ChargedParticles• Theoperationofallelectronicdevicesisbasedoncontrollingtheflowofchargedparticles
• Therearetwotypeofchargeinsolids– Electrons– Holes
• Therearetwomechanismthroughwhichchargecanbetransportedinamaterial– Drift (motionofchargecausedbyanelectricfield)– Diffusion (motionresultingfromanon-uniformchargedistribution)
ElectronicStructureoftheelements
• Atom’schemicalactivitydependsontheelectronsintheoutermostshells(orbits).TheseelectronsarecalledVALENCEelectrons.
• Inextremelypureelements,suchassilicon,theatomsarrangethemselvesinregularpatternscalledCRYSTALS.Thevalenceelectronsdeterminetheexactshape(=LATTICEstructure)
ElectronicorbitalsinsiliconElement AtomicNumber Configuration
Si 14 (1s)2(2s)2(2p)6(3s)2 (3p)2
[email protected] 4Source:Howe&Sodini
s,p,ddesignateorbitalshapesholdsupto2electronspholdsupto6electronsdholdsupto10electrons
SiliconCrystalLattice
Source:Howe&Sodini
concentration(atoms/cm3)ofatomsinsilicon
2D-RepresentationofSiliconCrystal
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
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.
EnergyBandStructure• Bandgapenergy(Eg)istheminimumenergytodislodgeanelectronfrom
itscovalentbond.• ForSiliconatroomtemp.(T=300°K)Eg =1.12eV=1.792× 10−19Joule
[email protected] 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!!!
𝑛" 𝑇 = 300𝐾 ≅ 1×10,-𝑐𝑚01
𝑛"(𝑇) ≅ 𝑛"(𝑇1--)×2505677,- 𝑐𝑚01
Intrinsiccarrierconcentrationasafunctionoftemperature
[email protected] 111000/300≅ 3.33
300°K= 27°Csource:Streetman
Foranintrinsicsemiconductor:n=p=ni
1.05☓1010
Intrinsiccarrierconcentration
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
(eV)
ExtrinsicSemiconductors(1)• Dopingwithdonorimpurities(N-typesemiconductor)
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)
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
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)
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
Mobilityoffreeelectronsandholes
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
ni2 = n ⋅ p
Dopingwithdonors(n-type)
• Chargeneutrality:
• Usingmass-actionlaw:
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
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
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)
vpdrift! "!!
= µp E→
vndrift! "!!
= −µn E→
Saturationofthedriftvelocity
• Eventuallythedriftvelocitysaturates:therearetoomanycollisionsamong`carriersandbetweencarriersandlattice
• vsat forsiliconis≈107cm/s=105 m/s
vdrift
sourceGray&Meyer:
𝑣FG"HI 𝐸 ≃𝜇𝐸
1 + 𝐸𝐸9
=𝜇𝐸
1 + 𝜇𝐸𝑣NOI
source:Razavi
Driftcurrentandcurrentdensity
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
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
source:Hu
• The thermal motion of an electron or a hole changes direction frequently by scattering off imperfections in the semiconductor crystal
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
qp = −qe ! q
Indiff ∝ Aqe
dndx
= −Aq dndx
I pdiff ∝ Aqp
dpdx
= Aq dpdx
Themorenonuniformistheconcentrationthemoreisthecurrent
Source:Razavi
Electronandholediffusioncurrent• Assumingthechargeconcentrationdecreaseswithincreasing
xitmeansthatdn/dxanddp/dxarenegativequantitiessotoconformwithconventionswehavetoputa– signinfrontoftheproportionalityconstantD
Indiff = −DnAqe
dndx
= DnAqdndx
I pdiff = −DpAqp
dpdx
= −DpAqdpdx
Source:Howe&Sodini
Diffusioncurrentdensities
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
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:
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