Semiconductor Device Modeling and Characterization – EE5342 Lecture 4 – Spring 2011

1. Semiconductor Device Modeling and Characterization – EE5342 Lecture 4 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. 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.

3. Second Assignment • Submit a signed copy of the document that is posted at www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf

4. Semiconductor Electronics - concepts thus far • Conduction and Valence states due to symmetry of lattice • “Free-elec.” dynamics near band edge • Band Gap • direct or indirect • effective mass in curvature • Thermal carrier generation • Chemical carrier gen (donors/accept)

5. Counting carriers - quantum density of 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/LxLyLz

6. QM 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)1/2 and E - Ec = h2k2/2mn* • Similar for the hole states where Ev - E = h2k2/2mp*

7. Fermi-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 1

8. Fermi-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 >> EF

9. Maxwell-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

10. Electron Conc. inthe MB approx. • Assuming the MB approx., the equilibrium electron concentration is

11. Electron 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 = 1E10/cm3

12. Calculating theequilibrium no • The ideal 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

13. 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

14. 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

15. Degenerate 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 ~/< Ev

16. 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

17. 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

18. Classes ofsemiconductors • 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

19. 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. • 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.