1. L04 24Jan021 Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. L04 24Jan022 Summary The concept of mobility introduced as a response function to the electric field in establishing a drift current Resistivity and conductivity defined Model equation def for  (N d,N a,T) Resistivity models developed for extrinsic and compensated materials

3. L04 24Jan023 Net silicon (ex- trinsic) resistivity Since  =  -1 = (nq  n + pq  p ) -1 The net conductivity can be obtained by using the model equation for the mobilities as functions of doping concentrations. The model function gives agreement with the measured  (N impur )

4. L04 24Jan024 Net silicon extr resistivity (cont.)

5. L04 24Jan025 Net silicon extr resistivity (cont.) Since  = (nq  n + pq  p ) -1, and  n >  p, (  = q  /m*) we have  p >  n Note that since 1.6(high conc.) <  p /  n < 3(low conc.), so 1.6(high conc.) <  n /  p < 3(low conc.)

6. L04 24Jan026 Net silicon (com- pensated) res. For an n-type (n >> p) compensated semiconductor,  = (nq  n ) -1 But now n = N = N d - N a, and the mobility must be considered to be determined by the total ionized impurity scattering N d + N a = N I Consequently, a good estimate is  = (nq  n ) -1 = [Nq  n (N I )] -1

7. L04 24Jan027 Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently,

8. L04 24Jan028 Carrier velocity saturation 1 The mobility relationship v =  E is limited to “low” fields v < v th = (3kT/m*) 1/2 defines “low” v =  o E[1+(E/E c )  ] -1/ ,  o = v 1 /E c for Si parameter electrons holes v 1 (cm/s) 1.53E9 T -0.87 1.62E8 T -0.52 E c (V/cm) 1.01 T 1.55 1.24 T 1.68  2.57E-2 T 0.66 0.46 T 0.17

9. L04 24Jan029 Carrier velocity 2 carrier velocity vs E for Si, Ge, and GaAs (after Sze 2 )

10. L04 24Jan0210 Carrier velocity saturation (cont.) At 300K, for electrons,  o = v 1 /E c = 1.53E9(300) -0.87 /1.01(300) 1.55 = 1504 cm 2 /V-s, the low-field mobility The maximum velocity (300K) is v sat =  o E c = v 1 = 1.53E9 (300) -0.87 = 1.07E7 cm/s

11. L04 24Jan0211 Diffusion of carriers In a gradient of electrons or holes, = p and = n are not zero Diffusion current,  J =  J p +  J n (note D p and D n are diffusion coefficients)

12. L04 24Jan0212 Diffusion of carriers (cont.) Note ( = p) x has the magnitude of dp/dx and points in the direction of increasing p (uphill) The diffusion current points in the direction of decreasing p or n (downhill) and hence the - sign in the definition of  J p and the + sign in the definition of  J n

13. L04 24Jan0213 Diffusion of Carriers (cont.)

14. L04 24Jan0214 Current density components

15. L04 24Jan0215 Total current density

16. L04 24Jan0216 Doping gradient induced E-field If N = N d -N a = N(x), then so is E f -E fi Define  = (E f -E fi )/q = (kT/q)ln(n o /n i ) For equilibrium, E fi = constant, but for dN/dx not equal to zero, E x = -d  /dx =- [d(E f -E fi )/dx](kT/q) = -(kT/q) d[ln(n o /n i )]/dx = -(kT/q) (1/n o )[dn o /dx] = -(kT/q) (1/N)[dN/dx], N > 0

17. L04 24Jan0217 Induced E-field (continued) Let V t = kT/q, then since n o p o = n i 2 gives n o /n i = n i /p o E x = - V t d[ln(n o /n i )]/dx = - V t d[ln(n i /p o )]/dx = - V t d[ln(n i /|N|)]/dx, N = -N a < 0 E x = - V t (-1/p o )dp o /dx = V t (1/p o )dp o /dx = V t (1/N a )dN a /dx

18. L04 24Jan0218 The Einstein relationship For E x = - V t (1/n o )dn o /dx, and J n,x = nq  n E x + qD n (dn/dx) = 0 This requires that nq  n [V t (1/n)dn/dx] = qD n (dn/dx) Which is satisfied if

19. L04 24Jan0219 Direct carrier gen/recomb gen rec - ++ - EvEv EcEc EfEf E fi E k EcEc EvEv (Excitation can be by light)

20. L04 24Jan0220 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

21. L04 24Jan0221 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

22. L04 24Jan0222 Shockley-Read- Hall Recomb EvEv EcEc EfEf E fi E k EcEc EvEv ETET Indirect, like Si, so intermediate state

23. L04 24Jan0223 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”

24. L04 24Jan0224 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

25. L04 24Jan0225 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

26. L04 24Jan0226 S-R-H recomb. (cont.) In the special case where  no =  po =  o the net recombination rate, U is

27. L04 24Jan0227 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

28. L04 24Jan0228 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

29. L04 24Jan0229 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

30. L04 24Jan0230 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)

31. L04 24Jan0231 Minority hole lifetimes. Taken from Shur 3, (p.101).

32. L04 24Jan0232 Minority electron lifetimes. Taken from Shur 3, (p.101).

33. L04 24Jan0233 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 ?

34. L04 24Jan0234 References 1 Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.