Semiconductor Device Modeling and Characterization – EE5342 Lecture 2 – Spring 2011

1. Semiconductor Device Modeling and Characterization – EE5342 Lecture 2 – 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. A Quick Review of Physics • Review of • Semiconductor Quantum Physics • Semiconductor carrier statistics • Semiconductor carrier dynamics

5. Bohr model H atom • Electron (-q) rev. around proton (+q) • Coulomb force, F=q2/4peor2, q=1.6E-19 Coul, eo=8.854E-14 Fd/cm • Quantization L = mvr = nh/2p • En= -(mq4)/[8eo2h2n2] ~ -13.6 eV/n2 • rn= [n2eoh]/[pmq2] ~ 0.05 nm = 1/2 Ao • for n=1, ground state

6. Quantum Concepts • Bohr Atom • Light Quanta (particle-like waves) • Wave-like properties of particles • Wave-Particle Duality

7. Energy Quanta for Light • Photoelectric Effect: • Tmax is the energy of the electron emitted from a material surface when light of frequency f is incident. • fo, frequency for zero KE, mat’l spec. • h is Planck’s (a universal) constant h = 6.625E-34 J-sec

8. Photon: A particle-like wave • E = hf, the quantum of energy for light. (PE effect & black body rad.) • f = c/l, c = 3E8m/sec, l = wavelength • From Poynting’s theorem (em waves), momentum density = energy density/c • Postulate a Photon “momentum” p = h/l = hk, h = h/2p wavenumber, k =2p /l

9. Wave-particle Duality • Compton showed Dp = hkinitial - hkfinal, so an photon (wave) is particle-like • DeBroglie hypothesized a particle could be wave-like, l = h/p • Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality model

10. Newtonian Mechanics • Kinetic energy, KE = mv2/2 = p2/2m Conservation of Energy Theorem • Momentum, p = mv Conservation of Momentum Thm • Newton’s second Law F = ma = m dv/dt = m d2x/dt2

11. Quantum Mechanics • Schrodinger’s wave equation developed to maintain consistence with wave-particle duality and other “quantum” effects • Position, mass, etc. of a particle replaced by a “wave function”, Y(x,t) • Prob. density = |Y(x,t)• Y*(x,t)|

12. 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.

13. Solutions for the Schrodinger Equation • Solutions of the form of y(x) = A exp(jKx) + B exp (-jKx) K = [8p2m(E-V)/h2]1/2 • Subj. to boundary conds. and norm. y(x) is finite, single-valued, conts. dy(x)/dx is finite, s-v, and conts.

14. Infinite Potential Well • V = 0, 0 < x < a • V --> inf. for x < 0 and x > a • Assume E is finite, so y(x) = 0 outside of well

15. Step Potential • V = 0, x < 0 (region 1) • V = Vo, x > 0 (region 2) • Region 1 has free particle solutions • Region 2 has free particle soln. for E > Vo , and evanescent solutions for E < Vo • A reflection coefficient can be def.

16. Finite Potential Barrier • Region 1: x < 0, V = 0 • Region 1: 0 < x < a, V = Vo • Region 3: x > a, V = 0 • Regions 1 and 3 are free particle solutions • Region 2 is evanescent for E < Vo • Reflection and Transmission coeffs. For all E

17. Kronig-Penney Model A simple one-dimensional model of a crystalline solid • V = 0, 0 < x < a, the ionic region • V = Vo, a < x < (a + b) = L, between ions • V(x+nL) = V(x), n = 0, +1, +2, +3, …, representing the symmetry of the assemblage of ions and requiring that y(x+L) = y(x) exp(jkL), Bloch’s Thm

18. K-P Potential Function*

19. 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

20. 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

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

22. K-P E(k) Relationship*

23. Analogy: a nearly-free electr. 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

24. 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.

25. 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

26. 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

27. 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.