1. EE 5340 Semiconductor Device Theory Lecture 14 - Fall 2009
Professor Ronald L. Carter

2. S-R-H rec for excess min carr
For n-type low-level injection and net excess minority carriers, (i.e., no > dn = dp > po = ni2/no),
U = dp/tp, (prop to exc min carr)
For p-type low-level injection and net excess minority carriers, (i.e., po > dn = dp > no = ni2/po),
U = dn/tn, (prop to exc min carr)
L 14 Oct 08

3. Minority hole lifetimes
Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991
The parameters used in the fit are
τo = 10 μs,
Nref = 1×1017/cm2, and
CA = 1.8×10-31cm6/s.
L 14 Oct 08

4. Minority electron lifetimes
Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991
The parameters used in the fit are
τo = 30 μs,
Nref = 1×1017/cm2, and
CA = 8.3×10-32 cm6/s.
L 14 Oct 08

5. References for Part A
Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003.
Mark E. Law, E. Solley, M. Liang, and Dorothea E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility, IEEE ELECTRON DEVICE LETTERS, VOL. 12, NO. 8, AUGUST 1991.
D.B.M. Klaassen; “A UNIFIED MOBILITY MODEL FOR DEVICE SIMULATION”, Electron Devices Meeting, Technical Digest., International 9-12 Dec Page(s):357 – 360.
David Roulston, Narain D. Arora, and Savvas G. Chamberlain “Modeling and Measurement of Minority-Carrier Lifetime versus Doping in Diffused Layers of n+-p Silicon Diodes”, IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-29, NO. 2, FEBRUARY 1982, pages
M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination in Heavily Doped Silicon”, Solid-State Electr. Vol. 26, pp , Download a copy at Tyagi.pdf.
L 14 Oct 08

6. S-R-H rec for deficient min carr
If n < ni and p < pi, then the S-R-H net recomb rate becomes (p < po, n < no):
U = R - G = - ni/(2t0cosh[(ET-Efi)/kT])
And with the substitution that the gen lifetime, tg = 2t0cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/tg
The intrinsic concentration drives the return to equilibrium
L 14 Oct 08

7. The Continuity Equation
The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives
L 14 Oct 08

8. The Continuity Equation (cont.)
L 14 Oct 08

9. The Continuity Equation (cont.)
L 14 Oct 08

10. The Continuity Equation (cont.)
L 14 Oct 08

11. The Continuity Equation (cont.)
L 14 Oct 08

12. The Continuity Equation (cont.)
L 14 Oct 08

13. The Continuity Equation (cont.)
L 14 Oct 08

14. Review of depletion approximation
pp << ppo, -xp < x < 0
nn << nno, 0 < x < xn
0 > Ex > -2Vbi/W, in DR (-xp < x < xn)
pp=ppo=Na & np=npo= ni2/Na, -xpc< x < -xp
nn=nno=Nd & pn=pno= ni2/Nd, xn < x < xnc x xn
xnc
-xpc
-xp Ev Ec
qVbi
EFi
EFn
EFp
L 14 Oct 9

15. Review of D. A. (cont.) Ex
-xpc
-xp xn
xnc x
-Emax
L 14 Oct 9

16. Forward Bias Energy BandsEv Ec
EFi xn
xnc
-xpc
-xp
q(Vbi-Va)
EFP
EFN
qVa x
Imref, EFn
Imref, EFp
L 14 Oct 9

17. Law of the junction: “Remember to follow the minority carriers”
L 14 Oct 9

18. Law of the junction (cont.)
L 14 Oct 9

19. Law of the junction (cont.)
L 14 Oct 9

20. Injection Conditions
L 14 Oct 9

21. Ideal Junction Theory Assumptions Ex = 0 in the chg neutral reg. (CNR)
MB statistics are applicable
Neglect gen/rec in depl reg (DR)
Low level injection applies so that dnp < ppo for -xpc < x < -xp, and dpn < nno for xn < x < xnc
Steady State conditions
L 14 Oct 9

22. Ideal Junction Theory (cont.)
L 14 Oct 9

23. Ideal Junction Theory (cont.)
L 14 Oct 9

24. Ideal Junction Theory (cont.)
L 14 Oct 9

25. Diffusion Length model
L = (Dt)1/2 Diffusion Coeff. is Pierret* model
L 14 Oct 9

26. Excess minority carrier distr fctn
L 14 Oct 9

27. Forward Bias Energy BandsEv Ec
EFi xn
xnc
-xpc
-xp
q(Vbi-Va)
EFP
EFN
qVa x
Imref, EFn
Imref, EFp
L 14 Oct 9

28. Carrier Injection ln(carrier conc) ln Na ln Nd ln ni ~Va/Vt ~Va/Vt
ln ni2/Nd
ln ni2/Na x
-xpc
-xp
xnc xn
L 14 Oct 9

29. Minority carrier currents
L 15 Oct 14

30. Evaluating the diode current
L 15 Oct 14

31. Special cases for the diode current
L 15 Oct 14

32. Ideal diode equation Assumptions: Current dens, Jx = Js expd(Va/Vt)
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]
L 15 Oct 14

33. Ideal diode equation (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
L 15 Oct 14

34. Diffnt’l, one-sided diode conductance
Static (steady-state) diode I-V characteristic IQ Va VQ
L 15 Oct 14

35. Diffnt’l, one-sided diode cond. (cont.)
L 15 Oct 14

36. Charge distr in a (1- sided) short diode
dpn
Assume Nd << Na
The sinh (see L10) 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
L 15 Oct 14

37. Charge distr in a 1- sided short diode
dpn
Assume Quasi-static charge distributions
Q’p = +qdpn(xn,Va)Wn/2
dQ’p =q(W/2) x {dpn(xn,Va+dV) dpn(xn,Va)}
Wn = xnc - xn (Va)
dpn(xn,Va+dV)
dpn(xn,Va)
dQ’p
Q’p x xn
xnc
L 15 Oct 14

38. References
* Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996
** Physics of Semiconductor Devices, M. Shur, Wiley.
L 14 Oct 9