Standing Friedel Waves: A Quantum Probe of Electronic States in Nanoscale Devices

We report a theoretical study of the dynamic response of electrons in a metallic nanowire or a two-dimensional electron gas under a capacitively coupled “spot gate” driven by an ac voltage. A dynamic standing Friedel wave (SFW) is formed near the spot gate and near edges and boundaries, analogous to the static Friedel oscillations near defects. The SFW wavelength is controlled by the ac voltage frequency and the device’s Fermi velocity, whereby the latter can be measured. In addition, the SFW amplitude exhibits resonant behavior at driving frequencies that are related to eigenenergy spacings in the device, allowing their direct measurement.

Figure 1

Schematic sketch of the SFW setup: (a) a quasi-1D nanowire and (b) a 2DES.

Figure 2

Charge response function $\Theta (\mathbf{q},\omega )$ of an infinite homogeneous nanowire at two different frequencies: (a) with ${\kappa }_{b}C=8.3e/(\mathrm{nm}\mathrm{V})$, (b) with ${\kappa }_b\to \infty$, and (c) for $\omega =0.23\mathrm{GHz}$ at different temperatures.

Figure 3

Standing Friedel wave in an infinite homogeneous nanowire generated by a sharp spot gate: (a) with ${\kappa }_{b}C=8.3e/(\mathrm{nm}\mathrm{V})$ (inset shows the imaginary part) and (b) with ${\kappa }_b\to \infty$.

Figure 4

Standing Friedel wave in (a) a nanowire with a finite length and (b) a 2DES with a rectangular geometry. Both calculations use the substrate screening ${\kappa }_{b}C=8.3e/(\mathrm{nm}\mathrm{V})$.

Figure 5

Resonances in a nanowire with ${\kappa }_b\to \infty$: (a) Fourier transform of $\delta \rho$ in the frequency domain at $q=2k_F$; (b), (c) spatial distributions of $\delta \rho$ just below (red curves) and above (blue curves) the resonances. The two resonances in (a) are at $\omega =14.5\mathrm{GHz}$ and $\omega =28.9\mathrm{GHz}$, and the arrows indicate the frequencies for the curves in (b) and (c).