Semiconductor Device Modeling and

Characterization – EE5342 Lecture 8 – Spring 2011

Professor Ronald L. Carterronc@uta.edu

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

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First Assignment

• e-mail to listserv@listserv.uta.edu– In the body of the message include

subscribe EE5342 • This will subscribe you to the

EE5342 list. Will receive all EE5342 messages

• If you have any questions, send to ronc@uta.edu, with EE5342 in subject line.

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Second Assignment

• Submit a signed copy of the document that is posted at

www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf

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Additional University Closure Means More Schedule

Changes• Plan to meet until noon some days in the next few weeks. This way we will make up for the lost time. The first extended class will be Monday, 2/14.

• The MT changed to Friday 2/18• The P1 test changed to Friday 3/11.• The P2 test is still Wednesday 4/13• The Final is still Wednesday 5/11.

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Shockley-Read-Hall Recomb

Ev

EcEfEfi

E

k

Ec

Ev

ET

Indirect, like Si, so intermediate state

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S-R-H trapcharacteristics1

• The Shockley-Read-Hall Theory requires an intermediate “trap” site in order to conserve both E and p

• If trap neutral when orbited (filled) by an excess electron - “donor-like”

• Gives up electron with energy Ec - ET

• “Donor-like” trap which has given up the extra electron is +q and “empty”

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S-R-H trapchar. (cont.)• If trap neutral when orbited (filled)

by an excess hole - “acceptor-like” • Gives up hole with energy ET - Ev

• “Acceptor-like” trap which has given up the extra hole is -q and “empty”

• Balance of 4 processes of electron capture/emission and hole capture/ emission gives the recomb rates

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S-R-H recombination• Recombination rate determined by:

Nt (trap conc.),vth (thermal vel of the carriers),sn (capture cross sect for electrons),sp (capture cross sect for holes), with

tno = (Ntvthsn)-1, and tpo = (Ntvthsn)-1, where sn~p(rBohr)2

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S-R-Hrecomb. (cont.)• In the special case where tno = tpo

= to the net recombination rate, U is

)pn( ,ppp and ,nnn wherekT

EfiEcoshn2npnpnU

dtpd

dtndGRU

oo

oTi

2i

t

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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 = ni

2) • For n-type (no > n = p > po =

ni2/no):

(np-ni2) = (no+n)(po+p)-ni

2 = nopo - ni

2 + nop + npo + np ~ nop (largest term)

• Similarly, for p-type, (np-ni2) ~ pon

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S-R-H “U” functioncharacteristics (cont)• For n-type, as above, the

denominator = to{no+n+po+p+2nicosh[(Et-Ei)kT]}, simplifies to the smallest value for Et~Ei, where the denom is tono, giving U = p/to as the largest (fastest)

• For p-type, the same argument gives U = n/to

• Rec rate, U, fixed by minority carrier

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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 wherekT

EfiEcoshn2npnpnU

dtpd

dtndGRU

oo

oTi

2i

t

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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 > n = p > po = ni

2/no), U = p/to, (prop to exc min carr)

• For p-type low-level injection and net excess minority carriers, (i.e., po > n = p > no = ni

2/po), U = n/to, (prop to exc min carr)

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Minority hole lifetimes. Taken from Shur3, (p.101).

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Minority electron lifetimes. Taken from Shur3, (p.101).

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Parameter example• tmin = (45 msec)

1+(7.7E-18cm3Ni+(4.5E-36cm6Ni

2

• For Nd = 1E17cm3, tp = 25 msec– Why Nd and tp ?

M. E. Law, E. Solley, M. Liang, and D. E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility,” IEEE Electron

Device Lett., vol. 12, pp. 401-403, 1991.

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M. E. Law, E. Solley, M. Liang, and D. E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility,” IEEE Electron

Device Lett., vol. 12, pp. 401-403, 1991.

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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,kzjyixn

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

.kdtdzjdt

dyidtdxv

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 ,Jq

1tn

dtdn

So .Jq1

tnvnt

ndtdn

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 ,tpor t

n RHS, the on term first The

z).y,(x, space in point particular a at p or n of Rate Generation Net the represents

Eq. Continuity the of -V,dtdp or dt

dn LHS, The

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The ContinuityEquation (cont.)

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

inflow of rate rate generation net change of rate Local

:as dinterprete be can Which

Jq1

dtdp

tp

:as holes the for equation continuity the write-re can we So,

p

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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.