Lattice Green function for extended defect calculations: Computation and error estimation with long-range forces

Computing the atomic geometry of lattice defects—e.g., point defects, dislocations, crack tips, surfaces, or boundaries—requires an accurate coupling of the local deformations to the long-range elastic field. Periodic or fixed boundary conditions used by classical potentials or density-functional theory may not accurately reproduce the correct bulk response to an isolated defect; this is especially true for dislocations. Flexible boundary conditions have been developed to produce the correct long-range strain field from a defect—effectively “embedding” a finite-sized defect with infinite bulk response, isolating it from either periodic images or free surfaces. Flexible boundary conditions require the calculation of the bulk response with the lattice Green function (LGF). While the LGF can be computed from the force-constant matrix, the force-constant matrix is only known to a maximum range. This paper illustrates how to accurately calculate the lattice Green function and estimate the error using a truncated force-constant matrix combined with knowledge of the long-range behavior of the lattice Green function. The effective range of deviation of the lattice Green function from the long-range elastic behavior provides an important length scale in multiscale quasicontinuum and flexible boundary-condition calculations, and measures the error introduced with periodic-boundary conditions.

Figure 1

(Color online) Separation of lattice Green function for a square lattice in two-dimensional reciprocal space into elastic Green function, discontinuity correction, and semicontinuum function (note different vertical scales). The lattice Green function has the periodicity of the reciprocal lattice and a second-order pole at the gamma point. The elastic Green function scales as $k^{-2}$ and is cut off to smoothly go to zero at the Brillouin-zone edges. The removal of the second-order pole creates a discontinuity independent of $\left|k\right|$ at the gamma point; the discontinuity correction removes the discontinuity and smoothly goes to zero at the Brillouin-zone edges. The remaining difference between the lattice Green function and the first two terms is the semicontinuum function, which is smooth everywhere in the Brillouin zone.

Figure 2

(Color online) Connection of solution of continuum differential equation mapped onto a lattice equation. The continuum differential equation that defines the solution on the left can be discretized by introducing a grid and approximating derivatives with finite differences over the grid to produce a lattice equation. In the limit that the grid spacing becomes small compared to the length scale of variation of the solution, the discrete approximation matches the continuum solution. This mapping can also be reversed by starting with a grid and a lattice equation, and then taking the limit of zero grid spacing to produce a corresponding continuum differential equation.

Figure 3

(Color online) Relative deviation between EGF and LGF for face-centered-cubic test case. The points give the deviation between the lattice Green function, computed with the full force-constant matrix, and the elastic Green function. The region 1 and 2 estimates are computed using the folded force-constant matrix in cubic supercells, and combined, as in Eq. (11), to produce the total deviation estimate. (a) Single random force-constant matrix shows an individual example of error estimation. (b) 100 different random force-constant matrices were computed along with their associated LGFs. The average deviation over the ensemble average shows accurate computation of the error even for the case of small supercells $\left(2\times 2\times 2\right)$ with the long-range force-constant matrix (cutoff at $15a_0$).