Neamen

270 C H A P T E R 7 The pn Junction

(a) Sketch the energy-band diagram for the pn junction. (b) Find the impurity doping
concentration in each region. (c) Determine Vbi.

7.7 Consider a uniformly doped GaAs pn junction with doping concentrations of
Na ϭ 2 ϫ 1015 cmϪ3 and Nd ϭ 4 ϫ 1016 cmϪ3. Plot the built-in potential barrier
Vbi versus temperature over the range 200 Յ T Յ 400 K.

7.8 (a) Consider a uniformly doped silicon pn junction at T ϭ 300 K. At zero bias, 25 per-
cent of the total space charge region is in the n-region. The built-in potential barrier is
Vbi ϭ 0.710 V. Determine (i) Na, (ii) Nd, (iii) xn, (iv) xp, and (v) ͉Emax͉. (b) Repeat part (a)
for a GaAs pn junction with Vbi ϭ 1.180 V.

7.9 Consider the impurity doping profile shown in Figure P7.9 in a silicon pn junction.
For zero applied voltage, (a) determine Vbi, (b) calculate xn and xp, (c) sketch the ther-
mal equilibrium energy-band diagram, and (d) plot the electric field versus distance
through the junction.

7.10 Consider a uniformly doped silicon pn junction with doping concentrations Na ϭ 2 ϫ
1017 cmϪ3 and Nd ϭ 4 ϫ 1016 cmϪ3. (a) Determine Vbi at T ϭ 300 K. (b) Determine the
temperature at which Vbi increases by 2 percent. (Trial and error may have to be used.)

7.11 The doping concentrations in a uniformly doped silicon pn junction are Na ϭ 4 ϫ
1016 cmϪ3 and Nd ϭ 2 ϫ 1015 cmϪ3. The measured built-in potential barrier is Vbi ϭ
0.550 V. Determine the temperature at which this result occurs.

7.12 An “isotype” step junction is one in which the same impurity type doping changes
from one concentration value to another value. An n-n isotype doping profile is shown
in Figure P7.12. (a) Sketch the thermal equilibrium energy-band diagram of the
isotype junction. (b) Using the energy-band diagram, determine the built-in potential
barrier. (c) Discuss the charge distribution through the junction.

7.13 A particular type of junction is an n region adjacent to an intrinsic region. This
junction can be modeled as an n-type region to a lightly doped p-type region. Assume
the doping concentrations in silicon at T ϭ 300 K are Nd ϭ 1016 cmϪ3 and Na ϭ
1012 cmϪ3. For zero applied bias, determine (a) Vbi, (b) xn, (c) xp, and (d) ͉Emax͉. Sketch
the electric field versus distance through the junction.

7.14 We are assuming an abrupt depletion approximation for the space charge region. That
is, no free carriers exist within the depletion region and the semiconductor abruptly
changes to a neutral region outside the space charge region. This approximation is

(Na Ϫ Nd) (cmϪ3) 2 ␮m Nd (cmϪ3)
1016 n type 1016

p type 1015

Ϫ1015 0

Ϫ4 ϫ 1015 0

Figure P7.9 | Figure for Problem 7.9. Figure P7.12 | Figure for Problem 7.12.

Problems 271

adequate for most applications, but the abrupt transition does not exist. The space
charge region changes over a distance of a few Debye lengths, where the Debye
length in the n region is given by

͓ ͔LD ϭ _⑀skT 1͞2
e2Nd

Calculate LD and find the ratio of LD /xn for the following conditions. The p-type
doping concentration is Na ϭ 8 ϫ 1017 cmϪ3 and the n-type doping concentration is
(a) Nd ϭ 8 ϫ 1014 cmϪ3, (b) Nd ϭ 2.2 ϫ 1016 cmϪ3, and (c) Nd ϭ 8 ϫ 1017 cmϪ3.

7.15 Examine the electric field versus distance through a uniformly doped silicon pn junction
at T ϭ 300 K as a function of doping concentrations. Assume zero applied bias. Sketch
the electric field versus distance through the space charge region and calculate ͦEmax ͦ
for: (a) Na ϭ 1017 cmϪ3 and 1014 Յ Nd Յ 1017 cmϪ3 and (b) Na ϭ 1014 cmϪ3 and 1014 Յ
Nd Յ 1017 cmϪ3. (c) What can be said about the results for Nd Ն 100 Na or Na Ն 100 Nd?

Section 7.3 Reverse Applied Bias

7.16 An abrupt silicon pn junction at T ϭ 300 K has impurity doping concentrations of
Na ϭ 5 ϫ 1016 cmϪ3 and Nd ϭ 1015 cmϪ3. Calculate (a) Vbi, (b) W at (i) VR ϭ 0 and
(ii) VR ϭ 5 V, and (c) ͦ Emax ͦ at (i) VR ϭ 0 and (ii) VR ϭ 5.

7.17 Consider the pn junction described in Problem 7.10 for T ϭ 300 K. The cross-
sectional area of the junction is 2 ϫ 10Ϫ4 cm2 and the applied reverse-biased voltage is
VR ϭ 2.5 V. Calculate (a) Vbi, (b) xn, xp, W, (c) ͦ Emax ͦ, and (d) the junction capacitance.

7.18 An ideal one-sided silicon pϩn junction at T ϭ 300 K is uniformly doped on both
sides of the metallurgical junction. It is found that the doping relation is Na ϭ 80 Nd
and the built-in potential barrier is Vbi ϭ 0.740 V. A reverse-biased voltage of VR ϭ 10 V
is applied. Determine (a) Na, Nd; (b) xp, xn; (c) ͦ Emax ͦ; and (d) Cj.

7.19 A silicon nϩp junction is biased at VR ϭ 5 V. (a) Determine the change in built-in
potential barrier if the doping concentration in the p region increases by a factor of 3.
(b) Determine the ratio of junction capacitance when the acceptor doping is 3Na com-
pared to that when the acceptor doping is Na. (c) Why does the junction capacitance
increase when the doping concentration increases?

7.20 (a) The peak electric field in a reverse-biased silicon pn junction is ͦEmaxͦ ϭ 3 ϫ 105 V/cm.
The doping concentrations are Nd ϭ 4 ϫ 1015 cmϪ3 and Na ϭ 4 ϫ 1017 cmϪ3. Find the
magnitude of the reverse-biased voltage. (b) Repeat part (a) for Nd ϭ 4 ϫ 1016 cmϪ3 and
Na ϭ 4 ϫ 1017 cmϪ3. (c) Repeat part (a) for Nd ϭ Na ϭ 4 ϫ 1017 cmϪ3.

7.21 Consider two pϩn silicon junctions at T ϭ 300 K reverse biased at VR ϭ 5 V. The
impurity doping concentrations in junction A are Na ϭ 1018 cmϪ3 and Nd ϭ 1015 cmϪ3,
and those in junction B are Na ϭ 1018 cmϪ3 and Nd ϭ 1016 cmϪ3. Calculate the ratio of
the following parameters for junction A to junction B: (a) W, (b) ͦ Emax ͦ, and (c) Cj.

7.22 Consider a uniformly doped GaAs pn junction at T ϭ 300 K. The junction capacitance
at zero bias is Cj(0) and the junction capacitance with a 10-V reverse-biased voltage is
Cj (10). The ratio of the capacitances is

_CCjj((1_00)) ϭ 3.13

Also under reverse bias, the space charge width into the p region is 0.2 of the total
space charge width. Determine (a) Vbi and (b) Na, Nd.