ElectronicsI- Semiconductors

talarico@gonzaga.edu 1

Chapter2

Fall2017

ChargedParticles• Theoperationofallelectronicdevicesisbasedoncontrollingtheflowofchargedparticles

• Therearetwotypeofchargeinsolids– Electrons– Holes

• Therearetwomechanismthroughwhichchargecanbetransportedinamaterial– Drift (motionofchargecausedbyanelectricfield)– Diffusion (motionresultingfromanon-uniformchargedistribution)

talarico@gonzaga.edu 2

ElectronicStructureoftheelements

• Atom’schemicalactivitydependsontheelectronsintheoutermostshells(orbits).TheseelectronsarecalledVALENCEelectrons.

• Inextremelypureelements,suchassilicon,theatomsarrangethemselvesinregularpatternscalledCRYSTALS.Thevalenceelectronsdeterminetheexactshape(=LATTICEstructure)

talarico@gonzaga.edu 3

ElectronicorbitalsinsiliconElement AtomicNumber Configuration

Si 14 (1s)2(2s)2(2p)6(3s)2 (3p)2

talarico@gonzaga.edu 4Source:Howe&Sodini

s,p,ddesignateorbitalshapesholdsupto2electronspholdsupto6electronsdholdsupto10electrons

SiliconCrystalLattice

talarico@gonzaga.edu 5

Source:Howe&Sodini

concentration(atoms/cm3)ofatomsinsilicon

2D-RepresentationofSiliconCrystal

talarico@gonzaga.edu 6

Figure 3.1 Two-dimensional representation of the silicon crystal. The circles represent the inner core of silicon atoms, with

+4 indicating its positive charge of +4q, which is neutralized by the charge of the four valence electrons. Observe how the

covalent bonds are formed by sharing of the valence electrons.

At 0 K, all bonds are intact and no free electrons are available for current conduction.

Source:Sedra &Smith

Freeelectronsandholes

talarico@gonzaga.edu 7

Source:Sedra &Smith

Figure 3.2 At room temperature, some of the covalent bonds are broken. Each broken bond gives rise to a

free electron and a hole, both of which become available for current conduction.

PeriodicTable

talarico@gonzaga.edu 8Source: Pierret

EnergyBandStructure• Bandgapenergy(Eg)istheminimumenergytodislodgeanelectronfrom

itscovalentbond.• ForSiliconatroomtemp.(T=300°K)Eg =1.12eV=1.792× 10−19Joule

talarico@gonzaga.edu 9Source:Millman &Halkias

Concentrationoffreeelectrons• Theconcentrationofelectrons(andholes)inpuresiliconatroom

temperatureisapproximately:

• Astemperatureincreases,theintrinsicconcentrationni approximatelydoublesevery10°Criseoverroomtemperature(source:Howe&Sodini)

• Giventhatthenumberofbondsis2×1023cm−3,atroomtemperatureonlyanextremelysmallfractionofthebondsarebroken(1in20×1012bonds,thatis1in5×1012 atoms)

• Toofew:weneedmore!!!

talarico@gonzaga.edu 10

𝑛" 𝑇 = 300𝐾 ≅ 1×10,-𝑐𝑚01

𝑛"(𝑇) ≅ 𝑛"(𝑇1--)×2505677,- 𝑐𝑚01

Intrinsiccarrierconcentrationasafunctionoftemperature

talarico@gonzaga.edu 111000/300≅ 3.33

300°K= 27°Csource:Streetman

Foranintrinsicsemiconductor:n=p=ni

1.05☓1010

Intrinsiccarrierconcentration

talarico@gonzaga.edu 12

𝑛" = 𝐴9𝐴:� 𝑇1/=𝑒0?@/(=A5) ≅ 5×10,C 𝑇1/=𝑒0?@/(=A5) (𝑐𝑚01)

𝑁9 = 𝐴9𝑇1/=

𝑁: = 𝐴:𝑇1/=

TheconstantsAC andAV canbederivedfromtheeffectivedensityofthestatesinconductionbandNC (cm-3)andvalencebandNV (cm-3).

ForsiliconatT=300K(sourcePierret):NC =3.22x1019 cm➖3 andNV=1.83x10 19 cm➖3

𝑛" 𝑇 = 300𝐾 ≅ 1×10,-𝑐𝑚01

Boltzmannconstant=K=8.617e-5eV/K=1.38×10−23 J/°KEnergyGapforsiliconatroomtemperature=1.12eV

EG alsodependsonT

• TheenergybandgapEg isaffectedbytemperatureaccordingtothefollowingVarshni equation:

• whereEg(0)isthebandgapenergyatabsolutezeroandaEandbE arematerialspecificconstants

EnergyBandGap

talarico@gonzaga.edu 13

(eV)

ExtrinsicSemiconductors(1)• Dopingwithdonorimpurities(N-typesemiconductor)

talarico@gonzaga.edu 14

Figure 3.3 A silicon crystal doped by a pentavalent element. Each dopant atom donates a free electron and is thus called a donor. The

doped semiconductor becomes n type.

Example:ND ≅10

17 cm-3source:Sedra &Smith

ExtrinsicSemiconductors(2)• Dopingwithacceptorimpurities(P-typesemiconductor)

talarico@gonzaga.edu 15

Figure 3.4 A silicon crystal doped with boron, a trivalent impurity. Each dopant atom gives rise to a hole, and the semiconductor becomes

p type.

Example:NA ≅10

19 cm-3source:Sedra &Smith

N-typesemiconductor

talarico@gonzaga.edu 16

EDEG

EC

EV

source:Millman &Halkias

source:Howe&Sodini

ρ = chargedensity[Cb / cm3]= 0 = (−qn)+ (qp)+ qND

electrons holes donors

q=1.6× 10−19Cb

P-typesemiconductor(1)

talarico@gonzaga.edu 17

ECEG

EV

EA

source:Millman &Halkias

source:Howe&Sodini

ρ = chargedensity[Cb / cm3]= 0 = (−qn)+ (qp)− qNA

electrons holes acceptors

q=1.6× 10−19Cb

P-typesemiconductor(2)

• Holescanbefilledbyabsorbingfreeelectrons,thereforethereisan“effective”flowofholes

• Holesareslowerthanfreeelectrons(duetotheprobabilityofaholetobefilled)

• Theeffectivemassofholesislargerthantheeffectivemassofthefreeelectrons:m*h >m*e

talarico@gonzaga.edu 18

Mobilityoffreeelectronsandholes

talarico@gonzaga.edu 19

source:Howe&Sodini

ForintrinsicsiliconatT=300K:μp ≈ 480cm2/(V·s)μn ≈ 1350cm2/(V·s)

≈2.8× μp

µ ∝T −3/2

• Electron and Hole mobilities for silicon at 300 K• Mobilities vary with doping level

Ntot =NA +ND=

MassActionLaw

• Themass-actionlawisvalidforbothintrinsic(pure)andextrinsic(doped)semiconductors

• Ifnthenp Alargernumberoffreeelectronscausestherecombinationrateoffreeelectronswithholestoincrease

talarico@gonzaga.edu 20

ni2 = n ⋅ p

Dopingwithdonors(n-type)

• Chargeneutrality:

• Usingmass-actionlaw:

talarico@gonzaga.edu 21

ρ = 0 = q(p − n + ND )

ni2

n− n + ND = 0⇔− ni

2

n+ n − ND = 0⇔ n2 − ND ⋅n − ni

2 = 0

flipsides multiplybothsidesbyn

n =ND ± ND

2 − 4ni2

2≈ ND

dopingwithND >>ni p ! ni

2

ND

HolesareMinorityCarriers

FreeelectronsareMajorityCarriers

Dopingwithacceptors(p-type)

• Chargeneutrality:

• DopingwithNA >>ni

talarico@gonzaga.edu 22

ρ = 0 = q(p − n + ND )

p ≈ NA

n ! ni

2

NA

Freeelectronsare MinorityCarriers

HolesareMajorityCarriers

Dopingwithbothdonorsandacceptors

• Chargeneutrality:

• Assumingthat|ND-NA|>>ni(nearlyalwaystrue)– ForND >NA

– ForNA >ND

talarico@gonzaga.edu 23

ρ = 0 = q(p − n + ND − NA )

n ! ND − NA and p ! ni

2

ND − NA

p ! NA − ND and n ! ni

2

NA − ND

FirstCarriersTransportMechanism:Drift

• Theprocessinwhichchargedparticlesmovebecauseofanelectricfieldiscalleddrift.

• Chargedparticleswillmoveatavelocitythatisproportionaltotheelectricfield(thisistrueaslongasthefielddoesn’tbecometoolarge)

talarico@gonzaga.edu 24

vpdrift! "!!

= µp E→

vndrift! "!!

= −µn E→

Driftvelocityinsilicon

talarico@gonzaga.edu 25

−vn,drift, vp,driftvelocitystarttosaturate

Saturationofthedriftvelocity

• Eventuallythedriftvelocitysaturates:therearetoomanycollisionsamong`carriersandbetweencarriersandlattice

• vsat forsiliconis≈107cm/s=105 m/s

talarico@gonzaga.edu 26

vdrift

sourceGray&Meyer:

𝑣FG"HI 𝐸 ≃𝜇𝐸

1 + 𝐸𝐸9

=𝜇𝐸

1 + 𝜇𝐸𝑣NOI

source:Razavi

DriftCurrent

talarico@gonzaga.edu 27

source:Streetman

H

drift

𝐸P=−F:RFP

≅ −S:RSP

= − :70T

=:7T

Driftcurrentandcurrentdensity

talarico@gonzaga.edu 28

J drift = I drift

W × H[A /m2 ]

cross-sectionArea[cm2]

volumeperunittime[cm3/s]

CurrentDensity

§ electriccurrent:amountofchargethatflowsthroughareferenceplaneperunittime

chargeperunitvolume

(akachargedensity)[Cb/cm3]

source:Howe&Sodini

+

+

Figure 4–16Current entering and leaving a volume Δx�A

source:Streetman

vdrift ! Δx

Δt

𝐼FG"HI =Δ𝑄Δ𝑡 =

Δ𝑥×𝑊×𝐻×𝑛×𝑞Δ𝑡 = 𝑣FG"HI×𝑊×𝐻×𝑛×𝑞

𝐶𝑏𝑠 = 𝐴

chargeperunitvolume[Cb/cm3]

DriftCurrentDensity

talarico@gonzaga.edu 29

Jndrift

vdriftn

Jpdrift

vpdrift

J drift = Jndrift + Jp

drift = vndriftqen+ vp

driftqp p = −µnEqen+µpEqhn =

= µnEqn+µpEqp = µnqn+µpqp( )E

=σ =1/ρconductivity[Ωm]−1

J drift =σE OhmLaw

qh = −qe ≡ q =1.6×10−19Cb

Source:Howe&Sodini

Conversion between resistivity and dopant density of silicon at room temperature

talarico@gonzaga.edu 30

source:Hu

• The thermal motion of an electron or a hole changes direction frequently by scattering off imperfections in the semiconductor crystal

talarico@gonzaga.edu 31

SecondCarriersTransportMechanism:Diffusion

Random thermal motion of an electron or hole in a solid.

source:Streetman

• Inamaterialwheretheconcentrationofparticlesisuniformtherandommotionbalancesoutandnonetmovementresult(drunksail-manwalk

Brownianwalks)

DiffusionCurrent• Ifthereisadifference(gradient)inconcentrationbetween

twopartsofamaterial,statisticallytherewillbemoreparticlescrossingfromthesidewithhigherconcentrationtothesidewithlowerconcentrationthanviceversa

• Thereforeweexpectanetfluxofparticles

talarico@gonzaga.edu 32

qp = −qe ! q

Indiff ∝ Aqe

dndx

= −Aq dndx

I pdiff ∝ Aqp

dpdx

= Aq dpdx

Themorenonuniformistheconcentrationthemoreisthecurrent

Source:Razavi

Electronandholediffusioncurrent• Assumingthechargeconcentrationdecreaseswithincreasing

xitmeansthatdn/dxanddp/dxarenegativequantitiessotoconformwithconventionswehavetoputa– signinfrontoftheproportionalityconstantD

talarico@gonzaga.edu 33

Indiff = −DnAqe

dndx

= DnAqdndx

I pdiff = −DpAqp

dpdx

= −DpAqdpdx

Source:Howe&Sodini

Diffusioncurrentdensities

talarico@gonzaga.edu 34

Source:Howe&Sodini

Jdiff = Jpdiff + Jn

diff

Jndiff = −Dnqe

dndx

= Dnqdndx

Jpdiff = −Dpqp

dpdx

= −Dpqdpdx

Einstein’sRelation• Sincebothμ andD aremanifestationofthermalrandom

motion(i.e.areduetostatisticalthermodynamicsphenomena)theyarenotindependent

talarico@gonzaga.edu 35

Dp

µp

= Dn

µn

= KTq

Einstein’sRelation

K=BoltzmannConstant=1.38×10−23 J/°K=8.62×10−5 eV/°KT=temperaturein°Kq=chargeofproton=1.602×10−19 Cb

VT !

KTq

ThermalVoltage AtroomtemperatureVT≈25.9mV

Totalcurrentdensity

• Theelectronandholetotalcurrentdensityis:

talarico@gonzaga.edu 36

J = Jp + Jn = Jpdrift + Jp

diff + Jndrift + Jn

diff

Jp = Jpdrift + Jp

diff = qpµpE − qDpdpdx

Jn = Jndrift + Jn

diff = qnµnE + qDndndx

J = qpµpE − qDpdpdx

+ qnµnE + qDndndx