Fermi Surface and Full-Brillouin-Zone Analysis

Part D branches by the observable required to answer the research question. This topic is chosen only when that quantity is needed; it is not a required step in every DFT study.

Rotate the full-zone object

Load dense-mesh eigenvalues into a Fermi-surface viewer such as FermiSurfer, XCrySDen, PyProcar, or an equivalent code-specific tool. Display the Brillouin-zone boundary, rotate the isosurface, identify sheets and pockets, and inspect whether apparent crossings survive changes in mesh and interpolation. Keep the exact energy reference and band selection visible.

A connected line through unrelated mesh points is not a Fermi surface, and a high-symmetry band path cannot exclude an off-path pocket. Find suitable electronic-property tools, visual tools, and specialist post-processing routes before making a topology or metallicity claim.

Use a full-Brillouin-zone calculation when the question depends on where a band reaches a chosen energy: a possible Fermi-surface sheet, pocket, neck, off-path crossing, or extremum. Start from an accepted structure and electronic reference state, then create a compatible uniform-zone eigenvalue field. Compare a Full-Zone Isovalue with a Band-Path Crossing reconstructs a real QE 7.5 aluminium mesh and path. Its current graphic is a two-dimensional projection of near-Fermi samples, not a three-dimensional isosurface; this repository contains no completed Fermi-surface export or topology result.

Produce the full-zone state

Prepare compatible scf.in and nscf.in files with the same structure, pseudopotentials, basis settings, charge, spin/SOC treatment, Hubbard definition, prefix, and accessible outdir. Choose a uniform mesh and enough bands for the energy window being searched.

pw.x -in scf.in > scf.out
grep -E '^[[:space:]]+convergence has been achieved in[[:space:]]+[0-9]+ iterations[[:space:]]*$' scf.out
grep -F "JOB DONE." scf.out

This establishes the parent density and checks, separately, the electronic solver message and normal program termination. It does not establish full-zone resolution.

pw.x -in nscf.in > nscf.out
grep -F "JOB DONE." nscf.out

The NSCF run supplies eigenvalues over the declared uniform mesh. Preserve the coordinates and weights actually represented; disabling symmetry may be useful when a downstream format needs every grid point, but it is not a universal requirement and does not make the mesh converged.

A high-symmetry path cannot establish a Fermi surface

A path calculation may be generated from the same accepted state for visual comparison:

pw.x -in bands.in > bands.out
grep -F "JOB DONE." bands.out

This produces eigenvalues only on the requested lines. A path crossing is not a substitute for the full-zone field and cannot prove metallicity, a pocket, or a fundamental gap.

The object is an equal-energy set over the full zone

For band $n$ , an equal-energy sheet at chemical potential $\mu$ satisfies

\varepsilon_n(\mathbf{k})=\mu.

Use $\mu$ and $\varepsilon_n(\mathbf{k})$ from the same electronic state. Keep the reciprocal basis, periodic-zone convention, mesh, weights, band and spin/SOC labels, occupations, and interpolation method with the surface or contour. In three dimensions the object is generally a surface; in two dimensions it is a contour. A slab calculation should not be presented as an ordinary three-dimensional bulk surface without addressing the intended dimensional reciprocal manifold.

Inspect point and band counts, energy range, finite eigenvalues, chemical-potential marker, coordinate order, and the scalar-field or vertex artifact used by the renderer. A screenshot alone cannot recover the band assignment, pocket volume, mesh, or isovalue.

Interpolation makes a dense mesh practical, not automatically reliable

If Wannier interpolation supplies a denser grid, compare direct and interpolated eigenvalues at held-out points near every claimed sheet. Preserve the subspace, projections, disentanglement and frozen windows, spinor treatment, and interpolation version. A dense smooth surface can still be wrong near a crossing.

Electron and hole labels require a declared reference

Electron- and hole-like labels require local curvature and filling in the declared band model, not the apparent convexity of one rendering. Luttinger’s relation connects the volume enclosed by a Fermi surface to particle density only under its stated translational and many-body assumptions; a Kohn—Sham isosurface is not automatically an experimental carrier density.

Test the feature before naming it

Refine the uniform mesh and vary the isovalue over the numerical and physical uncertainty relevant to the question. Repeat the check for basis settings, number of bands, occupations or smearing, structural and magnetic candidates, and SOC treatment. A pocket, neck, touching point, or topology that disappears under these tests is not established.

Supercells and magnetic order fold the zone; compare with a primitive-cell result only after an explicit mapping or unfolding. For noncollinear SOC, a “spin-up” sheet is generally a projection onto a declared axis rather than a conserved quantum number.

Geometry alone is not a transport calculation

The band velocity is

\mathbf{v}_n(\mathbf{k})=\frac{1}{\hbar}\nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k}),

but sheet geometry and velocity do not supply scattering times, interactions, contacts, or a conductivity tensor. A nesting-like geometry does not establish an instability, electron-phonon coupling, superconductivity, or density-wave response without the corresponding calculation.

Decide whether you have a field or only samples

An eigenvalue table or near-isovalue scatter is sampled evidence, not a surface. A converged full-zone field plus a reproducible three-dimensional export can support a state-specific equal-energy geometry for the declared model, mesh or validated interpolation, and isovalue. Without that object, stop at the sampled crossing statement. It does not establish an experimental Fermi surface, carrier concentration, effective mass, scattering rate, conductivity, quantum-oscillation frequency, instability, superconductivity, topology, material stability, or device performance. Preserve the parent state, full field, mesh/isovalue sensitivity, any path comparison, interpolation checks, rendering method, machine-readable surface data, and hashes.