Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2003 Professor Ronald L. Carter
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Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2003 Professor Ronald L. Carter [email protected] http://www.uta.edu/ronc/ L4 January 23 1 First Assignment • Send e-mail to [email protected] – 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 2 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 3 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 4 Exp. mobility model function for Si1max min mn, p mn, p Parameter mmin mmax Nref a L4 January 23 min mn, p mn, p Nd, a 1 N ref As 52.2 1417 9.68e16 0.680 a P 68.5 1414 9.20e16 0.711 B 44.9 470.5 2.23e17 0.719 5 Mobility (cm^2/V-sec) Exp. mobility model for P, As and B in Si 1500 1000 P As 500 B 0 1.E+13 1.E+15 1.E+17 1.E+19 Doping Concentration (cm^-3) L4 January 23 6 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 7 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 8 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 9 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 10 Net silicon (extrinsic) 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 11 Resistivity (ohm-cm) Net silicon extr resistivity (cont.) 1.00E+03 1.00E+02 P 1.00E+01 As 1.00E+00 B 1.00E-01 1.00E-02 1.E+13 1.E+15 1.E+17 1.E+19 Doping Concentration (cm^-3) L4 January 23 12 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 13 Net silicon (compensated) 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 14 Equipartition theorem • The thermodynamic energy per degree of freedom is kT/2 Consequently, 1 2 mv 2 vrms L4 January 23 thermal 3 kT, and 2 3kT 7 10 cm / sec m* 15 Carrier velocity 1 saturation • 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 16 vdrift L4 January 23 [cm/s] vs. E [V/cm] (Sze2, fig. 29a) 17 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 18 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) p p p Jp qDpp qDp i j k z x y n n n Jn qDn n qDn i j k x y z L4 January 23 19 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 20 Diffusion of Carriers (cont.) L4 January 23 21 Current density components Note, since E V Jp,drift s pE pqm pE pqm pV Jn,drift snE nqmnE nqmnV Jp,diffusion qDpp Jn,diffusion qDnn L4 January 23 22 Total current density The total current density is driven by the carrier gradients and the potential gradient Jtotal Jp,drift Jn,drift Jp,diff. Jn,diff. Jtotal s p sn V qDpp qDnn L4 January 23 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 24 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 25 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 Dp Dn kT Vt , likewise Vt mn q mp L4 January 23 26 Direct carrier gen/recomb (Excitation can be by light) - gen + L4 January 23 rec + E Ec Ef Efi Ec Ev Ev k 27 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 28 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 minoritycarrier lifetimes L4 January 23 29 Shockley-ReadHall Recomb Indirect, like Si, so intermediate state ET L4 January 23 E Ec Ef Efi Ec Ev Ev k 30 S-R-H trap 1 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 Ec - ET • “Donor-like” trap which has given up the extra electron is +q and “empty” L4 January 23 31 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 32 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 33 S-R-H recomb. (cont.) • In the special case where tno = tpo = to the net recombination rate, U is d dp ddn URG dt dt U pn ni2 ET Efi to p n 2ni cosh kT where n no dn, and p po dp, (dn dp) L4 January 23 34 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 35 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 36 S-R-H net recombination rate, U • In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is d dp ddn URG dt dt U pn ni2 ET Efi to p n 2ni cosh kT where n no dn, and p po dp, (dn dp) L4 January 23 37 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 38 Minority hole lifetimes. Taken from Shur3, (p.101). L4 January 23 39 Minority electron lifetimes. Taken from Shur3, (p.101). L4 January 23 40 Parameter example • tmin = (45 msec) 1+(7.7E-18cm3Ni+(4.5E-36cm6Ni2 • For Nd = 1E17cm3, tp = 25 msec – Why Nd and tp ? L4 January 23 41 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 42