Theoretical investigation of the effect of sample properties on the electron velocity in quantum Hall bars

We report on our theoretical investigation of the effects of the confining potential profile and sample size on the electron velocity distribution in (narrow) quantum Hall systems. The electrostatic properties of the electron system are obtained by the Thomas-Fermi-Poisson nonlinear screening theory. The electron velocity distribution as a function of the lateral coordinate is obtained from the slope of the screened potential at the Fermi level and within the incompressible strips. We compare our findings with the recent experiments.

Figure 1

(Color online) The cross section of the donor layer considering (a) ${\rho }_1\left(u\right)$ and (b) ${\rho }_2\left(u\right)$ for various values of the steepness parameter $(0.5⩽c⩽1)$, together with the calculated background potential profiles [lower panels, (c) and (d)]. The thin solid line represents a constant donor distribution $(c=1)$, whereas thick solid line corresponds to $c=0.5$. The line code denotes a gradual increase of $c$ with a step of 10%. The sample width $d$ and the depletion length $b$ are fixed and set to be $3\mu \mathrm{m}$ and $300\mathrm{nm}$, respectively. In both cases, the donor number density is kept constant and chosen to be $4\times {10}^{11}{\mathrm{cm}}^{-2}$. The inset depicts the variation of the background potential calculated at the center of the sample for ${\rho }_1$ (thin solid line) and ${\rho }_2$ (broken line).

Figure 2

(Color online) The screened potentials obtained from the bare confinement potentials shown in Fig. 1, at $B=0$ and $T=0$ (thick lines) and also the Fermi energy for vanishing field and temperature (thin horizontal lines). The inset depicts the variation of $E_F^0$ versus the steepness considering ${\rho }_1$ (solid line) and ${\rho }_2$ (broken line).

Figure 3

(Color online) The sample width dependence of the screened potential calculated at the center. The line code depicts the selected values of $c$ and two distribution functions.

Figure 4

(Color online) The numerical derivative of the screened potentials for different steepness values calculated at the edge of the sample for inverse parabolic donor distribution ${\rho }_2$. Horizontal axis essentially presents the Fermi level, i.e., increasing of the $u$ corresponds to increasing of the average electron density.

Figure 5

(Color online) The slopes of the screened potential calculated at the Fermi level (left panel) and within the IS (right panel), considering characteristic (half) sample widths of $d=1\mu \mathrm{m}$ (upper panel), $d=2\mu \mathrm{m}$ (middle panel) and $d=5\mu \mathrm{m}$ (lower panel). The electron temperatures are chosen to be $T=1\mathrm{K}$ (thick lines) and $T=5\mathrm{K}$ (thin lines) only in the upper panel.

Figure 6

(Color online) The sample width dependence of the IS thickness for $\nu \left(x\right)=2$. Calculations are performed at $1\mathrm{K}$ for three characteristic steepness values considering etched (left panel) and functionally doped (right panel) samples. Widths of the samples are selected to be $d=1\mu \mathrm{m}$ (top) $d=3\mu \mathrm{m}$ (middle), and $d=5\mu \mathrm{m}$ (bottom), whereas the electron depleted strips are fixed to be 10% of $d$.