Self-consistent calculation of the electron distribution near a quantum point contact in the integer quantum Hall effect

In this work we implement the self-consistent Thomas-Fermi-Poisson approach to a homogeneous two-dimensional electron system. We compute the electrostatic potential produced inside a semiconductor structure by a quantum point contact (QPC) placed at the surface of the semiconductor and biased with appropriate voltages. The model is based on a semianalytical solution of the Laplace equation. Starting from the calculated confining potential, the self-consistent (screened) potential and the electron densities are calculated for finite temperature and magnetic field. We observe that there are mainly three characteristic rearrangements of the incompressible edge states which will determine the current distribution near a QPC.

Figure 1

(Color online) External potential (left panel) experienced by a 2DES at different distances $z$ and the corresponding screened potentials (right panel). The separating dielectric material is assumed to be $\mathrm{GaAs}$ with $\kappa =12.4$ and the calculations are done at $T=0\mathrm{K}$.

Figure 2

(Color online) The image of the QPC (gray scale). The polygons are used to define the gates on the two-dimensional mesh. The 2DES resides under the dark (gray) regions, with a bulk electron density of $1.7\times {10}^{-11}{\mathrm{cm}}^{-2}$ (see Ref. 6).

Figure 3

(Color online) The bare confinement potential generated by the QPC, defined by the polygons shown in Fig. 2. The gray (color) scale indicates the strength of the potential. (b) Some characteristic cuts along the $x$ axis.

Figure 4

(Color online) The screened potential (upper panel) experienced by a 2DES at $85\mathrm{nm}$ below the surface and some characteristic cuts along the $x$ axis, together with an indication of the Fermi level $E_F$ (lower panel). The color scale represents the strength of the potential, and the cross sections are indicated by the same line code as in Fig. 3.

Figure 5

(Color online) The screened potential at $x=550\mathrm{nm}$ for five different tip distances. Note that the $x$ interval used for the calculation is smaller than in the previous figure, since we concentrate on the bulk structures rather than the edge ones.

Figure 6

(Color online) (a)–(d) Color-coded plot of the local filling factor versus position $(x,y)$ for a square sample of width $a_x=a_y=0.8\mathrm{\mu }\mathrm{m}$; white indicates $\nu (x,y)=2$. The average density is taken to be $3.0\times {10}^{11}{\mathrm{cm}}^{-2}$; $k_{B}T∕E_F=0.02$. (e) A sketch of the Hall resistance as a function of magnetic field.

Figure 7

(Color online) (a) The top view of the local filling factor $\nu (x,y)$ distribution of the 2DES, for average filling factor 1, in the plane located at $z=85\mathrm{nm}$ below the surface, at the “default” temperature $kT∕\hslash {\omega }_c=1∕50$, which will be used in all subsequent plots. The color scale depicts the density of electrons, whereas the dark shaded areas indicate the electron-depleted regions. (b) Side view of the local filling factor for $y=0$ (solid line), 468 (dashed line), 660 (dotted line), 750 (dash-dotted line), and $900\mathrm{nm}$ (dash-dot-dashed line). The horizontal lines in (a) show the positions of the cuts in (b), with the same line code. Note that the density has local minima at large and small $x$, which are finite size effects mentioned in the text.

Figure 8

(Color online) The local filling factor distribution for different average filling factors $\left(\overline{\nu }\right)$, which is defined by the number of the electrons in the unit cell. $\overline{\nu }=$ (a) 1.1, (b) $1.2$, (c) $1.4$, (d), $1.6$, (e) $1.8$ and (f) $2.2$. The color scale depicts the local electron concentration, whereas the abrupt color changes indicate the even-integer filling factors, i.e., incompressible strips [black for $\nu (x,y)=2$, magenta (gray) for $\nu (x,y)=4$].

Figure 9

(Color online) The local filling factor distribution for different average filling factors $\left(\overline{\nu }\right)$, for a tip separation $\delta y=300\mathrm{nm}$. Note that the number of electrons in the unit cell is changed, since the depleted areas are larger than in the previous figure. $\overline{\nu }=$ (a) 1.14, (b) $1.2$, (c) $1.34$, (d) $1.4$, (e) 1.6, and (f) 3.1. The color scale depicts the local electron concentration, whereas the high-contrast color regions indicate the even-integer filling factors, i.e., incompressible strips [black for $\nu (x,y)=2$, magenta (gray) for $\nu (x,y)=4$]. The calculations are done at $k_{B}T∕\hslash {\omega }_c=1∕50$ for an average electron density $1.7\times {10}^{-11}{\mathrm{cm}}^{-2}$.

Figure 10

(Color online) The self-consistent potential profile across the QPC, plotted for characteristic values of the average filling factor. Calculations are done at the default temperature and electron density.