lecture 23jan15

8/9/2019 Lecture 23Jan15

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Course: ECE 230LIntro to Microelectronic Devices and Circuits

Location: Teer 203 Instructor: Dr. Stiff-Roberts

Office: FCIEMAS 3511 Office Hours: Tu 1-4pm

E-mail: [email protected] Phone: 660-5560

Last Class This ClassDensity of States Function

Statistical Mechanics

Charge Carriers in SemiconductorsDopant Atoms and Energy Levels

The Extrinsic Semiconductor

Kronig-Penney Model

k-Space Diagram

Electrical Conduction in SolidsExtension to Three Dimensions

Density of States Function

ECE 230L, Spring 15, Stiff-Roberts Page 1January 23, 2015


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Density of States Function:

Quantum Theory of Solids


elsewhere,,

00

0

for0,,

zyxV

azay

ax

zyxV

Three-dimensional infinite potential well

Crystal is a cube with length a

Electrons are

allowed to move

relatively freely in

the conduction band

of a semiconductor,

but are confined tothe crystal.

Consider a free

electron confined to

a three-dimensional

infinite potential well,where the potential

well represents the

crystal.


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2

2

2222222

22

annnkkkkmE zyxzyx

where nx, ny, and nzare positive integers (negative values would not yield different

energy states)


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dEEgENE )()(

EhmEg3

23

24)(

Density of quantum states as a function of energytotal number of

quantum states between the energy E and E + dE per unit volume of

the crystal. Units are given as number of states per unit energy per unit volume (#

states/eV cm3)

To determine the total number of quantum states per unit volume, must

integrate the density of states over a given energy range.


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cv

v

p

v

ccnc

EEEEg

EEh

mEg

EEEEh

mEg

for,0)(

EEfor,24

)(

for,24

)(

v3

23*

3

23*

We can extend this model to a semiconductor in order to determine thedensity of quantum states in the conduction band and in the valence

band.

Electrons and holes are confined within the semiconductor, so the

infinite square well potential is still relevant.


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Statistical Mechanics:



There are three distribution laws

describing the distribution of

particles among available energystates (in each case the particles

are assumed to be noninteracting):

Maxwell-Boltzmann probability

function:Particles are

distinguishable, no limit to the

number of particles allowed ineach energy state (gas molecules

in a container at low pressure)

Bose-Einstein probability

function: Particles are

indistinguishable, no limit to the

number of particles allowed ineach energy state (photons)

Fermi-Dirac probability function:

Particles are indistinguishable,

only one particle is permitted in

each quantum state (electrons)


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)(

)(

exp1

1)(

Eg

EN

TkEE

Ef

B

F

F

Fermi-Dirac probability functionprobability that a quantum state at the

energy Ewill be occupied by an electron.

fF(E)Fermi-Dirac probability functionN(E)number density, number of particles per unit volume per unit energy

g(E)density of states (DOS), number of quantum states per unit volume

per unit energy

EFFermi energy


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This result shows that,

for T = 0K, the

electrons are in their

lowest possible energystates.

All states below EFare

filled and all states

above EFare empty.


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The Fermi energy

determines thestatistical distribution of

electrons and does not

have to correspond to

an allowed energy level.

The value of the Fermi

energy EFis critically

dependent on the

density of states

function of the system.

If g(E) and N0are

known for a given

system, then the Fermi

energy can be

determined.


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The probability of

a state being

occupied at E = EF

is for T > 0K.

At these

temperatures,there is a nonzero

probability that

some energy

states above EF

will be occupied

by electrons and

some energy

levels below EF

will be empty.


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Tk

EEEf

B

F

F

exp1

11)(1 Probability that a quantumstate is empty with no

electrons:


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TkEEEf

Tk

EEEf

Tk

B

FF

B

F

F

B

exp)(

exp1

1)(

E-EConsider F

The Fermi-Dirac distribution and the Maxwell-Boltzmann approximation

are within 5% of each other when the E-EF~ 3kBT.

Tk

EE

Tk

EE

Tk

EEEf

B

F

B

F

B

F

F exp

exp1

1

exp1

11)(1


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Charge Carriers in Semiconductors:

The Semiconductor in Equilibrium


)(1)()( EfEgEp Fv

The distribution of holesamong energy levels within the valence bandis givenby the density of allowed quantum states times the probability that a state is not

occupied by an electron.

The total hole concentration per unit volume in the valence band is found by

integrating the expression over the entire valence band energy range.

)()()( EfEgEn Fc

The distribution of electronsamong energy levels within the condu ct ion band

is given by the density of allowed quantum states times the probability that a

state is occupied by an electron.

The total electron concentration per unit volume in the conduction band is found

by integrating the expression over the entire conduction band energy range.


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*assume electron and hole

effective masses are equal

intrinsic semiconductor

pure semiconductor with no

impurity atoms and no lattice

defects

For an intrinsic

semiconductor*:

density of states functions

for electrons and holes

are symmetrical

Fermi energy is at the

midgap energy level

electron and holeconcentrations are equal


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Tk

EE

dETkEEEE

hmn

Tk

EE

Tk

EEEf

dEEfEgn

B

c

E B

Fc

n

B

F

B

F

F

Fc

c

Let

exp24

exp

exp1

1)(

)()(

3

23*

0

0

For the condition of thermal equilibrium (no external forces), the corresponding

electron and hole concentrations, in an intrinsic semiconductor, are:

n0thermal-equilibrium electron concentrationp0thermal-equilibrium hole concentration


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Tk

EE

h

Tkmn

d

dTk

EE

h

Tkmn

B

FcBn

B

FcBn

exp2

2

2

1exp:function,Gamma

expexp24

23

2

*

0

0

21

0

213

23*

0


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Tk

EENn

h

TkmN

B

Fc

c

B*n

c

exp

22:Define

0

23

2 effective density of states

function in the conduction

band


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20/30




ii pn

For an intrinsic semiconductor, the corresponding electron

and hole concentrations are:

niintrinsic electron concentration

piintrinsic hole concentration

EFiintrinsic Fermi energy (for intrinsic semiconductor)Egbandgap energy

S


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Tk

ENN

Tk

EENNn

Tk

EE

Tk

EENNpnn

Tk

EENnpp

Tk

EE

Nnn

B

g

vc

B

vcvci

B

vFi

B

Ficvciii

B

vFivii

B

Fic

ci

expexp

expexp

exp

exp

2

2

0

0


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Th S i d t i E ilib i


23/30




*

*

*

*

ln4

3

21

ln4

3

2

1

ln

2

1

2

1

expexp

n

p

BmidgapFi

midgapvc

n

p

BvcFi

c

vBvcFi

B

vFiv

B

Ficc

m

mTkEE

EEE

m

mTkEEE

N

NTkEEE

Tk

EEN

Tk

EEN

We can now calculate the intrinsic Fermi energy position, EFi



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Dopant Atoms and Energy Levels:



Add a group V element, like

phosphorus, as a substitutional

impurity atom in silicon.

Group V element has five

valence electrons.

The fifth valence electron is more

loosely bound to the phosphorusatom because not participating in

covalent bondingcalled the

donor electron.

As temperature increases, the

donor electron can break awayfrom the donor impurity atom and

enter the conduction band,

leaving behind a positively

charged donor ion.



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Edenergy state of the donor electron

n-type semiconductordonor atoms contribute electrons to the

conduction band without contributing holes to the valence band


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Eaenergy state of the acceptor

p-type semiconductoracceptor atoms contribute holes to the

valence band without contributing electrons to the conduction band



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Ionization energyapproximate energy

required to elevate the donor electron into the

conduction band, or to elevate a valence electron

into a discrete acceptor energy state.

Extrinsic semiconductorhas controlled amount of dopant atoms, either donors

or acceptors, so that the thermal-equilibrium electron and hole concentrations are

different from the intrinsic carrier concentration. One type of charge carrier is

dominant in an extrinsic semiconductor.



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The Extrinsic Semiconductor:



Tk

EEnp

Tk

EEnn

Tk

EE

Tk

EENn

Tk

EEEEN

Tk

EENn

B

FiFi

B

FiFi

B

FiF

B

Ficc

B

FiFFic

cB

Fc

c

exp

exp

expexp

expexp

0

0

0

0



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The Extrinsic Semiconductor:


ECE 230L Spring 15 Stiff Roberts Page 30January 23 2015

2

00

00

00

exp

expexp

i

B

g

vc

B

vF

B

Fcvc

npn

Tk

E

NNpn

TkEE

TkEENNpn

lecture 23jan15

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