Subfield: Electromagnetism

The fractional quantum Hall effect (FQHE) was discovered by D. C. Tsui and $\mathrm{H}$. Stormer at Bell Labs in 1981. In the experiment electrons were confined in two dimensions on the GaAs side by the interface potential of a GaAs/AlGaAs heterojunction fabricated by A. C. Gossard (here we neglect the thickness of the two-dimensional electron layer). A strong uniform magnetic field $B$ was applied perpendicular to the two-dimensional electron system. As illustrated in Figure 1, when a current $I$ was passing through the sample, the voltage $V_{\mathrm{H}}$ across the current path exhibited an unexpected quantized plateau (corresponding to a Hall resistance $R_{\mathrm{H}}=$ $3 h / e^{2}$ ) at sufficiently low temperatures. The appearance of the plateau would imply the presence of fractionally charged quasiparticles in the system, which we analyze below. For simplicity, we neglect the scattering of the electrons by random potential, as well as the electron spin.


Figure 1 (a)
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Figure 1 (b)
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Figure 1: (a) Sketch of the experimental setup for the observation of the FQHE. As indicated, a current $I$ is passing through a two-dimensional electron system in the longitudinal direction with an effective length $L$. The Hall voltage $V_{\mathrm{H}}$ is measured in the transverse direction with an effective width $W$. In addition, a uniform magnetic field $B$ is applied perpendicular to the plane. The direction of the current is given for illustrative purpose only, which may not be correct. (b) Hall resistance $R_{\mathrm{H}}$ versus $B$ at four different temperatures (curves shifted for clarity) in the original publication on the FQHE. The features at $R_{\mathrm{H}}=3 h / e^{2}$ are due to the FQHE.




(a) In a classical model, two-dimensional electrons behave like charged billiard balls on a table. In the GaAs/AlGaAs sample, however, the mass of the electrons is reduced to an effective mass $m^{*}$ due to their interaction with ions.
Context question:
(i)  Write down the equation of motion of an electron in perpendicular electric field $\vec{E}=-E_{y} \hat{y}$ and magnetic field $\vec{B}=B \hat{z}$.
Context answer:
$$
m^{*} \frac{d \vec{v}}{d t}=-e(\vec{v} \times \vec{B}+\vec{E})
$$
where $\vec{v}$ is the velocity of the electron.


Context question:
(ii)  Determine the magnitude of the velocity $v_{\mathrm{s}}$ of the electrons in the stationary case.
Context answer:
\boxed{$E / B$.}


Context question:
(iii)  Which direction is the velocity pointing at?
Context answer:
\boxed{-x}


Question: (b)  The Hall resistance is defined as $R_{\mathrm{H}}=V_{\mathrm{H}} / I$. In the classical model, find $R_{\mathrm{H}}$ as a function of the number of the electrons $N$ and the magnetic flux $\phi=B A=B W L$, where $A$ is the area of the sample, and $W$ and $L$ the effective width and length of the sample, respectively.

Solution:
The Hall voltage $V_{H}=E_{y} W$. The current in the $-x$ direction is

$$
I=\frac{\Delta Q}{\Delta t}=\frac{N e}{L / v_{s}}=\frac{N e}{L} \frac{E_{y}}{B}=e \frac{N}{\phi} V_{H}
$$

Therefore,

$$
R_{H}=\frac{V_{H}}{I}=\frac{1}{e} \frac{\phi}{N}
$$

Final answer: $R_{H}=\frac{1}{e} \frac{\phi}{N}$