4.5 Summary
203
a(InAs) = 6.0590 Å
a(InP) = 5.8696 Å
where 1 Å = 10
-10 m, the quaternary lattice spacing becomes
a(x, y) = 0.1894y − 0.4184x + 0.0130x y + 5.8696 Å
(b) For quaternary alloys that are lattice-matched to InP, the relation between x and
y can be determined by letting a(x, y) = a(InP). Show that because 0 ≤ x ≤ 0.47, the
resulting expression can be approximated by y 2.20x.
(c) A simple empirical relation that gives the bandgap energy in eV in terms of x
and y is
E g (x, y) = 1.35 + 0.668x − 1.17y + 0.758x
2
+ 0.18y
2
− 0.069x y − 0.322x
2 y + 0.33x y
2
.
Find the bandgap energy and the peak emission wavelength of In 0.74 Ga 0.26 As 0.56 P 0.44 .
[Answer: (c) E g = 0.956 eV; λ(μm) = 1.240/ E g (eV) = 1.297 μm.]
4.5 Using the expression E = hc/λ, show why the FWHM power spectral width
of LEDs becomes wider at longer wavelengths. [Answer: Differentiating the
expression for E yields λ =
λ
2
hc
E. For the same energy difference E, the
spectral width λ is proportional to the wavelength squared. Thus, for example,
λ 1550
λ 1310
=
1550
1310
2 = 1.40.]
4.6 A double-heterojunction InGaAsP LED emitting at a peak wavelength of
1310 nm has radiative and nonradiative recombination times of 25 and 90 ns,
respectively. The drive current is 35 mA.
(a) Find the internal quantum efficiency and the internal power level.
(b) If the refractive index of the light source material is n = 3.5, find the power
emitted from the device.
[Answer: (a) η int = 0.783, P int = 26 mW; (b) 0.37 mW].
4.7 Assume the injected minority carrier lifetime of an LED is 5 ns and that the
device has an optical output of 0.30 mW when a constant dc drive current
is applied. Using Eq. (4.18), plot the optical output power when the LED is
modulated at frequencies ranging from 20 to 100 MHz. Note what happens to
the LED output power at higher modulation frequencies.
4.8 Consider an LED having a minority carrier lifetime of 5 ns. Find the 3-dB
optical bandwidth and the 3-dB electrical bandwidth. [Answer: The 3-dB
optical bandwidth = 9.5 MHz; the 3-dB electrical bandwidth = 6.7 MHz.]
4.9 (a) A GaAlAs laser diode has a 500-μm cavity length, which has an effective
absorption coefficient of 10 cm
−1 . For uncoated facets the reflectivities are
203
a(InAs) = 6.0590 Å
a(InP) = 5.8696 Å
where 1 Å = 10
-10 m, the quaternary lattice spacing becomes
a(x, y) = 0.1894y − 0.4184x + 0.0130x y + 5.8696 Å
(b) For quaternary alloys that are lattice-matched to InP, the relation between x and
y can be determined by letting a(x, y) = a(InP). Show that because 0 ≤ x ≤ 0.47, the
resulting expression can be approximated by y 2.20x.
(c) A simple empirical relation that gives the bandgap energy in eV in terms of x
and y is
E g (x, y) = 1.35 + 0.668x − 1.17y + 0.758x
2
+ 0.18y
2
− 0.069x y − 0.322x
2 y + 0.33x y
2
.
Find the bandgap energy and the peak emission wavelength of In 0.74 Ga 0.26 As 0.56 P 0.44 .
[Answer: (c) E g = 0.956 eV; λ(μm) = 1.240/ E g (eV) = 1.297 μm.]
4.5 Using the expression E = hc/λ, show why the FWHM power spectral width
of LEDs becomes wider at longer wavelengths. [Answer: Differentiating the
expression for E yields λ =
λ
2
hc
E. For the same energy difference E, the
spectral width λ is proportional to the wavelength squared. Thus, for example,
λ 1550
λ 1310
=
1550
1310
2 = 1.40.]
4.6 A double-heterojunction InGaAsP LED emitting at a peak wavelength of
1310 nm has radiative and nonradiative recombination times of 25 and 90 ns,
respectively. The drive current is 35 mA.
(a) Find the internal quantum efficiency and the internal power level.
(b) If the refractive index of the light source material is n = 3.5, find the power
emitted from the device.
[Answer: (a) η int = 0.783, P int = 26 mW; (b) 0.37 mW].
4.7 Assume the injected minority carrier lifetime of an LED is 5 ns and that the
device has an optical output of 0.30 mW when a constant dc drive current
is applied. Using Eq. (4.18), plot the optical output power when the LED is
modulated at frequencies ranging from 20 to 100 MHz. Note what happens to
the LED output power at higher modulation frequencies.
4.8 Consider an LED having a minority carrier lifetime of 5 ns. Find the 3-dB
optical bandwidth and the 3-dB electrical bandwidth. [Answer: The 3-dB
optical bandwidth = 9.5 MHz; the 3-dB electrical bandwidth = 6.7 MHz.]
4.9 (a) A GaAlAs laser diode has a 500-μm cavity length, which has an effective
absorption coefficient of 10 cm
−1 . For uncoated facets the reflectivities are