1
Department of Electrical and Computer Engineering, Cornell University
ECE 407: Physics of Semiconductor and Nanostructures
Spring 2009
Homework 5
`
Due on Feb. 24, 2009 at 5:00 PM
Suggested Readings:
a) Lecture notes
Problem 5.1 (1D lattice energy bands outside the FBZ)
In the last homework, in problem 4.1, you found the exact solution for an electron in a periodic 1D lattice.
You will consider the same problem again here. Consider a 1D lattice of lattice constant
a
equal to 5
Angstroms. Suppose the potential has the form:
()
⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
=
x
a
V
x
a
V
x
V
π
4
cos
2
2
cos
2
2
1
Where
1
V
equals 0.3 eV and
2
V
also equals 0.3 eV. The exact solution for any wavevector
k
in the FBZ
can be written as a superposition of plane waves in the form:
∑
=
∑
=
∞
−∞
=
+
∞
−∞
=
+
m
x
G
k
i
m
m
G
k
m
k
k
m
m
e
L
G
c
G
c
1
φ
ψ
Where:
a
m
G
m
2
=
A good approximation to the exact solution can be obtained by terminating the series above at both ends,
as follows:
∑
=
−
=
+
N
N
m
G
k
m
k
k
m
G
c
Where
N
is some large number, say 10. The way you hopefully solved the problem in homework 4 was to
first choose a value of the wavevector
k
in the FBZ, then setup a matrix, and then find its three smallest
eigenvalues. The question is what if one chooses a value of the wavevector
k
that is not in the FBZ?
Would one end up with some new energy eigenvalues and new energy eigenfunctions? The goal of the
problem is to explore this point.
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 Spring '08
 RANA
 Atom, Atomic Orbitals, Electron, Chemical bond, Bloch wave

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