ECEPurdue-PSF-Lundstrom-L2.5v1.pdf.123456

dos

1. Primer on Semiconductors

Unit 2: Quantum Mechanics

Lecture 2.5:
Density of states

Mark Lundstrom
[email protected]
Electrical and Computer Engineering
Purdue University
West Lafayette, Indiana USA
1 Lundstrom: 2018

2. Conduction and valence bands
conduction “band”
valence “band”
mostly full states
mostly states empty
E
energy vs. position
It will be important for us to know
how the states are distributed in
energy within the conduction and
valence bands.
Most of the empty states (the
holes) in the valence band are
very near E
V.
Most of the filled states (the
electrons) in the conduction
band are very near E
C.
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3. Density of states
The number of states between E and E + dE is D(E)dE,
where D(E) is the “density of states” (DOS).
In this lecture, we will calculate the DOS. This calculation is
greatly simplified because only the region near the band
edges are important, and in that region, the bands are
nearly parabolic:
valence
band
conduction band
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3

4. States in a finite volume of semiconductor
4
ˆx
ˆy
ˆz
Finite volume, Ω
(part of an infinite volume)
Finite number of states
xyz
LLLΩ=
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Periodic boundary conditions:

5. x-direction
5
()()
x
ik x
k
x u xeψ=
0
()()01
xx
ik L
x
Leψψ= →=
2 1,2,3,...
xx
kL j jπ= =
2
x
x
kj
L
π
=
2
x
L
π
dk
( )
# of states 2
2
x
k
x
dk
N dk
L
π
= ×== = density of states in -space
x
k
L
Nk
π
“Brillouin zone”
spin

6. Density- of-states in k- space
6
1D:
2D:
3D:
dk
xy
dk dk
xyz
dk dk dk
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independent of E(k)

7. DOS: k-space vs. energy space
7
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x
dk
dE
E
k
2
x
Lπ
States are uniformly
distributed in k-space,
but non- uniformly
distributed in energy space.

Depends on E(k)
(e.g. different for parabolic
bands and linear bands)

8. Example 1: DOS(E) for 1D nanowire
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ˆx
ˆy
ˆz
Find DOS(E) per unit energy, per unit length, a
single subband assuming parabolic energy bands.

9. 1D (single subband)
9
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E
k
dk2Lπ
dE

10. 1D DOS
10
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DOS in subband, n. n = 1, 2, 3...

11. Don’t forget to multiply by 2
11
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2Lπ
dk
E
k
dk
dE
Multiply by 2 to
account for the
negative k-
states.
(parabolic
energy bands)

12. Multiple subbands
12
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13. Example 2: DOS(E ) for 2D electrons
13
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Find DOS(E) per unit energy, per unit area, for a
single subband assuming parabolic energy bands.
(large) normalization area
A

14. Example 2: DOS(E) for 2D electrons
14
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Area of each
state in k-space:

15. Example 3: DOS(E) for 3D electrons
15
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Volume of each
state in k-space:

16. Ellipsoidal band and valley degeneracy
16
Conduction band of Si:
6 equivalent ellipsoidal valleys:
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17. Comments on 3D DOS
17
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For a bulk semiconductor with parabolic bands:
Si:

GaAs:

18. Parabolic bands: 1D, 2D, and 3D
18
D
3D
E
D
2D
E
D
1D
E
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19. Lundstrom: 2018
19
Exercise
19
()Ek
“neutral point”
(“Dirac point”)
D
2D
E
Show that for graphene, the
2D DOS is:

20. 20
Summary
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The DOS is an important concept – one that we will use
frequently in the next unit.
The DOS depends on dimension.
DOS( k) is constant, but DOS(E) depends on the band
structure.
For 3D, bulk semiconductors with parabolic bands,
DOS ~ (m*)
3/2
and DOS ~ sqrt(E – E
C)