Optical Fiber Communications 4th Edition by Gerd Keiser

Exercise 14

The lattice spacing of In 1 x Ga x As y P₁y has been shown to obey Vegard's law. 21 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 ) = xya (BC) + x (1 y ) a (BD) + (1 x ) ya (AC) + (1 x ) (1 y ) a (AD)
where the a (IJ) are the lattice spacings of the binary compounds IJ.
( a ) Show that for In 1 x Ga x As y P₁y with
a (GaAs) = 5.6536
a (GaP) = 5.4512
a (InAs) = 6.0590
a (InP) = 5.8696
the quaternary lattice spacing becomes
a ( x, y ) = 0.1894 y 0.4184 x + 0.0130 xy + 5.8696
( 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 since 0 x 0.47, the resulting expression can be approximated by y 2.20 x.
( c ) A simple empirical relation that gives the bandgap energy in terms of x and y is 21 E g ( x, y ) = 1.35 + 0.668 x 1.17 y +0.758 x 2 + 0.18y 2 0.069 xy 0.322 x 2 y + 0.33 xy 2 eV. Find the bandgap energy and the peak emission wavelength of In 0 74 Ga 0 26 As 0.56 P 0.44.