Semiconductor Device Modeling and

Characterization – EE5342 Lecture 11 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

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

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2

Minority carriercurrents

1eLWsinhLxxcosh

LNDqn

xxx- for ,qDxJ

1eLWsinhLxxcosh

LNDqn

xxx for ,qDxJ

ta

p

ta

n

VV

npnpc

nan

2i

ppcdxnd

nn

VV

pnpnc

pd

p2i

ncndxpd

pp

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Evaluating thediode current

p/nn/pp/nd/a

p/n2isp/sn

spsns

VV

spnnp

LWcothLND

qnJ

sdefinition with JJJ where

1eJxJxJJ

then DR, in gen/rec no gminAssu

ta

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Special cases forthe diode current

nd

p2isp

pan2

isn

nppn

pd

p2isp

nan2

isn

nppn

WNDqnJ and ,WN

DqnJ

LW or ,LW :diode ShortLN

DqnJ and ,LNDqnJ

LW or ,LW :diode Long

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Ideal diodeequation• Assumptions:

– low-level injection– Maxwell Boltzman statistics– Depletion approximation– Neglect gen/rec effects in DR– Steady-state solution only

• Current dens, Jx = Js expd(Va/Vt)– where expd(x) = [exp(x) -1]

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Ideal diodeequation (cont.)• Js = Js,p + Js,n = hole curr + ele curr

Js,p = qni2Dp coth(Wn/Lp)/(NdLp) =

qni2Dp/(NdWn), Wn << Lp, “short” =

qni2Dp/(NdLp), Wn >> Lp, “long”

Js,n = qni2Dn coth(Wp/Ln)/(NaLn) =

qni2Dn/(NaWp), Wp << Ln, “short” =

qni2Dn/(NaLn), Wp >> Ln, “long”

Js,n << Js,p when Na >> Nd

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Diffnt’l, one-sided diode conductance

Va

IDStatic (steady-state) diode I-V characteristic

VQ

IQ QVaD

D dVdIg

tasD V

VdexpII

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Diffnt’l, one-sided diode cond. (cont.)

DQt

dQd

QDDQtDQ

Qd

tat

tQsVa

DQd

tastasD

IV

g1Vr ,resistance diode The

. VII where ,VI

Vg then

, VV If . VVVexpI

dVdIVg

VVdexpIVVdexpAJJAI

Q

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Charge distr in a (1-sided) short diode

• Assume Nd << Na

• The sinh (see L12) excess minority carrier distribution becomes linear for Wn << Lp

pn(xn)=pn0expd(Va/Vt)• Total chg = Q’p =

Q’p = qpn(xn)Wn/2xn

xxnc

pn(xn)

Wn = xnc- xn

Q’p

pn

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Charge distr in a 1-sided short diode

• Assume Quasi-static charge distributions

• Q’p = Q’p = qpn(xn)Wn/2

• dpn(xn) = (W/2)*

{pn(xn,Va+V) -

pn(xn,Va)}xn

xxnc

pn(xn,Va)

Q’p

pn pn(xn,Va+V)

Q’p

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Cap. of a (1-sided) short diode (cont.)

p

x

x p

ntransitQQ

transitt

DQ

pt

DQQ

taaa

a

Ddx

JpqVV

VI

DVI

V

VVddVdV

dVA

nc

n2WCr So,

. 2WC ,V V When

exp2WqApd

2)W(xpqAd

dQC Define area. diode A ,Q'Q

2n

dd

2n

dta

nn0nnn

pdpp

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General time-constant

npannnn

ap

ppp

pnVa

pn

VaD

Qd

CCC ecapacitanc diode total

the and ,dVdQCg and ,dV

dQCg

that so time sticcharacteri a always is There

ggdVJJdAdV

dIVg

econductanc the short, or long diodes, all For

QQ

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General time-constant (cont.)

times.-life carr. min. respective the, and side, diode long

the For times. transit charge physical

the ,D2W and ,D2

W

side, diode short the For

n0np0p

n

2p

transn,np

2n

transp,p

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General time-constant (cont.)

Fdd

transitminFgC

and 111 by given average

the is time transition effective Thesided-one usually are diodes Practical

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)pn( ,ppp and ,nnn wherekT

EfiEcoshn2npnpnU

dtpd

dtndGRU

oo

oTi

2i

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

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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

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eff,o

taieffavgrec

otaimaxfpfna

fnfii

fifni

x

xeffavgrec

2V2/VexpnqWxqUJ

2V2/VexpnU ,EEqV w/

,kT/EEexpnp and ,kT/EEexpnn cesin

xqUqUdxJ curr, ecRn

p

Effect of carrierrec. in DR (cont.)

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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).

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

form, [pnnn]xn=ni2exp(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)),

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High level injeffects (cont.)

KFKFKFsinj lh,siat

idtKFa

appdnna

tainj lh,sinj lh

VJJ ,JJJ :Note nNlnV2 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

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Diode Diffusion and Recombination Currents

ta

d

pnp

a

npn

pn

i

ac

aDiff

pppnncnnnnppcp

VV

pnpd

p

npna

niaDiff

VV

NLWL

NLWL

xxn

ViVi

The

DLxxWDLxxW

eLWLN

DLWLN

DAqnVi

The

ta

2exptanhtanh2:ratio current ionRecombinat to Diffusion

, , ,

1tanhtanh

:current Diffusion

Re

2

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Diode Diffusion and Recombination Currents – One

Sided Diode

minminminmin

min2

minminmin

min

min2

minmin

min2

22~

tanh2:current ionRecombinat

, , ,

~tanh

:density current Diffusion

Axqnn

xDND

NAqnISR

NDWD

xn

ISRIS

The

DLxxWDLxxW

DN

AqnDWDN

DAqnIS

The

di

i

dwafer

wafer

i

wafer

wafern

d

i

pppnncnnnnppcp

wafer

i

nwafernwaferi

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1N ,V2N

Vt

aexp~

1N ,VN

Vt

aexp~

Vext

ln(J)

data Effect of Rs

2NR ,VNR

Vt

aexp~

VKF

Plot of typical Va > 0 current density equations

Sexta RAJ-VV

KFS JJln

recsJln ,

SJln

KFJln

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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.