Semiconductor Device Modeling and Characterization – EE5342 Lecture 3 – Spring 2011

1. Semiconductor Device Modeling and Characterization – EE5342 Lecture 3 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

2. Web Pages • Bring the following to the first class • R. L. Carter’s web page • www.uta.edu/ronc/ • EE 5342 web page and syllabus • http://www.uta.edu/ronc/5342/syllabus.htm • University and College Ethics Policies www.uta.edu/studentaffairs/conduct/ www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf

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. Second 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.

5. Schrodinger Equation • Separation of variables gives Y(x,t) = y(x)• f(t) • The time-independent part of the Schrodinger equation for a single particle with KE = E and PE = V.

6. K-P Potential Function*

7. K-P Static Wavefunctions • Inside the ions, 0 < x < a y(x) = A exp(jbx) + B exp (-jbx) b = [8p2mE/h]1/2 • Between ions region, a < x < (a + b) = L y(x) = C exp(ax) + D exp (-ax) a = [8p2m(Vo-E)/h2]1/2

8. K-P Impulse Solution • Limiting case of Vo-> inf. and b -> 0, while a2b = 2P/a is finite • In this way a2b2 = 2Pb/a < 1, giving sinh(ab) ~ ab and cosh(ab) ~ 1 • The solution is expressed by P sin(ba)/(ba) + cos(ba) = cos(ka) • Allowed values of LHS bounded by +1 • k = free electron wave # = 2p/l

9. x x K-P Solutions* P sin(ba)/(ba) + cos(ba) vs.ba

10. K-P E(k) Relationship*

11. Analogy: a nearly-free X electron model • Solutions can be displaced by ka = 2np • Allowed and forbidden energies • Infinite well approximation by replacing the free electron mass with an “effective” mass (noting E = p2/2m = h2k2/2m) of

12. Silicon BandStructure** • Indirect Bandgap • Curvature (hence m*) is function of direction and band. [100] is x-dir, [111] is cube diagonal • Eg = 1.17-aT2/(T+b) a = 4.73E-4 eV/K b = 636K

13. Generalizationsand Conclusions • The symm. of the crystal struct. gives “allowed” and “forbidden” energies (sim to pass- and stop-band) • The curvature at band-edge (where k = (n+1)p) gives an “effective” mass.

14. Silicon Covalent Bond (2D Repr) • Each Si atom has 4 nearest neighbors • Si atom: 4 valence elec and 4+ ion core • 8 bond sites / atom • All bond sites filled • Bonding electrons shared 50/50 _= Bonding electron

15. Si Energy BandStructure at 0 K • Every valence site is occupied by an electron • No electrons allowed in band gap • No electrons with enough energy to populate the conduction band

16. Si Bond ModelAbove Zero Kelvin • Enough therm energy ~kT(k=8.62E-5eV/K) to break some bonds • Free electron and broken bond separate • One electron for every “hole” (absent electron of broken bond)

17. Band Model forthermal carriers • Thermal energy ~kT generates electron-hole pairs • At 300K Eg(Si) = 1.124 eV >> kT = 25.86 meV, Nc = 2.8E19/cm3 > Nv = 1.04E19/cm3 >> ni = 1.45E10/cm3

18. Donor: cond. electr.due to phosphorous • P atom: 5 valence elec and 5+ ion core • 5th valence electr has no avail bond • Each extra free el, -q, has one +q ion • # P atoms = # free elect, so neutral • H atom-like orbits

19. Bohr model H atom-like orbits at donor • Electron (-q) rev. around proton (+q) • Coulomb force, F=q2/4peSieo,q=1.6E-19 Coul, eSi=11.7, eo=8.854E-14 Fd/cm • Quantization L = mvr = nh/2p • En= -(Z2m*q4)/[8(eoeSi)2h2n2] ~-40meV • rn= [n2(eoeSi)h2]/[Zpm*q2] ~ 2 nm for Z=1, m*~mo/2, n=1, ground state

20. Band Model fordonor electrons • Ionization energy of donor Ei = Ec-Ed ~ 40 meV • Since Ec-Ed ~ kT, all donors are ionized, so ND ~ n • Electron “freeze-out” when kT is too small

21. Acceptor: Holedue to boron • B atom: 3 valence elec and 3+ ion core • 4th bond site has no avail el (=> hole) • Each hole, adds --q, has one -q ion • #B atoms = #holes, so neutral • H atom-like orbits

22. Hole orbits andacceptor states • Similar to free electrons and donor sites, there are hole orbits at acceptor sites • The ionization energy of these states is EA - EV ~ 40 meV, so NA ~ p and there is a hole “freeze-out” at low temperatures

23. Impurity Levels in Si: EG = 1,124 meV • Phosphorous, P: EC - ED = 44 meV • Arsenic, As: EC - ED = 49 meV • Boron, B: EA - EV = 45 meV • Aluminum, Al: EA - EV = 57 meV • Gallium, Ga: EA - EV = 65meV • Gold, Au: EA - EV = 584 meV EC - ED = 774 meV

24. 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)

25. 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.