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Chapter 5
Junctions
5.1 Introduction (chapter 3)
5.2 Equilibrium condition
5.2.1 Contact potential
5.2.2 Equilibrium Fermi level
5.2.3 Space charge at a junction
5.3 Forward bias
5.3.1
Irradiation
Mask/Shield/Pattern
(negative) Photoresist
Silicon Oxide
Silicon
Develop
Metal
Oxide
Lift off
Fermi Gas and Density of State
1 2 p2
E  mv 
2
2m
p
h

2
k

  h / 2
p  k
1 2 p 2  2 k F2
EF  mv 

2
2m 2m
EF
kF
Particle in a Infinite Well
2L

n
h nh
p 
 2L
1 2 p 2 n2h2
En  mv 

2
2m 8mL2
For three-dimensional box
(n12  n22  n32 )h 2
E
8mL2
Electron Energy Density
(n12  n22  n32 )h 2
E
8mL2
nz




n  n x i  n y j  nz k
ny
nx
Density of State ρ(E)
nz




n  n x i  n y j  nz k
ny
nx
(n12  n22  n32 )h 2
E
8mL2
Properties Dependent on Density of States
Specific heat Cel   2 D( EF )k B2T / 3
Susceptibility
 el   B2 D( EF )
Experiment provide information on density of state
EDS spectrum
Photoemission spectoscopy
Seebeck effect
Carrier concentration in semionductor
Optical absorption determination of dielectric constant
Fermi contact term in NMR
de Haas  van Alphen effect
Superconducting energy gap
Josephson junction tunneling in superconductors
N(E)f(E)
Ec
Ec
EF
Ev
Ev
N(E)[1-f(E)]
(a) Intrinsic
0
N (E )
0.5
1
f (E )
N(E): Density of state
N (E)  F (E)
f(E): Probability of occupation
(Fermi-Dirac distribution function)
N= N(E)dE: Total number of states per unit volume
N= N(E)f(E)dE: Concentration of
electrons in the conduction band
Ec
Ec
EF
Ev
Ev
Holes
(a) Intrinsic
0
N (E )
0.5
f (E )
1
Carrier concentration
Electrons
Ec
Ec
EF
Ev
Ev
Holes
(a) Intrinsic
Ec
Ec
EF
Ev
Ev
(b) n-type
Ec
Ec
EF
Ev
Ev
(c) p-type
0
N (E )
0.5
f (E )
1
Carrier concentration
EF
1
2m 3 / 2 1 / 2
 (E)  N (E)  2 ( 2 ) E
2 
This density of state equation is derived from
assumption of electron in the infinite well with
vacuum medium, where the E is proportional to k2.
We found that the free electron in the conduction band of
semiconductor has local minimum of energy E versus wave
number k. We can approximate the bottom portion of the curve
as if E is still proportional to k2 and write down the similar
energy-wave number equation as
 2 k F2
EF 
2m*n
to describe the behavior of the free electrons, where mn* is the
equivalent electron mass, which account for the electron
accommodation to medium change.
kF
 2 k F2
EF 
2m
EF
Eg
kF
If we prefer to the energy at the bottom of the conduction band as a nun-zero value of Ec
instead of Ec = 0, The density of state equation can be further modified as
2m*n 3 / 2
 ( E )  N ( E )  2 ( 2 ) ( E  Ec )1/ 2
2

1
2m*n 3 / 2
 ( E )  N ( E )  2 ( 2 ) ( E  Ec )1/ 2
2

1
f ( E )  ( E  EF ) / kT
 e ( EF  E ) / kT
e
1
1
n


0
2m 3 / 2 ( EF  Ec ) / kT  1 / 2  E / kT
N ( E ) f ( E )dE 
( 2) e
E e
dE
2

0
2 
1
2mkT 3 / 2 ( E F  Ec ) / kT
n  2(
) e
2
h
( given


0
x1/ 2e ax dx 
2m*n kT 3 / 2 ( EF  Ec ) / kT
-( Ec  E F ) / kT
no  2(
)
e

N
e
c
h2
po  2(
2m kT
h
*
p
2
)
3/ 2
e
-( E F  Ev ) / kT
 Nve
-( E F  Ev ) / kT

2a a
)
2m*n kT 3 / 2
N c  2(
)
2
h
N v  2(
2m*p kT
h2
)3 / 2
no  N c e -( Ec  EF ) / kT
ni  N c e -( Ec  Ei ) / kT
po  N v e -( E F  Ev ) / kT
pi  N v e -( Ei  Ev ) / kT
(general )
and
(intrinsic )
no po  ni pi
Nc: Effective density of state at bottom of C.B.
Nv: Effective density of state at top of V.B.
no: Concentration of electrons in the conduction band
po: Concentration of holes in the valence band
Ec: Conduction band edge
Ev: Valence band edge
EF: Fermi level
Ei: Fermi level for the undoped semiconductor (intrinsic)
no  ni e ( E F  Ei ) / kT
po  pi e
( Ei  E F ) / kT
where
ni  pi
Fermi Level and Carrier Concentration of Intrinsic
Semiconductor
ni  N c e -( Ec  Ei ) / kT
pi  N v e -( Ei  Ev ) / kT
Ei 
and
ni  pi
Ec  Ev
N
kT

ln v
2
2
Nc
m*p
Ec  Ev
3kT


ln *
2
4
mn
ni  2(
2kT 3 / 2 * * 3/4  Eg / 2 kT
) (m p mn ) e
2
h
Example 3-5
A Si sample is doped with 1017 As atoms/cm3. What is the equilibrium hole
concentration po at 300K? Where is EF relative to Ei?
5.1 Introduction
5.2 Equilibrium condition
5.2.1 Contact potential
5.2.2 Equilibrium Fermi level
5.2.3 Space charge at a junction
5.3 Forward bias
5.3.1
Electric field
Electric field
Einstein relationship
(explained later)
Einstein Relationship
J p ( x)  q n p( x)
drift
dp( x)
( x)  qD p
dx
diffusion
• At equilibrium, no net current flows in a semiconductor. Jp(x) = 0
• Any fluctuation which would begin a diffusion current also sets up an electric
field which redistributes carriers by drift.
• An examination of the requirements for equilibrium indicates that the diffusion
coefficient and mobility must be related.
Einstein Relationship
q
Drift
q c
vp 
 p
mp
q
( p  c )
mp
hole
Diffusion


 l
 c  mp v p
 l
1
dp
( p0  l )l
dx
 l  2
c
1
dp
( p0  l )l
dx
 l  2
c
l : mean free path  v p  c
dp
dp
l
l l
v p  c
dp
dp
x  l  l  dx  dx
 v p l
 Dp
c
c
dx
dx
l
0
x
l
J p ( x)  q x  qD p
dp
dx
(Dp  v p l )
Einstein Relationship
l  v p c
Drift and diffusion
q c
( p 
)
mp
(Dp  v p l )
drift
diffusion
1
1
2
m p v th  kT
2
2
Dp
kT

p
q
diffusion
Poisson's equation
The derivation of Poisson's equation in electrostatics follows. SI units are used and Euclidean
space is assumed.
Starting with Gauss' law for electricity (also part of Maxwell's equations) in a differential
control volume, we have:
 D  f

D
f
is the divergence operator.
is the electric displacement field.
is the free charge density (describing charges brought from outside).
Assuming the medium is linear, isotropic, and homogeneous (see polarization density), then:
D  E

is the permittivity of the medium.
E
is the electric field.
By substitution and division, we have:
f
E 

http://en.wikipedia.org/wiki/Poisson's_equation
F  qE 
kQq
r2
kQ
E 2
r
V  Ed 
kQ
r
U  qV  q E d 
kQq
r