Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/ Semiconductor Device Modeling and Characterization – EE5342 Lecture 2 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc/

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 ©rlc L02 21Jan2011

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. ©rlc L02 21Jan2011

A Quick Review of Physics Semiconductor Quantum Physics Semiconductor carrier statistics Semiconductor carrier dynamics ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

Quantum Concepts Bohr Atom Light Quanta (particle-like waves) Wave-like properties of particles Wave-Particle Duality ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

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)| ©rlc L02 21Jan2011

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. ©rlc L02 21Jan2011

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. ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

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. ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

K-P Potential Function* ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

K-P Solutions* x P sin(ba)/(ba) + cos(ba) vs. ba ©rlc L02 21Jan2011

K-P E(k) Relationship* ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

Generalizations and 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. ©rlc L02 21Jan2011

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 ©rlc L02 21Jan2011

Silicon Band Structure** 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 ©rlc L02 21Jan2011

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. ©rlc L02 21Jan2011