Semiconductor Physics and Devices

Donald A. Neamen

Chapter 9

Metal–Semiconductor and Semiconductor Heterojunctions - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider a contact between $\mathrm{Al}$ and $\mathrm{n}$ Si doped at $N_{d}=10^{16} \mathrm{~cm}^{-3} . T=300 \mathrm{~K}$.
(a) Draw the energy-band diagrams of the two materials before the junction is formed
(b) Draw the ideal energy band at zero bias after the junction is formed. ( $c$ ) Calculate $\phi_{B 0}, x_{d}$, and $\mathrm{E}_{\max }$ for part $(b) .(d)$ Repeat parts $(b)$ and $(c)$ using the data in Figure $9.5 .$

Chai Santi

Chai Santi

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Problem 2

(a) A Schottky barrier diode formed on n-type silicon has a doping concentration of $N_{d}=5 \times 10^{15} \mathrm{~cm}^{-3}$ and a barrier height of $\phi_{B 0}=0.65 \mathrm{~V}$. Determine the builtin potential barrier $V_{b i}$. $(b)$ If the doping concentration changes to $N_{d}=10^{16} \mathrm{~cm}^{-3}$, determine the values of $\phi_{B 0}$ and $V_{b i} .$ Do these values increase, decrease, or remain the same?
(c) Repeat part ( $b$ ) if the doping concentration is $N_{d}=10^{15} \mathrm{~cm}^{-3}$.

Chai Santi

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Problem 3

Gold is deposited on n-type silicon forming an ideal rectifying junction. The doping concentration is $N_{d}=10^{16} \mathrm{~cm}^{-3}$. Assume $T=300 \mathrm{~K}$. Determine the theoretical values of $(a) \phi_{B 0},(b) V_{b i}$, and $(c) x_{n}$ and $\left|\mathrm{E}_{\max }\right|$ at $(i) V_{R}=1 \mathrm{~V}$ and (ii) $V_{R}=5 \mathrm{~V}$.

Chai Santi

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Problem 4

A Schottky diode is formed by depositing gold on $\mathrm{n}$ -type GaAs that is doped at a concentration of $N_{d}=5 \times 10^{15} \mathrm{~cm}^{-3}$. For $T=300 \mathrm{~K}$, determine the theoretical values of
(a) $\phi_{B 0},(b) \phi_{n},(c) V_{b i}$, and $(d) x_{n}$ and $\left|\mathrm{E}_{\max }\right|$ at
(i) $V_{R}=1 \mathrm{~V}$ and
(ii) $V_{R}=5 \mathrm{~V}$.

Chai Santi

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Problem 5

Repeat Problem 9.4, parts (b) through $(d)$, if the experimentally determined barrier height is found to be $\phi_{B n}=0.88 \mathrm{~V}$.

Chai Santi

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Problem 6

(a) A Pt-n-type silicon junction with $N_{d}=10^{15} \mathrm{~cm}^{-3}$ has a cross-sectional area of $A=10^{-4} \mathrm{~cm}^{2}$. Let $T=300 \mathrm{~K}$. Using the data shown in Figure $9.5$, determine the junction capacitance at (i) $V_{R}=1 \mathrm{~V}$ and (ii) $V_{R}=5 \mathrm{~V} .(b)$ Repeat part $(a)$ for a doping concentration of $N_{d}=10^{16} \mathrm{~cm}^{-3}$.

Chai Santi

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Problem 7

A Schottky diode with $\mathrm{n}$ -type GaAs at $T=300 \mathrm{~K}$ yields the $1 / C^{\prime 2}$ versus $V_{R}$ plot shown in Figure $\mathrm{P} 9.7$, where $C^{\prime}$ is the capacitance per $\mathrm{cm}^{2}$. Determine $(a) V_{b j}(b) N_{d}$, (c) $\phi_{n}$, and $(d) \phi_{B 0}$.

Chai Santi

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Problem 8

Consider a W-n-type silicon Schottky barrier at $T=300 \mathrm{~K}$ with $N_{d}=5 \times 10^{15} \mathrm{~cm}^{-3}$. Use the data in Figure $9.5$ to determine the barrier height. ( $a$ ) Determine $V_{b i}, x_{n}$, and $\left|\mathrm{E}_{\max }\right|$ for $(i) V_{R}=1 \mathrm{~V}$ and $(i i) V_{R}=5 \mathrm{~V} .(b)$ Using the values of $\left|\mathrm{E}_{\max }\right|$ from part $(a)$, determine the Schottky barrier lowering parameters $\Delta \phi$ and $x_{m}$.

Chai Santi

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Problem 9

Starting with Equation (9.12), derive Equations (9.14) and (9.15).

Chai Santi

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Problem 10

A Au-n-GaAs Schottky diode at $T=300 \mathrm{~K}$ is doped at a concentration of $N_{d}=10^{16} \mathrm{~cm}^{-3}$. Use the data in Figure $9.5$ to determine the barrier height. Then determine (a) $V_{b i}, x_{n}$, and $\left|\mathrm{E}_{\max }\right|$ for zero bias. (b) Determine the reverse-biased voltage at which the Schottky barrier lowering, $\Delta \phi$, will be 5 percent of the barrier height.

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Problem 11

Consider n-type silicon doped at $N_{d}=10^{16} \mathrm{~cm}^{-3}$ with a gold contact to form a Schottky diode. Investigate the effect of Schottky barrier lowering. (a) Plot the Schottky barrier lowering $\Delta \phi$ versus reverse-biased voltage over the range $0 \leq V_{R} \leq 50 \mathrm{~V} .(b)$ Plot the ratio $J_{s T}\left(V_{R}\right) / J_{s T}\left(V_{R}=0\right)$ over the same range of reversebiased voltage.

Chai Santi

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Problem 12

The energy-band diagram of a Schottky diode is shown in Figure $9.6 .$ Assume the following parameters:
$$\begin{array}{rrrl}
\phi_{m}=5.2 \mathrm{~V} & \phi_{n}=0.10 \mathrm{~V} & \phi_{0} & =0.60 \mathrm{~V} \\
E_{g}=1.43 \mathrm{eV} & \delta=25& \epsilon_{i} & =\epsilon_{0} \\
\epsilon_{s}=(13.1) \epsilon_{0} & \chi=4.07 \mathrm{~V} & N_{d} & =10^{16} \mathrm{~cm}^{-3} \\
& & D_{i t} & =10^{13} \mathrm{eV}^{-1} \mathrm{~cm}^{-2}
\end{array}$$
(a) Determine the theoretical barrier height $\phi_{B 0}$ without interface states. $(b)$ Determine the barrier height with interface states. $(c)$ Repeat parts $(a)$ and $(b)$ if $\phi_{m}=4.5 \mathrm{~V}$.

Chai Santi

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Problem 13

A Schottky barrier diode contains interface states and an interfacial layer. Assume the following parameters:
$$\begin{array}{rrrl}
\phi_{m}=4.75 \mathrm{~V} & \phi_{n}=0.164 \mathrm{~V} & \phi_{0} & =0.230 \mathrm{~V} \\
E_{g}=1.12 \mathrm{eV} & \delta=20 A & \epsilon_{i} & =\epsilon_{0} \\
\epsilon_{s}=(11.7) \epsilon_{0} & \chi=4.01 \mathrm{~V} & N_{d} & =5 \times 10^{16} \mathrm{~cm}^{-3} \\
& & \phi_{B 0} & =0.60 \mathrm{~V}
\end{array}$$
Determine the interface state density, $D_{i t}$, in units of $\mathrm{eV}^{-1} \mathrm{~cm}^{-2}$.

Chai Santi

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Problem 14

A Schottky diode at $T=300 \mathrm{~K}$ is formed with $\mathrm{Pt}$ on $\mathrm{n}$ -type silicon with a doping concentration of $N_{d}=5 \times 10^{15} \mathrm{~cm}^{-3}$. The barrier height is found to be $\phi_{B n}=0.89 \mathrm{~V}$. Determine $($ a $) \phi_{n},(b) V_{b i},(c) J_{s T}$, and $(d) V_{a}$ such that $J_{n}=5 \mathrm{~A} / \mathrm{cm}^{2}$. (Neglect the barrier lowering effect.)

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Problem 15

(a) Consider a Schottky diode at $T=300 \mathrm{~K}$ that is formed with tungsten on $\mathrm{n}$ -type silicon. Use Figure $9.5$ to determine the barrier height. Assume a doping concentration of $N_{d}=10^{16} \mathrm{~cm}^{-3}$ and assume a cross-sectional area $A=10^{-4} \mathrm{~cm}^{2}$. Determine the forward-bias voltage required to induce a current of $(i) 10 \mu \mathrm{A},(i i) 100 \mu \mathrm{A}$, and
(iii) $1 \mathrm{~mA}$. $(b)$ Repeat part $(a)$ for a temperature of $T=350 \mathrm{~K}$. (Neglect the barrier lowering effect.)

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Problem 16

An Au-n-GaAs Schottky diode at $T=300 \mathrm{~K}$ has a doping concentration of $N_{d}=10^{16} \mathrm{~cm}^{-3} \cdot$ (a) Using Figure $9.5$, determine the barrier height. (b) Calculate the reverse-biased saturation current $J_{s T} .(c)$ Determine the forward-bias voltage required to induce a current density of $J_{n}=10 \mathrm{~A} / \mathrm{cm}^{2} .(d)$ What is the change in forwardbias voltage necessary to double the current density? (Neglect the Schottky barrier lowering.)

Chai Santi

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Problem 17

(a) Consider an Au-n-type GaAs Schottky diode with a cross-sectional area of $10^{-4} \mathrm{~cm}^{2}$. Plot the forward-bias current-voltage characteristics over a voltage range of $0 \leq V_{D} \leq 0.5 \mathrm{~V}$. Plot the current on a log scale. $(b)$ Repeat part $(a)$ for an Au-n-type silicon Schottky diode. $(c)$ What conclusions can be drawn from these results?

Chai Santi

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Problem 18

A Schottky diode at $T=300 \mathrm{~K}$ is formed between tungsten and $\mathrm{n}$ -type silicon doped at $N_{d}=10^{16} \mathrm{~cm}^{-3}$. The cross-sectional area is $A=10^{-4} \mathrm{~cm}^{2}$. Determine the reverse-biased saturation current at $(a) V_{R}=2 \mathrm{~V}$ and $(b) V_{R}=4 \mathrm{~V}$. (Take into account the Schottky barrier lowering.)

Chai Santi

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Problem 19

Starting with the basic current equation given by Equation (9.18), derive the relation given by Equation (9.23).

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Problem 20

The reverse-saturation current densities in a pn junction diode and a Schottky diode are $10^{-11} \mathrm{~A} / \mathrm{cm}^{2}$ and $6 \times 10^{-8} \mathrm{~A} / \mathrm{cm}^{2}$, respectively, at $T=300 \mathrm{~K}$. The cross-sectional area of the Schottky diode is $A=10^{-4} \mathrm{~cm}^{2}$. The current in each diode is $0.80 \mathrm{~mA}$. The difference in forward-bias voltages between the two diodes is $0.285 \mathrm{~V}$. Determine (a) the voltage applied to each diode and $(b)$ the cross-sectional area of the pn junction diode.

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Problem 21

A pn junction diode and a Schottky diode each have cross-sectional areas of $A=8 \times 10^{-4} \mathrm{~cm}^{2}$. The reverse saturation current densities at $T=300 \mathrm{~K}$ for the pn junction diode and Schottky diode are $8 \times 10^{-13} \mathrm{~A} / \mathrm{cm}^{2}$ and $6 \times 10^{-9} \mathrm{~A} / \mathrm{cm}^{2}$, respectively. Determine the required forward-bias voltage in each diode to yields currents of (a) $150 \mu \mathrm{A},(b) 700 \mu \mathrm{A}$, and (c) $1.2 \mathrm{~mA}$.

Kratika Bhadauria

Kratika Bhadauria

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Problem 22

(a) The two diodes described in Problem $9.21$ are connected in series and are driven by a constant current source of $0.80 \mathrm{~mA}$. Determine $(i)$ the current in each diode and (ii) the voltage across each diode. ( $b$ ) Repeat part $(a)$ for the case when the diodes are connected in parallel.

Chai Santi

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Problem 23

A Schottky diode and a pn junction diode have cross-sectional areas of $A=7 \times 10^{-4} \mathrm{~cm}^{2}$. The reverse-saturation current densities at $T=300 \mathrm{~K}$ of the Schottky diode and pn junction are $4 \times 10^{-8} \mathrm{~A} / \mathrm{cm}^{2}$ and $3 \times 10^{-12} \mathrm{~A} / \mathrm{cm}^{2}$, respectively. A forward-bias current of $0.8 \mathrm{~mA}$ is required in each diode. $(a)$ Determine the forward-bias voltage required across each diode. $(b)$ If the voltage from part $(a)$ is maintained across each diode, determine the current in each diode if the temperature is increased to $400 \mathrm{~K}$. (Take into account the temperature dependence of the reversesaturation currents. Assume $E_{g}=1.12 \mathrm{eV}$ for the pn junction diode and $\phi_{B 0}=0.82 \mathrm{~V}$ for the Schottky diode.)

Chai Santi

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Problem 24

Compare the current-voltage characteristics of a Schottky barrier diode and a pn junction diode. Use the results of Example $9.5$ and assume diode areas of $5 \times 10^{-4} \mathrm{~cm}^{2}$. Plot the current-voltage characteristics on a linear scale over a current range of $0 \leq I_{D} \leq 10 \mathrm{~mA}$

Chai Santi

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Problem 25

The contact resistance of an ohmic contact is $R_{c}=10^{-4} \Omega-\mathrm{cm}^{2}$. Determine the junction resistance if the cross-sectional area is ( $a$ ) $10^{-3} \mathrm{~cm}^{2}$, (b) $10^{-4} \mathrm{~cm}^{2}$, and $(c) 10^{-5} \mathrm{~cm}^{2}$.

Chai Santi

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Problem 26

(a) The contact resistance of an ohmic contact is $R_{c}=5 \times 10^{-5} \Omega-\mathrm{cm}^{2}$. The crosssectional area of the junction is $10^{-5} \mathrm{~cm}^{2}$. Determine the voltage across the junction if the current is $(i) I=1 \mathrm{~mA}$ and $(i i) I=100 \mu \mathrm{A} .(b)$ Repeat part $(a)$ if the crosssectional area is $10^{-6} \mathrm{~cm}^{2}$.

Chai Santi

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Problem 27

An ohmic contact between a metal and silicon may be formed that has a very low barrier height. (a) Determine the value of $\phi_{B n}$ that will produce a contact resistance of $R_{c}=5 \times 10^{-5} \Omega-\mathrm{cm}^{2}$ at $T=300 \mathrm{~K} .(b)$ Repeat part $(a)$ for a contact resistance of $R_{c}=5 \times 10^{-6} \Omega-\mathrm{cm}^{2}$.

Chai Santi

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Problem 28

A metal, with a work function $\phi_{m}=4.2 \mathrm{~V}$, is deposited on an $\mathrm{n}$ -type silicon semiconductor with $\chi_{s}=4.0 \mathrm{~V}$ and $E_{g}=1.12 \mathrm{eV}$. Assume no interface states exist at the junction. Let $T=300 \mathrm{~K}$. (a) Sketch the energy-band diagram for zero bias for the case when no space charge region exists at the junction. (b) Determine $N_{d}$ so that the condition in part $(a)$ is satisfied. $(c)$ What is the potential barrier height seen by electrons in the metal moving into the semiconductor?

Chai Santi

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Problem 29

Consider the energy-band diagram of a silicon Schottky junction under zero bias shown in Figure P9.29. Let $\phi_{B 0}=0.7 \mathrm{~V}$ and $T=300 \mathrm{~K}$. Determine the doping required so that $x_{d}=50 \AA$ at the point where the potential is $\phi_{B 0} / 2$ below the peak value. (Neglect the barrier lowering effect.)

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Problem 30

A metal-semiconductor junction is formed between a metal with a work function of $4.3 \mathrm{eV}$ and $\mathrm{p}$ -type silicon with an electron affinity of $4.0 \mathrm{eV}$. The acceptor doping concentration in the silicon is $N_{a}=5 \times 10^{16} \mathrm{~cm}^{-3}$. Assume $T=300 \mathrm{~K}$. $(a)$ Sketch the thermal equilibrium energy-band diagram. (b) Determine the height of the Schottky barrier. (c) Sketch the energy-band diagram with an applied reverse-biased voltage of $V_{R}=3 \mathrm{~V} .(d)$ Sketch the energy-band diagram with an applied forward-bias voltage of $V_{a}=0.25 \mathrm{~V}$

Chai Santi

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Problem 31

(a) Consider a metal-semiconductor junction formed between a metal with a work function of $4.65 \mathrm{eV}$ and $\mathrm{Ge}$ with an electron affinity of $4.13 \mathrm{eV}$. The doping concentration in the Ge material is $N_{d}=6 \times 10^{13} \mathrm{~cm}^{-3}$ and $N_{a}=3 \times 10^{13} \mathrm{~cm}^{-3}$. Assume $T=300 \mathrm{~K}$. Sketch the zero bias energy-band diagram and determine the Schottky barrier height. $(b)$ Repeat part $(a)$ if the metal work function is $4.35 \mathrm{eV}$.

Chai Santi

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Problem 32

Sketch the energy-band diagrams of an abrupt $\mathrm{Al}_{0.3} \mathrm{Ga}_{0.7}$ As-GaAs heterojunction for:
(a) N $^{+}$ -AlGaAs, intrinsic GaAs, (b) N $^{+}-$ AlGaAs, p-GaAs, and $\left(\right.$ c) $\mathrm{P}^{+}-\mathrm{AlGaAs}, \mathrm{n}^{+}-$ GaAs. Assume $E_{g}=1.85 \mathrm{eV}$ for $\mathrm{Al}_{03} \mathrm{Ga}_{0.7}$ As and assume $\Delta E_{c}=\frac{2}{3} \Delta E_{\mathrm{g}}$

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Problem 33

Repeat Problem $9.32$ assuming the ideal electron affinity rule. Determine $\Delta E_{c}$ and $\Delta E_{v}$.

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Problem 34

Starting with Poisson's equation, derive Equation $(9.48)$ for an abrupt heterojunction.

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Problem 35

(a) Derive an expression for $d V_{a} / d T$ as a function of current density in a Schottky diode. Assume the minority carrier current is negligible. (b) Compare $d V_{a} / d T$ for a GaAs Schottky diode to that for a Si Schottky diode. (c) Compare $d V_{a} / d T$ for a Si Schottky diode to that for a Si pn junction diode.

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Problem 36

The $\left(1 / C_{j}\right)^{2}$ versus $V_{R}$ data are measured for two Schottky diodes with equal areas. One diode is fabricated with $1 \Omega-\mathrm{cm}$ silicon and the other diode with $5 \Omega$ -cm silicon. The plots intersect the voltage axis as $V_{R}=-0.5 \mathrm{~V}$ for diode $\mathrm{A}$ and at $V_{R}=-1.0 \mathrm{~V}$ for diode $\mathrm{B}$. The slope of the plot for diode $\mathrm{A}$ is $1.5 \times 10^{18}\left(\mathrm{~F}^{2}-\mathrm{V}\right)^{-1}$ and that for diode B is $1.5 \times 10^{17}\left(\mathrm{~F}^{2}-\mathrm{V}\right)^{-1}$. Determine which diode has the higher metal work function and which diode has the lower resistivity silicon.

Chai Santi

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Problem 37

Both Schottky barrier diodes and ohmic contacts are to be fabricated by depositing a particular metal on a silicon integrated circuit. The work function of the metal is 4.5 V. Considering the ideal metal-semiconductor contact, determine the allowable range of doping concentrations for each type of contact. Consider both p- and n-type silicon regions.

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Problem 38

Consider an n-GaAs-p-AlGaAs heterojunction in which the bandgap offsets are $\Delta E_{c}=0.3 \mathrm{eV}$ and $\Delta E_{v}=0.15 \mathrm{eV}$. Discuss the difference in the expected electron and hole currents when the junction is forward biased.

Chai Santi

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