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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

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Forward Bias Energy Bands

1eppkT/EEexpnp ta VV0nnFpFiiequilnon

1/exp 0 ta VV

ppFiFniequilnon ennkTEEnn

Ev

Ec

EFi

xn xnc-xpc -xp 0

q(Vbi-Va)

EFPEFNqVa

x

Imref, EFn

Imref, EFp

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Law of the junction(follow the min. carr.)

t

bia

n

p

p

na

t

bi

no

po

po

no

po

not

no

pot2

i

datbi

V

V-Vexp

n

n

pp

,0V when and

,V

V-exp

n

n

pp

get to Invert

.nn

lnVp

plnV

n

NNlnVV

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Law of the junction (cont.)

t

a

p

t

a

n

t

a

t

a

t

bi

t

bia

Vixpp

Vixnn

V

no

iVp

no

pon

Vnopo

Vpn

ennpennp

en

nep

n

np

ennaepp

V

2

V

2

V2V

VV-V

also ,

Junction theof Law the

have We

nd for So

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Law of the junction (cont.)

dnonapop

ppnn

ppopppop

nnonnnon

a

Nnn and Npp

injection level- low Assume

.pn and pn Assume

.ppp ,nnn and

,nnn ,ppp So

. 0V for nnot' eq.-non to Switched

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pt

apop

nt

anon

V

V-

pononoV

V-V

pon

t

biaponno

xx at ,1VV

expnn sim.

xx at ,1VV

exppp so

,epp ,pepp

giving V

V-Vexpppp

t

bi

t

bia

InjectionConditions

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Ideal JunctionTheory (cont.)

Apply the Continuity Eqn in CNR

ncnn

ppcp

xxx ,Jq1

dtdn

tn

0

and

xxx- ,Jq1

dtdp

tp

0

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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

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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

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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

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Carrier Injection

-xp

xn-xpc 0

ln(carrier conc)ln Naln Nd

ln ni

ln ni2/Nd

ln ni2/Na

xnc

x

~Va/Vt~Va/Vt

1enxn t

aV

V

popp

1epxp t

aV

V

nonn

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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

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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

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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

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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 QVa

DD dV

dIg

t

asD V

VdexpII

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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

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Charge distr in a (1-sided) short diode

• Assume Nd << Na

• The sinh excess minority carrier distribution becomes linear for Wn << Lp

pn(xn)=pn0expd(Va/Vt)

• Total chg = Q’p = Q’p = qpn(xn)Wn/2x

n

x

xnc

pn(xn

)

Wn = xnc-

xn

Q’p

pn

20

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

21

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

1epxp t

aV

V

nonn

nd

pisp

pa

nisn WN

DqnJ

WN

DqnJ 22 and ,

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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

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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 ,D2

W and ,

D2W

side, diode short the For

n0np0p

n

2p

transn,np

2n

transp,p

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General time-constant (cont.)

Fdd

transitminF

gC

and 111

by given average

the is time transition effective The

sided-one usually are diodes Practical

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References *Fundamentals of Semiconductor Theory and Device

Physics, by Shyh Wang, Prentice Hall, 1989. **Semiconductor Physics & Devices, by Donald A.

Neamen, 2nd ed., Irwin, Chicago. M&K = Device Electronics for Integrated Circuits, 3rd

ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003.

• 1Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986.

• 2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.

• 3 Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.