EE 5340 Semiconductor Device Theory Lecture 12 - Fall 2009 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

Soln to Poisson’s Eq in the D.R. Ex W(Va-dV) W(Va) -xp xn x -xpc xnc -Emax(V) -Emax(V-dV) L 12 Oct 01

Effect of V  0 L 12 Oct 01

Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn Junction C (cont.) r +Qn’=qNdxn +qNd dQn’=qNddxn -xp x -xpc xn xnc -qNa Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn dQp’=-qNadxp Qp’=-qNaxp L 12 Oct 01

Junction Capacitance The junction has +Q’n=qNdxn (exposed donors), and (exposed acceptors) Q’p=-qNaxp = -Q’n, forming a parallel sheet charge capacitor. L 12 Oct 01

Junction C (cont.) So this definition of the capacitance gives a parallel plate capacitor with charges dQ’n and dQ’p(=-dQ’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 Va=0. L 12 Oct 01

Junction C (cont.) The C-V relationship simplifies to L 12 Oct 01

Junction C (cont.) If one plots [Cj]-2 vs. Va Slope = -[(Cj0)2Vbi]-1 vertical axis intercept = [Cj0]-2 horizontal axis intercept = Vbi Cj-2 Vbi Va Cj0-2 L 12 Oct 01

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 L 12 Oct 01

dy/dx - Numerical Differentiation L 12 Oct 01

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 r=qax in the region of the junction. L 12 Oct 01

Practical Jctns (cont.) Na(x) N N Shallow (steep) implant Na(x) Linear approx. Box or step junction approx. Nd Nd Uniform wafer con x (depth) x (depth) xj L 12 Oct 01

Linear graded junction Let the net donor concentration, N(x) = Nd(x) - Na(x) = ax, so r =qax, -xp < x < xn = xp = xo, (chg neu) r = qa x r Q’n=qaxo2/2 -xo x xo Q’p=-qaxo2/2 L 12 Oct 01

Linear graded junction (cont.) Let Ex(-xo) = 0, since this is the edge of the DR (also true at +xo) L 12 Oct 01

Linear graded junction (cont.) Ex -xo xo x -Emax |area| = Vbi-Va L 12 Oct 01

Linear graded junction (cont.) L 12 Oct 01

Linear graded junction, etc. L 12 Oct 01

Doping Profile If the net donor conc, N = N(x), then at x, the extra charge put into the DR when Va->Va+dVa is dQ’=-qN(x)dx The increase in field, dEx =-(qN/e)dx, by Gauss’ Law (at x, but also all DR). So dVa=-xddEx= (W/e) dQ’ Further, since qN(x)dx, for both xn and xn, we have the dC/dx as ... L 12 Oct 01

Arbitrary doping profile (cont.) L 12 Oct 01

Arbitrary doping profile (cont.) L 12 Oct 01

Arbitrary doping profile (cont.) L 12 Oct 01

Arbitrary doping profile (cont.) L 12 Oct 01

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=0.33mm What is C’j0? = 31.9 nFd/cm2 What is LD? = 0.04 mm L 12 Oct 01

Reverse bias junction breakdown Avalanche breakdown Electric field accelerates electrons to sufficient energy to initiate multiplication of impact ionization of valence bonding electrons field dependence shown on next slide Heavily doped narrow junction will allow tunneling - see Neamen*, p. 274 Zener breakdown L 12 Oct 01

Effect of V  0 L 12 Oct 01

Ecrit for reverse breakdown [M&K] Taken from p. 198, M&K** L 12 Oct 01

Reverse bias junction breakdown L 12 Oct 01

Ecrit for reverse breakdown [M&K] Taken from p. 198, M&K** Casey Model for Ecrit L 12 Oct 01

Reverse bias junction breakdown Assume -Va = VR >> Vbi, so Vbi-Va-->VR Since Emax~ 2VR/W = (2qN-VR/(e))1/2, and VR = BV when Emax = Ecrit (N- is doping of lightly doped side ~ Neff) BV = e (Ecrit )2/(2qN-) Remember, this is a 1-dim calculation L 12 Oct 01

Junction curvature effect on breakdown The field due to a sphere, R, with charge, Q is Er = Q/(4per2) for (r > R) V(R) = Q/(4peR), (V at the surface) So, for constant potential, V, the field, Er(R) = V/R (E field at surface increases for smaller spheres) Note: corners of a jctn of depth xj are like 1/8 spheres of radius ~ xj L 12 Oct 01

BV for reverse breakdown (M&K**) Taken from Figure 4.13, p. 198, M&K** Breakdown voltage of a one-sided, plan, silicon step junction showing the effect of junction curvature.4,5 L 12 Oct 01

References [M&K] Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, Wiley, New York, 1986. [2] Devices for Integrated Circuits: Silicon and III-V Compound Semiconductors, by H. Craig Casey, Jr., John Wiley & Sons, New York, 1999. Bipolar Semiconductor Devices, by David J. Roulston, McGraw-Hill, Inc., New York, 1990. L 12 Oct 01