Precise quantum-geometric electronic properties from first principles

TL;DR

This work presents a first-principles method to compute quantum-geometric properties of Bloch electrons—Berry curvature $\Omega$, quantum metric $g$, orbital magnetic moment, and inverse effective masses—from derivatives of cell-periodic Bloch functions with respect to ${\bf k}$, using perturbation theory solved via a Sternheimer equation in a pseudopotential plane-wave framework. Degenerate perturbation theory is carefully incorporated to handle band crossings, enabling accurate, direction-dependent transport masses alongside geometric tensors in a single, efficient pipeline. The method is validated on Si and GaAs against finite-difference benchmarks and contrasted with a two-band model in gapped graphene, while trigonal tellurium demonstrates sign changes tied to chirality, illustrating the physical impact of quantum geometry. Overall, the approach provides numerically precise, unified ab initio access to key geometric and transport properties across diverse materials, with an open-source implementation for broad use.

Abstract

The calculation of quantum-geometric properties of Bloch electrons -- Berry curvature, quantum metric, orbital magnetic moment and effective mass -- was implemented in a pseudopotential plane-wave code. The starting point was the first derivative of the periodic part of the wavefunction psi_k with respect to the wavevector k. This was evaluated with perturbation theory by solving a Sternheimer equation. Comparison of effective masses obtained from perturbation theory for silicon and gallium arsenide with carefully-converged numerical second derivatives of band energies confirms the high precision of the method. Calculations of quantum-geometric quantities for gapped graphene were performed by adding a bespoke symmetry-breaking potential to first-principles graphene. As the two bands near the opened gap are reasonably isolated, the results could be compared with those obtained from an analytical two-band model, allowing to assess the strengths and limitations of such widely-used models. The final application was trigonal tellurium, where quantum-geometric quantities flip sign with chirality.

Figures (12)

• Figure 1: A flowchart of the procedure for calculating the quantum-geometric quantities and transport equivalent effective masses. The draft user guide, with the instructions to perform these calculations and interpret the results, is available from the same website as the code cpw2000.
• Figure 2: The dependence on direction of the second derivative of the band energy (inverse effective mass) is shown for the top of the valence band of Si at the $\Gamma$ point. The distance from the center of each figure to the surface is proportional to the second derivative of the band energy in that direction. The inverse effective masses are clearly anisotropic, and cubic symmetry is respected. The scale of the two figures is not the same.
• Figure 3: The valence bands (blue) and conduction bands (red) of GaAs are shown for the main symmetry directions of the fcc lattice. The highly magnified behavior of the hole bands near the top of the valence band is shown in the insets, where the energy range is $10^{-5}\,\,\text{eV}$ and the wavevector range is $1.18 \times 10^{-3}$ atomic units. The magenta lines connect the insets to the regions of the main plot that were magnified.
• Figure 4: The effective masses near the top of the valence bands of GaAs are shown as a function of the deviation from the $\Gamma$ point. The horizontal scale is "shifted logarithmic" as described in the text, since the system has several energy and wavevector scales. The split-off hole mass is almost constant. The light- and heavy-hole masses are constant over a sizeable wavevector range, but converge to the same value at $\Gamma$.
• Figure 5: The energy bands of gapped graphene are shown as a function of wavevector distance from (BZ corner) in the direction of the BZ center. The blue squares and red dots are the valence and conduction bands obtained by adding a bespoke potential to the DFT calculation. The dashed-black lines and solid-green lines both pertain to the low-energy model, and are obtained by inserting slightly different sets of parameters into Eq. \ref{['eq:twoband']}.