EE 5340 Semiconductor Device Theory Lecture 12 – Spring 2011 Professor Ronald L. Carter

©rlc L12-01Mar2011 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 2

©rlc L12-01Mar2011 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 Due to Charge neutrality Q p ’ + Q n ’ = 0, => N a x p = N d x n [Coul/cm 2 ] 3

©rlc L12-01Mar2011 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 ≈0≈0 4

©rlc L12-01Mar2011 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 5

©rlc L12-01Mar2011 Induced E-field in the D.R. (cont.) Poisson’s Equation  E =  / , has the one-dimensional form, dE x /dx =  / , which must be satisfied for  = -qN a, -x p < x < 0, and  = +qN d, 0 < x < x n, with E x = 0 for the remaining range 6

©rlc L12-01Mar2011 Soln to Poisson’s Eq in the D.R. xnxn x -x p -x pc x nc ExEx -E max 7

©rlc L12-01Mar2011 Soln to Poisson’s Eq in the D.R. (cont.) 8

©rlc L12-01Mar2011 Soln to Poisson’s Eq in the D.R. (cont.) 9

©rlc L12-01Mar2011 Comments on the E x and V bi V bi is not measurable externally since E x is zero at both contacts The effect of E x does not extend beyond the depletion region The lever rule [N a x p =N d x n ] was obtained assuming charge neutrality. It could also be obtained by requiring E x (x=0  x  E x (x=0  x)  E max 10

©rlc L12-01Mar2011 Sample calculations V t  mV at 300K  =  r  o = 11.7*8.85E-14 Fd/cm = 1.035E-12 Fd/cm If N a  5E17/cm 3, and N d  2E15 /cm 3, then for n i  1.4E10/cm 3, then what is V bi = 757 mV 11

©rlc L12-01Mar2011 Sample calculations What are N eff, W ? N eff, = 1.97E15/cm 3 W = micron What is x n ? = micron What is E max ? 2.14E4 V/cm 12

©rlc L12-01Mar2011 Soln to Poisson’s Eq in the D.R. xnxn x -x p -x pc x nc ExEx -E max (V) -E max (V-  V) W(V a ) W(V a -  V) 13

©rlc L12-01Mar2011 Effect of V  0 14

©rlc L12-01Mar2011 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 15

©rlc L12-01Mar2011 Junction Capacitance The junction has +Q’ n =qN d x n (exposed donors), and (exposed acceptors) Q’ p =-qN a x p = -Q’ n, forming a parallel sheet charge capacitor. 16

©rlc L12-01Mar2011 Junction C (cont.) So this definition of the capacitance gives a parallel plate capacitor with charges  Q’ n and  Q’ p (=-  Q’ n ), separated by, L (=W), with an area A and the capacitance is then the ideal parallel plate capacitance. Still non-linear and Q is not zero at V a =0. 17

©rlc L12-01Mar2011 Junction C (cont.) The C-V relationship simplifies to 18

©rlc L12-01Mar2011 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 j

©rlc L12-01Mar2011 Junction Capacitance Estimate CJO Define y  Cj/CJO Calculate y/(dy/dV) = {d[ln(y)]/dV} -1 A plot of r  y/(dy/dV) vs. V has slope = -1/M, and intercept = VJ/M 20

©rlc L12-01Mar2011 dy/dx - Numerical Differentiation 21

©rlc L12-01Mar2011 Practical Junctions Junctions are formed by diffusion or implantation into a uniform concentration wafer. The profile can be approximated by a step or linear function in the region of the junction. If a step, then previous models OK. If linear, let the local charge density  =qax in the region of the junction. 22

©rlc L12-01Mar2011 Practical Jctns (cont.) Shallow (steep) implant N x (depth) Box or step junction approx. N x (depth) N a (x) xjxj Linear approx. NdNd NdNd N a (x) Uniform wafer con 23

©rlc L12-01Mar2011 Linear graded junction Let the net donor concentration, N(x) = N d (x) - N a (x) = ax, so  =qax, -x p < x < x n = x p = x o, (chg neu) xoxo -x o   = qa x Q’ n =qax o 2 /2 Q’ p =-qax o 2 /2 x 24

©rlc L12-01Mar2011 Linear graded junction (cont.) Let E x (-x o ) = 0, since this is the edge of the DR (also true at +x o ) 25

©rlc L12-01Mar2011 Linear graded junction (cont.) x ExEx -E max xoxo -x o |area| = V bi -V a 26

©rlc L12-01Mar2011 Linear graded junction (cont.) 27

©rlc L12-01Mar2011 Linear graded junction, etc. 28

©rlc L12-01Mar References 1 and M&K Device Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996, for another treatment of the  model. 2 Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, and ** Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.