Predicting interface structures: From $\mathrm{SrTiO}{}_3$ to graphene

We present here a fully first-principles method for predicting the atomic structure of interfaces. Our method is based on the ab initio random structure searching (AIRSS) approach, applied here to treat two-dimensional defects. The method relies on repeatedly generating random structures in the vicinity of the interface and relaxing them within the framework of density functional theory (DFT). The method is simple, requiring only a small set of parameters that can be easily connected to the chemistry of the system of interest, and efficient, ideally adapted to high-throughput first-principles calculations on modern parallel architectures. Being first-principles, our method is transferable, an important requirement for a generic computational method for the determination of the structure of interfaces. Results for two structurally and chemically very different interfaces are presented here, grain boundaries in graphene and grain boundaries in strontium titanate $\left(\mathrm{SrTiO}{}_3\right)$. We successfully find a previously unknown low energy grain boundary structure for the graphene system, as well as recover the previously known higher energy structures. For the $\mathrm{SrTiO}{}_3$ system we study both stoichiometric and nonstoichiometric compositions near the grain boundary and find previously unknown low energy structures for all stoichiometries. We predict that these low energy structures have long-range distortions to the ground state crystal structure emanating into the bulk from the interface.

Figure 1

Graphene grain boundary structures between armchair and zigzag regions. The three lowest energy grain boundary structures are shown, with both GB-II and GB-III significantly lower in energy than GB-I. Also shown is the setup for searching for interface structures using AIRSS.

Figure 2

Grain boundary interface energy as a function of ${\mu }_{{\text{TiO}}_2}$, for the $\Sigma 3\left(111\right)$ grain boundary in $\mathrm{SrTiO}{}_3$. The interface energy for five different stoichiometries, $\Gamma =0;\pm 1;\pm 2$, are shown as solid lines for our results, with the overall lowest energy structure being the $\mathrm{SrO}{}_3$-terminated stoichiometric structure. Also shown in dotted lines are the previously known lowest energy structures for $\Gamma =\pm 1$ from Ref. [8] and the ideal Ti-terminated structure in the $Pm\overline{3}m$ phase.

Figure 3

Predicted atomic structure of the $\mathrm{SrTiO}{}_3$ grain boundaries for stoichiometries $\Gamma =0,\pm 1,\pm 2$. Three structures for the stoichiometric condition $\left(\Gamma =0\right)$ are shown, the two degenerate low energy $\mathrm{SrO}{}_3$-terminated structures and the higher energy Ti-terminated structure. For the lowest energy structure for nonstoichiometric, SrO-rich conditions $\left(\Gamma =+1\right)$, the two adjacent grains are sheared by 0.5 Å along the $\langle 110\rangle$ direction. All structures show significant oxygen displacements from the bulk low temperature $I4/mcm$ phase. The randomization region used for interface prediction is indicated for the $\mathrm{SrO}{}_3$-terminated $\Gamma =0$ structure in gray; all atoms in this region are randomized. The extent of the randomization region is $9.5\text{Å}\times 9.2\text{Å}\times 5.5\text{Å}$ along $\langle 11\overline{2}\rangle ,\langle 111\rangle$, and $\langle 110\rangle$ directions. All structures have two symmetric grain boundaries in the supercell due to periodicity. Red, green, and blue circles represent O, Sr, and Ti atoms, respectively.

Figure 4

Stoichiometric $\left(\Gamma =0\right)$ $\mathrm{SrO}{}_3$-terminated $\mathrm{SrTiO}{}_3$ grain boundary with 240 atoms in the unit cell. The crystal structure of the bulk part is of distorted $I4/mcm$ type. The distortions reach far into the bulk material. The angle $\delta$ that the vector between two O atoms makes with the normal of the grain boundary plane is shown for one set of O atoms at the midpoint between the two periodic grain boundaries. The midpoint between the grain boundaries is indicated by a dashed red line, while the center grain boundary plane is indicated by a dashed black line and its periodic image by solid vertical lines at the edges of the cell. Red, green, and blue circles represent O, Sr, and Ti atoms, respectively.

Figure 5

Distortion angle $\delta$ as a function of distance perpendicular to the grain boundary of Fig. 4. The distortions reach far into the bulk material, where at 14 Å from the grain boundary a significant distortion of $\delta \sim {20}^{\circ }$ can still be observed, as opposed to ${\delta }^0=35.2^{\circ }$ for the bulk $I4/mcm$ bulk without a grain boundary defect. The ideal $I4/mcm$ structure is shown in the inset.