1. EE 5340 Semiconductor Device Theory Lecture 14 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

2. ©rlc L14-08Mar20112 S-R-H net recom- bination rate, U In the special case where  no =  po =  o = (N t v th  o ) -1 the net rec. rate, U is

3. ©rlc L14-08Mar20113 S-R-H “U” function characteristics The numerator, (np-n i 2 ) simplifies in the case of extrinsic material at low level injection (for equil., n o p o = n i 2 ) For n-type (n o >  n =  p > p o = n i 2 /n o ): (np-n i 2 ) = (n o +  n)(p o +  p)-n i 2 = n o p o - n i 2 + n o  p +  np o +  n  p ~ n o  p (largest term) Similarly, for p-type, (np-n i 2 ) ~ p o  n

4. ©rlc L14-08Mar20114 S-R-H rec for excess min carr For n-type low-level injection and net excess minority carriers, (i.e., n o >  n =  p > p o = n i 2 /n o ), U =  p/  p, (prop to exc min carr) For p-type low-level injection and net excess minority carriers, (i.e., p o >  n =  p > n o = n i 2 /p o ), U =  n/  n, (prop to exc min carr)

5. 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 τ o = 10 μs, N ref = 1×10 17 /cm 2, and C A = 1.8×10 -31 cm 6 /s. ©rlc L14-08Mar20115

6. 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 τ o = 30 μs, N ref = 1×10 17 /cm 2, and C A = 8.3×10 -32 cm 6 /s. ©rlc L14-08Mar20116

7. Minority Carrier Lifetime, Diffusion Length and Mobility Models in Silicon A. [40%] Write a review of the model equations for minority carrier (both electrons in p-type and holes in n-type material) lifetime, mobility and diffusion length in silicon. Any references may be used. At a minimum the material given in the following references should be used. Based on the information in these resources, decide which model formulae and parameters are the most accurate for D n and L n for electrons in p-type material, and D p and L p holes in n-type material. B. [60%] This part of the assignment will be given by 10/12/09. Current-voltage data will be given for a diode, and the project will be to determine the material parameters (Nd, Na, charge-neutral region width, etc.) of the diode. ©rlc L14-08Mar20117

8. References for Part A Device Electronics for Integrated Circuits, 3 rd 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, 1990. Technical Digest., International 9-12 Dec. 1990 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 284-291. M. S. Tyagi and R. Van Overstraeten, “Minority Carrier Recombination in Heavily Doped Silicon”, Solid-State Electr. Vol. 26, pp. 577-597, 1983. Download a copy at Tyagi.pdf.Tyagi.pdf ©rlc L14-08Mar20118

9. 9 S-R-H rec for deficient min carr If n < n i and p < p i, then the S-R-H net recomb rate becomes (p < p o, n < n o ): U = R - G = - n i /(2  0 cosh[(E T -E fi )/kT]) And with the substitution that the gen lifetime,  g = 2  0 cosh[(E T -E fi )/kT], and net gen rate U = R - G = - n i /  g The intrinsic concentration drives the return to equilibrium

10. ©rlc L14-08Mar201110 The Continuity Equation The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives

11. ©rlc L14-08Mar201111 The Continuity Equation (cont.)

12. ©rlc L14-08Mar201112 The Continuity Equation (cont.)

13. ©rlc L14-08Mar201113 The Continuity Equation (cont.)

14. ©rlc L14-08Mar201114 The Continuity Equation (cont.)

15. ©rlc L14-08Mar201115 The Continuity Equation (cont.)

16. ©rlc L14-08Mar201116 The Continuity Equation (cont.)

17. ©rlc L14-08Mar201117 Review of depletion approximation Depletion Approx. p p << p po, -x p < x < 0 n n << n no, 0 < x < x n 0 > E x > -2V bi /W, in DR (-x p < x < x n ) p p =p po =N a & n p =n po = n i 2 /N a, -x pc < x < -x p n n =n no =N d & p n =p no = n i 2 /N d, x n < x < x nc x xnxn x nc -x pc -x p 0 EvEv EcEc qV bi E Fi E Fn E Fp

18. ©rlc L14-08Mar201118 Review of D. A. (cont.) x xnxn x nc -x pc -x p ExEx -E max

19. ©rlc L14-08Mar201119 Forward Bias Energy Bands EvEv EcEc E Fi xnxn x nc -x pc -x p 0 q(V bi -V a ) E FP E FN qV a x Imref, E Fn Imref, E Fp

20. ©rlc L14-08Mar201120 References 1 and M&K Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996, for another treatment of the  model. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981. 3 and ** Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997. Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.