Semiconductor Device Modeling and Characterization EE5342, Lecture 5-Spring 2002

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Semiconductor Device Modeling and CharacterizationEE5342, Lecture 5-Spring 2002

Professor Ronald L. [email protected]

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

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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

• min = (45 sec) 1+(7.7E-18cm3Ni+(4.5E-

36cm6Ni2

• For Nd = 1E17cm3, p = 25 sec

– Why Nd and p ?

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Direct rec forexcess min carr• Define low-level injection as

n = p < no, for n-type, andn = p < po, for p-type

• The recombination rates then areR’n = R’p = n(t)/n0, for p-type,

and R’n = R’p = p(t)/p0, for n-type

• Where n0 and p0 are the minority-carrier lifetimes

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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/(20cosh[(ET-Efi)/kT])

• And with the substitution that the gen lifetime, g = 20cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/g

• 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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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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.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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z).y,(x, at p

or n of Change of Rate Local explicit"" the

is ,tp

or tn

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

dtdn LHS, The

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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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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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Poisson’sEquation• The electric field at (x,y,z) is

related to the charge density =q(Nd-Na-p-n) by the Poisson Equation:

silicon for 7.11

andFd/cm, ,14E85.8

with , ypermitivit the is

xE

E where, ,E

r

o

ro

x

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Poisson’sEquation• For n-type material, N = (Nd - Na) > 0,

no = N, and (Nd-Na+p-n)=-n +p +ni

2/N

• For p-type material, N = (Nd - Na) < 0, po = -N, and (Nd-Na+p-n) = p-n-ni

2/N

• So neglecting ni2/N, [=(Nd-Na+p-n)]

carriers. excess with material type-p

and type-n for ,npq

E

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Quasi-FermiEnergy

used. be must level

Fermi-quasi the then ,nnn i.e.,

m,equilibriu not in ionconcentrat the If

kT

EEexp

nn and ,

nn

lnkTEE

:by given are level Energy Fermi the and

conc carrier mequilibriu the m,equilibriu In

o

fif

i

o

i

ofif

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Quasi-FermiEnergy (cont.)

kT

EE

nnn

nnn

kTEE

fifn

i

o

i

ofifn

exp

:is density carrier the and

, ln

:defined is (Imref) level Fermi-Quasi The

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Quasi-FermiEnergy (cont.)

kT

EE

npp

npp

kTEE

fpfi

i

o

i

ofpfi

exp

:is density carrier the and

, ln

:as defined is

(Imref) level Fermi-Quasi the holes, For

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Energy bands forp- and n-type s/c

p-type

Ec

Ev

EFi

EFp

qp= kT ln(ni/Na)

Ev

Ec

EFi

EFnqn= kT ln(Nd/ni)

n-type

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JunctionC (cont.)

xn

x-xp

-xpc xnc

+qNd

-qNa

+Qn’=qNdxn

Qp’=-qNaxp

Charge neutrality => Qp’ + Qn’ = 0,

=> Naxp =

Ndxn

Qn’=qNdxn

Qp’=-qNaxp

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JunctionC (cont.)• The C-V relationship simplifies to

][Fd/cm ,NNV2

NqN'C herew

equation model a ,VV

1'C'C

2

dabi

da0j

21

bi

a0jj

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JunctionC (cont.)• If one plots [C’j]

-2 vs. Va

Slope = -[(C’j0)2Vbi]-1

vertical axis intercept = [C’j0]-2 horizontal axis intercept = Vbi

C’j-2

Vbi

Va

C’j0-2

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Arbitrary dopingprofile• If the net donor conc, N = N(x), then at xn,

the extra charge put into the DR when Va->Va+Va is Q’=-qN(xn)xn

• The increase in field, Ex =-(qN/)xn, by Gauss’ Law (at xn, but also const).

• So Va=-(xn+xp)Ex= (W/) Q’

• Further, since N(xn)xn = N(xp)xp gives, the dC/dxn as ...

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Arbitrary dopingprofile (cont.)

p

n3j

nn

p

n2j

n

p2n

xNxN

1

dVdC

q

'C

dCVd

qC

dxCd

N with

,dVCd

dCxd

qNdVxd

qNdVdQ

C further

,xN

xN1

'C

dx

dx1

WdxdC

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,VV2

qN'C where , junctionstep

sided-one to apply Now .

dVdC

q

'C xN

profile doping the ,xN xN orF

abij

3j

n

pn

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bi0j

bi

23

bi

a0j

23

bi

a30j

V2qN

'C when ,N

V1

VV

121

'qC

VV

1'C

N so

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Example

• An assymetrical p+ n junction has a lightly doped concentration of 1E16 and with p+ = 1E18. What is W(V=0)?

Vbi=0.816 V, Neff=9.9E15, W=0.33m

• What is C’j? = 31.9 nFd/cm2

• What is LD? = 0.04 m

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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.)

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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Law of the junction (cont.)

t

a

pt

a

n

t

a

t

a

t

bi

t

bia

VV

2ixpp

VV

2ixnn

VV

no

2iV

V

pono

pon

VV

nopoVV-V

pn

ennp also ,ennp

Junction the of Law the

enn

epn

np have We

enn nda epp for So

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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

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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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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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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)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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References

• 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.

Semiconductor Device Modeling and Characterization EE5342, Lecture 5-Spring 2002

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