Chapter 4: p-n Junctions

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  1. An abrupt silicon p-n junction (Na = 1016 cm-3 and Nd = 4 x 1016 cm-3) is biased with Va = -3 V. Calculate the built-in potential, the depletion layer width and the maximum electric field of the junction.
  2. An abrupt silicon p-n junction consists of a p-type region containing 1016 cm-3 acceptors and an n-type region containing also 1016 cm-3 acceptors in addition to 1017 cm-3 donors.
    1. Calculate the thermal equilibrium density of electrons and holes in the p-type region as well as both densities in the n-type region.
    2. Calculate the built-in potential of the p-n junction.
    3. Calculate the built-in potential of the p-n junction at 100C.
  3. Consider an abrupt silicon p-n junction with a built-in potential of 0.62 V.
    1. What is the potential across the depletion region at an applied voltage, Va, of 0, 0.5 and -2 Volt?
    2. If the depletion layer is 1 micrometer at Va = 0 Volt, find the maximum electric field in the depletion region.
    3. Assuming that the net doping density |Nd - Na| is the same in the n-type and p-type region of the diode, carefully sketch the electric field and the potential as a function of position throughout the depletion region. Add numeric values wherever possible.
  4. An abrupt silicon (ni = 1010 cm-3) p-n junction consists of a p-type region containing 1016 cm-3 acceptors and an n-type region containing 5 x 1016 cm-3 donors.
    1. Calculate the built-in potential of this p-n junction.
    2. Calculate the total width of the depletion region if the applied voltage, Va equals 0, 0.5 and -2.5 V.
    3. Calculate maximum electric field in the depletion region at 0, 0.5 and -2.5 V.
    4. Calculate the potential across the depletion region in the n-type semiconductor at 0, 0.5 and -2.5 V.
  5. Consider an abrupt p-n diode in thermal equilibrium with as many donors in the n-type region as acceptors in the p-type region and a maximum electric field of -13 kV/cm and a total depletion layer width of 1 mm. (assume es/ e0 = 12)
    1. What is the applied voltage, Va?
    2. What is the built-in potential of the diode?
    3. What is the donor density in the n-type region and the acceptor density in the p-type region?
    4. What is the intrinsic carrier density of the semiconductor if the temperature is 300 K ?
  6. A silicon (nI = 1010 cm-3) p-n diode with Na = 1018 cm-3 has a capacitance of 10-8 F/cm2 at an applied voltage of 0.5 V. Find the donor density.
  7. A silicon (ni = 1010 cm-3) p-n diode has a maximum electric field of -106 V/cm and a depletion layer width of 1 mm. The acceptor density in the p-type region is four times larger than the donor density in the n-type region. Calculate both doping densities.
  8. Consider a symmetric silicon p-n diode (Na = Nd)
    1. Calculate the built-in potential if Na = 1013, 1015 and 1017 cm-3. Also, calculate the doping densities corresponding to a built-in potential of 0.7 V.
    2. For the same as in part a), calculate the total depletion layer widths, the capacitance per unit area and the maximum electric field in thermal equilibrium.
    3. For the same as in part a), calculate the total depletion layer widths, the capacitance per unit area and the maximum electric field in thermal equilibrium.
    4. Repeat part a) and b) with Na = 3 Nd.
  9. A one-sided silicon diode has a breakdown voltage of 1000 V for which the maximum electric field at breakdown is 100 kV/cm. What is the maximum possible doping density in the low doped region, the built-in potential, the depletion layer width and the capacitance per unit area? Assume that bulk potential of the highly doped region is Eg/2 (= 0.56 V).
  10. A silicon p-n junction (Na = 1016 cm-3 and Nd = 4 x 1016 cm-3) is biased with Va = 0.6 V. Calculate the ideal diode current assuming that the n-type region is much smaller than the diffusion length with wn' = 1 mm and assuming a "long" p-type region. Use mn = 1000 cm2/V-s and mp = 300 cm2/V-s. The minority carrier lifetime is 10 ms and the diode area is 100 mm by 100 mm.
  11. Derive equation 4.4.28 from 4.4.14.
  12. Calculate the relative error when using the "short diode" approximation if Ln = 2 wp' and Lp = 2 wn'.
  13. A silicon p-n junction (Na = 1015 cm-3, wp = 1 mm and Nd = 4 x 1016 cm-3, wn = 1 mm) is biased with Va = 0.5 V. Use mn = 1000 cm2/V-s and mp = 300 cm2/V-s. The minority carrier lifetime is 10 ms and the diode area is 100 mm by 100 mm.
    1. Calculate the built-in potential of the diode.
    2. Calculate the depletion layer widths, xn and xp, and the widths of the quasi-neutral regions.
    3. Compare the width of the quasi-neutral regions with the minority-carrier diffusion-lengths and decide whether to use the "long" or "short" diode approximation. Calculate the current through the diode.
    4. Compare the result of part c) with the current obtained by using the general solution (equation 4.4.24)
    5. Using the approximation chosen in part c) calculate the ratio of the electron current to the hole current traversing the depletion region.
  14. An abrupt silicon p-n diode consists of a p-type region containing 1018 cm-3 acceptors and an n-type region containing 1015 cm-3 donors.
    1. Calculate the breakdown field in the n-type region.
    2. Using the breakdown field from part a), calculate the breakdown voltage of the diode.
    3. What is the depletion layer width at breakdown?
    4. Discuss edge effects and specify the minimum junction depth needed to avoid these effects.
  15. A 1 cm2 solar cell consists of a p-type region containing 1018 cm-3 acceptors and an n-type region containing 1015 cm-3 donors. wp' = 0.1 mm and wn >> Lp. Use mn = 1000 cm2/V-s and mp = 300 cm2/V-s. The minority carrier lifetime is 10 ms . The diode is illuminated with sun light, yielding a photocurrent density of 30 mA/cm2.
    1. Calculate the open circuit voltage and short-circuit current of the solar cell.
    2. Calculate the maximum power generated by the call and the corresponding voltage and current.
    3. Calculate the fill factor of the solar cell.
    4. Calculate the fill factor for the same cell when it is illuminated by a concentrator so that the photocurrent density equals 300 A/cm2.
  16. A semiconductor device made of silicon has, under thermal equilibrium, an M-shaped electric field distribution as shown in the figure below.
    1. Find the total potential across the semiconductor as a function of Emax with a = 0.1 mm.
    2. Find the total potential across the semiconductor as a function of Emax with a = 0.1 mm.
    3. Find Emax and the built-in voltage fi.
    4. Plot Nd - Na for -a > x > a and indicate numeric values. Specify whether the different regions are p-type or n-type.
  17. Design an abrupt silicon p-n diode with a capacitance per unit area of 10 nF/cm2 in thermal equilibrium and a maximum electric field of 105 V/cm at a reverse bias of 10 Volt. Provide values of the acceptor and donor density, the built-in potential and the depletion layer width in thermal equilibrium and at a reverse bias of 10 Volt.
  18. A silicon p-n junction consists of a half-sphere with one-micron radius and a doping density of 1018 cm-3 embedded in an n-type substrate with a donor density of 1016 cm-3. Breakdown occurs in the diode when the maximum field reaches 6x105 V/cm. Calculate the breakdown voltage. Justify any assumptions you make.
  19. Calculate the built-in voltage for a silicon p-n junction with Na = Nd = 1015 cm-3 at T = 500 K. Do not assume the electron and hole concentration to equal the donor or acceptor concentration.
  20. Derive the minority electron density in a silicon p-n junction at the edge of the depletion region as a function of the acceptor density and the applied voltage. State the approximations made. Calculate the minority electron density for Na = 1017 cm-3, Nd = 1016 cm-3 and Va = -2 V.
  21. An abrupt silicon p-n diode has a maximum electric field of -106 V/cm and a depletion layer width of 10 mm. The acceptor density in the p-type region is three times larger than the donor density in the n-type region. Calculate both doping densities.
  22. A one-sided abrupt silicon p-n junction with Na = 1018 cm-3 and Nd = 1015 cm-3 is biased with Va = 0.7 V. Calculate the ideal diode current using the following parameters: wp' = wn' = 100 mm, mn = 500 cm2/V-s and mp = 300 cm2/V-s. The minority carrier lifetime is 1 ms and the diode area is 100 mm by 100 mm. Use either the "long" or "short" diode equation and justify your choice.
  23. A capacitance measurement of a one sided p+-n diode resulted in the following plot of 1/C2 versus the applied voltage. Calculate and plot the doping profile Nd as a function of the distance from the metallurgical interface. The diode are is 10-4 cm2 and the relative dielectric constant of the semiconductor is 12. If the electron and hole masses equal the free electron mass, m0, what is the bandgap of the semiconductor?
  24. A "long" abrupt p-n diode consists of a p-type region with a four times higher resistivity than the n-type region, while the depletion layer width in the p-type region is twice that in the n-type region. What is the ratio of the maximum electron current to the maximum hole current? Assume the minority carrier lifetime to be the same in both regions
  25. For a silicon p-n diode find the maximum built-in voltage at 300K, assuming non-degenerate material. Repeat at 300oC.
  26. An n+-n-p+ diode has the following field distribution:
    1. Calculate the voltage applied to the diode.
    2. Plot then charge distribution throughout the depletion region at that bias voltage.
    3. Calculate the donor density in the middle region.
  27. A one-sided p+-n diode has an n-type region width wn which is much larger than the hole diffusion length Lp. Derive an expression for the current I(Va) through the diode, taking into account the modulation of the depletion region due to the applied voltage. Ignore the injection of electrons into the p-type region.
  28. An abrupt p-n diode consists of a p-type region containing 1016 cm-3 shallow acceptors and an n-type region containing also 1016 cm-3 shallow acceptors in addition to 1017 cm-3 shallow donors.
    1. Calculate the thermal equilibrium density of electrons and holes in the p-type region as well as both densities in the n-type region.
    2. Calculate the built-in potential of the p-n diode.
    3. Calculate the built-in potential of the p-n diode at 100oC.
  29. same as 4.3
  30. Consider a one-sided silicon p+-n junction with Va = 0.6 Volt, Nd = 1015 cm-3, tn = tp = 10 ms.
    1. Calculate the electron and hole density at the edge of the n-type quasi-neutrral region.
    2. Calculate the hole current density at the edge of the n-type quasi-neutral region and 10 microns away from that edge in to quasi-neutral region. Assume the quasi-neutral region to be "long".
  31. An abrupt p-n diode has a built-in potential of 0.75 Volt and a depletion layer width of 1 micron at a forward bias of 0.5 Volt. What is the width of the depletion layer at a reverse bias voltage of 1 Volt?
  32. An abrupt p-n diode has a built-in potential of 0.7 V and an n-type region doped with 1016 cm-3 shallow donors. Calculate the acceptor density, Na, in the p-type region, the depletion layer width in both regions, xn and xp, and the maximum electric field, Emax, if the diode is in thermal equilibrium.
  33. Consider an abrupt silicon p-i-n diode with Na = Nd = 1017 cm-3 and a 2 micron wide intrinsic region. Va = 0 Volt.
    1. Calculate the electric field using the full depletion approximation.
    2. Calculate n(x) and p(x) in the intrinsic material from n = ni exp[(EF - Ei)/kT] and p = ni exp[(Ei - EFkT]
    3. Calculate Jn(x) in the intrinsic material
    4. Does the drift current equal the diffusion current in the intrinsic material? Why? Is the full depletion approximation valid? Why?
  34. The field distribution of an abrupt silicon p-n diode is shown below. The electron density at x = -a = -0.1 mm equals 1017 cm-3. Calculate the electron density at x = a = 0.1 mm. The maximum electric field equals Emax = 1.5 x 105 V/cm. Note that the diode is not in thermal equilibrium.
  35. An abrupt silicon p-n diode is uniformly doped with a donor density, Nd = 1017 cm-3, in the n-type and an unknown acceptor density in the p-type region. The depletion layer width in the n-type region is three times the depletion layer width in the p-type region and the maximum electric field in the junction is 105 V/cm. Find the acceptor density, the built-in potential, the applied voltage and the corresponding junction capacitance per unit area.
  36. A silicon p-n with a saturation current of 10-10 A is used as a solar cell. The diode is illuminated with sunlight, yielding a photocurrent of 1 mA. Find the maximum power, which can be generated by this diode. Calculate the corresponding voltage and current.

Boulder, November 2007