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)