Semiconductor Device Modeling and Characterization – EE5342 Lecture 5 – Spring 2011 Professor Ronald L. Carter

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©rlc L05-28Jan20114 Classes of semiconductors Intrinsic: n o = p o = n i, since N a &N d << n i =[N c N v exp(E g /kT)] 1/2,(not easy to get) n-type: n o > p o, since N d > N a p-type: n o < p o, since N d < N a Compensated: n o =p o =n i, w/ N a - = N d + > 0 Note: n-type and p-type are usually partially compensated since there are usually some opposite-type dopants

©rlc L05-28Jan20115 Equilibrium concentrations Charge neutrality requires q(p o + N d + ) + (-q)(n o + N a - ) = 0 Assuming complete ionization, so N d + = N d and N a - = N a Gives two equations to be solved simultaneously 1. Mass action, n o p o = n i 2, and 2. Neutralityp o + N d = n o + N a

©rlc L05-28Jan20116 For N d > N a >Let N = N d -N a, and (taking the + root) n o = (N)/2 + {[N/2] 2 +n i 2 } 1/2 For N d+ = N d >> n i >> N a we have >n o = N d, and >p o = n i 2 /N d Equilibrium conc n-type

©rlc L05-28Jan20117 For N a > N d >Let N = N d -N a, and (taking the + root) p o = (-N)/2 + {[-N/2] 2 +n i 2 } 1/2 For N a- = N a >> n i >> N d we have >p o = N a, and >n o = n i 2 /N a Equilibrium conc p-type

©rlc L05-28Jan20118 Position of the Fermi Level E fi is the Fermi level when n o = p o E f shown is a Fermi level for n o > p o E f < E fi when n o < p o E fi < (E c + E v )/2, which is the mid- band

©rlc L05-28Jan20119 E F relative to E c and E v Inverting n o = N c exp[-(E c -E F )/kT] gives E c - E F = kT ln(N c /n o ) For n-type material: E c - E F =kTln(N c /N d )=kTln[(N c P o )/n i 2 ] Inverting p o = N v exp[-(E F -E v )/kT] givesE F - E v = kT ln(N v /p o ) For p-type material: E F - E v = kT ln(N v /N a )

©rlc L05-28Jan E F relative to E fi Letting n i = n o gives  E f = E fi n i = N c exp[-(E c -E fi )/kT], so E c - E fi = kT ln(N c /n i ). Thus E F - E fi = kT ln(n o /n i ) and for n- typeE F - E fi = kT ln(N d /n i ) Likewise E fi - E F = kT ln(p o /n i ) and for p- type E fi - E F = kT ln(N a /n i )

©rlc L05-28Jan Locating E fi in the bandgap Since E c - E fi = kT ln(N c /n i ), and E fi - E v = kT ln(N v /n i ) The sum of the two equations gives E fi = (E c + E v )/2 - (kT/2) ln(N c /N v ) Since N c = 2.8E19cm -3 > 1.04E19cm -3 = N v, the intrinsic Fermi level lies below the middle of the band gap

©rlc L05-28Jan Sample calculations E fi = (E c + E v )/2 - (kT/2) ln(N c /N v ), so at 300K, kT = meV and N c /N v = 2.8/1.04, E fi is 12.8 meV or 1.1% below mid-band For N d = 3E17cm -3, given that E c - E F = kT ln(N c /N d ), we have E c - E F = meV ln(280/3), E c - E F = eV =117meV ~3x(E c - E D ) what N d gives E c -E F =E c /3

©rlc L05-28Jan Equilibrium electron conc. and energies

©rlc L05-28Jan Equilibrium hole conc. and energies

©rlc L05-28Jan Carrier Mobility In an electric field, E x, the velocity (since a x = F x /m* = qE x /m*) is v x = a x t = (qE x /m*)t, and the displ x = (qE x /m*)t 2 /2 If every  coll, a collision occurs which “resets” the velocity to = 0, then = qE x  coll /m* =  E x

©rlc L05-28Jan Carrier mobility (cont.) The response function  is the mobility. The mean time between collisions,  coll, may has several important causal events: Thermal vibrations, donor- or acceptor-like traps and lattice imperfections to name a few. Hence  thermal = q  thermal /m*, etc.

©rlc L05-28Jan Carrier mobility (cont.) If the rate of a single contribution to the scattering is 1/  i, then the total scattering rate, 1/  coll is

©rlc L05-28Jan Drift Current The drift current density (amp/cm 2 ) is given by the point form of Ohm Law J = (nq  n +pq  p )(E x i+ E y j+ E z k), so J = (  n +  p )E =  E, where  = nq  n +pq  p defines the conductivity The net current is

©rlc L05-28Jan Drift current resistance Given: a semiconductor resistor with length, l, and cross-section, A. What is the resistance? As stated previously, the conductivity,  = nq  n + pq  p So the resistivity,  = 1/  = 1/(nq  n + pq  p )

©rlc L05-28Jan Drift current resistance (cont.) Consequently, since R =  l/A R = (nq  n + pq  p ) -1 (l/A) For n >> p, (an n-type extrinsic s/c) R = l/(nq  n A) For p >> n, (a p-type extrinsic s/c) R = l/(pq  p A)

©rlc L05-28Jan 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.