Length Scale of Bulk Quantum Hall Effect

Quantum hall effects (QHE) are consequences of the condensation of the charge carriers into a novel macroscopic quantum state. Condensation produces steps in the hall resistance (R$_{xy})$ in fundamental units of h/e$^{2}$ (25812.8 Ohms), which correlate with Shubnikov-deHass oscillations in R$_{xx}$ and is represented graphically by the von Klitzing plot; which is a ``double-y'' graph of the hall resistance R$_{xy}$ and magnetoresistance R$_{xx}$ isotherms as functions of B. In two-dimensions electrical resistance per square is scale independent so the steps in R$_{xy}$ are in ohms. QHE condensation occurs in bulk systems as well. However, resistance of three-dimensional conductors depends on the sample geometry and the relevant transport coefficient, resistivity ($\rho _{xy})$, which is dimensionally different from resistance. Hence the QHE plateaus in bulk samples are not directly expressed in ohms. In this case the macroscopic $\rho _{xy}$ can be related to the R$_{xy}$ by a quantum length factor ``L'', we define L such that R$_{xy}$ = L$^{-1}(\rho _{xy). }$By analyzing literature data [Kul'bachinskii, V.A. et al., JETP Let., \textbf{70}, 767, 1999] we determine that for Sb$_{2}$Te$_{3 , }$L is equal to 1.7 nm a remarkable microscopic scale. This factor L is not just the ratio of the macroscopic length to the cross-sectional area of the conductor; instead, it is an effective length associated with the quantum hall states.