Semiconductor Device Modeling and Characterization – EE5342 Lecture 8 – Spring 2011 Professor Ronald L. Carter

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©rlc L08-11Feb20114 Additional University Closure Means More Schedule Changes Plan to meet until noon some days in the next few weeks. This way we will make up for the lost time. The first extended class will be Monday, 2/14. The MT changed to Friday 2/18 The P1 test changed to Friday 3/11. The P2 test is still Wednesday 4/13 The Final is still Wednesday 5/11.

©rlc L08-11Feb20115 Shockley-Read- Hall Recomb EvEv EcEc EfEf E fi E k EcEc EvEv ETET Indirect, like Si, so intermediate state

©rlc L08-11Feb20116 S-R-H trap characteristics 1 The Shockley-Read-Hall Theory requires an intermediate “trap” site in order to conserve both E and p If trap neutral when orbited (filled) by an excess electron - “donor-like” Gives up electron with energy E c - E T “Donor-like” trap which has given up the extra electron is +q and “empty”

©rlc L08-11Feb20117 S-R-H trap char. (cont.) If trap neutral when orbited (filled) by an excess hole - “acceptor-like” Gives up hole with energy E T - E v “Acceptor-like” trap which has given up the extra hole is -q and “empty” Balance of 4 processes of electron capture/emission and hole capture/ emission gives the recomb rates

©rlc L08-11Feb20118 S-R-H recombination Recombination rate determined by: N t (trap conc.), v th (thermal vel of the carriers),  n (capture cross sect for electrons),  p (capture cross sect for holes), with  no = (N t v th  n ) -1, and  po = (N t v th  n ) -1, where  n ~  (r Bohr ) 2

©rlc L08-11Feb20119 S-R-H recomb. (cont.) In the special case where  no =  po =  o the net recombination rate, U is

©rlc L08-11Feb 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

©rlc L08-11Feb S-R-H “U” function characteristics (cont) For n-type, as above, the denominator =  o {n o +  n+p o +  p+2n i cosh[(E t -E i )kT]}, simplifies to the smallest value for E t ~E i, where the denom is  o n o, giving U =  p/  o as the largest (fastest) For p-type, the same argument gives U =  n/  o Rec rate, U, fixed by minority carrier

©rlc L08-11Feb 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

©rlc L08-11Feb 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/  o, (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/  o, (prop to exc min carr)

©rlc L08-11Feb Minority hole lifetimes. Taken from Shur 3, (p.101).

©rlc L08-11Feb Minority electron lifetimes. Taken from Shur 3, (p.101).

©rlc L08-11Feb Parameter example  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 ?

M. E. Law, E. Solley, M. Liang, and D. E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility,” IEEE Electron Device Lett., vol. 12, pp , ©rlc L08-11Feb201117

©rlc L08-11Feb M. E. Law, E. Solley, M. Liang, and D. E. Burk, “Self-Consistent Model of Minority-Carrier Lifetime, Diffusion Length, and Mobility,” IEEE Electron Device Lett., vol. 12, pp , 1991.

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©rlc L08-11Feb 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

©rlc L08-11Feb The Continuity Equation The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives

©rlc L08-11Feb The Continuity Equation (cont.)

©rlc L08-11Feb The Continuity Equation (cont.)

©rlc L08-11Feb The Continuity Equation (cont.)

©rlc L08-11Feb The Continuity Equation (cont.)

©rlc L08-11Feb The Continuity Equation (cont.)

©rlc L08-11Feb The Continuity Equation (cont.)

©rlc L08-11Feb References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, **Semiconductor Physics & Devices, by Donald A. Neamen, 2nd ed., Irwin, Chicago. M&K = Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, Physics of Semiconductor Devices, Shur, Prentice- Hall, 1990.