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EE 5340 Semiconductor Device Theory Lecture 12 – Spring 2011

EE 5340 Semiconductor Device Theory Lecture 12 – Spring 2011. Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc. Depletion Approximation.

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EE 5340 Semiconductor Device Theory Lecture 12 – Spring 2011

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  1. EE 5340Semiconductor Device TheoryLecture 12 – Spring 2011 Professor Ronald L. Carter ronc@uta.edu http://www.uta.edu/ronc

  2. DepletionApproximation • Assume p << po = Nafor -xp < x < 0, so r = q(Nd-Na+p-n) = -qNa, -xp < x < 0, and p = po = Nafor -xpc < x < -xp, so r = q(Nd-Na+p-n) = 0, -xpc < x < -xp • Assume n << no = Ndfor 0 < x < xn, so r = q(Nd-Na+p-n) = qNd, 0 < x < xn, and n = no = Ndfor xn < x < xnc, so r = q(Nd-Na+p-n) = 0, xn < x < xnc

  3. Depletion approx.charge distribution r +Qn’=qNdxn +qNd [Coul/cm2] -xp x -xpc xn xnc Due to Charge neutrality Qp’ + Qn’ = 0, => Naxp = Ndxn -qNa Qp’=-qNaxp [Coul/cm2]

  4. Induced E-fieldin the D.R. • The sheet dipole of charge, due to Qp’ and Qn’ induces an electric field which must satisfy the conditions • Charge neutrality and Gauss’ Law* require thatEx = 0 for -xpc < x < -xp and Ex = 0 for -xn < x < xnc ≈0

  5. O O O O O O + + + - - - Induced E-fieldin the D.R. Ex N-contact p-contact p-type CNR n-type chg neutral reg Depletion region (DR) Exposed Donor ions Exposed Acceptor Ions W x -xpc -xp xn xnc 0

  6. Induced E-fieldin the D.R. (cont.) • Poisson’s Equation E = r/e, has the one-dimensional form, dEx/dx = r/e, which must be satisfied for r = -qNa, -xp < x < 0, and r = +qNd, 0 < x < xn, with Ex = 0 for the remaining range

  7. Soln to Poisson’sEq in the D.R. Ex xn -xp x -xpc xnc -Emax

  8. Soln to Poisson’sEq in the D.R. (cont.)

  9. Soln to Poisson’sEq in the D.R. (cont.)

  10. Comments on theEx and Vbi • Vbi is not measurable externally since Ex is zero at both contacts • The effect of Ex does not extend beyond the depletion region • The lever rule [Naxp=Ndxn] was obtained assuming charge neutrality. It could also be obtained by requiring Ex(x=0-dx) = Ex(x=0+dx) = Emax

  11. Sample calculations • Vt = 25.86 mV at 300K • e = ereo = 11.7*8.85E-14 Fd/cm = 1.035E-12 Fd/cm • If Na=5E17/cm3, and Nd=2E15 /cm3, then for ni=1.4E10/cm3, then what is Vbi = 757 mV

  12. Sample calculations • What are Neff, W ? Neff, = 1.97E15/cm3 W = 0.707 micron • What is xn ? = 0.704 micron • What is Emax ? 2.14E4 V/cm

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

  14. Effect of V  0

  15. 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

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

  17. JunctionC (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.

  18. JunctionC (cont.) • The C-V relationship simplifies to

  19. 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

  20. 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

  21. dy/dx - Numerical Differentiation

  22. 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.

  23. 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

  24. 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

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

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

  27. Linear gradedjunction (cont.)

  28. Linear gradedjunction, etc.

  29. References 1 and M&KDevice Electronics for Integrated Circuits, 2 ed., by Muller and Kamins, Wiley, New York, 1986. See Semiconductor Device Fundamentals, by Pierret, Addison-Wesley, 1996, for another treatment of the m model. 2Physics of Semiconductor Devices, by S. M. Sze, Wiley, New York, 1981. 3 and **Semiconductor Physics & Devices, 2nd ed., by Neamen, Irwin, Chicago, 1997. Fundamentals of Semiconductor Theory and Device Physics, by Shyh Wang, Prentice Hall, 1989.

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