CapacitanceofForwardBiasedDiodeWhen a diode changes from reverse biased (with little current through it) to forward biased (with significant current across it) the charge being stored near and across the junction changesPart of the change in charges is due to the change in the width of the depletion region and therefore the amount of immobile charge stored in it ( - Cj)An additional change in the charge storage is necessary to account for the excess of minority carriers close to the depletion region edges required for the diffusion current to exists. This component is modeled by another capacitance, called the diffusion capacitance ( - Cd)As a diode is turned off (changes from forward biased to reverse biased) for a short period of time a current will flow in the negative direction until the minority charge is removed

ChargeofForwardBiasedDiodeDirection of positive currentholediffusionimmobile chargeelectrondiffusion----++++++++++++------PIn the P region we have a lot of holesthat will diffuse toward the N regionNIn the N region we have a lot of electronsthat will diffuse toward the P regionDepletion Region

TotalCapacitanceofForwardBiasedDiodeIt is the sum of the diffusion capacitance Cd and thecapacitance CjC total -::C d +C jdepletionFor a forward biased diode the junction capacitanceapproximated by:C j 2 C j0is roughlyC j 2 C j0for V D 0.75 0 The approximation is not critical since the diffusion capacitance istypically much larger than than the depletion capacitanceC d C j

DiffusionCapacitanceTo find the diffusion capacitance we first find the minority chargeclose the depletion edgesQd andthendifferentiate it withrespectto the voltageappliedVd.d Q dI d (@V D )Small signalTdV dV DTThe diffusion capacitance of a forward biased diodeis proportional to the diode currentdiffusion capacitanceC d =[] =T V

excess of minoritycharge (electrons)stored in the P regionDiffusionChargeexcess of minoritycharge (holes)stored in the NregionQ d =Q p +Q n-x' Q p q AA n p p x' dx ' "' q A n p 0O eL pdx ' 0 0V d-x' e V T -1)V2q A ni LndV TL pq AA n p0 O e 1l edx ' 'Q :::nN0AV d-x' cJJV TL p'q A n p0 e-1)J edx ' =0oo(e)[(e-1)V dV dJ- x'22niqA nLVLipV=qA NT 1 Le=pT0pNDDpn( =0)pn ()= pn (=0)e L n-x' L pn p (x' )=n p (x' =0 )enp(x'=0) V d0V dVpn0 eTV Tn p0 epx'n00np0xxn-x0p

DiffusionChargeThe excess hole charge stored in the N regionis given by:pn xn pn0 e 1 V d /V TQ p =A q [ pn (x n )- p n0 ] L p ==Aq L p p n0 eV d IV T 1 qD ppn0 e 1 \V V2p2J p LL pdTL AJ p I:::IIppppDDppL pL p D p ppn0 O e 1 l)JV d V TpqD pSimilarly, the excesselectron charge storedinthe P region is:Q n = I n Tn

TotalDiffusionChargeThus, the total excess minority carrier charge is:Q d =Q p +Q n =I p T p + I n TnSince the diode current isit is moreconvenientI d =I p +I nto express the excess charge as:Q d =TTI d(where TTis called mean transit time )

DiffusionCapacitance]V DdQ dd -rT I d dI d-rTV DV Ddddd[V d1[V d1VVr d [dV1[][]dI dd I ss e 1llIeI+II1TTsdsd R;dV dV TV TV TV DV DV DV DV DdI dV TVDC d T =[ dV] =TT [dV] = r

TransitionTimecumbersome:Thegeneralexpression for ,;T isquiteTT =C dr dV d( 22i(]V D)V DdI dqA DnniqA Dn1VpwherewithI== e-1 )+ Tdr ddVLNLNnApDd)()V dQd =(22C d =[d Q dq A n iL p q A ni Ln e V T -1withN DN AdV d

TransitionTimeIn practice, since usually diodes are single sided (i.e. one side willbemuch more heavily doped than the otherside) the minority chargeignoredstorage in the heavily dopedsidecanbe eV d)N eV d-1)22q A ni Lnq A nLV T -1VipQ n ==Q ==TpNN DAAssumingthe P sideis moreheavily doped than theside:N A N DQ p QnQ d Q pI d I p L pp2 T p DNOTE:Holes (Qp) are minority carriers on the N side Electrons (Qn) are minority carriers on the P side

SingleSidedDiodesOne side of the diode is more heavily doped than the otherMany of the junctions encountered in integrated circuits are one-sided junctionsthe well.with the lightly doped side being the substrate orFoe single sided diodes the depletion region will extend mostly onthe lightly doped side.The depletion capacitance is almost independent of the doping concentration on the heavily doped side

SingleSidedDiodesThe PN junctions inside CMOS ICs are single-sidedNMOS transistors havedoped than the P side:PMOS transistors have doped than the N side:parasitic diodes with theN D > N Aparasitic diodes with theN A N DN sidemore heavilyP sidemore heavilyNOTE: in general within an MOS transistor, itis undesirable tohave a forward biased junction, it usually means there is aproblem.

SchottkyDiodesA different type of diode, can be realized by contacting a metal to a lightly doped semiconductor region.The use of a lightly doped semiconductor, causes a depletion region to form at the interface between the aluminum anode and the n+ silicon region

SchottkyDiodesThe voltage drop of a forward biased Schottky diode is smaller. Thevalue depends on the metal used. For aluminum is approx 0.5 VWhen the diode is forward biased there is no minority charge storage in the lightly doped n+ region. Thus Cd = 0The absence of diffusion capacitance makes the diode much faster.

Diodesrealizedt--t:+--oinCMOStechnologySideViewn-wellp-substratep-substrateTopView()Figure 14.54C 10.DiodeNOTE:For the case of Fig 14.54(a) the anode is inevitably grounded. techno] gyre lized inb-,-,

DiodeSPICEmodelJParameterSPICEDescriptionIsISSaturation CurrentRsRSOhmic Series Resistance of the p-n regions and the contactsnNEmission Coefficientq)oVJ, PB, PHIBuilt-in VoltageCJoCJO, CJZero Bias Junction CapacitanceM.MJ, MGrading Coefficient1rTTTransit TimeVKBVReverse Breakdown Knee VoltageIKIBVReverse Breakdown Knee Current

DiodeSPICEModeling I d =I s [exp (nV)-1 +GMINV d V dTConvergence Aid]BV +V dBVV TT=- IS (BV/VT)I d =-I s +GMINV d

MOSphysicalp-dunnel U11JlStstortastructuren- ellp ubslmlcPhy ical .tntcturen n-\ ellfan n-channel. nd p-ch:mnGateMetal(AI)Bulk or substrateFig. 1.6A cross section of o typical n ni the electrostatic potential at equilibrium is positivei cp0 ( x )n ( x )=n e V TBy convention the reference for the potentialconcentration is the intrinsic concentration

ThermalEquilibriumAsimilar derivation for theresult:holeconcentrationleads tothefollowingp0 (x )VT0iniNOTE:for P typesilicon since p0 >ni theelectrostatic potential at equilibrium is negativecp0 ( x )=-V T ln () 0 x p (x )=n e

MOSCapacitorinThermalEquilibriumsource/drain/bulkgateFigure. Using the MOSFET as a capacitorThe gate and the substrate of the MOS transistor form a parallel plate capacitor with the SiO2 as dielectricSince source and drain are separated by back-to- back junctions, the resistance between the source and the drain is very high (> 1012 ohm)At equilibrium the p- substrate and the n+ source and drain form a pn junction.Therefore a depletion region exists betweenthe n+ source and drain and the p- substrate

MOSstructureinThermalEquilibriumGBE0 n+- pcpn +=V T ln ()N Dcp p=-V T ln ()N Aequilibrium potential inthe polysilicon (gate)equilibrium potential inthe silicon (bulk=substrate)nnii17-319-310-3N A 10cmN D v3310cmni 10cm n+ p 550 mV 420 mV 970 mV V= 0

MOSinThermalEquilibriumGBE 0 n+ p 970 mVFromthe sign of the potential drop across the MOS structure itfollows that the electric field points from gate to bulk.Therefore, a positive charge must be present on the polysilicon gate and there must be a balancing negative charge in the p-type silicon substrate (the oxide will be considered a charge-free perfect insulator)V= 0

ChargeontheMOSinTESince the gate is highly conductive n+ polysilicon the gate chargeQG0can bethought as a sheet charge located at the bottom surface of the polysilicon gateThe charge on the p-type silicon substrateQB0is formed by the immobilenegatively charged acceptor ions (to a depletion depth ofXd0)leftbehindbythe mobileholesrepelledbythepositivechargeonthegate.GB00MOSV= 0

ChargeontheMOSin TE,lc char c,d Jllt1 nn \llh8Iul'Ci ure 3.21Qualitative picture of charge distribu ion in an MOS capacitorhp- ype subs rate in thermal equilibrium.2[units:Cicm ]000 =-0Bo=qN A X doichar-G

PotentialacrosstheMOSinTEVoltage dropacross the oxideVoltage drop acrossthe depletion regionIn the siliconMOSV ox , 0 +V B, 0BUILT-IN-cpn+ -cp p=the MOS structureBuilt in Voltage acrossSurface Potential Potential at the SiO2-siliconinterface (x=0)

PotentialacrosstheThe equilibrium potential of a given referred as its Fermi potentialMOSinTEmaterial iscommonlyN AN Dn+F-gateTniniThe Built in voltageacross the MOS structure is oftenexpressed in term of the work function between thematerial and the bulk silicongateN D N An2iNOTE: This term is referred as the metal-to-silicon work function even though thegate terminal is something other than metal (i.e. polysilicon)-cpMS =cpF-gate-cpF-bulk =cpn+ -cp p :=BUILT-IN =V T ln ()cp== cp=V ln ( )cp p :=cpF-bulk =-V T ln ( )

MOSinTE:fixingKVLGB mn+ V ox , 0 V B0 pm 0V ox , 0 V B0 mn+ pm =( cpn+-cp p ):=cpBUILT-INV= 0

MOSinTE n+ p mn+ pm NOTE:for the case of TE the gate and the bulk metal contacts are at the same potential

MOSinTE:QuantitativeAnalysisE oxIn the oxide (-tox < x < 0) the charge density is zero thus thefield is constant (Eox):E (0+ )0dE = pGauss ' s Law :dxEBoundary condition at the oxide/silicon interface (0-x 0+):::3EoxEs+++ ox0E ox =Es0E (0 )E (0 )=E oxE ox = E (0 )-t-t s oxThe total excess charge in the region-tox x Xd0 is zero (neutrality of charge)The electric field is confined in the region-tox < x < Xd0

MOSinTE:QuantitativeAnalysisE oxWithin the same material theelectric field will not jump !In the charged region of the silicon oxide/silicon interface (0+sconstant:x