Semiconductor Device Modeling and Characterization – EE5342 Lecture 09– Spring 2011 Professor...

Semiconductor Device Modeling and

Characterization – EE5342 Lecture 09– Spring 2011

Professor Ronald L. [email protected]

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

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First Assignment

• e-mail to [email protected]– 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 [email protected], 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.

MT and P1 Assignment on Friday, 2/18/11

• Quizzes and tests are open book – must have a legally obtained copy-no

Xerox copies.– OR one handwritten page of notes.– Calculator allowed.

• A cover sheet will be published by Wednesday, 2/16/11.

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Energy bands forp- and n-type s/c

p-typeEc

Ev

EFi

EFp

qfp= kT ln(ni/Na)

Ev

Ec

EFi

EFnqfn= kT ln(Nd/ni)

n-type

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Making contactin a p-n junction• Equate the EF in

the p- and n-type materials far from the junction

• Eo(the free level), Ec, Efi and Ev must be continuous

N.B.: qc = 4.05 eV (Si),

and qf = qc + Ec - EF

Eo

EcEf Efi

Ev

qc (electron affinity)

qfF

qf(work function)

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Band diagram forp+-n jctn* at Va = 0

Ec

EfNEfi

Ev

Ec

EfP

Efi

Ev

0 xn

x-xp

-xpc xnc

qfp < 0

qfn > 0

qVbi = q(fn - fp)

*Na > Nd -> |fp| > fn

p-type for x<0 n-type for x>0

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• A total band bending of qVbi = q(fn-fp) = kT ln(NdNa/ni

2) is

necessary to set EfP = EfN

• For -xp < x < 0, Efi - EfP < -qfp, = |qfp|

so p < Na = po, (depleted of maj. carr.)

• For 0 < x < xn, EfN - Efi < qfn,

so n < Nd = no, (depleted of maj. carr.)

-xp < x < xn is the Depletion Region

Band diagram forp+-n at Va=0 (cont.)

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DepletionApproximation• Assume p << po = Na for -xp < x <

0, so r = q(Nd-Na+p-n) = -qNa, -xp

< x < 0, and p = po = Na for -xpc <

x < -xp, so r = q(Nd-Na+p-n) =

0, -xpc < x < -xp

• Assume n << no = Nd for 0 < x <

xn, so r = q(Nd-Na+p-n) = qNd, 0 <

x < xn, and n = no = Nd for xn < x

< xnc, so r = q(Nd-Na+p-n) =

0, xn < x < xnc

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

related to the charge density r=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)=-dn +dp +ni

2/N

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

2/N

• So neglecting ni2/N, [r=(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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Induced E-fieldin the D.R.• The sheet dipole of charge, due to

Qp’ and Qn’ induces an electric field which must satisfy the conditions

• Charge neutrality and Gauss’ Law* require that Ex = 0 for -xpc < x < -

xp and Ex = 0 for -xn < x < xnc QQAdxEAdVdSE 'p

'n

xx

xxx

VS

n

p

h 0

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Induced E-fieldin the D.R.

xn

x-xp-xpc xnc

O-O-O-

O+O+

O+

Depletion region (DR)

p-type CNR

Ex

Exposed Donor ions

Exposed Acceptor Ions

n-type chg neutral reg

p-contact N-contact

W

0

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Depletion approx.charge distribution

xn

x-xp

-xpc xnc

r+qNd

-qNa

+Qn’=qNdxn

Qp’=-qNaxp

Charge neutrality => Qp’ + Qn’ = 0,

=> Naxp = Ndxn

[Coul/cm2]

[Coul/cm2]

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1-dim soln. ofGauss’ law

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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Depletion Approxi-mation (Summary)• For the step junction defined by

doping Na (p-type) for x < 0 and

Nd, (n-type) for x > 0, the

depletion width W = {2 (Vbi-

Va)/qNeff}1/2, where Vbi = Vt

ln{NaNd/ni2}, and

Neff=NaNd/(Na+Nd). Since

Naxp=Ndxn, xn = W/(1 + Nd/Na),

and xp = W/(1 +

Na/Nd).

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One-sided p+n or n+p jctns• If p+n, then Na >> Nd, and

NaNd/(Na + Nd) = Neff --> Nd,

and W --> xn, DR is all on lightly d. side

• If n+p, then Nd >> Na, and

NaNd/(Na + Nd) = Neff --> Na,

and W --> xp, DR is all on lightly d. side

• The net effect is that Neff --> N-, (- = lightly doped side) and W --> x-

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JunctionC (cont.)

xn

x-xp

-xpc xnc

r+qNd

-qNa

+Qn’=qNdxn

Qp’=-qNaxp

Charge neutrality => Qp’ + Qn’ = 0,

=> Naxp =

Ndxn

dQn’=qNddxn

dQp’=-qNadxp

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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+dVa is

dQ’=-qN(xn)dxn

• The increase in field, dEx

=-(qN/e)dxn, by Gauss’ Law (at xn, but also const).

• So dVa=-(xn+xp)dEx= (W/e) dQ’

• Further, since N(xn)dxn = N(xp)dxp

gives, the dC/dxn as ...

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Arbitrary dopingprofile (cont.)

p

n

j

3j

j

j

n

j

nd

ndj

p

n2j

n

p2

n

j

xNxN

1

dV

'dCq

'C

'CdVd

q

'C

xd

'Cd N with

, dV

'CddC'xd

qNdVxd

qNdVdQ'

'C further

,xN

xN1

'C

dx

dx1

Wdx

'dC

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,VV2

qN'C where , junctionstep

sided-one to apply Now .

dV'dC

q

'C xN

profile doping the ,xN xN orF

abij

3j

n

pn

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)( and ,

12

and

when area),(A and V, , '

,quantities measured of in terms So,

22

0

VCxN

dV

CdqA

NxNxNN

CAC

jnd

j

rapnd

jj

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Debye length• The DA assumes n changes from

Nd to 0 discontinuously at xn,

likewise, p changes from Na to 0

discontinuously at -xp.

• In the region of xn, the 1-dim

Poisson equation is dEx/dx = q(Nd -

n), and since Ex = -df/dx, the

potential is the solution to -d2f/dx2

= q(Nd - n)/e

n

xxn

Nd

0

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Debye length (cont)• Since the level EFi is a reference

for equil, we set f = Vt ln(n/ni)

• In the region of xn, n = ni exp(f/Vt),

so d2f/dx2 = -q(Nd - ni ef/Vt), let

f = fo + f’, where fo = Vt

ln(Nd/ni) so Nd - ni ef/Vt = Nd[1 -

ef/Vt-fo/Vt], for f - fo = f’ << fo, the

DE becomes d2f’/dx2 = (q2Nd/ekT)f’, f’ << fo

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Debye length (cont)• So f’ = f’(xn) exp[+(x-xn)/LD]

+con. and n = Nd ef’/Vt, x ~ xn,

where LD is the “Debye length”

material. intrinsic for 2n and type-p

for N type,-n for N pn :Note

length. transition a ,q

kTV ,

pnqV

L

i

ad

tt

D

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Debye length (cont)• LD estimates the transition length

of a step-junction DR (concentrations

Na and Nd with Neff = NaNd/(Na +Nd)). Thus,

bi

efft

da0V

dDaDV2

NV

N1

N1

W

NLNL

a

• For Va=0, & 1E13 < Na,Nd < 1E19

cm-3

13% < d < 28% => DA is OK

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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=0.33mm

• What is C’j? = 31.9 nFd/cm2

• What is LD? = 0.04 mm

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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.

Semiconductor Device Modeling and Characterization – EE5342 Lecture 09– Spring 2011 Professor...

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