Scanning gate experiments: From strongly to weakly invasive probes

An open resonator fabricated in a two-dimensional electron gas is used to explore the transition from strongly invasive scanning gate microscopy to the perturbative regime of weak tip-induced potentials. With the help of numerical simulations that faithfully reproduce the main experimental findings, we quantify the extent of the perturbative regime in which the tip-induced conductance change is unambiguously determined by properties of the unperturbed system. The correspondence between the experimental and numerical results is established by analyzing the characteristic length scale and the amplitude modulation of the conductance change. In the perturbative regime, the former is shown to assume a disorder-dependent maximum value, while the latter linearly increases with the strength of a weak tip potential.

Figure 1

(a) Scanning electron micrograph of the sample. Schottky gates of the quantum point contact and the cavity gate in light gray, GaAs surface in dark gray. Scan area outlined by the red rectangle. Black labels symbolize Ohmic contacts. (b) Conductance $G$ as a function of the cavity-gate voltage $V_{\mathrm{cav}}$. The vertical dashed black line marks voltage of 2DEG depletion below the gate. Inset: $G$ as a function of $V_{\mathrm{QPC}}$ at $V_{\mathrm{cav}}=0$. The red line marks the working point $V_{\mathrm{QPC}}=-0.5\mathrm{V}$. (c) Numerical derivative $dG/d{V}_{\mathrm{SD}}$ as function of $V_{\mathrm{QPC}}$ and $V_{\mathrm{SD}}$. Cavity modes highlighted with white arrows. (d) Numerical simulation of the conductance as a function of the height ${\gamma }_{\mathrm{cav}}$ of the cavity-gate potential. Green-blue lines: no disorder. Red lines: realistic disorder. Dashed and solid lines: $T=0$ and $T=500\mathrm{m}\mathrm{K}$, respectively. Inset: simulation of $G$ for ${\gamma }_{\mathrm{cav}}=0$ as a function of the QPC gate potential ${\gamma }_{\mathrm{QPC}}$. The vertical line indicates the value used in the calculations as a function of ${\gamma }_{\mathrm{cav}}$.

Figure 2

Numerically calculated PLDOS at the Fermi energy corresponding to the scattering states entering the cavity through the QPC, for a depleting cavity gate. The left panel depicts the PLDOS for an ideal clean system, the right panel is for a sample with realistic disorder strength.

Figure 3

Two cavity-gate voltage scan series showing the derivative $dG/dx$ of the conductance with respect to tip position for tip voltage $V_{\mathrm{tip}}=-6\mathrm{V}$ in the upper panels (a)–(c), and for $V_{\mathrm{tip}}=-1\mathrm{V}$ in the lower panels (d)–(f). The cavity voltage $V_{\mathrm{cav}}$ is zero in the left panels (a) and (d). In the central panels (b) and (e), $V_{\mathrm{cav}}=-0.2\mathrm{V}$ does not deplete the 2DEG under the gate, while the value $V_{\mathrm{cav}}=-0.3\mathrm{V}$ used in the right panels (c) and (f) is beyond the depletion threshold. The dashed gray, solid gray, and solid black lines depict the outline of the cavity gate that becomes more important from left to right.

Figure 4

Conductance change $\mathrm{\Delta }G$ as a function of tip position $(x,y)$ for four different tip voltages $V_{\mathrm{tip}}$ for a cavity (indicated by solid lines) created by a voltage $V_{\mathrm{cav}}=-0.5V$ which depletes the 2DEG underneath. The dashed horizontal line in panel (d) is the place chosen to study the characteristic spatial scale of the conductance map (see text).

Figure 5

Numerically calculated tip-induced conductance corrections for two different disorder configurations (left and right panels) as a function of tip position $(x,y)$, for the QPC operating on the third plateau and in the presence of a strong reflector cavity potential (indicated by solid lines). The upper panels (a) and (f) correspond to a large tip strength, while the panels (b)–(d) and (g)–(i) depict the conductance change due to a weakly invasive tip of decreasing strength. The lowest panels (e) and (j) show the prediction of first-order perturbation theory (2) that describes well the effect of a weak tip.

Figure 6

The correlation $C$ of the numerically calculated conductance correction $G(x,y)-G_0$ in the cavity area with the prediction of first-order perturbation theory (2), as a function of the tip strength $u_{\mathrm{t}}$. Olive and red dots are results for the realizations used in the left and right columns in Fig. 5, respectively.

Figure 7

(a) Black triangles: characteristic length scale $l_{\mathrm{c}}$ as a function of $V_{\mathrm{tip}}$ (lower axis), extracted from the experimental SGM response $P\left(k\right)$ as described in the text and shown in the insets. Colored lines and points: $l_{\mathrm{c}}$ extracted from the numerically generated SGM response at $T=500\mathrm{m}\mathrm{K}$ as a function of the corresponding tip strength $u_{\mathrm{t}}$ (upper axis) for four different impurity realizations. The dashed horizontal lines show intrinsic length scales of the system: the Fermi wavelength ${\lambda }_{\mathrm{F}}$, and its half ${\lambda }_{\mathrm{F}}/2$. (b) The standard deviation ${\sigma }_G$ of the conductance data as a function of the tip voltage from the experimental measurements (black triangles) and ${\sigma }_G/2$ for the same four disorder configurations used in (a) (colored lines and points). Three characteristic voltages are marked: $V_{\mathrm{li}}^{}$ is the least-invasive voltage; $V_{\mathrm{depl}}^{}$ the depletion voltage of the tip; $V_{\mathrm{wi}}^{}$ identifies the transition from the weakly to the strongly invasive $V_{\mathrm{tip}}$. The disorder broadening $\mathrm{\Delta }{V}_{\mathrm{dis}}^{}$ is indicated by dotted black lines. The conversion between the theoretical tip strength parameter $u_{\mathrm{t}}$ and $V_{\mathrm{tip}}$ has been assumed to be linear with parameters chosen such that the experimental data in the weakly invasive regime are similar to the average of the numerically obtained ones. (c) The power spectrum (blue line) of the tip-induced conductance change measured at $V_{\mathrm{tip}}=-0.2V$ along the dashed line in Fig. 4. The red line is the fit (see text) to extract the characteristic value $k_{\mathrm{c}}$. (d) The power spectrum found in the theoretical simulation for the disorder realization that is represented by blue lines and points in (a) and (b). Dotted and dashed lines are for different weak tip strengths, the solid line is for a moderately strong tip.