EE 5340 Semiconductor Device Theory Lecture 13 – Spring 2011 Professor Ronald L. Carter

©rlc L13-03Mar2011 Doping Profile If the net donor conc, N = N(x), then at x, the extra charge put into the DR when V a ->V a +  V a is  Q’=-qN(x)  x The increase in field,  E x =-(qN/  )  x, by Gauss’ Law (at x, but also all DR). So  V a =-x d  E x = (W/  )  Q’ Further, since qN(x)  x, for both x n and x n, we have the dC/dx as... 2

©rlc L13-03Mar2011 Arbitrary doping profile (cont.) 3

©rlc L13-03Mar2011 Arbitrary doping profile (cont.) 4

©rlc L13-03Mar2011 Arbitrary doping profile (cont.) 5

©rlc L13-03Mar2011 Arbitrary doping profile (cont.) 6

©rlc L13-03Mar2011 Example An assymetrical p+ n junction has a lightly doped concentration of 1E16 and with p+ = 1E18. What is W(V=0)? Vbi=0.816 V, Neff=9.9E15, W=0.33  m What is C’ j0 ? = 31.9 nFd/cm2 What is L D ? = 0.04  m 7

©rlc L13-03Mar2011 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 8

©rlc L13-03Mar2011 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 9

©rlc L13-03Mar2011 Effect of V  0 10

©rlc L13-03Mar2011 Reverse bias junction breakdown 11

©rlc L13-03Mar2011 E crit for reverse breakdown [M&K] Taken from p. 198, M&K** Casey 2 model for E crit 12

©rlc L13-03Mar2011 Table 4.1 (M&K* p. 186) Nomograph for silicon uniformly doped, one-sided, step junctions (300 K). (See Figure 4.15 to correct for junction curvature.) (Courtesy Bell Laboratories). 13

©rlc L13-03Mar2011 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 14

©rlc L13-03Mar201115

©rlc L13-03Mar Direct carrier gen/recomb gen rec EvEv EcEc EfEf E fi E k EcEc EvEv (Excitation can be by light)

©rlc L13-03Mar Direct gen/rec of excess carriers Generation rates, G n0 = G p0 Recombination rates, R n0 = R p0 In equilibrium: G n0 = G p0 = R n0 = R p0 In non-equilibrium condition: n = n o +  n and p = p o +  p, where n o p o =n i 2 and for  n and  p > 0, the recombination rates increase to R’ n and R’ p

©rlc L13-03Mar Direct rec for low-level injection Define low-level injection as  n =  p < n o, for n-type, and  n =  p < p o, for p-type The recombination rates then are R’ n = R’ p =  n(t)/  n0, for p-type, and R’ n = R’ p =  p(t)/  p0, for n-type Where  n0 and  p0 are the minority- carrier lifetimes

©rlc L13-03Mar Shockley-Read- Hall Recomb EvEv EcEc EfEf E fi E k EcEc EvEv ETET Indirect, like Si, so intermediate state

©rlc L13-03Mar S-R-H trap characteristics* 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 L13-03Mar 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 L13-03Mar 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  p ) -1, where  n,p ~  (r Bohr,n.p ) 2

©rlc L13-03Mar 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 L13-03Mar 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 L13-03Mar References 1 and M&K Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 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, and ** Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.