1. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
EE5342 – Semiconductor Device Modeling and Characterization Lecture 05-Spring 2010
Professor Ronald L. Carter

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4. Direct carrier gen/recombk Ec Ev
(Excitation can be by light)
gen
rec - + Ev Ec Ef
Efi
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5. Direct gen/rec of excess carriers
Generation rates, Gn0 = Gp0
Recombination rates, Rn0 = Rp0
In equilibrium: Gn0 = Gp0 = Rn0 = Rp0
In non-equilibrium condition:
n = no + dn and p = po + dp, where nopo=ni2
and for dn and dp > 0, the recombination rates increase to R’n and R’p
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6. Direct rec for low-level injection
Define low-level injection as dn = dp < no, for n-type, and dn = dp < po, for p-type
The recombination rates then are R’n = R’p = dn(t)/tn0, for p-type, and R’n = R’p = dp(t)/tp0, for n-type
Where tn0 and tp0 are the minority-carrier lifetimes
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7. Shockley-Read- Hall Recomb
Indirect, like Si, so intermediate state Ec Ec ET Ef
Efi Ev Ev k
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8. S-R-H trap characteristics1
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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9. S-R-H trap char. (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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10. tpo = (Ntvthsn)-1, where sn~p(rBohr)2
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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11. S-R-H recomb. (cont.)
In the special case where tno = tpo = to the net recombination rate, U is
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12. S-R-H “U” function characteristics
The numerator, (np-ni2) simplifies in the case of extrinsic material at low level injection (for equil., nopo = ni2)
For n-type (no > dn = dp > po = ni2/no):
(np-ni2) = (no+dn)(po+dp)-ni2 = nopo - ni2 + nodp + dnpo + dndp ~ nodp (largest term)
Similarly, for p-type, (np-ni2) ~ podn
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13. S-R-H “U” function characteristics (cont)
For n-type, as above, the denominator = to{no+dn+po+dp+2nicosh[(Et-Ei)kT]}, simplifies to the smallest value for Et~Ei, where the denom is tono, giving U = dp/to as the largest (fastest)
For p-type, the same argument gives U = dn/to
Rec rate, U, fixed by minority carrier
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14. S-R-H net recom- bination rate, U
In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is
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15. S-R-H rec for excess min carr
For n-type low-level injection and net excess minority carriers, (i.e., no > dn = dp > po = ni2/no),
U = dp/to, (prop to exc min carr)
For p-type low-level injection and net excess minority carriers, (i.e., po > dn = dp > no = ni2/po),
U = dn/to, (prop to exc min carr)
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16. Minority hole lifetimes. Taken from Shur3, (p.101).
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17. Minority electron lifetimes. Taken from Shur3, (p.101).
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18. Parameter example tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2
For Nd = 1E17cm3, tp = 25 msec
Why Nd and tp ?
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19. S-R-H rec for deficient 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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20. The Continuity Equation
The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives
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21. The Continuity Equation (cont.)
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22. The Continuity Equation (cont.)
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23. The Continuity Equation (cont.)
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24. The Continuity Equation (cont.)
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25. The Continuity Equation (cont.)
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26. The Continuity Equation (cont.)
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27. 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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28. 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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29. 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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30. 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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31. 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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32. 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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33. 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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34. Quasi-Fermi Energy
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35. Quasi-Fermi Energy (cont.)
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36. Quasi-Fermi Energy (cont.)
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37. 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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38. 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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39. 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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40. 1-dim soln. of Gauss’ law Ex
-xpc
-xp xn
xnc x
-Emax
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41. 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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42. 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.
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