Libro2 P

BR
Wiley/Razavi/Fundamentals of Microelectronics
Sec. 2.4
[Razavi.cls v. 2006]
June 30, 2007 at 13:42
55 (1)
Chapter Summary
55
The drift current density is proportional to the electric field and the mobility of the carriers
and is given by
.
The diffusion current density is proportional to the gradient of the carrier concentration
and given by
.
A
junction is a piece of semiconductor that receives -type doping in one section and
-type doping in an adjacent section.
The
junction can be considered in three modes: equilibrium, reverse bias, and forward
bias.
Upon formation of the
junction, sharp gradients of carrier densities across the junction
result in a high current of electrons and holes. As the carriers cross, they leave ionized
atoms behind, and a depletion resgion is formed. The electric field created in the depletion region eventually stops the current flow. This condition is called equilibrium.
The electric field in the depletion results in a built-in potential across the region equal to
, typically in the range of 700 to 800 mV.
Under reverse bias, the junction carries negligible current and operates as a capacitor. The
capacitance itself is a function of the voltage applied across the device.
Under forward bias, the junction carries a current that is an exponential function of the
applie voltage:
.
Since the exponential model often makes the analysis of circuits difficult, a constantvoltage model may be used in some cases to estimate the circuits response with less
mathematical labor.
Under a high reverse bias voltage,
junctions break down, conducting a very high current. Depending on the structure and doping levels of the device, Zener or avalanche
breakdown may occur.
Problems
1. The intrinsic carrier concentration of germanium (GE) is expressed as
(2.127)
where
eV.
(a) Calculate at
K and
K and compare the results with those obtained in Example
2.1 for Si.
(b) Determine the electron and hole concentrations if Ge is doped with P at a density of
.
2. An -type piece of silicon experiences an electric field equal to 0.1 V/ m.
(a) Calculate the velocity of electrons and holes in this material.
(b) What doping level is necessary to provide a current density of 1 mA/
under these
conditions? Assume the hole current is negligible.
3. A -type piece of silicon with a length of
m and a cross section area of
m
m sustains a voltage difference of 1 V.
(a) If the doping level is
cm , calculate the total current flowing through the device at
K.
(b) Repeat (a) for
K assuming for simplicity that mobility does not change with
temperature. (This is not a good assumption.)
BR
June 30, 2007 at 13:42
56
Chap. 2
56 (1)
Basic Physics of Semiconductors
4. From the data in Problem 1, repeat Problem 3 for Ge. Assume

and
.
5. Figure 2.37 shows a -type bar of silicon that is subjected to electron injection from the left
16
5 x 10
16
2 x 10
Electrons
Holes
Figure 2.37
and hole injection from the right. Determine the total current flowing through the device if the
cross section area is equal to 1
m.
6. In Example 2.9, compute the total number of electrons stored in the material from
to
. Assume the cross section area of the bar is equal to .
7. Repeat Problem 6 for Example 2.10 but for
to
. Compare the results for linear
and exponential profiles.
8. Repeat Problem 7 if the electron and hole profiles are sharp exponentials, i.e., they fall to
negligible values at
m and
, respectively (Fig. 2.38).
16
5 x 10
16
2 x 10
Electrons
Figure 2.38
Holes
9. How do you explain the phenomenon of drift to a high school student?

10. A junction employs
and
.
(a) Determine the majority and minority carrier concentrations on both sides.
(b) Calculate the built-in potential at
K,
K, and
K. Explain the trend.
11. Due to a manufacturing error, the -side of a
junction has not been doped. If
, calculate the built-in potential at
K.
12. A
junction with
and
experiences a reverse
bias voltage of 1.6 V.
(a) Determine the junction capacitance per unit area.
(b) By what factor should
be increased to double the junction capacitance?
13. An oscillator application requires a variable capacitance with the characteristic shown in Fig.
2.39. Determine the required
if
/cm .
14. Consider a
junction in forward bias.
(a) To obtain a current of 1 mA with a voltage of 750 mV, how should
be chosen?
(b) If the diode cross section area is now doubled, what voltage yields a current of 1 mA?
15. Figure 2.40 shows two diodes with reverse saturation currents of
and
placed in parallel.
(a) Prove that the parallel combination operates as an exponential device.
BR
Sec. 2.4
June 30, 2007 at 13:42
57 (1)
Chapter Summary
57
C j (fF/ m 2 )
2.2
1.3
1.5
0.5
V R (V)
Figure 2.39
I tot
VB
D1
D2
Figure 2.40
(b) If the total current is

, determine the current carried by each diode.
16. Two identical
junctions are placed in series.
(a) Prove that this combination can be viewed as a single two-terminal device having an
exponential characteristic.
(b) For a tenfold change in the current, how much voltage change does such a device require?
17. Figure 2.41 shows two diodes with reverse saturation currents of
and
placed in series.
IB
D1
D1
V D1
V D2
VB
Figure 2.41
Calculate ,
, and
in terms of
,
, and
.
18. In the circuit of Problem 17, we wish to increase
by a factor of 10. What is the required
change in
?
19. Consider the circuit shown in Fig. 2.42, where
A. Calculate
and
for
IX
VX
R1
2 k
D1
Figure 2.42
V, 0.8 V, 1 V, and 1.2 V. Note that

changes little for
V.
20. In the circuit of Fig. 2.42, the cross section area of
is increased by a factor of 10. Determine
and
for
V and 1.2 V. Compare the results with those obtained in Problem
19.
BR
June 30, 2007 at 13:42
58
58 (1)
Chap. 2
21. Suppose
in Fig. 2.42 must sustain a voltage of 850 mV for
required .
22. For what value of
in Fig. 2.42, does
sustain a voltage equal to
A.
23. We have received the circuit shown in Fig. 2.43 and wish to determine
V. Calculate the
? Assume
and
. We note
IX
R1
VX
D1
Figure 2.43
that
mA and
24. Figure 2.44 depicts a parallel resistor-diode combination. If
for
mA, 2 mA, and 4 mA.
IX
R1
mA. Calculate
and .
A, calculate
D1
1 k
Figure 2.44
25. In the circuit of Fig. 2.44, we wish

to carry a current of 0.5 mA for
mA.
Determine the required .
26. For what value of
in Fig. 2.44, does
carry a current equal to
? Assume
A.
27. We have received the circuit shown in Fig. 2.45 and wish to determine
and . Measure-
IX
VX
D1
R1
Figure 2.45
ments indicate that

V and
and .
28. The circuit illustrated in Fig. 2.46 employs two identical diodes with
IX
V. Calculate
A.
D1
R1
2 k
D2
Figure 2.46
Calculate the voltage across

for
mA.
29. In the circuit of Fig. 2.47, determine the value of
such that this resistor carries 0.5 mA.
BR
Sec. 2.4
June 30, 2007 at 13:42
59 (1)
Chapter Summary
59
D1
IX
1 mA R 1
D2
Figure 2.47
Assume
A for each diode.
30. Sketch
as a function of
for the circuit shown in Fig. 2.48. Assume (a) a constantvoltage model, (b) an exponential model.
IX
VX
D1
R1
Figure 2.48
SPICE Problems
In the following problems, assume
31. For the circuit shown in Fig. 2.49, plot
2mA.
A.
as a function of
I in
D1
. Assume
varies from 0 to
Vout
Figure 2.49
32. Repeat Problem 31 for the circuit depicted in Fig. 2.50, where
are the currents flowing through
and
equal?
I in
D1
R1
k . At what value of
Vout
Figure 2.50
33. Using SPICE, determine the value of

in Fig. 2.50 such that
carries 1 mA if
mA.
34. In the circuit of Fig. 2.51,
. Plot
as a function of
if
varies from
V to
V. At what value of
are the voltage drops across
and
equal?
R1
V in
Figure 2.51
Vout
D1
BR
60
June 30, 2007 at 13:42
Chap. 2
60 (1)
35. In the circuit of Fig. 2.51, use SPICE to select the value of
V. We say the circuit limits the output.
such that
References
1. B. Streetman and S. Banerjee, Solid-State Electronic Device, fifth edition, Prentice-Hall, 1999.
V for

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BR

June 30, 2007 at 13:42

June 30, 2007 at 13:42

Basic Physics of Semiconductors

4. From the data in Problem 1, repeat Problem 3 for Ge. Assume

9. How do you explain the phenomenon of drift to a high school student?

June 30, 2007 at 13:42

(b) If the total current is

V, 0.8 V, 1 V, and 1.2 V. Note that

June 30, 2007 at 13:42

Basic Physics of Semiconductors

25. In the circuit of Fig. 2.44, we wish

ments indicate that

Calculate the voltage across

such that this resistor carries 0.5 mA.

June 30, 2007 at 13:42

33. Using SPICE, determine the value of

June 30, 2007 at 13:42

Basic Physics of Semiconductors

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