Hall effect - semiclassical approach

[armandowww]

I'm perplexed by semiclassical treatment of Hall effect.
I'm referring to Aschroft Mermin text. At page 233 is my doubt:
why is it necessary the introduction of w drift velocity vector and why does it take that form?

 

[Gokul43201]

Recall the trajectory of a free charged particle in crossed E and B fields (from Jackson, or some SR course) - it is a cycloid with a drift velocity along E X B.

You can derive this by transforming to the frame where E' = 0.

 

[charng]

The Hall effect is a phenomenon in which a voltage difference is generated perpendicular to the direction of current flow in a conducting material when a magnetic field is applied. The semiclassical approach to understanding the Hall effect involves considering the behavior of both electrons and holes (absence of electrons) in a material under the influence of a magnetic field.

In this approach, the introduction of the drift velocity vector is necessary because it helps to explain the observed Hall effect. The drift velocity is the average velocity of charge carriers (electrons or holes) in a material, which is influenced by both the electric field and the magnetic field. In the presence of a magnetic field, the charge carriers experience a Lorentz force that causes them to deflect in a direction perpendicular to both the electric field and the magnetic field. This deflection results in a net motion of charge carriers towards one side of the material, which leads to the observed Hall voltage.

As for why the drift velocity takes a specific form, this can be understood by considering the classical equations of motion for a charged particle in a magnetic field. The resulting trajectory of the particle is a helix, with a component of motion perpendicular to the electric field. This leads to a drift velocity that is proportional to both the electric field and the magnetic field, with a constant of proportionality that depends on the charge and mass of the particle. This is the form of the drift velocity vector that is used in the semiclassical approach to the Hall effect.

In summary, the semiclassical approach to the Hall effect is a useful way to understand the behavior of charge carriers in a material under the influence of a magnetic field. The introduction of the drift velocity vector is necessary to explain the observed Hall voltage, and its specific form can be derived from classical equations of motion.