Lecture #4

OUTLINE

• Energy band model (revisited)

• Thermal equilibrium

• Fermi-Dirac distribution– Boltzmann approximation

• Relationship between EF and n, p

Read: Chapter 2 (Section 2.4)

EE130 Lecture 4, Slide 2Spring 2007

Important Constants

• Electronic charge, q = 1.610-19 C

• Permittivity of free space, o = 8.85410-14 F/cm

• Boltzmann constant, k = 8.6210-5 eV/K

• Planck constant, h = 4.1410-15 eVs

• Free electron mass, mo = 9.110-31 kg

• Thermal voltage kT/q = 26 mV

EE130 Lecture 4, Slide 3Spring 2007

Dopant Ionization (Band Model)

EE130 Lecture 4, Slide 4Spring 2007

Carrier Concentration vs. Temperature

EE130 Lecture 4, Slide 5Spring 2007

Electrons and Holes (Band Model)

• Electrons and holes tend to seek lowest-energy positions– Electrons tend to fall

– Holes tend to float up (like bubbles in water)

electron kinetic energy

hole kinetic energy

Incr

easi

ng

ele

ctro

n e

ner

gy

Incr

easi

ng

ho

le e

ner

gy

Ec

Ev

EE130 Lecture 4, Slide 6Spring 2007

Thermal Equilibrium

• No external forces are applied:– electric field = 0, magnetic field = 0

– mechanical stress = 0

– no light

• Dynamic situation in which every process is balanced by its inverse process– Electron-hole pair (EHP) generation rate = EHP recombination rate

• Thermal agitation electrons and holes exchange energy with the crystal lattice and each other Every energy state in the conduction band and valence

band has a certain probability of being occupied by an electron

EE130 Lecture 4, Slide 7Spring 2007

Analogy for Thermal Equilibrium

• There is a certain probability for the electrons in the conduction band to occupy high-energy states under the agitation of thermal energy (vibrating atoms)

Dish

Vibrating Table

Sand particles

EE130 Lecture 4, Slide 8Spring 2007

Fermi Function• Probability that an available state at energy E is occupied:

• EF is called the Fermi energy or the Fermi level

There is only one Fermi level in a system at equilibrium.

If E >> EF :

If E << EF :

If E = EF :

kTEE FeEf /)(1

1)(

EE130 Lecture 4, Slide 9Spring 2007

Effect of Temperature on f(E)

EE130 Lecture 4, Slide 10Spring 2007

Boltzmann Approximation

Probability that a state is empty (occupied by a hole):

kTEEF

FeEfkTEE /)()( ,3 If

kTEEF

FeEfkTEE /)(1)( ,3 If

kTEEkTEE FF eeEf /)(/)()(1

EE130 Lecture 4, Slide 11Spring 2007

Equilibrium Distribution of Carriers

• Obtain n(E) by multiplying gc(E) and f(E)

Energy banddiagram

Density ofStates

Probabilityof occupancy

Carrier distribution

EE130 Lecture 4, Slide 12Spring 2007

• Obtain p(E) by multiplying gv(E) and 1-f(E)

Energy banddiagram

Density ofStates

Probabilityof occupancy

Carrier distribution

EE130 Lecture 4, Slide 13Spring 2007

Equilibrium Carrier Concentrations

• Integrate n(E) over all the energies in the conduction band to obtain n:

• By using the Boltzmann approximation, and extending the integration limit to , we obtain

band conduction of top

cE

c(E)f(E)dEgn

2/3

2

* /)( 2

2 where

h

kTmNeNn nc

kTEEc

Fc

EE130 Lecture 4, Slide 14Spring 2007

• Integrate p(E) over all the energies in the valence band to obtain p:

• By using the Boltzmann approximation, and extending the integration limit to -, we obtain

1band valenceof bottom vE

v dEf(E)(E)gp

2/3

2

* /)( 2

2 where

h

kTmNeNp pv

kTEEv

vF

EE130 Lecture 4, Slide 15Spring 2007

Intrinsic Carrier Concentration

2

/ /)(

/)( /)(

i

kTEvc

kTEEvc

kTEEv

kTEEc

n

eNNeNN

eNeNnpGvc

vFFc

2/ kTEvci

GeNNn

EE130 Lecture 4, Slide 16Spring 2007

N-type Material

Energy banddiagram

Density ofStates

Probabilityof occupancy

Carrier distribution

EE130 Lecture 4, Slide 17Spring 2007

P-type Material

Energy banddiagram

Density ofStates

Probabilityof occupancy

Carrier distribution

EE130 Lecture 4, Slide 18Spring 2007

Dependence of EF on Temperature

1013 1014 1015 1016 1017 1018 1019 1020

300K

400K

400K300K

kTEE

cFceNn /)( nNkTEE ccF ln

Net Dopant Concentration (cm-3)

Ec

Ev

EF for donor-doped

EF for acceptor-doped

EE130 Lecture 4, Slide 19Spring 2007

Summary• Thermal equilibrium:

– Balance between internal processes with no external stimulus (no electric field, no light, etc.)

– Fermi function

• Probability that a state at energy E is filled with an electron, under equilibrium conditions.

• Boltzmann approximation:

For high E, i.e. E – EF > 3kT:

For low E, i.e. EF – E > 3kT:

kTEE FeEf /)(1

1)(

kTEE FeEf /)()(

kTEEFeEf /)()(1

EE130 Lecture 4, Slide 20Spring 2007

• Relationship between EF and n, p :

• Intrinsic carrier concentration :

/)( kTEEv

vFeNp

/)( kTEEc

FceNn

2/ kTEvci

GeNNn