Charge equilibration in integer and fractional quantum Hall edge channels in a generalized Hall-bar device

Charge equilibration between quantum Hall edge states can be studied to reveal the geometric structure of edge channels not only in the integer quantum Hall (IQH) regime but also in the fractional quantum Hall (FQH) regime, particularly for hole-conjugate states. Here we report on a systematic study of charge equilibration in both IQH and FQH regimes by using a generalized Hall bar, in which a quantum Hall state is nested in another quantum Hall state with different Landau filling factors. This provides a feasible way to evaluate equilibration in various conditions even in the presence of scattering in the bulk region. The validity of the analysis is tested in the IQH regime by confirming consistency with previous works. In the FQH regime, we find that the equilibration length for counterpropagating $\delta \nu =1$ and $\delta \nu =-1/3$ channels along a hole-conjugate state at Landau filling factor $\nu =2/3$ is much shorter than that for copropagating $\delta \nu =1$ and $\delta \nu =1/3$ channels along a particle state at $\nu =4/3$. The difference can be associated with the distinct geometric structures of the edge channels. Our analysis with generalized Hall-bar devices would be useful in studying edge equilibration and edge structures.

Figure 1

Edge equilibration between (a) copropagating, and (b) counterpropagating inner and outer channels, $C_{\mathrm{i}}$ and $C_{\mathrm{o}}$, in an interaction region of length $\ell$. Bulk equilibration between the right- and left-going parts of $C_{\mathrm{i}}$ may coexist with edge equilibration. (c) Schematic sketch of a generalized Hall bar, which is composed of six quasi-Corbino-type Ohmic contacts, ${\mathrm{\Omega }}_1–{\mathrm{\Omega }}_6$, with etched trenches and a Hall-bar-shaped metal gate. The drawn edge channels are formed at ${\nu }_{\mathrm{G}}$ (= 2) $>{\nu }_{\mathrm{B}}$ (= 1), where closed edge channel(s) $C_{\mathrm{i}}$ formed underneath the gate is coupled to the outer channel(s) $C_{\mathrm{o}}$ by edge equilibration. The measurement setup is also drawn. (d) An optical image for the central part of the device. Relevant length scales for the metal gate are labeled.

Figure 2

(a) Color plot of $V_{\mathrm{xx}}$ measured as a function of gate voltage $V_{\mathrm{g}}$ and magnetic field $B$ for the $\ell =100 \mu \mathrm{m}$ device. Dashed lines indicate the filling factor ${\nu }_{\mathrm{B}}$ for the ungated region and ${\nu }_{\mathrm{G}}$ for the gated region. The data around ${\nu }_{\mathrm{B}}={\nu }_{\mathrm{G}}$ showing too small $G<0.1e^2/h$ to measure $V_{\mathrm{xx}}$ are grayed out. Representative edge channel structures at several conditions marked by open and solid symbols are sketched in (b)–(g). (b) Edge structure at ${\nu }_{\mathrm{G}}=1$ and ${\nu }_{\mathrm{B}}=2$, where the transport is dominated by the spin-down edge channel from the spin-resolved second-lowest Landau level. (c) Edge structure at ${\nu }_{\mathrm{G}}=2$ and ${\nu }_{\mathrm{B}}=1$, where a closed channel formed underneath the gate coupled to the outer channel through edge equilibration, marked by short bars. (d) Counterpropagating $\delta \nu =1$ and $\delta \nu =-1/3$ and (e) copropagating $\delta \nu =1$ and $\delta \nu =1/3$ edge channels are formed for equilibration at ${\nu }_{\mathrm{G}}=2/3$ and ${\nu }_{\mathrm{G}}=4/3$, respectively, in ${\nu }_{\mathrm{B}}=1$. (f) Edge structure at ${\nu }_{\mathrm{G}}=1/3$ and ${\nu }_{\mathrm{B}}=2/3$, where complicated edge equilibration between two $\delta \nu =-1/3$ and one $\delta \nu =1$ channels is involved. (g) Edge structure at ${\nu }_{\mathrm{G}}=1$ and ${\nu }_{\mathrm{B}}=2/3$, where a sole $\delta \nu =-1/3$ hole channel from the 2/3 state contributes to the transport.

Figure 3

A model for edge and bulk equilibration. (a) Edge channels in a generalized Hall bar, where edge and bulk equilibrations are marked by dashed lines. The edge and bulk equilibrations are characterized by $P$ and $P^{\prime }$, respectively. (b) Parallel channels with conductance $n_1{e}^2/h$ and $n_2{e}^2/h$ along the $x$ axis. Uniform scattering conductivity $g{e}^2/h$ per unit length can be used to relate voltages $V_{1/2}^{(0/x)}$ along the channel. The calculated color plots of (c) conductance $G(P,P^{\prime })$ normalized by $e^2/3h$ between the source (S) and drain (D) contacts and (d) voltage $V_{\mathrm{xx}}$ normalized by source voltage $V_{\mathrm{AC}}$ are shown as a function of the edge equilibration rate $P$ and the bulk equilibration rate $P^{\prime }$ for $n_{\mathrm{o}}=1$ and $n_{\mathrm{i}}=-1/3$.

Figure 4

Gate voltage $V_{\mathrm{g}}$ dependence of (a) $G$, (b) $V_{\mathrm{xx}}$, (c) $V_{\mathrm{xy}}$, and (d) $P$ and (e) $P^{\prime }$ at ${\nu }_{\mathrm{B}}=4$ for the $\ell =100 \mu \mathrm{m}$ device. The corresponding filling factor ${\nu }_{\mathrm{G}}$ is shown in the top scale. Reference levels with $P=1$ and $P^{\prime }=0$ are marked as horizontal bars. (f) The $\ell$ dependence of $P$ and $P^{\prime }$ at ${\nu }_{\mathrm{G}}=5$ and ${\nu }_{\mathrm{B}}=4$. The solid red line in $P$ shows a fit to Eq. (3) for extracting the equilibration length ${\ell }_{\mathrm{e}}$. The blue dashed line in $P^{\prime }$ is a guided for the eye. (g) $V_{\mathrm{g}}$ dependence of ${\ell }_{\mathrm{e}}$. (h) Landau levels gained by the edge potential for ${\nu }_{\mathrm{G}}=5$ and ${\nu }_{\mathrm{B}}=4$ in the left panel and ${\nu }_{\mathrm{G}}=6$ and ${\nu }_{\mathrm{B}}=4$ in the right panel. The distance $a$ between the outer bundle with $n_{\mathrm{o}}$ ($={\nu }_{\mathrm{B}}$) and the inner bundle with $n_{\mathrm{i}}$ ($={\nu }_{\mathrm{G}}-{\nu }_{\mathrm{B}}$) is shorter for larger ${\nu }_{\mathrm{G}}$.

Figure 5

Gate voltage $V_{\mathrm{g}}$ dependence of (a) $G$, (b) $V_{\mathrm{xx}}$, (c) $V_{\mathrm{xy}}$, and (d) $P$ and (e) $P^{\prime }$ at ${\nu }_{\mathrm{B}}=1$ for the $\ell =100 \mu \mathrm{m}$ device. The corresponding filling factor ${\nu }_{\mathrm{G}}$ is shown in the top scale. Reference levels with $P=1$ and $P^{\prime }=0$ are marked as horizontal bars for comparison with the experiment data. (f) The $\ell$ dependence of $P$ and $P^{\prime }$ at ${\nu }_{\mathrm{G}}=2/3$ and ${\nu }_{\mathrm{G}}=4/3$. The solid red lines in $P$ are the fit to Eq. (2) for extracting the equilibration length ${\ell }_{\mathrm{e}}$. The blue dashed lines in $P^{\prime }$ are a guided for the eye. (g) $V_{\mathrm{g}}$ dependence of ${\ell }_{\mathrm{e}}$. (h) The band structure for ${\nu }_{\mathrm{G}}=2/3$ (left) and ${\nu }_{\mathrm{G}}=4/3$ (right) formed in the same magnetic field. At ${\nu }_{\mathrm{G}}=2/3$, the counterpropagating channels for edge equilibration originate from the up-bending $\delta \nu =1$ band and the down-bending $\delta \nu =-1/3$ band with a band gap of $E_{2/3}$. However, at ${\nu }_{\mathrm{G}}=4/3$, the copropagating channels for edge equilibration originate from both the up-bending spin-up band with $\delta \nu =1$ and the spin-down band with $\delta \nu =1/3$. The two bands are separated by Zeeman energy $E_Z$. The interchannel distance is labeled as $a^{\prime }$ for ${\nu }_{\mathrm{G}}=2/3$ and $a$ for ${\nu }_{\mathrm{G}}=4/3$.

Figure 6

Gate voltage $V_{\mathrm{g}}$ dependence of (a) $G$, (b) $V_{\mathrm{xx}}$, and (c) $V_{\mathrm{xy}}$ at ${\nu }_{\mathrm{B}}=2/3$ for the $\ell =100$ and $10\mu \mathrm{m}$ devices. The corresponding filling factor ${\nu }_{\mathrm{G}}$ is shown in the top scale. Reference levels with $P=1$ and $P^{\prime }=0$ are marked as horizontal bars for comparison with the experiment data.