Engineering & Science
Integer Quantum Hall Resistance (RK / nu) & Landau Level Cyclotron Gap Formula
Calculate quantized Hall plateau resistance R_xy = R_K / nu (where von Klitzing's constant R_K = h / e^2 = 25,812.80745 Ohms) and Landau level cyclotron spacing Delta_E = hbar * e * B / m_eff (in meV).
Anchor the SI Ohm to topological quantum plateaus accurate to 1 part in 10^10!
Cool a 2D electron gas (such as a GaAs/AlGaAs heterostructure or graphene) to cryogenic temperatures in a strong perpendicular magnetic field B, and classical linear Hall resistance suddenly locks onto flat, razor-sharp topological plateaus: R_xy = h / (nu * e^2) = R_K / nu, where integer filling factor nu = 1, 2, 3, ... counts filled Landau levels and Klaus von Klitzing's constant (1985 Physics Nobel Prize; exact since the 2019 SI redefinition!) is R_K = h / e^2 = 25,812.80745... Ohms!
Why do these plateaus survive crystal defects and sample geometry? Because perpendicular field B quantizes 2D electron orbits into discrete Landau Levels separated by cyclotron energy gap Delta_E = hbar * omega_c = (hbar * e * B) / (m_eff * m_e) = (0.1157676 * B_Tesla) / m_eff_ratio (in meV)! In GaAs (effective mass ratio m*/m_e = 0.067) at B = 10.0 T, the Landau gap is a huge 17.28 meV (~200 K equivalent, crushing 4.2 K thermal noise!), locking the nu = 2 metrological resistance standard plateau at R_xy = 12,906.40 Ohms!
| Quantum Hall Plateau / Semiconductor | Filling Factor nu & Field B | Quantized Hall R_xy (Ohms) | Landau Gap hbar*omega_c (meV) |
|---|---|---|---|
| Fundamental Plateau (nu = 1) | B = 10 T, GaAs m* = 0.067 | 25,812.81 Ohms (Exact R_K) | 17.28 meV |
| Metrological Standard (nu = 2) | B = 10 T, GaAs m* = 0.067 | 12,906.40 Ohms (R_K / 2) | 17.28 meV |
| Higher Filling Plateau (nu = 4) | B = 10 T, Silicon m* = 0.19 | 6,453.20 Ohms (R_K / 4) | 6.09 meV |
Quantized Integer Hall Plateau Resistance R_xy = R_K / nu (Ohms)
Divides exact SI von Klitzing constant 25,812.80745 Ohms by positive integer filling factor B2.
=IF(OR(B2 < 1, INT(B2) <> B2), "Filling Factor Must Be Positive Integer", ROUND(25812.80745 / B2, 2))Landau Level Cyclotron Energy Spacing Delta_E = hbar * omega_c (meV)
Evaluates (0.1157676 * B_Tesla) / (m_eff / m_e) in milli-electron-volts.
=IF(OR(B3 <= 0, B4 <= 0), "Invalid Field/Effective Mass", ROUND((0.1157676 * B3) / B4, 2))Try the example
Use the original values or change them to test your understanding.