Quantised cyclotron motion of electrons in a magnetic field, and the quantum Hall effect.
-Level spacing hbar w_c (meV)
-Energy E_n (meV)
-Magnetic length l_B (nm)
-Orbit radius r_c (nm)
-States per level (1e14 /m2)
-Filling factor nu
-Hall resistance (kOhm)
-k_B T / hbar w_c
The science
A free electron of effective mass m* in a field B circles at the cyclotron frequency w_c = e B / m*. Quantum mechanics allows only discrete energies E_n = hbar w_c (n + 1/2), the Landau levels. The natural length is the magnetic length l_B = sqrt(hbar / (e B)), and the classical orbit radius at level n is r_c = l_B sqrt(2n + 1).
In the symmetric gauge the states carry angular momentum m: psi(n, m) ~ r^m exp(-r^2 / 4 l_B^2) L_n^m(r^2 / 2 l_B^2), with mean square radius 2 l_B^2 (2n + m + 1). Every level holds e B / h states per unit area, so the filling factor of a sheet of electrons with density n_s is nu = n_s h / (e B).
When nu is an integer the Hall resistance locks onto R_xy = h / (nu e^2) and the longitudinal resistance vanishes, provided k_B T is well below hbar w_c. The plateau curve here is drawn where nu is within 0.18 of an integer; that window is a visual choice, because real plateau widths depend on disorder.
Honest simplifications: non-interacting electrons, no spin or Zeeman splitting, no disorder-broadened levels, no edge states, and no fractional quantum Hall effect. The Hall plateau curve is schematic.