Slide 1Review the FollowingFirst AssignmentSecond AssignmentQuantum density of states functionQM density of states (cont.)Fermi-Dirac distribution fctnFermi-Dirac DF (continued)Maxwell-Boltzman ApproximationElectron Conc. in the MB approx.Electron and Hole Conc in MB approxCalculating the equilibrium noEquilibrium con- centration for noEquilibrium con- centration for poSlide 15Degenerate and nondegenerate casesSlide 17Donor ionizationDonor ionization (continued)Slide 20ReferencesEE 5340Semiconductor Device TheoryLecture 04 – Spring 2011Professor Ronald L. [email protected]://www.uta.edu/ronc©rlc L04-28Jan20112Review the Following•R. L. Carter’s web page:–www.uta.edu/ronc/•EE 5340 web page and syllabus. (Refresh all EE 5340 pages when downloading to assure the latest version.) All links at:–www.uta.edu/ronc/5340/syllabus.htm•University and College Ethics Policies–www.uta.edu/studentaffairs/conduct/•Makeup lecture at noon Friday (1/28) in 108 Nedderman Hall. This will be available on the web.©rlc L04-28Jan20113First Assignment•Send e-mail to [email protected]–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©rlc L04-28Jan20114Second Assignment•Submit a signed copy of the document posted at www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf©rlc L04-28Jan2011Quantum densityof states function•1 dim electron wave #s range for n+1 “atoms” is 2p/L < k < 2p/a where a is “interatomic” distance and L = na is the length of the assembly (k = 2p/l)•Shorter ls, would “oversample”•if n increases by 1, dp is h/L•Extn 3D: E = p2/2m = h2k2/2m so a vol of p-space of 4pp2dp has h3/LxLyLz5©rlc L04-28Jan2011QM density of states (cont.)•So density of states, gc(E) is (Vol in p-sp)/(Vol per state*V) =4pp2dp/[(h3/LxLyLz)*V]•Noting that p2 = 2mE, this becomes gc(E) = {4p(2mn*)3/2/h3}(E-Ec)1/2and E - Ec = h2k2/2mn*•Similar for the hole states whereEv - E = h2k2/2mp*6©rlc L04-28Jan2011Fermi-Diracdistribution fctn•The probability of an electron having an energy, E, is given by the F-D distr fF(E) = {1+exp[(E-EF)/kT]}-1•Note: fF (EF) = 1/2•EF is the equilibrium energy of the system•The sum of the hole probability and the electron probability is 17©rlc L04-28Jan2011Fermi-DiracDF (continued)•So the probability of a hole having energy E is 1 - fF(E)•At T = 0 K, fF (E) becomes a step function and 0 probability of E > EF•At T >> 0 K, there is a finite probability of E >> EF8©rlc L04-28Jan2011Maxwell-BoltzmanApproximation•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 9©rlc L04-28Jan2011Electron Conc. inthe MB approx.•Assuming the MB approx., the equilibrium electron concentration is    kTEEexpNndEEfEgnFccoEEcoFmaxc10©rlc L04-28Jan2011Electron and HoleConc 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/cm311©rlc L04-28Jan2011Calculating theequilibrium no•The idea is to calculate the equilibrium electron concentration no for the FD distribution, wherefF(E) = {1+exp[(E-EF)/kT]}-1gc(E) = [4p(2mn*)3/2(E-Ec)1/2]/h3   dEEfEgnFmaxcEEco12©rlc L04-28Jan2011Equilibrium 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, 213©rlc L04-28Jan2011Equilibrium 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 po14©rlc L04-28Jan2011Figure 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) 15©rlc L04-28Jan2011Degenerate andnondegenerate 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 ~/< Ev16©rlc L04-28Jan2011Figure 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)17©rlc L04-28Jan2011Donor 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 18©rlc L04-28Jan2011Donor 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 19©rlc L04-28Jan2011Figure 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) 20©rlc L04-28Jan201121References *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