L07 04Feb02 1

Semiconductor Device Modeling and CharacterizationEE5342, Lecture 7-Spring 2002

Professor Ronald L. Carterronc@uta.edu

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

L07 04Feb02 2

Ideal JunctionTheory

Assumptions

• Ex = 0 in the chg neutral reg. (CNR)

• MB statistics are applicable• Neglect gen/rec in depl reg (DR)• Low level injections apply so that

np < ppo for -xpc < x < -xp, and pn

< nno for xn < x < xnc

• Steady State conditions

L07 04Feb02 3

Ideal JunctionTheory (cont.)

Apply the Continuity Eqn in CNR

ncnn

ppcp

xxx ,Jq1

dtdn

tn

0

and

xxx- ,Jq1

dtdp

tp

0

L07 04Feb02 4

Ideal JunctionTheory (cont.)

ppc

nn

p2p

2

ncnpp

n2n

2

ppx

nnxx

xxx- for ,0D

n

dx

nd

and ,xxx for ,0D

p

dx

pd

giving dxdp

qDJ and

dxdn

qDJ CNR, the in 0E Since

L07 04Feb02 5

Ideal JunctionTheory (cont.)

)contacts( ,0xnxp and

,1en

xn

pxp

B.C. with

.xxx- ,DeCexn

xxx ,BeAexp

So .D L and D L Define

pcpncn

VV

po

pp

no

nn

ppcL

xL

x

p

ncnL

xL

x

n

pp2pnn

2n

ta

nn

pp

L07 04Feb02 6

Excess minoritycarrier distr fctn

1eLWsinh

Lxxsinhnxn

,xxW ,xxx- for and

1eLWsinh

Lxxsinhpxp

,xxW ,xxx For

ta

ta

VV

np

npcpop

ppcpppc

VV

pn

pncnon

nncnncn

L07 04Feb02 7

Forward Bias Energy Bands

xn xnc-xpc -xp 0

Ev

Ec

q(Vbi-Va)

EF

EF

EFi

qVa

x

Imref, EFn

Imref, EFp

1ennkT/EEexpnn ta VV0ppFiFniequilnon

1eppkT/EEexpnp ta VV0nnFpFiiequilnon

L07 04Feb02 8

CarrierInjection

xn-xpc 0

ln(carrier conc)

ln Naln Nd

ln ni

ln ni2/Nd

ln ni2/Na

xnc-xp

x

~Va/Vt~Va/Vt

1enxn t

aV

V

popp

1epxp t

aV

V

nonn

L07 04Feb02 9

Minority carriercurrents

1eLWsinh

Lxxcosh

LNDqn

xxx- for ,qDxJ

1eLWsinh

Lxxcosh

LN

Dqn

xxx for ,qDxJ

ta

p

ta

n

VV

np

npc

na

n2i

ppcdx

ndnn

VV

pn

pnc

pd

p2i

ncndxpd

pp

L07 04Feb02 10

Evaluating thediode current

p/nn/pp/nd/a

p/n2isp/sn

spsns

VV

spnnp

LWcothLN

DqnJ

sdefinition with JJJ where

1eJxJxJJ

then DR, in gen/rec no gminAssu

ta

L07 04Feb02 11

Special cases forthe diode current

nd

p2isp

pa

n2isn

nppn

pd

p2isp

na

n2isn

nppn

WN

DqnJ and ,

WND

qnJ

LW or ,LW :diode Short

LN

DqnJ and ,

LND

qnJ

LW or ,LW :diode Long

L07 04Feb02 12

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]

L07 04Feb02 13

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

L07 04Feb02 14

Diffnt’l, one-sided diode conductance

Va

IDStatic (steady-state) diode I-V characteristic

VQ

IQ QVa

DD dV

dIg

t

asD V

VdexpII

L07 04Feb02 15

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

DQ

t

dQd

QDDQt

DQQd

tat

tQs

Va

DQd

tastasD

IV

g1

Vr ,resistance diode The

. VII where ,V

IVg then

, VV If . V

VVexpI

dV

dIVg

VVdexpIVVdexpAJJAI

Q

L07 04Feb02 16

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/2

x

n

x

xnc

pn(xn

)

Wn = xnc-

xn

Q’p

pn

L07 04Feb02 17

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)}

x

n

xxnc

pn(xn,Va)

Q’p

pn pn(xn,Va+V)

Q’p

L07 04Feb02 18

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

p

x

x p

ntransitQQ

transitt

DQ

pt

DQQ

taaa

a

Ddx

Jp

qVV

V

I

DV

IV

VVddVdV

dVA

nc

n2W

Cr So,

. 2W

C ,V V When

exp2

WqApd2

)W(xpqAd

dQC Define area. diode A ,Q'Q

2n

dd

2n

dta

nn0nnn

pdpp

L07 04Feb02 19

General time-constant

np

a

nnnn

a

pppp

pnVa

pn

Va

DQd

CCC ecapacitanc diode total

the and ,dVdQ

Cg and ,dV

dQCg

that so time sticcharacteri a always is There

ggdV

JJdA

dVdI

Vg

econductanc the short, or long diodes, all For

QQ

L07 04Feb02 20

General time-constant (cont.)

times.-life carr. min. respective the

, and side, diode long

the For times. transit charge physical

the ,D2

W and ,

D2W

side, diode short the For

n0np0p

n

2p

transn,np

2n

transp,p

L07 04Feb02 21

General time-constant (cont.)

Fdd

transitminF

gC

and 111

by given average

the is time transition effective The

sided-one usually are diodes Practical

L07 04Feb02 22

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).

L07 04Feb02 23

)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

L07 04Feb02 24

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

L07 04Feb02 25

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.