1. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Semiconductor Device Modeling and Characterization – EE5342 Lecture 09– Spring 2011
Professor Ronald L. Carter

2. 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 with EE5342 in subject line.
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3. Second Assignment
Submit a signed copy of the document that is posted at
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4. 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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5. 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.
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6. Energy bands for p- and n-type s/c
p-type
n-type Ec Ev Ec
EFi
EFn
qfn= kT ln(Nd/ni)
EFi
qfp= kT ln(ni/Na)
EFp Ev
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7. Making contact in 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
qc (electron affinity) qf
(work function) Ec Ef
Efi
qfF Ev
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8. Band diagram for p+-n jctn* at Va = 0Ec
qVbi = q(fn - fp)
qfp < 0
Efi Ec
EfP
EfN Ev
Efi
qfn > 0
*Na > Nd -> |fp| > fn Ev
p-type for x<0
n-type for x>0 x
-xpc
-xp xn
xnc
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9. Band diagram for p+-n at Va=0 (cont.)
A total band bending of qVbi = q(fn-fp) = kT ln(NdNa/ni2) 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
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10. Depletion Approximation
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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11. Poisson’s Equation
The electric field at (x,y,z) is related to the charge density r=q(Nd-Na-p-n) by the Poisson Equation:
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12. Poisson’s Equation
For n-type material, N = (Nd - Na) > 0, no = N, and (Nd-Na+p-n)=-dn +dp +ni2/N
For p-type material, N = (Nd - Na) < 0, po = -N, and (Nd-Na+p-n) = dp-dn-ni2/N
So neglecting ni2/N, [r=(Nd-Na+p-n)]
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13. Quasi-Fermi Energy
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14. Quasi-Fermi Energy (cont.)
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15. Quasi-Fermi Energy (cont.)
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16. Induced E-field in 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
h 0
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17. Induced E-field in the D.R.Ex
p-contact
N-contact O - O +
p-type CNR
n-type chg neutral reg O - O + O - O +
Exposed Acceptor Ions
Depletion region (DR)
Exposed Donor ions W x
-xpc
-xp xn
xnc
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18. Depletion approx. charge distribution
+Qn’=qNdxn
+qNd
[Coul/cm2]
-xp x
-xpc xn
xnc
-qNa
Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn
Qp’=-qNaxp
[Coul/cm2]
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19. 1-dim soln. of Gauss’ law Ex -xpc -xp xn xnc x -Emax
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20. 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 = {2e(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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21. 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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22. Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn
Junction C (cont.) r
+Qn’=qNdxn
+qNd
dQn’=qNddxn
-xp x
-xpc xn
xnc
-qNa
Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn
dQp’=-qNadxp
Qp’=-qNaxp
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23. Junction C (cont.) The C-V relationship simplifies to
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24. Junction C (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
C’j0-2 Va
Vbi
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25. Arbitrary doping profile
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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26. Arbitrary doping profile (cont.)
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27. Arbitrary doping profile (cont.)
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28. Arbitrary doping profile (cont.)
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29. Arbitrary doping profile (cont.)
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30. n x xn Nd
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
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31. 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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32. 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”
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33. 13% < d < 28% => DA is OK
Debye length (cont)
LD estimates the transition length of a step-junction DR (concentrations Na and Nd with Neff = NaNd/(Na +Nd)). Thus,
For Va=0, & 1E13 < Na,Nd < 1E19 cm-3
13% < d < 28% => DA is OK
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34. 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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35. 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.
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