Slide 1Web PagesFirst AssignmentQuantum ConceptsBohr model for the H atom (cont.)Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Quantum MechanicsSchrodinger EquationSolutions for the Schrodinger EquationInfinite Potential WellStep PotentialFinite Potential BarrierKronig-Penney ModelK-P Potential Function*K-P Static WavefunctionsK-P Impulse SolutionK-P Solutions*K-P E(k) Relationship*Analogy: a nearly -free electr. modelGeneralizations and ConclusionsSilicon Band Structure**ReferencesEE 5340Semiconductor Device TheoryLecture 02 – Spring 2011Professor Ronald L. [email protected]://www.uta.edu/ronc©rlc L02 20Jan20112Web 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 20Jan20113First Assignment•Send e-mail to [email protected]–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 20Jan20114Quantum Concepts•Bohr Atom•Light Quanta (particle-like waves)•Wave-like properties of particles•Wave-Particle Duality©rlc L02 20Jan20115Bohr 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 20Jan20116Energy 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) constanth = 6.625E-34 J-sec stopomaxqVffhmvT 221©rlc L02 20Jan20117Photon: 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 20Jan20118Wave-particle duality•Compton showed Dp = hkinitial - hkfinal, so an photon (wave) is particle-like©rlc L02 20Jan20119Wave-particle duality•DeBroglie hypothesized a particle could be wave-like, l = h/p©rlc L02 20Jan201110Wave-particle duality•Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality model©rlc L02 20Jan201111Newtonian Mechanics•Kinetic energy, KE = mv2/2 = p2/2m Conservation of Energy Theorem•Momentum, p = mvConservation of Momentum Thm•Newton’s second Law F = ma = m dv/dt = m d2x/dt2©rlc L02 20Jan201112Quantum 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 20Jan201113Schrodinger Equation•Separation of variables givesY(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  22280xxmE V x x  h2( )©rlc L02 20Jan201114Solutions 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.    1dxxx©rlc L02 20Jan201115Infinite 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 2,88E1,2,3,...=n ,sin2222222nhkhpmkhmanhaxnax©rlc L02 20Jan201116Step 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 , andevanescent solutions for E < Vo •A reflection coefficient can be def.©rlc L02 20Jan201117Finite 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 20Jan201118Kronig-Penney ModelA 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 20Jan201119K-P Potential Function*©rlc L02 20Jan201120K-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 20Jan201121K-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 20Jan201122K-P Solutions*©rlc L02 20Jan201123K-P E(k) Relationship*©rlc L02 20Jan201124Analogy: 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) of122224kEhm©rlc L02 20Jan201125Generalizationsand 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 20Jan201126Silicon 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©rlc L02 20Jan201127References *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,