Slide 1Review the FollowingFirst AssignmentSecond AssignmentKronig-Penney ModelK-P Potential Function*K-P Impulse SolutionK-P Solutions*K-P E(k) Relationship*Analogy: a nearly -free electr. modelGeneralizations and ConclusionsSilicon Band Structure**Generalizations and ConclusionsAnalogy: a nearly -free electr. modelSilicon Covalent Bond (2D Repr)Si Energy Band Structure at 0 KSi Bond Model Above Zero KelvinBand Model for thermal carriersDonor: cond. electr. due to phosphorousBohr model H atom- like orbits at donorBand Model for donor electronsAcceptor: Hole due to boronHole orbits and acceptor statesImpurity Levels in Si: EG = 1,124 meVReferencesEE 5340Semiconductor Device TheoryLecture 03 – Spring 2011Professor Ronald L. [email protected]://www.uta.edu/ronc©rlc L03 27Jan20112Review the Following•R. L. Carter’s web page:–www.uta.edu/ronc/•EE 5340 web page and syllabus. (Refresh all EE 5340 pages when downloading to assure the latest version.) All links at:–www.uta.edu/ronc/5340/syllabus.htm•University and College Ethics Policies–www.uta.edu/studentaffairs/conduct/•Makeup lecture at noon Friday (1/28) in 108 Nedderman Hall. This will be available on the web.©rlc L03 27Jan20113First 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 L03 27Jan20114Second Assignment•Submit a signed copy of the document posted at www.uta.edu/ee/COE%20Ethics%20Statement%20Fall%2007.pdf©rlc L03 27Jan20115Kronig-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 L03 27Jan20116K-P Potential Function*©rlc L03 27Jan20117K-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 L03 27Jan20118K-P Solutions*©rlc L03 27Jan20119K-P E(k) Relationship*©rlc L03 27Jan201110Analogy: 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 L03 27Jan201111Generalizationsand 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 L03 27Jan201112Silicon 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 L03 27Jan2011Generalizationsand 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.13©rlc L03 27Jan2011Analogy: 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14©rlc L03 27Jan2011Silicon 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 electron15©rlc L03 27Jan2011Si Energy BandStructure 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 band16©rlc L03 27Jan2011Si Bond ModelAbove 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)17©rlc L03 27Jan2011Band Model forthermal 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/cm318©rlc L03 27Jan2011Donor: 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 orbits19©rlc L03 27Jan2011Bohr 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 nmfor Z=1, m*~mo/2, n=1, ground state20©rlc L03 27Jan2011Band Model fordonor 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 small21©rlc L03 27Jan2011Acceptor: Holedue 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 orbits22©rlc L03 27Jan2011Hole orbits andacceptor 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 temperatures23©rlc L03 27Jan2011Impurity 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 meVEC - ED = 774 meV24©rlc L03 27Jan201125References *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,