Semiconductor Physics and Devices

Donald A. Neamen

Chapter 14

Optical Devices - all with Video Answers

Educators

Chapter Questions

Problem 1

Determine the maximum wavelength $\lambda$ of a light source that can generate electronhole pairs in (a) $\mathrm{Si},(b) \mathrm{Ge},(c) \mathrm{GaAs}$, and $(d) \operatorname{InP} .$

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Problem 2

(a) Two sources generate light at wavelengths of $\lambda=480 \mathrm{~nm}$ and $\lambda=725 \mathrm{~nm}$, respectively. What are the corresponding photon energies? ( $b$ ) Three sources generate light with photon energies of $E=0.87 \mathrm{eV}, E=1.32 \mathrm{eV}$, and $E=1.90 \mathrm{eV}$, respectively. What are the corresponding wavelengths?

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Problem 3

(a) A sample of GaAs is $1.2 \mu \mathrm{m}$ thick. The sample is illuminated with a light source that generates photons with energies of $h \nu=1.65 \mathrm{eV}$. Determine the $(i)$ absorption coefficient and (ii) fraction of energy that is absorbed in the material. ( $b$ ) Repeat part $(a)$ for a sample of GaAs that is $0.80 \mu \mathrm{m}$ thick and is illuminated with photons with energies of $h v=1.90 \mathrm{eV}$.

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Problem 4

A light source with $h v=1.3 \mathrm{eV}$ and at a power density of $10^{-2} \mathrm{~W} / \mathrm{cm}^{2}$ is incident on a thin slab of silicon. The excess minority carrier lifetime is $10^{-6} \mathrm{~s}$. Determine the electron-hole generation rate and the steady-state excess carrier concentration. Neglect surface effects.

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Problem 5

An n-type GaAs sample has a minority carrier lifetime of $\tau_{p}=2 \times 10^{-7} \mathrm{~s}$. Incident photons with energies $h \nu=1.65 \mathrm{eV}$ generate an excess carrier concentration of $\delta p=5 \times 10^{15} \mathrm{~cm}^{-3}$ at the surface of the semiconductor. (a) Determine the incident power required. ( $b$ ) At what distance in the semiconductor does the generation rate drop to 10 percent of that at the surface?

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Problem 6

Consider a silicon semiconductor that is illuminated with photons with energies $h \nu=1.40 \mathrm{eV} .(a)$ Determine the thickness of the material such that 90 percent of the energy is absorbed. (b) Determine the thickness of the material such that 30 percent of the energy is transmitted through the material.

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Problem 7

If the thickness of a GaAs semiconductor is $1 \mu \mathrm{m}$ and 50 percent of the incident monochromic photon energy is absorbed, determine the incident photon energy and wavelength.

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Problem 8

Consider monochromatic light at an intensity $I_{v 0}$ incident on the surface at $x=0$ of an $\mathrm{n}$ -type semiconductor that extends to $x=\infty .$ Assume the electric field is zero in the semiconductor and assume a surface recombination velocity, $s$. Taking into account the absorption coefficient, determine the steady-state excess hole concentration as a function of $x$.

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Problem 9

Monochromatic light with intensity $I_{\nu 0}$ is incident on a p-type semiconductor as shown in Figure $\mathrm{P} 14.9$. Assume the surface recombination velocity at $x=0$ is $s=\infty$ and assume the surface recombination velocity at $x=W$ is $s=s_{0}$. Derive the expression for the steady-state excess electron concentration as a function of $x$.

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Problem 10

A long silicon pn junction solar cell at $T=300 \mathrm{~K}$ has the following parameters:
$N_{a}=10^{16} \mathrm{~cm}^{-3}, N_{d}=10^{15} \mathrm{~cm}^{-3}, D_{n}=25 \mathrm{~cm}^{2} / \mathrm{s}, D_{p}=10 \mathrm{~cm}^{2} / \mathrm{s}, \tau_{n 0}=10^{-6} \mathrm{~s}$, and
$\tau_{p 0}=5 \times 10^{-7} \mathrm{~s}$. The cross-sectional area of the solar cell is $5 \mathrm{~cm}^{2}$. The entire junction is uniformly illuminated such that the generation rate of electron-hole pairs is $G_{L}=5 \times 10^{21} \mathrm{~cm}^{-3} \mathrm{~s}^{-1} \cdot(a)$ Calculate the short circuit photocurrent generated in the space charge region. (b) Using the results of part $(a)$, calculate the open-circuit voltage. ( $c$ ) Determine the ratio of $V_{a c}$ to $V_{b i}$

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Problem 11

A long silicon pn junction solar cell has the same parameters as described in Problem 14.10. The generated photocurrent in the cell is $I_{L}=120 \mathrm{~mA}$. Determine the $(a)$ open-circuit voltage, $(b)$ the voltage across the junction that will produce a total solar cell current of $I=100 \mathrm{~mA},(c)$ the maximum power output of the solar cell, and $(d)$ the external load resistance that will produce the maximum
power.

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Problem 12

Consider the solar cell described in Problem 14.10. (a) The generated photocurrent is $I_{L}=10 \mathrm{~mA}$. Determine (i) the open-circuit voltage and (ii) the maximum power output. (b) The solar cell now uses a solar concentrator such that the photocurrent increases by a factor of $10 .$ Determine the new values of $(i)$ open circuit voltage and ( $(i i)$ maximum power output. ( $c$ ) Determine the ratio of maximum power from part $(b)$ to that from part $(a)$.

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Problem 13

Consider an ideal long $\mathrm{n}^{+} \mathrm{p}$ junction GaAs solar cell at $T=300 \mathrm{~K}$ in which excess carriers are uniformly generated. The parameters of the diode are as follows:
$$
\begin{array}{ll}
N_{d}=10^{19} \mathrm{~cm}^{-3} & D_{n}=225 \mathrm{~cm}^{2} / \mathrm{s} \\
\tau_{n 0}=\tau_{p 0}=5 \times 10^{-8} \mathrm{~s} & D_{p}=7 \mathrm{~cm}^{2} / \mathrm{s}
\end{array}
$$
The generated photocurrent density is $J_{L}=30 \mathrm{~mA} / \mathrm{cm}^{2} .$ Plot the open-circuit voltage as a function of the acceptor doping concentration for $10^{15} \leq N_{a} \leq 10^{18} \mathrm{~cm}^{-3}$.

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Problem 14

A long silicon pn junction solar cell with an area of $2 \mathrm{~cm}^{2}$ has the following parameters:
$$
\begin{array}{ll}
N_{d}=10^{19} \mathrm{~cm}^{-3} & N_{a}=3 \times 10^{16} \mathrm{~cm}^{-3} \\
D_{p}=6 \mathrm{~cm}^{2} / \mathrm{s} & D_{n}=18 \mathrm{~cm}^{2} / \mathrm{s} \\
\tau_{p 0}=5 \times 10^{-7} \mathrm{~s} & \tau_{n 0}=5 \times 10^{-6} \mathrm{~s}
\end{array}
$$
Assume that excess carriers are uniformly generated in the solar cell and that $J_{L}=25 \mathrm{~mA} / \mathrm{cm}^{2}$. Let $T=300 \mathrm{~K}$. (a) Plot the $I-V$ characteristics of the diode,
(b) determine the maximum power output of the solar cell, and $(c)$ calculate the external load resistance that will produce the maximum power.

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Problem 15

A silicon solar cell at $T=300 \mathrm{~K}$ has a cross-sectional area of $6 \mathrm{~cm}^{2}$ and a reverse saturation current of $I_{S}=2 \times 10^{-9} \mathrm{~A}$. The induced short-circuit photocurrent is $I_{L}=180 \mathrm{~mA}$. Determine the $(a)$ open-circuit voltage, $(b)$ maximum power output, and $(c)$ load resistance that will produce the maximum output power. $(d)$ If the load resistance determined in part $(c)$ is increased by 50 percent, what is the new value of the maximum output power?

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Problem 16

Consider a silicon solar cell at $T=300 \mathrm{~K}$ with a reverse saturation current of $I_{S}=10^{-10} \mathrm{~A}$ and an induced short-circuit photocurrent of $I_{L}=100 \mathrm{~mA} .(a)$ Determine $V_{o c} .(b)$ Find $V_{m}, I_{m}$, and $P_{m} .(c)$ How many cells, operating at the maximum output power, must be connected in series to produce an output voltage of at least $10 \mathrm{~V} ?(d)$ How many of the $10 \mathrm{~V}$ cells in part $(c)$ must be connected in parallel to produce an output power of at least $5.2 \mathrm{~W} ?(e)$ Considering the results of part $(d)$, what must be the load resistance connected across the solar cell system to produce the maximum output power?

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Problem 17

Consider the pn junction solar cell with nonuniform absorption. Derive the expression for the excess minority carrier electron concentration for the short-circuit condition and for the case when the p region is very long and the $\mathrm{n}$ region is short.

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Problem 18

The absorption coefficient in amorphous silicon is approximately $10^{4} \mathrm{~cm}^{-1}$ at $h v=1.7 \mathrm{eV}$ and $10^{5} \mathrm{~cm}^{-1}$ at $h v=2.0 \mathrm{eV}$. Determine the amorphous silicon thickness for each case so that 90 percent of the photons are absorbed.

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Problem 19

Consider an $\mathrm{n}$ -type silicon photoconductor at $T=300 \mathrm{~K}$ doped at $N_{d}=5 \times 10^{15} \mathrm{~cm}^{-3}$. The cross-sectional area is $A=5 \times 10^{-4} \mathrm{~cm}^{2}$ and the length is $L=120 \mu \mathrm{m}$. The carrier parameters are $\mu_{n}=1200 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}, \mu_{p}=400 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$,
$\tau_{n 0}=5 \times 10^{-7} \mathrm{~s}$, and $\tau_{p 0}=10^{-7} \mathrm{~s}$. The photoconductor is uniformly illuminated such that the generation rate of electron-hole pairs is $G_{L}=10^{21} \mathrm{~cm}^{-3} \mathrm{~s}^{-1}$. For 3 volts applied to the photoconductor, determine ( $a$ ) the thermal equilibrium current, $(b)$ the steady-state excess carrier concentration, $(c)$ the photoconductivity,
( $d$ ) the steady-state photocurrent, and $(e)$ the photocurrent gain.

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Problem 20

Excess carriers are uniformly generated in a GaAs photoconductor at a rate of $G_{L}=10^{21} \mathrm{~cm}^{-3}-\mathrm{s}^{-1}$. The area is $A=10^{-4} \mathrm{~cm}^{2}$ and the length is $L=100 \mu \mathrm{m}$. The
other parameters are:
$$
\begin{array}{ll}
N_{d}=5 \times 10^{16} \mathrm{~cm}^{-3} & N_{a}=0 \\
\mu_{n}=8000 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s} & \mu_{p}=250 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s} \\
\tau_{n 0}=10^{-7} \mathrm{~s} & \tau_{p 0}=10^{-8} \mathrm{~s}
\end{array}
If a voltage of 5 volts is applied, calculate $(a)$ the steady-state excess carrier concentration, $(b)$ the photoconductivity, $(c)$ the steady-state photocurrent, and $(d)$ the photoconductor gain.
$$

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Problem 21

Consider an $\mathrm{n}$ -type silicon photoconductor that is $1 \mu \mathrm{m}$ thick, $50 \mu \mathrm{m}$ wide, and has an applied electric field in the longitudinal dimension of $50 \mathrm{~V} / \mathrm{cm}$. If the incident photon flux is $\Phi_{0}=10^{16} \mathrm{~cm}^{-2}-\mathrm{s}^{-1}$ and the absorption coefficient is $\alpha=5 \times 10^{4} \mathrm{~cm}^{-1}$, calculate the steady-state photocurrent if $\mu_{n}=1200 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$,
$\mu_{p}=450 \mathrm{~cm}^{2} / \mathrm{V}-\mathrm{s}$, and $\tau_{p 0}=2 \times 10^{-7} \mathrm{~s}$

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Problem 22

A long silicon pn junction photodiode has the following parameters at $T=300 \mathrm{~K}$ :
$N_{a}=10^{16} \mathrm{~cm}^{-3}, N_{d}=2 \times 10^{15} \mathrm{~cm}^{-3}, D_{p}=10 \mathrm{~cm}^{2} / \mathrm{s}, D_{n}=25 \mathrm{~cm}^{2} / \mathrm{s}, \tau_{p 0}=10^{-7} \mathrm{~s}$
and $\tau_{n 0}=5 \times 10^{-7} \mathrm{~s}$. The cross-sectional area of the diode is $A=10^{-3} \mathrm{~cm}^{2}$. Assume that a reverse-biased voltage of 5 volts is applied and that a uniform generation rate for electron-hole pairs of $G_{L}=10^{21} \mathrm{~cm}^{-3} \mathrm{~s}^{-1}$ exists throughout the entire photodiode. (a) Determine the prompt component of photocurrent. (b) Find the steady-state excess carrier concentrations in the $\mathrm{p}$ and $\mathrm{n}$ regions far from the junction. ( $c$ ) Determine the total steady-state photocurrent.

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Problem 23

Starting with the ambipolar transport equation for minority carrier holes, derive Equation (14.41) using the geometry shown in Figure $14.17$.

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Problem 24

Three silicon PIN photodiodes A, B, and C, at $T=300 \mathrm{~K}$ have intrinsic region widths of 2,10 , and $80 \mu \mathrm{m}$, respectively. A photon flux of $\Phi_{0}=5 \times 10^{17} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}$ is incident on the surface of each diode as shown in Figure 14.19. ( $a$ ) For an absorption coefficient of $\alpha=10^{4} \mathrm{~cm}^{-1}$, calculate the prompt photocurrent density in each diode. $(b)$ Repeat part $(a)$ for an absorption coefficient of $\alpha=5 \times 10^{2} \mathrm{~cm}^{-1}$

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Problem 25

Consider a silicon PIN photodiode at $T=300 \mathrm{~K}$ with the geometry shown in Figure 14.19. The intrinsic region width is $100 \mu \mathrm{m}$. Assume that a reverse-biased voltage is applied such that the intrinsic region is completely depleted. The incident photon power is $I_{v 0}=0.080 \mathrm{~W} / \mathrm{cm}^{2}$, the absorption coefficient is $\alpha=10^{3} \mathrm{~cm}^{-1}$, and the photon energy is $1.5 \mathrm{eV}$. Neglect any absorption in the $\mathrm{p}^{+}$ top layer of the photodiode. (a) Determine the steady-state electron-hole generation rate, $G_{L}$, versus distance in the intrinsic region. ( $b$ ) Determine the steady-state photocurrent density.

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Problem 26

A silicon PIN photodiode at $T=300 \mathrm{~K}$ has the geometry shown in Figure $14.19 .$ The intrinsic region width is $20 \mu \mathrm{m}$ and is fully depleted. (a) The electron-hole pair generation rate in the intrinsic region is $G_{L}=10^{21} \mathrm{~cm}^{-3} \mathrm{~s}^{-1}$ and is uniform throughout the intrinsic region. Calculate the steady-state photocurrent density for this condition. ( $b$ ) The generation rate of electron-hole pairs is $G_{L}=10^{21} \mathrm{~cm}^{-3} \mathrm{~s}^{-1}$ at $x=0$ and the absorption coefficient is $\alpha=10^{3} \mathrm{~cm}^{-1}$. Determine the steady-state photocurrent density for this situation.

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Problem 27

Consider a silicon PIN photodiode exposed to sunlight. Calculate the intrinsic region width so that at least 90 percent of all photons with wavelengths $\lambda \leq 1 \mu \mathrm{m}$ are absorbed in the intrinsic region. Neglect any absorption in the $\mathrm{p}^{+}$ or $\mathrm{n}^{+}$ regions.

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Problem 28

Consider the $\mathrm{Al}_{x} \mathrm{Ga}_{1-x}$ As system. Determine the range of the direct bandgap energies possible and the corresponding range of wavelengths.

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Problem 29

Consider the $\mathrm{GaAs}_{1-x} \mathrm{P}_{x}$ system. $(a)$ For a mole fraction $x=0.2$, determine the
(i) bandgap energy and (ii) corresponding photon wavelength. ( $b$ ) Repeat part ( $a$ ) for a mole fraction $x=0.32$.

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Problem 30

Using Figure $14.23$, determine the mole fraction $x$ in $\mathrm{Al}_{x} \mathrm{Ga}_{1-x}$ As such that the material would emit light at a wavelength of $\lambda=0.670 \mu \mathrm{m} .$ What is the corresponding bandgap energy?

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Problem 31

Repeat Problem $14.30$ for the $\mathrm{GaAs}_{1-{ }_{x}} \mathrm{P}_{x}$ system.

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Problem 32

Consider a pn junction GaAs LED. Assume that photons are generated uniformly in all directions in a plane perpendicular to the junction at a distance of $0.50 \mu \mathrm{m}$ from the surface. ( $a$ ) Taking into account total internal reflection, calculate the fraction of photons that have the potential of being emitted from the semiconductor. ( $b$ ) Using the results of part $(a)$ and including Fresnel loss, determine the fraction of generated photons that will be emitted from the semiconductor into air (neglect absorption losses).

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Problem 33

In a pn junction LED, consider a point source in the semiconductor at the junction and assume that photons are emitted uniformly in all directions. Show that (neglecting photon absorption) the external quantum efficiency of the LED is given by
$$
\eta_{\mathrm{ext}}=\frac{2 \bar{n}_{1} \bar{n}_{2}}{\left(\bar{n}_{1}+\bar{n}_{2}\right)^{2}}\left(1-\cos \theta_{c}\right)
$$
where $\bar{n}_{1}$ and $\bar{n}_{2}$ are the index of refraction parameters for the air and semiconductor, respectively, and $\theta_{c}$ is the critical angle.

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Problem 34

Consider an optical cavity. If $N \gg 1$, show that the wavelength separation between two adjacent resonant modes is $\Delta \lambda=\lambda^{2} / 2 L$.

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Problem 35

If the photon output of a laser diode is equal to the bandgap energy, find the wavelength separation between adjacent resonant modes in a GaAs laser with $L=75 \mu \mathrm{m} .$

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