Chemistry Reference
In-Depth Information
Mott, N. F. (1974)
Metal-Insulator Transitions
, Taylor and Francis, London.
Vogl, P., H. J. Hjalmarson and J. D. Dow (1983)
J. Phys. Chem. Solids
44
365.
Vogl, P. (1984)
Advances in Electronics and Electron Physics
62
101.
Wei, S. H. and A. Zunger, (1996)
Appl. Phys. Lett.
69
2719.
Problems
4.1 Use eq. (4.9a) to estimate the value of the energy parameter
E
p
for the
direct gap semiconductors listed in Table 4.1. Can you observe any
trends in the calculated
E
p
values?
4.2 Use eq. (4.9b) and the values of
E
p
calculated in problem 4.1 to
estimate the light-hole relative effectivemasses of the direct gap semi-
conductors listed in Table 4.1. Compare the calculated masses with
the experimentally determined values listed in the table.
4.3 Use the data in Table 4.1 to calculate the donor ground state binding
energy,
E
imp
1
, and defect electron effective Bohr radius,
a
∗
, in InSb,
InP and GaN. Hence estimate the donor doping density
N
d
required
to achieve impurity band conduction at low temperatures in each of
these compounds.
4.4 We use the
k
p
method in Section 4.2 and Appendix E to determine
the band structure in the neighbourhood of
k
·
=
0 using the zone
centre
wavefunctions and energies, and the
k
-dependent per-
turbation Hamiltonian of eq. (4.3). Show that we can generalise the
k
(
k
=
0
)
p
method to determine the band structure in the neighbourhood of
an arbitrary wavevector
k
0
by introducing the wavevector
q
·
k
0
and then re-arrangingSchrödinger's equation so that the
q
-dependent
Hamiltonian
H
q
=
k
−
e
−
i
q
·
r
H
0
e
i
q
·
r
acts on the
q
-independent wavefunc-
=
tion, exp
(
i
k
0
·
r
)
u
n
k
(
r
)
. Show that the dispersion along direction
i
near
the state at energy
E
n
(
k
0
)
is then given in second order perturbation
theory by
2
q
2
2
q
2
m
2
p
nn
|
2
|
i
·
k
0
)
+
2
m
+
E
n
(
k
)
=
E
n
(
(
)
−
E
n
(
)
E
n
k
0
k
0
n
=
n
4.5 Consider the Kronig-Penney (K-P) potential of fig. 3.8 with period
L
and a thin barrier of height
V
0
and width
b
centred on the ori-
gin. We saw in Chapter 3 that the NFE wavefunction for the lower
state at the
n
th energy gap is given by
/
1
2
sin
ψ
(
x
)
=
(
2
/
L
)
(
n
π
x
/
L
)
,
L
n
ψ
U
n
(
)
=
while the NFE wavefunction for the upper state is given by
x
1
/
2
cos
(
. Show that the magnitude of the momentum
matrix element
P
LU
n
linking the
n
th upper and lower state varies
2
/
L
)
(
n
π
x
/
L
)
