Question 1

What is the quantum Hall effect?

Accepted Answer

When a two-dimensional electron gas is placed in a strong perpendicular magnetic field at low temperature, its Hall conductance becomes quantised in exact integer multiples of e²/h. The Hall resistance forms flat plateaus while the longitudinal resistance drops to zero on those plateaus.

Question 2

What are Landau levels?

Accepted Answer

In a magnetic field the continuous energy spectrum of electrons collapses into discrete, highly degenerate levels called Landau levels, spaced by the cyclotron energy ħω_c. Each level can hold eB/h electrons per unit area.

Question 3

Why is the conductance quantised so precisely?

Accepted Answer

The quantisation is topological. Each filled Landau level contributes exactly one quantum e²/h of Hall conductance, an integer (the Chern number) that cannot change continuously. This makes the plateaus immune to sample disorder and shape, accurate to better than one part in a billion.

Question 4

What is the filling factor ν?

Accepted Answer

The filling factor ν = n·h/(eB) is the number of filled Landau levels, i.e. the electron density n divided by the magnetic flux density of states eB/h. On an integer plateau ν is a whole number and σ_xy = ν·e²/h.

Question 5

What are edge states?

Accepted Answer

At the boundary of the sample the Landau levels bend upward, crossing the Fermi energy and creating one-dimensional conducting channels along the edges. These edge states carry current in one direction only (chiral), which is why they cannot back-scatter and dissipate no energy.

Question 6

Why does the longitudinal resistance R_xx peak between plateaus?

Accepted Answer

On a plateau the Fermi level sits in a gap between Landau levels, so there are no bulk states to scatter into and R_xx → 0. Between plateaus the Fermi level passes through a Landau level, bulk states conduct and scatter, producing a sharp peak in R_xx.

Question 7

What is the von Klitzing constant?

Accepted Answer

R_K = h/e² ≈ 25 812.807 ohms is the quantum of Hall resistance. Since 2019 it has an exactly defined value used as the international resistance standard, named after Klaus von Klitzing who discovered the integer quantum Hall effect in 1980.

Question 8

What conditions are needed to observe it?

Accepted Answer

You need a clean two-dimensional electron system (such as a GaAs heterostructure or graphene), a strong magnetic field of several tesla, and low temperatures so that thermal energy kT is much smaller than the Landau-level spacing ħω_c.

Question 9

What is the fractional quantum Hall effect?

Accepted Answer

At even higher fields and purer samples, plateaus appear at fractional filling such as ν = 1/3. These arise from strong electron–electron interactions forming a new correlated state with fractionally charged quasiparticles. This simulation focuses on the integer effect.

Question 10

How does this simulation work?

Accepted Answer

It computes the Landau-level fan as the magnetic field changes, fills levels up to the Fermi energy to find the filling factor ν, draws the σ_xy = ν·e²/h staircase and the R_xx peaks, and animates chiral electrons circulating along the sample edges.

📶 Quantum Hall Effect — Quantised Conductance

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