Subfield: Modern Physics

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}


Context 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.
Context answer:
\boxed{$R_{H}=\frac{1}{e} \frac{\phi}{N}$}


Context question:
(c)  We know that electrons move in circular orbits in the magnetic field. In the quantum mechanical picture, the impinging magnetic field $B$ could be viewed as creating tiny whirlpools, so-called vortices, in the sea of electrons-one whirlpool for each flux quantum $h / e$ of the magnetic field, where $h$ is the Planck's constant and $e$ the elementary charge of an electron. For the case of $R_{\mathrm{H}}=3 h / e^{2}$, which was discovered by Tsui and Stormer, derive the ratio of the number of the electrons $N$ to the number of the flux quanta $N_{\phi}$,
Context answer:
\boxed{$\frac{1}{3}$}


Question: (d)  It turns out that binding an integer number of vortices $(n>1)$ with each electron generates a bigger surrounding whirlpool, hence pushes away all other electrons. Therefore, the system can considerably reduce its electrostatic Coulomb energy at the corresponding filling factor. Determine the scaling exponent $\alpha$ of the amount of energy gain for each electron $\Delta U(B) \propto B^{\alpha}$.

Solution:
The average distance between electrons can be written as $f l_{0}$, where

$$
l_{0}=\sqrt{\frac{L W}{N}}=\sqrt{\frac{\phi}{N B}}=\sqrt{\frac{h}{\nu e B}}
$$

and $f$ is a dimensionless constant that is determined by the electron distribution (or, quantum mechanically, wave function). Binding multiple vortices with an electron effectively reduces the probability of other electrons getting close. Therefore, the electrons optimize their distribution in such a way that their average distance increases from $f_{1} l_{0}$ to $f_{2} l_{0}\left(f_{1}<f_{2}\right)$. One expect the Coulomb energy gain per electron is proportional to

$$
\frac{e^{2}}{4 \pi \varepsilon_{0} \varepsilon_{r}\left(f_{1} l_{0}\right)}-\frac{e^{2}}{4 \pi \varepsilon_{0} \varepsilon_{r}\left(f_{2} l_{0}\right)}=\left(\frac{1}{f_{1}}-\frac{1}{f_{2}}\right) \frac{e^{2}}{4 \pi \varepsilon_{0} \varepsilon_{r} l_{0}}
$$

Therefore, $\Delta(B) \propto 1 / l_{0} \propto \sqrt{B}$, or $\alpha=1 / 2$.

Final answer: $\frac{1}{2}$