Pseudomorphic epitaxy and strain tensors

When a layer is grown on a substrate with a different lattice constant, it distorts to match the substrate in-plane spacing. This pseudomorphic strain is not a nuisance to be avoided: it is a design tool that shifts band edges, splits the heavy- and light-hole bands and tunes the emission wavelength and polarisation. This page derives the strain tensor that OghmaNano uses and explains how it feeds the k·p Hamiltonians.

1. Pseudomorphic growth

Below a critical thickness a thin layer grows coherently: its in-plane atomic rows line up with the substrate, so its in-plane lattice constant is forced to that of the substrate while it relaxes freely along the growth direction. The in-plane strain is therefore

\[ \varepsilon_{xx} = \varepsilon_{yy} = \frac{a_{\text{sub}} - a_{\text{layer}}}{a_{\text{layer}}}, \]

where \(a_{\text{sub}}\) and \(a_{\text{layer}}\) are the unstrained substrate and layer lattice constants. If the layer's natural spacing exceeds the substrate's (\(a_{\text{layer}} > a_{\text{sub}}\)) the in-plane strain is negative and the layer is compressively strained; the reverse gives tensile strain. OghmaNano flags an error if the lattice constants or elastic constants are missing, since strain cannot then be defined.

2. The out-of-plane strain

Along the growth direction the layer is free, so there is no stress normal to the surface: \(\sigma_{zz}=0\). For a cubic crystal grown on the [001] plane, Hooke's law relates the normal stress to the strains through the elastic constants \(C_{11}\) and \(C_{12}\),

\[ \sigma_{zz} = C_{11}\varepsilon_{zz} + C_{12}(\varepsilon_{xx}+\varepsilon_{yy}) = 0. \]

Setting \(\varepsilon_{xx}=\varepsilon_{yy}\) and solving gives the relation OghmaNano uses,

\[ \varepsilon_{zz} = -\,\frac{2 C_{12}}{C_{11}}\,\varepsilon_{xx}. \]

The layer therefore distorts oppositely along \(z\): a compressively strained layer (\(\varepsilon_{xx}<0\)) stretches along the growth direction (\(\varepsilon_{zz}>0\)), and vice versa. For [001] growth the shear components \(\varepsilon_{xy}, \varepsilon_{xz}, \varepsilon_{yz}\) are all zero, so the strain is purely biaxial.

3. Hydrostatic and biaxial parts

It is useful to split the strain into a hydrostatic part, which changes the volume, and a biaxial (shear) part, which changes the shape. The hydrostatic part is the trace

\[ \mathrm{Tr}(\varepsilon) = \varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz} = 2\varepsilon_{xx}\left(1 - \frac{C_{12}}{C_{11}}\right), \]

which shifts the band gap through the deformation potentials. The biaxial part, proportional to \(\varepsilon_{xx}-\varepsilon_{zz}\), does not change the volume but lifts the degeneracy of the valence bands and, in silicon, splits the conduction valleys. The two parts act through different Hamiltonian terms, so they can be discussed independently.

4. Effect on the band edges

The hydrostatic component shifts the conduction band by \(a_c\,\mathrm{Tr}(\varepsilon)\) and the average valence band by \(a_v\,\mathrm{Tr}(\varepsilon)\), changing the gap. The biaxial component enters the valence band through the shear deformation potentials \(b\) and \(d\) (zincblende) and splits the heavy and light holes. Compressive strain typically pushes the heavy hole uppermost, favouring TE emission; tensile strain can push the light hole up, favouring TM emission. The precise operators are the subject of the Bir–Pikus page, and how each solver consumes the strain tensor is summarised there and on the model pages (zincblende, wurtzite, silicon, Luttinger–Kohn).