EE 5340 Semiconductor Device Theory Lecture 02 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

Web Pages Review the following R. L. Carter’s web page www.uta.edu/ronc/ EE 5340 web page and syllabus www.uta.edu/ronc/5340/syllabus.htm University and College Ethics Policies www.uta.edu/studentaffairs/conduct/ www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf ©rlc L02 20Jan2011

First Assignment Send e-mail to ronc@uta.edu 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 L02 20Jan2011

Quantum Concepts Bohr Atom Light Quanta (particle-like waves) Wave-like properties of particles Wave-Particle Duality ©rlc L02 20Jan2011

Bohr model for the H atom (cont.) En= - (mq4)/[8eo2h2n2] ~ -13.6 eV/n2 * rn= [n2eoh2]/[pmq2] ~ 0.05 nm = 1/2 Ao * *for n=1, ground state ©rlc L02 20Jan2011

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 20Jan2011

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 20Jan2011

Wave-particle duality Compton showed Dp = hkinitial - hkfinal, so an photon (wave) is particle-like ©rlc L02 20Jan2011

Wave-particle duality DeBroglie hypothesized a particle could be wave-like, l = h/p ©rlc L02 20Jan2011

Wave-particle duality Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality model ©rlc L02 20Jan2011

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 20Jan2011

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 20Jan2011

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 Total E = E and PE = V. The Kinetic Energy, KE = E - V ©rlc L02 20Jan2011

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 20Jan2011

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 20Jan2011

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 20Jan2011

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 20Jan2011

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 20Jan2011

K-P Potential Function* ©rlc L02 20Jan2011

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 20Jan2011

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 valued of LHS bounded by +1 k = free electron wave # = 2p/l ©rlc L02 20Jan2011

K-P Solutions* ©rlc L02 20Jan2011

K-P E(k) Relationship* ©rlc L02 20Jan2011

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 20Jan2011

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 20Jan2011

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 20Jan2011

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. ©rlc L02 20Jan2011