Semiconductor Device Modeling and Characterization EE5342, Lecture 7-Spring 2004

1. Semiconductor Device Modeling and CharacterizationEE5342, Lecture 7-Spring 2004 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. MidTerm andProject Tests • MidTerm on Thursday 2/12 • Cover sheet to be posted at http://www.uta.edu/ronc/5342/tests/ • Project 1 draft assignment will be posted 2/13. • Project report to be used in doing: • Project 1 Test on Thursday 3/11 • Cover sheet will be posted as above

3. Ideal diodeequation • Assumptions: • low-level injection • Maxwell Boltzman statistics • Depletion approximation • Neglect gen/rec effects in DR • Steady-state solution only • Current dens, Jx = Js expd(Va/Vt) • where expd(x) = [exp(x) -1]

4. Ideal diodeequation (cont.) • Js = Js,p + Js,n = hole curr + ele curr Js,p = qni2Dp coth(Wn/Lp)/(NdLp) = qni2Dp/(NdWn), Wn << Lp, “short” = qni2Dp/(NdLp), Wn >> Lp, “long” Js,n = qni2Dn coth(Wp/Ln)/(NaLn) = qni2Dn/(NaWp), Wp << Ln, “short” = qni2Dn/(NaLn), Wp >> Ln, “long” Js,n << Js,p when Na >> Nd

5. Diffnt’l, one-sided diode cond. (cont.)

6. Cap. of a (1-sided) short diode (cont.)

7. General time-constant

8. General time-constant (cont.)

9. General time-constant (cont.)

10. Effect of non-zero E in the CNR • This is usually not a factor in a short diode, but when E is finite -> resistor • In a long diode, there is an additional ohmic resistance (usually called the parasitic diode series resistance, Rs) • Rs = L/(nqmnA) for a p+n long diode. • L=Wn-Lp (so the current is diode-like for Lp and the resistive otherwise).

11. Effect of carrierrecombination in DR • The S-R-H rate (tno = tpo = to) is

12. Effect of carrierrec. in DR (cont.) • For low Va ~ 10 Vt • In DR, n and p are still > ni • The net recombination rate, U, is still finite so there is net carrier recomb. • reduces the carriers available for the ideal diode current • adds an additional current component

13. Effect of carrierrec. in DR (cont.)

14. High level injection effects • Law of the junction remains in the same form, [pnnn]xn=ni2exp(Va/Vt), etc. • However, now dpn = dnn become >> nno = Nd, etc. • Consequently, the l.o.t.j. reaches the limiting form dpndnn = ni2exp(Va/Vt) • Giving, dpn(xn) = niexp(Va/(2Vt)), or dnp(-xp) = niexp(Va/(2Vt)),

15. High level injeffects (cont.)

16. Summary of Va > 0 current density eqns. • Ideal diode, Jsexpd(Va/(hVt)) • ideality factor, h • Recombination, Js,recexp(Va/(2hVt)) • appears in parallel with ideal term • High-level injection, (Js*JKF)1/2exp(Va/(2hVt)) • SPICE model by modulating ideal Js term • Va = Vext - J*A*Rs = Vext - Idiode*Rs

17. ln(J) Plot of typical Va > 0 current density equations data Effect of Rs Vext VKF

18. Reverse bias (Va<0)=> carrier gen in DR • Va< 0 gives the net rec rate, U = -ni/2t0, t0 = mean min carr g/r l.t.

19. Reverse bias (Va< 0),carr gen in DR (cont.)

20. Reverse biasjunction 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

21. Reverse biasjunction 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

22. Reverse biasjunction breakdown

23. Ecrit for reverse breakdown (M&K**) Taken from p. 198, M&K** Casey Model for Ecrit

24. Junction curvatureeffect 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

25. BV for reverse breakdown (M&K**) Taken from Figure 4.13, p. 198, M&K** Breakdown voltage of a one-sided, plan, silicon step junction showing the effect of junction curvature.4,5

26. rp rpc rj rn rnc Gauss’ Law

27. Spherical DiodeFields calculations For rj < ro ≤ rn, Setting Er = 0 at r= rn, we get Note that the equivalent of the lever law for this spherical diode is

28. Spherical DiodeFields calculations Assume Na >> Nd, so rn – rj d >> rj – rp. Further, setting the usual definition for the potential difference, and evaluating the potential difference at breakdown, we have PHIi – Va = BV and Emax = Em = Ecrit = Ec. We also define a = 3eEm/qNd[cm].

29. Showing therj  ∞ limit C1. Solve for rn – rj = d as a function of Emax and solve for the value of d in the limit of rj. The solution for rn is given below. .

30. Solving for theBreakdown (BV) Solve for BV = [fi – Va]Emax = Ecrit, and solve for the value of BV in the limit of rj. The solution for BV is given below .

31. Spherical diodeBreakdown Voltage

32. Example calculations • Assume throughout that p+n jctn with Na = 3e19cm-3 and Nd = 1e17cm-3 • From graph of Pierret mobility model, mp = 331 cm2/V-sec and Dp = Vtmp = ? • Why mp and Dp? • Neff = ? • Vbi = ?

34. Parameters forexamples • Get tmin from the model used in Project 2 tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2 • For Nd = 1E17cm3, tp = 25 msec • Why Nd and tp ? • Lp = ?

35. Hole lifetimes, taken from Shur***, p. 101.

36. Example • Js,long, = ? • If xnc, = 2 micron, Js,short, = ?

37. Example(cont.) • Estimate VKF • Estimate IKF

38. Example(cont.) • Estimate Js,rec • Estimate Rs if xnc is 100 micron

39. Example(cont.) • Estimate Jgen for 10 V reverse bias • Estimate BV

40. Diode equivalentcircuit (small sig) ID h is the practical “ideality factor” IQ VD VQ

41. Small-signal eqcircuit Cdiff and Cdepl are both charged by Va = VQ Va rdiff Cdepl Cdiff

42. Diode Switching • Consider the charging and discharging of a Pn diode • (Na > Nd) • Wd << Lp • For t < 0, apply the Thevenin pair VF and RF, so that in steady state • IF = (VF - Va)/RF, VF >> Va , so current source • For t > 0, apply VR and RR • IR = (VR + Va)/RR, VR >> Va, so current source

43. Diode switching(cont.) VF,VR >> Va F: t < 0 Sw RF R: t > 0 VF + RR D VR +

44. Diode chargefor t < 0 pn pno x xn xnc

45. Diode charge fort >>> 0 (long times) pn pno x xn xnc

46. Equationsummary

47. Snapshot for tbarely > 0 pn Total charge removed, Qdis=IRt pno x xn xnc

48. I(t) for diodeswitching ID IF ts ts+trr t - 0.1 IR -IR

49. References * Semiconductor Physics and Devices, 2nd ed., by Neamen, Irwin, Boston, 1997. **Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, John Wiley, New York, 1986. ***Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990.