ECEE 302: Electronic Devices

7 October 2002

ECEE 302 ElectronicDevices

Drexel UniversityECE Department

BMF-Lecture 3-100702-Page -1Copyright © 2002 Barry Fell

ECEE 302: Electronic Devices

Lecture 3. Physical Foundations of Solid State Physics, Semi-Conductor Energy Bands, and Charge Carriers

7 October 2002

7 October 2002




Outline (1 of 2)

• Atomic Bonding in Solids– ionic– covalent

• Energy Bands in a Solid– Formation– Band Theory of Solids– Direct and Indirect Transitions in Semiconductors– Variation of Energy Band Structure through alloying

• Plane Waves

• Charge Carriers– Electons and Holes– Effective Mass of the charge carrier– Intrinsic and Extrinsic Semi-conductors– Quantum Wells

7 October 2002




Outline (2 of 2)

• Charge Carrier Concentrations (Density)– Fermi Level

– Electron and Hole Concentrations at Equilibrium

– n-Type and p-Type semi-conductors

– Temperature Dependence

– Compensation and Space Charge Neutrality

• Carrier Drift in Electro-Magnetic Fields– Conductivity and Mobility

– Resistivity (Theory of Resistance)

– Effect of Temperature and Doping on Mobility

– Hall Effect

• Invariance of the Fermi Level at Equilibrium

7 October 2002




Ionic Bond• Ionic Bonding

– exchange of an electron between two atoms so each acheives a closed shell

– result is a positive (electron donor) and negative (electron acceptor) ion

– ions attract forming a bond

• Examples: NaCl, KCl, KFl, NaFl

NaValence = +1

ClValence = -1

Na+

looses electronto Cl

Cl-

gains electronfrom Na

Na+

Cl-

ElectrostaticAttraction forms basis forAtomic Bonding into Crystal Structure

3s12p63p5 3p6

7 October 2002




Valance Bond• Valance Bond: Bonding due to two atoms of complementary

valance combining chemically to share electrons across the bond– Valance Band 4 (and 4): C, Si, Ge, SiC

– Valance Band 3 and 5: GaAs, InP,

– Valance Band 2 and 6: CdS, CdTe

• Examples– Face Centered Cubic: Diamond (C), Silicon (Si)

7 October 2002




Energy Bands in a Solid

• Formation of the Energy Bands Text Figure 3-3• Band Theory of Solids Text Figure 3-4

– Insulator– Semi - Conductor – Conductor

– Energy Gap Eg

• Direct and Indirect Transitions in Solids Text Figure 3-5– E=h or =E/h (the Einstein-Planck Relationship)– Plane Waves, Expectation Value, and Momentum

• Variation of Energy Band Structure – Alloying Sze, Figure,

p299– Band Gap Behavior Text Figure 3-6

7 October 2002




Plane Waves

right the to moving is wave the vk

vt

x and

txk Thus

txk0 and

e1 and

eee e or

tt,xxt,x Then

moves.it as wave the onpoint a followingconsider Now,

v2

22

2

k Then

wave) the of velocity phase(vfrequencywavelength Recall,

2frequencyangular and 2

number wavek where

et,x

wave plane aConsider

p

txkj

txkjtkxjttxxkjtkxj

p

p

tkxj

x

Wave, x, at time t

Wave, x+x,t+t at time t+t

7 October 2002




Expectation value of an operator• Operators represent measured quantities

• The expectation value of an operator is the average value associated with the operator.

dxxp

xdxxp

x

find we ,x"" value the hasx that xp (weight) yprobabilit with b) to

a from values on (takes b to a from variesthat x quantity a if Then

"x" value the taking x of yprobabilit relative the as w interpret We

w

xwx

by given is ,w,...,w,w,w

weights with x,...,x,x,x numbers ofset a ofx average weighted A

x N

1x

is x,...,x,x,x numbers ofset a of xxx average The

b

a

b

a weighted

nnn

N

1nn

N

1nnn

weighted

N321

N321 weighted

N

1nn

N321avg

7 October 2002




Relationship between momentumand wave number (Textbook, p64)

dxxx

dxxxj

x

p and

xjp

p momentum,-x the of average)(or nexpectatio the takemust

we crystal the in electron the of momentum-x average the find To

crystal the in electron the of position the x

crystal the in electron the of 2

number wavek

structure crystal the on dependsthat constant U

x positionat electron the finding of ypropabilit x where

Uex

function wave a by described are crystal a in electrons The

*

*

x

x

x

x

xjk x

7 October 2002




Relationship between momentumand wave number (Textbook, p64) (2 of 2)

k

UdxU

UdxUk

UdxU

dxUekeU

UdxU

dxUejkj

eU

dxUeeU

dxUexj

eU

dxxx

dxxxj

x

p and

x

*

*x

*

xjkx

xjk*

*

xjkx

xjk*

xjkxjk*

xjkxjk*

*

*

x

xxxx

xx

xx

7 October 2002




Charge Carriers in a Crystal

• Electrons and Holes Text Figures 3-7 & 3-8

• Electron Hole Pairs - EHP

• Effective Mass of the charge carriers Text Figure 3-9– Derivation of effective mass formula

• Realistic Band Structures Text Figure 3-10

• Intrinsic and Extrinsic Semi-conductors Text Figure 3-12– donors ==yield electrons to the crystal (n-material)

– acceptors == take electrons from the band and form holes (p-material)

– intrinsic electron and hole concentrations ni

– intrinsic semi-conductor electron-hole generation rates gi

– intrinsic semi-conductor recombination rates ri

– Use of Bohr model to calculate Binding Energy of an electron in the solid

• Quantum Wells Text Figure 3-13

7 October 2002




Derivation of Effective Mass Formula• Effective Mass of electron or hole depends on structure

(curvature) of the Energy Band Surface within the solid

• We will determine the effective mass formula from consideration of a free electron

dkEd

*m and

*mdk

Ed

be to defined is m*, mass, effective the ,Thus

mm

k

dk

d

dk

dE

dk

d

dk

Ed and

m

k

m2

k2

m2

k

dk

d

dk

dE Now

m2

k

m2

pE

electron free afor so

ip)relationsh (deBroglie kp mechanics, quantumfor Recall

mv)(p m2

pmv

2

1E particle free a For

2

2

2

2

2

2

22

2

2

222

22

22

7 October 2002




Intrinsic Semi-Conductor Carrier Concentrations

Tnn calculate we whenlater result this need will We

gnpnr Thus

.p holes, and

,n electrons, of ionconcentrat e)temperatur (constant mequilibriu

the to alproportion is ,r rate, ncombinatio-re thethat assume We

EHP) of rate ationenerg(g EHP) of rate ncombinatio-re(r

e)temperatur (constant mequilibriu In

conductor?-semi intrinsic anfor npn does Why

conductor-semi the within

holes/cm ofnumber the is which np and

conductor-semi the within

cmelectrons/ ofnumber the is which nn

by denoted be conductor -semi a in carriers of ionconcentrat theLet

ii

i2ir00ri

0

0

i

ii

i

3i

3i

7 October 2002




Use of Bohr Model to calculate Binding Energy of Electron in a Solid (1 of 2)

220

4

electronionizationH energy binding

2220

4

ionization

220

4

2220

4

electron

42

mqE-EE

is electronouter the separate to required energy the Hence

0n

1

42

mq-E

or )(n by given is Hydrogen of level atom)donor the from

electron the remove completely to (energy energy ionization The

42

mq-

n

1

42

mq-E

is atom Hydrogen the of level 1)(n state ground theFor

theoryBohr the by calculated levels energy atomic the using by

atomdonor a from electronouter the grab to needed energy the

i.e. energy, bindingdonor the of estimate an make can We

7 October 2002




Use of Bohr Model to calculate Binding Energy of Electron in a Solid (2 of 2)

22r0

4

solid energy binding

r

r00

42

q*mE

Thus

*m mass effective the by m mass the and 524) page III, Appendix - Table

Textbook (see material the ofconstant dielectric the is where

, by space free of ypremitivit replacemust we solid a In

7 October 2002




Example (1 of 3)• Calculate the approximate donor binding energy for GaAs

(Textbook example 3-3)

222

2422

234214

41931-

22r0

4


r

31-0

22r0

4


sJouleFarad

mcoulombskg 108.34

2sJoule1063.6

2.13m/Farads1085.842

coulombs106.1kg109.110.67

42

q*mE

Thus

13.2GaAs ofconstant dielectric the is and

kg109.11m masses,

electron 0.067 GaAs in electron an of mass effective the is *m where

42

q*mE

expression the by given is GaAs of energy binding heT

7 October 2002




Example (2 of 3)

Joules108.34sJouleFarad

mcoulombskg 108.34E

and

joulemnewtonm s

mkg so

s

m-kgnewtons ma,F law, sNewton' From

m s

mkg

s

mkg

sJoule

mmnewtonskg so

meter-newtonJoule But

sJoule

mmnewtonskg

sJouleFarad

mmfaradnewtonskg

sJouleFarad

mcoulombskg Hence

mfaradnewtonscoulomb and

m-farad

coulombs

mm

faradcoulombs

newtons llydimentiona or

r

q

4

1F law sCoulomb' From

sJouleFarad

mcoulombskg consider Now

22

222

2422


2

2

22

2

22

22

22

22

222

22

222

24

2

2

2

2

2

2

0

222

24

7 October 2002




Example (3 of 3)

ev105.2E

Hence

ev105.2

Joule

ev100.625Joules 108.34

Joules 108.34

and

ev100.625Joule 1

,Joules106.1ev 1 since But

3-solid energy binding

3-

1922-

22-

19

19

7 October 2002




Charge Carrier Density (or Concentration) (1 of 2)• Fermi Level Text Figure 3-14

– Distributions• Maxwell Boltzman

• Bose-Einstein

• Fermi-Dirac

– Properties of the Fermi-Dirac Distribution

– The Fermi Level Energy is the energy level in the solid where there is a probability of 0.5=1/2 that an electron will be present in that state

• Application of the Fermi distribution to the description of intrinsic semi-conductions Text Figure 3-15

• Density of States Text Appendix IV, page 525

• Electron and Hole Concentrations (densities) at Equilibrium (T=constant) Text Figure 3-16

– Electron and Hole Concentration Calculation as a function of temperature

7 October 2002




Quantum Statistics• Maxwell Boltzmann Distribution Function

– Classical Distribution of particles– Multiple particles– Distinguishable particles

• Bose-Einstein Distribution – Quantum Mechanical Distribution– Multiple, Indistinguishable particles, in same state– integer spin particles– Examples: photons, phonons, mesons

• Fermi-Dirac Distribution (Quantum Mechanical, indistinguishable particles

– Quantum Mechanical Distribution– Single, indistinguishable Particle, in same state– half-integer spin particles– Examples: electrons, protons, neutrons

7 October 2002




Comparison of the Distribution Functions

holes electrons, mesons photons, molecules gas Examples

ishableindistingu ishableindistingu habledistinguis Particles of Types

quantum quantum classical Theory of Type

Exclusion Pauli State Each

1or 0 number any number any in Particles

of Number1e

1 AT)f(E,

1e

1 AT)f(E, AeT)f(E,

Dirac-Fermi Einstein-Bose Boltzmann onDistributi

Maxwell

kT

E-

kT

E-

kT

E-

7 October 2002




Maxwell-Boltzmann (Classical) Distribution Law

kTs

s

ssss

1ss

s1s

s

1s s

ns

s21

ss

s

s

s

eAg

n),g,nf(

when value probablemost its has ondistributi This

constant a ,nN particles, ofNumber Total

constant a ,nE Energy Total

that conditions the to subject

!n

g!N,...)n,...,n, W(n

by given is ondistributi the ofweight lstatistica The

energy of states gover

ddistribute particles habledistinguisn on based ondistributi Classical

7 October 2002




Bose-Einstein Distribution Law

Radiation. Body Black of Law sPlanck'for basis the isIt light). of

(particles photons ofbehavior the describes ondistributi This

to 0 fromnumber any equal can n1e

1A

g

n),g,nf(





!1g!n

!1gn,...)n,...,n, W(n


energy of states gover

ddistribute particles habledistinguis-inn on based ondistributi Mechanical Quantum

s

kT

ss

ssss

1ss

s1s

s

1s ss

sss21

ss

s

s

7 October 2002




Physical Foundation of Planck’s Black Body Radiation Formula

• Classical Description: All state share the same amount of energy. This results in the ultra-violet catastrophe

• Quantum Mechanical Description: The states are weighted by the Bose-Einstein distribution. States of higher energy are less likely to occur. The energy is determined the overall thermal energy kT. Most photons (radiation) is of lower energy

7 October 2002




Fermi-Dirac Distribution Law

Solids in ondistributi Fermi thefor basis the isIt

etc neutrons, protons, electrons, ofbehavior the describes ondistributi This

occuppied) is (state 1r o occuppied)not (state 0 values the on take only can n1e

1A

g

n),g,nf(





!ng!n

!g,...)n,...,n, W(n


Principle) Exclusion (Pauli only 1 and

0 values the to restricted is n where energy of states gover

ddistribute particles habledistinguis-inn on based ondistributi Mechanical Quantum

s

kT

ss

ssss

1ss

s1s

s

1s sss

ss21

sss

s

s

7 October 2002




Fermi Energy

83)-81 pp. textbook, (See level. energythat at

present is electron anthat 1/20.5 of yprobabilit the has

which solid the in level energy het isE level energy The

e1

1Ef

relation the by given isIt on.distributi Dirac-Fermi the on

based is Level Energy Fermi thefor ondistributi The

F

kT

EE F

7 October 2002




Properties of the Fermi-Dirac Distribution and the Fermi Energy, EF (1 of 3)

Kelvin in solid the of etemperatur absoluteT

energy to etemperatur absolute relates k

.Kelvin/Joules101.38 constant sBoltzmann'k

electron an by

occupied being of 1/2 of yprobabilit the has E level

energyat solid the within state AEnergy. FermiE

state energy the of energy E

T etemperatur absoluteat is solid the if electron

an by occuppied be will E energy hasthat

solid the in state energy anthat yprobabilitEf

wheree1

1Ef

by given is onDistributiFermi The

23-

F

F

kT

EE F

7 October 2002





2

1

11

1

e1

1

e1

1Ef EEFor

01

e1

1

e1

1Ef EEFor

101

1

e1

1

e1

1Ef EEFor

zero) (absolute K0 T at

e1

1Ef

Consider

kT

0

kT

EEF

0

EEF

0

EEF

kT

EE

F

F

F

F

7 October 2002





0)(x 2

1

11

1

e1

1

e1

1Ef EEFor

0)(x 2

1

11

1

e1

1

e1

1Ef EEFor

2

1

11

1

e1

1

e1

1Ef EEFor

zero) (absolute K0 Tor f

e1

1Ef

Consider

x

kT

EEF

x

kT

EEF

0

kT

EEF

kT

EE

F

F

F

F

7 October 2002




Calculation of electron (hole) concentration (1 of 7)• Electron or Hole Concentration = number of electrons (holes) per

unit volume that will be found at an energy level, E, within the conduction (valance) band of a solid at temperature T

• The electron concentration in the conduction band depends on three factors

– the fermi distribution, f(E,T)=probability an electron will be found with energy E within a solid at temperature T

– density of states N(E)=number of available states at Energy E within the solid

– E is an allowed energy (the electron cannot be at any energy within the band gap). For an electron to be a carrier it must be in the conduction band

• The hole concentration in the valance band depends on three factors– the fermi distribution, [1- f(E,T)]=probability a hole will be found with energy E

within a solid at temperature T

– density of states N(E)=number of available states at Energy E within the solid

– E is an allowed energy (the hole cannot be at any energy within the band gap). For a hole to be a carrier it must be in the conduction band

7 October 2002




Calculation of electron concentration (2 of 7)

same. the is pairs) hole (electron EHP

of rate ionrecombinat and generation the --- rg i.e. Crystal,

the within condition mequilibriu an represents "0" subscript The

dE

e1

1)E(NdET)(E,nTn

energyover (1) gintegratin by given is carriers) ofnumber

the (i.e., ,n band, conduction the within ionconcentrat electron The

(1)

e1

1)E(N)T,E(f)E(NT)(E,n

expression

the by determined is T),(E,n T, eTemperatur absolute and E energyat

conductor-semi a in m)equilibriu(at electrons of ionconcentrat The

ii

E kT

EEE

00

0

kT

EE0

0

C

FC

C

FC

7 October 2002




Calculation of hole concentration (3 of 7)

same. the is pairs) hole (electron EHP

of rate ionrecombinat and generation the --- rg i.e. Crystal,

the within condition mequilibriu an represents "0" subscript The

dE

e1

e)E(NdET)(E,pTp

energyover (2) gintegratin by given is carriers) ofnumber

the (i.e., ,p band, valance the within ionconcentrat hole The

(2)

e1

e)E(N

e1

11)E(N)T,E(f1)E(NT)(E,p

expression

the by determined is T),(E,p T, eTemperatur absolute and E energyat

conductor-semi a in m)equilibriu(at holes of ionconcentrat The

ii

E

0 kT

EE

kT

EEE

0

00

0

kT

EE

kT

EE

kT

EE0

0

V

FV

FVV

FV

FV

FV

7 October 2002




Calculation of hole concentration (4 of 7)Density of States N(E)

crystal. the in cellunit the of volume the is

V where ,V

2

L

2 is state) momentum (allowed state-k each by

occupied volume the find Wek. quantizing in results p quantizing Thus

.2

k where kh

p iprelationsh deBroglie the by Recall 2.

L by denoted be willIt

crystal. the of structure cellunit the by determined is period The

theory. quantum the by allowed being momentum of values certain only

in results This - crystal the on conditions boundary periodic Impose 1.

steps following the perform we N(E)

529)-525 pp. IV, Appendix (Textbook, states of density the determine To

cc

3

3

3

7 October 2002





crystal. the of

volumeE/unit energy with states electron ofnumber the is This

2

k2 is volumeunit per states electron ofnumber the Thus 4.

holeor electron anfor 1/2)-or 1/2(s

spin of states possible 2 theaccount into takes 2factor the where

2

Vk2

V2

k2

are k in states momentum

ofnumber The .dkk4 volume has this scoordinate spherical In

k. volume with space) (momentum space-k of region aConsider 3.

3

3c

c

3

2

7 October 2002





dEEm2

N(E)

and

dEEm2

2

dkk42

2

k2N(E)dE

crystal the of volumeunit per E energy havethat

states momentum ofnumber N(E) g,calculatin inresult this Using .6E

dE2mdk and

E2mk or

2m

k

2m

pE Then energy.

kinetic has only(it free is electron the assuming by done is This

E. energy to k of terms in momentum relatemust weext N 5.

212

3

2

*

2

212

3

2

*

23

2

3

2

*

2

*

*

2

*

2

7 October 2002




Calculation of electron & hole concentration (7 of 7)

2

*p

VkT

EE

2

*p

0

2

*n

CkT

EE

2

*n

E kT

EE2

123

2

*

2

E kT

EEE

00

kTm22N where e

kTm22Tp

by given is band

conduction the in holes of volume)it (number/un ionconcentrat the Similarly

kTm22N where e

kTm22

dE

e1

1dEE

m2

dE

e1

1(E)N dET)(E,nTn

Therefore

VF

FC

C

FC

C

FC

C

7 October 2002




Calculation of electron & hole concentration for Semi-Conductors (1 of 2)

eNNeNN

eNNeeNN

eNeNTpTn

have Welevel. Fermi the ,E of position the

changesjust Doping varied. is doping the if evenconstant is n and

p ofproduct the T, etemperaturat material particular a For

ekTm2

2Tp and ekTm2

2Tn

be will ionsconcentrat hole and electron intrinsic The

Gap Band the of middle thenear level energy

intrinsic an ,EE m,equilibriuat conductors-semi intrinsic For

kT

E

VCkT

EE

VC

kT

EEEE

VCkT

EE

kT

EE

VC

kT

EE

VkT

EE

C00

F

0

0

kT

EE

2

*p

ikT

EE

2

*n

i

iF

gVC

VFFCVFFC

VFFC

VIIC

7 October 2002




Calculation of electron & hole concentration for Semi-Conductors (2 of 2)

TnTpTn

iprelationsh the have also we so

eNNTpTn

,Why? TpTn sincebut

eNNTpTn

doping) (no conductor -semi intrinsic anor f Similarly

2i00

kT2

E

VCii

ii

kT

E

VCii

g

g

7 October 2002




Relationship between intrinsic and doped Semi-conductors (1 of 2)

densitiescarrier intrinsic

the of terms in densitiescarrier mequilibriu the write can we Therefore

eNTp and eNTn

by given is densitycarrier hole and electron mequilibriu The

eNTp and eNTn

by given are ionsconcentrat hole and electron conductor -semi intrinsic The

densitiescarrier of types two the between iprelationsh a write can We

mequilibriuat

conductors-semi undopedor doped torefer densitiescarrier mEquilibriu

mequilibriuat conductors-semi undoped torefer densitiescarrier Intrinsic

kT

EE

V0kT

EE

C0

kT

EE

VikT

EE

Ci

VFFC

NIIC

7 October 2002




Relationship between intrinsic and doped Semi-conductors (2 of 2)

eTpeeTpTp and

eTneeTnTn

results the have We

eNTp and eNTn

by given is densitycarrier hole and electron mequilibriu the Since

eTpN and eTnN

eNTp and eNTn

Since

kT

EE

ikT

EE

kT

EE

i0

kT

EE

ikT

EE

kT

EE

i0

kT

EE

V0kT

EE

C0

kT

EE

iVkT

EE

iC

kT

EE

VikT

EE

Ci

FIVFVI

IFFCIC

VFFC

VIIC

VIIC

7 October 2002




Example: Textbook, example 3-5, p88

4.07evEE

ev10.625Joule 1 Since

Joules106.5015.71Joules104.14

1066.lnJoules104.14

cm/101.5

cm/10ln K003

K

Joules101.38

n

n ln kTEE and

.EEfor solved be can This

eTnTn

iprelationsh theConsider ,n and n know we Since .EE determine to wish We)b(

cm/1025.2cm/10

cm/101.5

N

n

n

np

,p calculate can we

npn From

(Why?) Nnset we ,nN Since

number) this calculate you would How 87. page of middle textbook,

the (from cm/101.5 Si in electrons intrinsic of ionconcentrat The

1017 atomsdopant of ionconcentrat the be NLet (a)

:Solution

?E to relative E is where (b)

K? 300at ionconcentrat hole mequilibriu the iswhat (a)

.atoms/cm As10 with Si of sample a Dope :Problem

IF

19

-1920-

7e

20-

310

3 17

e23-

i

0eIF

IF

kT

EE

i0

0iIF

333 17

2310

d

2i

0

2i

0

0

2i00

d0id

310

d

IF

3 17

IF

7 October 2002




Charge Carrier Density (or Concentration) (2 of 2)• Temperature Dependence of Carrier Text Figure 3-17 & 18

Concentrations– Intrinsic Carrier Concentration as function of temperature

• Space Charge

• Compensation Text Figure 3-19

• Space Charge Neutrality

kT2

TE

43

*p

*n2i

g

emmh

kT22)T(n

7 October 2002




Carrier Drift in Electro-Magnetic Fields• Conductivity and Mobility

– Conductivity = property of solid to carry current– Mobility = ease with which an electron (or hole) can move in a solid

• Atomic Model of Resistance (conductivity)– electrons collide (are scattered) by the atomic centers– mean free path = average length an electron (or hole) travels before it is scattered– conductivity effective mass– Resistance and Ohm’s Law

• Effects of Temperature and Doping on Text Figure 3-22 & 23

Electron and Hole Mobility– Dependence of scattering on mobility

• High Field Effects Text Figure 3-24– Breakdown of Ohm’s Law

• Hall Effect Text Figure 3-25– Hall Voltage– Hall Coefficient– Method of determining charge carriers (electrons or holes)

7 October 2002




Conductivity Model

dt

dpFneE-

by given is voltage) applied an to (due field electric an of influence

theunder electrons of motion the crystal the in level atomic an On

EJ J

E or

A

lAJlE becomesIR V and

A

lR A,JI l,EV

e)(ResistancR describesthat solid the of theory physical the isWhat

IRV Law sOhm' Recall

xxx

7 October 2002




Scattering of Electrons in a Solid (1 of 2)

t

dtpdp

as drepresente be can change This occurs.

scattering the when changes (hole) electron the of momentum thethat Note

llyexponentia

decreases (holes) electrons dunscattere ofnumber the increases time As

eNN(t)

is yourself)it (try equation this to solution The

t events, scattering between time mean the by divided

scatterednot arethat electrons ofnumber total the to alproportion

is scatterednot arethat electrons ofnumber in decrease the means This

)t(Nt

1

dt

)t(dN-

as zedcharacteri be can

scattered] been havethat electrons of[number N(t) of decrease of rate The

t time the by scatterednot arethat electrons ofnumber the be N(t)Let

xx

t

t

0

7 October 2002




Scattering of Electrons in a Solid (2 of 2)

Solid

the of )resistance (inverse tyconductivi the to related is and solid the in

(holes) electrons the of velocitydrift the called is velocity average This*m

Etq

*m

pv

is (hole) electronper velocity average thethat states This

Etqn

pp

is E field the in electron an by carried momentum average theor

nqEt

p

dt

dp have we and

t

p

dt

dp Then

xxx

xx

x

x

xxx

xx

7 October 2002




Current Density and Conductivity

*m

tnq

solid the of tyconductivi thefor have We

EJ since But

E*m

tnqvqn

A

IJ

area the to normal direction a

in solid the of areaunit per current the isJ densitycurrent The

2

x

2

xx

7 October 2002




Mobility (electrons and holes)

xxpnx

p

n

xnx

x

x

2

EEpnqJ

by given is solid the by conductedcurrent total the Then

holes,for mobility a define

can Wesolid. the in electrons of mobility the to refers where

EqnJ as mobility of terms in written be can densitycurrent The

E

v- find we sdefinition the From

solid the

within move can (holes) electrons easily how describes mobility The*m

tq Then mobility. (hole) electron the called is where

qn*m

tnq written be can solid the of tyconductivi The

7 October 2002




Hall Effect (Figure 3-25 of Text) (1 of 2)

tVq

BI

wtVq

BwtI

qE

BJ

qR

1p

expression

the from ,V and ,I ,B of tsmeasuremenour from p find can We

t.coefficien Hall the called is R

qp

1R and ,BJRB

qp

JE

have we then ,wEV voltage Hall The

)0v( BvE

where direction y the in field electric an induce weEffect Hall the In

BvBvEqF be to direction y the in

force the calculate can we BvEqF Force Lorentz the From

conductor-semi a in carriers

charge theabout sticscharacteri determine to us permitseffect Hall The

AB

zx

AB

zx

y

zx

H0

ABxz0

H

0HzxHz

0

xy

yAB

zzxy

zxxzyy

7 October 2002




Hall Effect (Figure 3-25 of Text) (2 of 2)

H

H

0p

0p

x

CD

R

qR1q

1

qp

find we

pq1

Sincewt

lI

V

l

Rwt

l

RA1cm-

y,resistivit calculate can we R, material,

the of resistance the measure we if effect, Hall the ofpart asthat Note

7 October 2002




Invariance of the Fermi Level (Figure 5-11)

)E(f)E(f or

)E(f)E(f)E(f)E(f)E(f)E(f

)E(N)E(N by through dividing

)E(f)E(N)E(f)E(N)E(N)E(f)E(N)E(f)E(N)E(f)E(N)E(N)E(f)E(N

)E(f1)E(N)E(f)E(N)E(f1)E(N)E(f)E(N

so m,equilibriuat equal are These

)E(f1)E(N)E(f)E(N 1 to 2 fromtransfer of rate

and

)E(f1)E(N)E(f)E(N 2 to 1 fromtransfer of rate

E, energyAt

1. to 2 material from carriers charge of flow ngcompensati a

by balanced bemust 2 material to 1 material from holes)or (electrons

carriers charge of flow anythat means This Voltage).(or EMF of source

external anwithout conductor -semi the in flowcurrent no is This function.

ondistributi Dirac-Fermi a by described is junction this of portion Each

junction a intogether joined conductors-semi dissimilar two Consider

21

122211

21

11221222211211

11222211

1122

2211

7 October 2002




Preview of Next Lecture:Excess Carriers in Semi-Conductors (Chapter 4)

• Creation of Excess Carriers– Optical Absorption

– Photoluminescence

– Photo-conductivity

• Electron - Hole Recombination

• Carrier (Electron or Hole) Trapping

• Diffusion of Excess Carriers

Documents

ECEE 302: Electronic Devices