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EE 5340 Semiconductor Device Theory Lecture 11 - Fall 2003

EE 5340 Semiconductor Device Theory Lecture 11 - Fall 2003. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc. E FN. Band diagram for p + -n jctn* at V a = 0. E c. qV bi = q( f n - f p ). q f p < 0. E c. E Fi. E FP. E v. E Fi. q f n > 0.

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EE 5340 Semiconductor Device Theory Lecture 11 - Fall 2003

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  1. EE 5340Semiconductor Device TheoryLecture 11 - Fall 2003 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

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

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

  4. JunctionC (cont.) r +Qn’=qNdxn +qNd dQn’=qNddxn -xp x -xpc xn xnc Charge neutrality => Qp’ + Qn’ = 0, => Naxp = Ndxn -qNa dQp’=-qNadxp Qp’=-qNaxp

  5. 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).

  6. Soln to Poisson’sEq in the D.R. Ex W(Va-dV) W(Va) xn -xp x -xpc xnc -Emax(V) -Emax(V-dV)

  7. Effect of V  0

  8. JunctionCapacitance • The junction has +Q’n=qNdxn (exposed donors), and (exposed acceptors) Q’p=-qNaxp = -Q’n, forming a parallel sheet charge capacitor.

  9. JunctionC (cont.) • This Q ~ (Vbi-Va)1/2 is clearly non-linear, and Q is not zero at Va = 0. • Redefining the capacitance,

  10. Cj-2 Cj0-2 Va Vbi JunctionC (cont.) • If one plots [Cj]-2vs. Va Slope = -[(Cj0)2Vbi]-1 vertical axis intercept = [Cj0]-2 horizontal axis intercept = Vbi

  11. 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

  12. dy/dx - Numerical Differentiation

  13. 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.

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

  15. Linear gradedjunction • 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

  16. Linear gradedjunction (cont.) • Let Ex(-xo) = 0, since this is the edge of the DR (also true at +xo)

  17. Linear gradedjunction (cont.) Ex -xo xo x -Emax |area| = Vbi-Va

  18. Linear gradedjunction (cont.)

  19. Linear gradedjunction, etc.

  20. 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 ...

  21. Arbitrary dopingprofile (cont.)

  22. Arbitrary dopingprofile (cont.)

  23. Arbitrary dopingprofile (cont.)

  24. Arbitrary dopingprofile (cont.)

  25. 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

  26. Reverse biasjunction 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

  27. Effect of V  0

  28. Ecrit for reverse breakdown (M&K**) Taken from p. 198, M&K**

  29. Reverse biasjunction 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

  30. Junction curvatureeffect 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

  31. 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

  32. 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.

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