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