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Fundamentals of
Fundamentals of
Nanoelectronics
Nanoelectronics
Prof. Supriyo Datta
ECE 453
Purdue University
Network for Computational Nanotechnology
10.08.2004
Lecture 19: Bandstructure
Ref. Chapter 5.1 & 5.2
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• The bandstructure principle helps us to
understand the energy levels of large
solids and serves as a background to
understand the energy levels of
nanostructures. For electrical conduction
we need to understand the energy levels
and we need to know where the Fermi
level lies because the energy levels around
the Fermi level determine the IV
characteristics.
• We always start from Schrödinger
equation:
• As long as there is spherical symmetry
for the potential we can reduce this
equation to three 1D equations. Solving
each equation numerically then involves
turning the equation to a matrix equation.
To so we first set up a lattice of points.
• By doing this we have discretized the
quantities we are dealing with. For
example the wavefunction will have a value
at each lattice point.
• Notice that without the spherical
symmetry the problem is not separable;
hence we have to deal with the 3D
equation which easily gets out of hand.
• To solve the problem we turn into another
method, namely the idea of basis
functions.
)
(
)
(
2
)
(
2
2
r
r
U
m
r
E
G
G
=
G
Ψ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
∇
−
=
Ψ
x
1
2
3
n1
n
n+1
N
a
n
Ψ
∑
=
Ψ
m
m
m
r
u
r
)
(
)
(
G
G
ψ
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This note was uploaded on 01/12/2011 for the course ME 305 taught by Professor Wright,j during the Spring '10 term at Birla Institute of Technology & Science, Pilani  Hyderabad.
 Spring '10
 WRIGHT,J

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