EE 5340 Semiconductor Device Theory Lecture 4 - Fall 2009 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

First Assignment Send e-mail to ronc@uta.edu On the subject line, put “5340 e-mail” In the body of message include email address: ______________________ Your Name*: _______________________ Last four digits of your Student ID: _____ * Your name as it appears in the UTA Record - no more, no less L 04 Sept 04

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 L 04 Sept 04

Maxwell-Boltzman Approximation fF(E) = {1+exp[(E-EF)/kT]}-1 For E - EF > 3 kT, the exp > 20, so within a 5% error, fF(E) ~ exp[-(E-EF)/kT] This is the MB distribution function MB used when E-EF>75 meV (T=300K) For electrons when Ec - EF > 75 meV and for holes when EF - Ev > 75 meV L 04 Sept 04

Electron Conc. in the MB approx. Assuming the MB approx., the equilibrium electron concentration is L 04 Sept 04

Electron and Hole Conc in MB approx Similarly, the equilibrium hole concentration is po = Nv exp[-(EF-Ev)/kT] So that nopo = NcNv exp[-Eg/kT] ni2 = nopo, Nc,v = 2{2pm*n,pkT/h2}3/2 Nc = 2.8E19/cm3, Nv = 1.04E19/cm3 and ni = 1.45E10/cm3 L 04 Sept 04

Calculating the equilibrium no The idea is to calculate the equilibrium electron concentration no for the FD distribution, where fF(E) = {1+exp[(E-EF)/kT]}-1 gc(E) = [4p(2mn*)3/2(E-Ec)1/2]/h3 L 04 Sept 04

Equilibrium con- centration for no Earlier quoted the MB approximation no = Nc exp[-(Ec - EF)/kT],(=Nc exp hF) The exact solution is no = 2NcF1/2(hF)/p1/2 Where F1/2(hF) is the Fermi integral of order 1/2, and hF = (EF - Ec)/kT Error in no, e, is smaller than for the DF: e = 31%, 12%, 5% for -hF = 0, 1, 2 L 04 Sept 04

Equilibrium con- centration for po Earlier quoted the MB approximation po = Nv exp[-(EF - Ev)/kT],(=Nv exp h’F) The exact solution is po = 2NvF1/2(h’F)/p1/2 Note: F1/2(0) = 0.678, (p1/2/2) = 0.886 Where F1/2(h’F) is the Fermi integral of order 1/2, and h’F = (Ev - EF)/kT Errors are the same as for po L 04 Sept 04

Figure 1.10 (a) Fermi-Dirac distribution function describing the probability that an allowed state at energy E is occupied by an electron. (b) The density of allowed states for a semiconductor as a function of energy; note that g(E) is zero in the forbidden gap between Ev and Ec. (c) The product of the distribution function and the density-of-states function. (p. 17 - M&K) L 04 Sept 04

Degenerate and nondegenerate cases Bohr-like doping model assumes no interaction between dopant sites If adjacent dopant atoms are within 2 Bohr radii, then orbits overlap This happens when Nd ~ Nc (EF ~ Ec), or when Na ~ Nv (EF ~ Ev) The degenerate semiconductor is defined by EF ~/> Ec or EF ~/< Ev L 04 Sept 04

Figure 1.13 Energy-gap narrowing Eg as a function of electron concentration. [A. Neugroschel, S. C. Pao, and F. A. Lindhold, IEEE Trans. Electr. Devices, ED-29, 894 (May 1982).] taken from p. 25 - M&K) L 04 Sept 04

Donor ionization The density of elec trapped at donors is nd = Nd/{1+[exp((Ed-EF)/kT)/2]} Similar to FD DF except for factor of 2 due to degeneracy (4 for holes) Furthermore nd = Nd - Nd+, also For a shallow donor, can have Ed-EF >> kT AND Ec-EF >> kT: Typically EF-Ed ~ 2kT L 04 Sept 04

Donor ionization (continued) Further, if Ed - EF > 2kT, then nd ~ 2Nd exp[-(Ed-EF)/kT], e < 5% If the above is true, Ec - EF > 4kT, so no ~ Nc exp[-(Ec-EF)/kT], e < 2% Consequently the fraction of un-ionized donors is nd/no = 2Nd exp[(Ec-Ed)/kT]/Nc = 0.4% for Nd(P) = 1e16/cm3 L 04 Sept 04

Figure 1. 9 Electron concentration vs Figure 1.9 Electron concentration vs. temperature for two n-type doped semiconductors: (a) Silicon doped with 1.15 X 1016 arsenic atoms cm-3[1], (b) Germanium doped with 7.5 X 1015 arsenic atoms cm-3[2]. (p.12 in M&K) L 04 Sept 04

Classes of semiconductors Intrinsic: no = po = ni, since Na&Nd << ni =[NcNvexp(-Eg/kT)]1/2, (not easy to get) 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 L 04 Sept 04

Equilibrium concentrations 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 L 04 Sept 04

Equilibrium conc (cont.) For Nd > Na (taking the + root) no = (Nd-Na)/2 + {[(Nd-Na)/2]2+ni2}1/2 For Nd >> Na and Nd >> ni, can use the binomial expansion, giving no = Nd/2 + Nd/2[1 + 2ni2/Nd2 + … ] So no = Nd, and po = ni2/Nd in the limit of Nd >> Na and Nd >> ni L 04 Sept 04

Example calculations For Nd = 3.2E16/cm3, ni = 1.4E10/cm3 no = Nd = 3.2E16/cm3 po = ni2/Nd , (po is always ni2/no) = (1.4E10/cm3)2/3.2E16/cm3 = 6.125E3/cm3 (comp to ~1E23 Si) For po = Na = 4E17/cm3, no = ni2/Na = (1.4E10/cm3)2/4E17/cm3 = 490/cm3 L 04 Sept 04

Position of the Fermi 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 L 04 Sept 04

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) L 04 Sept 04

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) L 04 Sept 04

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 L 04 Sept 04

Sample calculations 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 L 04 Sept 04

Equilibrium electron conc. and energies L 04 Sept 04

Equilibrium hole conc. and energies L 04 Sept 04

vx = axt = (qEx/m*)t, and the displ 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 L 04 Sept 04

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. L 04 Sept 04

Carrier mobility (cont.) If the rate of a single contribution to the scattering is 1/ti, then the total scattering rate, 1/tcoll is L 04 Sept 04

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). L 04 Sept 04

Parameter Arsenic Phosphorus Boron μmin 52.2 68.5 44.9 μmax 1417 1414 470.5 Nref 9.68 X 1016 9.20 X 1016 2.23 X 1017 α 0.680 0.711 0.719 Figure 1.16 (cont. M&K) L 04 Sept 04

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 L 04 Sept 04

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.) L 04 Sept 04

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).] L 04 Sept 04

References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989. **Semiconductor Physics & Devices, by Donald A. Neamen, 2nd ed., Irwin, Chicago. M&K = Device Electronics for Integrated Circuits, 3rd ed., by Richard S. Muller, Theodore I. Kamins, and Mansun Chan, John Wiley and Sons, New York, 2003. L 04 Sept 04