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Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2004

Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2004. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/. Web Pages. You should be aware of information at R. L. Carter’s web page www.uta.edu/ronc/ EE 5342 web page and syllabus

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Semiconductor Device Modeling and Characterization EE5342, Lecture 4-Spring 2004

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  1. Semiconductor Device Modeling and CharacterizationEE5342, Lecture 4-Spring 2004 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

  2. Web Pages • You should be aware of information at • R. L. Carter’s web page • www.uta.edu/ronc/ • EE 5342 web page and syllabus • www.uta.edu/ronc/5342/syllabus.htm • University and College Ethics Policies • www2.uta.edu/discipline/ • www.uta.edu/ronc/5342/Ethics.htm • Submit a signed copy to Dr. Carter

  3. First Assignment • e-mail to listserv@listserv.uta.edu • In the body of the message include subscribe EE5342 • This will subscribe you to the EE5342 list. Will receive all EE5342 messages • If you have any questions, send to ronc@uta.edu, with EE5342 in subject line.

  4. Equilibriumconcentrations • Charge neutrality requires q(po + Nd+) + (-q)(no + Na-) = 0 • Assuming complete ionization, so Nd+ = Nd and Na- = Na • Gives two equations to be solved simultaneously 1. Mass action, no po = ni2, and 2. Neutrality po + Nd = no + Na

  5. Equilibrium electronconc. and energies

  6. Equilibrium hole conc. and energies

  7. Mobility 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

  8. Drift currentresistance • 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)

  9. Exp. mobility modelfunction 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

  10. Exp. mobility modelfor P, As and B in Si

  11. Carrier mobilityfunctions (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

  12. Carrier mobilityfunctions (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

  13. 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

  14. Ionized impuritymobility 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

  15. 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)

  16. Net silicon extrresistivity (cont.)

  17. Net silicon extrresistivity (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.)

  18. 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

  19. Equipartitiontheorem • The thermodynamic energy per degree of freedom is kT/2 Consequently,

  20. 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

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

  22. 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

  23. 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)

  24. 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

  25. Diffusion ofCarriers (cont.)

  26. Current densitycomponents

  27. Total currentdensity

  28. 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

  29. 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

  30. 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

  31. E - - Ec Ec Ef Efi gen rec Ev Ev + + k Direct carriergen/recomb (Excitation can be by light)

  32. 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

  33. 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

  34. Shockley-Read-Hall Recomb E Indirect, like Si, so intermediate state Ec Ec ET Ef Efi Ev Ev k

  35. 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”

  36. 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

  37. 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

  38. S-R-Hrecomb. (cont.) • In the special case where tno = tpo = to the net recombination rate, U is

  39. 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

  40. 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

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

  42. 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)

  43. Minority hole lifetimes. Taken from Shur3, (p.101).

  44. Minority electron lifetimes. Taken from Shur3, (p.101).

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

  46. 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.

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