Set of supplementary notes.

20 CHAPTER 1. CLASSICAL MODELS FOR ELECTRONS IN SOLIDS
1.2.4 Low frequency (DC) transport properties in the Drude model
We now use the differential equation we obtained earlier for the current density in applied
electric and magnetic fields, (1.14), in order to study the electrical transport in a transverse
magnetic field. We consider an arrangement, in which a static magnetic field B is applied along
the ẑ direction, and static currents and electrical fields are constrained to the x − y plane.
The equations of motion for charge carriers with charge q are now 2 .
(
∂t + τ −1) j x = q2 n
m (E x + v y B)
(
∂t + τ −1) j y = q2 n
m (E y − Bv x ) (1.23)
(
∂t + τ −1) j z = q2 n
m E z
In steady state , we set the time derivatives ∂ t = d/dt = 0, and get the three components
of the current density
( qτ
)
j x = qn
m E x + βv y
( qτ
)
j y = qn
m E y − βv x (1.24)
j z = qn qτ m E z
with the dimensionless parameter β = qB m τ = ω cτ = µB the product of the cyclotron
frequency (ω c = qB/m) and the relaxation time, or of the mobility and the applied field.
Hall effect
Consider now the rod-shaped geometry of Fig. 1.9. The current is forced by geometry to flow
only in the x-direction, and so j y = 0, v y = 0. Since there is no flow in the normal direction,
there must be an electric field E = −v × B, which exactly counterbalances the Lorentz force
on the carriers. This is the Hall effect. We find
and
v x = qτ m E x , (1.25)
E y = βE x (1.26)
(remember β = qB τ). It turns out that for high mobility materials, and large magnetic
m
fields, it is not hard to reach large values of |β| ≫ 1, so that the electric fields are largely
normal to the electrical currents.
2 Again, we define e to be the magnitude of the charge of an electron. Note, also, that the particle mass m,
may in general differ from the mass of an electron in vacuum, m e . We will see later that in solids the effective
charge carrier mass depends on details of the electronic structure.