Hall Resistance in the Hopping Regime: A "Hall Insulator"?

Abstract

The Hall conductivity and resistivity of strongly localized electrons at low temperatures and at small magnetic fields are obtained. It is found that the results depend on whether the conductivity or the resistivity tensors are averaged to obtain the macroscopic Hall resistivity. In the second case the Hall resistivity always diverges exponentially as the temperature tends to zero. But when the Hall resistivity is derived from the averaged conductivity, the resulting temperature dependence is sensitive to the disorder configuration. Then the Hall resistivity may approach a constant value as T → 0. This is the Hall insulating behavior. It is argued that for strictly dc conditions, the transport quantity that should be averaged is the resistivity.72.20. My, 71.50+t Recently it has been stated [1][2][3] that the zero-temperature Hall resistivity ρ xy (ω) of noninteracting electrons in the insulating regime remains finite as the frequency ω → 0. This result was derived from the Kubo formula for the frequency-dependent conductivity. It was found that at zero temperature the disorder-averaged σ xy (ω) of the Anderson insulator vanishes at low frequencies proportionally to ω 2 . Since to leading order in ω the longitudinal conductivity σ xx ∼ iωε 0 , where ε 0 is the dielectric constant, this yielded that the Hall resistivity ρ xy ∼ σ xy /(σ 2 xx + σ 2 xy ) approaches a constant in the small-frequency, zero-temperature limit.Obviously one would like to examine the Hall resistance at finite temperatures. Here we address this question for strongly localized electrons. We consider the problem using the Holstein model [4] for the Hall effect in a system with localized states. The conductivity tensor for a triangle of three sites is obtained first. To get the macroscopic Hall resistivity one may attempt to apply two averaging procedures. We find that the result strongly depends on whether the conductivity tensor is averaged before inverting it or vice versa. In the first case a Hall insulator behavior is possible, while in the second the macroscopic Hall resistivity increases with decreasing temperature. The latter is similar to the finding of Friedman and Pollak [5]. The two behaviors are argued to be related with "ac" and "dc" conditions.The previous [1-3] discussions of the zero-temperature Hall resistivity were all based on the ensemble averaged Kubo formula, which yields thatσ xx =σ yy and thatσ xy vanishes proportionally to the magnetic field (the bar above the quantity indicates ensemble averaging). However, before averaging, σ xy includes a field-independent term. We show that in the strongly localized regime this term is comparable in magnitude to σ xx and σ yy . This leads to delicate cancellations when the local (unaveraged) conductivity tensor is inverted to obtain the resistivity, and consequently to the very different temperature dependences described above.It is convenient to investigate the transport properties of electrons in the hopping regime by constructing the rate equations for the electron distribu...

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