EE 5340 Semiconductor Device Theory Lecture 2 - Fall 2009 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

Quantum Concepts Bohr Atom Light Quanta (particle-like waves) Wave-like properties of particles Wave-Particle Duality L02 Aug 27

Wave-particle duality Davisson and Germer demonstrated wave-like interference phenomena for electrons to complete the duality model L02 Aug 27

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 L02 Aug 27

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)| L02 Aug 27

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 L02 Aug 27

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. L02 Aug 27

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 ©L02 Aug 28

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. ©L02 Aug 28

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 ©L02 Aug 28

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 ©L02 Aug 28

K-P Potential Function* ©L02 Aug 28

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 ©L02 Aug 28

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 ©L02 Aug 28

K-P Solutions* ©L02 Aug 28

K-P E(k) Relationship* ©L02 Aug 28

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 ©L02 Aug 28

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. ©L02 Aug 28

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 ©L02 Aug 28

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 ©L02 Aug 28

Si Energy Band Structure at 0 K Every valence site is occupied by an electron No electrons allowed in band gap No electrons with enough energy to populate the conduction band ©L02 Aug 28

Si Bond Model Above Zero Kelvin Enough therm energy ~kT(k=8.62E-5eV/K) to break some bonds Free electron and broken bond separate One electron for every “hole” (absent electron of broken bond) ©L02 Aug 28

Band Model for thermal carriers Thermal energy ~kT generates electron-hole pairs At 300K Eg(Si) = 1.124 eV >> kT = 25.86 meV, Nc = 2.8E19/cm3 > Nv = 1.04E19/cm3 >> ni = 1.45E10/cm3 ©L03 Sept 02

Donor: cond. electr. due to phosphorous P atom: 5 valence elec and 5+ ion core 5th valence electr has no avail bond Each extra free el, -q, has one +q ion # P atoms = # free elect, so neutral H atom-like orbits ©L03 Sept 02

Bohr model H atom- like orbits at donor Electron (-q) rev. around proton (+q) Coulomb force, F=q2/4peSieo,q=1.6E-19 Coul, eSi=11.7, eo=8.854E-14 Fd/cm Quantization L = mvr = nh/2p En= -(Z2m*q4)/[8(eoeSi)2h2n2] ~-40meV rn= [n2(eoeSi)h2]/[Zpm*q2] ~ 2 nm for Z=1, m*~mo/2, n=1, ground state ©L03 Sept 02

Band Model for donor electrons Ionization energy of donor Ei = Ec-Ed ~ 40 meV Since Ec-Ed ~ kT, all donors are ionized, so ND ~ n Electron “freeze-out” when kT is too small ©L03 Sept 02

Acceptor: Hole due to boron B atom: 3 valence elec and 3+ ion core 4th bond site has no avail el (=> hole) Each hole, adds --q, has one -q ion #B atoms = #holes, so neutral H atom-like orbits ©L03 Sept 02

Hole orbits and acceptor states Similar to free electrons and donor sites, there are hole orbits at acceptor sites The ionization energy of these states is EA - EV ~ 40 meV, so NA ~ p and there is a hole “freeze-out” at low temperatures ©L03 Sept 02

Impurity Levels in Si: EG = 1,124 meV Phosphorous, P: EC - ED = 44 meV Arsenic, As: EC - ED = 49 meV Boron, B: EA - EV = 45 meV Aluminum, Al: EA - EV = 57 meV Gallium, Ga: EA - EV = 65meV Gold, Au: EA - EV = 584 meV EC - ED = 774 meV ©L03 Sept 02

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. L02 Aug 27