Semiconductor Device Modeling and Characterization EE5342, Lecture 1-Spring 2011 Professor Ronald L. Carter

©rlc L01 19Jan20112 Web Pages *Bring the following to the first class R. L. Carter’s web page – EE 5342 web page and syllabus – University and College Ethics Policies

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©rlc L01 19Jan20114 A Quick Review of Physics Review of –Semiconductor Quantum Physics –Semiconductor carrier statistics –Semiconductor carrier dynamics

©rlc L01 19Jan20115 Bohr model H atom Electron (-q) rev. around proton (+q) Coulomb force, F=q 2 /4  o r 2, q=1.6E-19 Coul,  o =8.854E-14 Fd/cm Quantization L = mvr = nh/2  E n = -(mq 4 )/[8  o 2 h 2 n 2 ] ~ eV/n 2 r n = [n 2  o h]/[  mq 2 ] ~ 0.05 nm = 1/2 A o for n=1, ground state

©rlc L01 19Jan20116 Quantum Concepts Bohr Atom Light Quanta (particle-like waves) Wave-like properties of particles Wave-Particle Duality

©rlc L01 19Jan20117 Energy Quanta for Light Photoelectric Effect: T max is the energy of the electron emitted from a material surface when light of frequency f is incident. f o, frequency for zero KE, mat’l spec. h is Planck’s (a universal) constant h = 6.625E-34 J-sec

©rlc L01 19Jan20118 Photon: A particle -like wave E = hf, the quantum of energy for light. (PE effect & black body rad.) f = c/, c = 3E8m/sec, = wavelength From Poynting’s theorem (em waves), momentum density = energy density/c Postulate a Photon “momentum” p = h/  = hk, h = h/2  wavenumber, k =  2  /

©rlc L01 19Jan20119 Wave-particle Duality Compton showed  p = hk initial - hk final, so an photon (wave) is particle-like DeBroglie hypothesized a particle could be wave-like,  = h/p Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality model

©rlc L01 19Jan Newtonian Mechanics Kinetic energy, KE = mv 2 /2 = p 2 /2m Conservation of Energy Theorem Momentum, p = mv Conservation of Momentum Thm Newton’s second Law F = ma = m dv/dt = m d 2 x/dt 2

©rlc L01 19Jan 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”,  (x,t) Prob. density = |  (x,t)   (x,t)|

©rlc L01 19Jan Schrodinger Equation Separation of variables gives  (x,t) =  (x)  (t) The time-independent part of the Schrodinger equation for a single particle with KE = E and PE = V.

©rlc L01 19Jan Solutions for the Schrodinger Equation Solutions of the form of  (x) = A exp(jKx) + B exp (-jKx) K = [8  2 m(E-V)/h 2 ] 1/2 Subj. to boundary conds. and norm.  (x) is finite, single-valued, conts. d  (x)/dx is finite, s-v, and conts.

©rlc L01 19Jan Infinite Potential Well V = 0, 0 < x < a V --> inf. for x a Assume E is finite, so  (x) = 0 outside of well

©rlc L01 19Jan Step Potential V = 0, x < 0 (region 1) V = V o, x > 0 (region 2) Region 1 has free particle solutions Region 2 has free particle soln. for E > V o, and evanescent solutions for E < V o A reflection coefficient can be def.

©rlc L01 19Jan Finite Potential Barrier Region 1: x < 0, V = 0 Region 1: 0 < x < a, V = V o Region 3: x > a, V = 0 Regions 1 and 3 are free particle solutions Region 2 is evanescent for E < V o Reflection and Transmission coeffs. For all E

©rlc L01 19Jan Kronig-Penney Model A simple one-dimensional model of a crystalline solid V = 0, 0 < x < a, the ionic region V = V o, 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  (x+L) =  (x) exp(jkL), Bloch’s Thm

©rlc L01 19Jan K-P Potential Function*

©rlc L01 19Jan K-P Static Wavefunctions Inside the ions, 0 < x < a  (x) = A exp(j  x) + B exp (-j  x)  = [8  2 mE/h] 1/2 Between ions region, a < x < (a + b) = L  (x) = C exp(  x) + D exp (-  x)  = [8  2 m(V o -E)/h 2 ] 1/2

©rlc L01 19Jan K-P Impulse Solution Limiting case of V o -> inf. and b -> 0, while  2 b = 2P/a is finite In this way  2 b 2 = 2Pb/a < 1, giving sinh(  b) ~  b and cosh(  b) ~ 1 The solution is expressed by P sin(  a)/(  a) + cos(  a) = cos(ka) Allowed values of LHS bounded by +1 k = free electron wave # = 2  /

©rlc L01 19Jan K-P Solutions* P sin(  a)/(  a) + cos(  a) vs.  a x x

©rlc L01 19Jan K-P E(k) Relationship*

©rlc L01 19Jan References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, **Semiconductor Physics & Devices, by Donald A. Neamen, 2nd ed., Irwin, Chicago.