EE 5340Semiconductor Device TheoryLecture 15 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

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2

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: “Rememberto follow the minority carriers”

tbia

np

pna

tbi

nopo

pono

ponot

nopo

t2i

datbi

VV-Vexpn

npp ,0V when and

,VV-expn

npp get to Invert

.nnlnVp

plnV

nNNlnVV

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Law of the junction (cont.)

dnonapop

ppnnppopppop

nnonnnona

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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Law of the junction (cont.)

ta

pta

n

ta

ta

tbi

tbia

VV

2ixpp

VV

2ixnn

VV

no

2iV

V

ponopo

n

VV

nopoVV-V

pn

ennp also ,ennp

Junction the of Law the

ennepn

np have We

enn nda epp for So

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ptapop

ntanon

VV-

pononoVV-V

pon

tbiaponno

xx at ,1VVexpnn sim.

xx at ,1VVexppp so

,epp ,pepp

giving VV-Vexpppp

tbi

tbia

InjectionConditions

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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 injection applies so that

np < ppo for -xpc < x < -xp, and pn < nno for xn < x < xnc

• Steady State conditions

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Ideal Junction Theory (cont.)

ppcn

ncnp

xxx- ,Jq1

dtdn

tn0

and , xxx ,Jq1

dtdp

tp0

CNR the to Equation Continuity the applying

and , 0tn

tp case, (static) state steady the In

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Ideal JunctionTheory (cont.)

ppc

nnp

2p2

ncnpp

n2n2

ppx

nnxx

xxx- for ,0Dn

dxnd

and ,xxx for ,0Dp

dxpd

giving dxdpqDJ and

dxdnqDJ CNR, the in 0E Since

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Ideal JunctionTheory (cont.)

)contacts( ,0xnxp and

,1enxn

pxp B.C. with

.xxx- ,DeCexn

xxx ,BeAexp

So .D L and D L Define

pcpncn

VVpo

ppno

nn

ppcLxLxp

ncnLxLx

n

pp2pnn2n

ta

nn

pp

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0.1

1.0

10.0

100.0

1000.0

1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20Doping Concentration (cm̂ - 3)

Diff

usio

n Le

ngth

, L

(mic

rons

)electrons holes

Diffusion Length model

2imim

min N36E5.4N18E7.71sec45

L = (D)1/2 Diffusion Coeff. is Pierret* model

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Minority hole lifetimesMark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991

The parameters used in the fit are

τo = 10 μs, Nref = 1×1017/cm2, and CA = 1.8×10-31cm6/s.

2DAorefD

op NCNN1 τ

ττ

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Minority electron lifetimesMark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991

The parameters used in the fit are

τo = 30 μs, Nref = 1×1017/cm2, and CA = 8.3×10-32 cm6/s.

2DAorefD

on NCNN1 τ

ττ

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Excess minoritycarrier distr fctn

1eLWsinhLxxsinhnxn

,xxW ,xxx- for and

1eLWsinhLxxsinhpxp

,xxW ,xxx For

ta

ta

VV

npnpc

pop

ppcpppc

VV

pnpnc

non

nncnncn

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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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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

aVV

popp

1epxp t

aVV

nonn

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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 = qni

2Dn coth(Wp/Ln)/(NaLn) = qni

2Dn/(NaWp), Wp << Ln, “short” = qni

2Dn/(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

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 L10)

excess minority carrier distribution becomes linear for Wn << Lp

pn(xn)=pn0expd(Va/Vt)

• Total chg = Q’p = Q’p = qpn(xn)Wn/2

xn

x

xnc

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 = +qpn(xn,Va)Wn/2

• Q’p =q(W/2) x

{pn(xn,Va+V) -

pn(xn,Va)}• Wn = xnc - 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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Evaluating the diode current density

pnpd

p2isp

npna

n2isn

spsns

VV

spnnpaD

LWcothLN

DqnJ

,LWcothLN

DqnJ

sdefinition the with JJJ where

1eAJAxJxJVi

then DR, in gen/rec no gminAssu

ta

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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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References1 and M&KDevice Electronics for Integrated

Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996, for another treatment of the model.

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

3 and **Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997.

Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.