L08 07Feb02 1

Semiconductor Device Modeling and CharacterizationEE5342, Lecture 8-Spring 2002

Professor Ronald L. Carterronc@uta.edu

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

L08 07Feb02 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

L08 07Feb02 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

L08 07Feb02 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.)

L08 07Feb02 5

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)),

L08 07Feb02 6

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

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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

L08 07Feb02 8

Plot of typical Va > 0 current density eqns.

Vext

ln J

data

ln(JKF)

ln(Js)

ln[(Js*JKF) 1/2]

Effect

of Rs

t

aV

Vexp~

t

aV2

Vexp~

VKF

ln(Jsrec)

Effect of high level injection

low level injection

recomb. current

Vext-Vd=JARs

L08 07Feb02 9

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

L08 07Feb02 10

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

L08 07Feb02 11

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

L08 07Feb02 12

Ecrit for reverse breakdown (M&K**)

Taken from p. 198, M&K**

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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

L08 07Feb02 14

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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Example calculations• Assume throughout that p+n jctn with Na

= 3e19cm-3 and Nd = 1e17cm-3

• From graph of Pierret mobility model, p

= 331 cm2/V-sec and Dp = Vtp = ? • Why p and Dp?

• Neff = ?

• Vbi = ?

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0

500

1000

1500

1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20

Doping Concentration (cm̂ - 3)

Mob

ility

(cm̂

2/V

-se

c)P As B n(Pierret) p(Pierret)

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Parameters forexamples• Get min from the model used in Project

2 min = (45 sec) 1+(7.7E-18cm3Ni+(4.5E-36cm6Ni

2

• For Nd = 1E17cm3, p = 25 sec

– Why Nd and p ?

• Lp = ?

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Hole lifetimes, taken from Shur***, p. 101.

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Example

• Js,long, = ?

• If xnc, = 2 micron, Js,short, = ?

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Example(cont.)• Estimate VKF

• Estimate IKF

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Example(cont.)• Estimate Js,rec

• Estimate Rs if xnc is 100 micron

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Example(cont.)• Estimate Jgen for 10 V reverse bias

• Estimate BV

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Diode equivalentcircuit (small sig)

ID

VDVQ

IQ

t

Q

dd

VD

D

V

I

r1

gdVdI

Q

is the practical

“ideality factor”

Q

tdiff

t

Qdiffusion

mintrdd

IV

r , V

IC

long) for short, for ( , Cr

L08 07Feb02 25

Small-signal eqcircuit

CdiffCdep

l

rdiff

Cdiff and Cdepl are both charged by

Va = VQ

Qa

2/1

bi

ajojdepl VV ,

VV

1CCC

Va

L08 07Feb02 26

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

L08 07Feb02 27

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

L08 07Feb02 28

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

L08 07Feb02 29

Diode charge fort >>> 0 (long times)

xn xncx

pn

pno

tF V/Vnon ep0t,xp

t,xp

sppp

S Jdxdp

qDJ since ,qAD

Idxdp

L08 07Feb02 30

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

L08 07Feb02 31

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

L08 07Feb02 32

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

L08 07Feb02 33

References

* Semiconductor Physics and Devices, 2nd ed., by Neamen, Irwin, Boston, 1997.

**Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, John Wiley, New York, 1986.

***Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.