Engineering & Science
Shubnikov-de Haas (SdH) Onsager Frequency to 2DEG Carrier Density Formula
Extract two-dimensional electron gas (2DEG) sheet carrier density n_2D = 0.2418 * g_s * g_v * B_F (× 10^11 cm^-2) directly from 1/B Shubnikov-de Haas oscillation frequency B_F (Tesla).
Measuring quantum-well and graphene sheet density purely from 1/B magnetoresistance oscillations!
At cryogenic temperatures in high-mobility GaAs/AlGaAs heterostructures, AlGaN/GaN HEMTs, and monolayer graphene, longitudinal resistance R_xx oscillates periodically in inverse magnetic field Δ(1/B) = 1 / B_F as quantized Landau levels cross the Fermi energy! By Lars Onsager’s 1952 flux-quantization theorem, the oscillation frequency B_F (in Tesla) depends ONLY on fundamental constants (e / h = 2.4180 × 10^10 cm^-2·T^-1) and spin/valley degeneracies (g_s, g_v): n_2D = g_s * g_v * (e / h) * B_F = 0.2418 * g_s * g_v * B_F × 10^11 cm^-2!
Unlike classical Hall-bar density—which can be distorted by parallel parasitic conduction channels—SdH frequency isolates each quantum subband cleanly: in a GaAs quantum well (spin degeneracy g_s = 2, single Γ-valley g_v = 1) with B_F = 8.50 T, n_2D = 4.111 × 10^11 cm^-2; in monolayer graphene (g_s = 2, K/K′ valley degeneracy g_v = 2) with B_F = 12.0 T, n_2D = 11.606 × 10^11 cm^-2 (= 1.161 × 10^12 cm^-2)!
| 2D Quantum System (g_s, g_v) | SdH Frequency B_F (Tesla) | Period Δ(1/B) (T^-1) | Sheet Density n_2D (× 10^11 cm^-2) |
|---|---|---|---|
| GaAs/AlGaAs 2DEG (gs = 2, gv = 1) | 8.50 T | 0.1176 T^-1 | 4.111 × 10^11 cm^-2 |
| Monolayer Graphene (gs = 2, gv = 2) | 12.00 T | 0.0833 T^-1 | 11.606 × 10^11 cm^-2 |
| Invalid B_F <= 0 or Degeneracy < 1 | Undefined | Error Guard | "Invalid SdH Parameters" |
2DEG Sheet Carrier Density n_2D (in units of 10^11 cm^-2)
Evaluates 0.2418 * g_s * g_v * B_F from spin degeneracy B2, valley degeneracy B3, and 1/B FFT peak frequency B4 (Tesla).
=IF(OR(B2 < 1, B3 < 1, B4 <= 0), "Invalid SdH Parameters", ROUND(0.2418 * B2 * B3 * B4, 3))Inverse-Field Oscillation Period Δ(1/B) = 1 / B_F (Tesla^-1)
Computes the spacing between adjacent magnetoresistance minima when plotted against 1/B.
=IF(B4 <= 0, "Invalid SdH Parameters", ROUND(1 / B4, 4))Try the example
Use the original values or change them to test your understanding.