Physics 927
E.Y.Tsymbal
1
Section 10
Metals: Electron Dynamics and Fermi Surfaces
Electron dynamics
The next important subject we address is electron dynamics in metals. Our consideration will be
based on a semiclassical model. The term “semiclassical” comes from the fact that within this
model the electronic structure is described quantummechanically but electron dynamics itself is
considered in a classical way, i.e. using classical equations of motion. Within the semiclassical
model we assume that we know the electronic structure of metal, which determines the energy
band as a function of the wave vector. The aim of the model is to relate the band structure to the
transport properties as a response to the applied electric field.
Given the functions
E
n
(
k
) the semiclassical model associates with each electron a position, a
wave vector and a band index
n
. In the presence of applied fields the position, the wave vector,
and the index are taken to evolve according to the following rules:
(1) The band index is a constant of the motion. The semiclassical model ignores the possibility
of interband transitions. This implies that within this model it assumed that the applied electric
field is small.
(2) The time evolution of the position and the wave vector of an electron with band index
n
are
determined by the equations of motion:
( )
1
( )
n
n
dE
d
dt
d
=
=
k
r
v
k
k
(1)
( , )
( , )
d
t
e
t
dt
=
=
−
k
F r
E r
(2)
Strictly speaking Eq.(2) has to be proved. It is identical to the Newton’s second law if we
assume that the electron momentum is equal to
k
. The fact that electrons belong to particular
bands makes their movement in the applied electric field different from that of free electrons.
For example, if the applied electric field is independent of time, according to Eq.(2) the wave
vector of the electron increases uniformly with time.
( )
(0)
e
t
t
=
−
E
k
k
(3)
Since velocity and energy are periodic in the reciprocal lattice, the velocity and the energy will
be oscillatory. This is in striking contrast to the free electron case, where
v
is proportional to
k
and grows linearly in time.
The
k
dependence (and, to within a scale factor, the
t
dependence) of the velocity is illustrated in
Fig.2, where both
E
(
k
) and v(
k
)
are plotted in one dimension. Although the velocity is linear in
k
near the band minimum, it reaches a maximum as the zone boundary is approached, and then
drops back down, going to zero at the zone edge. In the region between the maximum of v
and
the zone edge the velocity actually decreases with increasing
k
,
so that the acceleration of the
electron is opposite to the externally applied electric force!
This extraordinary behavior is a consequence of the additional force exerted by the periodic
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 Fall '11
 STAFF
 Physics

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