1. EE 5340 Semiconductor Device Theory Lecture 13 - Fall 2003
Professor Ronald L. Carter
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2. Direct carrier gen/recombk Ec Ev
(Excitation can be by light)
gen
rec - + Ev Ec Ef
Efi
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3. 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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4. 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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5. Shockley-Read- Hall Recomb
Indirect, like Si, so intermediate state Ec Ec ET Ef
Efi Ev Ev k
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6. 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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7. 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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8. 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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9. 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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10. 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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11. 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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12. Minority electron lifetimes, taken from Shur** p. 101.
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13. Minority hole lifetimes, taken from Shur** p. 101.
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14. 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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15. 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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16. The Continuity Equation
The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives
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17. The Continuity Equation (cont.)
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18. The Continuity Equation (cont.)
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19. The Continuity Equation (cont.)
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20. The Continuity Equation (cont.)
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21. The Continuity Equation (cont.)
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22. The Continuity Equation (cont.)
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23. References
* Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, Wiley, New York, 1986.
** Physics of Semiconductor Devices, M. Shur, Wiley.
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