Lec. 31
The Fermi-Dirac Distribution Function
Ex.) at T=0 Ki) If E > EF , exp() fF(E) = 0ii) If E < EF , exp(-) 0 fF(E) = 1 iii) If E = EF , exp(0) 1 fF(E) =1/2
Carrier Concentration
• Carrier distribution statistical approach
kTEE FeEf /)(1
1)(
f(E) : The probability of occupancy of an available state at E by an electronEF : Fermi energyk : Boltzmann constant ( = 8.62 10-5 eV/K = 1.38 10-5 J/K)
Symmetrical about EF
Lec. 32
n
p
1
Lec. 33
• Density of states (DOS) in Silicon conduction and valence bands:
Counting states (stadium seats) in 3-D, see Appendix IV:
Where the most important thing to remember is E1/2 (more states at higher energies).
CE
dEENEfn )()(
mn* (effective mass of electron)
DOS in the conduction band:1/2
3
3/2*n
h)(2m4N(E) )(π
cEE
DOS in the valence band:
1/23
3/2*
h)(2m4N(E) h )(π EvE mh* (effective mass of hole)
• Total electron concentration in conduction band
1/23
3/2*n
h)(2m4 )(π
cEE
CE
dEkTEE Fe /)(11
1/23
3/2*n
h)(2m4 )(π
cEE
CE
dEkTEE Fe /)(1
(Boltzmann approximation)
2/3
2
*22).(
hkTmBCofdensityeffectiveNwhere n
CkTEE
CFCeN /)(
(at 300 K, NC = 2.8 1019 cm-3)
(high- low)
Lec. 34
• Fermi-Dirac distribution function vs Boltzmann approximation
The difference between Fermi-Dirac and Boltzmann aproximation becomes within 1/20.
Boltzmann approximation
Fermi-Dirac function
EF E
1/2
1.0 3kTEE F
vE
dEENEfp )())(1(
• Total hole concentration in valence band
1/23
3/2*h
h)(2m4 )(π EE v
vE
dE)1
11( /)( kTEE Fe
1/23
3/2*h
h)(2m4 )(π EE v )1
1
1( X
XdE
vE
)(
/)( kTEE Fe
(Boltzmann approximation)
2/3
2
*22).(
hkTmBVofdensityeffectiveNwhere h
vkTEE
vvFeN /)(
(high- low)
∵
Lec. 35
Acceptor Dopingp0 > n0 : p-type
n
p
Q3) What about high temp case?
Q1) What is the meaning of N(E)dE?
DOS at dE
CE
dEENEfn )()(
dEENEf )()(
Donor Dopingn > p : n-type
Lec. 36
Students (electrons)
Chairs (N(E))
Massage Chairs (higher f(E))
No chairs (Eg)
Why zero at Ec? N(E)
1. Electrons does not follow statistical equation.
But scientists observed phenomenon (N(E), electron distribution) and fit into statistical equation.
Then why do we learn this? Amazingly, using statitical equation, every electronic phenomenon can be explained.
N(E)f(E)f(E)
Why zero? f(E)
2.
Lec. 37
kTEgVC
kTEvEcVC
kTEvEEEcVC eNNeNNeNNnp FF //)(/))((
kTEEv
vFeNp /)( kTEEC
FCeNn /)(
kTEgVC
kTEvEcVC
kTEvEiEiEcVCii eNNeNNeNNpn //)(/))((
202 1025.2 nppnn iiiii pn
kTEEi
iFenn /)(
kTEEC
FCeNn /)(
•
•kTEE
CiiCeNn /)(
kTEEi
Fienp /)(
at RT
iiF n
nkTEE lni
Fi npkTEE ln
•
d
CFC
NNkTEE ln
(At RT, Nv = 1.04 1019 cm-3
NC = 2.8 1019 cm-3)
a
VVF N
NkTEE ln
FE
iE
Lec. 38
Ex) Si sample is doped with As 1017 /cm3, What is the equilibrium hole concentration at RT?
3317
2020
1025.210
1025.21025.2
cmn
p
eVnnkTEE
iiF 407.0
105.110ln026.0ln 10
17
Where is EF relative to Ei?1.1eV
0.407eV
What is the relationship of Eg, T, n, band diagram, and EF,
This case is only for F-D function with constant Ef.
kTEE FeEf /)(1
1)(
cf)
Lec. 39
Reference: Sze “Physics of Semiconductor Device”
g=2
Nd is total dopantNd
+ is ionized donors
g=4
Ionization energy (eV)
Donor in Si P As Sb
Ionization energy (eV) 0.045 0.054 0.039
Lec. 310
Carrier concentration vs T
Low tempHigh temp
Low temp High temp
1015
Lec. 311
Keep in mind ni is very temperature-sensitive!
Ex) in Silicon:
While T = 300 → 330 K (10% increase)
ni = 1010 → 1011 cm-3 (10x increase!)
)exp()()( 2/3**3/2
kT
EmmkTAeNNpnn gpn
kTEgVCiii
,22(2/3
2
*
hkTmN h
v
)222/3
2
*
hkTmN n
C
Eg & ni vs T
∵
Lec. 312
Compensation
What if a piece of silicon contains both dopant types?
1017 dopant
5x1016 acceptor
Lec. 313
The Intrinsic Fermi-Level, EFi , Position
Because of ni = pi,
In case of Si, mn* = 1.18 m0 , mp
* = 0.81 m0
EFi - Emidgap = - 0.0128 eV
However, 0.0128 eV is small value so it can be said that EFi Emidgap for intrinsic Si.
*n
*p
vc
c
vvcFi
vFiv
Ficc
m
mkTln
43)E(E
21
NN
kTln21)E(E
21E
kT)E(E
expNkT
)E(EexpN
Lec. 314
The density of negative charge = The density of positive charge.- Example ;
When both donors and acceptors are added to the same region to form a compensated semiconductor.
Charge Neutrality
da NpNn
For n-doped S/C dNpn
2innp
2inN-n(n d )
0 2inN-n d
2 n
]4[21 22 did NnNn
At RT, ni2=2.25x1020 << Nd
2=1034 dNnIn case of high temp, ni value can not be negligible.
What if Nd=1x1017, Na=7x1016?
nnp
2i
+e-
Lec. 315
C-band electrons (or V-band holes) are essentially free to move around at finite temperature & doping. So what do they do?
Instantaneous velocity given by thermal energy:Scattering time (with what?) is of the order ~ 0.1 ps.So average distance travelled between scattering: L ~ mean free path
But no electric field = not useful = boring materials.
Conductivity and Mobility
Then turn on an electric field:
E on
EEmqtv n
nn *
So average velocity in E-field is:
*,
,pn
pn mqt
We call the proportionality constant mobility:
(cm2/Vs)
EEmqtv p
pp *
(t is mean time between scattering events)
Lec. 316
Scattering Mechanism
There are two collision or scattering mechanisms that dominate in a semiconductor and affect the carrier mobility.
1. lattice scattering or phonon scattering
The lattice vibration cause a disruption in the perfect periodic potential function. scattering of electrons
3/2ph Tμ
i
3/2
i NT
μ
iphT 111
2. Ionized impurity scattering- Ionized impurity (dopant atom) scattering
- Electron-electron or electron-hole scattering
+
e-
Lec. 317
• At RT, mobility of Si is dominated by phonon scattering.High doping
Electron mobility versus temperature for different doping levels
1. Si (Nd< 10^12 cm-3)2. Si (Nd< 4x10^13 cm-3)3. Nd= 1.75x10^16 cm-3; 4. Nd= 1.3x10^17 cm-3;
Mobilities of intrinsic semiconductors at RT
Si Ge GaAs InAs n (cm2/Vꞏs) 1400 3900 8500 30000 p (cm2/Vꞏs) 470 1900 400 500
Mobility Data
Lec. 318
Mobilities vs Impurity concentration
Lec. 319
Low-field slope = mobilityHigh-field effect = velocity saturation due to very strong lattice scattering. Any additional energy from the E-field is transferred to the lattice (phonons) rather than increasing the carrier velocity.
Result: constant velocity (current) at very high fields!
Velocity vs Electric Field
(electrons in silicon)
Current Density
Drift current density ∝ net carrier drift velocity
∝ carrier concentration
∝ carrier charge
EqnqnvJ ndndriftn
EqpqpvJ pdpdriftp
(charge crossing plane of area A in time t)
(unit ?)
(sign)
+ -
EpnqJJJ pndriftp
driftn
drift )(
Lec. 320
Resistivity & Conductivity
EpnqJ pndrift )(
EEJ
)(11
pn pnq
Resistivity of a semiconductor:
n-type:nnq
11
p-type:ppq
11
wtL
wtLR
1
JAI cf)