Material parameters and where they enter the solvers
Learning objectives. After reading this page you should be able to:
- Understand the physical meaning of each quantum-well material parameter.
- Know which Hamiltonian term each parameter controls.
- Recognise the common energy-reference and sign conventions.
- Distinguish zincblende, wurtzite and silicon parameter sets.
- Know where authoritative parameter values are compiled.
Prerequisites. The model pages you intend to use; this page is a reference to read alongside them.
Estimated reading time. 25 minutes.
Every quantum-well solver is only as good as the material parameters it is given. This page explains what each parameter means physically, which Hamiltonian term it controls, its units and the common sign or convention traps — rather than listing numerical values, which belong in the material database and in the compilations cited on the references page. Parameters are grouped by the solver that uses them.
1. Energies and energy references
The most common source of error is the energy reference. OghmaNano measures energies from the structure valence-band maximum (placed at zero), and each material's valence edge \(E_v\) is supplied on a common absolute scale so that band offsets are correct. The multiband solvers derive the conduction edge as \(E_c = E_v + E_g\) and, internally, an average valence energy that the spin-orbit and crystal-field terms then split back to the physical band edge.
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(E_g\) | Fundamental band gap | Conduction–valence separation; \(E_c=E_v+E_g\) | eV |
| \(E_v\) | Valence-band edge on the common scale | Diagonal valence energy; sets band offsets | eV |
| \(E_c\) | Conduction-band edge (derived) | Conduction diagonal | eV |
| \(E_{v,\text{av}}\) | Spin-orbit/crystal-field-averaged valence energy | Internal diagonal before SO/CF splitting | eV |
| \(\Delta_{\text{so}}\) | Spin-orbit splitting | Split-off band position; SO coupling | eV |
2. Effective masses
The band-edge electron mass fixes the conduction curvature and, in the multiband models, is used to back out the remote-band parameter. Hole masses appear directly only in the parabolic solver; in the k·p models the valence masses are encoded in the Luttinger (or wurtzite \(A_i\)) parameters instead. Silicon's valleys are anisotropic and need two masses each.
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(m_e\) | Band-edge electron mass | Conduction curvature; remote parameter \(A_c\) | \(m_0\) |
| \(m_h\) | Hole mass (parabolic solver) | Valence curvature in the scalar model | \(m_0\) |
| \(m_l\) | Longitudinal valley mass (Si) | Mass along a valley axis | \(m_0\) |
| \(m_t\) | Transverse valley mass (Si) | Mass across a valley axis | \(m_0\) |
3. Zincblende valence and coupling parameters
The Luttinger parameters describe the coupled heavy-, light- and split-off holes; the Kane energy sets the conduction–valence coupling. In the 8-band model OghmaNano combines these into the internal \(A_c, L, M, N, P\) parameters, renormalising \(L\) and \(N\) (and, in the standard mode, \(A_c\)) to avoid double-counting the explicit Kane coupling.
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(\gamma_1,\gamma_2,\gamma_3\) | Luttinger parameters | Valence \(L, M, N\) kinetic terms | dimensionless |
| \(E_p\) | Kane energy | Kane coupling \(P=\sqrt{(\hbar^2/2m_0)E_p}\) | eV |
| \(P\) | Kane momentum matrix element | Conduction–valence coupling | eV·m (internally) |
Implementation note. The Luttinger parameters used inside the 8-band Hamiltonian are the full bulk values; the code performs the \(+E_p/E_g\) subtraction internally when forming \(L\) and \(N\). Do not pre-subtract the coupling from the Luttinger parameters in the material database, or the valence masses will be wrong. This is a frequent source of discrepancy when porting parameters between codes that use different conventions (full versus modified Luttinger parameters).
4. Zincblende deformation potentials and elastic constants
Strain enters through the deformation potentials and requires the elastic constants and lattice constants to compute the strain tensor. The conduction and valence hydrostatic potentials combine into the gap deformation potential; the shear potentials \(b\) and \(d\) split the valence bands.
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(a_c\) | Conduction hydrostatic deformation potential | \(a_c\,\mathrm{Tr}\varepsilon\) shift | eV |
| \(a_v\) | Valence hydrostatic deformation potential | Bir–Pikus \(P=a_v\mathrm{Tr}\varepsilon\) | eV |
| \(b\) | Tetragonal shear deformation potential | Bir–Pikus \(Q\); HH/LH splitting | eV |
| \(d\) | Rhombohedral shear deformation potential | Bir–Pikus \(R, S\) shear coupling | eV |
| \(a\), \(a_{\text{sub}}\) | Layer and substrate lattice constants | In-plane mismatch strain | m |
| \(C_{11}, C_{12}\) | Elastic stiffness constants | \(\varepsilon_{zz}=-2(C_{12}/C_{11})\varepsilon_{xx}\) | Pa |
5. Wurtzite parameters
Wurtzite needs more parameters because of its lower symmetry: six valence parameters, three splitting energies, two Kane energies, two conduction-mass parameters and separate hydrostatic and shear deformation potentials.
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(A_1\dots A_6\) | Valence effective-mass parameters | Valence \(L_i, M_i, N_i\) kinetic terms | dimensionless |
| \(\delta_1\) | Crystal-field splitting | Raises \(X,Y\) relative to \(Z\) | eV |
| \(\delta_2, \delta_3\) | Spin-orbit parameters | Valence SO coupling | eV |
| \(E_{p1}, E_{p2}\) | c-axis and basal Kane energies | \(P_1\) (S–Z), \(P_2\) (S–X,Y) | eV |
| \(S_1, S_2\) | c-axis and basal conduction-mass parameters | Anisotropic conduction dispersion | dimensionless |
| \(a_1, a_2\) | c-axis and basal conduction deformation potentials | Conduction strain shift | eV |
| \(D_1\dots D_6\) | Valence deformation potentials | Valence strain (\(D_6\) shear only) | eV |
| \(C_{ij}\) | Hexagonal elastic constants | c-plane strain tensor | Pa |
6. Silicon Δ-valley parameters
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(\Xi_d\) | Dilatational deformation potential | \(\Xi_d\,\mathrm{Tr}\varepsilon\): shifts all valleys | eV |
| \(\Xi_u\) | Uniaxial deformation potential | \(\Xi_u\,\varepsilon_{\text{valley}}\): splits \(\Delta_2/\Delta_4\) | eV |
| \(m_l, m_t\) | Longitudinal / transverse masses | Anisotropic valley dispersion | \(m_0\) |
7. Dilute-nitride (10-band) parameters
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(x_N\) | Nitrogen mole fraction | Coupling strength \(V_N=\beta_N\sqrt{x_N}\) | dimensionless |
| \(E_N\) | Nitrogen resonant-level energy | Nitrogen diagonal \(H_{NN}\) | eV |
| \(\beta_N\) | Nitrogen coupling constant | Conduction–nitrogen coupling | eV |
8. Optical and transport parameters
The gain and scattering calculations need a few further inputs: the refractive index and Lorentzian broadening for gain, and the mass density, sound velocity and acoustic deformation potential for scattering. These are properties of the medium and the phonons rather than of the band structure.
| Parameter | Meaning | Controls | Units |
|---|---|---|---|
| \(n_r\) | Refractive index | Gain prefactor | dimensionless |
| \(\gamma\) | Lorentzian half-width | Lineshape broadening | eV |
| \(\rho\) | Mass density | Acoustic scattering prefactor | kg·m\(^{-3}\) |
| \(v_s\) | Sound velocity | Acoustic scattering prefactor | m·s\(^{-1}\) |
| \(D\) | Acoustic deformation potential | Scattering matrix element | eV |
Key points
- OghmaNano references energies to the structure valence-band maximum; supply \(E_v\) on a common scale.
- Provide full (unmodified) Luttinger parameters; the code performs the Kane renormalisation internally.
- Wurtzite needs \(A_1\dots A_6\), \(\delta_1,\delta_2,\delta_3\), \(E_{p1},E_{p2}\), \(S_1,S_2\) and \(a_1,a_2,D_1\dots D_6\).
- Silicon uses \(\Xi_d,\Xi_u,m_l,m_t\); dilute nitrides add \(x_N,E_N,\beta_N\).