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PNJunction(Diode)
• WhenN-typeandP-typedopantsareintroducedside-by-sideinasemiconductor,aPNjunctionoradiodeisformed.
source:RazaviJUNCTION
DiodeI/Vcharacteristics• Ourgoalistofindoutthecurrent-voltagecharacteristicsofthe
device
• Diodeoperationregions:– Equilibrium=opencircuitterminals=nobias– Forwardbias– Reversebias– (Breakdown)
D
DEquilibrium
D
D
“smoke”
ΦB
PNJunctioninEquilibrium(noBiasapplied)
• Becauseeachsideofthejunctioncontainsanexcessofholesorelectronscomparedtotheotherside,thereexistsalargeconcentrationgradient.Therefore,adiffusioncurrent flowsacrossthejunctionfromeachside.
[email protected] 4source:Razavi
nn0
pn0 np0
pp0+ND < NA
nn0≈ND
pn0≈ni2/ND
pp0≈NA
np0≈ni2/NA
PNjunctioninEquilibrium(nobiasapplied)• Diffusionacrossthejunctioncausesalotofelectronsandholestorecombineleavinga
depletionregioninproximityofthejunction• Thefixedionsleftbehindcreateanelectricfield• TheelectricfieldopposesthediffusionofholesintheNregionandfreeelectronsintheP
region• Oncetheelectricfieldisstrongenoughtostopthediffusioncurrentsthejunctionreaches
equilibrium– A“barrier”opposingdiffusionforms
5
−−−−−−−−−−−−−−−−−−
++++++++++++++++++++++++++++++++++++
−−
−
−
++
++
FreeElectrons
PositiveDonorIons
NegativeAcceptorIons
FreeHoles
DepletionRegion
PNND <NA
Depletionapproximation• WhathappensinsidethePNjunctioninequilibriumisnotquiteasblackandwhiteasdepicted
ρ(x) = q p0 (x)− n0 (x)+ ND − NA( )
• OntheN-sideofthetransitionregion(0<x<xn0),sincetherearenoacceptors(NA =0)andtheholeconcentrationisnegligible(p0 ≈0)
• Sincen0(x)<ND ontheN-sideofthetransition(andevenmoresoontheP-sideofthetransition)thereisaroll-offinelectronconcentration
ρ(x) = q p0 (x)− n0 (x)+ ND − NA( ) ≈ q −n0 (x)+ ND( )
source:Howe&Sodini
Depletionapproximation
• Thedepletionapproximation assumesthatthemobilecarriers(nandp)aresmallinnumbercomparedtothedonorandacceptorionconcentrations(ND andNA)inthedepletionregion(SCR),andthedeviceischarge-neutralelsewhere(QNR)
• Depletionapproximation
– NA >>pp henceρ ≈−qNA for−xp ≤x≤0– ND >>nn henceρ ≈qND for0≤x≤xn– Thechargedensityisnegligibleinthebulkregions;thatisρ ≈0forx>xn andx<−xp
• OntheP-sideofthetransitionregion(−xp <x<0),sincetherearenodonors(ND =0)andtheelectronsconcentrationisnegligible(n0 ≈0)
• Sincep0(x)<NA ontheP-sideofthetransitionregion(andevenmoresoontheN-sideofthetransition)thereisaroll-offinholeconcentration
ρ(x) = q p0 (x)− n0 (x)+ ND − NA( ) ≈ q p0 (x)− NA( )
source:Howe&Sodini
Thatis,therearenofreemobilechargesindepletionregion
Onemoretime…
• aa
source:Sodini
(P-side)
(N-side)=0
=0 ≈0(minoritycarriers)
≈0(minoritycarriers)
Na >>p0(depletionapproximation)
Nd >>n0(depletionapproximation)
P-type N-type
NOTE:fortheexampleinthispictureNA<ND
Depletionapproximation
Depletionapproximationmeansweassumethechargedensityineachtransitionregiontobeuniform
Chargedensity
x
qND
−qNA
−
+
source:G.W.Neudeck
PNjunctioninEquilibrium(nobiasapplied)
• Sincethecircuitisopennocurrentsexists• Thereforeinequilibriumthedriftcurrents
resultingfromtheelectricfieldgeneratedbythefixedionsinthedepletionregionmustexactlycancelthediffusioncurrents
• Thismustholdforbothtypesofchargecarriers:Jn =Jp =0
• Let’stakeoneofthetwoequationsandtrytosolveit:
10
−−−−−−−−−−−−−−−−−−
++++++++++++++++++++++++++++++++++++
−−
−
−
++
++
FreeElectrons
PositiveDonorIons
NegativeAcceptorIons
FreeHoles
DepletionRegion
PN
ND <NA
Jp = 0 = pqµpEx − qDpdpdx
Jn = 0 = nqµnEx + qDndndx
0
nn0pn0
pp0
np0
JUNCTIONx
xp0−xn0−pµp
dVdx
= Dpdpdx
pqµpEx = qDpdpdx
Ex = − dVdx
Built-involtage(equilibrium)
−pµpdVdx
= Dpdpdx
− dVV (−xn0)
V (xp0)∫ =
Dp
µp
dppp(−xn0)
p(xp0)∫
V (−xn0 )−V (xp0 ) =VT lnpp0pn0
ΦB =VT lnNA
ni2 / ND
!
ΦB =VT lnNAND
ni2
nn0pn0
pp0
np0
xxp0−xn0
PN
++−−
Ex
0
ΦB+ −
++ −−
ExampleFindbuiltinvoltageofasiliconPNjunctionatroomtemperatureandwithNA ≈1018 andND ≈1017
ΦB ≅ 26mV × ln10181017
1020≅ 26mV × ln(10)× Log10
10181017
1020≅
≅ 60mV ×15≅ 900mV
depletionregion
PNJunctioninequilibrium(nobiasapplied)
−−−−−−−−−−−−−−−−−−
++++++++++++++++++++++++++++++++++++
−−−−
++
++
DepletionRegion
PN
+
−
• SomeofthethermallygeneratedholesintheNregion(minoritycarrier)movestowardthejunctionandreachtheedgeofthedepletionregion.There,theyexperiencethestrongelectricforcepresentindepletionregion,andgetsweptallthewayintothePregion(wheretheybecomemajoritycarriers).
• Thesamehappenforthethermallygeneratedminoritycarriers (freeelectrons)inthePregion.
• Atequilibriumtheflowofcarrierssweptbytheelectricfield(drift)andtheflowofcarriersduethegradientincarriersconcentration(diffusion)cancelout
ID =Idiff – Idrift =0
Ex
IdriftIdiff+ −ΦB
PN ID
−−−−−−
+++++++++
VD
WidthoftheDepletionregioninequilibrium
13
−−−−−−−−−−−−
++++++++++++++++++++++++
−−−−
++
++
PN
−xn0 +xp00E
metalcontact
AK
metalcontact
−xn0 +xp0x
concentration[cm−3]pp0=NA
np0≈ni2/NA
nn0=ND
pn0≈ni2/ND
W=Xd0
IdriftIdiff
−xn0 x+xp0
ρ =chargedensity[Cb/cm−3]
−qNA
+qND
|Q−|=qNAAxp0|Q+|=qNDAxn0
WidthoftheDepletionregioninequilibrium
−xn0 x+xp0
ρ =chargedensity[Cb/cm−3]
−qNA
+qND
|Q−|=qNAAxp0|Q+|=qNDAxn0
Chargeconservation:
|Q+ |= |Q− | !QJ 0
"AqNDxn0 = AqNAxp0"
NDxn0 = NAxp0E(x)
x−xn0 +xp0
E(0) =
−qNAxp0ε si
= −qNDxn0ε si
! Emax
PoissonEquation:dEdx
= ρε si
!
dE = ρε sidx Aslongasweknowhowtocompute
theareaofarectanglewearefinewiththemath!!!Permittivityofcrystallinesilicon:
εsi ≈11.7×8.85×10−12F/mεr ε0
• Let’sgofromtheelectricfieldtothevoltage
WidthoftheDepletionregioninequilibrium
−E(x)
x−xp0 +xn0
qNAxp0ε si
= qNDxn0ε si
E(x) = − dΦ0
dx⇔ − E(x) ⋅dx = dΦ0
Φ0(x)
xArea“red”triangle
12qNDxn0
2
ε siArea“blue”triangle=
Area“blue”triangle
12qNAxp0
2
ε siArea“red”triangle=
0
Aswemoveinthexdirection(from–xp0 toxn)theareaunder–E(x)keepsgrowing
NOTE: forconvenienceIflippedtheorientationofthep-sideandthen-sidewithrespecttothexdirectionsothenameofthedepletionregion’sedgeatthep-sidebecomes–xp0 andthenameofthedepletionregion’sedgeatthen-sidebecomes+xn0
ΦB=Φ0 (xn0)−Φ0(−xp0)
ΦB =12qNAxp0
2
ε si+ 12qNDxn0
2
ε si
ΦB+−
E
− ++
parabolaN
parabolaP
source:Howe&Sodini
−
Metal-SemiconductorContactPotential
• Atfirstglancewouldseemstrangethatthepotentialdropsatthecontactswouldhavetoadduptothebuiltinpotentialofthejunction.Werethisisnotthecase,KVLwouldimplythatcurrentwouldflowinthermalequilibriumwhenaresistorisconnectedacrossthemetalcontactstothepandnregion.Thisconclusionviolateourbasicunderstandingofthemeaningofequilibrium:namelythenetcurrentmustbezeroinequilibrium
source:Howe&Sodini
WidthoftheDepletionregioninequilibrium
W = xp0 + xn0W = xn0 1+
ND
NA
⎛⎝⎜
⎞⎠⎟= xn0
NA + ND
NA
⎛⎝⎜
⎞⎠⎟
W = xp0 1+NA
ND
⎛⎝⎜
⎞⎠⎟= xp0
NA + ND
ND
⎛⎝⎜
⎞⎠⎟NAxp0 = NDxn0
Chargeneutrality:
xn0 =WNA
NA + ND
⎛⎝⎜
⎞⎠⎟
xp0 =WND
NA + ND
⎛⎝⎜
⎞⎠⎟
ΦB =qNA
2ε sixp0
2 + qND
2ε sixn0
2 ΦB =qNA
2ε siW 2 ND
NA + ND
⎛⎝⎜
⎞⎠⎟
2
+ qND
2ε siW 2 NA
NA + ND
⎛⎝⎜
⎞⎠⎟
2
ΦB =q2ε si
W 2 NAND2 + NDNA
2
NA + ND( )2W 2 = 2ε si
qΦB
NA + ND( )2NAND
2 + NDNA2 =
2ε siq
ΦBNA + ND( )2
NAND (ND + NA )
W = 2ε si
qΦB
NA + ND
NAND
! Xdo xn0 =2ε siq
ΦB
NA + ND( )NAND
NA2
(NA + ND )2 = 2ε si
qΦB
NA
ND (NA + ND )
xp0 =2ε siq
ΦBND
NA(NA + ND )
Chargeoneithersideofthedepletionregion(inequilibrium)
QJ 0 ! |Q+ |= |Q− |= AqNDxn0
xn0 x−xp0
ρ =chargedensity[Cb/cm−3]
−qNA
+qND
|Q−|=qNAAxp0
|Q+|=qNDAxn0QJ 0 = AqNDW
NA
NA + ND
= Aq NAND
NA + ND
W
QJ 0 = AqNAND
NA + ND
2ε siq
ΦBNA + ND
NAND
= A 2ε siqNAND
NA + ND
ΦB
JunctionCharge
xn0 =WNA
NA + ND
⎛⎝⎜
⎞⎠⎟
xp0 =WND
NA + ND
⎛⎝⎜
⎞⎠⎟
=xn0
=W=Xd0
EffectofbiasvoltageonthePNJunction
source:Streetman
ΦB
Φp
Φn ΦJ=ΦB−VF
ΦJ=ΦB+VR
VF=VD VR=−VD
+VD− +VD−
+VD=0−
IdiffIdrift
IdiffIdrift Idrift
Idiff
ID =0 ID>0 ID<0
ID =Idiff −Idrift
Thedriftcurrenthasthesamedirection oftheelectricfield(anditisproportionaltoit)
(a) Opencircuit(b)ForwardBias(c)ReverseBias(Equilibrium)
Jp,drift = pqµpEJn,drift = nqµnE
ΦB ΦJ ΦJ
PNJunctionunderReverseBias
• TheexternalvoltageVR isdirectedfromtheNsidetothePside(sameasΦB),thereforetheelectricfield(ER)duetothebiasVR enhancesthebuiltinfieldE.ThedepletionregionWiswidened
• Sincethe“barrier”potentialriseshigherthaninequilibriumitgetsharderforthemajoritycarrierstocrossthejunctionandeasierfortheminoritycarriertobeswept(drifted)acrossthejunction.
• Thecurrentcarriedunderreversebiasisnegligibletheminoritycarriersaretoofew!!
VR=−VD VR=−VD
ID
−ΦB + −ΦB +
source:Millman &Halkias
source:Razavi
Equilibrium(opencircuit)
ReverseBias
+ ΦB −E
E
ER
+ ΦJ − ΦJ =ΦB +VR
PNjunctionunderreversebias
KVL:VD +Φmn +ΦJ +Φmn =0
ΦJ=−VD −Φmn −Φmp =ΦB−VD
=ΦBpotentialbarrierinequilibrium
potentialbarrierinnon-equilibrium
source:Howe&Sodini
PNjunctionunderreversebias
22
Example:
PNjunctionunderreversebiasesVD=−2.4VandVD=−6.4V
AssumingthepotentialbarrierinequilibriumisΦB =0.8V
AsthePNjunctiongetsmorereversebiased• thedepletionregion Xd =xp +xn becomes
wider• thepotentialbarrier becomesstronger
(a)chargedensity(b)electricfield(c)potential
source:Howe&Sodini
ApplicationofReverseBiasedPNJunction:VoltageDependentcapacitor
• AreversebiasedPNjunctioncanbeusedasacapacitor.ByvaryingVR,thedepletionwidthchanges,andsodoesthecapacitanceassociated.ThereforeareversebiasedPNjunctioncanbeseenasavoltage-dependentcapacitor.
Varactor
p
n
Qualitativelywe seethatifVR goesupthenCJ goesdownsource:Razavi
CapacitanceofReverseBiasedDiode
Xd @VD= 2ε si
qΦB −VD( )NA + ND
NAND
= 2ε siq
ΦB 1−VDΦB
⎛⎝⎜
⎞⎠⎟NA + ND
NAND
= Xd0 1−VDΦB
CJ @VD= ε si ⋅AXd @VD
= qε si2
NAND
NA + ND( )1
ΦB −VD( ) =qε si2
NAND
NA + ND( )1ΦB
11−VD /ΦB( ) =
= CJ01−VD /ΦB
widthofdepletionregioninequilibrium
CJ0=junctioncapacitanceatzerobias(inequilibrium)
= CJ0 = ε siAXdo
careful:capperunitarea
Junctioncapacitanceofreversebiaseddiode
CJ0
VR=−VD VD=VF
ReverseVoltage ForwardVoltage
CJsource:StreetmanCJ (VD ) =
CJ0
1− VDΦB
⎛⎝⎜
⎞⎠⎟
M = CJ0
1+ VRΦB
⎛⎝⎜
⎞⎠⎟
M
• M=0.5fortheabruptPNjunctionwemodeled
• TypicalvaluesforMrangefrom0.3to0.5
• SPICEusesVJ(orPB)todenotethebuiltinvoltageinequilibrium(ΦB)
Thejunctioncapacitanceisnonlinear
QJ @VD= A 2ε siq
NAND
NA + ND
ΦB −VD( ) = A 2ε siqNAND
NA + ND
ΦB 1−VDΦB
⎛⎝⎜
⎞⎠⎟=QJ 0 1−
VDΦB
=
=QJ 0 1+VRΦB
chargestoredineithersideofthedepletionregioninequilibrium
Junctioncapacitanceofreversebiaseddiode
• TherelationbetweenchargeQJ andvoltage(VR=−VD) acrossthediodeisnotlinear• ItisdifficulttodefineacapacitancethatchangeswheneverVR changes
source:Sedra &Smith
P
VP
QJ = A 2ε siqNAND
NA + ND
ΦB +VR( ) = ς ΦB +VR
= ς
2 ΦB 1+VRΦB
⎛⎝⎜
⎞⎠⎟
= CJO
1+ VRΦB
CJ @VP= dQJ
dVR VR=VP
= ddVR
ς ΦB +VR( )@VP
= ς2 ΦB +VR @VP
=
=ς
@VP @VP
squarerootlaw
𝐶"|@%&=1𝐴𝑑𝑄"𝑑𝑉-
./01/&
/A
=23
/A
/A
Junctioncapacitanceofreversebiaseddiode
source:Sedra &Smith
P
VP
CJ @VP= dQJ
dVR VR=VP
≠ QJ (VR )VR VR=VP
Importantresulttorememberwhenusingnonlineardevices
/A
PNJunctioninForwardBias
ID
VD=VF VD=VF
−ΦB + −ΦB +
source:Millman &Halkias
• Theelectricfield(EF)duetothebias(VF)isoppositetothebuiltinfieldE.ThedepletionregionWisshortened
• Sincethe“barrier”potentialbecomeslowerthaninequilibriumitgetseasierforthemajoritycarrierstodiffusethroughthejunctionandharderfortheminoritycarriertobeswept(drifted)acrossthejunction.
• Themajoritycarriersarealot,soweexpectaconsiderablecurrentflow:ID =Idiff −Idrift
Equilibrium(opencircuit)
+ ΦB −
+ ΦJ −ForwardBias
EFE
ΦJ =ΦB −VF
source:Razavi
p-nID
Diode’sCurrent/VoltageCharacteristic
ID = IS (eVDVT −1)
≈−IS
NOTE:Consider|VD/VT|≈2.3exp(2.3)≈9.97≈10
exp(−2.3)≈0.1
AtroomtemperatureVD =2.3× VT≈59.8mV≈60mV
source:Razavi
source:Streetman
−I(drift)
ID
VD
VD
ID
ID = I(drift) eqVDKT −1
⎛
⎝⎜
⎞
⎠⎟
verysteep!
Diode’sCurrent/VoltageCharacteristic
31
• HowcanwequantifytheI/Vbehaviorofthediode?• Weneedtolookathowelectronsandholesmoveundertheeffectofthebias
voltage• Inequilibrium:
• WithanexternalbiasvoltageappliedthesituationissimilarbutnowthepotentialbarrierisΦB – VD insteadofonlyΦB
ΦB =VT lnpp0pn0
⇔ pn0 = pp0 ⋅e−ΦB /VT ⇔ pn0 =
pp0eΦB /VT
holesatxn0holesat−xp0
−xp0 andxn0aretheedgesofthedepletionregioninequilibrium
pn = pp ⋅eVD /VT
eΦB /VT
holesatxnholesat−xp
−xp andxn aretheedgesofthedepletionregionwithnonzerobias
x
−Wp −xp 0 xn Wn
WP WN
P-side N-side
+ +−
−VD GNDA K
quasi-neutralregion
quasi-neutralregion
LawoftheJunction
np = nn ⋅eVD /VT
eΦB /VT
electrons at−xp
electronsatxnTheLawoftheJunctionrelatestheminoritycarrierconcentrationononesideofthejunctiontothemajoritycarrierconcentrationontheothersideofthejunction
np0 =nn0eΦB /VT
electronsat−xp0
electronsatxn0
Diode’sCurrent/VoltageCharacteristic
• WhenweapplyaforwardbiasVDthepotentialbarrierisreducedfromΦB toΦB−VD.Thisreducesthedriftfieldandupsetsthebalancebetweendiffusionanddriftthatexistsatzerobias.
• ElectronscannowdiffusefromtheNside(wheretheyaremajoritycarriers)tothePside(wheretheybecomeminoritycarriers).Similarly,holesdiffusefromthePsideintotheNside.Thismigrationofcarriersfromonesidetotheotheriscalledinjection.
• Asaresultofinjection,moreelectronsarenowpresentat–xp andmoreholesappearatxn thanwhenthebarrierwashigher
– Thereisanexcessofminoritycarriersattheedgesofthedepletionregion
• Let’smakesomesimplifyingassumptionstohelpquantifyingpn(xn),pp(−xp),nn(xn)andnp(xp)
– Thereiselectricfieldonlyinthespacechargeregion(thatisbetween–xp andxn).Theregionsfrom–Wp to–xp (a.k.a.Pbulkregion)andfromxn toWn (a.k.a.Nbulkregion)arequasi-neutral(itisliketheywereperfectconductorsandinperfectconductorsthereisnoelectricfieldinside)
Diode’sCurrent/VoltageCharacteristic
• Therequirementthatthebulkregionsoneithersideremaincharge-neutralevenunderappliedbias(remember:E(x)=ρ(x)/ε,sonofieldmeansnochargedensity)impliesthatanyincreasesintheminoritycarriersduetoinjectionmustbecounterbalancedbyinanincreaseofmajoritycarriers
– OnthePsideρ(x=−xp)=0impliesthat:
– OntheNsideρ(x=xn)=0impliesthat:
• Assumingthelevelofinjectionislowwecanneglecttheslightincreaseinmajoritycarrierconcentrationduetocarrierinjectionandwrite:
pp (−xp ) = NA + np (−xp ) ≈ pp0 + np (−xp )
nn (xn ) = ND + pn (xn ) ≈ nn0 + pn (xn )
pp (−xp ) ≈ pp0 ≈ NA nn (xn ) ≈ nn0 ≈ ND
np (−xp ) << pp0 pn (xn ) << nn0
MinorityandMajoritycarriersconcentrationsinaforwardbiaseddiode
• ConcentrationsHowe&Sodini
xpn0np0
Quasi-neutralitydemandsthatateverypointinthequasi-neutralregions:excessminoritycarrierconcentration=excessmajoritycarriersconcentration
source:Howe
&Sodini
MinorityandMajoritycarriersconcentrationsinaforwardbiaseddiode
Quasi-neutralitydemandsthatateverypointinthequasi-neutralregions:excessminoritycarrierconcentration=excessmajoritycarriersconcentration
NOTE:atthispointwedidnotputmuchthinkingintheprofileoftheminoritycarriersdiffusingintheQNRs,howeverifweassumethediffusionoccursoverashortdistancewecanapproximatethecurveswithstraightlines!
0−xp−Wp
p(−xp)
pp-QNR
Na
qPp
qNp
n(−xp)ni2
Na
x
source:Sodini
n
• UnderthelowinjectionconditionwecanapproximatetheLawoftheJunctionasfollows:
pn (xn ) =pp (−xp )eΦB /VT
⋅eVD /VT ≈pp0eΦB /VT
⋅eVD /VT = pn0 ⋅eVD /VT
np (−xp ) =nn (xn )eΦB /VT
⋅eVD /VT ≈ nn0eΦB /VT
⋅eVD /VT = np0 ⋅eVD /VT
pn (xn ) ≈ pn0 ⋅eVD /VT ≈ ni
2
ND
⋅eVD /VT
np (−xp ) ≈ np0 ⋅eVD /VT ≈ ni
2
NA
⋅eVD /VT
Theconcentrationofminoritycarriersattheedgesofthespacechargeregionareanexponentialfunctionoftheappliedbias
Diode’sCurrent/VoltageCharacteristic
source:G.W.Neudeck
MinoritycarriersinaforwardbiasedPNjunction
• Theexcessconcentrationofminoritycarriersattheedgesis:
• Astheinjectedcarriersdiffusethroughthequasi-neutralregionstheyrecombinewiththemajoritycarriersandeventuallydisappear.Theexcessminoritycarriersdecayexponentiallywithdistance,andthedecayconstantiscalleddiffusionlengthL.
Δnp ! np0 ⋅e
VD /VT − np0 = np0 1− eVD /VT( )
Δpn ! pn0 ⋅eVD /VT − pn0 = pn0 1− e
VD /VT( )
np (x) = Δnpe(x+xp )Ln
pn (x) = Δpne−(x−xn )
Lp
source:Sedra &Smith p+ n
Itcanbeshownthroughthecurrentcontinuityequationthatunderthelowinjectionconditiontheconcentration’sprofileoftheminoritycarriersdiffusinginthequasineutralregionsisexponential
Aside(asimpletricktosimplifythemath)
• Changetheaxisselectionforthebulkregionsasfollows:
• Thiswaywecanwritetheminoritycarriersconcentrationsasfollows:np(x '') = Δnpe
−x ''Ln
pn (x ') = Δpne−x 'Lp
x '' ! 0'' ≡ −xpx ' ! 0' ≡ xn
Ln = Dnτ nLp = Dpτ p
source:G.W.Neudeck
source:G.W.Neudeck
Ln =Diffusionlengthforelectrons[m]Lp =Diffusionlengthforholes[m]Dn =Diffusioncoefficientforelectron[m2/s]Dp =Diffusioncoefficientforholes[m2/s]τn =meanlifetimeforelectrons[s]τp =meanlifetimeforholes[s]
VA = voltage applied ≡VD
PhysicalinterpretationofL,D,andτ
• PhysicallyLp andLn representstheaveragedistanceminoritycarrierscandiffuseintoa“sea”ofmajoritycarriersbeforebeingannihilated
• Physicallyτp andτn representstheaveragetimeminoritycarrierscandiffuseintoa“sea”ofmajoritycarriersbeforebeingannihilated
• PhysicallytheDp andDn representstheeasewithwhichholesandelectronsdiffuse
Lp !xΔpn (x)
0
∞
∫ dx
Δpn (x)0
∞
∫ dx
Ln !xΔnp (x)
0
∞
∫ dx
Δnp (x)0
∞
∫ dx
Dp ! µp
KTq
Dn ! µnKTq
PNJunctioncurrent• Thecurrentthatflowsthroughany“transverse”sectionxmustbethe
sameandisformedbybothholesandelectrons
• Ifwelookatthecurrentthroughasectionxinthequasi-neutralregions,sinceinthequasi-neutralregionsthereisnoelectricfieldtherewillbenodrift
J(x) = Jn (x)+ Jp (x)Jn (x) = Jn,drift (x)+ Jn,diff (x)Jp (x) = Jp,drift (x)+ Jp,diff (x)
Jn (x) = Jn,diff (x) = qDn
dnp (x)dx
= q Dn
Lnnp0 e
VD /VT −1( )e(x+xp )/Lp for −Wp ≤ x ≤ −xp
J p (x) = Jp,diff (x) = −qDpdpn (x)dx
= qDp
Lp
pn0 eVD /VT −1( )e−(x−xn )/Lp for xn ≤ x ≤Wn
injectedelectronsdiffusingonP-side(minoritycarriers)
injectedholesdiffusingonN-side (minoritycarriers)
+(x+xp)/Ln
PNJunctioncurrents• Theeasiersectionsforevaluatingthediffusioncurrentsaretheedgesofthespace
chargeregion(x=−xp andx=xn)
• Thetotalcurrentthroughasectionofthediodecanbefoundasfollows:
• Ifwemakethesimplifyingassumptionthatthereisnorecombination-generationinthespace-chargeregion,thismeansthattheelectrons/holescurrentatx=−xp andatx=xn donotchangeinthespace-chargeregionandthereforewecanwrite:
Jn (−xp ) = Jn,diff (−xp ) = qDn
Lnnp0 e
VD /VT −1( )
Jp (xn ) = Jp,diff (xn ) = qDp
Lp
pn0 eVD /VT −1( )
J = Jn (−xp )+ Jp (−xp ) = Jn (xn )+ Jp (xn )
Jn (−xp ) = Jn (xn )Jp (xn ) = Jp (−xp )
Wedonotknowhavethisvalue(yet)!
Wedonotknowhavethisvalue(yet)!
PNJunctioncurrent• Thereforewecanaddtogetherthediffusioncurrentsatx=−xp andx=xn
43
J = Jn (−xp )+ Jp (xn ) = qDn
Lnnp0 e
VD /VT −1( ) + q Dp
Lp
pn0 eVD /VT −1( )
J = q Dn
Lnnp0 + q
Dp
Lp
pn0⎛
⎝⎜⎞
⎠⎟eVD /VT −1( )
ID = J × A = In,diff + I p,diff = Aq Dn
Lnnp0 + Aq
Dp
Lp
pn0⎛
⎝⎜⎞
⎠⎟eVD /VT −1( )
ID = Aq Dn
Lnni2
NA
+Dp
Lp
ni2
ND
⎛
⎝⎜⎞
⎠⎟eVD /VT −1( ) = IS eVD /VT −1( )
scalecurrent(a.k.a.saturationcurrent)
Minorityconcentrationsforaforwardbiasedshortbasediode
• Ifthediodeisshorttheexponentialdecayscanbeapproximatedwithstraightlines
−xp 0xn
np0
pn0
−Wp Wn
VDA GNDK
WidthNWidthP
x
ID = AqDn
WidthPnp0 +
Dp
WidthNpn0
⎛
⎝⎜
⎞
⎠⎟ eVD /VT −1( ) = IS eVD /VT −1( )
dnpdx
≈np(−xp )− n(−Wp )
WidthP=np0e
VD /VT − np0WidthP
dpndx
≈pn (xn )− pn (Wn )
−WidthN= −
pn0eVD /VT − pn0WidthN
Currentdensitiesintheforwardbiascase
source:G.W.Neudeck
NomatterwhichsectionxweconsiderJpandJn mustaddtothesametotalvalueJ
AssumptionsusedtoderivetheIdealdiodeequation
• Depletionapproximation(indepletionregion:NA >>pp0(x),ND >>nn0(x))• Negligiblefieldsinthequasineutralregions(QNR).Thismeansthatthe
appliedvoltageVD isdroppedentirelyacrossthespacechargeregion(SCR)
• Quasi-neutralregionsthataresolongthattheinjectedcarriersallrecombinebeforereachingtheendcontacts
• Lowlevelsofinjection• Norecombination-generationinthespace-chargeregion.Thismeans
thattheelectroncurrentandtheholecurrentareconstantinthespace-chargeregion(thatisJn(−xp)=Jn(xn)andJp(xn)=Jp(−xp))
• Realdiodesmaydeviateconsiderablyfromtheidealequation.
46ID = I*S (e
VDnVT −1)
n =emissioncoefficient(a.k.a.idealityfactor)Typicallytheidealityfactornvariesfrom1to2
Idealdiode
ID
ISVD VD
log10|ID|
IS
y = log10 IS ⋅eVDVT
⎛
⎝⎜⎜
⎞
⎠⎟⎟= log10 IS( )+ log10 (e) ⋅
VDVT
≈ log10 IS( )+ 0.43VDVT
x =VD
dydx
≈0.43VT
at room temperature (300K ) :VT0.43
≈26mV0.43
≈60mV10
source:Sodini
every60mVthecurrentchangesbyadecade
Idealdiodeequation• Althoughwederivedtheidealdiodeequationbysubjectingthediodetoa
forwardbias,thestepsfollowedcanalsobeappliedforthecaseofreversebias(theonlydifferenceisthatnowVD <0).
• Theequationworksforbothforwardandreverseregion.
ID
VD
ID = IS eVD /VT −1( )
source:Streetman ≈ ISeVD /VT
≈ −IS
Minorityconcentrationsforreversebiaseddiode
• Inforwardbias,minoritycarriersareinjectedintoquasi-neutralregions• Inreversebias,minoritycarriersareextracted(sweptaway)fromthequasi-
neutral regions• Inreversebiasthereisaverysmallamountofcarriersavailableforextraction
– ID saturatestosmallvalue
Minority-carrierdensitydistributionasafunctionofthedistancefromthejunction.(a)Forwardbiasedjunction(b)Reversebiasedjunction(Thedepletionregionisassumedsosmallrelativetothediffusionlengththatisnotindicatedinthefigure)
source:Millman &Halkias
Small-signalequivalentmodelfordiode
• Examinetheeffectofasmallsignaloverlappingthebias:
[email protected] 52Figure 4.14 Development of the diode small-signal model.source:Sedra &Smith
vD (t) ≡VD + vd (t)
iD = IS evDVT −1
⎛
⎝⎜⎜
⎞
⎠⎟⎟= IS e
VD+vdVT −1
⎛
⎝⎜⎜
⎞
⎠⎟⎟
iD ≅ iD VD+diDdvD VD
(vD −VD ) = ID +diDdvD VD
vd
iD isafunctionofvD:
Q ≡ (ID,VD )
diDdvD VD
=ddvD
IS evDVT −1
⎛
⎝⎜⎜
⎞
⎠⎟⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥VD
=ISe
VDVT
VT=ISe
VDVT + IS − ISVT
=
=ID + ISVT
iD − ID ≅ gdvd ⇔ id ≅ gdvd
≡ gd
*AppliedTaylor
*
Small-signalequivalentmodelfordiode
• Fromasmallsignalpointofviewthediodebehavesasa(nonlinear)resistanceofvalue:
• rdiode dependsonbias:
• Inforwardbias:Inreversebias:
• Undertheassumptionthatthesignaloverlappingthebiasissmallthei-vrelationshipofthediodecanbelinearized:
[email protected] 53Figure 4.14 Development of the diode small-signal model.source:Sedra &Smith
rdiode =1gd=
VTID + IS
iD ≅ ID + gdvd = ID + idvD =VD + vd
gd =ID + ISVT
gd ≈IDVT
gd ≈ 0
rdiode =1/gd
Small-signalequivalentmodelfordiode• Besideshavingcurrentflowingthroughit(resistivebehavior)thediode
alsoexhibitchargedisplacementmechanismsthatcanbemodeledasacapacitivebehavior
• Thechargeresponsetothevoltageappliedacrossthediodeisnon-linear
• Aslongasthevoltagesignalsuperimposedtothebiasvoltageissmall,therelationshipbetweenchargeandvoltagecanbelinearized.
• Completesmallsignalmodelfordiode
vD =VD + vd ≡Cdiode
qD ≈QD VD+dqDdvD VD
(vD −VD ) =QD VD+Cdiode ⋅ vd
rdiode Cdiode
CapacitiveeffectsinthePNjunction• Diodeshavetwocapacitiveeffects
• But…pleasecareful:therearewaytoomanyletters“D”:
– depletioncapacitance=junctioncapacitance=CJ– diffusioncapacitance=storagecapacitance=CS– Cdiode =CJ+CS
• Wealreadyknowquiteabitaboutthejunctioncap.thatdevelopswhenthediodeisinreversebias(…butwhataboutforwardbias?)
• Sofarwedidnoteventhinkabouttheexistenceofstorage(diffusion)capacitance
• Weneedtodigmoreintothecapacitiveeffects…
Cdiode
A
K
CJ
A
K
CS
JunctionCapacitance
[email protected] 56BehaviorofjunctioncapacitanceasafunctionofbiasvoltageVD
(Chawla-Gummel)
≡ΦB≡ΦB
2
SPICE
≈ 2×CJ 0
ForHandCalculation
source:Gray,Hurst,LewisandMeyer
JunctionCapacitance• Earlier,wederivedthecapacitanceassociatedwithdepletionregion
(junctioncapacitance)forareversebiased diodeas:
• UndersignificantforwardbiasVD >ΦB/2theapproximationthattherearenomobilecarriersinthedepletionregionbecomesinvalid(thiswasthebasicassumptionbehindthederivation!!)
• Therefore,underforwardbias acommonapproximationistotakethevalueofCJ atVD =ΦB/2(orbeingslightlymorepessimistic)
Cj VD=
CJO
1− VDΦB
CJO = zero-biasjunctioncapacitance = A εsiq2⋅NAND
NA + ND
⋅1ΦB
Cj VD=ΦB2=
CJO1−1/ 2
= 2 ×CJO ≈ 2×CJO
Diffusioncapacitance• Thediffusioncapacitanceisduetochargecarrierstorage
occurringinQNRs(tomaintainquasi-neutrality)
0−xp−Wp
p(−xp)
pp-QNR
Na
qPp
qNp
n(−xp)ni2
Na
x
Majority(+)
Minority(−)
Majority(−)
Minority(+)
qPn =qA2
Wn − xn( ) ni2
Nd
eVDVT −
ni2
Nd
⎛
⎝⎜⎜
⎞
⎠⎟⎟= −qNn
n(−xp ) =ni2
Na
eVDVT
WhathappentothemajoritycarriersintheQNRs?
minoritycarrierschargeinQNRs?
p(xn ) =ni2
Nd
eVDVT
Let’scomputechargeinminoritycarriers“triangles”:
qNp = −qA2
Wp − xp( ) ni2
Na
eVDVT −
ni2
Na
⎛
⎝⎜⎜
⎞
⎠⎟⎟= −qPp
Quasi-neutralitydemandsthatateverypointinthequasi-neutralregions:excessminoritycarrierconcentration=excessmajoritycarriersconcentration
n
Diffusioncapacitance• Let’sexaminewhathappenwhenweapplyasmallincreasein
voltage(vD=VD+vd=VD+ΔvD)
SmallincreaseinVD
SmallincreaseinqPn
SmallincreaseinqNn
Behavesasacapacitorofcapacitance:
n(xn)[forVD+vd]n(xn)[forVD]
n
pp(xn)[forVD+vd]
p(xn)[forVD]Cdn =
dqPndvD vD=VD
=qA2
Wn − xn( ) ni2
Nd
1VTeVDVT
• Ifwerecalltheequationoftheshort-basediode:
• usingsomemathmanipulationwecanrewritethediffusioncapacitanceofthen-QSRregionasafunctionoftheportionofdiodecurrentduetotheholesinn-QNR(ID,p) :
• Wedefinetransittime oftheholesthroughthen-QNRtheaveragetimeforaholetodiffusethroughN-QNR:
Cdn =qA2
Wn − xn( ) ni2
Nd
1VTeVDVT
Wn − xn( )Wn − xn( )
Dp
Dp
=1VT
Wn − xn( )2
2Dp
qAWn − xn( )Dp
ni2
Nd
eVDVT ≅
1VT
Wn − xn( )2
2Dp
ID,p
ID = AqDn
WP − xp
ni2
Na
+Dp
WN − xn
ni2
Nd
⎛
⎝⎜⎜
⎞
⎠⎟⎟ e
VD /VT −1( ) = ID,n (−xp )+ ID,p(xn )
Diffusioncapacitance
τTp ≡Wn − xn( )2
2Dp
• Theninsummary:• Aside:
– wealreadyhadanamefortheaveragetimeittakesaholetodiffusethroughann-region.Itisthehole’slifetimeτp
– Infact,ifwecomputetheamountofexcesschargegivenbytheholesinn-QSRaccurately(seeslide39):
Diffusioncapacitance
τ p =Lp2
Dp
Cdn ≅ τTpID,pVT
qPn = qA pn (x ')0
∞
∫ ⋅dx ' = qA Δpne−x 'Lp
0
∞
∫ ⋅dx ' = qA pn0 eVD /VT −1( )e
−x 'Lp
0
∞
∫ dx ' =
= qApn0 eVD /VT −1( ) e
−x 'Lp
0
∞
∫ dx ' = qApn0 eVD /VT −1( )Lp
– Therefore:
– Similarlyfortheexcesschargeandcapacitanceinthep-QNR
– Thetotalexcesschargestoredandtheassociatedtotalcapacitanceis:
Diffusioncapacitance τ p ≡Lp2
Dp
Cdn =dqPndvD vD=VD
= qA ni2
Nd
Lp
VTeVD /VT =
Lp2
Dp
1VTqA ni
2
Nd
Dp
Lp
eVD /VT ≈ τ pID,pVT
qNp = −qA np(x '')0
∞
∫ ⋅dx '' = −qA Δnpe−x ''Ln
0
∞
∫ ⋅dx '' = −qAnp0 eVD /VT −1( )Ln
Cdp =−dqNpdvD vD=VD
=dqNpdvD vD=VD
=Ln2
Dn
1VTqA ni
2
Na
Dn
LneVD /VT ≈ τn
ID,nVT
qS = qPn + qNp = τ pID,p +τnID,n
Note:Theincrementinstoredelectronchargeisnegative,butresultsfromelectronsthatareinjectedfromthecathode(N-side)totheanode(P-SIDE)
CS =dqS
dvD vD=VD
=Cdn +Cdp ≈ τnID.nVT
+τ pID. pVT
Note:Theincrementinthestoredholechargecomesfrominjected holesfromtheP-SIDE(anode)totheN-side(cathode)
Diffusioncapacitance
0−xp−Wp
p(−xp)forVD
pp-QNR
Na
−dqNp n(−xp)forVD
ni2
Na
x
n
p(−xp)forVD+vd+dqPp
n(−xp)forVD+vd
n(xn)[forVD+vd]n(xn)[forVD]
n
pp(xn)[forVD+vd]
p(xn)[forVD]
−dqNn
+dqPn
Diffusioncapacitance:physicalintuition
• IfwehadlookedmorecarefullyattheexpressionsofthechargestoredintheQNRsratherthanimmediatelystartingtocrunchequationswewouldhavenoticedthat:
• Therefore,thestoredchargeqS,isproportionaltothediode’scurrentID,whichisalsoproportionaltoexp(VD/VT)−1
• Theconstantofproportionalityiscalledcharge-storagetimeortransittime
−qNp = qPp ∝ eVDVT −1
⎛
⎝⎜⎜
⎞
⎠⎟⎟
qPn = −qNn ∝ eVDVT −1
⎛
⎝⎜⎜
⎞
⎠⎟⎟
qS = qPn + qNp ∝ eVDVT −1
⎛
⎝⎜⎜
⎞
⎠⎟⎟∝ ID ⇔ qS = τT ID
CS ≅ τTIDVT
≅ τTgd =τTrdiode
• Thereisaverysimpleexplanationtothisproportionality,ID istherateofminoritychargeinjectionintothediode.
• Insteadystate.Thisratemustbeequaltotherateofchargerecombination,whichisqS/τT
• Inaone-sidedjunctionthetransittimeτT istherecombinationlifetimeonthelighter-dopingside,wheremostofthechargeinjectionandrecombinationtakeplace
• Ingeneral,τT isanaverageoftherecombinationlifetimesontheN-sideandP-side
• Inanyevent,ID andqS aresimplylinkedthroughachargestoragetimeτT(a.k.a. τS)
Diffusioncapacitance:physicalintuition
Example
• NA >>ND– N-sideismorelightlydopedthantheP-side– DiffusionofminoritycarriersonN-sidedominates
– Theexcesschargestoredis:
– Andfinally:
Diffusioncapacitance:physicalintuition
ID ≈ ID,p(xn ) ≈ qADp
Lp
pn0eVDVT = qA
Dp
Lp
ni2
Nd
eVDVT
τ p ≡ hole lifetime ≡Lp2
Dp
τ S ≈ τ p
CS ≈ τ pIDVT
≈ τ p × gD
qS ≈ qPn ≈ τ pID,p(xn ) ≈ τ pID
BiasDependenceofCJ andCS• CJ (associatedwith
chargemodulationindepletionregion)dominatesinreversebiasandsmallforwardbias
• CS (associatedwithchargestorageinQNRstomaintainquasi-neutrality)dominatesinstrongforwardbias
[email protected] 67source:Sodini
Cdiode
Cdiode
CS
CJ
VdiodeCS ∝ e
VDVT
CJ ∝1
ΦB −VD
source:Sedra &Smith
Summary:B=5×1015cm−3K−3/2
ni =1×1010cm−3
Eg =1.12eV
Summary:
source:Sedra &Smith
TheFiveSemiconductorEquations• Poisson’sEquation:
• Electroncurrentdensity:
• Holecurrentdensity:
• Continuityofelectrons:
• Continuityofholes:
d 2Φdx2
= − ρε= − q
εp − n + ND − NA( ) E(x) = − dΦ
dxGauss ' Law
Jn = qµnE + qDndndx
Jp = qµpE − qDpdpdx
∂n∂t
= 1q∂Jn∂x
+ Gn − Rn( )
∂p∂t
= 1q∂Jp∂x
+ Gp − Rp( )
d 2Φdx2
= − dEdx
= − ρε
SPICEcharacteristicofdiode
[email protected] 72SPICEsource:Antognetti &Massobrio
Example:D1 1 2 diode_1N4004.model diode_1N4004 D (IS=18.8n RS=0 BV=400 + IBV=5.00u CJO=30 M=0.333 N=2)
D2 3 4 idealmod_d.model idealmod_d D + IS=10f+ n = 0.01 + IBV=0.1n * By default BV = inf
GeneralSxntax:D[name] <anode_node> <cathode_node> <modelname>.model <modelname> D (<paramX>=<valueX> <paramY>=<valueY> ...)
