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Chapter 5
Junctions
5.1 Introduction (chapter 3)5.2 Equilibrium condition
5.2.1 Contact potential5.2.2 Equilibrium Fermi level5.2.3 Space charge at a junction
5.3 Forward bias5.3.1
(negative) Photoresist
Silicon Oxide
Silicon
Mask/Shield/Pattern
Irradiation
Metal
Oxide
Lift off
Develop
Fermi Gas and Density of State
m
pmvE
22
1 22
m
k
m
pmvE F
F 222
1 2222
kp
2/h
h
p
2
kFE
Fk
Particle in a Infinite Well
n
L2
L
nhhp
2
2
2222
822
1
mL
hn
m
pmvEn
2
223
22
21
8
)(
mL
hnnnE
For three-dimensional box
Electron Energy Density
2
223
22
21
8
)(
mL
hnnnE
xn
yn
zn
knjninn zyx
Density of State ρ(E)
xn
yn
zn
knjninn zyx
2
223
22
21
8
)(
mL
hnnnE
Properties Dependent on Density of States
3/)( 22 TkEDCheatSpecific BFel
)(2FBel EDlitySusceptibi
Experiment provide information on density of state
spectrumEDSyspectoscopionPhotoemiss
effectSeebeckorsemionductinionconcentratCarrier
constantdielectricofiondeterminatabsorptionOpticalNMRintermcontactFermi
effectAlphenvanHaasde gapenergyctingSupercondu
ctorssuperconduintunnelingjunctionJosephson
)(EN )(Ef )()( EFEN 0 0.5 1
cE
vE
cE
vE
N(E)[1-f(E)]
N(E)f(E)
(a) Intrinsic
FE
N(E): Density of state f(E): Probability of occupation (Fermi-Dirac distribution function)
)(EN )(Ef ionconcentratCarrier0 0.5 1
cE
vE
cE
vE
Holes(a) Intrinsic
FE
N= N(E)dE: Total number of states per unit volume N= N(E)f(E)dE: Concentration of electrons in the conduction band
)(EN )(Ef ionconcentratCarrier0 0.5 1
cE
vE
cE
vE
(c) p-type
FE
cE
vE
cE
vE
(b) n-type
FE
cE
vE
cE
vE
Holes
Electrons
(a) Intrinsic
FE
212322
)2
(2
1)()( // E
mENE
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.
FE
Fk
m
kE F
F 2
22
FE
FkgE
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
*n
FF m
kE
2
22
to describe the behavior of the free electrons, where mn* is the
equivalent electron mass, which account for the electron accommodation to medium change.
212322
)()2
(2
1)()( //
*
cn EE
mENE
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
kTEEkTEE
F
Fe
eEf /
/)(
)( 1
1)(
212322
)()2
(2
1)()( //
*
cn EE
mENE
0
21)(23220
)2
(2
1)()( dEeEe
mdEEfENn kTEkTEE cF ////
kTEE cFeh
mkTn // )(23
2)
2(2
)2
(0
21
aadxexgiven ax
/
kTEEc
kTEEno
FccF eNeh
kTmn ///
*)(-)(23
2)
2(2
232
)2
(2 /*
h
kTmN n
c
232
)2
(2 /*
h
kTmN p
v
kTEE
vkTEEp
ovFvF eNe
h
kTmp ///
*)(-)(-23
2)
2(2
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 bandpo: Concentration of holes in the valence bandEc: Conduction band edgeEv: Valence band edgeEF: Fermi levelEi: Fermi level for the undoped semiconductor (intrinsic)
kTEEco
FceNn /)-( kTEE
vovFeNp /)-(
kTEEci
iceNn /)-( kTEE
vivieNp /)-(
)(general )(intrinsic
kTEEio
iFenn /)( kTEE
ioFiepp /)(
ii pnwhere
iioo pnpnand
Fermi Level and Carrier Concentration of Intrinsic Semiconductor
kTEEci
iceNn /)-( kTEE
vivieNp /)-(
ii pnand
*
*
ln
ln
n
pvc
c
vvci
m
mkTEE
N
NkTEEE
4
3
2
22
kTEgnpi emm
h
kTn 23/423
2)()
2(2 /**/
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 Introduction5.2 Equilibrium condition
5.2.1 Contact potential5.2.2 Equilibrium Fermi level5.2.3 Space charge at a junction
5.3 Forward bias5.3.1
Electric field
Einstein relationship(explained later)
Electric field
Einstein Relationship
dx
xdpqDxxpqxJ pnp
)()()()(
drift 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.
cl
ldxdp
lp
)(21
0
cl
ldxdp
lp
)(21
0
dx
dpD
dx
dpldx
dpll
dxdp
l
ppc
cp
cllx
v
v
l l0
x
l
l�
dx
dpqDqxJ pxp )(
cppathfreemeanl v:
)( lD pp v
Einstein Relationship
ppc mq v
)(p
cp m
q
pp
cp m
q vhole
Drift
Diffusion
cpl v
kTm thp 2
1
2
1 2 v
)( lD pp v)(p
cp m
q
drift diffusion
q
kTD
p
p
Einstein Relationship
Drift and diffusion
diffusion
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:
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:
is the permittivity of the medium.
is the electric field.
By substitution and division, we have:
fD
D
f
ED E
fE
http://en.wikipedia.org/wiki/Poisson's_equation
Poisson's equation
2r
kQqqEF
2r
kQE
r
kQEdV
r
kQqdEqqVU