Strained quantum well

A strained quantum well is a quantum well made of an alloy whose relaxed lattice constant differs from the substrate's, grown thinner than the critical thickness so that it takes the substrate's in-plane lattice constant instead of forming dislocations. The mismatch, typically 0.5–1.5% in laser active regions, is stored as biaxial strain: compressive when the well's natural lattice constant is larger than the substrate's, tensile when it is smaller. The strain shifts and splits the valence bands, which changes the threshold, the differential gain and the polarization of the gain. Compressively strained wells are used in most current semiconductor lasers.

Mismatch and strain

The mismatch of a layer of relaxed lattice constant $a_L$ on a substrate of lattice constant $a_S$ is defined as in lattice matching,

$$f = \frac{a_L - a_S}{a_S}.$$

For In₀.₂Ga₀.₈As, Vegard's law with $a_\text{GaAs} = 5.6533$ Å and $a_\text{InAs} = 6.0583$ Å gives $a_L = 5.7343$ Å, so $f = +1.43\%$ on GaAs. A coherent layer is squeezed in the plane to $a_S$, an in-plane strain of

$$\varepsilon_\parallel = \frac{a_S - a_L}{a_L} = -1.41\%,$$

and, by the Poisson response of the crystal, it expands along the growth direction by $\varepsilon_\perp = -2(C_{12}/C_{11})\,\varepsilon_\parallel = +1.34\%$, using elastic constants interpolated between GaAs and InAs.

Critical thickness

A strained layer stays coherent only up to a critical thickness, beyond which misfit dislocations at the interface relieve the strain. The Matthews-Blakeslee force-balance model sets the thickness at which a misfit dislocation first lowers the strain energy. For a well buried between unstrained barriers, where each dislocation must leave a misfit segment at both interfaces, it gives about 22 nm at 1% mismatch and about 14 nm at the 1.43% of In₀.₂Ga₀.₈As; a single uncapped layer reaches only about 40% of these values (9 nm and 6 nm). The 7–8 nm wells used in 980 nm lasers lie within the buried-well limit. The model is conservative, since dislocation formation is kinetically limited and layers often exceed it before relaxing, although such a layer can relax later under the heat and current of operation. In practice the product of strain and thickness, summed over all the wells, is kept below a limit established for the material system. A relaxed well shows up as broadened, weaker photoluminescence and as a cross-hatch pattern on the surface under a Nomarski microscope; high-resolution X-ray diffraction measures the strain and period of a coherent stack directly from its satellite peaks.

Heavy and light holes

In unstrained bulk III-V material the heavy-hole and light-hole bands are degenerate at the top of the valence band. Biaxial strain has a hydrostatic part, which shifts the bandgap, and a shear part, which splits the two hole bands by

$$\Delta E_{hh\text{-}lh} \approx |b|\left(1 + \frac{2C_{12}}{C_{11}}\right)|\varepsilon_\parallel|,$$

where $b$ is the shear deformation potential, about −2 eV in GaAs. For In₀.₂Ga₀.₈As on GaAs this is about 54 meV from strain alone; quantum confinement, which also lifts the heavy hole above the light hole, adds to it. Under compression the heavy hole is the top level; under tension the light hole moves up and can become the top level.

Two consequences follow. Compression lightens the in-plane effective mass of the top hole band, bringing the valence-band density of states closer to that of the conduction band, so population inversion is reached at a lower carrier density: lower transparency current, higher differential gain, and reduced loss from hole-related processes such as intervalence-band absorption and some Auger channels. Adams, and Yablonovitch and Kane, proposed this in 1986. The second consequence is polarization: for light traveling in the plane of the well, the electron to heavy-hole transition couples only to TE light, while the electron to light-hole transition couples to TM light with four times the strength it has for TE (relative matrix elements 2/3 and 1/6). Compressive wells therefore give TE gain, and tensile wells with the light hole on top give predominantly TM gain (TE and TM polarization).

Strain compensation

A multi-well stack accumulates strain, and the total can exceed the critical thickness even when each well is within it. Strain compensation places barriers of the opposite sign between the wells, so that the average strain is near zero. GaAs₀.₈P₀.₂ has a lattice constant of 5.6127 Å and is under +0.72% tensile strain on GaAs; to balance a 7 nm In₀.₂Ga₀.₈As well at −1.41%, a thickness-weighted average requires about 13.7 nm of barrier. More exact criteria also weight each layer by its elastic constants. Mixing tensile and compressive wells in one stack is also the usual route to the polarization-insensitive gain of a semiconductor optical amplifier.

Common questions

Why are laser quantum wells strained?

Because compressive strain lowers the transparency density and raises the differential gain, which lowers the threshold and improves the modulation response. It also extends the wavelengths a substrate can reach: InGaAs wells on GaAs cover roughly 900–1100 nm, including the 980 nm pump band.

Does compressive strain favor TE or TM?

TE. The heavy-hole level is on top, and the electron to heavy-hole transition has no TM component for light propagating in the plane of the well. Tensile strain raises the light hole and gives TM gain.

What happens if a strained well exceeds the critical thickness?

It relaxes by forming misfit dislocations, which act as non-radiative recombination centers and can grow into the active region during operation, lowering efficiency and reliability.

References: J. W. Matthews and A. E. Blakeslee, "Defects in epitaxial multilayers: I. Misfit dislocations," Journal of Crystal Growth 27, 118 (1974). A. R. Adams, "Band-structure engineering for low-threshold high-efficiency semiconductor lasers," Electronics Letters 22, 249 (1986). E. Yablonovitch and E. O. Kane, "Reduction of lasing threshold current density by the lowering of valence band effective mass," Journal of Lightwave Technology 4, 504 (1986). I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, "Band parameters for III-V compound semiconductors and their alloys," Journal of Applied Physics 89, 5815 (2001). L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).