Vogl-Hjalmarson-Dow Tight-Binding (material-specific parameters)
sp3s* tight-binding Hamiltonian builder using material-specific fitted
parameters -- far more accurate than harrison_tb's
universal table, at the cost of needing a separately fitted parameter row
per material instead of one table for everything.
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 (MATERIALS) are
transcribed from an independent open-source implementation,
github.com/rpmuller/TightBinding
(TB.py), which cites the same paper. 16 zinc-blende/diamond
semiconductors included: C, Si, Ge, Sn, SiC, AlP, AlAs, AlSb, GaP, GaAs,
GaSb, InP, InAs, InSb, ZnSe, ZnTe.
The method
Same Slater-Koster sp3 machinery as harrison_tb, plus
one addition: an extra s* orbital per atom (5 orbitals/atom instead of
4, 10x10 Hamiltonian per unit cell instead of 8x8). s* has no literal
physical meaning -- it exists purely as a fitting degree of freedom to pull
the lowest conduction band down to the right energy, something a bare sp3
basis structurally cannot do (see harrison_tb's accuracy notes: it
routinely misplaces the conduction-band minimum for indirect-gap
materials). Each element's Material row (Esa/Epa/Esc/Epc/Essa/Essc +
7 hopping integrals Vss/Vxx/Vxy/Vsapc/Vscpa/Vssapc/Vsscpa) is fitted as a
whole per material, not derived from a universal formula the way
harrison_tb.ETA is.
sp3s_star_hamiltonian(kxyz, material) builds the periodic Bloch
Hamiltonian directly (k in units of $2\pi/a$ along the conventional cubic
axes: $\Gamma=(0,0,0)$, $X=(1,0,0)$, $L=(0.5,0.5,0.5)$).
direct_gap_at_gamma reads off the gap at $\Gamma$ -- only the true
fundamental gap for direct-gap materials. For indirect-gap materials
(Si, Ge), the real conduction-band minimum sits off-$\Gamma$, so
band_extrema_along_path(material, k_start, k_end) scans a k-space line
and finds the true valence-band max / conduction-band min instead of
reading only $\Gamma$.
Accuracy
Validated against real experimental gaps (GaAs, Si, Ge) -- see the Dense-Evolution-Discovery validation page for the full writeup and numbers.
::: dense_evolution.solvers.vhd_tb
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.