Semiconductor Device Modeling and Characterization – EE5342 Lecture 09– Spring 2011 Professor Ronald L. Carter
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©rlc L09-14Feb20114 Additional University Closure Means More Schedule Changes Plan to meet until noon some days in the next few weeks. This way we will make up for the lost time. The first extended class will be Monday, 2/14. The MT changed to Friday 2/18 The P1 test changed to Friday 3/11. The P2 test is still Wednesday 4/13 The Final is still Wednesday 5/11.
MT and P1 Assignment on Friday, 2/18/11 Quizzes and tests are open book –must have a legally obtained copy-no Xerox copies. –OR one handwritten page of notes. –Calculator allowed. A cover sheet will be published by Wednesday, 2/16/11. ©rlc L09-14Feb20115
6 Energy bands for p- and n-type s/c p-type EcEc EvEv E Fi E Fp q p = kT ln(n i /N a ) EvEv EcEc E Fi E Fn q n = kT ln(N d /n i ) n-type
©rlc L09-14Feb20117 Making contact in a p-n junction Equate the E F in the p- and n-type materials far from the junction E o (the free level), E c, E fi and E v must be continuous N.B.: q = 4.05 eV (Si), and q = q E c - E F EoEo EcEc EfEf E fi EvEv q (electron affinity) qFqF qq (work function)
©rlc L09-14Feb20118 Band diagram for p + -n jctn* at V a = 0 EcEc E fN E fi EvEv EcEc E fP E fi EvEv 0 xnxn x -x p -x pc x nc q p < 0 q n > 0 qV bi = q( n - p ) *N a > N d -> | p | > n p-type for x<0 n-type for x>0
©rlc L09-14Feb20119 A total band bending of qV bi = q( n - p ) = kT ln(N d N a /n i 2 )is necessary to set E fP = E fN For -x p < x < 0, E fi - E fP < -q p, = |q p | so p < N a = p o, (depleted of maj. carr.) For 0 < x < x n, E fN - E fi < q n, so n < N d = n o, (depleted of maj. carr.) -x p < x < x n is the Depletion Region Band diagram for p + -n at V a =0 (cont.)
©rlc L09-14Feb Depletion Approximation Assume p << p o = N a for -x p < x < 0, so = q(N d -N a +p-n) = -qN a, -x p < x < 0, and p = p o = N a for -x pc < x < -x p, so = q(N d -N a +p-n) = 0, -x pc < x < -x p Assume n << n o = N d for 0 < x < x n, so = q(N d -N a +p-n) = qN d, 0 < x < x n, and n = n o = N d for x n < x < x nc, so = q(N d -N a +p-n) = 0, x n < x < x nc
©rlc L09-14Feb Poisson’s Equation The electric field at (x,y,z) is related to the charge density =q(N d -N a -p-n) by the Poisson Equation:
©rlc L09-14Feb Poisson’s Equation For n-type material, N = (N d - N a ) > 0, n o = N, and (N d -N a +p-n)=- n + p +n i 2 /N For p-type material, N = (N d - N a ) < 0, p o = -N, and (N d -N a +p-n) = p- n-n i 2 /N So neglecting n i 2 /N, [ =(N d -N a +p-n)]
©rlc L09-14Feb Quasi-Fermi Energy
©rlc L09-14Feb Quasi-Fermi Energy (cont.)
©rlc L09-14Feb Quasi-Fermi Energy (cont.)
©rlc L09-14Feb Induced E-field in the D.R. The sheet dipole of charge, due to Q p ’ and Q n ’ induces an electric field which must satisfy the conditions Charge neutrality and Gauss’ Law* require thatE x = 0 for -x pc < x < -x p and E x = 0 for -x n < x < x nc h 0h 0
©rlc L09-14Feb Induced E-field in the D.R. xnxn x -x p -x pc x nc O - O - O - O + O + O + Depletion region (DR) p-type CNR ExEx Exposed Donor ions Exposed Acceptor Ions n-type chg neutral reg p-contact N-contact W 0
©rlc L09-14Feb Depletion approx. charge distribution xnxn x -x p -x pc x nc +qN d -qN a +Q n ’=qN d x n Q p ’=-qN a x p Charge neutrality => Q p ’ + Q n ’ = 0, => N a x p = N d x n [Coul/cm 2 ]
©rlc L09-14Feb dim soln. of Gauss’ law x xnxn x nc -x pc -x p ExEx -E max
©rlc L09-14Feb Depletion Approxi- mation (Summary) For the step junction defined by doping N a (p-type) for x 0, the depletion width W = {2 (V bi -V a )/qN eff } 1/2, where V bi = V t ln{N a N d /n i 2 }, and N eff =N a N d /(N a +N d ). Since N a x p =N d x n, x n = W/(1 + N d /N a ), and x p = W/(1 + N a /N d ).
©rlc L09-14Feb One-sided p+n or n+p jctns If p + n, then N a >> N d, and N a N d /(N a + N d ) = N eff --> N d, and W --> x n, DR is all on lightly d. side If n + p, then N d >> N a, and N a N d /(N a + N d ) = N eff --> N a, and W --> x p, DR is all on lightly d. side The net effect is that N eff --> N -, (- = lightly doped side) and W --> x -
©rlc L09-14Feb Junction C (cont.) xnxn x -x p -x pc x nc +qN d -qN a +Q n ’=qN d x n Q p ’=-qN a x p Charge neutrality => Q p ’ + Q n ’ = 0, => N a x p = N d x n Q n ’=qN d x n Q p ’=-qN a x p
©rlc L09-14Feb Junction C (cont.) The C-V relationship simplifies to
©rlc L09-14Feb Junction C (cont.) If one plots [C’ j ] -2 vs. V a Slope = -[(C’ j0 ) 2 V bi ] -1 vertical axis intercept = [C’ j0 ] -2 horizontal axis intercept = V bi C’ j -2 V bi VaVa C’ j0 -2
©rlc L09-14Feb Arbitrary doping profile If the net donor conc, N = N(x), then at x n, the extra charge put into the DR when V a ->V a + V a is Q’=-qN(x n ) x n The increase in field, E x =-(qN/ ) x n, by Gauss’ Law (at x n, but also const). So V a =-(x n +x p ) E x = (W/ ) Q’ Further, since N(x n ) x n = N(x p ) x p gives, the dC/dx n as...
©rlc L09-14Feb Arbitrary doping profile (cont.)
©rlc L09-14Feb Arbitrary doping profile (cont.)
©rlc L09-14Feb Arbitrary doping profile (cont.)
©rlc L09-14Feb Arbitrary doping profile (cont.)
©rlc L09-14Feb Debye length The DA assumes n changes from N d to 0 discontinuously at x n, likewise, p changes from N a to 0 discontinuously at -x p. In the region of x n, the 1-dim Poisson equation is dE x /dx = q(N d - n), and since E x = -d /dx, the potential is the solution to -d 2 /dx 2 = q(N d - n)/ n x xnxn NdNd 0
©rlc L09-14Feb Debye length (cont) Since the level E Fi is a reference for equil, we set = V t ln(n/n i ) In the region of x n, n = n i exp( /V t ), so d 2 /dx 2 = -q(N d - n i e /Vt ), let = o + ’, where o = V t ln(N d /n i ) soN d - n i e /Vt = N d [1 - e /Vt- o/Vt ], for - o = ’ << o, the DE becomes d 2 ’/dx 2 = (q 2 N d / kT) ’, ’ << o
©rlc L09-14Feb Debye length (cont) So ’ = ’(x n ) exp[+(x-x n )/L D ]+con. and n = N d e ’/Vt, x ~ x n, where L D is the “Debye length”
©rlc L09-14Feb Debye length (cont) L D estimates the transition length of a step-junction DR (concentrations N a and N d with N eff = N a N d /(N a +N d )). Thus, For V a =0, & 1E13 < N a,N d < 1E19 cm -3 13% DA is OK
©rlc L09-14Feb Example An assymetrical p+ n junction has a lightly doped concentration of 1E16 and with p+ = 1E18. What is W(V=0)? Vbi=0.816 V, Neff=9.9E15, W=0.33 m What is C’ j ? = 31.9 nFd/cm2 What is L D ? = 0.04 m
©rlc L09-14Feb 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, Physics of Semiconductor Devices, Shur, Prentice- Hall, 1990.