EE 5340 Semiconductor Device Theory Lecture 13 - Fall 2003 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc L 13 Oct 7

Direct carrier gen/recomb k Ec Ev (Excitation can be by light) gen rec - + Ev Ec Ef Efi L 13 Oct 7

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 L 13 Oct 7

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 L 13 Oct 7

Shockley-Read- Hall Recomb Indirect, like Si, so intermediate state Ec Ec ET Ef Efi Ev Ev k L 13 Oct 7

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” L 13 Oct 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 L 13 Oct 7

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 L 13 Oct 7

S-R-H net recom- bination rate, U In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is L 13 Oct 7

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 L 13 Oct 7

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 L 13 Oct 7

Minority electron lifetimes, taken from Shur** p. 101. L 13 Oct 7

Minority hole lifetimes, taken from Shur** p. 101. L 13 Oct 7

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) L 13 Oct 7

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 L 13 Oct 7

The Continuity Equation The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives L 13 Oct 7

The Continuity Equation (cont.) L 13 Oct 7

The Continuity Equation (cont.) L 13 Oct 7

The Continuity Equation (cont.) L 13 Oct 7

The Continuity Equation (cont.) L 13 Oct 7

The Continuity Equation (cont.) L 13 Oct 7

The Continuity Equation (cont.) L 13 Oct 7

References * Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, Wiley, New York, 1986. ** Physics of Semiconductor Devices, M. Shur, Wiley. L 13 Oct 7