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2.57 NanotoMacro Transport Processes
Fall 2004
Lecture 9
3.4 Density of states
(1) Electron in a quantum well
∞
=
U
U=0
x
ENERGY AND
n=
1
n=
2
n=
3
WAVEFUNCTION
For electrons in a quantum well, the energy has discrete levels as
2
hn
2
E
=
(n=1,2,…)
8
mD
2
For wavefunction
Ψ
, we have degeneracy g(n)=2 due to the spin.
,
ns
(2) Harmonic oscillator
The energy is
1
K
E
=
h
ν
(
n
+
1/ 2);
=
(n=0,1,2…)
n
2
π
m
The wavefunction is
Ψ
, and the degeneracy is g(n)=1.
n
(3) Rigid rotation
The energy eigenvalues are
2
=
(
(
E
l
=
A
A
+
1)
=
hB
A
A
+
(for m
≤
A
,
A
=0,1,2, …).
I
2
For wavefunction
Ψ
nlm
, the degeneracy is g(
l
)=2
l
+1.
(4) Hydrogen atom
E
n
el
=−
Mc
1
2
13.6
eV
(
n
≥
,
1
n
≥
A
+
1 and
m
≤
A
,
A
=0, 1, 2, …)
22
2
=
n
n
2
The wavefunction
Ψ
nlms
corresponds to degeneracy g(n)=2n
2
.
Now let us consider electrons in a solid. The parabolic approximation at the band edge
gives
2
2
)
=
22
=
2
(
k
+
k
y
2
+
k
k
x
z
EE
=
=
−
c
2
m
*
2
m
*
,
2.57 Fall 2004 – Lecture 9
1
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π
na
x
,,
k
=±
. In different directions, n values can be different. For
where
kk
y
z
L
wavefunction
Ψ
(,
y
,
k
)
, we have three quantum numbers.
x
z
15
10
Debye
k
ω
π
/a
Normalized Electron Energy E/Eo
5
0
1
0
1
k / (
π
/a)
In the Debye approximation, we have energy dispersion as
ω
=
vk
=
v
2
2
2
x
y
z
k
k
k
+
+
;
E
n
=
h
ν
(
n
+
1
2
)
,
where v is sound velocity.
Density of (quantum mechanical) states (DOS):
(a) Electron
Volume of
L
2
0
L
4
L
6
L
2
L
4
dk
k+dk
One
k
x
k
y
k
k
x
k
y
k
z
Unit Cell
2.57 Fall 2004 – Lecture 9
2
3
In above figure, we can find the volume of one state is
V
=
(2
π
/
L
) . In the above sphere,
1
the number of states within k and k+dk is
2
4
kk
V
k
∆
2
∆
N
∆=
=
V
1
2
2
,
in which V=L
3
is the crystal volume.
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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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