_{\text{g}}(V_{\text{QPC}},V_{\text{g}})$, accounting for the lead resistance (see the text). Local maxima of $\text{d}g/\text{d}{V}_{\text{g}}$ (white lines) indicate transitions between adjacent conductance plateaus. (d) SEM picture of a screen gate equivalent to that of ${\mathrm{QPC}}_2$. As shown in the sketch, the actual device is covered with a 130-nm-thick layer of cross-linked PMMA which carries a global top gate. (e) Pinch-off curves of ${\mathrm{QPC}}_2$ corrected for a gate-voltage-dependent lead resistance, including a constant 4.4-$\mathrm{k}\mathrm{\Omega }$ resistance of external RC filters (cf. Fig. 2): $G\left(V_{\text{t}}\right)/G_{\text{Q}}$ for $V_{\text{s}}=0.5\mathrm{V}$ and $G\left(V_{\text{s}}\right)/G_{\text{Q}}$ for $V_{\text{t}}=-3.4\mathrm{V}$ at $V=-0.1\mathrm{mV}$. (f) $\text{d}g/\text{d}{V}_{\text{g}}(V_{\text{QPC}},V_{\text{t}})$ of ${\mathrm{QPC}}_2$, accounting for the lead resistance. Additional lines and symbols in (c) and (f) are explained in the text.

Figure 2

Resistances $R_{\text{lead}}$ of the leads to the QPCs [cf. sketch in Fig. 1]. For the split-gate design of ${\mathrm{QPC}}_1{R}_{\text{lead}}$ is constant, while it is a function of gate voltages for ${\mathrm{QPC}}_2$.

Figure 3

Subband spacings $\delta \epsilon \left(N\right)$ of both QPCs for the three pinch-off curves presented in Fig. 1. Lines are guides for the eyes. At the intersection of the two lines of ${\mathrm{QPC}}_2$ the gate voltages $V_{\text{s}}$ and $V_{\text{t}}$ will be identical for both measurements. Error bars reflect the uncertainties of the red lines in Figs. 1 and 1.

Figure 4

Comparison between hard-wall (a–c) and parabolic (d–f) potential models of the lateral confinement. (a) Width of the hard-wall potential $W\left(N\right)$. (b) Offset of the hard-wall potential ${\mathrm{\Phi }}_0\left(N\right)$. (c) Shape of the hard-wall potential for $1\le N\le 9$, only for ${\mathrm{QPC}}_1$. (d) Curvature of the parabolic potential ${\omega }_y\left(N\right)$. (e) Offset of the parabolic potential ${\mathrm{\Phi }}_0\left(N\right)$. (f) Shape of the parabolic potential for $1\le N\le 9$, only for ${\mathrm{QPC}}_1$. Error bars in (a), (b), (d), and (e) are calculated by error propagation from the error of $\delta \varepsilon \left(N\right)$ (cf. Fig. 3).

Figure 5

One-dimensional carrier density $n_{\text{1D}}\left(N\right)$, assuming an infinitely long hard-wall 1D channel of width $W\left(N\right)$ and depth $E_{\text{F}}-{\mathrm{\Phi }}_0\left(N\right)$ of ${\mathrm{QPC}}_1$ (red squares; right-side axis) and the corresponding 1D capacitance density $c_{\text{1D}}\left(N\right)$ (blue triangles; left-side axis).

Figure 6

Width of the hard-wall potential $W\left(V_{\text{g}}\right)$ for ${\mathrm{QPC}}_1$ [same data as the $W\left(N\right)$ in Fig. 4]. The slope of the red line is $\text{d}W/\text{d}{V}_{\text{g}}=300\text{nm}/\mathrm{V}$ (cf. the text).

Figure 7

(a) QPC nominally identical to ${\mathrm{QPC}}_1$ for $N=4$, coupled to a hemispherical mirror which is defined by a negative gate voltage $V_{\text{m}}$. (b) Conductance of the QPC as a function of $V_{\text{m}}$. (c) Fourier transform of the conductance. From the peak value, we determine the period $\delta V_{\text{m}}\simeq 150\mathrm{mV}$ of the oscillations in (b).