Suppose that a band
dispersion in one dimensions (1D) is described by
*
E
*
(
*
k
*
) = -2
*
b
*
cos(
*
ka
*
)
,
where
*
a
*
is the lattice constant.

(a)
Plot the
*
E
*
(
*
k
*
) versus
*
k
*
band dispersion for just the
first Brillouin Zone (1
^{
st
}
BZ).

(b)
Using
the expression for the group velocity
*
v
_{
g
}
*
= (1/

*
x
*
(
*
t
*
) =
*
x
*
(0) + (2
*
b
*
*
*
/
*
eE
*
){cos[
*
k
*
(0)
*
a
*
–
*
eaEt
*
*
*
/
*
ħ
*
] – cos[(
*
k
*
(0)
*
a
*
)]}

(c)
Setting
*
k
*
(
0) = 0, show that a Taylor expansion of the motion
*
x
*
(
*
t
*
)
agrees with the ballistic result to order
*
t
*
^{
2
}
, if the
ballistic mass is taken to be the effective mass
*
m
*
*
at
*
k
*
= 0, where
*
m
*
* is given by the external force
*
F
*
_{
ext
}
such that
*
F
*
_{
ext
}
=
*
m
*
*
*
d
*
^{
2
}
*
x
*
/
*
dt
*
^{
2
}
and 1/
*
m
*
*=
(1/
*
ħ
*
^{
2
}
) (
*
d
*
^{
2
}
*
E
*
/
*
dk
*
^{
2
}
).

(d) The result
in part (b) illustrates a phenomenon called “Bloch Oscillations” which is
characterized by electrons oscillating in position and in
**
k
**