Semiconductor Device Modeling and

Characterization – EE5342 Lecture 12 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc/

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• Dinj– N~1, rd~N*Vt/iD– rd*Cd = TT =– Cdepl given by

CJO, VJ and M

• Drec– N~2, rd~N*Vt/iD– rd*Cd = ?– Cdepl =?

SPICE DiodeModel

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** The diode is modeled as an ohmic resistance (RS/area) in series with an intrinsic diode. <(+) node> is the anode and <(-) node> is the cathode. Positive current is current flowing from the anode through the diode to the cathode. [area value] scales IS, ISR, IKF,RS, CJO, and IBV, and defaults to 1. IBV and BV are both specified as positive values.In the following equations:Vd = voltage across the intrinsic diode onlyVt = k·T/q (thermal voltage)

k = Boltzmann’s constantq = electron chargeT = analysis temperature (°K)Tnom = nom. temp. (set with TNOM option

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D Diode **General FormD<name> <(+) node> <(-) node> <model name> [area value]ExamplesDCLAMP 14 0 DMODD13 15 17 SWITCH 1.5Model Form.MODEL <model name> D [model parameters] .model D1N4148-X D(Is=2.682n N=1.836 Rs=.5664 Ikf=44.17m Xti=3 Eg=1.11 Cjo=4p M=.3333 Vj=.5 Fc=.5 Isr=1.565n Nr=2 Bv=100 Ibv=10 0uTt=11.54n)*$

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Diode Model Parameters **Model Parameters (see .MODEL statement)

Description UnitDefault

IS Saturation current amp 1E-14N Emission coefficient 1ISR Recombination current parameter amp 0NR Emission coefficient for ISR 1IKF High-injection “knee” current amp infiniteBV Reverse breakdown “knee” voltage volt infiniteIBV Reverse breakdown “knee” current amp 1E-10NBV Reverse breakdown ideality factor 1RS Parasitic resistance ohm 0TT Transit time sec 0CJO Zero-bias p-n capacitance farad 0VJ p-n potential volt 1M p-n grading coefficient 0.5FC Forward-bias depletion cap. coef, 0.5EG Bandgap voltage (barrier height) eV 1.11

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Diode Model Parameters **Model Parameters (see .MODEL statement)

Description UnitDefault

XTI IS temperature exponent 3TIKF IKF temperature coefficient (linear) °C-1 0TBV1 BV temperature coefficient (linear) °C-1 0TBV2 BV temperature coefficient (quadratic) °C-2 0TRS1 RS temperature coefficient (linear) °C-1 0TRS2 RS temperature coefficient (quadratic) °C-2 0

T_MEASURED Measured temperature °CT_ABS Absolute temperature °CT_REL_GLOBAL Rel. to curr. Temp. °CT_REL_LOCAL Relative to AKO model temperature

°C

For information on T_MEASURED, T_ABS, T_REL_GLOBAL, and T_REL_LOCAL, see the .MODEL statement.

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**DC CurrentId = area(Ifwd - Irev) Ifwd = forward current = InrmKinj + IrecKgen Inrm = normal current = IS(exp ( Vd/(NVt))-1)

Kinj = high-injection factorFor: IKF > 0, Kinj = (IKF/(IKF+Inrm))1/2otherwise, Kinj = 1

Irec = rec. cur. = ISR(exp (Vd/(NR·Vt))- 1)

Kgen = generation factor = ((1-Vd/VJ)2+0.005)M/2

Irev = reverse current = Irevhigh + Irevlow

Irevhigh = IBVexp[-(Vd+BV)/(NBV·Vt)]Irevlow = IBVLexp[-(Vd+BV)/(NBVL·Vt)}

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vD=Vext

ln iD

Data

ln(IKF)

ln(IS)

ln[(IS*IKF) 1/2]

Effect

of Rs

t

a

VNFV

exp~

t

a

VNRV

exp~

VKF

ln(ISR)

Effect of high level injection

low level injection

recomb. current

Vext-

Va=iD*Rs

t

a

VNV

2exp~

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Interpreting a plotof log(iD) vs. VdIn the region where Irec < Inrm < IKF, and iD*RS << Vd.

iD ~ Inrm = IS(exp (Vd/(NVt)) - 1)

For N = 1 and Vt = 25.852 mV, the slope of the plot of log(iD) vs. Vd is evaluated as

{dlog(iD)/dVd} = log (e)/(NVt) = 16.799 decades/V = 1decade/59.526mV

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Static Model Eqns.Parameter ExtractionIn the region where Irec < Inrm < IKF, and iD*RS << Vd.

iD ~ Inrm = IS(exp (Vd/(NVt)) - 1)

{diD/dVd}/iD = d[ln(iD)]/dVd = 1/(NVt)

so N ~ {dVd/d[ln(iD)]}/Vt Neff,

and ln(IS) ~ ln(iD) - Vd/(NVt) ln(ISeff).

Note: iD, Vt, etc., are normalized to 1A, 1V, resp.

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Static Model Eqns.Parameter ExtractionIn the region where Irec > Inrm, and iD*RS << Vd.

iD ~ Irec = ISR(exp (Vd/(NRVt)) - 1)

{diD/dVd}/iD = d[ln(iD)]/dVd ~ 1/(NRVt)

so NR ~ {dVd/d[ln(iD)]}/Vt Neff,

& ln(ISR) ~ln(iD) -Vd/(NRVt )

ln(ISReff).

Note: iD, Vt, etc., are normalized to 1A, 1V, resp.

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Static Model Eqns.Parameter ExtractionIn the region where IKF > Inrm, and iD*RS << Vd.

iD ~ [ISIKF]1/2(exp (Vd/(2NVt)) - 1)

{diD/dVd}/iD = d[ln(iD)]/dVd ~ (2NVt)-1

so 2N ~ {dVd/d[ln(iD)]}/Vt 2Neff,

and ln(iD) -Vd/(NRVt) ½ln(ISIKFeff).

Note: iD, Vt, etc., are normalized to 1A, 1V, resp.

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Static Model Eqns.Parameter Extraction

In the region where iD*RS >> Vd.

diD/Vd ~ 1/RSeff

dVd/diD RSeff

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Getting Diode Data forParameter Extraction• The model

used .model Dbreak D( Is=1e-13 N=1 Rs=.5 Ikf=5m Isr=.11n Nr=2)

• Analysis has V1 swept, and IPRINT has V1 swept

• iD, Vd data in Output

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diD/dVd - Numerical Differentiation

Vd iD diD/ dVd(central diff erence)

Vd(n-1) iD(n-1) … etc. …

Vd(n) iD(n) (iD(n+1) - iD(n-1))/ (Vd(n+1) - Vd(n-1))

Vd(n+1) iD(n+1) (iD(n+2) - iD(n))/ (Vd(n+2) - Vd(n))

Vd(n+2) iD(n+2) … etc. …

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dln(iD)/dVd - Numerical Differentiation

Vd iD dln (iD)/ dVd (central diff erence)

Vd(n-1) iD(n-1) … etc. …

Vd(n) iD(n) ln (iD(n+1)/ iD(n-1))/ (Vd(n+1)-Vd(n-1))

Vd(n+1) iD(n+1) ln (iD(n+2)/ iD(n))/ (Vd(n+2) - Vd(n))

Vd(n+2) iD(n+2) … etc. …

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1.E-13

1.E-11

1.E-09

1.E-07

1.E-05

1.E-03

1.E-01

1.E+01

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

iD(A), Iseff(A), and 1/Reff(mho) vs. Vext(V)

Diode Par.Extraction 1

2345

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Neff vs. Vext

1/Reff

iD

ISeff

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Results ofParameter Extraction• At Vd = 0.2 V, NReff = 1.97,

ISReff = 8.99E-11 A.• At Vd = 0.515 V, Neff = 1.01,

ISeff = 1.35 E-13 A.• At Vd = 0.9 V, RSeff = 0.725 Ohm• Compare to

.model Dbreak D( Is=1e-13 N=1 Rs=.5 Ikf=5m Isr=.11n Nr=2)

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Hints for RS and NFparameter extractionIn the region where vD > VKF. Defining

vD = vDext - iD*RS and IHLI = [ISIKF]1/2.

iD = IHLIexp (vD/2NVt) + ISRexp (vD/NRVt)

diD/diD = 1 (iD/2NVt)(dvDext/diD - RS) + …

Thus, for vD > VKF (highest voltages only)

plot iD-1 vs. (dvDext/diD) to get a line with

slope = (2NVt)-1, intercept = - RS/(2NVt)

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Application of RS tolower current dataIn the region where vD < VKF. We still have

vD = vDext - iD*RS and since.

iD = ISexp (vD/NVt) + ISRexp (vD/NRVt) Try applying the derivatives for methods

described to the variables iD and vD (using RS and vDext).

You also might try comparing the N value from the regular N extraction procedure to the value from the previous slide.

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Reverse bias (Va<0)=> carrier gen in DR• Va < 0 gives the net rec rate,

U = -ni/, = mean min carr g/r l.t.

NNN/NNN and

qN

VV2W where ,

2Wqn

J

(const.) U- G where ,qGdxJ

dadaeff

eff

abi

0

igen

x

xgen

n

p

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Reverse bias (Va< 0),carr gen in DR (cont.)

gens

gen

gengensrev

JJJ

JSPICE

JJJJJ

or of largest the set then ,0

V when 0 since :note model

VV where ,

current generation the plus bias negative

for current diode ideal the of value The

current the to components two are there

bias, reverse ,)0V(V for lyConsequent

a

abi

ra

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Reverse biasjunction breakdown• Avalanche breakdown

– Electric field accelerates electrons to sufficient energy to initiate multiplication of impact ionization of valence bonding electrons

– field dependence shown on next slide

• Heavily doped narrow junction will allow tunneling - see Neamen*, p. 274– Zener breakdown

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Reverse biasjunction breakdown• Assume -Va = VR >> Vbi, so Vbi-Va--

>VR

• Since Emax~ 2VR/W =

(2qN-VR/())1/2, and VR = BV when

Emax = Ecrit (N- is doping of lightly

doped side ~ Neff)

BV = (Ecrit )2/(2qN-)

• Remember, this is a 1-dim calculation

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Reverse biasjunction breakdown

8/3

4/3

0

4/3

2/3

20

161/

1.1/ 120 so

,161/

1.1/ 60 gives *,***

usually , 2

D.A. theand diode sided-one a Assuming

EN

EqNVE

EN

EVBVCasey

BVqN

EBV

g

Sicrit

B

g

icritSi

i

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Ecrit for reverse breakdown (M&K**)

Taken from p. 198, M&K**

Casey Model for Ecrit

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Junction curvatureeffect on breakdown• The field due to a sphere, R, with

charge, Q is Er = Q/(4r2) for (r > R)

• V(R) = Q/(4R), (V at the surface)• So, for constant potential, V, the field,

Er(R) = V/R (E field at surface increases for smaller spheres)

Note: corners of a jctn of depth xj are like 1/8 spheres of radius ~ xj

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BV for reverse breakdown (M&K**)

Taken from Figure 4.13, p. 198, M&K**

Breakdown voltage of a one-sided, plan, silicon step junction showing the effect of junction curvature.4,5

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Diode Switching

• Consider the charging and discharging of a Pn diode – (Na > Nd)

– Wd << Lp

– For t < 0, apply the Thevenin pair VF and RF, so that in steady state • IF = (VF - Va)/RF, VF >> Va , so current source

– For t > 0, apply VR and RR

• IR = (VR + Va)/RR, VR >> Va, so current source

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Diode switching(cont.)

+

+ VF

VR

DRR

RF

Sw

R: t > 0

F: t < 0

ItI s

F

FF R

VI0tI

VF,VR >>

Va

F

F

F

aFQ R

VR

VVI

0,t for

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Diode chargefor t < 0

xn xncx

pn

pno

Dp2W

,IWV,xqp'Q

2N

TR

TRFnFnndiff,p

D

2i

noV/V

noFn Nn

p ,epV,xp tF

dxdp

qDJ since ,qAD

Idxdp

ppp

F

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Diode charge fort >>> 0 (long times)

xn xncx

pn

pno

tF V/Vnon ep0t,xp

t,xp

sppp

S Jdxdp

qDJ since ,qADI

dxdp

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Equationsummary

Q discharge to flows

R/VI current, a 0, but small, t For

RV

I ,qAD

Idxdp

AJI ,AqD

I

JqD1

dxdp

RRR

F

FF

p

F

0t,F

ssp

s

,ppt,R

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Snapshot for tbarely > 0

xn xncx

pn

pno

p

F

qADI

dxdp

p

RqAD

Idxdp

tF V/Vnon ep0t,xp

0t,xp Total charge removed, Qdis=IRt

st,xp

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I(t) for diodeswitching

ID

t

IF

-IR

ts ts+trr

- 0.1 IR

sRdischarge

p

Rs

tIQ

constant, a is qAD

Idxdp

,tt 0 For

pnp

p2is L/WtanhL

DqnI

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References

*Semiconductor Device Modeling with SPICE, 2nd ed., by Massobrio and Antognetti, McGraw Hill, NY, 1993.

**MicroSim OnLine Manual, MicroSim Corporation, 1996.