Ternary Compound Semiconductor's Material Values

Alloy	Bandgap $E_g(x)$ (eV, 300 K)	Bowing parameter $b$ (eV)	Lattice constant $a(x)$ (Å, Vegard’s law)	Notes
${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$	$E_g^{\Gamma }(x)=1.424+1.247x(x\le 0.45)$ $E_g^X(x)\approx 1.900+0.125x+0.143x^2(x>0.45)$	$\approx 0$ (direct branch, ~linear)	$a(x)=5.653+0.0078x$	Direct for $x\lesssim 0.45$, indirect ($\Gamma \to X$) beyond that. Near-perfect lattice match to GaAs across the whole composition range ($a_{\mathrm{AlAs}}\approx a_{\mathrm{GaAs}}$), which is why AlGaAs/GaAs heterostructures can be grown nearly strain-free at any $x$.
${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$	$E_g(x)=(1-x)(1.424)+x(0.354)-bx(1-x)$	$0.477$	$a(x)=(1-x)(5.653)+x(6.058)$	Direct gap throughout. Lattice-matched to InP substrates at $x\approx 0.53$ ($\mathrm{I}{\mathrm{n}}_{0.53}\mathrm{G}{\mathrm{a}}_{0.47}\mathrm{As}$, $a\approx 5.869\text{A˚}$), the composition most used in HEMTs and near-IR photodetectors.
${\mathrm{GaAs}}_{1-x}{\mathrm{P}}_x$	$E_g(x)=(1-x)(1.424)+x(2.78)-bx(1-x)$	$\approx 0.19$	$a(x)=(1-x)(5.653)+x(5.451)$	The GaP endpoint is the $\Gamma$-valley gap (2.78 eV); GaP itself is indirect (X gap 2.26 eV). Direct-to-indirect crossover near $x\approx 0.45\text{–}0.49$. Historically important for early red/yellow LEDs before III-nitride LEDs became dominant.
${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$	$E_g(x)=(1-x)(3.44)+x(0.7)-bx(1-x)$	$\approx 1.4(\text{reported}1.0\text{–}3.8)$	$a(x)=(1-x)(3.189)+x(3.545)$	Direct throughout. Spans the entire visible range from UV (GaN) to near-IR (InN) with composition, making it the workhorse material for blue/green LEDs and laser diodes. The bowing parameter is one of the most disputed in III-V literature — reported values vary widely with growth quality and phase separation at high $x$.
${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$	$E_g(x)=(1-x)(3.44)+x(6.2)-bx(1-x)$	$0.7$	$a(x)=(1-x)(3.189)+x(3.112)$	Direct throughout. The equivalent linear form is $a(x)=(-0.077x+3.189)\times 10^{-10}\text{m}$ (Ambacher et al., J. Appl. Phys. 85, 3222 (1999)). The bowing parameter $b=0.7$ eV is the recommended value of Vurgaftman and Meyer, J. Appl. Phys. 94, 3675 (2003). Used as the barrier layer in AlGaN/GaN HEMTs; larger $x$ increases polarization-induced 2DEG density but also strain.
${\mathrm{In}}_x{\mathrm{Al}}_{1-x}\mathrm{As}$	$E_g(x)=(1-x)(3.0)+x(0.36)-bx(1-x)$	$\approx 0.70$	$a(x)=(1-x)(5.661)+x(6.058)$	The AlAs endpoint is the $\Gamma$-valley gap (3.0 eV); AlAs itself is indirect (X gap 2.16 eV). Direct for most of the composition range. Lattice-matched to InP at $x\approx 0.52$ ($\mathrm{I}{\mathrm{n}}_{0.52}\mathrm{A}{\mathrm{l}}_{0.48}\mathrm{As}$), commonly paired with $\mathrm{I}{\mathrm{n}}_{0.53}\mathrm{G}{\mathrm{a}}_{0.47}\mathrm{As}$ as the barrier layer in InP-based HEMTs.
${\mathrm{Hg}}_{1-x}{\mathrm{Cd}}_x\mathrm{Te}$	$E_g(x,T)=-0.302+1.93x-0.810x^2+0.832x^3+5.35\times 10^{-4}T(1-2x)$	(built into Hansen’s empirical formula above)	$a(x)\approx (1-x)(6.461)+x(6.482)$	Here $x$ is the Cd fraction (Hansen et al., J. Appl. Phys. 53, 7099 (1982)). Bandgap tunable from $\approx 1.5\mathrm{eV}$ ($x=1$, CdTe) down through zero and into the negative (semimetallic HgTe, $x=0$), used to engineer detectors across the entire infrared spectrum (SWIR through VLWIR). Nearly constant lattice constant across all $x$ is what allows compositionally graded structures without significant strain.

Formula Reference

General bandgap bowing form used above:

$E_g(x)=(1-x)E_g(A)+x{E}_g(B)-bx(1-x)$

$a(x)=(1-x)a(A)+xa(B)\text{(Vegard’s law, linear interpolation)}$

Variables: $x$: alloy composition fraction of the second-listed constituent; $E_g(A)$, $E_g(B)$: bandgaps of the two binary endpoint compounds; $b$: bowing parameter, a positive number (material-specific) capturing the deviation of the real alloy bandgap from a naive linear interpolation; $a(A)$, $a(B)$: lattice constants of the two endpoint binaries.

Important Questions

Ques: Why does the bandgap of an alloy deviate from a straight-line interpolation between its two binary endpoints?

A purely linear interpolation (bowing parameter $b=0$) would assume the alloy’s electronic structure is a simple average of its constituents, but in reality alloying introduces compositional disorder that perturbs the band structure beyond what a virtual-crystal average predicts — this deviation is captured empirically by the bowing parameter $b$. Bowing tends to be small when the two endpoint compounds are chemically and structurally similar (e.g. AlGaAs, where Al and Ga are nearly interchangeable in the zinc blende lattice) and large when they differ more (e.g. InGaN, where the large size mismatch between In and Ga contributes to both bowing and phase-separation effects at high indium content). Reference: I. Vurgaftman, J.R. Meyer, and L.R. Ram-Mohan, “Band parameters for III-V compound semiconductors and their alloys,” J. Appl. Phys. 89, 5815 (2001); I. Vurgaftman and J.R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94, 3675 (2003).

Ques: What does “lattice-matched” mean and why does it matter for these alloys?

An alloy composition is lattice-matched to a given substrate when $a(x)$ of the alloy equals the substrate’s lattice constant, so that an epitaxial layer can be grown on top of it with no misfit strain. Growing off this composition still works but introduces strain that accumulates with thickness; beyond a critical thickness this strain relaxes by forming dislocations, which degrade carrier mobility and device reliability. This is why compositions like $\mathrm{I}{\mathrm{n}}_{0.53}\mathrm{G}{\mathrm{a}}_{0.47}\mathrm{As}$ on InP or $\mathrm{I}{\mathrm{n}}_{0.52}\mathrm{A}{\mathrm{l}}_{0.48}\mathrm{As}$ on InP are so commonly used in practice — they are the specific points where $a(x)$ crosses the substrate’s lattice constant. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.