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Lattice constant for ternary compound $mathrm{In}_{0.532}mathrm{Ga}_{0.468}mathrm{As}$

Physics Asked on April 15, 2021

I am currently studying Physics of Photonic Devices, second edition, by Shun Lien Chuang. Chapter 1.3 The Field of Optoelectronics says the following when discussing semiconductor physics:

For example, by controlling the mole fraction of gallium and indium in an $mathrm{In}_{1 – x}mathrm{Ga}_{x}mathrm{As}$ material, a wide tunable range of band gap is possible because $mathrm{InAs}$ has a $0.354 text{eV}$ band gap and $mathrm{GaAs}$ has a $1.424 text{eV}$ band gap at room temperature. The lattice constant of the ternary alloy has a linear dependence on the mole fraction
$$a(mathrm{A}_xmathrm{B}_{1 – x}mathrm{C}) = xa(mathrm{AC}) + (1 – x)a(mathrm{BC}) tag{1.3.1}$$
where $a(mathrm{AC})$ is the lattice constant of the binary compound $mathrm{AC}$ and $a(mathrm{BC})$ is that of the compound $mathrm{BC}$. This linear interpolation formula works very well for the lattice constant, but not for the band gap. For the band-gap dependence, a quadratic dependence on the mole fraction $x$ is usually required
$$E_g(mathrm{A}_xmathrm{B}_{1 – x}mathrm{C} = x E_g(mathrm{AC}) + (1 – x)E_g(mathrm{BC}) – bx(1 – x) tag{1.3.2}$$
where $b$ is called the bowing parameter because it causes a deviation of the ternary band-gap energy away from a linear interpolation of the two band-gap energies of the binary compounds. Figure 1.9 plots the band-gap energy at $T = 0 text{K}$ as a function of the lattice constant for many binary and ternary compound semiconductors. For example, $mathrm{GaAs}$ has a band gap of $1.519 text{eV}$ at low temperature and a lattice constant of $5.6533 mathring{mathrm{A}}$, whereas $mathrm{InAs}$ has a band gap of $0.417 text{eV}$ and a lattice constant of $6.0584 mathring{mathrm{A}}$, as indicated. A ternary $mathrm{In}_{1 – x} mathrm{Ga}_x mathrm{As}$ compound has the two end points at $mathrm{GaAs}$ ($x = 0$) and $mathrm{InAs}$ ($x = 1$) and its band gap has a slight downward bowing below a linear interpolation. At $x = 0.468$, the $mathrm{In}_{0.532}mathrm{Ga}_{0.468}mathrm{As}$ alloy has a lattice constant matched to that of the $mathrm{InP}$ ($5.8688 mathring{mathrm{A}}$).
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For the ternary compound $mathrm{In}_{0.532}mathrm{Ga}_{0.468}mathrm{As}$, how did the author ascertain the lattice constant of $5.8688 mathring{mathrm{A}}$? It is clear how this is done for the binary compounds using figure 1.9, but I don’t understand how it was done for $mathrm{In}_{0.532}mathrm{Ga}_{0.468}mathrm{As}$. Did the author look this up using some other (not presented) graph?

I would greatly appreciate it if people would please take the time to clarify this.

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