EE 5340 Semiconductor Device Theory Lecture 11 - Fall 2003 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc L 11 Sept 30

Band diagram for p+-n jctn* at Va = 0 Ec qVbi = q(fn - fp) qfp < 0 EFi Ec EFP EFN Ev EFi qfn > 0 *Na > Nd -> |fp| > fn Ev p-type for x<0 n-type for x>0 x -xpc -xp xn xnc L 11 Sept 30

Band diagram for p+-n jctn* at Va  0 Ec q(Va) q(Vbi-Va) qfp < 0 EFi Ec EFP EFN Ev EFi qfn > 0 *Na > Nd -> |fp| > fn Ev p-type for x<0 n-type for x>0 x -xpc -xp xn xnc L 11 Sept 30

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 11 Sept 30

Depletion Approxi- mation (Summary) For the step junction defined by doping Na (p-type) for x < 0 and Nd, (n-type) for x > 0, the depletion width W = {2e(Vbi-Va)/qNeff}1/2, where Vbi = Vt ln{NaNd/ni2}, and Neff=NaNd/(Na+Nd). Since Naxp=Ndxn, xn = W/(1 + Nd/Na), and xp = W/(1 + Na/Nd). L 11 Sept 30

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 11 Sept 30

Effect of V  0 L 11 Sept 30

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 11 Sept 30

Junction C (cont.) This Q ~ (Vbi-Va)1/2 is clearly non-linear, and Q is not zero at Va = 0. Redefining the capacitance, L 11 Sept 30

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 11 Sept 30

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 11 Sept 30

dy/dx - Numerical Differentiation L 11 Sept 30

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 11 Sept 30

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 11 Sept 30

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 11 Sept 30

Linear graded junction (cont.) Let Ex(-xo) = 0, since this is the edge of the DR (also true at +xo) L 11 Sept 30

Linear graded junction (cont.) Ex -xo xo x -Emax |area| = Vbi-Va L 11 Sept 30

Linear graded junction (cont.) L 11 Sept 30

Linear graded junction, etc. L 11 Sept 30

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 11 Sept 30

Arbitrary doping profile (cont.) L 11 Sept 30

Arbitrary doping profile (cont.) L 11 Sept 30

Arbitrary doping profile (cont.) L 11 Sept 30

Arbitrary doping profile (cont.) L 11 Sept 30

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 11 Sept 30

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 11 Sept 30

Effect of V  0 L 11 Sept 30

Ecrit for reverse breakdown (M&K**) Taken from p. 198, M&K** L 11 Sept 30

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 11 Sept 30

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 11 Sept 30

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 11 Sept 30

References * Semiconductor Physics and Devices, 2nd ed., by Neamen, Irwin, Boston, 1997. **Device Electronics for Integrated Circuits, 2nd ed., by Muller and Kamins, Wiley, New York, 1986. L 11 Sept 30