Semiconductor Device Modeling and Characterization – EE5342 Lecture 3 – Spring 2011 Professor Ronald L. Carter

©rlc L03 24Jan20112 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 L03 24Jan20115 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 L03 24Jan20116 K-P Potential Function*

©rlc L03 24Jan20117 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 L03 24Jan20118 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 L03 24Jan20119 K-P Solutions* P sin(  a)/(  a) + cos(  a) vs.  a x x

©rlc L03 24Jan K-P E(k) Relationship*

©rlc L03 24Jan Analogy: a nearly-free X electron 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 L03 24Jan 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 L03 24Jan 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 L03 24Jan 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 L03 24Jan 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

©rlc L03 24Jan 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)

©rlc L03 24Jan Band Model for thermal carriers Thermal energy ~kT generates electron-hole pairs At 300K Eg(Si) = eV >> kT = meV, Nc = 2.8E19/cm3 > Nv = 1.04E19/cm3 >> ni = 1.45E10/cm3

©rlc L03 24Jan 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

©rlc L03 24Jan Bohr model H atom- like orbits at donor Electron (-q) rev. around proton (+q) Coulomb force, F=q 2 /4  Si  o,q=1.6E-19 Coul,  Si =11.7,  o =8.854E-14 Fd/cm Quantization L = mvr = nh/2  E n = -(Z 2 m*q 4 )/[8(  o  Si ) 2 h 2 n 2 ] ~-40meV r n = [n 2 (  o  Si )h 2 ]/[Z  m*q 2 ] ~ 2 nm for Z=1, m*~m o /2, n=1, ground state

©rlc L03 24Jan Band Model for donor electrons Ionization energy of donor E i = E c -E d ~ 40 meV Since E c -E d ~ kT, all donors are ionized, so N D ~ n Electron “freeze- out” when kT is too small

©rlc L03 24Jan 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

©rlc L03 24Jan 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 E A - E V ~ 40 meV, so N A ~ p and there is a hole “freeze-out” at low temperatures

©rlc L03 24Jan Impurity Levels in Si: E G = 1,124 meV Phosphorous, P: E C - E D = 44 meV Arsenic, As:E C - E D = 49 meV Boron, B: E A - E V = 45 meV Aluminum, Al: E A - E V = 57 meV Gallium, Ga: E A - E V = 65meV Gold, Au: E A - E V = 584 meV E C - E D = 774 meV

©rlc L03 24Jan2011 Semiconductor Electronics - concepts thus far Conduction and Valence states due to symmetry of lattice “Free-elec.” dynamics near band edge Band Gap –direct or indirect –effective mass in curvature Thermal carrier generation Chemical carrier gen (donors/accept) 24

©rlc L03 24Jan References *Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, **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, Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981.