The Bir–Pikus strain Hamiltonian

Strain deforms the crystal potential and therefore shifts and mixes the valence bands. The Bir–Pikus Hamiltonian is the strain analogue of the kinetic k·p valence Hamiltonian: it has exactly the same structure, with strain components replacing products of wavevectors. OghmaNano implements it in two equivalent forms — a compact \(P, Q, R, S\) form for the Luttinger–Kohn solver and an explicit Cartesian form for the 8-band solvers.

1. The P, Q, R, S form

In the total-angular-momentum basis the valence strain Hamiltonian is built from four quantities. Using the deformation potentials \(a_v\) (hydrostatic) and \(b, d\) (shear), OghmaNano computes

\[ \begin{aligned} P &= a_v\,\mathrm{Tr}(\varepsilon), \\ Q &= \tfrac{1}{2}\,b\,(\varepsilon_{xx}+\varepsilon_{yy}-2\varepsilon_{zz}), \\ R &= -\tfrac{\sqrt{3}}{2}\,b\,(\varepsilon_{xx}-\varepsilon_{yy}) + i\,d\,\varepsilon_{xy}, \\ S &= -\,d\,(\varepsilon_{xz} - i\,\varepsilon_{yz}), \end{aligned} \]

with \(\mathrm{Tr}(\varepsilon)=\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}\). \(P\) is the hydrostatic shift of the average valence band; \(Q\) is the biaxial splitting of heavy and light holes; and \(R, S\) are shear couplings that vanish for unsheared [001] strain. These four strain quantities are added directly to the kinetic \(P, Q, R, S\) of the Luttinger–Kohn Hamiltonian, so that the same \(6\times6\) matrix describes kinetic and strain physics together.

2. The Cartesian X/Y/Z form

The 8-band zincblende model works in the Cartesian orbital basis, so OghmaNano expresses the same physics as a \(3\times3\) matrix acting on \(|X\rangle, |Y\rangle, |Z\rangle\):

\[ H^{\varepsilon} = \begin{pmatrix} a_v\mathrm{Tr}\varepsilon + b(2\varepsilon_{xx}-\varepsilon_{yy}-\varepsilon_{zz}) & \sqrt{3}\,d\,\varepsilon_{xy} & \sqrt{3}\,d\,\varepsilon_{xz} \\[3pt] \sqrt{3}\,d\,\varepsilon_{xy} & a_v\mathrm{Tr}\varepsilon + b(2\varepsilon_{yy}-\varepsilon_{xx}-\varepsilon_{zz}) & \sqrt{3}\,d\,\varepsilon_{yz} \\[3pt] \sqrt{3}\,d\,\varepsilon_{xz} & \sqrt{3}\,d\,\varepsilon_{yz} & a_v\mathrm{Tr}\varepsilon + b(2\varepsilon_{zz}-\varepsilon_{xx}-\varepsilon_{yy}) \end{pmatrix}. \]

The diagonal combines the hydrostatic shift \(a_v\mathrm{Tr}(\varepsilon)\) with the axial term \(b(2\varepsilon_{ii}-\varepsilon_{jj}-\varepsilon_{kk})\); the off-diagonal shear terms \(\sqrt{3}\,d\,\varepsilon_{ij}\) vanish for [001] biaxial strain. The strain potential is spin independent, so OghmaNano applies this \(3\times3\) block identically to the spin-up and spin-down valence orbitals.

3. Why the two forms agree

The Cartesian and \(P, Q, R, S\) forms are related by the same unitary transformation that connects the \(|X\rangle,|Y\rangle,|Z\rangle\) orbitals to the \(|J,m_j\rangle\) states. For example, the Cartesian diagonal difference \(b(2\varepsilon_{zz}-\varepsilon_{xx}- \varepsilon_{yy})\) is exactly \(-2Q\) with \(Q\) as defined above, and the shear entries map onto \(R\) and \(S\). Both forms use the same three deformation potentials and give identical valence energies; the choice is purely one of convenience for the basis each solver uses.

4. The deformation potentials

Three numbers control the valence strain response. The hydrostatic potential \(a_v\) sets how the average valence energy moves with volume; combined with the conduction potential \(a_c\) it gives the gap deformation potential \(a = a_c - a_v\). The axial potential \(b\) controls the tetragonal (biaxial) splitting of the heavy and light holes under [001] strain, and \(d\) controls the response to shear strains along other directions. All three have units of energy (eV) and are typically negative for the valence band. Their values and sign conventions are discussed on the material-parameters page.