Semiconductor Device Modeling and Characterization EE5342, Lecture 6-Spring 2010

1. Semiconductor Device Modeling and CharacterizationEE5342, Lecture 6-Spring 2010 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. Project 1A – Diode parameters to use

3. 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.

4. Example

5. 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

6. 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]

7. 1-dim soln. ofGauss’ law Ex -xp xn xnc -xpc x -Emax

8. 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).

9. One-sided p+n or n+p jctns • If p+n, then Na >>Nd, and NaNd/(Na +Nd) = Neff --> Nd, and W --> xn, DR is all on lightly d. side • If n+p, then Nd >>Na, and NaNd/(Na +Nd) = Neff --> Na, and W --> xp, DR is all on lightly d. side • The net effect is that Neff --> N-, (- = lightly doped side) and W --> x-

10. JunctionC (cont.) r +Qn’=qNdxn +qNd dQn’=qNddxn -xp x -xpc xn xnc Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn -qNa dQp’=-qNadxp Qp’=-qNaxp

11. JunctionC (cont.) • The C-V relationship simplifies to

12. JunctionC (cont.) • If one plots [C’j]-2vs. Va Slope = -[(C’j0)2Vbi]-1 vertical axis intercept = [C’j0]-2 horizontal axis intercept = Vbi C’j-2 C’j0-2 Va Vbi

13. Arbitrary dopingprofile • If the net donor conc, N = N(x), then at xn, the extra charge put into the DR when Va->Va+dVa is dQ’=-qN(xn)dxn • The increase in field, dEx =-(qN/e)dxn, by Gauss’ Law (at xn, but also const). • So dVa=-(xn+xp)dEx= (W/e) dQ’ • Further, since N(xn)dxn = N(xp)dxp gives, the dC/dxn as ...

14. Arbitrary dopingprofile (cont.)

15. Arbitrary dopingprofile (cont.)

16. Arbitrary dopingprofile (cont.)

17. Arbitrary dopingprofile (cont.)

18. n Nd 0 xn x Debye length • The DA assumes n changes from Nd to 0 discontinuously at xn, likewise, p changes from Na to 0 discontinuously at -xp. • In the region of xn, the 1-dim Poisson equation is dEx/dx = q(Nd - n), and since Ex = -df/dx, the potential is the solution to -d2f/dx2 = q(Nd - n)/e

19. Debye length (cont) • Since the level EFi is a reference for equil, we set f = Vt ln(n/ni) • In the region of xn, n = ni exp(f/Vt), so d2f/dx2 = -q(Nd - ni ef/Vt), let f = fo + f’, where fo = Vt ln(Nd/ni) so Nd - ni ef/Vt = Nd[1 - ef/Vt-fo/Vt], for f - fo = f’ << fo, the DE becomes d2f’/dx2 = (q2Nd/ekT)f’, f’ << fo

20. Debye length (cont) • So f’= f’(xn) exp[+(x-xn)/LD]+con. and n = Nd ef’/Vt, x ~ xn, where LD is the “Debye length”

21. Debye length (cont) • LD estimates the transition length of a step-junction DR (concentrations Na and Nd with Neff = NaNd/(Na +Nd)). Thus, • For Va=0, & 1E13 <Na,Nd< 1E19 cm-3 • 13% <d< 28% => DA is OK

22. Example • An assymetrical p+ n junction has a lightly doped concentration of 1E16 and with p+ = 1E18. What is W(V=0)? Vbi=0.816 V, Neff=9.9E15, W=0.33mm • What is C’j? = 31.9 nFd/cm2 • What is LD? = 0.04 mm

23. Ideal JunctionTheory Assumptions • Ex = 0 in the chg neutral reg. (CNR) • MB statistics are applicable • Neglect gen/rec in depl reg (DR) • Low level injections apply so that dnp < ppo for -xpc < x < -xp, and dpn < nno for xn < x < xnc • Steady State conditions

24. q(Vbi-Va) Imref, EFn Ec EFN qVa EFP EFi Imref, EFp Ev x -xpc -xp xn xnc 0 Forward Bias Energy Bands

25. Law of the junction(follow the min. carr.)

26. Law of the junction (cont.)

27. Law of the junction (cont.)

28. InjectionConditions

29. Ideal JunctionTheory (cont.) Apply the Continuity Eqn in CNR

30. Ideal JunctionTheory (cont.)

31. Ideal JunctionTheory (cont.)

32. Excess minoritycarrier distr fctn

33. CarrierInjection ln(carrier conc) ln Na ln Nd ln ni ~Va/Vt ~Va/Vt ln ni2/Nd ln ni2/Na x xnc -xpc -xp xn 0

34. Minority carriercurrents

35. Evaluating thediode current

36. Special cases forthe diode current

37. 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]

38. 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

39. Diffnt’l, one-sided diode conductance ID Static (steady-state) diode I-V characteristic IQ Va VQ

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

41. Charge distr in a (1-sided) short diode dpn • Assume Nd << Na • The sinh (see L12) excess minority carrier distribution becomes linear for Wn << Lp • dpn(xn)=pn0expd(Va/Vt) • Total chg = Q’p = Q’p = qdpn(xn)Wn/2 Wn = xnc- xn dpn(xn) Q’p x xn xnc

42. Charge distr in a 1-sided short diode dpn • Assume Quasi-static charge distributions • Q’p = Q’p = qdpn(xn)Wn/2 • ddpn(xn) = (W/2)* {dpn(xn,Va+dV) - dpn(xn,Va)} dpn(xn,Va+dV) dpn(xn,Va) dQ’p Q’p x xnc xn

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

44. General time-constant

45. General time-constant (cont.)

46. General time-constant (cont.)