Semiconductor Device Modeling and CharacterizationEE5342, Lecture 9 -Spring 2010

Professor Ronald L. Carterronc@uta.edu

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

L09 February 15 2

)pn( ,ppp and ,nnn where

kTEfiE

coshn2np

npnU

dtpd

dtnd

GRU

oo

oT

i

2i

Effect of carrierrecombination in DR• The S-R-H rate (no = po = o) is

L09 February 15 3

Effect of carrierrec. in DR (cont.)• For low Va ~ 10 Vt

• In DR, n and p are still > ni

• The net recombination rate, U, is still finite so there is net carrier recomb.– reduces the carriers available for the

ideal diode current– adds an additional current component

L09 February 15 4

eff,o

taieffavgrec

o

taimaxfpfna

fnfii

fifni

x

xeffavgrec

2V2/Vexpn

qWxqUJ

2V2/Vexpn

U ,EEqV w/

,kT/EEexpnp

and ,kT/EEexpnn cesin

xqUqUdxJ curr, ecRn

p

Effect of carrierrec. in DR (cont.)

L09 February 15 5

Effect of non-zero E in the CNR• This is usually not a factor in a short

diode, but when E is finite -> resistor• In a long diode, there is an additional

ohmic resistance (usually called the parasitic diode series resistance, Rs)

• Rs = L/(nqnA) for a p+n long diode.

• L=Wn-Lp (so the current is diode-like for Lp and the resistive otherwise).

L09 February 15 6

High level injection effects• Law of the junction remains in the same

form, [pnnn]xn=ni

2exp(Va/Vt), etc.

• However, now pn = nn become >> nno = Nd, etc.

• Consequently, the l.o.t.j. reaches the limiting form pnnn = ni

2exp(Va/Vt)

• Giving, pn(xn) = niexp(Va/(2Vt)), or np(-xp) = niexp(Va/(2Vt)),

L09 February 15 7

High level injeffects (cont.)

KFKFKFsinj lh,s

i

at

i

dtKFa

appdnn

a

tainj lh,sinj lh

VJJ ,JJJ :Note

nN

lnV2 or ,n

NlnV2VV Thus

Nx-n or ,Nxp giving

V of range the for important is This

V2/VexpJJ

:is density current injection level-High

L09 February 15 8

Summary of Va > 0 current density eqns.• Ideal diode, Jsexpd(Va/(Vt))

– ideality factor,

• Recombination, Js,recexp(Va/(2Vt))– appears in parallel with ideal term

• High-level injection, (Js*JKF)

1/2exp(Va/(2Vt))

– SPICE model by modulating ideal Js term

• Va = Vext - J*A*Rs = Vext - Idiode*Rs

L09 February 15 9

Diode Diffusion and Recombination Currents

t

a

d

pnp

a

npn

pn

i

ac

aDiff

pppnncnnnnppcp

VV

pnpd

p

npna

niaDiff

VV

N

LWL

N

LWL

xxn

Vi

Vi

The

DLxxWDLxxW

eLWLN

D

LWLND

AqnVi

The

ta

2exptanhtanh2

:ratio current ionRecombinat to Diffusion

, , ,

1tanhtanh

:current Diffusion

Re

2

L09 February 15 10

Diode Diffusion and Recombination Currents – One

Sided Diode

minminminmin

min2

minminmin

min

min2

minmin

min2

22~

tanh2

:current ionRecombinat

, , ,

~tanh

:density current Diffusion

Axqn

n

x

D

NDN

AqnISR

N

DWD

xn

ISRIS

The

DLxxWDLxxW

DN

Aqn

DWDN

DAqnIS

The

di

i

dwafer

wafer

i

wafer

wafern

d

i

pppnncnnnnppcp

wafer

i

nwafernwafer

i

L09 February 15 11

1N ,

V2NV

t

aexp~

1N ,

VNV

t

aexp~

Vext

ln(J)

data Effect of Rs

2NR ,

VNRV

t

aexp~

VKF

Plot of typical Va > 0 current density equations

Sexta RAJ-VV

KFS JJln

recsJln ,

SJln

KFJln

L09 February 15 12

• 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

Project 1A – Diode parameters to use

L09 February 15 13

Param Value UnitsIS 3.608E-14 AN 1IKF 10.00 ARS 10 OhmISR 2.422E-12 ANR 2M 0.5VJ 755 mVCJ0 3.316E-13 FdTNOM 27 CRTH 500

Tasks• Using PSpice or any simulator, plot the i-v curve for

this diode, assuming Rth = 0, for several temperatures in the range 300 K < TEMP = TAMB < 304 K.

• Using this data, determine what the i-v plot would be for Rth = 500 K/W.

• Using this data, determine the maximum operating temperature for which the diode conductance is within 1% of the Rth = 0 value at 300 K.

• Do the same for a 10% tolerance.• Propose a SPICE macro which would give the Rth =

500 K/W i-v relationship.

L09 February 15 14

Example

L09 February 15 15

Approaches• Phenomenological

• Theoretical

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L09 February 15 17

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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

L09 February 15 20

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)*$

L09 February 15 21

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

L09 February 15 22

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.

L09 February 15 23

**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)}

L09 February 15 24

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~

L09 February 15 25

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

L09 February 15 26

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.

L09 February 15 27

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.

L09 February 15 28

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.

L09 February 15 29

Static Model Eqns.Parameter Extraction

In the region where iD*RS >> Vd.

diD/Vd ~ 1/RSeff

dVd/diD RSeff

L09 February 15 30

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

L09 February 15 31

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. …

L09 February 15 32

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. …

L09 February 15 33

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

L09 February 15 34

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)

L09 February 15 35

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)

L09 February 15 36

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.

L09 February 15 37

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

L09 February 15 38

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

L09 February 15 39

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

L09 February 15 40

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

L09 February 15 41

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

L09 February 15 42

Ecrit for reverse breakdown (M&K**)

Taken from p. 198, M&K**

Casey Model for Ecrit

L09 February 15 43

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

L09 February 15 44

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

L09 February 15 45

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

L09 February 15 46

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

L09 February 15 47

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

L09 February 15 48

Diode charge fort >>> 0 (long times)

xn xncx

pn

pno

tF V/Vnon ep0t,xp

t,xp

sppp

S Jdxdp

qDJ since ,qADI

dxdp

L09 February 15 49

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

L09 February 15 50

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

L09 February 15 51

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

L09 February 15 52

References

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

**MicroSim OnLine Manual, MicroSim Corporation, 1996.