EE5342 – Semiconductor Device Modeling and Characterization Lecture 05-Spring 2010

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

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4. E - - Ec Ec Ef Efi gen rec Ev Ev + + k Direct carriergen/recomb (Excitation can be by light)

5. Direct gen/recof 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

6. Direct rec forlow-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

7. Shockley-Read-Hall Recomb E Indirect, like Si, so intermediate state Ec Ec ET Ef Efi Ev Ev k

8. S-R-H trapcharacteristics1 • 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”

9. S-R-H trapchar. (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

10. 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

11. S-R-Hrecomb. (cont.) • In the special case where tno = tpo = to the net recombination rate, U is

12. S-R-H “U” functioncharacteristics • 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

13. S-R-H “U” functioncharacteristics (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

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

15. S-R-H rec forexcess 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)

16. Minority hole lifetimes. Taken from Shur3, (p.101).

17. Minority electron lifetimes. Taken from Shur3, (p.101).

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 ?

19. S-R-H rec fordeficient 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

20. The ContinuityEquation • The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives

21. The ContinuityEquation (cont.)

22. The ContinuityEquation (cont.)

23. The ContinuityEquation (cont.)

24. The ContinuityEquation (cont.)

25. The ContinuityEquation (cont.)

26. The ContinuityEquation (cont.)

27. p-type Ec Ec Ev EFn qfn= kT ln(Nd/ni) EFi Ev Energy bands forp- and n-type s/c n-type EFi qfp= kT ln(ni/Na) EFp

28. Eo Making contactin 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 qc(electron affinity) qf (work function) Ec Ef Efi qfF Ev

29. EfN Band diagram forp+-n jctn* at Va = 0 Ec qVbi = q(fn -fp) qfp < 0 Ec Efi EfP Ev Efi qfn > 0 *Na > Nd -> |fp|> fn Ev p-type for x<0 n-type for x>0 x -xpc xn 0 -xp xnc

30. Band diagram forp+-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

31. DepletionApproximation • Assume p << po = Nafor -xp < x < 0, so r = q(Nd-Na+p-n) = -qNa, -xp < x < 0, and p = po = Nafor -xpc < x < -xp, so r = q(Nd-Na+p-n) = 0, -xpc < x < -xp • Assume n << no = Ndfor 0 < x < xn, so r = q(Nd-Na+p-n) = qNd, 0 < x < xn, and n = no = Ndfor xn < x < xnc, so r = q(Nd-Na+p-n) = 0, xn < x < xnc

32. Poisson’sEquation • The electric field at (x,y,z) is related to the charge density r=q(Nd-Na-p-n) by the Poisson Equation:

33. Poisson’sEquation • 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)]

34. Quasi-FermiEnergy

35. Quasi-FermiEnergy (cont.)

36. Quasi-FermiEnergy (cont.)

37. Depletion approx.charge distribution r +Qn’=qNdxn +qNd [Coul/cm2] -xp x -xpc xn xnc Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn -qNa Qp’=-qNaxp [Coul/cm2]

38. Induced E-fieldin 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 thatEx = 0 for -xpc < x < -xp and Ex = 0 for -xn < x < xnc h0

39. O O O O O O + + + - - - Induced E-fieldin the D.R. Ex N-contact p-contact p-type CNR n-type chg neutral reg Depletion region (DR) Exposed Donor ions Exposed Acceptor Ions W x -xpc -xp xn xnc 0

40. 1-dim soln. ofGauss’ law Ex -xp xn xnc -xpc x -Emax

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).

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.