Density of States and Band

Structure

Shi Chen
Electrical Engineering
SMU
 Band Structure
In insulators, E g > 10eV , empty conduction band
In metals, conduction bands are partly filled or
overlaped with valence bands.
In semicondutors, E g is smaller than that of matals
so that electrons can possiblely jump to conduction band
In dopped semicondutors. There is an additional
donner level(n doped) near the bottom of condution
band( E c ) or an acceptor level(p doped) near the
valence band(E v )
 ∞

n = g (E ) f F (E )dE
EV

p = g (E ) f F (E )dE
EC

−∞
3
2 m*n 2
1
g (E ) = 4π
h2
E1 2
f F (E ) =
e( E − E F ) kT
+1
 Density of states
• is the number of states per volume in a
g (E )

small energy range.

gC (E) = 4π 2 (E − E )
n 12
C
h
3
2m*p 2

gV (E) = 4π 2 (E − E) 12
V
h
Effective density of states

3 2
2π m *
p kT
N v = 2 2
For holes
h
 Fermi Level
• The distribution of electron/holes satisfy Fermi-
Dirac distribution
1
f (E) = ( E − E F ) kT
1+ e
• Fermi Level can be defined by the occupation
probability of electrons at 0K
 Example:Density of states, distribution
function and electron density for degenerate
and non-degenerate n-type semiconductor
Basic Properties of Fermi Level
• Fermi Level is an intrinsic property of the
material, it is sufficient to describe the
carrier occupation function by Fermi Level
• Only the available bands can have
electrons/holes even when the occupy
function f(E) is not zero.
• Intrinsic carrier density is a strong function
of temperature
 3
π 8m * 2

(E − E )
∞ 1
n0 = n
C
2
e −( E − E F ) kT
dE = N C e −( E C − EF ) kT

2 h2 EC

3
π 8m
* 2

(E − E ) 2 e −( E
EV 1
p0 = p
V
F − E ) kT
dE = NV e −( E F − EV ) kT

2 h2 −∞

− ( EC − EV ) kT − E g kT
n0 p0 = N C NV e = N C NV e

! "

3
2πkT
(m m )
2 3 − E g 2 kT
n0 p0 = n 2
i
ni = 2 *
n
*
p
4
e
h2
EC + EV 3 m*n
# EF = − kT ln * = EFi
2 4 mp
The non-Boltzmann approx. hole
Concentration
(E
− E ) 2 dE
1
4π
p0 = 3 (2 m*p ) 2
3 EV
V

h −∞
1 + exp((EF − E ) kT )
EV − E EV − EF
$ η= ηF =
kT kT
3 1
2 m kT η 2 dη
* 2
∞
p0 = 4π p

h2 0
1 + exp (η − η F )
1
η 2 dη
F1 2 (η F ) =
∞
%
0
1 + exp (η − η F )
2
& p0 = NV F1 2 (η F )
π
 Why do we need non-Boltzmann
model
• The available situation for Boltzmann
approximation if that that the Fermi level is
far from band edges.
• When highly doped, Fermi Levels are very
near band edges.
• Most laser devices are highly doped.
• The 3-D integration is a hard work. That is
the challenge of using Fermi-Dirac Model.
 Doping
• N - type • P - type

EC
ED

EV
EA
# !

'

n0 p0 = ni2
(
!
N A+

"
'

& ! ) "
* + , - ) " ' n0 + N −A = p0 + N D+
! N A− N D+ - ) "
2
n
n0 + N A− = i + N D+
n0
 Temperature dependence

http://touch.caltech.edu/courses/EE40%20Web%20Files/Thermoelectric%20Notes.pdf
 Steady state vs. Equilibrium State
• Equilibrium refers to a condition of no
external excitation except for temperature,
and no net motion of charge.
• Steady state refers to a nonequilibrium
condition in which all processes are
constant and are balanced by opposing
process.
 EFn − EFi
)! n = n0 + ∆n = ni exp
kT
' EFn , EFp EFi − EFp
p = p0 + ∆p = ni exp
E F − E Fi kT
n0 = ni exp
kT
(
E − EF
p0 = ni exp Fi
kT
. ")! /
' ) ' "
. ! ! $
)! ! 0 )
! " ) 0 "
' 0 "
 References
• Ben G. Streetman, Sanjay Banerjee Solid
state electronic devices, Fifth edition,
Chapter 3,4,5
• Chuang Optielectronics,Chapter 2
• http://nina.ecse.rpi.edu/shur/SDM1/Notes/
Noteshtm/07Concentr/Index.htm
•