L04 24Jan021 Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2002 Professor Ronald L. Carter

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

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 )

L04 24Jan024 Net silicon extr resistivity (cont.)

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.)

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

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

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 E8 T E c (V/cm) 1.01 T T 1.68  2.57E-2 T T 0.17

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

L04 24Jan0210 Carrier velocity saturation (cont.) At 300K, for electrons,  o = v 1 /E c = 1.53E9(300) /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) = 1.07E7 cm/s

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)

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

L04 24Jan0213 Diffusion of Carriers (cont.)

L04 24Jan0214 Current density components

L04 24Jan0215 Total current density

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

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

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

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

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

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

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

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”

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

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

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

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

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

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

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)

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

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

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 ?

L04 24Jan0234 References 1 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, 1981.