Lattice matching

Lattice matching means growing an epitaxial crystalline layer whose natural lattice constant equals that of the substrate beneath it. The atoms of the layer then sit directly on the sites of the substrate lattice, with no strain and no misfit dislocations at the interface, and layers of any thickness can be stacked. Nearly every semiconductor laser and detector for telecom wavelengths depends on it: In₀.₅₃Ga₀.₄₇As has the lattice constant of InP, 5.8687 Å, and a bandgap of about 0.75 eV, which places its absorption edge at 1.65 µm. Al$_x$Ga$_{1-x}$As on GaAs is matched within 0.14% at every aluminum fraction, and InGaAsP on InP can be matched at any gap from 0.75 eV to 1.34 eV, wavelengths from 1.65 µm to 0.92 µm.

Mismatch

The lattice mismatch of a layer on a substrate is

$$f = \frac{a_\text{layer} - a_\text{sub}}{a_\text{sub}}$$

using the relaxed, unstrained lattice constants. With the room-temperature values GaAs 5.6533 Å, InP 5.8687 Å, InAs 6.0583 Å, Si 5.431 Å and Ge 5.658 Å:

A positive $f$ means the layer is compressed in the plane when it grows coherently, and a negative $f$ means it is stretched. The heterojunction entry uses below 0.1% as the working criterion for a matched layer.

Vegard's law and the In fraction

The lattice constant of an alloy is close to the linear interpolation between its end compounds, Vegard's law. For In$_x$Ga$_{1-x}$As,

$$a(x) = x\,a_\text{InAs} + (1-x)\,a_\text{GaAs} .$$

Setting $a(x) = a_\text{InP}$ gives

$$x = \frac{a_\text{InP} - a_\text{GaAs}}{a_\text{InAs} - a_\text{GaAs}} = 0.532 ,$$

the In₀.₅₃Ga₀.₄₇As of every InGaAs photodetector. The slope of $a(x)$ is 0.405 Å per unit $x$, so an error of 0.01 in the indium fraction produces a mismatch of 0.07%, which is why growers calibrate composition so carefully. Quaternary alloys such as In$_{1-x}$Ga$_x$As$_y$P$_{1-y}$ have two composition parameters, so one can be used to hold the lattice match while the other sets the gap; that freedom is what lets III-V semiconductors cover the telecom bands on a single InP substrate.

Strain and critical thickness

A mismatched layer can still grow without defects if it is thin: it adopts the substrate's in-plane lattice constant and stores the difference as elastic strain, a pseudomorphic or coherent layer. Beyond a critical thickness the stored energy is relieved by misfit dislocations at the interface. The Matthews-Blakeslee model gives roughly 9 nm for a 1% mismatch and about 1 nm for germanium on silicon; layers grown at low temperature often exceed these values before relaxing, because dislocation formation is kinetically limited.

Below the critical thickness, strain is a design tool. Compressively strained quantum wells reshape the valence band, lowering transparency density and raising differential gain, and they extend the wavelength range of a substrate, as with InGaAs wells on GaAs at 980 nm.

Measuring it

High-resolution X-ray diffraction is the standard check: the layer and substrate give separate Bragg peaks, and their angular separation gives the mismatch. A coherent layer is distorted tetragonally, its vertical spacing changing more than the relaxed mismatch, by a factor $1 + 2C_{12}/C_{11}$, about 1.9 for GaAs-like materials; the measured perpendicular mismatch must be corrected by this factor.

Why dislocations matter

Threading dislocations that run up from a relaxed interface act as non-radiative recombination centers and as generation centers. In lasers and LEDs they reduce efficiency and grow into dark-line defects that cause rapid failure; in detectors they raise dark current. III-V layers grown directly on silicon carry threading-dislocation densities of $10^7$–$10^8$ cm⁻², against below $10^4$ cm⁻² in native InP. Germanium-on-silicon photodetectors work despite the 4.2% mismatch because a low-temperature seed layer and annealing confine most defects near the interface, though their dark current stays higher than lattice-matched InGaAs. The alternative to growth is bonding: heterogeneous integration bonds lattice-matched III-V material grown on its own substrate onto silicon, avoiding mismatch altogether.

Pitfalls

Lattice constants change with temperature, and the substrate and layer expand at different rates; a layer matched at the growth temperature, several hundred degrees Celsius, can be slightly mismatched at room temperature, and large thermal-expansion differences crack thick GaAs on silicon on cooling. Vegard's law is approximate, and bandgaps need an added bowing term.

Common questions

Why is InGaAs grown on InP?

At 53% indium InGaAs has the lattice constant of InP, so thick defect-free absorbers can be grown, and InP is transparent at 1.3 and 1.55 µm, serving as window and contact layers.

Is AlGaAs lattice matched to GaAs?

Nearly: AlAs differs from GaAs by 0.14%, so every Al$_x$Ga$_{1-x}$As composition lies within that, small enough for thick layers in practice.

What is the lattice mismatch of germanium on silicon?

About 4.2% ($a$ = 5.658 Å against 5.431 Å), with gallium arsenide on silicon close behind at 4.1%.

References: J. W. Matthews and A. E. Blakeslee, "Defects in epitaxial multilayers: I. Misfit dislocations," J. Cryst. Growth 27, 118 (1974); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).