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/
ħ
)(
dE
/
dk
) =
dx
/
dt
, show that
x
(
t
) can be integrated
to yield a
semi-classical
expression for the position of the electron
wavepacket
:
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
-
space
under the influence of applied electric field
E
. Let
E
=
1
V/ cm,
a
= 3
Å
,
and the scattering time
t
c
=
2.5 x 10
-14
s at temperature
T
= 300 K. Determine the period
of oscillation in (b) by inserting these numbers and compare the result with
t
c
.
How far will an electron traverse in
k
-space before it will be
scattered? How does this compare with the width of the
1
st
BZ?