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4 Light Sources for Fiber Links
For high-speed long-distance communications one needs single-mode lasers,
which must contain only a single longitudinal mode and a single transverse mode.
Consequently, the spectral width of the optical emission pattern is very narrow.
Popular single-mode lasers include the distributed-feedback (DFB) laser and the
vertical-cavity surface-emitting laser (VCSEL).
Problems
4.1 Measurements show that the bandgap energy E g for GaAs varies with temperature according to the empirical formula
E g (T ) ≈ 1.55 − 4.3 × 10
−4 T
where E g is given in electron volts (eV). Using this expression, show that the
temperature dependence of the intrinsic electron concentration n i is
n i = 5 × 10
15 T
3/2 e
−8991/T
4.2 Repeat the steps given in Example 4.2 for a p-type semiconductor. In particular,
show that when the net acceptor concentration is much greater than n i , then p p
= N A and n p = n
2
i /N A .
4.3 An engineer has two Ga 1−x Al x As LEDs: one has a bandgap energy of 1.540 eV
and the other has x = 0.015.
(a) Find the aluminum mole fraction x and the emission wavelength for the first LED.
(b) Find the bandgap energy and the emission
wavelength of the other LED.
[Answer:
(a) x = 0.090, λ = 805 nm; (b) E g = 1.620 eV, λ = 766 nm.]
4.4 The lattice spacing of In 1−x Ga x As y P 1−y has been shown to obey Vegard’s law.
This states that for quaternary alloys of the form A 1−x B x C y D 1−y , where A and B
are group III elements (e.g., Al, In, and Ga) and C and D are group V elements
(e.g., As, P, and Sb), the lattice spacing a(x, y) of the quaternary alloy can be
approximated by
a(x, y) = x ya(BC) + x(1 − y)a(BD)
+ (1 − x)ya(AC) + (1 − x) (1 − y)a(AD)
where the a(IJ) parameters are the lattice spacings of the binary compounds IJ.
(a) Show that for In 1−x Ga x As y P 1−y with
a(GaAs) = 5.6536 Å
a(GaP) = 5.4512 Å
4 Light Sources for Fiber Links
For high-speed long-distance communications one needs single-mode lasers,
which must contain only a single longitudinal mode and a single transverse mode.
Consequently, the spectral width of the optical emission pattern is very narrow.
Popular single-mode lasers include the distributed-feedback (DFB) laser and the
vertical-cavity surface-emitting laser (VCSEL).
Problems
4.1 Measurements show that the bandgap energy E g for GaAs varies with temperature according to the empirical formula
E g (T ) ≈ 1.55 − 4.3 × 10
−4 T
where E g is given in electron volts (eV). Using this expression, show that the
temperature dependence of the intrinsic electron concentration n i is
n i = 5 × 10
15 T
3/2 e
−8991/T
4.2 Repeat the steps given in Example 4.2 for a p-type semiconductor. In particular,
show that when the net acceptor concentration is much greater than n i , then p p
= N A and n p = n
2
i /N A .
4.3 An engineer has two Ga 1−x Al x As LEDs: one has a bandgap energy of 1.540 eV
and the other has x = 0.015.
(a) Find the aluminum mole fraction x and the emission wavelength for the first LED.
(b) Find the bandgap energy and the emission
wavelength of the other LED.
[Answer:
(a) x = 0.090, λ = 805 nm; (b) E g = 1.620 eV, λ = 766 nm.]
4.4 The lattice spacing of In 1−x Ga x As y P 1−y has been shown to obey Vegard’s law.
This states that for quaternary alloys of the form A 1−x B x C y D 1−y , where A and B
are group III elements (e.g., Al, In, and Ga) and C and D are group V elements
(e.g., As, P, and Sb), the lattice spacing a(x, y) of the quaternary alloy can be
approximated by
a(x, y) = x ya(BC) + x(1 − y)a(BD)
+ (1 − x)ya(AC) + (1 − x) (1 − y)a(AD)
where the a(IJ) parameters are the lattice spacings of the binary compounds IJ.
(a) Show that for In 1−x Ga x As y P 1−y with
a(GaAs) = 5.6536 Å
a(GaP) = 5.4512 Å