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Semiconductor Device Modeling and Characterization EE5342, Lecture 5-Spring 2005. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/. Equipartition theorem. The thermodynamic energy per degree of freedom is kT/2 Consequently,. Carrier velocity saturation 1.
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Semiconductor Device Modeling and CharacterizationEE5342, Lecture 5-Spring 2005 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/
Equipartitiontheorem • The thermodynamic energy per degree of freedom is kT/2 Consequently,
Carrier velocitysaturation1 • 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
Carrier velocitysaturation (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
Diffusion ofcarriers • 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)
Diffusion ofcarriers (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
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
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
The Einsteinrelationship • 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
E - - Ec Ec Ef Efi gen rec Ev Ev + + k Direct carriergen/recomb (Excitation can be by light)
Direct gen/recof 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
Direct rec forlow-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
Shockley-Read-Hall Recomb E Indirect, like Si, so intermediate state Ec Ec ET Ef Efi Ev Ev k
S-R-H trapcharacteristics1 • 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”
S-R-H trapchar. (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
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
S-R-Hrecomb. (cont.) • In the special case where tno = tpo = to the net recombination rate, U is
S-R-H “U” functioncharacteristics • 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
S-R-H “U” functioncharacteristics (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
S-R-H net recom-bination rate, U • In the special case where tno = tpo = to = (Ntvthso)-1 the net rec. rate, U is
S-R-H rec forexcess 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)
Parameter example • tmin = (45 msec) 1+(7.7E-18cm3)Ni+(4.5E-36cm6)Ni2 • For Nd = 1E17cm3, tp = 25 msec • Why Nd and tp ?
S-R-H rec fordeficient 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
The ContinuityEquation • The chain rule for the total time derivative dn/dt (the net generation rate of electrons) gives
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.
