1. EE 5340 Semiconductor Device Theory Lecture 13 – Spring 2011
Professor Ronald L. Carter

2. Doping Profile
If the net donor conc, N = N(x), then at x, the extra charge put into the DR when Va->Va+dVa is dQ’=-qN(x)dx
The increase in field, dEx =-(qN/e)dx, by Gauss’ Law (at x, but also all DR).
So dVa=-xddEx= (W/e) dQ’
Further, since qN(x)dx, for both xn and xn, we have the dC/dx as ...
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3. Arbitrary doping profile (cont.)
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4. Arbitrary doping profile (cont.)
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5. Arbitrary doping profile (cont.)
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6. Arbitrary doping profile (cont.)
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7. 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’j0? = 31.9 nFd/cm2
What is LD? = 0.04 mm
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8. Reverse bias junction breakdown
Avalanche breakdown
Electric field accelerates electrons to sufficient energy to initiate multiplication of impact ionization of valence bonding electrons
field dependence shown on next slide
Heavily doped narrow junction will allow tunneling - see Neamen*, p. 274
Zener breakdown
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9. Reverse bias junction breakdown
Assume -Va = VR >> Vbi, so Vbi-Va-->VR
Since Emax~ 2VR/W = (2qN-VR/(e))1/2, and VR = BV when Emax = Ecrit (N- is doping of lightly doped side ~ Neff)
BV = e (Ecrit )2/(2qN-)
Remember, this is a 1-dim calculation
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10. Effect of V  0
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11. Reverse bias junction breakdown
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12. Ecrit for reverse breakdown [M&K]
Taken from p. 198, M&K**
Casey 2model for Ecrit
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13. Table 4.1 (M&K* p. 186) Nomograph for silicon uniformly doped, one-sided, step junctions (300 K). (See Figure 4.15 to correct for junction curvature.) (Courtesy Bell Laboratories).
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14. Junction curvature effect on breakdown
The field due to a sphere, R, with charge, Q is Er = Q/(4per2) for (r > R)
V(R) = Q/(4peR), (V at the surface)
So, for constant potential, V, the field, Er(R) = V/R (E field at surface increases for smaller spheres)
Note: corners of a jctn of depth xj are like 1/8 spheres of radius ~ xj
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16. Direct carrier gen/recombk Ec Ev
(Excitation can be by light)
gen
rec - + Ev Ec Ef
Efi
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17. 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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18. 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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19. Shockley-Read- Hall Recomb
Indirect, like Si, so intermediate state Ec Ec ET Ef
Efi Ev Ev k
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20. S-R-H trap characteristics*
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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21. 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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22. tpo = (Ntvthsp)-1, where sn,p~p(rBohr,n.p)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 = (Ntvthsp)-1, where sn,p~p(rBohr,n.p)2
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23. 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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24. 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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25. References
1 and M&KDevice Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996, for another treatment of the m model.
2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.
3 and **Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997.
Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.
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