Vogl-Hjalmarson-Dow Tight-Binding (material-specific parameters)¶

Same Slater-Koster sp3 machinery as harrison_tb, plus one addition -- an extra s* orbital per atom, a pure fitting degree of freedom with no physical meaning of its own, whose only job is pulling the lowest conduction band down to the right energy. harrison_tb's universal table structurally cannot do that (see its own Accuracy section); this module trades the "one table fits everything" convenience for a parameter row fitted per material, and gets real accuracy back in return.

Step 1. The gap at Gamma¶

import dense_evolution as de

Si = de.VHD_MATERIALS['Si']
de.direct_gap_at_gamma(Si)

3.4299999999999984

VHD_MATERIALS['Si'] is silicon's real fitted parameter row (Vogl, Hjalmarson & Dow 1983). direct_gap_at_gamma(material) reads the gap at the Gamma point (k=(0,0,0)) -- 3.43 eV, matching real silicon's actual direct gap almost exactly. That number alone would be misleading, though: silicon's fundamental gap -- the one that actually sets its optical/electronic behavior -- is not this one.

Step 2. Why Gamma alone isn't enough¶

vbm, vbm_k, cbm, cbm_k, gap = de.band_extrema_along_path(
    Si, k_start=(0, 0, 0), k_end=(1, 0, 0)
)
cbm_k, gap

(array([0.732, 0.   , 0.   ]), np.float64(1.171344217906544))

band_extrema_along_path(material, k_start, k_end) scans a k-space line (here Gamma to X, in units of 2*pi/a) and finds the real valence-band maximum and conduction-band minimum -- which don't have to sit at the same k-point at all. For silicon, they don't: the conduction-band minimum lands at k=0.732 along Gamma-X, not at Gamma, and the true gap between it and the valence-band maximum is 1.17 eV -- close to real silicon's well-known experimental indirect gap (1.12 eV), and nothing like Step 1's 3.43 eV Gamma-only number. This is exactly what makes silicon an indirect-gap material, and exactly the failure mode harrison_tb's simpler universal model can't resolve on its own.

Details¶

16 materials included: C, Si, Ge, Sn, SiC, AlP, AlAs, AlSb, GaP, GaAs, GaSb, InP, InAs, InSb, ZnSe, ZnTe -- all zinc-blende/diamond semiconductors, each with its own fitted VHD_MATERIALS row (Esa/Epa/Esc/Epc/Essa/Essc on-site energies plus 7 hopping integrals), not derived from a universal formula the way harrison_tb's ETA table is.

sp3s_star_hamiltonian(kxyz, material): the lower-level Bloch Hamiltonian builder both direct_gap_at_gamma and band_extrema_along_path call internally -- a 10x10 matrix per unit cell (5 orbitals/atom, including s*), not the 8x8 harrison_tb builds. k is in units of 2*pi/a along the conventional cubic axes: Gamma=(0,0,0), X=(1,0,0), L=(0.5,0.5,0.5).

Source: P. Vogl, H. P. Hjalmarson, J. D. Dow, "A Semi-empirical tight-binding theory of the electronic structure of semiconductors," J. Phys. Chem. Solids 44 (5), 365-378 (1983). Parameter values transcribed from an independent open-source implementation, github.com/rpmuller/TightBinding (TB.py), which cites the same paper.

Accuracy: validated against real experimental gaps (GaAs, Si, Ge) -- see the Dense-Evolution-Discovery validation page for the full writeup and numbers.

vhd_tb ¶

Vogl-Hjalmarson-Dow (1983) material-specific sp3s* tight-binding parameters -- a more accurate alternative to harrison_tb's universal (materials-independent) parameters, at the cost of needing a separately fitted parameter set per material.

Source: P. Vogl, H. P. Hjalmarson, J. D. Dow, "A Semi-empirical tight-binding theory of the electronic structure of semiconductors", J. Phys. Chem. Solids 44 (5), 365-378 (1983). Parameter values below are transcribed from an independent open-source implementation of the same table, github.com/rpmuller/TightBinding (TB.py, fetched 2026-08-05), which cites the same paper; not reproduced from the paper itself (not open access). Only materials appearing in that source are included.

sp3s basis: s, px, py, pz, and an extra excited s orbital per atom (5 orbitals/atom, 10x10 Hamiltonian per zinc-blende/diamond 2-atom cell) -- the s* orbital exists purely to push the lowest conduction band down to the right energy without needing d-orbitals; it has no literal physical excited state behind it.

Hermiticity note: the reference implementation's own H-builder forms the lower-left (cation-anion) block as Hac.conjugate() without transposing it, which does not in general produce a Hermitian matrix (the block mixes Vsapc/Vscpa asymmetrically). _cation_anion_block below is built the same way as that source's get_Hac, but assembled into the full Hamiltonian with a proper conjugate transpose (.conj().T) -- verified Hermitian for every material in MATERIALS (see the module's own smoke test).

Validated against experiment (run 2026-08-05): GaAs direct gap at Gamma (k=0) computes to 1.55 eV vs. the experimental 1.42 eV (~9% high) -- compare to harrison_tb.zincblende_hamiltonian's 2.91 eV (~105% high) for the same material using Harrison's universal parameters. The improvement is expected: these parameters are fitted per material specifically to reproduce band-edge energies, unlike Harrison's one-table-fits-all approach.

sp3s_star_hamiltonian ¶

sp3s_star_hamiltonian(kxyz, material)

10x10 sp3s Bloch Hamiltonian, basis order [anion: s,px,py,pz,s, cation: s,px,py,pz,s*].

material: a Material namedtuple, or a key into MATERIALS. kxyz: crystal momentum in units of 2*pi/a along the conventional cubic axes (Gamma=(0,0,0), X=(1,0,0), L=(0.5,0.5,0.5)).

Source code in dense_evolution/solvers/vhd_tb.py

def sp3s_star_hamiltonian(kxyz, material):
    """
    10x10 sp3s* Bloch Hamiltonian, basis order
    [anion: s,px,py,pz,s*, cation: s,px,py,pz,s*].

    material: a Material namedtuple, or a key into MATERIALS.
    kxyz: crystal momentum in units of 2*pi/a along the conventional
    cubic axes (Gamma=(0,0,0), X=(1,0,0), L=(0.5,0.5,0.5)).
    """
    if isinstance(material, str):
        if material not in MATERIALS:
            raise ValueError(f"no VHD parameters for {material!r}; available: {sorted(MATERIALS)}")
        material = MATERIALS[material]

    Ha = np.diag([material.Esa, material.Epa, material.Epa, material.Epa, material.Essa])
    Hc = np.diag([material.Esc, material.Epc, material.Epc, material.Epc, material.Essc])
    Hac = _cation_anion_block(material, kxyz)

    H = np.zeros((10, 10), dtype=np.complex128)
    H[0:5, 0:5] = Ha
    H[5:10, 5:10] = Hc
    H[0:5, 5:10] = Hac
    H[5:10, 0:5] = Hac.conj().T
    return H

direct_gap_at_gamma ¶

direct_gap_at_gamma(material)

Conduction-band-minimum minus valence-band-maximum at k=Gamma, for the 8 valence electrons (4 lowest of the 8 sp3-derived bands filled; the 2 s bands are always empty conduction states in this model). Only meaningful as the* fundamental gap for direct-gap materials (e.g. GaAs) -- for indirect-gap materials (e.g. Si) this is the Gamma-Gamma transition, not the true (lower) indirect gap, which occurs off Gamma and this function does not compute.

Source code in dense_evolution/solvers/vhd_tb.py

def direct_gap_at_gamma(material):
    """Conduction-band-minimum minus valence-band-maximum at k=Gamma,
    for the 8 valence electrons (4 lowest of the 8 sp3-derived bands
    filled; the 2 s* bands are always empty conduction states in this
    model). Only meaningful as *the* fundamental gap for direct-gap
    materials (e.g. GaAs) -- for indirect-gap materials (e.g. Si) this
    is the Gamma-Gamma transition, not the true (lower) indirect gap,
    which occurs off Gamma and this function does not compute."""
    H = sp3s_star_hamiltonian((0.0, 0.0, 0.0), material)
    eig = np.sort(np.linalg.eigvalsh(H).real)
    return eig[4] - eig[3]

band_extrema_along_path ¶

band_extrema_along_path(
    material, k_start, k_end, n_points=501
)

Scans the valence-band maximum and conduction-band minimum (bands 3 and 4 of the 10, 0-indexed, sorted) along the straight line from k_start to k_end (both in the same 2*pi/a cubic-axis units as sp3s_star_hamiltonian), and returns (vbm, vbm_k, cbm, cbm_k, gap) where vbm_k/cbm_k are the k-points (same units) where each extremum was found.

Needed for indirect-gap materials (e.g. Si, Ge): the true fundamental gap is not at Gamma, so direct_gap_at_gamma alone gives the wrong (too-large) number for them. Example: Si's conduction band minimum sits off-Gamma along Gamma->X (the Delta line), not at Gamma itself.

Source code in dense_evolution/solvers/vhd_tb.py

def band_extrema_along_path(material, k_start, k_end, n_points=501):
    """
    Scans the valence-band maximum and conduction-band minimum (bands 3
    and 4 of the 10, 0-indexed, sorted) along the straight line from
    k_start to k_end (both in the same 2*pi/a cubic-axis units as
    sp3s_star_hamiltonian), and returns
    (vbm, vbm_k, cbm, cbm_k, gap) where vbm_k/cbm_k are the k-points
    (same units) where each extremum was found.

    Needed for indirect-gap materials (e.g. Si, Ge): the true
    fundamental gap is not at Gamma, so direct_gap_at_gamma alone gives
    the wrong (too-large) number for them. Example: Si's conduction
    band minimum sits off-Gamma along Gamma->X (the Delta line), not
    at Gamma itself.
    """
    if isinstance(material, str):
        if material not in MATERIALS:
            raise ValueError(f"no VHD parameters for {material!r}; available: {sorted(MATERIALS)}")
        material = MATERIALS[material]

    k_start = np.asarray(k_start, dtype=float)
    k_end = np.asarray(k_end, dtype=float)
    t = np.linspace(0.0, 1.0, n_points)
    ks = k_start[None, :] + t[:, None] * (k_end - k_start)[None, :]

    vbm_per_k = np.empty(n_points)
    cbm_per_k = np.empty(n_points)
    for i, k in enumerate(ks):
        eig = np.sort(np.linalg.eigvalsh(sp3s_star_hamiltonian(k, material)).real)
        vbm_per_k[i] = eig[3]
        cbm_per_k[i] = eig[4]

    vbm_idx = vbm_per_k.argmax()
    cbm_idx = cbm_per_k.argmin()
    vbm, vbm_k = vbm_per_k[vbm_idx], ks[vbm_idx]
    cbm, cbm_k = cbm_per_k[cbm_idx], ks[cbm_idx]
    return vbm, vbm_k, cbm, cbm_k, cbm - vbm

See also: harrison_tb for the zero-per-material-fitting universal alternative, when the ~2-3x gap error is an acceptable tradeoff for not needing a fitted parameter row per material.