Semiconductor Physics and Devices

Donald A. Neamen

Chapter 13

The Junction Field-Effect Transistor - all with Video Answers

Educators

Chapter Questions

Problem 1

(a) Draw the structure of a p-channel JFET similar to the structure shown in Figure 13.2. (b) Qualitatively discuss the $I-V$ characteristics, including current directions and voltage polarities, similar to those shown in Figures $13.3$ and $13.4$.

Chai Santi

Chai Santi

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Problem 2

Consider the n-channel JFET in Figure P13.2. The p-type substrate is connected to the n-type source terminal. Sketch the space charge regions for various $V_{G S}$ values when $V_{D S}=0$ and for various $V_{D S}$ values when $V_{G S}=0$.

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Problem 3

An $\mathrm{n}$ -channel GaAs pn JFET at $T=300 \mathrm{~K}$ has parameters $N_{d}=3 \times 10^{16} \mathrm{~cm}^{-3}$, $N_{a}=2 \times 10^{18} \mathrm{~cm}^{-3}$, and $a=0.40 \mu \mathrm{m} .(a)$ Calculate the $(i)$ internal pinchoff voltage $V_{p o}$ and (ii) pinchoff voltage $V_{p}$. (b) Determine the minimum undepleted channel thickness, $a-h$, for $V_{G S}=-0.5 \mathrm{~V}$ and for $(i) V_{D S}=0,($ ii $) V_{D S}=0.5 \mathrm{~V}$,
and (iii) $V_{D S}=2.5 \mathrm{~V} .(c)$ Find $V_{D S}($ sat $)$ for $(i) V_{G S}=0$ and $(i i) V_{G S}=-1.0 \mathrm{~V}$.

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Problem 4

Repeat Problem $13.3$ for an n-channel silicon pn JFET with the same geometrical and electrical parameters.

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Problem 5

Consider a p-channel GaAs pn JFET at $T=300 \mathrm{~K}$. The parameters are $N_{d}=10^{18} \mathrm{~cm}^{-3}$ and $a=0.65 \mu \mathrm{m} .(a)$ Determine the channel doping concentration such that the internal pinchoff voltage is $V_{p o}=2.75 \mathrm{~V} .(b)$ Using the results of part
(a), what is the pinchoff voltage $V_{p} ?(c)$ For $V_{S D}=0$, determine the value of $V_{G S}$ such that the minimum undepleted channel thickness is $0.15 \mu \mathrm{m} .(d)$ For $V_{G S}=0$, find the value of $V_{S D}$ such that the channel is just pinched off at the drain terminal.

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Problem 6

Repeat Problem $13.5$ for a p-channel silicon pn JFET with the same geometrical and electrical parameters.

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Problem 7

The parameters of a p-channel silicon pn JFET are $N_{d}=3 \times 10^{18} \mathrm{~cm}^{-3}$ and $N_{a}=2 \times 10^{16} \mathrm{~cm}^{-3} .(a)$ Determine the metallurgical channel thickness, $a$, such that the pinchoff voltage is $V_{p}=+3.0 \mathrm{~V} .(b)$ Using the results of part $(a)$, determine the internal pinchoff voltage $V_{p 0} .(c)$ Determine $V_{S D}$ (sat) for (i) $V_{G S}=0$ and
(ii) $V_{G S}=1.5 \mathrm{~V}$.

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Problem 8

A p-channel GaAs pn JFET has the same parameters as given in Problem 13.7. Repeat the calculations for parts $(a),(b)$, and $(c)$.

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Problem 9

The doping concentrations in a silicon $\mathrm{n}$ -channel pn JFET are $N_{a}=4 \times 10^{18} \mathrm{~cm}^{-3}$ and $N_{d}=4 \times 10^{16} \mathrm{~cm}^{-3} .$ (a) Design the channel metallurgical thickness, $a$, such that $V_{D S}($ sat $)=5.0 \mathrm{~V}$ for $V_{G s}=0 .(b)$ Using the results of part $(a)$, find the $(i)$ internal pinchoff voltage $V_{p o}$ and ( $(i)$ pinchoff voltage $V_{p}$.

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Problem 10

Consider a p-channel GaAs pn JFET. The doping concentrations are $N_{d}=10^{18} \mathrm{~cm}^{-3}$ and $N_{a}=5 \times 10^{15} \mathrm{~cm}^{-3}$. (a) Design the channel metallurgical thickness, a, such that $V_{S D}($ sat $)=3.5 \mathrm{~V}$ for $V_{G S}=+1.0 \mathrm{~V} .(b)$ Using the results of part $(a)$, determine the ( $i$ ) internal pinchoff voltage $V_{p O}$ and (ii) pinchoff voltage $V_{p}$.

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Problem 11

An $\mathrm{n}$ -channel silicon JFET at $T=300 \mathrm{~K}$ has the following parameters:
$$
\begin{array}{rlrl}
N_{a} & =10^{19} \mathrm{~cm}^{-3} & N_{d} & =10^{16} \mathrm{~cm}^{-3} \\
a & =0.50 \mu \mathrm{m} & L & =20 \mu \mathrm{m} \\
W & =400 \mu \mathrm{m} & \mu_{n} & =1000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}
\end{array}
$$
Ignoring velocity saturation effects, calculate $(a) I_{P I} ;$ (b) $V_{D S}$ (sat) for (i) $V_{G S}=0$,
(ii) $V_{G S}=V_{p} / 4,($ iii $) V_{G S}=V_{p} / 2$, and (iv) $V_{G S}=3 V_{p} / 4 ;$ and $(c) I_{D 1}$ (sat) for the same $V_{G S}$ values in part $(b) .(d)$ Using the results from parts $(b)$ and $(c)$, plot the $I-V$ characteristics.

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Problem 12

Consider the JFET described in Problem 13.11. Compute and plot the channel conductance, $g_{d}$, as a function of $V_{G S}$ for $0<\left|V_{G S}\right|<\left|V_{p}\right|$

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Problem 13

Consider an n-channel GaAs JFET at $T=300 \mathrm{~K}$ with the following parameters:
$$
\begin{array}{rlrl}
N_{a} & =5 \times 10^{18} \mathrm{~cm}^{-3} & N_{d} & =2 \times 10^{16} \mathrm{~cm}^{-3} \\
a & =0.35 \mu \mathrm{m} & L & =10 \mu \mathrm{m} \\
W & =30 \mu \mathrm{m} & \mu_{n} & =8000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}
\end{array}
$$
Ignoring velocity saturation effects, calculate $(a) G_{01} ;(b) V_{D S}($ sat $)$ for $V_{G S}=0$ and $V_{G S}=V_{p} / 2 ;$ and $(c) I_{D 1}($ sat $)$ for $V_{G S}=0$ and $V_{G S}=V_{p} / 2 .(d)$ Sketch the $I-V$ characteristics using the results from parts $(b)$ and $(c)$.

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Problem 14

Using the parameters from Problem 13.11, calculate the maximum transconductance in the saturation region. Normalize this transconductance to millisiemens per unit width, or $\mathrm{mS} / \mathrm{mm}$.

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Problem 15

(a) Calculate the maximum transconductance for the transistor described in Problem $13.13$ (b) Determine the maximum transconductance if the channel length is reduced to $2 \mu \mathrm{m}$.

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Problem 16

The Schottky barrier height, $\phi_{B n}$, of a metal-n-GaAs MESFET is $0.90 \mathrm{~V}$. The channel doping is $N_{d}=1.5 \times 10^{16} \mathrm{~cm}^{-3}$, and the channel thickness is $a=0.5 \mu \mathrm{m}$. $T=300 \mathrm{~K}$. (a) Calculate the internal pinchoff voltage $V_{p 0}$ and the threshold voltage $V_{T}$. (b) Determine whether the MESFET is depletion type or enhancement type.

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Problem 17

Consider an $\mathrm{n}$ -channel GaAs MESFET at $T=300 \mathrm{~K}$ with a gold Schottky barrier contact. Assume $\phi_{B n}=0.89 \mathrm{~V}$. The channel thickness is $a=0.35 \mu \mathrm{m} .(a)$ Determine the uniform channel doping so that the threshold voltage is $V_{T}=0.10 \mathrm{~V}$. (b) Using the results of part $(a)$, determine the threshold voltage at $T=400 \mathrm{~K}$.

Chai Santi

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Problem 18

The barrier height of the metal contact in an n-channel GaAs MESFET is $\phi_{B n}=0.87 \mathrm{~V}$. The channel doping concentration is $N_{d}=2 \times 10^{16} \mathrm{~cm}^{-3} \cdot$ (a) Determine the channel thickness, $a$, such that the internal pinchoff voltage is $V_{p o}=1.5 \mathrm{~V}$. (b) Using the results of part $(a)$, find the threshold voltage $V_{T} .(c)$ Calculate the minimum undepleted channel width for $V_{G S}=+0.4 \mathrm{~V}$ when $(i) V_{D S}=0,(i i) V_{D S}=1 \mathrm{~V}$, and (iii) $V_{D S}=4 \mathrm{~V}$.

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Problem 19

Two n-channel GaAs MESFETs have barrier heights of $\phi_{B n}=0.87 \mathrm{~V}$. (a) The channel doping concentration in device 1 is $N_{d}=5 \times 10^{15} \mathrm{~cm}^{-3}$ and the channel metallurgical thickness is $a=0.50 \mu \mathrm{m}$. Determine the threshold voltage. $(b)$ The channel doping concentration in device 2 is $N_{d}=3 \times 10^{16} \mathrm{~cm}^{-3} .$ Find the metallurgical channel thickness, $a$, such that the threshold voltage is the same as that of device 1 .

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Problem 20

Consider an n-channel GaAs MESFET at $T=300 \mathrm{~K}$ with $\phi_{B n}=0.85 \mathrm{~V}$ and $a=0.25 \mu \mathrm{m}$. Determine the channel doping concentration such that $V_{T}=0.5 \mathrm{~V}$.

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Problem 21

An n-channel silicon MESFET is fabricated using a gold contact. The $\mathrm{n}$ -channel doping is $N_{d}=10^{16} \mathrm{~cm}^{-3}$ and the temperature is $T=300 \mathrm{~K}$. When a gate voltage of $V_{G S}=0.35 \mathrm{~V}$ is applied with $V_{D S}=0$, the undepleted channel thickness is $0.075 \mu \mathrm{m} .$ (a) Determine the channel thickness dimension $a$ and the threshold voltage $V_{T}$. $(b)$ Determine the value of $V_{D S}($ sat $)$ for $V_{G S}=0.35 \mathrm{~V}$.

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Problem 22

The barrier height of an n-channel GaAs MESFET is $\phi_{B n}=0.90 \mathrm{~V}$. The metallurgical channel thickness is $a=0.65 \mu \mathrm{m}$ and the channel doping concentration is $N_{d}=2 \times 10^{16} \mathrm{~cm}^{-3} \cdot(a)$ Determine (i) $V_{b i}$
(ii) $V_{p o}$, and (iii) $V_{T}$. $(b)$ Find $V_{D S}$ (sat) for
(i) $V_{G S}=-1.0 \mathrm{~V},($ ii $) V_{G S}=-2.0 \mathrm{~V}$, and
(iii) $V_{G S}=-3.0 \mathrm{~V}$.

Chai Santi

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Problem 23

The parameters of an n-channel GaAs MESFET are $V_{T}=0.15 \mathrm{~V}, a=0.25 \mu \mathrm{m}$, $L=1.5 \mu \mathrm{m}, W=12 \mu \mathrm{m}$, and $\mu_{n}=6500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s} .(a)$ Determine the conduction
parameter $k_{n} .$ (b) Find $I_{D 1}$ (sat) for (i) $V_{G S}=0.25 \mathrm{~V}$ and $($ ii $) V_{G S}=0.45 \mathrm{~V}$. $(c)$ Find $V_{D S}$ (sat) for (i) $V_{G S}=0.25 \mathrm{~V}$ and $($ ii $) V_{G S}=0.45 \mathrm{~V}$.

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Problem 24

An n-channel GaAs MESFET has the same parameters as described in Problem 13,23 except for the channel width. (a) The maximum transconductance is to be $\mathrm{g}_{m s}=$ $1.25 \mathrm{~mA} / \mathrm{V}$ at $V_{G S}=0.45 \mathrm{~V}$. Determine the required channel width $W .(b)$ Using the results of part $(a)$, find $I_{D 1}$ (sat) for (i) $V_{G S}=0.25 \mathrm{~V}$ and $($ ii $) V_{G S}=0.45 \mathrm{~V}$.

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Problem 25

Use Equation (13.27) to plot $I_{D 1}$ versus $V_{D S}$ for a given value of $V_{G S}$. If $V_{D S}$ is allowed to become larger than $V_{D S}($ sat $)$, then $I_{D 1}$ decreases from a peak value that occurs at $V_{D S}($ sat $) .$ From these plots, determine $V_{D S}($ sat $)$ at several values of $V_{G S}$ Compare these values with those determined from Equation (13.12).

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Problem 26

This problem is to compare the JFET drain current as given by Equation (13.14) with that given by Equation (13.35). Choose device parameters such that the drain currents at $V_{G S}=0$ are the same from the two equations.

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Problem 27

A uniformly doped n-channel silicon pn JFET has the following parameters: $N_{a}=$ $10^{18} \mathrm{~cm}^{-3}, N_{d}=3 \times 10^{16} \mathrm{~cm}^{-3}, a=0.50 \mu \mathrm{m}$, and $\mu_{n}=850 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$. The maxi-
mum drain-to-source voltage is $V_{D S}=10 \mathrm{~V} .(a)$ For $V_{G S}=0$, the effective channel length $L^{\prime}$ is to be no less than 90 percent of the original channel length. Determine the minimum value of $\mathrm{L}$. $(b)$ Repeat part $(a)$ for $V_{G S}=-3 \mathrm{~V}$.

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Problem 28

If the change in the channel length, $\Delta L$, is assumed small, derive an approximate expression in terms of channel parameters for $\lambda$ given in Equation (13.53). (Note: The parameter $\lambda$ may not be a constant. However, justify using Equation (13.53) by plotting the expression for $\lambda$ over the range $1.5 V_{D S}($ sat $) \leq V_{D S} \leq 3.0 V_{D S}($ sat $) .$ Use typical parameter values.)

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Problem 29

As a first approximation, assume that the electric field in the channel of an $\mathrm{n}$ -channel silicon JFET is uniform through the channel. Also, assume that the drift velocity versus electric field for the electrons is given by the piecewise linear approximation given in Figure P13.29. Let:
$$
\begin{array}{rlrl}
N_{a} & =5 \times 10^{18} \mathrm{~cm}^{-3} & N_{d} & =4 \times 10^{16} \mathrm{~cm}^{-3} \\
L & =2 \mu \mathrm{m} & W & =30 \mu \mathrm{m} \\
a & =0.50 \mu \mathrm{m} & &
\end{array}
$$
(a) Determine $V_{D S}$ at which velocity saturation occurs. Let $V_{G S}=0 .(b)$ For $V_{G S}=0$, determine $h_{\text {sat. }}$ ( $c$ ) Calculate $I_{D 1}$ (sat) if velocity saturation occurs. ( $d$ ) If the electron mobility is a constant and equal to $\mu_{n}=1000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, calculate $I_{D 1}$ (sat) if velocity saturation did not occur.

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Problem 30

(a) Repeat Problem $13.29$ if $L=1 \mu \mathrm{m}$ and all other parameters remain the same.
(b) If velocity saturation occurs, does the relation $I_{D 1}($ sat $) \propto L^{-1}$ still apply? Explain.

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Problem 31

The channel length of an $\mathrm{n}$ -channel GaAs MESFET is $L=2 \mu \mathrm{m}$. Assume that the average horizontal electric field in the channel is $\mathrm{E}=5 \mathrm{kV} / \mathrm{cm}$. Calculate the transit time of an electron through the channel assuming $(a)$ a constant mobility of $\mu_{n}=8000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$ applies and $(b)$ velocity saturation applies.

Chai Santi

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Problem 32

The channel length of an n-channel silicon MESFET is $L=2 \mu \mathrm{m}$. Assume that the average horizontal electric field in the channel is $\mathrm{E}=10 \mathrm{kV} / \mathrm{cm} .$ Calculate the transit time of an electron through the channel assuming $(a)$ a constant mobility of $\mu_{n}=1000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$ applies and $(b)$ velocity saturation applies.

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Problem 33

Consider a one-sided silicon n-channel JFET at $T=300 \mathrm{~K}$, pinched off as shown in Figure P13.33. The source-to-gate and drain-to-gate reverse-biased currents are split geometrically as shown. Assume that the reverse-biased currents are dominated by the generation current. Assume the following parameters:
$$
\begin{array}{rlrl}
N_{a} & =5 \times 10^{18} \mathrm{~cm}^{-3} & N_{d} & =3 \times 10^{16} \mathrm{~cm}^{-3} \\
\tau_{0} & =5 \times 10^{-8} \mathrm{~s} & a & =0.30 \mu \mathrm{m} \\
W & =30 \mu \mathrm{m} & L & =2.4 \mu \mathrm{m}
\end{array}
$$
Calculate $I_{D G}$ for $(a) V_{D S}=0,(b) V_{D S}=1 \mathrm{~V}$, and $(c) V_{D S}=5 \mathrm{~V} .[$ Use Equation $(8.42)$ and consider the volume of the depletion region.]

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Problem 34

The source series resistance of a MESFET will reduce the value of transconductance, $g_{m s} .$ Assume the doping in the source region of a GaAs MESFET is $N_{d}=7 \times 10^{16} \mathrm{~cm}^{-3}$ and the dimensions are $a=0.3 \mu \mathrm{m}, L=1.5 \mu \mathrm{m}$, and
$W=5.0 \mu \mathrm{m}$. Let $\mu_{n}=4500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$ and $\phi_{B n}=0.89 \mathrm{~V}$. $(a)$ Determine the ideal
value of $g_{m s}$ for $V_{G S}=0 .(b)$ Determine the value of $r_{s}$ for which the value of $g_{m s}^{\prime}$ is 80 percent of the ideal value. ( c) Determine the maximum effective distance from the edge of the channel to the source terminal so that $r_{s}$ is no larger than the value determined in part $(b)$.

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Problem 35

Estimate the cutoff frequency of the MESFET in Problem $13.34$.

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Problem 36

An n-channel GaAs MESFET at $T=300 \mathrm{~K}$ has the following parameters:
$\phi_{B n}=0.90 \mathrm{~V}, N_{d}=4 \times 10^{16} \mathrm{~cm}^{-3}, \mu_{n}=7500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}, a=0.30 \mu \mathrm{m}, W=5 \mu \mathrm{m}$ and $L=1.2 \mu \mathrm{m}$. Calculate the cutoff frequency using $(a)$ the constant mobility model and $(b)$ the saturation velocity model.

Chai Santi

Numerade Educator

Problem 37

Consider a silicon n-channel pn JFET. The parameters are $a=0.40 \mu \mathrm{m}$, $\mu_{n}=1000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, and $N_{d}=2 \times 10^{16} \mathrm{~cm}^{-3}$. Determine the cutoff frequency for (a) $L=3 \mu \mathrm{m}$ and (b) $L=1.5 \mu \mathrm{m}$.

Chai Santi

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Problem 38

A silicon p-channel pn JFET has parameters $\mu_{p}=420 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}, a=0.40 \mu \mathrm{m}$, and $N_{a}=2 \times 10^{16} \mathrm{~cm}^{-3} .$ Determine the maximum channel length such that the cutoff frequency is $(a) f_{T}=5 \mathrm{GHz}$ and $(b) f_{r}=12 \mathrm{GHz}$

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Problem 39

Consider an $\mathrm{N}-\mathrm{Al}_{0.3} \mathrm{Ga}_{0.7}$ As-intrinsic GaAs abrupt heterojunction. Assume that the AlGaAs is doped to $N_{d}=3 \times 10^{18} \mathrm{~cm}^{-3}$ and has a thickness of $350 \AA$. Let $\phi_{B n}=0.89 \mathrm{~V}$, and assume that $\Delta E_{c}=0.24 \mathrm{eV} .(a)$ Calculate $V_{\text {off }}$ and $(b)$ calculate $n_{s}$ for $V_{g}=0$

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Problem 40

If the electrons in the channel of the JFET in Problem $13.39$ are traveling at a saturation velocity of $2 \times 10^{7} \mathrm{~cm} / \mathrm{s}$, determine $(a)$ the transconductance per unit width at $V_{g}=0$ and $(b)$ the saturation current per unit width at $V_{g}=0 .$ (Assume $\left.V_{0}=1 \mathrm{~V} .\right)$

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Problem 41

Consider an abrupt $\mathrm{N}-\mathrm{Al}_{0.3} \mathrm{Ga}_{0.7}$ As-intrinsic GaAs heterojunction. The N-AlGaAs is doped to $N_{d}=2 \times 10^{18} \mathrm{~cm}^{-3}$. The Schottky barrier height is $0.85 \mathrm{~V}$ and the heterojunction conduction-band edge discontinuity is $\Delta E_{c}=0.22 \mathrm{eV}$. Determine the thickness of the AlGaAs layer so that $V_{\text {off }}=-0.3 \mathrm{~V}$.

Chai Santi

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Problem 42

Design a one-sided silicon p-channel pn JFET such that $V_{p}=3.2 \mathrm{~V}, I_{D 1}($ sat $)=$ $1.2 \mathrm{~mA}$ at $V_{G S}=0$, and $f_{T}=10 \mathrm{GHz}$. Determine the required values of $L, W$, and $N_{a}$

Chai Santi

Numerade Educator

Problem 43

Design a one-sided GaAs n-channel MESFET with a barrier height of $\phi_{B n}=0.89 \mathrm{~V}$ such that $V_{T}=+0.12 \mathrm{~V}, I_{D S S}=2.0 \mu \mathrm{A}$ at $V_{G S}=0.45 \mathrm{~V}$, and $f_{T}=50 \mathrm{GHz}$. Assume $\mu_{n}=7500 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$.

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Problem 44

Design a pair of complementary n-channel and p-channel silicon JFETs so that $I_{D S s}=1 \mathrm{~mA}$ and $\left|V_{p}\right|=3.2 \mathrm{~V}$ for each device at $T=300 \mathrm{~K}$. If the devices are to operate for $0 \leq V_{D S} \leq 5 \mathrm{~V}$, comment on velocity saturation and channel length modulation effects in your design.

Chai Santi

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