Extending Solid-State Calculations to Ultra-Long-Range Length Scales

We present a method that enables solid-state density functional theory calculations to be applied to systems of almost unlimited size. Computations of physical effects up to the micron length scale but which nevertheless depend on the microscopic details of the electronic structure, are made possible. Our approach is based on a generalization of the Bloch state, which involves an additional sum over a finer grid in reciprocal space around each $\mathbf{k}$ point. We show that this allows for modulations in the density and magnetization of arbitrary length on top of a lattice-periodic solution. Based on this, we derive a set of ultra-long-range Kohn-Sham equations. We demonstrate our method with a sample calculation of bulk LiF subjected to an arbitrary external potential containing nearly 3500 atoms. We also confirm the accuracy of the method by comparing the spin density wave state of bcc Cr against a direct supercell calculation starting from a random magnetization density. Furthermore, the spin spiral state of $\gamma$-Fe is correctly reproduced.

Figure 1

(a) Schematic of the $\mathit{\kappa }$-point grid. For each $\mathbf{k}$ point (black dashed line) all bands (blue) are augmented with a fine grid of $\mathit{\kappa }$ points (green). Three different types of couplings between $\mathit{\kappa }$ points corresponding to different length scales are possible. (i) A coupling between two identical $\mathit{\kappa }$ points but with different band indices (ii) a coupling between different $\mathit{\kappa }$ points sharing the same band index, and (iii) a coupling between different $\mathit{\kappa }$ points with different band indices. The maximum length scale of the calculation may be chosen by adjusting the $\mathit{\kappa }$-point grid. (b) A schematic of the long-range approach. The red lines indicate unit cells. The lattice periodic density ${\rho }_{\mathbf{Q}}$ (blue) is altered by a $\mathbf{Q}$-dependent modulation (orange) with a different periodicity. The result (lower graph) depends on both, the long-range modulation and the lattice periodic solution. $a$ is the lattice constant of a unit cell and $A$ is the lattice constant of the ultracell, which is the smallest cell that contains the full long-range solution.

Figure 2

(a) Ultra-long-range magnetization density of $\gamma$-Fe plotted in the plane perpendicular to $[001]$. The color indicates the magnitude of the magnetization and the arrows indicate the direction. The modulation encompasses 32 unit cells in the $[100]$ direction. (b) Plot of moment against unit cell volume for both the long-range and spin-spiral ansatz.

Figure 3

(a) Magnetization density for bcc Cr over 21 unit cells. (b) Change in density over the same range. For the ultracell, this was generated by setting ${\rho }_{\mathbf{Q}=0}(\mathbf{r})$ in Eq. (6) to zero. For the supercell, the lattice-periodic density was subtracted leaving just the modulated density.

Figure 4

Self-consistent density without the ${\rho }_{\mathbf{Q}=0}(\mathbf{r})$ term for a 3456 atom ultracell of LiF with an artificial external potential. The plotting plane is perpendicular to $[001]$ and contains $48\times 36$ unit cells.