Optimization of Ab-Initio Based Tight-Binding Models

TL;DR

This work tackles automating tight-binding model construction directly from ab-initio band structures to enable large-scale, inhomogeneous material simulations. It introduces a conjugate-gradient–based optimization scheme, grounded in first-order perturbation theory, to fit TB parameters with a minimal, progressively extended neighbor set and a preconditioned, least-squares update, achieving accurate bandstructures with fewer orbitals than conventional Wannier approaches. The method demonstrates comparable accuracy to Wannier90 with far fewer parameters, avoids disentanglement errors in metals, and yields smooth Fermi surfaces suitable for transport analyses. Overall, the approach offers a fast, automated pathway to transferable tight-binding models that are well-suited for high-throughput and large-scale materials calculations.

Abstract

The electronic structure of solids can routinely be calculated by standard methods like density functional theory. However, in complicated situations like interfaces, grain boundaries or contact geometries one needs to resort to more simplified models of the electronic structure. Tight-binding models are using a reduced set of orbitals and aim to approximate the electronic structure by short range hopping processes. For example, maximally localized Wannier functions are often used for that purpose. However, their accuracy is limited by the need to disentangle the electronic bands. Here, we develop and investigate a different procedure to obtain tight-binding models inspired by machine-learning techniques. The model parameters are optimized in such a way as to reproduce ab-initio band structure data as accurately as possible using an as small as possible number of model parameters. The procedure is shown to result in models with smaller ranges and fewer orbitals than maximally localized Wannier functions but same or even better accuracy. We argue that such a procedure is more useful for automated construction of tight-binding models particularly for large-scale materials calculations.

Figures (7)

• Figure 1: Convergence of Gradient Descent and the improved method based on least squares. The iterations of the x-axis are those of the whole procedure. The CG iterations have been limited to 4.
• Figure 2: Graphene and $\mathrm{ MoS}_{2}$ with neighbors $(1,0,0)$, $(3/2,\sqrt{ 3 }/2,0)$. The reference DFT data is dashed. The top two bands are fitted with a low weight.
• Figure 3: Bandstructures for Cu and Si with neighbors $(0,1/2,1/2)$, $(0,0,1)$, $(1/2,1/2,1)$, K with neighbors $(1/2,1/2,1/2)$, $(0,0,1)$, $(0,1,1)$, Bi with neighbors $(0,0.958,0)$, $(0.553,0,0.833)$, $(1.106,0,-0.833)$, NiTi with neighbors $(0,0,1)$, $(0,1,1)$, $(1,1,1)$, $(0,0,2)$ and TaAs with cubic symmetrized normalized neighbors $(1/2,1/2,1/2)$, $(0,0,1)$, $(1/2,1/2,3/2)$. The reference DFT data is dashed.
• Figure 4: Maximal errors of the tight-binding models for wannier90 with all neighbors, our model best model with 16 neighbors and cut-off Wannier90 to match our model in neighbor count. All models are fit on a $16^{3}$ grid, but evaluated on a $24^{3}$ grid. The vertical line is the Fermi energy and the dashed vertical line is the upper edge of the frozen energy window.
• Figure 5: Error of Wannier90 ($\times$) compared to our method ($\square$), when used for interpolation with a finite number of neighbors up to some maximal magnitude. The open squares show the result with randomized restarts. The diamonds are results with increased band count.