Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ Semiconductor Device Modeling and Characterization EE5342, Lecture 5-Spring 2005 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ L5 February 02

Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently, L5 February 02

Carrier velocity saturation1 The mobility relationship v = mE is limited to “low” fields v < vth = (3kT/m*)1/2 defines “low” v = moE[1+(E/Ec)b]-1/b, mo = v1/Ec for Si parameter electrons holes v1 (cm/s) 1.53E9 T-0.87 1.62E8 T-0.52 Ec (V/cm) 1.01 T1.55 1.24 T1.68 b 2.57E-2 T0.66 0.46 T0.17 L5 February 02

vdrift [cm/s] vs. E [V/cm] (Sze2, fig. 29a) L5 February 02

Carrier velocity saturation (cont.) At 300K, for electrons, mo = v1/Ec = 1.53E9(300)-0.87/1.01(300)1.55 = 1504 cm2/V-s, the low-field mobility The maximum velocity (300K) is vsat = moEc = v1 = 1.53E9 (300)-0.87 = 1.07E7 cm/s L5 February 02

Diffusion of carriers In a gradient of electrons or holes, p and n are not zero Diffusion current,`J =`Jp +`Jn (note Dp and Dn are diffusion coefficients) L5 February 02

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`Jp and the + sign in the definition of`Jn L5 February 02

Diffusion of Carriers (cont.) L5 February 02

Current density components L5 February 02

Total current density L5 February 02

Doping gradient induced E-field If N = Nd-Na = N(x), then so is Ef-Efi Define f = (Ef-Efi)/q = (kT/q)ln(no/ni) For equilibrium, Efi = constant, but for dN/dx not equal to zero, Ex = -df/dx =- [d(Ef-Efi)/dx](kT/q) = -(kT/q) d[ln(no/ni)]/dx = -(kT/q) (1/no)[dno/dx] = -(kT/q) (1/N)[dN/dx], N > 0 L5 February 02

Induced E-field (continued) Let Vt = kT/q, then since nopo = ni2 gives no/ni = ni/po Ex = - Vt d[ln(no/ni)]/dx = - Vt d[ln(ni/po)]/dx = - Vt d[ln(ni/|N|)]/dx, N = -Na < 0 Ex = - Vt (-1/po)dpo/dx = Vt(1/po)dpo/dx = Vt(1/Na)dNa/dx L5 February 02

The Einstein relationship For Ex = - Vt (1/no)dno/dx, and Jn,x = nqmnEx + qDn(dn/dx) = 0 This requires that nqmn[Vt (1/n)dn/dx] = qDn(dn/dx) Which is satisfied if L5 February 02

Direct carrier gen/recomb k Ec Ev (Excitation can be by light) gen rec - + Ev Ec Ef Efi L5 February 02

Direct gen/rec of excess carriers Generation rates, Gn0 = Gp0 Recombination rates, Rn0 = Rp0 In equilibrium: Gn0 = Gp0 = Rn0 = Rp0 In non-equilibrium condition: n = no + dn and p = po + dp, where nopo=ni2 and for dn and dp > 0, the recombination rates increase to R’n and R’p L5 February 02

Direct rec for low-level injection Define low-level injection as dn = dp < no, for n-type, and dn = dp < po, for p-type The recombination rates then are R’n = R’p = dn(t)/tn0, for p-type, and R’n = R’p = dp(t)/tp0, for n-type Where tn0 and tp0 are the minority-carrier lifetimes L5 February 02

Shockley-Read- Hall Recomb Indirect, like Si, so intermediate state Ec Ec ET Ef Efi Ev Ev k L5 February 02

S-R-H trap characteristics1 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 Ec - ET “Donor-like” trap which has given up the extra electron is +q and “empty” L5 February 02

S-R-H trap char. (cont.) If trap neutral when orbited (filled) by an excess hole - “acceptor-like” Gives up hole with energy ET - Ev “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 L5 February 02

tpo = (Ntvthsn)-1, where sn~p(rBohr)2 S-R-H recombination Recombination rate determined by: Nt (trap conc.), vth (thermal vel of the carriers), sn (capture cross sect for electrons), sp (capture cross sect for holes), with tno = (Ntvthsn)-1, and tpo = (Ntvthsn)-1, where sn~p(rBohr)2 L5 February 02

S-R-H recomb. (cont.) In the special case where tno = tpo = to the net recombination rate, U is L5 February 02

S-R-H “U” function characteristics The numerator, (np-ni2) simplifies in the case of extrinsic material at low level injection (for equil., nopo = ni2) For n-type (no > dn = dp > po = ni2/no): (np-ni2) = (no+dn)(po+dp)-ni2 = nopo - ni2 + nodp + dnpo + dndp ~ nodp (largest term) Similarly, for p-type, (np-ni2) ~ podn L5 February 02

S-R-H “U” function characteristics (cont) For n-type, as above, the denominator = to{no+dn+po+dp+2nicosh[(Et-Ei)kT]}, simplifies to the smallest value for Et~Ei, where the denom is tono, giving U = dp/to as the largest (fastest) For p-type, the same argument gives U = dn/to Rec rate, U, fixed by minority carrier L5 February 02

S-R-H net recom- bination rate, U In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is L5 February 02

S-R-H rec for excess min carr For n-type low-level injection and net excess minority carriers, (i.e., no > dn = dp > po = ni2/no), U = dp/to, (prop to exc min carr) For p-type low-level injection and net excess minority carriers, (i.e., po > dn = dp > no = ni2/po), U = dn/to, (prop to exc min carr) L5 February 02

Minority hole lifetimes. Taken from Shur3, (p.101). L5 February 02

Minority electron lifetimes. Taken from Shur3, (p.101). L5 February 02

Parameter example tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2 For Nd = 1E17cm3, tp = 25 msec Why Nd and tp ? L5 February 02

S-R-H rec for deficient min carr If n < ni and p < pi, then the S-R-H net recomb rate becomes (p < po, n < no): U = R - G = - ni/(2t0cosh[(ET-Efi)/kT]) And with the substitution that the gen lifetime, tg = 2t0cosh[(ET-Efi)/kT], and net gen rate U = R - G = - ni/tg The intrinsic concentration drives the return to equilibrium L5 February 02

The Continuity Equation The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives L5 February 02

The Continuity Equation (cont.) L5 February 02

The Continuity Equation (cont.) L5 February 02

The Continuity Equation (cont.) L5 February 02

The Continuity Equation (cont.) L5 February 02

The Continuity Equation (cont.) L5 February 02

The Continuity Equation (cont.) L5 February 02

References 1Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. 2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981. 3 Physics of Semiconductor Devices, Shur, Prentice-Hall, 1990. L5 February 02