1. ECE 663 AC Diode Characteristics Resistor network supplies DC bias set point Capacitor provides AC signal input V out =I diode R3 R3

2. ECE 663 AC small signal resistance

3. ECE 663 Small signal AC conductance Small Signal AC resistance Resistance depends on DC set point – voltage controlled resistor

4. ECE 663 Reactance: An equivalent circuit Let us work out Y for reverse bias first

5. Reverse Bias Conductance I V I constant with reverse bias voltage  g d ≈ 0 I = I 0 (e -qV BR /kT -1) ≈ -I 0

6. ECE 663 Reverse Bias (‘Depletion’) Capacitance AC voltage modifies depletion width Depletion width changes small Looks like adding charges to parallel plates AC capacitance

7. ECE 663 RB capacitance C-V measurements Plot of 1/C 2 vs V is a straight line (constant doping) and the slope gives doping profile. Y-intercept gives built-in voltage

8. ECE 663

9. So reverse bias equivalent circuit RsRs Notice that for reverse bias, circuit parameters are frequency independent, as if we’re in DC characteristics. Why?

10. ECE 663 Reverse bias p-n junction is a majority carrier device Very few minority carriers have made it to the opposite side Depletion width change requires flow of majority carriers (n from n-side and p from p-side flow in and out) Since majority carriers move very fast by drift, they can follow the AC field instantly, so the response is ‘quasi-static ’

11. ECE 663 Just how fast are majority carriers? Drifting charges, with fields in turn determined by charge ∂ n/ ∂ t = -(1/q) ∂ J n / ∂ x + (g N - r N ) J n = qn  n E + qD N ∂ n/ ∂ x ≈ qn  n E ≈  n E K s  0 ∂ E / ∂ x = q(p - n + N D + - N A - ) ≈ -qn ∂ n/ ∂ t = -n/   = K s   /  n (Dielectric Relaxation Time)

12. ECE 663 How fast is it?  = K s    n K s = 11.9  0 = 8.854 x 10 -12 F/m  n (@ doping 10 15 /cm 3 ) ~ 4  -cm  ≈ 5 ps ! As long as fields are not too fast ( < 10 GHz), charges follow field quasi-statically

13. ECE 663 Let’s now go to forward bias

14. ECE 663 Capacitance in Forward Bias Stored charge = excess minority carriers

15. ECE 663 AC field varies minority carrier pile- up (recall law of the junction) p(x n ) = (n i 2 /N D )[e q(V + v ac )/kT – 1]

16. ECE 663 Also, minority carriers are slow and may not follow AC field quasi-statically Thus we expect circuit parameters to be frequency-dependent !

17. ECE 663 How fast are minority carriers?  ≈ 1/N T  T v t (Minority carrier lifetime) N T ~ 10 12 /cm 3 (for N A ~ 10 14 /cm 3 )  T ~  (10 -10 m) 3 v t =  3kT/m ~ 10 5 m/s  ≈ 300  s So for fast fields (  >> 1/  ), expect carriers to go out of phase, leading to freq-dependent circuit parameters

18. ECE 663 But how do we include such phase lag effects?

19. Back to MCDE ∂  n/ ∂ t = D N ∂ 2  n/ ∂ x 2 –  n/  n Can’t drop this at AC fields !! j  n

20. Back to MCDE 0 = D N ∂ 2  n/ ∂ x 2 –  n(1+j  n )/  n  n   n /(1+j  n )

21. So in Shockley equation I = qA(n i 2 / N D )  D N (1+j  n ) /  n x [e q(V + v ac )/kT – 1] i diff = G 0  (1+j  n ) v ac i diff = (G d + j  C d ) v ac

22. Square root of (1+j  n ) 1 + j  = Ae j  A =  (1 +  2  2 )  = tan -1 (  ) Real(1+j  ) = A 1/2 cos(  /2) Im(1+j  ) = A 1/2 sin(  /2) cos  = 1/  (1+  2  2 ) = 2cos 2 (  /2) - 1 = 1 – 2sin 2 (  /2) Re  (1+j  ) = G d Im  (1+j  ) = j  C d

23. 1 G d (  )/G 0 ~  C d (  )/C 0 ~ 1/   For high frequency (  >> 1), minority carriers can’t follow fields, so capacitance goes down and the p-n junction becomes ‘leaky’ so its conductance goes up

24. In summary Reverse bias is a depletion capacitance, zero conductance It looks like a DC capacitance, except its width depends on voltage Forward bias looks like a frequency dependent diffusion capacitance and a diffusion conductance to give an overall admittance