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EE 5340 Semiconductor Device Theory Lecture 5 - Fall 2010

EE 5340 Semiconductor Device Theory Lecture 5 - Fall 2010. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc. Second Assignment. Please print and bring to class a signed copy of the document appearing at http://www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf.

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EE 5340 Semiconductor Device Theory Lecture 5 - Fall 2010

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  1. EE 5340Semiconductor Device TheoryLecture 5 - Fall 2010 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

  2. Second Assignment • Please print and bring to class a signed copy of the document appearing at http://www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf

  3. Intrinsic carrierconc. (MB limit) • ni2 = no po = Nc Nv e-Eg/kT • Nc = 2{2pm*nkT/h2}3/2 • Nv = 2{2pm*pkT/h2}3/2 • Eg = 1.17 eV - aT2/(T+b) a = 4.73E-4 eV/K b = 636K

  4. Classes ofsemiconductors • Intrinsic: no = po = ni, since Na&Nd << ni, ni2 = NcNve-Eg/kT, ~1E-13 dopant level ! • n-type: no > po, since Nd > Na • p-type: no < po, since Nd < Na • Compensated: no=po=ni, w/ Na- = Nd+ > 0 • Note: n-type and p-type are usually partially compensated since there are usually some opposite- type dopants

  5. n-type equilibriumconcentrations • N ≡Nd- Na , n type N > 0 • For all N, no = N/2 + {[N/2]2+ni2}1/2 • In most cases, N >> ni, so no = N, and po= ni2/no = ni2/N, (Law of Mass Action is al- ways true in equilibrium)

  6. p-type equilibriumconcentrations • N ≡Nd - Na , p type  N < 0 • For all N, po = |N|/2 + {[|N|/2]2+ni2}1/2 • In most cases, |N| >> ni, so po = |N|, and no = ni2/po = ni2/|N|, (Law of Mass Action is al- ways true in equilibrium)

  7. Position of theFermi Level • Efi is the Fermi level when no = po • Ef shown is a Fermi level for no > po • Ef < Efi when no < po • Efi < (Ec + Ev)/2, which is the mid-band

  8. EF relative to Ec and Ev • Inverting no = Nc exp[-(Ec-EF)/kT] gives Ec - EF = kT ln(Nc/no) For n-type material: Ec - EF =kTln(Nc/Nd)=kTln[(Ncpo)/ni2] • Inverting po = Nv exp[-(EF-Ev)/kT] gives EF - Ev = kT ln(Nv/po) For p-type material: EF - Ev = kT ln(Nv/Na)

  9. EF relative to Efi • Letting ni = no gives  Ef = Efi ni = Nc exp[-(Ec-Efi)/kT], so Ec - Efi = kT ln(Nc/ni). Thus EF - Efi = kT ln(no/ni) and for n-type EF - Efi = kT ln(Nd/ni) • Likewise Efi - EF = kT ln(po/ni) and for p-type Efi - EF = kT ln(Na/ni)

  10. Locating Efi in the bandgap • Since Ec - Efi = kT ln(Nc/ni), and Efi - Ev = kT ln(Nv/ni) • The 1st equation minus the 2nd gives Efi = (Ec + Ev)/2 - (kT/2) ln(Nc/Nv) • Since Nc = 2.8E19cm-3 > 1.04E19cm-3 = Nv, the intrinsic Fermi level lies below the middle of the band gap

  11. Samplecalculations • Efi = (Ec + Ev)/2 - (kT/2) ln(Nc/Nv), so at 300K, kT = 25.86 meV and Nc/Nv = 2.8/1.04, Efi is 12.8 meV or 1.1% below mid-band • For Nd = 3E17cm-3, given that Ec - EF = kT ln(Nc/Nd), we have Ec - EF = 25.86 meV ln(280/3), Ec - EF = 0.117 eV =117meV ~3x(Ec - ED) what Nd gives Ec-EF =Ec/3

  12. Equilibrium electronconc. and energies

  13. Equilibrium hole conc. and energies

  14. Carrier Mobility • In an electric field, Ex, the velocity (since ax = Fx/m* = qEx/m*) is vx = axt = (qEx/m*)t, and the displ x = (qEx/m*)t2/2 • If every tcoll, a collision occurs which “resets” the velocity to <vx(tcoll)> = 0, then <vx> = qExtcoll/m* = mEx

  15. Carrier mobility (cont.) • The response function m is the mobility. • The mean time between collisions, tcoll, may has several important causal events: Thermal vibrations, donor- or acceptor-like traps and lattice imperfections to name a few. • Hence mthermal = qtthermal/m*, etc.

  16. Carrier mobility (cont.) • If the rate of a single contribution to the scattering is 1/ti, then the total scattering rate, 1/tcoll is

  17. Figure 1.16 (p. 31 M&K) Electron and hole mobilities in silicon at 300 K as functions of the total dopant concentration. The values plotted are the results of curve fitting measurements from several sources. The mobility curves can be generated using Equation 1.2.10 with the following values of the parameters [3] (see table on next slide).

  18. Figure 1.16 (cont. M&K)

  19. Drift Current • The drift current density (amp/cm2) is given by the point form of Ohm Law J = (nqmn+pqmp)(Exi+ Eyj+ Ezk), so J = (sn + sp)E =sE, where s = nqmn+pqmp defines the conductivity • The net current is

  20. 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)

  21. Drift currentresistance (cont.) • Consequently, since R = rl/A R = (nqmn + pqmp)-1(l/A) • For n >> p, (an n-type extrinsic s/c) R = l/(nqmnA) • For p >> n, (a p-type extrinsic s/c) R = l/(pqmpA)

  22. Drift currentresistance (cont.) • Note: for an extrinsic semiconductor and multiple scattering mechanisms, since R = l/(nqmnA) or l/(pqmpA), and (mn or p total)-1 = Smi-1, then Rtotal = S Ri (series Rs) • The individual scattering mechanisms are: Lattice, ionized impurity, etc.

  23. Net intrinsicmobility • Considering only lattice scattering

  24. 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

  25. Net extrinsicmobility • Considering only lattice and impurity scattering

  26. 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.)

  27. 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

  28. Figure 1.17 (p. 32 in M&K1) Low-field mobility in silicon as a function of temperature for electrons (a), and for holes (b). The solid lines represent the theoretical predictions for pure lattice scattering [5].

  29. Exp. m(T=300K) modelfor P, As and B in Si1

  30. 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

  31. 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

  32. 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

  33. 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)

  34. Figure 1.15 (p. 29) M&K Dopant density versus resistivity at 23°C (296 K) for silicon doped with phosphorus and with boron. The curves can be used with little error to represent conditions at 300 K. [W. R. Thurber, R. L. Mattis, and Y. M. Liu, National Bureau of Standards Special Publication 400–64, 42 (May 1981).]

  35. Net silicon extrresistivity (cont.) • Since r = (nqmn + pqmp)-1, and mn > mp, (m = qt/m*) we have rp > rn, for the same NI • Note that since 1.6(high conc.) < rp/rn < 3(low conc.), so 1.6(high conc.) < mn/mp < 3(low conc.)

  36. 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

  37. Figure 1.16 (p. 31 M&K) Electron and hole mobilities in silicon at 300 K as functions of the total dopant concentration. The values plotted are the results of curve fitting measurements from several sources. The mobility curves can be generated using Equation 1.2.10 with the following values of the parameters [3] (see table on next slide).

  38. 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

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

  40. 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

  41. Carrier velocity2 carrier velocity vs E for Si, Ge, and GaAs (after Sze2)

  42. 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

  43. 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)

  44. 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

  45. Diffusion ofCarriers (cont.)

  46. Current densitycomponents

  47. Total currentdensity

  48. 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

  49. 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

  50. 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

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