Question

In the integer quantum Hall effect, with the applied magnetic field reduced, more and more LLs get filled and one can observe higher and higher plateaus in the Hall conductivity σH(B). Superficially, σ_H(B→0) will simply diverge roughly as 1/B because the LL spacing ∝B. But this must not be the case in reality. How should σ_H(B) behave as B approaches zero?

One way I can think of is introducing the relaxation time τ, but not so clear how it regularizes the behaviour.

Answer 1

In the integer quantum Hall effect, as the magnetic field approaches zero, the Hall conductivity (σH(B)) does not diverge as simply as 1/B. Instead, it approaches a value known as the quantum of conductivity (e^2/h). The behavior is regulated by the relaxation time (τ) which accounts for electron scattering by impurities or defects.

Step-by-step explanation:

As the magnetic field approaches zero in the integer quantum Hall effect, the behavior of the Hall conductivity (σH(B)) is not simply a diverging function like 1/B. In reality, σH(B) approaches a value known as the quantum of conductivity (e^2/h), where e is the electron charge and h is Planck's constant. This value represents the fundamental unit of conductivity for the system. The regularizing factor in this behavior is the relaxation time (τ). The relaxation time accounts for the scattering of electrons by impurities or defects in the material, preventing the complete divergence of σH(B) as B approaches zero.