1. L08 Feb 081 Lecture 08 Semiconductor Device Modeling and Characterization EE5342 - Spring 2001 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. L08 Feb 082 Ideal diode equation (cont.) J s = J s,p + J s,n = hole curr + ele curr J s,p = qn i 2 D p coth(W n /L p )/(N d L p ) = qn i 2 D p /(N d W n ), W n > L p, “long” J s,n = qn i 2 D n coth(W p /L n )/(N a L n ) = qn i 2 D n /(N a W p ), W p > L n, “long” J s,n > N d

3. L08 Feb 083 Diffnt’l, one-sided diode conductance VaVa IDID Static (steady- state) diode I-V characteristic VQVQ IQIQ

4. L08 Feb 084 Diffnt’l, one-sided diode cond. (cont.)

5. L08 Feb 085 Charge distr in a (1- sided) short diode Assume N d << N a The sinh (see L12) excess minority carrier distribution becomes linear for W n << L p  p n (x n )=p n0 expd( V a /V t ) Total chg = Q’ p = Q’ p = q  p n (x n )W n /2 xnxn x x nc  p n (x n ) W n = x nc - x n Q’ p pnpn

6. L08 Feb 086 Charge distr in a 1- sided short diode Assume Quasi- static charge distributions Q’ p = Q’ p = q  p n (x n )W n /2 d  p n (x n ) = (W/2)* {  p n (x n,V a +  V) -  p n (x n,V a )} xnxn x x nc  p n (x n,V a ) Q’ p pnpn  p n (x n,V a +  V)  Q’ p

7. L08 Feb 087 Cap. of a (1-sided) short diode (cont.)

8. L08 Feb 088 General time- constant

9. L08 Feb 089 General time- constant (cont.)

10. L08 Feb 0810 General time- constant (cont.)

11. L08 Feb 0811 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, R s ) R s = L/(nq  n A) for a p+n long diode. L=W n -L p (so the current is diode-like for L p and the resistive otherwise).

12. L08 Feb 0812 Effect of carrier recombination in DR The S-R-H rate (  no =  po =  o ) is

13. L08 Feb 0813 Effect of carrier rec. in DR (cont.) For low V a ~ 10 V t In DR, n and p are still > n i 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

14. L08 Feb 0814 Effect of carrier rec. in DR (cont.)

15. L08 Feb 0815 High level injection effects Law of the junction remains in the same form, [p n n n ] x n =n i 2 exp (V a /V t ), etc. However, now  p n =  n n become >> n no = N d, etc. Consequently, the l.o.t.j. reaches the limiting form  p n  n n = n i 2 exp(V a /V t ) Giving,  p n (x n ) = n i exp(V a /(2V t )), or  n p (-x p ) = n i exp(V a /(2V t )),

16. L08 Feb 0816 High level inj effects (cont.)

17. L08 Feb 0817 Summary of V a > 0 current density eqns. Ideal diode, J s expd(V a /(  V t )) –ideality factor,  Recombination, J s,rec exp(V a /(2  V t )) –appears in parallel with ideal term High-level injection, (J s *J KF ) 1/2 exp(V a /(2  V t )) –SPICE model by modulating ideal J s term V a = V ext - J*A*R s = V ext - I diode *R s

18. L08 Feb 0818 Plot of typical V a > 0 current density eqns. V ext ln J data ln(J KF ) ln(J s ) ln[(J s *J KF ) 1/2 ] Effect of R s V KF ln(J srec ) Effect of high level injection low level injection recomb. current V ext -V d =JAR s

19. L08 Feb 0819 Reverse bias (V a carrier gen in DR V a < 0 gives the net rec rate, U = -n i /  ,   = mean min carr g/r l.t.

20. L08 Feb 0820 Reverse bias (V a < 0), carr gen in DR (cont.)

21. L08 Feb 0821 Reverse bias junction 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

22. L08 Feb 0822 E crit for reverse breakdown (M&K**) Taken from p. 198, M&K**

23. L08 Feb 0823 Reverse bias junction breakdown Assume -V a = V R >> V bi, so V bi -V a -->V R Since E max ~ 2V R /W = (2qN - V R /(  )) 1/2, and V R = BV when E max = E crit (N - is doping of lightly doped side ~ N eff ) BV =  (E crit ) 2 /(2qN - ) Remember, this is a 1-dim calculation

24. L08 Feb 0824 Junction curvature effect on breakdown The field due to a sphere, R, with charge, Q is E r = Q/(4  r 2 ) for (r > R) V(R) = Q/(4  R), (V at the surface) So, for constant potential, V, the field, E r (R) = V/R (E field at surface increases for smaller spheres) Note: corners of a jctn of depth x j are like 1/8 spheres of radius ~ x j

25. L08 Feb 0825 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. L08 Feb 0826 Example calculations Assume throughout that p+n jctn with N a = 3e19cm -3 and N d = 1e17cm - 3 From graph of Pierret mobility model,  p = 331 cm 2 /V-sec and D p = V t  p = ? Why  p and D p ? N eff = ? V bi = ?

27. L08 Feb 0827

28. L08 Feb 0828 Parameters for examples Get  min from the model used in Project 2  min = (45  sec) 1+(7.7E-18cm 3  N i +(4.5E-36cm 6  N i 2 For N d = 1E17cm 3,  p = 25  sec –Why N d and  p ? L p = ?

29. L08 Feb 0829 Hole lifetimes, taken from Shur***, p. 101.

30. L08 Feb 0830 Example J s,long, = ? If x nc, = 2 micron, J s,short, = ?

31. L08 Feb 0831 Example (cont.) Estimate V KF Estimate I KF

32. L08 Feb 0832 Example (cont.) Estimate J s,rec Estimate R s if x nc is 100 micron

33. L08 Feb 0833 Example (cont.) Estimate J gen for 10 V reverse bias Estimate BV

34. L08 Feb 0834 Diode equivalent circuit (small sig) IDID VDVD VQVQ IQIQ  is the practical “ideality factor”

35. L08 Feb 0835 Small-signal eq circuit C diff C depl r diff C diff and C depl are both charged by V a = V Q VaVa

36. L08 Feb 0836 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.