I am trying to understand parts of this webpage on the quantum hall effect, and I am stuck on the part where they talk about Corbino geometry. So we have a conductive 2D annulus and a changing magnetic flux confined to the middle of the annulus.
We will also try to do the experiment in reverse i.e. apply an electric field along the circumference of the disk and measure the current $I$ in the radial direction, as shown in the figure. The radial current is easy to measure – we just measure the amount of charge $\Delta Q$ transferred between the inner and outer edges of the Corbino geometry and obtain the radial current $I = \Delta Q/\Delta T$, where $\Delta T$ is the time over which this is done.
But how do we apply an electric field in the tangential direction? The easiest way to do this is to apply a time-dependent magnetic field in the centre of the disc and use the Faraday effect.
We can calculate the electric field from the changing magnetic field using Faraday’s law as $\oint d{\bf{r}\cdot\bf{E}}=\partial_t \Phi$, where $\Phi$ is the magnetic flux resulting from the field in the center of the disk. Assuming that the electric field depends only on the radius $R$ we find that the resulting tangential electric field is given by
$$ E(R,t)=\frac{1}{2\pi R}\,\partial_t \Phi. $$
Given $I$, we can also calculate the other component of the measurement of the Hall conductance $\sigma_{H}$ i.e. the radial current density $j=I/(2\pi R)$ at the same radius $R$ as we calculated the electric field.
What I don't understand is, what is the origin of the radial current? The magnetic field is confined to the center of the disk, so the electrons are not affected by the Lorentz force due to this $B$-field. The electric field or the EMF due to the $B$-field will cause the electrons to go in one of the azimuthal directions, but not radially, so it's completely unclear why there would be any radial current here at all.
My understanding is that in the usual 2D strip scenario, the quantum Hall effect occurs when there is both an electric field in the longitudinal direction and a magnetic field perpendicular to the strip to provide the Lorentz force (as in the classical Hall effect). Based on the quote above, it doesn't seem like there is a magnetic field perpendicular to the annulus itself (it's only in the center).
Can anyone clarify this to me? Or is the claim that there is radial current false?
Best Answer
That webpage does not describe the Corbino disc experiment very well:
If you want to read an account of a real Corbino disc experiment, with all the details, you can have a look to the following paper (paywalled):
B. Jeanneret et al., "Observation of the integer quantum Hall effect by magnetic coupling to a Corbino ring", Phys. Rev. B 51 9752
In the above paper instead of measuring the radial short-circuit current, the authors measure the open-circuit voltage, but the principle is the same.