Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2003 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ L4 January 23

First Assignment Send e-mail to ronc@uta.edu On the subject line, put “5342 e-mail” In the body of message include email address: ______________________ Last Name*: _______________________ First Name*: _______________________ Last four digits of your Student ID: _____ * As it appears in the UTA Record - no more, no less L4 January 23

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 m(Nd,Na,T) Resistivity models developed for extrinsic and compensated materials L4 January 23

Drift current resistance Given: a semiconductor resistor with length, l, and cross-section, A. What is the resistance? As stated previously, the conductivity, s = nqmn + pqmp So the resistivity, r = 1/s = 1/(nqmn + pqmp) L4 January 23

Exp. mobility model function for Si1 Parameter As P B mmin 52.2 68.5 44.9 mmax 1417 1414 470.5 Nref 9.68e16 9.20e16 2.23e17 a 0.680 0.711 0.719 L4 January 23

Exp. mobility model for P, As and B in Si L4 January 23

Carrier mobility functions (cont.) The parameter mmax models 1/tlattice the thermal collision rate The parameters mmin, Nref and a model 1/timpur the impurity collision rate The function is approximately of the ideal theoretical form: 1/mtotal = 1/mthermal + 1/mimpurity L4 January 23

Carrier mobility functions (ex.) Let Nd = 1.78E17/cm3 of phosphorous, so mmin = 68.5, mmax = 1414, Nref = 9.20e16 and a = 0.711. Thus mn = 586 cm2/V-s Let Na = 5.62E17/cm3 of boron, so mmin = 44.9, mmax = 470.5, Nref = 9.68e16 and a = 0.680. Thus mp = 189 cm2/V-s L4 January 23

Lattice mobility The mlattice is the lattice scattering mobility due to thermal vibrations Simple theory gives mlattice ~ T-3/2 Experimentally mn,lattice ~ T-n where n = 2.42 for electrons and 2.2 for holes Consequently, the model equation is mlattice(T) = mlattice(300)(T/300)-n L4 January 23

Ionized impurity mobility function The mimpur is the scattering mobility due to ionized impurities Simple theory gives mimpur ~ T3/2/Nimpur Consequently, the model equation is mimpur(T) = mimpur(300)(T/300)3/2 L4 January 23

Net silicon (ex- trinsic) resistivity Since r = s-1 = (nqmn + pqmp)-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 s(Nimpur) L4 January 23

Net silicon extr resistivity (cont.) L4 January 23

Net silicon extr resistivity (cont.) Since r = (nqmn + pqmp)-1, and mn > mp, (m = qt/m*) we have rp > rn Note that since 1.6(high conc.) < rp/rn < 3(low conc.), so 1.6(high conc.) < mn/mp < 3(low conc.) L4 January 23

Net silicon (com- pensated) res. For an n-type (n >> p) compensated semiconductor, r = (nqmn)-1 But now n = N = Nd - Na, and the mobility must be considered to be determined by the total ionized impurity scattering Nd + Na = NI Consequently, a good estimate is r = (nqmn)-1 = [Nqmn(NI)]-1 L4 January 23

Equipartition theorem The thermodynamic energy per degree of freedom is kT/2 Consequently, L4 January 23

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 L4 January 23

vdrift [cm/s] vs. E [V/cm] (Sze2, fig. 29a) L4 January 23

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 L4 January 23

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) L4 January 23

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 L4 January 23

Diffusion of Carriers (cont.) L4 January 23

Current density components L4 January 23

Total current density L4 January 23

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 L4 January 23

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 L4 January 23

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 L4 January 23

Direct carrier gen/recomb k Ec Ev (Excitation can be by light) gen rec - + Ev Ec Ef Efi L4 January 23

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 L4 January 23

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 L4 January 23

Shockley-Read- Hall Recomb Indirect, like Si, so intermediate state Ec Ec ET Ef Efi Ev Ev k L4 January 23

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” L4 January 23

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 L4 January 23

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 L4 January 23

S-R-H recomb. (cont.) In the special case where tno = tpo = to the net recombination rate, U is L4 January 23

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 L4 January 23

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 L4 January 23

S-R-H net recom- bination rate, U In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is L4 January 23

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) L4 January 23

Minority hole lifetimes. Taken from Shur3, (p.101). L4 January 23

Minority electron lifetimes. Taken from Shur3, (p.101). L4 January 23

Parameter example tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2 For Nd = 1E17cm3, tp = 25 msec Why Nd and tp ? L4 January 23

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. L4 January 23