kpenvelope

A six-band k.p envelope-function solver for wurtzite heterostructures, solved self-consistently with Poisson's equation on a 1D grid. Built for polarization-induced two-dimensional hole gases (GaN/AlN and related systems), where the confining potential is not imposed but emerges from the balance between the fixed polarization charge and the gas itself.

Status

v0.3.0 (alpha). Implemented and tested:

• six-band wurtzite valence Hamiltonian (standard Chuang-Chang form), discretized with symmetrized operator ordering so the matrix is exactly Hermitian for position-dependent parameters
• envelope eigen-solution at arbitrary in-plane k
• Gauss-law hole potential from the density profile
• self-consistent loop with T = 0 subband filling using numeric in-plane edge masses, converging to charge neutrality
• cited GaN and AlN parameter sets with closed-form verification (new in v0.2, see below)
• dispersion and mass utilities (new in v0.3): subband dispersions along an in-plane path (subband_dispersion) and the local finite-difference effective mass of any dispersion (local_mass), because a hole mass is not one number. Asserted in the test suite: the decoupled demo set returns exactly 1/|A| m0 at every momentum; applied to the bulk Rinke 2008 GaN bands the utility reproduces the quasi-cubic asymptotic masses 1.89 and 0.180 m0; and a locally flat branch reports an infinite mass rather than an error, because a diverging mass is physics.

Not yet implemented, stated plainly because they matter physically:

• finite barriers. Hard walls one grid spacing outside the grid ends. A hard wall forces the envelope to vanish at the interface and pushes the gas outward, which shifts subband energies and masses; a finite AlN barrier treated as position-dependent material parameters is the v0.3 gate before research-grade use.
• strain terms, full k-grid (non-parabolic) filling, spin splitting analysis, transport lifetimes.

Cited parameter sets (new in v0.2)

• gan_rinke2008(): the consistent GW-based GaN valence set of Rinke et al., Phys. Rev. B 77, 075202 (2008) (A1..A6, Delta_CR = 10 meV, Delta_SO = 17 meV, delta2 = delta3 = Delta_SO/3), as tabulated in Extended Data Table 1 of Chang et al., Nature Electronics 9, 346 (2026) and used in the source paper below. eps_r = 10.4 (E parallel to c) from Barker and Ilegems, Phys. Rev. B 7, 743 (1973).
• aln_rinke2008(): the matching AlN set (Delta_CR = -295 meV from Rinke et al.; Delta_SO = 22 meV from de Carvalho et al., Appl. Phys. Lett. 97, 232101 (2010)), intended as a barrier material. Its permittivity is deliberately NaN, and the self-consistent solver refuses to run on it, because no vetted value is shipped and none is needed for a barrier.

The test-suite locks every number and checks the GaN set against closed forms: the zone-center splittings come out at 5.20 and 21.80 meV (against accepted experimental values near 5-6 and 22 meV), and the asymptotic in-plane masses at m0/|A2+A4-A5| = 1.89 m0 and m0/|A2+A4+A5| = 0.18 m0, the quasi-cubic values quoted in the source paper's Supplemental Material.

For any other material or parameterization, populate WurtziteParameters from the literature (e.g. Vurgaftman and Meyer, J. Appl. Phys. 94, 3675 (2003)) and record the source in the mandatory reference field. The shipped demo_single_band() set is a decoupled, non-physical configuration used by the test-suite, chosen because it has closed-form well solutions to test against.

Install and use

pip install kpenvelope

For development, clone the repository and pip install -e .[test].

import numpy as np
from kpenvelope import gan_rinke2008, solve_self_consistent

p = gan_rinke2008()                      # cited set; or your own WurtziteParameters
z = np.linspace(0.0, 6.0, 97)            # nm, from the interface
res = solve_self_consistent(p, z, ps=0.46)   # ps in nm^-2; 4.6e13 cm^-2 = 0.46
# res.energies, res.masses, res.density, res.potential, res.occupations

Convention: valence-electron energies, holes occupy the highest eigenvalues; energies in eV, lengths in nm, sheet densities in nm^-2.

Verification

The test-suite checks Hermiticity with every coupling switched on, the decoupled square-well limit against the analytic spectrum, the uniform slab against the analytic Gauss-law potential, convergence plus exact charge neutrality of the self-consistent loop, and (new in v0.2) the cited GaN set against the closed-form quasi-cubic splittings and masses above.

One comparison against the source paper is on record and stated honestly: a hard-wall self-consistent run at the measured sheet density (4.6e13 cm^-2, 97 points over 6 nm) puts the gas centroid at 0.62 nm against 0.568 nm for the hard-wall row of the paper's Table S1, with the same subband structure (two heavy branches filled, the light branch a minority). The difference comes from the filling model: this package fills parabolic edge-mass subbands, while the paper fills the computed non-parabolic dispersions, and the light branch is strongly non-parabolic. Together with the hard-wall boundary this is why the same warning stands: do not use these numbers in publications before the finite-barrier, dispersion-filled v0.3.

Methodological basis

T. M. Mahim, A. S. M. Mohsin and M. M. Rahman, "Origin of the conflicting hole masses in the GaN/AlN two-dimensional hole gas" (under review); code for the paper: https://github.com/Tanvir-Mahmud-Mahim/gan-2dhg-masses-lifetimes

and S. L. Chuang and C. S. Chang, Phys. Rev. B 54, 2491 (1996). This package is the general-purpose tool; the paper repository reproduces the specific published study, including the finite-barrier physics that this package does not yet have.

License

Apache-2.0