Modeling of quantum point contacts in high magnetic fields and with current bias outside the linear response regime

The electron and current-density distributions in the close proximity of quantum point contacts (QPCs) are investigated. A three-dimensional Poisson equation is solved self-consistently to obtain the electron density and potential profile in the absence of an external magnetic field for gate and etching defined devices. We observe the surface charges and their apparent effect on the confinement potential, when considering the (deeply) etched QPCs. In the presence of an external magnetic field, we investigate the formation of the incompressible strips and their influence on the current distribution both in the linear response and out of linear response regime. A spatial asymmetry of the current carrying incompressible strips, induced by the large source drain voltages, is reported for such devices in the nonlinear regime.

Figure 1

(Color online) (a) The silicon doped heterostructure, the 2DES is formed at the interface of the GaAs/AlGaAs (denoted by minus signs) and the metallic gates are deposited on the surface. At zero gate bias, the electron density is determined by the number of donors, which we chose to be $4\times {10}^{16}{\text{m}}^{-2}$. Charge distribution at different layers obtained by considering a gate voltage $-1.7\text{V}$, at the gates (b), the dopant layer (c) and the 2DES (d). It is clearly seen that not all the excess electrons are captured by the 2DES, rather a significant amount is accumulated on the surface. The electrostatic quantities are normalized with the relevant scales; i.e., charge (density) is normalized with the average electron density [i.e., $Q_{2\text{DES}}\left(x,y\right)=n_{\text{el}}\left(x,y\right)/{\overline{n}}_{\text{el}}$] and electrostatic potential (energy) with the potential energy of a single electron, $-\text{eV}$.

Figure 2

(Color online) Spatial distribution of the electrons as a function of the gate voltage. At zero bias (a) more electrons are populated under the gate which changes when depletion starts (b)–(d), where the gate voltage is set to be (b) $-0.3\text{V}$, (c) $-0.7\text{V}$, (d) $-1.0\text{V}$. Almost no electrons are left beyond $-1.5\text{V}$ applied to the gates, (e) $-1.5\text{V}$, (f) $-2.0\text{V}$, pinch-off.

Figure 3

(Color online) Potential profile across the QPCs for two different geometrical patterns. The insets depict the smooth (C1) and the sharp edged (C2) patterns. The dark (blue) regions are electron depleted; i.e., the local potential is larger than the $E_F$, denoted by the solid thick horizontal line. The white contours denote the depletion boundary, where the width of the QPC is set to be 200 nm and the curvature is changed. The dashed lines stand for the first configuration and two different $W$ values [$=200\text{nm}$ (black) and 300 nm (red)], whereas for the sharper configuration four $W$ values are shown.

Figure 4

(Color online) Potential cross section of (a) gated and (b) etched QPCs calculated at selected gate biases and etching depths for C1 for a fixed $W=150\text{nm}$. The insets focus on the high bias or shallow etching profiles

Figure 5

(Color online) Spatial distribution of the electron density [(a) and (c)] and screened potential [(b) and (d)] for etched (upper panel) and gated samples (lower panel), $y=450\text{nm}$.

Figure 6

(Color online) The distribution of the spin degenerate LBES at (a) $\nu \approx 4$ and (b) $\nu \approx 8$. The potential cross section at $\nu =8$ plateau (c). Sketch of the conductance $\left(G\right)$ and expected coherence $\Xi$ (d) for C1 considering $W=150\text{nm}$ with an applied gate potential $V_g=2.0\text{V}$, so that the 2DES is depleted beneath the gates. Since spin degeneracy is assumed each edge state carries two units of quantized current.

Figure 7

(Color online) Spatial distribution of the incompressible strips (white areas) for characteristic $B$ values (a) 6.8 T, (b) 7.3 T, (d) 8.3 T, (e) 8.8 T calculated at temperatures $\hslash {\omega }_c/k_{B}T⪡1$, together with the sketch of Hall resistance and the coherence $\Xi$ (c). The QPC configuration considered here is C1 with $W=150\text{nm}$; the gate voltage is chosen such that all the electrons beneath the gates are depleted.

Figure 8

(Color online) The local current density calculated at different field strengths, same as in Fig. 7. The intensity of the current density is chosen such that the applied current does not effect the density distribution, i.e., $\left|j\left(x,y\right)\right|\sim 0.4\times {10}^{-3}\text{A}/\text{m}$.

Figure 9

(Color online) The local filling factor distribution and the corresponding local current density calculated at the default temperature. The current intensity is sufficiently high $\left(\left|j\left(x,y\right)\right|=2.0\times {10}^{-2}\text{A}/\text{m}\right)$ to induce an asymmetry on the density distribution via position dependent electrochemical potential.