F.3 Concept of Effective Mass
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so that the electron group velocity v g is given by
v g (k) =
1
¯
h
d
dk
=
2Aa
¯
h
sin(ka) ,
(F.3.14)
with a corresponding reduced mass m ∗ , which is also a function of k, given by
m
∗ (k) =
¯
h 2
2Aa 2
1
cos(ka)
.
(F.3.15)
Let us briefly take a side trip to see how electrons in an energy band in a solid
react to the application of an electric field of magnitude E in order to determine the
effective number of electrons available for carrying an electric current. An applied
electric field will impose a force eE on each electron, so that the equation of motion
(F.3.11) for an electron will become
dv g
dt
=
1
¯
h
d 2
dk 2 eE .
(F.3.16)
The effect of accelerating all available electrons, each carrying charge e gives the
rate of change of electric current I 0 that flows initially when an electric field is
applied. 1 We may therefore write figuratively
dI 0
dt
=
d
dt
(ev g )
=
1
¯
h 2 e
2 E
d 2
dk 2 ,
from which, if we replace the summation by an equivalent integration over a
continuum of wave vectors, we find that
dI 0
dt
=
1
¯
h 2 e
2 E
1
π
d 2
dk 2 dk .
(F.3.17)
For N noninteracting free electrons, we obtain simply
dI 0
dt
=
e 2 E
m e
N ,
(F.3.18)
1 This is not the rate of change of the electric current under stationary conditions: to apply this to
the steady-state, collisions must also be considered, as the flow of electric current is inherently a
nonequilibrium phenomenon.