Incompressible strips in dissipative Hall bars as origin of quantized Hall plateaus

We study the current and charge distribution in a two-dimensional electron system, under the conditions of the integer quantized Hall effect, on the basis of a quasilocal transport model, that includes nonlinear screening effects on the conductivity via the self-consistently calculated density profile. The existence of “incompressible strips” with integer Landau level filling factor is investigated within a Hartree-type approximation, and nonlocal effects on the conductivity along those strips are simulated by a suitable averaging procedure. This allows us to calculate the Hall and the longitudinal resistance as continuous functions of the magnetic field $B$, with plateaus of finite widths and the well-known, exactly quantized values. We emphasize the close relation between these plateaus and the existence of incompressible strips, and we show that for $B$ values within these plateaus the potential variation across the Hall bar is very different from that for $B$ values between adjacent plateaus, in agreement with recent experiments.

Figure 1

(Color online) Electron density profiles for two values of the magnetic field ($\hslash {\omega }_c∕E_F^0=0.94$ and 0.65) and different approximations: Thomas-Fermi (solid lines), Hartree (dashed), and quasi-Hartree (dash-dotted). The insets show the enlarged plateau regions for both cases. $\alpha =0.01,k_{B}T∕E_F^0=0.0124$. Kinks in the upper inset indicate mesh points.

Figure 2

(Color online) Gray scale plot of filling factor versus position $x$ and magnetic field $B$, calculated within the TFPA. The regions of incompressible strips with $\nu \left(x\right)=2$ and $\nu \left(x\right)=4$ are indicated. For sufficiently large $B({\Omega }_c\equiv \hslash {\omega }_c>E_F^0)$ the system is compressible (indicated by “CS”), while for the lower $B$ values included in the figure it always contains incompressible strips (“IS”). The dashed horizontal line refers to Fig. 5 below; $\alpha =0.01$, $k_{B}T∕E_F^0=0.0124$.

Figure 3

(Color online) Magnetic-field dependence of the width of the $\nu =2$ incompressible strips, for three different approximations: the analytical result of Ref. 9 (CSG), the TFPA (TFA), and the quasi-Hartree approximation (QHA). Note the inverted $B$ scale. Inset: ratio of the incompressible strip width to the magnetic length in QHA.

Figure 4

(Color online) Filling factor $\nu \left(x\right)\approx h{\sigma }_H\left(x\right)∕e^2$ versus position in the left half of a symmetric high-mobility $(R=20\mathrm{nm},{\gamma }_I=0.1)$ Hall bar of width $d=1.5\mu \mathrm{m}$, for three values of the magnetic field, $\hslash {\omega }_c∕E_F^0=0.6$, 0.8 and 0.95, and without ($\lambda =0$, solid black lines) and with ($\lambda =20\mathrm{nm}$, long-dashed green lines) averaging according to Eq. (22). The insets show the plateau regions (incompressible strips) enlarged and include in addition two results for larger collision broadening but no averaging $(\lambda =0)$. Other specifications in the text.

Figure 5

(Color online) (a) Filling factor and conductivity profiles for the left half of a symmetric sample with $d=1.5\mu \mathrm{m}$, $n_0=4\times {10}^{11}{\mathrm{cm}}^{-2}$, $b=0.9d$, calculated within the TFPA-SCBA, for $R=0.1\mathrm{nm}$ and ${\gamma }_I=0.1$, $\hslash {\omega }_c∕E_F^0=0.48$, and $k_{B}T∕E_F^0=0.01$. (b) and (c) repeat the data of (a) (solid lines) near the incompressible strips with filling factors $\nu \left(x\right)=4$ and $\nu \left(x\right)=2$, respectively. The dash-dotted lines demonstrate the effect of spatial averaging, according to Eq. (22), with $\lambda =30\mathrm{nm}$.

Figure 6

(Color online) Calculated Hall and longitudinal resistances versus scaled magnetic field $\hslash {\omega }_c∕E_F^0$, for different values of the averaging length $\lambda$. The sample parameters are $d=1.5\mu \mathrm{m}$, $n_0=4\times {10}^{11}{\mathrm{cm}}^{-2}$, $b=0.952d$, $R=10\mathrm{nm}$, ${\gamma }_I=0.1$, and $k_{B}T∕E_F^0=0.01$.

Figure 7

(Color online) Calculated Hall resistance (light solid line) and (scaled) longitudinal resistance (black solid line) versus magnetic field, measured in units of $\hslash {\omega }_c∕E_F^0$. The underlying gray scale plot shows the averaged filling factor profile, as in Fig. 2. The crescentlike areas indicate the regions of incompressible strips with local filling factors 2 (right) and 4 (left).

Figure 8

(Color online) Hall and longitudinal resistances versus magnetic field, calculated for different temperatures, $t=k_{B}T∕E_F^0$. Sample parameters: $d=1.5\mu \mathrm{m}$, $n_0=4\times {10}^{11}{\mathrm{cm}}^{-2}$, $b=0.952d$, $R=10\mathrm{nm}$ and ${\gamma }_I=0.1$, $\lambda =30\mathrm{nm}$.

Figure 9

(Color online) Hall and longitudinal resistances versus magnetic field, calculated for different values of the collision broadening. Sample parameters: $d=1.5\mu \mathrm{m}$, $n_0=4\times {10}^{11}{\mathrm{cm}}^{-2}$, $b=0.952d$, $\lambda =30\mathrm{nm}$, and $k_{B}T∕E_F^0=0.01$ (i.e., $T=1.57\mathrm{K}$).

Figure 10

(Color online) Hall potential profile $V_H\left(x\right)={\int }_0^{x}d{x}^{\prime }{E}_x\left(x^{\prime }\right)$ across the sample, for varying $B∕B_0\equiv \hslash {\omega }_c∕E_F^0$ and constant applied current $I$. Normalization: $[V_H\left(d\right)-V_H(-d)]∕I=R_H$; sample parameters: $d=1.5\mu \mathrm{m}$, $n_0=4\times {10}^{11}{\mathrm{cm}}^{-2}$, $b=0.952d$, $\lambda =30\mathrm{nm}$, and $k_{B}T∕E_F^0=0.01$.

Figure 11

(Color online) SCBA results for longitudinal $\left({\sigma }_l\right)$ and Hall $\left({\sigma }_H\right)$ conductivity, in units of $e^2∕h$, versus filling factor, at fixed magnetic field for different values of impurity range $(\alpha =R∕l)$ and strength $(\gamma =\Gamma ∕\hslash {\omega }_c)$, and of temperature $(t=k_{B}T∕\hslash {\omega }_c)$. For the two lower panels the correction $-\Delta {\sigma }_H$ to ${\sigma }_H^0=e^2\nu ∕h$ is negligible, and ${\sigma }_H$ is not shown.