Understanding Carrier Recombination Effects in Diode Modeling and Characterization

1. Semiconductor Device Modeling and CharacterizationEE5342, Lecture 9 -Spring 2010 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

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

3. 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

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

5. 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).

6. 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)),

7. High level injeffects (cont.)

8. 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

9. Diode Diffusion and Recombination Currents

10. Diode Diffusion and Recombination Currents – One Sided Diode

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

12. SPICE DiodeModel • Dinj • N~1, rd~N*Vt/iD • rd*Cd = TT = • Cdepl given by CJO, VJ and M • Drec • N~2, rd~N*Vt/iD • rd*Cd = ? • Cdepl =? t

13. Project 1A – Diode parameters to use

14. Tasks • Using PSpice or any simulator, plot the i-v curve for this diode, assuming Rth = 0, for several temperatures in the range 300 K < TEMP = TAMB < 304 K. • Using this data, determine what the i-v plot would be for Rth = 500 K/W. • Using this data, determine the maximum operating temperature for which the diode conductance is within 1% of the Rth = 0 value at 300 K. • Do the same for a 10% tolerance. • Propose a SPICE macro which would give the Rth = 500 K/W i-v relationship.

15. Example

16. Approaches • Phenomenological • Theoretical

19. ** The diode is modeled as an ohmic resistance (RS/area) in series with an intrinsic diode. <(+) node> is the anode and <(-) node> is the cathode. Positive current is current flowing from the anode through the diode to the cathode. [area value] scales IS, ISR, IKF,RS, CJO, and IBV, and defaults to 1. IBV and BV are both specified as positive values. In the following equations: Vd = voltage across the intrinsic diode onlyVt = k·T/q (thermal voltage)k = Boltzmann’s constantq = electron charge T = analysis temperature (°K) Tnom = nom. temp. (set with TNOM option)

20. D Diode ** General Form D<name> <(+) node> <(-) node> <model name> [area value] Examples DCLAMP 14 0 DMODD13 15 17 SWITCH 1.5 Model Form .MODEL <model name> D [model parameters] .model D1N4148-X D(Is=2.682n N=1.836 Rs=.5664 Ikf=44.17m Xti=3 Eg=1.11 Cjo=4p M=.3333 Vj=.5 Fc=.5 Isr=1.565n Nr=2 Bv=100 Ibv=10 0u Tt=11.54n) *$

21. Diode Model Parameters ** • Model Parameters (see .MODEL statement) • Description Unit Default • IS Saturation current amp 1E-14 • N Emission coefficient 1 • ISR Recombination current parameter amp 0 • NR Emission coefficient for ISR 1 • IKF High-injection “knee” current amp infinite • BV Reverse breakdown “knee” voltage volt infinite • IBV Reverse breakdown “knee” current amp 1E-10 • NBV Reverse breakdown ideality factor 1 • RS Parasitic resistance ohm 0 • TT Transit time sec 0 • CJO Zero-bias p-n capacitance farad 0 • VJ p-n potential volt 1 • M p-n grading coefficient 0.5 • FC Forward-bias depletion cap. coef, 0.5 • EG Bandgap voltage (barrier height) eV 1.11

22. Diode Model Parameters ** • Model Parameters (see .MODEL statement) • Description Unit Default • XTI IS temperature exponent 3 • TIKF IKF temperature coefficient (linear) °C-1 0 • TBV1 BV temperature coefficient (linear) °C-1 0 • TBV2 BV temperature coefficient (quadratic) °C-2 0 • TRS1 RS temperature coefficient (linear) °C-1 0 • TRS2 RS temperature coefficient (quadratic) °C-2 0 • T_MEASURED Measured temperature °C • T_ABS Absolute temperature °C • T_REL_GLOBAL Rel. to curr. Temp. °C • T_REL_LOCAL Relative to AKO model temperature °C • For information on T_MEASURED, T_ABS, T_REL_GLOBAL, and T_REL_LOCAL, see the .MODEL statement.

23. ** DC Current Id = area(Ifwd - Irev)Ifwd = forward current = InrmKinj + IrecKgenInrm = normal current = IS(exp (Vd/(NVt))-1) Kinj = high-injection factor For: IKF > 0, Kinj = (IKF/(IKF+Inrm))1/2 otherwise, Kinj = 1 Irec = rec. cur. = ISR(exp (Vd/(NR·Vt))- 1) Kgen = generation factor = ((1-Vd/VJ)2+0.005)M/2Irev = reverse current = Irevhigh + IrevlowIrevhigh = IBVexp[-(Vd+BV)/(NBV·Vt)]Irevlow = IBVLexp[-(Vd+BV)/(NBVL·Vt)}

24. Vext-Va=iD*Rs low level injection ln iD ln(IKF) Effect ofRs ln[(IS*IKF) 1/2] Effect of high level injection ln(ISR) Data ln(IS) vD= Vext recomb. current VKF

25. Interpreting a plotof log(iD) vs. Vd In the region where Irec < Inrm < IKF, and iD*RS << Vd. iD ~ Inrm = IS(exp (Vd/(NVt)) - 1) For N = 1 and Vt = 25.852 mV, the slope of the plot of log(iD) vs. Vd is evaluated as {dlog(iD)/dVd} = log (e)/(NVt) = 16.799 decades/V = 1decade/59.526mV

26. Static Model Eqns.Parameter Extraction In the region where Irec < Inrm < IKF, and iD*RS << Vd. iD ~ Inrm = IS(exp (Vd/(NVt)) - 1) {diD/dVd}/iD = d[ln(iD)]/dVd = 1/(NVt) so N ~ {dVd/d[ln(iD)]}/Vt  Neff, and ln(IS) ~ ln(iD) - Vd/(NVt) ln(ISeff). Note: iD, Vt, etc., are normalized to 1A, 1V, resp.

27. Static Model Eqns.Parameter Extraction In the region where Irec > Inrm, and iD*RS << Vd. iD ~ Irec = ISR(exp (Vd/(NRVt)) - 1) {diD/dVd}/iD = d[ln(iD)]/dVd ~ 1/(NRVt) so NR ~ {dVd/d[ln(iD)]}/Vt Neff, & ln(ISR) ~ln(iD) -Vd/(NRVt ) ln(ISReff). Note: iD, Vt, etc., are normalized to 1A, 1V, resp.

28. Static Model Eqns.Parameter Extraction In the region where IKF > Inrm, and iD*RS << Vd. iD ~ [ISIKF]1/2(exp (Vd/(2NVt)) - 1) {diD/dVd}/iD = d[ln(iD)]/dVd ~ (2NVt)-1 so 2N ~ {dVd/d[ln(iD)]}/Vt  2Neff, and ln(iD) -Vd/(NRVt)  ½ln(ISIKFeff). Note: iD, Vt, etc., are normalized to 1A, 1V, resp.

29. Static Model Eqns.Parameter Extraction In the region where iD*RS >> Vd. diD/Vd ~ 1/RSeff dVd/diD  RSeff

30. Getting Diode Data forParameter Extraction • The model used .model Dbreak D( Is=1e-13 N=1 Rs=.5 Ikf=5m Isr=.11n Nr=2) • Analysis has V1 swept, and IPRINT has V1 swept • iD, Vd data in Output

31. diD/dVd - Numerical Differentiation

32. dln(iD)/dVd - Numerical Differentiation

33. Diode Par.Extraction 1/Reff iD ISeff

34. Results ofParameter Extraction • At Vd = 0.2 V, NReff = 1.97, ISReff = 8.99E-11 A. • At Vd = 0.515 V, Neff = 1.01, ISeff = 1.35 E-13 A. • At Vd = 0.9 V, RSeff = 0.725 Ohm • Compare to .model Dbreak D( Is=1e-13 N=1 Rs=.5 Ikf=5m Isr=.11n Nr=2)

35. Hints for RS and NFparameter extraction In the region where vD > VKF. Defining vD = vDext - iD*RS and IHLI = [ISIKF]1/2. iD = IHLIexp (vD/2NVt) + ISRexp (vD/NRVt) diD/diD = 1  (iD/2NVt)(dvDext/diD - RS) + … Thus, for vD > VKF (highest voltages only) • plot iD-1vs. (dvDext/diD) to get a line with • slope = (2NVt)-1, intercept = - RS/(2NVt)

36. Application of RS tolower current data In the region where vD < VKF. We still have vD = vDext - iD*RS and since. iD = ISexp (vD/NVt) + ISRexp (vD/NRVt) • Try applying the derivatives for methods described to the variables iD and vD (using RS and vDext). • You also might try comparing the N value from the regular N extraction procedure to the value from the previous slide.

37. 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.

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

39. 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

40. 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

41. Reverse biasjunction breakdown

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

43. 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

44. 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

45. 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

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

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

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

49. Equationsummary

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