The Hall effect and quantized plateaus
Send a current along a narrow sample and connect a voltmeter across its width. Why does a perpendicular magnetic field produce a voltage there? And why, in some samples, does that voltage stay constant over a range of fields?
A voltage across the sample
A magnetic field deflects moving charges sideways. Charge accumulates at the edges until the resulting electric field balances the transverse magnetic force. The potential difference between the edges is the Hall voltage. In the ordinary Hall effect, reversing the field at fixed current reverses the voltage.
Dividing the Hall voltage by the current gives the Hall resistance in ohms. In the low-field, single-carrier model it varies linearly with field. For a sheet, the slope gives the carrier density per unit area, and its sign identifies electrons or holes once the contact polarity is known. If both contribute, their mobilities also enter the interpretation.
When do steps appear?
A perpendicular magnetic field separates the orbital energies of a two-dimensional electron system into discrete Landau levels. At sufficiently low temperature, when these levels are well resolved, sweeping the field can reveal plateaus in Hall resistance. On an integer quantum Hall plateau, the longitudinal resistance measured along the current approaches zero.
Why does a plateau have a finite width? Disorder localises some electronic states in the interior of the sample. As the field varies slightly, their occupation can change while the number of conducting edge channels stays fixed. The Hall resistance can therefore stay at one value over an interval of field.
What is a thermoelectric Hall plateau?
A temperature difference provides another way to drive transport. In a three-dimensional sample, the thermoelectric Hall conductivity αₓᵧ relates a temperature gradient to a transverse charge-current density and has units of A/(K·m). One can obtain it from open-circuit thermovoltages together with electrical resistivity measurements.
Measurement schematic and relation
Magnitude of the Hall resistance on a two-dimensional integer plateau. Here ν is the positive integer plateau index, h is Planck’s constant and e is the elementary charge. At ν = 2 the resistance is about 12.9 kΩ.
Reading a plot: start with the vertical axis
In the group's 2020 ZrTe₅ study, strong fields confined carriers to the lowest Landau level. Figure 3 compares αₓᵧ with αₓᵧ/T. At fixed temperature, αₓᵧ approaches a plateau as the field increases. Division by temperature lets the authors compare different temperature runs. The plateau height belongs to a thermoelectric transport coefficient; the resistance quantum for a two-dimensional sheet has different units.
Open the original experiment ↗Papers from the group
- Observation of a thermoelectric Hall plateau in the extreme quantum limitNature Communications · 2020
- Two-Dimensional Quantum Hall Effect and Zero Energy State in Few-Layer ZrTe₅Nano Letters · 2021