When a semiconductor layer’s physical thickness is reduced below the de Broglie wavelength of an electron ((L_z lesssim 20 ext{ nm})), continuous band motion collapses into discrete quantized subbands along the confinement direction. In two dimensions, this yields a step-function density of states and, under extreme magnetic fields, the exact topological quantization of Hall resistance.
1. The 1D Quantum Well and 2D Subband Formation
Consider a thin semiconductor well layer (material A, e.g., GaAs) sandwiched between two wider bandgap barrier layers (material B, e.g., ( ext{Al}_x ext{Ga}_{1-x} ext{As})). Let the potential profile along the growth axis (z) be a finite square well of width (L_z) and barrier height (V_0 = Delta E_c). The single-electron Schrödinger equation separates into transverse plane wave motion and longitudinal confinement:
[
psi(mathbf{r}) = chi_n(z) e^{i (k_x x + k_y y)}
]
The longitudinal envelope function (chi_n(z)) satisfies:
[
left[ -rac{hbar^2}{2m^*} rac{d^2}{dz^2} + V(z)
ight] chi_n(z) = E_n chi_n(z)
]
For an infinite potential well ((V_0 o infty)), the quantized eigenenergies are:
[
E_n = rac{hbar^2 pi^2 n^2}{2m^* L_z^2}, quad n = 1, 2, 3, dots
]
Total energy in the two-dimensional electron gas (2DEG) is the sum of subband energy and in-plane kinetic dispersion:
[
E(n, mathbf{k}_perp) = E_n + rac{hbar^2 (k_x^2 + k_y^2)}{2m^*}
]
2. The Two-Dimensional Density of States (2D DOS)
In two dimensions, the phase-space area per (mathbf{k})-state is ((2pi/L)^2). Counting spin-degenerate states within an annular ring of radius (k_perp) and thickness (dk_perp):
[
dN = 2 imes rac{pi k_perp dk_perp}{(2pi/L)^2} = rac{L^2}{2pi} k_perp dk_perp
]
Using (E – E_n = rac{hbar^2 k_perp^2}{2m^*} implies dE = rac{hbar^2 k_perp dk_perp}{m^*}), the density of states per unit area for a single subband is entirely independent of energy:
[
g_{2D}(E) = rac{1}{L^2} rac{dN}{dE} = rac{m^*}{pi hbar^2}
]
Summing over all accessible subbands yields the characteristic staircase density of states:
[
g_{2D}(E) = rac{m^*}{pi hbar^2} sum_{n} Theta(E – E_n)
]
where (Theta(cdot)) is the Heaviside step function. This sharp step-like density of states dramatically sharpens optical absorption edges and reduces threshold currents in semiconductor quantum-well lasers.
3. Landau Levels and the Integer Quantum Hall Effect (IQHE)
When a perpendicular magnetic field (mathbf{B} = B hat{mathbf{z}}) is applied to a 2DEG, the continuous in-plane parabolic dispersion collapses into discrete, highly degenerate Landau levels. In the Landau gauge (mathbf{A} = (0, Bx, 0)), the Hamiltonian maps identically to a 1D quantum harmonic oscillator with cyclotron frequency (omega_c = rac{e B}{m^*}):
[
E_N = left( N + rac{1}{2}
ight) hbar omega_c pm rac{1}{2} g^* mu_B B, quad N = 0, 1, 2, dots
]
The macroscopic degeneracy of each Landau level per unit area is proportional to the magnetic flux density:
[
n_L = rac{e B}{h} = rac{1}{2pi ll_B^2}
]
where (ll_B = sqrt{rac{hbar}{e B}}) is the magnetic length. When the Fermi level lies in the localized mobility gap between the (
u)-th and ((
u+1))-th Landau level, bulk current vanishes ((sigma_{xx} o 0)), and transport is mediated exclusively by dissipationless topological chiral edge states.
Discovered by Klaus von Klitzing in 1980, the transverse Hall resistance (R_{xy}) forms extraordinarily flat, universal plateaus quantized in terms of fundamental physical constants:
[
R_H = rac{h}{
u e^2} = rac{R_K}{
u}, quad
u in mathbb{Z}^+
]
where (R_K = rac{h}{e^2} approx 25812.807 Omega) is the von Klitzing constant, now defining the international metrological standard of electrical resistance.
