Semiconductor Device Modeling and Characterization – EE5342 Lecture 2 – Spring 2011 Professor Ronald L. Carter

©rlc L02 21Jan20112 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

©rlc L02 21Jan20113 First Assignment to –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 with EE5342 in subject line.

©rlc L02 21Jan20114 A Quick Review of Physics Review of –Semiconductor Quantum Physics –Semiconductor carrier statistics –Semiconductor carrier dynamics

©rlc L02 21Jan20115 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 L02 21Jan20116 Quantum Concepts Bohr Atom Light Quanta (particle-like waves) Wave-like properties of particles Wave-Particle Duality

©rlc L02 21Jan20117 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 L02 21Jan20118 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 L02 21Jan20119 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 L02 21Jan 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 L02 21Jan 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 L02 21Jan 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 L02 21Jan 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 L02 21Jan Infinite Potential Well V = 0, 0 < x < a V --> inf. for x a Assume E is finite, so  (x) = 0 outside of well

©rlc L02 21Jan 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 L02 21Jan 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 L02 21Jan 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 L02 21Jan K-P Potential Function*

©rlc L02 21Jan 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 L02 21Jan 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 L02 21Jan K-P Solutions* P sin(  a)/(  a) + cos(  a) vs.  a x x

©rlc L02 21Jan K-P E(k) Relationship*

©rlc L02 21Jan Analogy: a nearly -free electr. model Solutions can be displaced by ka = 2n  Allowed and forbidden energies Infinite well approximation by replacing the free electron mass with an “effective” mass (noting E = p 2 /2m = h 2 k 2 /2m) of

©rlc L02 21Jan 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)  ) gives an “effective” mass.

©rlc L02 21Jan 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 21Jan Silicon Band Structure** Indirect Bandgap Curvature (hence m*) is function of direction and band. [100] is x-dir, [111] is cube diagonal E g =  T 2 /(T+  )  = 4.73E-4 eV/K  = 636K

©rlc L02 21Jan References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, **Semiconductor Physics & Devices, by Donald A. Neamen, 2nd ed., Irwin, Chicago.