Semiconductor Physics and Devices

Donald A. Neamen

Chapter 12

The Bipolar Transistor - all with Video Answers

Educators

Chapter Questions

Problem 1

For a uniformly doped $\mathrm{n}^{++} \mathrm{p}^{+} \mathrm{n}$ bipolar transistor in thermal equilibrium, $(a)$ sketch the energy-band diagram, $(b)$ sketch the electric field through the device, and
(c) repeat parts $(a)$ and $(b)$ for the transistor biased in the forward-active region.

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Problem 2

Consider $\mathrm{a} \mathrm{p}^{++} \mathrm{n}^{+} \mathrm{p}$ bipolar transistor, uniformly doped in each region. Sketch the energy-band diagram for the case when the transistor is $(a)$ in thermal equilibrium,
(b) biased in the forward-active mode, $(c)$ biased in the inverse-active region, and
$(d)$ biased in cutoft with both the $\mathrm{B}-\mathrm{E}$ and $\mathrm{B}-\mathrm{C}$ junctions reverse biased.

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Problem 3

The parameters of the base region in a silicon npn bipolar transistor are $D_{n}=18 \mathrm{~cm}^{2} / \mathrm{s}$, $n_{B 0}=4 \times 10^{3} \mathrm{~cm}^{-3}, x_{B}=0.80 \mu \mathrm{m}$, and $A_{B E}=5 \times 10^{-5} \mathrm{~cm}^{2} .(a)$ Comparing Equa-
tions (12.1) and (12.2), calculate the magnitude of $I_{S .}$ (b) Calculate the collector current for $(i) v_{B E}=0.58 \mathrm{~V},(i i) v_{\bar{H} E}=0.65 \mathrm{~V}$, and (iii) $v_{\mathrm{BE}}=0.72 \mathrm{~V}$.

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Problem 4

An npn silicon bipolar transistor has the following base parameters: $D_{n}=22 \mathrm{~cm}^{-} / \mathrm{s}$, $x_{E}=0.80 \mu \mathrm{m}$, and $n_{B 0}=2 \times 10^{4} \mathrm{~cm}^{-3} \cdot(a)$ The collector current is to be $\left|i_{c}\right|=2 \mathrm{~mA}$
when biased at $v_{B E}=0.60 \mathrm{~V}$. What is the required cross-sectional area $A_{B E}$ ?
(b) Using the results of part $(a)$, what is the value of $v_{B E}$ such that $\left|i_{c}\right|=5 \mathrm{~mA} ?$

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Problem 5

Consider the transistor described in Problem 12.3. (a) For a common-base current gain of $\alpha=0.9850$, determine the common-emitter current gain [note: $\beta=\alpha /(1-\alpha)]$.
(b) Determine the emitter and base currents corresponding to the collector currents determined in Problem 12.3. ( $c$ ) Repeats parts ( $a$ ) and ( $b$ ) for a common-base current gain of $\alpha=0.9940$.

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Problem 6

A bipolar transistor is biased in the forward-active region. ( $a$ ) For a base current of $I_{\bar{B}}=4.2 \mu \mathrm{A}$ and a collector current of $I_{C}=0.625 \mathrm{~mA}$, determine $(i) \beta,(i i) \alpha$, and (iii) $I_{E \cdot}(b)$ For a collector current of $I_{C}=1.254 \mathrm{~mA}$ and an emitter current of $I_{E}=1.273 \mathrm{~mA}$, determine (i) $\beta,(i i) \alpha$, and (iii) $I_{B .}(c)$ For a base current of $I_{B}=.065 \mu \mathrm{A}$ and a common-emitter current gain of $\beta=150$, determine (i) $\alpha$,
(ii) $I_{C}$, and (iii) $I_{E \text { - }}$

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Problem 7

Assume that an npn bipolar transistor has a common-emitter current gain of $\beta=100 .(a)$ Sketch the ideal current-voltage characteristics $\left(i_{C}\right.$ versus $\left.v_{C R}\right)$, like those in Figure $12.9$, as $i_{B}$ varies from zero to $0.1 \mathrm{~mA}$ in $0.01-\mathrm{mA}$ increments. Let $\nu_{C E}$ vary over the range $0 \leq \nu_{C E} \leq 10 \mathrm{~V} .(b)$ Assuming $V_{c c}=10 \mathrm{~V}$ and $R_{c}=1 \mathrm{k} \Omega$
in the circuit in Figure $12.8$, superimpose the load line on the transistor characteristics in part $(a) .(c)$ Plot, on the resulting graph, the value of $i_{c}$ and $\nu_{C E}$ corresponding to $i_{6}=0.05 \mathrm{~mA}$

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Problem 8

Consider Figure 12.8. Assume $V_{C C}=3 \mathrm{~V}$ and $V_{\mathrm{BE}}=0.65 \mathrm{~V} .(a)$ For $R_{C}=25 \mathrm{k} \Omega$,
(i) plot $I_{C}$ versus $V_{C E}$ over the range $0.20 \leq V_{C E} \leq 3 \mathrm{~V} .($ ii $)$ At what value of $I_{C}$ does $V_{C B}=0 ?(b)$ Repeat part $(a)$ for $R_{C}=10 \mathrm{k} \Omega .$

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Problem 9

A uniformly doped silicon npn bipolar transistor at $T=300 \mathrm{~K}$ is biased in the forward-active mode. The doping concentrations are $N_{E}=8 \times 10^{17} \mathrm{~cm}^{-3}$, $N_{E}=2 \times 10^{16} \mathrm{~cm}^{-3}$, and $N_{c}=10^{15} \mathrm{~cm}^{-3} \cdot(a)$ Determine the thermal-equilibrium
values $p_{E 0}, n_{B 0}$, and $p_{c 0} \cdot(b)$ For $V_{B E}=0.640 \mathrm{~V}$, calculate the values of $n_{B}$ at $x=0$ and $p_{E}$ at $x^{\prime}=0 .(c)$ Sketch the minority carrier concentrations through the device and label each curve.

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Problem 10

A silicon pnp bipolar transistor at $T=300 \mathrm{~K}$ is uniformly doped and is biased in the forward-active mode. The doping concentrations are $N_{E}=5 \times 10^{17} \mathrm{~cm}^{-3}$, $N_{B}=10^{16} \mathrm{~cm}^{-3}$, and $N_{C}=10^{15} \mathrm{~cm}^{-3} .(a)$ Find the thermal-equilibrium values $n_{E 0}, p_{B 0,}$ and $n_{C \mathrm{C}} .(b)$ Determine the values of $p_{B}$ at $x=0$ and $n_{E}$ at $x^{\prime}=0$ for $V_{E B}-0.615 \mathrm{~V} .(c)$ Sketch the minority carrier concentrations through the device and label each curve.

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Problem 11

Consider a uniformly doped silicon npn bipolar transistor at $T=300 \mathrm{~K}$. The device is biased in the forward-active mode with $V_{C B}=2.5 \mathrm{~V}$. The metallurgical base width is $x_{B 0}=1.0 \mu \mathrm{m} .$ The doping concentrations are $N_{E}=8 \times 10^{17} \mathrm{~cm}^{-3}, N_{B}=2 \times 10^{16} \mathrm{~cm}^{-3}$,
and $N_{c}=10^{15} \mathrm{~cm}^{-3} .(a)$ Determine the B-E voltage such that the minority carrier electron concentration, $n_{B}$, at $x=0$ is 10 percent of the majority carrier hole concentration. ( $b$ ) At this bias, determine the minority carrier hole concentration at $x^{\prime}=0$.

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Problem 12

Consider the minority carrier electron concentration in the base of an npn bipolar transistor as given by Equation (12.15a). In this problem, we want to compare the gradient of the electron concentration evaluated at the $\mathrm{B}$ - $\mathrm{C}$ junction to that evaluated at the $\mathrm{B}-\mathrm{E}$ junction. In particular, calculate the ratio of $d\left(\delta n_{\bar{B}}\right) / d x$ at $x=x_{B}$ to $d\left(\delta n_{B}\right) / d x$ at $x=0$ for $(a) x_{B} / L_{B}=0.1,(b) x_{B} / L_{B}=1.0$, and $(c) x_{B} / L_{\bar{B}}=10 .$

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Problem 13

Derive the expressions for the coefficients given by Equations $(12.14 a)$ and (12.14b).

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Problem 14

Derive the expression for the excess minority carrier hole concentration in the base region of a uniformly doped pnp bipolar transistor operating in the forward-active region.

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Problem 15

The excess electron concentration in the base of an npn bipolar transistor is given by Equation (12.15a). The linear approximation is given by Equation (12.15b). If $\delta n_{b 0}(x)$ is the linear approximation given by Equation $(12.15 \mathrm{~b})$ and $\delta n_{B}(x)$ is the actual distribution given by Equation ( $12.15 a$ ), determine
$$
\frac{\delta n_{B o}(x)-\delta n_{E}(x)}{\delta n_{D 0}(x)} \times 100 \%
$$
at $x=x_{B} / 2$ for $(a) x_{B} / L_{B}=0.1$ and $(b) x_{B} / L_{b}=1.0 .$ Assume $V_{B E} \gg k T / e$.

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Problem 16

Consider a uniformly doped silicon pnp bipolar transistor biased in the forward-active mode at low injection. The excess minority carrier hole concentration at $x=0$ is $\delta p_{E}(0)=10^{15} \mathrm{~cm}^{-3}$ and the excess minority carrier hole concentration at $x=x_{B}$ is $\delta p_{\bar{B}}\left(x_{B}\right)=-5 \times 10^{3} \mathrm{~cm}^{-3} \cdot(a)$ What is the majority carrier electron concentration in the base region and what is the $\mathrm{E}-\mathrm{B}$ voltage? $(b)$ Assuming $x_{\bar{n}}=0.80 \mu \mathrm{m}$ and $\mathrm{D}_{B}=10 \mathrm{~cm}^{2} / \mathrm{s}$, calculate the magnitude of diffusion current density at $(i) x=0$ and $(i i) x=x_{B}$ for the case when $x_{B} \ll L_{\mathrm{B}} .$ (See Equation (12.15b).) ( $c$ ) Repeat part
(b) for the case when $x_{B}=L_{s}=12 \mu \mathrm{m}$. (See Equation (12.15a).) (d) Determine the ratio $J\left(x=x_{B}\right) / J(x=0)$ for parts $(b)$ and $(c)$.

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Problem 17

(a) A uniformly doped npn bipolar transistor at $T=300 \mathrm{~K}$ is biased in saturation. Starting with the continuity equation for minority carriers, show that the excess electron concentration in the base region can be expressed as
$$
\delta n_{B}(x)=n_{B 0}\left\{\left[\exp \left(\frac{e V_{B E}}{k T}\right)-1\right]\left(1-\frac{x}{x_{B}}\right)+\left[\exp \left(\frac{e V_{B C}}{k T}\right)-1\right]\left(\frac{x}{x_{B}}\right)\right\}
$$
for $x_{B} / L_{B} \ll 1$ where $x_{B}$ is the neutral base width. (b) Show that the minority carrier diffusion current in the base is then given by
$$
J_{n}=-\frac{e D_{i} n_{B D}}{x_{B}}\left[\exp \left(\frac{e V_{B E}}{k T}\right)-\exp \left(\frac{e V_{B C}}{k T}\right)\right]
$$
(c) Show that the total excess minority carrier charge (C/cm $^{2}$ ) in the base region is given by
$$
\delta Q_{n B}=\frac{-e n_{\mathrm{Bn}} x_{\mathrm{F}}}{2}\left\{\left[\exp \left(\frac{e V_{\mathrm{BE}}}{k T}\right)-1\right]+\left[\exp \left(\frac{e V_{B C}}{k T}\right)-1\right]\right\}
$$

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Problem 18

Consider a silicon npn bipolar transistor at $T=300 \mathrm{~K}$ with uniform doping concentrations of $N_{E}=10^{18} \mathrm{~cm}^{-3}, N_{B}=5 \times 10^{16} \mathrm{~cm}^{-3}$, and $N_{C}=10^{15} \mathrm{~cm}^{-3}$. Let
$D_{B}=25 \mathrm{~cm}^{2} / \mathrm{s}, x_{B}=0.70 \mu \mathrm{m}$, and assume $x_{\bar{f}} \ll L_{\mathrm{B}}$. The transistor is operating in
saturation with $\left|J_{n}\right|=125 \mathrm{~A} / \mathrm{cm}^{2}$ and $V_{B E}=0.70 \mathrm{~V}$. Determine $\left(\right.$ a) $V_{B C},(b) V_{C E}(\mathrm{sat})$,
(c) the $\# / \mathrm{cm}^{2}$ of excess minority carrier electrons in the base region, and $(d)$ the $\# / \mathrm{cm}^{2}$ of excess minority carrier holes in the long collector. Let $L_{c}=35 \mu \mathrm{m}$.

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Problem 19

An npn silicon bipolar transistor at $T=300 \mathrm{~K}$ has uniform dopings of $N_{E}=$ $10^{19} \mathrm{~cm}^{-3}, N_{B}=10^{17} \mathrm{~cm}^{-3}$, and $N_{C}=7 \times 10^{15} \mathrm{~cm}^{-3}$. The transistor is operating
in the inverse-active mode with $V_{B E}=-2 \mathrm{~V}$ and $V_{\mathrm{AC}}=0.565 \mathrm{~V} .(a)$ Sketch the minority carrier distribution through the device. (b) Determine the minority carrier concentrations at $x=x_{\bar{B}}$ and $x^{\prime \prime}=0 .(c)$ If the metallurgical base width is $1.2 \mu \mathrm{m}$, determine the neutral base width.

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Problem 20

A uniformly doped silicon pnp bipolar transistor at $T=300 \mathrm{~K}$ with dopings of $N_{E}=5 \times 10^{17} \mathrm{~cm}^{-3}, N_{B}=10^{16} \mathrm{~cm}^{-3}$, and $N_{C}=5 \times 10^{14} \mathrm{~cm}^{-3}$ is biased in the
inverse-active mode. What is the maximum B-C voltage so that the low-injection condition applies?

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Problem 21

( $a$ ) The following currents are measured in a uniformly doped npn bipolar transistor.
$$
\begin{aligned}
I_{n E} &=0.50 \mathrm{~mA} & I_{p E} &=3.5 \mu \mathrm{A} \\
I_{n C} &=0.495 \mathrm{~mA} & I_{R} &=5.0 \mu \mathrm{A} \\
I_{O} &=0.50 \mu \mathrm{A} & I_{\mathrm{po} 0} &=0.50 \mu \mathrm{A}
\end{aligned}
$$
Determine the following current gain parameters: (i) $\gamma,(i i) \alpha_{\bar{T}},(i i i) \delta,(i v) \alpha$, and $(v) \beta .(b)$ If the required value of common-emitter current gain is $\beta=120$, determine new values of $I_{n C}, I_{p E,}$ and $I_{B}$ to meet this specification assuming $\gamma=\alpha_{T}=\delta$

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Problem 22

A silicon pnp bipolar transistor at $T=300 \mathrm{~K}$ has a $\mathrm{B}-\mathrm{E}$ cross-sectional area of $A_{\overline{B E}}-5 \times 10^{-4} \mathrm{~cm}^{2}$, neutral base width of $x_{B}-0.70 \mu \mathrm{m}$, a neutral emitter width of $x_{E}=0.50 \mu \mathrm{m}$, and uniform doping concentrations of $N_{E}=5 \times 10^{17} \mathrm{~cm}^{-3}$, $N_{B}=10^{16} \mathrm{~cm}^{-3}$, and $N_{C}=10^{15} \mathrm{~cm}^{-3} .$ Other transistor parameters are $D_{B}=10 \mathrm{~cm}^{2} / \mathrm{s}$
$D_{E}=15 \mathrm{~cm}^{2} / \mathrm{s}, \tau_{E 0}=\tau_{B 0}=5 \times 10^{-7} \mathrm{~s}$, and $\tau_{c 0}=2 \times 10^{-6} \mathrm{~s}$. The transistor is
biased in the forward-active mode and the recombination factor is $\delta=0.995 .$ Determine the collector current for $(a) V_{E B}=0.550 \mathrm{~V},(b) I_{B}=0.80 \mu \mathrm{A}$, and
(c) $I_{E}=125 \mu \mathrm{A}$.

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Problem 23

Consider a uniformly doped npn bipolar transistor at $T=300 \mathrm{~K}$ with the following parameters:

For $V_{B E}=0.60 \mathrm{~V}$ and $V_{C E}=5 \mathrm{~V}$, calculate $(a)$ the currents $J_{n E}, J_{p E}, J_{n} c$, and $J_{R}$ and
(b) the current gain factors $\gamma, \alpha_{T}, \delta, \alpha$, and $\beta$.

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Problem 24

Three npn bipolar transistors have identical parameters except for the base doping concentrations and neutral base widths. The base parameters for the three devices are as follows:
\begin{tabular}{lll}
\hline Device & Base doping & Base width \\
\hline $\mathrm{A}$ & $N_{B}=N_{\mathrm{se}}$ & $x_{B}=x_{B 0}$ \\
$\mathrm{~B}$ & $N_{\mathrm{B}}=2 N_{50}$ & $x_{B}=x_{B 0}$ \\
$\mathrm{C}$ & $N_{B}=N_{\mathrm{so}}$ & $x_{B}=x_{B 0} / 2$ \\
\hline
\end{tabular}
(The base doping concentration for the $\mathrm{B}$ device is twice that of $\mathrm{A}$ and $\mathrm{C}$, and the neutral base width for the $\mathrm{C}$ device is half that of $\mathrm{A}$ and $\mathrm{B}$.)
(a) Determine the ratio of the emitter injection efficiency of $(i)$ device $\mathrm{B}$ to device A and $(i i)$ device $\mathrm{C}$ to device $\mathrm{A}$.
(b) Repeat part ( $a$ ) for the base transport factor.
(c) Repeat part ( $a$ ) for the recombination factor.
(d) Which device has the largest common-emitter current $\operatorname{gain} \beta ?$

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Problem 25

Repeat Problem $12.24$ for three devices in which the emitter parameters vary. The emitter parameters for the three devices are as follows:
\begin{tabular}{lll}
\hline Device & Emitter doping & Emitter width \\
\hline A & $N_{E}=N_{E 0}$ & $x_{E}=x_{E 0}$ \\
B & $N_{E}=2 N_{E n}$ & $x_{E}=x_{E 0}$ \\
C & $N_{E}=N_{\text {En }}$ & $x_{E}=x_{E n} / 2$ \\
\hline
\end{tabular}

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Problem 26

An npn silicon transistor is biased in the inverse-active mode with $V_{B E}=-3 \mathrm{~V}$ and $V_{\mathrm{HC}}=0.6 \mathrm{~V}$. The doping concentrations are $N_{E}=10^{18} \mathrm{~cm}^{-3}, N_{B}=10^{17} \mathrm{~cm}^{-3}$, and
$N_{C}=10^{16} \mathrm{~cm}^{-3} .$ Other parameters are $x_{B}=1 \mu \mathrm{m}, \tau_{E 0}=\tau_{\mathrm{BO}}=\tau_{\mathrm{Cl}}=2 \times 10^{-7} \mathrm{~s}$
$D_{E}=10 \mathrm{~cm}^{2} / \mathrm{s}, D_{B}=20 \mathrm{~cm}^{2} / \mathrm{s}, D_{c}=15 \mathrm{~cm}^{2} / \mathrm{s}$, and $A=10^{-3} \mathrm{~cm}^{2} .(a)$ Calculate
and plot the minority carrier distribution in the device. (b) Calculate the collector and emitter currents. (Neglect geometry factors and assume the recombination factor is unity.)

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Problem 27

(a) Calculate the base transport factor, $\alpha_{T}$, for $x_{E} / L_{B}=0.01,0.10,1.0$, and $10 .$ Assuming that $\gamma$ and $\delta$ are unity, determine $\beta$ for each case. (b) Calculate the emitter injection efficiency, $\gamma$, for $N_{B} / N_{E}=0.01,0.10,1.0$, and $10 .$ Assuming that $\alpha_{T}$ and $\delta$ are unity, determine $\alpha$ for each case. (c) Considering the results of parts $(a)$ and $(b)$, what conclusions can be made concerning when the base transport factor or when the emitter injection efficiency are the limiting factors for the commonemitter current gain?

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Problem 28

(a) Calculate the recombination factor for $V_{B E}=0.2,0.4$, and $0.6 \mathrm{~V}$. Assume the following parameters:
$$
\begin{aligned}
D_{E} &=25 \mathrm{~cm}^{2} / \mathrm{s} & D_{E} &=10 \mathrm{~cm}^{2} / \mathrm{s} \\
N_{E} &=5 \times 10^{18} \mathrm{~cm}^{-3} & & N_{B}=1 \times 10^{17} \mathrm{~cm}^{-3} \\
N_{C} &=5 \times 10^{15} \mathrm{~cm}^{-3} & & x_{B} &=0.7 \mu \mathrm{m} \\
\tau_{B 0} &=\tau_{\mathrm{E} 0}=10^{-7} \mathrm{~s} & J_{r 0} &=2 \times 10^{-9} \mathrm{~A} / \mathrm{cm}^{2} \\
n_{i} &=1.5 \times 10^{10} \mathrm{~cm}^{-3} & &
\end{aligned}
$$
(b) Assuming the base transport and emitter injection efficiency factors are unity, calculate the common-emitter current gain for the conditions in part $(a)$.
(c) Considering the results of part $(b)$, what can be said about the recombination factor being the limiting factor in the common-emitter current gain.

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Problem 29

Consider a uniformly dope silicon npn bipolar transistor at $T=300 \mathrm{~K}$ with the following parameters: $D_{B}=23 \mathrm{~cm}^{2} / \mathrm{s}, D_{E}=8 \mathrm{~cm}^{2} / \mathrm{s}, \tau_{\mathrm{Bo}}=2 \times 10^{-7} \mathrm{~s}, \tau_{\mathrm{EO}}=8 \times$
$10^{-8} \mathrm{~s}, N_{B}=2 \times 10^{16} \mathrm{~cm}^{-3}$, and $x_{E}=0.35 \mu \mathrm{m}$. The recombination factor has been
determined to be $\delta=0.9975 .$ The required common-emitter current gain is $\beta=150$. A minimum neutral base width of $x_{8}=0.80 \mu \mathrm{m}$ can be fabricated. $(a)$ Determine an appropriate neutral base width and the minimum emitter doping concentration, $N_{E}$, to meet this specification. (b) Using the results of part $(a)$, what are the values of $\alpha_{T}$ and $\gamma$ ?

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Problem 30

( $a$ ) The recombination current density, $J_{r 0}$, in an npn silicon bipolar transistor at $T=300 \mathrm{~K}$ is $J_{r 0}=5 \times 10^{-8} \mathrm{~A} / \mathrm{cm}^{2}$. The uniform dopings are $N_{E}=10^{18} \mathrm{~cm}^{-3}$,
$N_{B}=5 \times 10^{16} \mathrm{~cm}^{-3}$, and $N_{C}=10^{15} \mathrm{~cm}^{-3} .$ Other parameters are $D_{E}=10 \mathrm{~cm}^{2} / \mathrm{s}$
$D_{B}=25 \mathrm{~cm}^{2} / \mathrm{s}, \tau_{E 0}=10^{-8} \mathrm{~s}$, and $\tau_{\mathrm{E} 0}=10^{-7} \mathrm{~s} .$ Determine the neutral base width so
that the recombination factor is $\delta=0.995$ when $V_{B E}=0.55 \mathrm{~V}(b)$ If $J_{\mathrm{ro}}$ remains constant with temperature, what is the value of $\delta$ when $V_{B E}=0.55 \mathrm{~V}$ for the case when the temperature is $T=400 \mathrm{~K} ?$ Use the value of $x_{8}$ determined in part $(a) .$

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Problem 31

(a) Plot, for a bipolar transistor, the base transport factor, $\alpha_{T}$, as a function of $\left(x_{B} / L_{B}\right)$ over the range $0.01 \leq\left(x_{B} / L_{B}\right) \leq 10 .$ (Use a log scale on the horizontal axis.) ( $b$ ) Assuming that the emitter injection efficiency and recombination factors are unity, plot the common-emitter gain for the conditions in part $(a) .(c)$ Considering the results of part $(b)$, what can be said about the base transport factor being the limiting factor in the common-emitter current gain?

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Problem 32

(a) Plot the emitter injection efficiency as a function of the doping ratio, $N_{B} / N_{E}$, over the range $0.01 \leq N_{B} / N_{E} \leq 10 .$ Assume that $D_{E}=D_{B}, L_{B}=L_{E}$, and $x_{B}=x_{E}$ (Use a log scale on the horizontal axis.) Neglect bandgap narrowing effects.
(b) Assuming that the base transport factor and recombination factors are unity, plot the common-emitter current gain for the conditions in part $(a) .(c)$ Considering the results of part $(b)$, what can be said about the emitter injection efficiency being the limiting factor in the common-emitter current gain.

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Problem 33

(a) Plot the recombination factor as a function of the forward-bias $B$ - $E$ voltage for $0.1 \leq V_{B E} \leq 0.6 .$ Assume the following parameters:
$$
\begin{aligned}
D_{B} &=25 \mathrm{~cm}^{2} / \mathrm{s} & D_{E} &=10 \mathrm{~cm}^{2} / \mathrm{s} \\
N_{E} &=5 \times 10^{18} \mathrm{~cm}^{-3} & & N_{B}=1 \times 10^{17} \mathrm{~cm}^{-3} \\
N_{C} &=5 \times 10^{15} \mathrm{~cm}^{-3} & & x_{B} &=0.7 \mu \mathrm{m} \\
\tau_{B 0} &=\tau_{\mathrm{E} 0}=10^{-7} \mathrm{~s} & & J_{r 0}=2 \times 10^{-9} \mathrm{~A} / \mathrm{cm}^{2} \\
n_{i} &=1.5 \times 10^{10} \mathrm{~cm}^{-3} & &
\end{aligned}
$$
(b) Assuming the base transport and emitter injection efficiency factors are unity, plot the common-emitter current gain for the conditions in part $(a) .(c)$ Considering the results of part (b), what can be said about the recombination factor being the limiting factor in the common-emitter current gain.

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Problem 34

The emitter in a BJT is often made very thin to achieve high operating speed. In this problem, we investigate the effect of emitter width on current gain. Consider the emitter injection efficiency given by Equation (12.35a). Assume that $N_{E}=100$ $N_{\bar{B}}, D_{E}=D_{B}$, and $L_{E}=L_{\bar{B}} .$ Also let $x_{B}=0.1 L_{b} .$ Plot the emitter injection efficiency for $0.01 L_{E} \leq x_{E} \leq 10 L_{E} .$ From these results, discuss the effect of emitter width on the current gain.

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Problem 35

An npn bipolar transistor is biased in the forward-active mode. $(a)$ The collector current is $I_{C}=1.2 \mathrm{~mA}$ when biased at $V_{C E}=2 \mathrm{~V}$. The Early voltage is $V_{A}=120 \mathrm{~V}$. Determine $(i)$ the output resistance $\mathrm{r}_{w},(i i)$ the output conductance $g_{e}$, and $(\mathrm{iii})$ the collector current when biased at $V_{C E}=4 \mathrm{~V} .(b)$ Repeat part $(a)$ if the collector current is $I_{C}=0.25 \mathrm{~mA}$ when biased at $V_{C E}=2 \mathrm{~V}$ and the Early voltage is $V_{A}=160 \mathrm{~V}$.

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Problem 36

The output resistance of a pnp bipolar transistor is $r_{a}=180 \mathrm{k} \Omega$. The Early voltage is $V_{A}=80 \mathrm{~V}$. Determine the change in collector current if $V_{E C}$ increases from 2 to $5 \mathrm{~V}$.

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Problem 37

A uniformly doped silicon npn bipolar transistor at $T=300 \mathrm{~K}$ has parameters $N_{E}=2 \times 10^{18} \mathrm{~cm}^{-3}, N_{B}=2 \times 10^{16} \mathrm{~cm}^{-3}, N_{C}=2 \times 10^{15} \mathrm{~cm}^{-3}, x_{10}=0.85 \mu \mathrm{m}$
and $D_{B}=25 \mathrm{~cm}^{2} / \mathrm{s}$. Assume $x_{B o} \ll L_{s}$ and let $V_{B E}=0.650 \mathrm{~V}$. $($ a $)$ Determine the electron diffusion current density in the base for (i) $V_{C B}=4 \mathrm{~V},(i i) V_{C B}=8 \mathrm{~V}$, and
(iii) $V_{C B}=12 \mathrm{~V}_{.}(b)$ Estimate the Early voltage.

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Problem 38

The base width of a bipolar transistor is normally small to provide a large current gain and increased speed. The base width also affects the Early voltage. In a silicon npn bipolar transistor at $T=300 \mathrm{~K}$, the doping concentrations are $N_{E}=10^{18} \mathrm{~cm}^{-3}$, $N_{E}=3 \times 10^{16} \mathrm{~cm}^{-3}$, and $N_{C}=5 \times 10^{15} \mathrm{~cm}^{-3} .$ Assume $D_{B}=20 \mathrm{~cm}^{2} / \mathrm{s}$ and
$\tau_{\mathrm{Bn}}=5 \times 10^{-7} \mathrm{~s}$, and let $V_{B E}=0.70 \mathrm{~V} .$ Using voltages $V_{C B}=5 \mathrm{~V}$ and $V_{C B}=10 \mathrm{~V}$
as two data points, estimate the Early voltage for metallurgical base widths of
(a) $1.0 \mu \mathrm{m}$
(b) $0.80 \mu \mathrm{m}$, and
(c) $0.60 \mu \mathrm{m}$

Chai Santi

Numerade Educator

Problem 39

A uniformly doped pnp silicon bipolar transistor has a base doping of $N_{E}=10^{16} \mathrm{~cm}^{-3}$, a collector doping of $N_{C}=10^{15} \mathrm{~cm}^{-3}$, a metallurgical base width of $x_{\overline{B 0}}=0.70 \mu \mathrm{m}$, a base minority carrier diffusion coefficient of $D_{E}=10 \mathrm{~cm}^{2} / \mathrm{s}$, and a B-E cross-sectional area of $\mathrm{A}_{B \mathcal{E}}=10^{-4} \mathrm{~cm}^{2}$. The transistor is biased in the forward-active mode with $V_{E B}=0.625 \mathrm{~V}$. Neglecting the $\mathrm{B}-\mathrm{E}$ space charge width and assuming $x_{E} \ll L_{6},(a)$ detemine the change in neutral base width as $V_{\text {ic }}$ changes from 1 to $5 \mathrm{~V},(b)$ find the corresponding change in collector current,
( $c$ ) estimate the Early voltage, and $(d)$ find the output resistance.

Chai Santi

Numerade Educator

Problem 40

Consider a uniformly doped silicon npn bipolar transistor in which $x_{E}=x_{\mathrm{B}}$, $L_{E}=L_{B}$, and $D_{E}=D_{B} .$ Assume that $\alpha_{T}=\delta=0.995$ and let $N_{B}=10^{17} \mathrm{~cm}^{-3}$, Cal-
culate and plot the common-emitter current gain $\beta$ for $N_{E}=10^{17}, 10^{18}, 10^{19}$, and $10^{20} \mathrm{~cm}^{-3}$, and for the case $(a)$ when the bandgap narrowing effect is neglected, and
(b) when the bandgap narrowing effect is taken into account.

Chai Santi

Numerade Educator

Problem 41

A silicon pnp bipolar transistor at $T=300 \mathrm{~K}$ is to be designed so that the emitter injection efficiency is $\gamma=0.996$. Assume that $x_{E}=x_{B}, L_{E}=L_{B}, D_{E}=D_{B}$, and let $N_{E}=10^{19} \mathrm{~cm}^{-3} .$ (a) Determine the maximum base doping, taking into account bandgap narrowing. (b) If bandgap narrowing were neglected, what would be the maximum base doping required?

Chai Santi

Numerade Educator

Problem 42

The current crowding effect, to a first approximation, can be determined by using the geometry shown in Figure P12.42. Assume that one-half of the base current enters from each side of the emitter strip and flows uniformly to the center of the emitter. The base is p type with the following parameters: $N_{B}=2 \times 10^{16} \mathrm{~cm}^{-3}$, $x_{\bar{B}}=0.65 \mu \mathrm{m}, \mu_{\mathrm{p}}=250 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, and $L=25 \mu \mathrm{m} .(a)$ Assume $S=10 \mu \mathrm{m} .$
(i) Calculate the resistance between $x=0$ and $x=S / 2 .(i i)$ If $I_{B} / 2=5 \mu \mathrm{A}$, determine the voltage drop between $x=0$ and $x=S / 2 .$ (iii) For $V_{B E}=0.60 \mathrm{~V}$ at $x=S / 2$, determine the ratio of electrons being injected into the base at $x=S / 2$ compared to $x=0 .(b)$ Repeat part $(a)$ for $S=3 \mu \mathrm{m}$.

Chai Santi

Numerade Educator

Problem 43

Consider the geometry shown in Figure $\mathrm{P} 12.42$ and the device parameters given in Problem $12.42$ except for the emitter width $S .$ Determine the maximum value of $S$ such that the ratio of electrons being injected into the base at $x=S / 2$ compared to $x=0$ is no less than $0.90$.

Chai Santi

Numerade Educator

Problem 44

The base doping in a diffused $\mathrm{n}^{+}$ pn bipolar transistor can be approximated by an exponential as
$$
N_{B}=N_{E}(0) \exp \left(\frac{-a x}{x_{B}}\right)
$$
where $a$ is a constant and is given by
$$
a=\ln \left(\frac{N_{B}(0)}{N_{B}\left(x_{E}\right)}\right)
$$
(a) Show that, in thermal equilibrium, the electric field in the neutral base region is a constant. $(b)$ Indicate the direction of the electric field. Does this electric field aid or retard the flow of minority carrier electrons across the base? (c) Derive an expression for the steady-state minority carrier electron concentration in the base under forward bias. Assume no recombination occurs in the base. (Express the electron concentration in terms of the electron current density.)

Chai Santi

Numerade Educator

Problem 45

Consider a uniformly doped pnp silicon bipolar transistor with doping concentrations of $N_{E}=10^{18} \mathrm{~cm}^{-3}, N_{B}=5 \times 10^{16} \mathrm{~cm}^{-3}$, and $N_{C}=2 \times 10^{15} \mathrm{~cm}^{-3} .$ The
common-base current gain is $\alpha=0.9930 .$ Determine (a) $B V_{B C 0},(b) B V_{B C 1}$, and
(c) the emitter-base breakdown voltage. (Assume $N=3$ for the empirical constant.)

Chai Santi

Numerade Educator

Problem 46

A high-voltage silicon npn bipolar transistor is to be designed such that the uniform base doping is $N_{B}=10^{16} \mathrm{~cm}^{-3}$ and the common-emitter current grain is $\beta=50$. The breakdown voltage $B V_{C E O}$ is to be at least $60 \mathrm{~V}$. Determine the maximum collector doping and the minimum collector length to support this voltage. (Assume $n=3 .$ )

Chai Santi

Numerade Educator

Problem 47

A silicon npn bipolar transistor is uniformly doped with $N_{B}=5 \times 10^{16} \mathrm{~cm}^{-3}$ and $N_{C}=8 \times 10^{15} \mathrm{~cm}^{-3}$. The metallurgical base width is $x_{\mathrm{so}}=0.50 \mu \mathrm{m}$ with $V_{B E}=0$
and $V_{C}=0 .(a)$ Determine the expected avalanche $\mathrm{B}-\mathrm{C}$ breakdown voltage.
(b) Calculate the value of $V_{C B}$ at which punch-through occurs.

Chai Santi

Numerade Educator

Problem 48

Consider an npn silicon bipolar transistor with doping concentrations of $N_{B}=2 \times 10^{16} \mathrm{~cm}^{-3}$ and $N_{C}=5 \times 10^{15} \mathrm{~cm}^{-3}$, and with a metallurgical base width
of $x_{50}=0.65 \mu \mathrm{m}$. Let $V_{B E}=0.625 \mathrm{~V}$. $(a)$ Determine $V_{C E}$ at punch-through. (b) Calculate the magnitude of the maximum electric field in the $\mathrm{B}-\mathrm{C}$ space charge region at punch-through.

Chai Santi

Numerade Educator

Problem 49

A uniformly doped silicon pnp bipolar transistor has doping concentrations of $N_{E}=10^{18} \mathrm{~cm}^{-3}, N_{\mathrm{B}}=5 \times 10^{16} \mathrm{~cm}^{-3}$, and $N_{c}=3 \times 10^{15} \mathrm{~cm}^{-3} .$ Determine the mini-
mum metallurgical base width such that the punch-through voltage is $V_{p v}=15 \mathrm{~V}$.

Chai Santi

Numerade Educator

Problem 50

The $V_{C E}$ (sat) voltage of an npn transistor in saturation continues to decrease slowly as the base current increases. In the Ebers-Moll model, assume $\alpha_{F}=0.99$, $\alpha_{R}=0.20$, and $I_{C}=1 \mathrm{~mA}$. For $T=300 \mathrm{~K}$, determine the base current, $I_{B}$, necessary to give $(a) V_{c E}(\mathrm{sat})=0.30 \mathrm{~V},(b) V_{C E}(\mathrm{sat})=0.20 \mathrm{~V}$, and $(c) V_{c E}(\mathrm{sat})=0.10 \mathrm{~V}$

Chai Santi

Numerade Educator

Problem 51

Consider an npn bipolar transistor biased in the active mode. Using the Ebers-Moll model, derive the equation for the base current, $I_{B}$, in terms of $\alpha_{F}, \alpha_{R}, I_{E S}, I_{C S}$, and $\underline{V_{B E}}$ -

Chai Santi

Numerade Educator

Problem 52

Consider the Ebers-Moll model and let the base terminal be open so $I_{B}=0 .$ Show that, when a collector-emitter voltage is applied, we have
$$
I_{C}=I_{C E O}=I_{C S} \frac{\left(1-\alpha_{F} \alpha_{R}\right)}{\left(1-\alpha_{F}\right)}
$$

Chai Santi

Numerade Educator

Problem 53

The parameters in the Ebers-Moll model are $\alpha_{F}=0.9920, I_{E S}=5 \times 10^{-14} \mathrm{~A}$, and $I_{C S}=10^{-13} \mathrm{~A} .$ Let $T=300 \mathrm{~K} .$ Plot $I_{C}$ versus $V_{C B}$ for $-0.5<V_{C B}<2 \mathrm{~V}$ and for
(a) $V_{\overline{B E}}=0.2 \mathrm{~V}$
(b) $V_{\overline{B E}}=0.4 \mathrm{~V}$, and
(c) $V_{B E}=0.6 \mathrm{~V} .$ (Note that $\left.V_{C B}=-V_{B C} .\right)$

Chai Santi

Numerade Educator

Problem 54

The collector-emitter saturation voltage, from the Ebers-Moll model, is given by Equation (12.77). Consider a power BJT in which $\alpha_{F}=0.975, \alpha_{R}=0.150$, and $I_{C}=5 \mathrm{~A} .$ Plot $V_{C E}$ (sat) versus $I_{B}$ over the range $0.15 \leq I_{B} \leq 1 \mathrm{~A}$.

Chai Santi

Numerade Educator

Problem 55

Consider a uniformly doped silicon bipolar transistor at $T=300 \mathrm{~K}$ with the following parameters:
$$
\begin{array}{ll}
I_{E}=0.25 \mathrm{~mA} & C_{j e}=0.35 \mathrm{pF} \\
x_{B}=0.65 \mu \mathrm{m} & D_{n}=25 \mathrm{~cm}^{2} / \mathrm{s} \\
x_{d k}=2.2 \mu \mathrm{m} & r_{c}=18 \Omega \\
C_{x}=C_{\mu}=0.020 \mathrm{pF} & \beta=125
\end{array}
$$
(a) Determine the transit time factors (i) $\tau_{e},(i i) \tau_{b},($ iii $) \tau_{d}$, and (iv) $\tau_{c}$.
(b) Find the total transit time $\tau_{c e}$ (c) Calculate the cutoff frequency $f_{T}$.
(c) Find the beta cutoff frequency $f_{\beta}$.

Chai Santi

Numerade Educator

Problem 56

In a particular bipolar transistor, the base transit time is 20 percent of the total delay time. The base width is $0.5 \mu \mathrm{m}$ and the base diffusion coefficient is $D_{B}=20 \mathrm{~cm}^{2} / \mathrm{s}$. Determine the cutoff frequency.

Chai Santi

Numerade Educator

Problem 57

Assume the base transit time of a BJT is $100 \mathrm{ps}$ and carriers cross the $1.2 \mu \mathrm{m} \mathrm{B}-\mathrm{C}$ space charge region at a speed of $10^{7} \mathrm{~cm} / \mathrm{s}$. The emitter-base junction charging time is $25 \mathrm{ps}$ and the collector capacitance and resistance are $0.10 \mathrm{pF}$ and $10 \Omega$, respectively. Determine the cutoff frequency.

Chai Santi

Numerade Educator

Problem 58

(a) A silicon npn bipolar transistor at $T=300 \mathrm{~K}$ is to be designed such that the common-emitter current gain is at least $\beta=120$ and the Early voltage is at least $V_{A}=140 \mathrm{~V} .(b)$ Repeat part $(a)$ for a pnp silicon bipolar transistor.

Chai Santi

Numerade Educator

Problem 59

Design a uniformly doped silicon npn bipolar transistor so that $\beta=100$ at $T=$ $300 \mathrm{~K}$. The maximum CE voltage is to be $15 \mathrm{~V}$ and any breakdown voltage is to be at least three times this value. Assume the recombination factor is constant at $\delta=$ $0.995 .$ The transistor is to be operated in low injection with a maximum collector current of $I_{C}=5 \mathrm{~mA}$. Bandgap narrowing effects and base width modulation effects are to be minimized. Let $D_{E}=6 \mathrm{~cm}^{2} / \mathrm{s}, D_{B}=25 \mathrm{~cm}^{2} / \mathrm{s}, \tau_{E 0}=10^{-8} \mathrm{~s}$, and $\tau_{\mathrm{fa}}=$
$10^{-7}$ s. Determine doping concentrations, the metallurgical base width, the active area, and the maximum allowable $V_{B E-}$

Chai Santi

Numerade Educator

Problem 60

Design a pair of complementary npn and pnp bipolar transistors. The transistors are to have the same metallurgical base and emitter widths of $W_{B}=0.75 \mu \mathrm{m}$ and $x_{E}=0.5 \mu \mathrm{m} .$ Assume that the following minority carrier parameters apply to each device.
$$
\begin{array}{ll}
D_{n}=23 \mathrm{~cm}^{2} / \mathrm{s} & \tau_{m 0}=10^{-7} \mathrm{~s} \\
D_{p}=8 \mathrm{~cm}^{2} / \mathrm{s} & \tau_{p 0}=5 \times 10^{-8} \mathrm{~s}
\end{array}
$$
The collector doping concentration in each device is $5 \times 10^{15} \mathrm{~cm}^{-3}$ and the recombination factor in each device is constant at $\delta=0.9950 .(a)$ Design, if possible, the devices so that $\beta=100$ in each device. If this is not possible, how close a match can be obtained? ( $b$ ) With equal forward-bias base-emitter voltages applied, the collector currents are to be $I_{C}=5 \mathrm{~mA}$ with each device operating in low injection. Determine the active cross-sectional areas.

Chai Santi

Numerade Educator