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ECE-656: Fall 2011
Lecture 3: Density of States
Professor Mark Lundstrom Electrical and Computer Engineering
Purdue University, West Lafayette, IN USA
1 8/25/11
2
k-space vs. energy-space
N
3D(k) d
3k =
!
4" 3d
3k = D
3DE( )dE
N(k): independent of bandstructure
D(E): depends on E(k)
N(k) and D(E) are proportional to the volume, !, but it is common to
express D(E) per unit energy and per unit volume. We will use the
D3D(E) to mean the DOS per unit energy-volume.
Lundstrom ECE-656 F11
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about the limits of the integrals
Lundstrom ECE-656 F11
BW >> k
BT
E
F
f
0! 0
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outline
1) Density of states
2) Example: graphene
3) Discussion
4) Summary
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Lundstrom ECE-656 F11
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example: 1D DOS
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example: 1D DOS for parabolic bands
E = EC+!
2k
2
2m*
! =1
!
dE
dk=
2 E " EC
( )m
*
D1D
(E) =1
!!
2m*
E " EC
independent of E(k)
parabolic E(k)
Lundstrom ECE-656 F11
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density of states in a nanowire
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2D density of states
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density of states in a film
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effective mass vs. tight binding
sp3s*d5 tight binding calculation by
Yang Liu, Purdue University, 2007
TSi = 3 nm
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effective mass vs. tight binding
sp3s*d5 tight binding calculation by Yang Liu, Purdue University, 2007
near subband edge well above subband edge
Lundstrom ECE-656 F11
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exercise
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how does non-parabolicity affect DOS(E)?
non-parabolicity increases DOS (E)
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alternative approach
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proof
in k-space, we know:
nL=
1
L k
! f0
E( )" E # Ek( ) dE$
can also work in energy-space:
nL= f
0E( )
1
L! E " E
k( )k
# dE$
Lundstrom ECE-656 F11
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interpretation
counts the states between E and E +dE
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outline
1) Density of states
2) Example: graphene
3) Discussion
4) Summary
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graphene
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Graphene is a one-atom-thick planar carbon sheet with a
honeycomb lattice.
Graphene has an unusual bandstructure that leads to
interesting effects and potentially to useful electronic devices.
source: CNTBands 2.0 on nanoHUB.org
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graphene
E(k) Brillouin zone
Datta: ECE 495N – fall 2008:
https://nanohub.org/resources/5710 (Lecture 21) https://nanohub.org/resources/5721 (Lecture 22)
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simplified bandstructure near E = 0 We will use a very simple description of the graphene bandstructure,
which is a good approximation near the Fermi level.
We will refer to the EF > 0 case, as
“n-type graphene” and to the EF < 0
case as “p-type graphene.”
k
y
“neutral point” (“Dirac point”)
(valley degeneracy)
k
x
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DOS for graphene: method 2
D2 D
E( ) =1
A! E " E
k||
( )k
||
# =1
A
A
2$( )2% 2 ! (E " E
k||
)2$k||dk
||
0
&
'
D2 D
E( ) =g
V
!!2"F
2# (E $ E
k||
)Ek
||
dEk
||
0
%
&
D2 D
E( ) =2E
!!2"
F
2E > 0
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D2 D
E( ) =2 E
!!2"
F
2
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DOS for graphene: method 1
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DOS for graphene: method 1
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N (k) dk = AgV
kdk
!
= AgV
EdE
! !"F( )
2
= AD2 D
E( )dE
D2 D
E( ) =2 E
!!2"
F
2
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outline
1) Density of states
2) Example: graphene
3) Discussion
4) Summary
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density of states
D3D
E
D2D
E
D1D
E
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density of states for bulk silicon
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–5 –4 –3 –2 –1 0 1 2 3 4 5 60
2
4
6
ENERGY (eV)
DO
S (
1022
cm
–1 e
V–1
)
The DOS is calculated with nonlocal empirical pseudopotentials
including the spin-orbit interaction. (Courtesy Massimo Fischetti, August, 2011.)
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computing the density of states
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–5 –4 –3 –2 –1 0 1 2 3 4 5 60
2
4
6
ENERGY (eV)
DO
S (
1022
cm
–1 e
V–1
)
Courtesy Massimo Fischetti, August, 2011.
no. of states =!k( )
3
2" #( )$ 2
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density of states for bulk silicon (near the band edge)
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(Courtesy Massimo Fischetti, August, 2011)
0.0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
me,d1=0.3288 me (gc=6)!1=–1.0 eV–1
me,d2=0.2577 me (gc=6)!2= 0.0 eV–1
ELECTRON KINETIC ENERGY (eV)
DO
S (
1021
cm
–1 e
V–1
)
0.0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
mh,d1=0.8076 me (gv=1)!1=–0.5 eV–1
mh,d2=0.7528 me (gv=1)!2=–0.25 eV–1
HOLE KINETIC ENERGY (eV)
DO
S (
1021
cm
–1 e
V–1
)
0.0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
10
me,d1=0.3288 me (gc=6)!1=–1.0 eV–1
me,d2=0.2577 me (gc=6)!2= 0.0 eV–1
ELECTRON KINETIC ENERGY (eV)
DO
S (
1021
cm
–1 e
V–1
)
0.0 0.1 0.2 0.3 0.4 0.50
2
4
6
8
mh,d1=0.8076 me (gv=1)!1=–0.5 eV–1
mh,d2=0.7528 me (gv=1)!2=–0.25 eV–1
HOLE KINETIC ENERGY (eV)
DO
S (
1021
cm
–1 e
V–1
)
conduction band valence band
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outline
1) Density of states
2) Example: graphene
3) Discussion
4) Summary
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summary
1) When computing the carrier density, the important
quantity is the density of states, D(E).
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2) The DOS depends on dimension (1D, 2D, 3D) and
bandstructure.
3) If E(k) can be described analytically, then we can
obtain analytical expressions for DOS(E). If not, we
can compute it numerically.
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questions
1) Density of states
2) Example: graphene
3) Discussion
4) Summary
