EE 5340Semiconductor Device TheoryLecture 14 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

http://www.uta.edu/ronc

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S-R-H net recom-bination rate, U• In the special case where tno = tpo

= to = (Ntvthso)-1 the net rec. rate, U is

)pn( ,ppp and ,nnn where

kTEfiE

coshn2np

npnU

dtpd

dtnd

GRU

oo

oT

i

2i

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S-R-H “U” functioncharacteristics• The numerator, (np-ni

2) simplifies in the case of extrinsic material at low level injection (for equil., nopo

= ni2)

• For n-type (no > dn = dp > po = ni

2/no):

(np-ni2) = (no+dn)(po+dp)-ni

2 = nopo - ni

2 + nodp + dnpo + dndp ~ nodp (largest term)

• Similarly, for p-type, (np-ni2) ~

podn

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S-R-H rec forexcess min carr• For n-type low-level injection and

net excess minority carriers, (i.e., no > dn = dp > po = ni

2/no),

U = dp/tp, (prop to exc min carr)• For p-type low-level injection and

net excess minority carriers, (i.e., po > dn = dp > no = ni

2/po),

U = dn/tn, (prop to exc min carr)

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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Minority Carrier Lifetime, Diffusion Length and Mobility Models in Silicon

A. [40%] Write a review of the model equations for minority carrier (both electrons in p-type and holes in n-type material) lifetime, mobility and diffusion length in silicon. Any references may be used. At a minimum the material given in the following references should be used.Based on the information in these resources, decide

which model formulae and parameters are the most accurate for Dn and Ln for electrons in p-type material, and Dp and Lp holes in n-type material.

 B. [60%] This part of the assignment will be given by

10/12/09. Current-voltage data will be given for a diode, and the project will be to determine the material parameters (Nd, Na, charge-neutral region width, etc.) of the diode.©rlc L14-

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References for Part ADevice Electronics for Integrated Circuits, 3rd ed., by Richard S.

Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003.

Mark 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.

D.B.M. Klaassen; “A UNIFIED MOBILITY MODEL FOR DEVICE SIMULATION”, Electron Devices Meeting, 1990. Technical Digest., International 9-12 Dec. 1990 Page(s):357 – 360.

David Roulston, Narain D. Arora, and Savvas G. Chamberlain “Modeling and Measurement of Minority-Carrier Lifetime versus Doping in Diffused Layers of n+-p Silicon Diodes”, IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-29, NO. 2, FEBRUARY 1982, pages 284-291.

M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination in Heavily Doped Silicon”, Solid-State Electr. Vol. 26, pp. 577-597, 1983. Download a copy at Tyagi.pdf.

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S-R-H rec fordeficient min carr• If n < ni and p < pi, then the S-R-H

net recomb rate becomes (p < po, n < no):

U = R - G = - ni/(2t0cosh[(ET-Efi)/kT])• And with the substitution that the

gen lifetime, tg = 2t0cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/tg

• The intrinsic concentration drives the return to equilibrium

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The ContinuityEquation• The chain rule for the total time

derivative dn/dt (the net generation rate of electrons) gives

n,kz

jy

ix

n

is gradient the of definition The

.dtdz

zn

dtdy

yn

dtdx

xn

tn

dtdn

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The ContinuityEquation (cont.)

vntn

dtdn then

,BABABABA Since

.kdtdz

jdtdy

idtdx

v

is velocity vector the of definition The

zzyyxx

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The ContinuityEquation (cont.)

etc. ,0xx

dtd

dtdx

x

since ,0dtdz

zdtdy

ydtdx

xv

RHS, the on term second the gConsiderin

.vnvnvn as

ddistribute be can operator gradient The

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The ContinuityEquation (cont.)

.Equations" Continuity" the are

Jq1

tp

dtdp and ,J

q1

tn

dtdn

So .Jq1

tn

vntn

dtdn

have we ,vqnJ since ly,Consequent

pn

n

n

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The ContinuityEquation (cont.)

z).y,(x,at p

or n of Change of Rate Local explicit"" the

is , RHS, on the first term The

z).y,(x, spacein point particular aat por

n of Rate GenerationNet therepresents

Eq. Continuity theof -U,or LHS, The

t

port

n

dt

dp

dt

dn

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The ContinuityEquation (cont.)

q).( holes and (-q) electrons for signs

in difference the Note z).y,(x, point

the of" out" flowing ionsconcentrat

p or n of rate local the is Jq1

or

Jq1

RHS, the on term second The

p

n

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The ContinuityEquation (cont.)

inflowof rate rate generation net changeof rate Local

:as dinterprete be can Which

n

Udtdn

where , Jq1

dtdn

tn

and p

Udtdp

where , Jq1

dtdp

tp

:as equations continuity the write-re can we So,

n

n

p

p

τ

δ

τ

δ

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Review of depletion approximation

Depletion Approx.• pp << ppo, -xp < x

< 0• nn << nno, 0 < x

< xn

• 0 > Ex > -2Vbi/W,

in DR (-xp < x <

xn)

• pp=ppo=Na &

np=npo= ni2/Na, -

xpc< x < -xp

• nn=nno=Nd &

pn=pno= ni2/Nd, xn

< x < xnc

xxn xnc-xpc-xp

0

Ev

Ec

qVbi

EFi

EFn

EFp

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Review of D. A. (cont.)

nx

nnax

ppax

px

ndpada

daeff

npeff

bi

xx ,0E

,xx0 ,xxNq E

,0xx ,xxNq

- E

xx ,0E

,xNxN ,NN

NNN

,xxW ,qN

VaV2W

xxn xn

c

-xpc-xp

Ex

-Emax

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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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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 m 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.