Microscopic first-principles model of strain-induced interaction in concentrated size-mismatched alloys

The harmonic Kanzaki-Krivoglaz-Khachaturyan model of strain-induced interaction is generalized to concentrated size-mismatched alloys and adapted to first-principles calculations. The configuration dependence of both Kanzaki forces and force constants is represented by real-space cluster expansions that can be constructed based on the calculated forces. The model is implemented for the fcc lattice and applied to ${\mathrm{Cu}}_{1-x}{\mathrm{Au}}_x$ and ${\mathrm{Fe}}_{1-x}{\mathrm{Pt}}_x$ alloys for concentrations $x=0.25$, 0.5, and 0.75. The asymmetry between the $3d$ and $5d$ elements leads to large quadratic terms in the occupation-number expansion of the Kanzaki forces and thereby to strongly non-pairwise long-range interaction. The main advantage of the full configuration-dependent lattice deformation model is its ability to capture this singular many-body interaction. The roles of ordering striction and anharmonicity in Cu-Au and Fe-Pt alloys are assessed. Although the harmonic force constants defined with respect to the unrelaxed lattice are unsuitable for the calculation of the vibrational entropies, the phonon spectra for ordered and disordered alloys are found to be in good agreement with experimental data. The model is further adapted to concentration wave analysis and Monte Carlo simulations by means of an auxiliary multiparametric real-space cluster expansion, which is used to find the ordering temperatures. Good agreement with experiment is found for all systems except ${\mathrm{CuAu}}_3$ (due to the known failure of the generalized gradient approximation) and ${\mathrm{FePt}}_3$, where the discrepancy is likely due to the neglect of magnetic disorder.

Figure 1

Fitting of the calculated forces to many-body Kanzaki force models. Each Cartesian component for each site is represented by one data point. Data sets for different concentrations are shifted by 2 units along the $y$ axis. Red open squares: Two-parameter model with nearest-neighbor Kanzaki forces. Blue opaque circles: Complete model; see text and Table 1.

Figure 2

Fitting of the force constants referenced from the ideal fcc lattice for (a) Cu-Au and (b) Fe-Pt alloys. The forces predicted by the fit are plotted against the calculated forces. Each Cartesian component for each site is represented by one data point. Data sets for different concentrations are shifted along the $y$ axis. Filled (blue) circles and open squares correspond to the models with configuration-dependent and configuration-independent force constants, respectively. See text for details about these models.

Figure 3

Relaxation energies predicted using the lattice-deformation model with configuration-dependent (filled circles) or fixed (open squares) force constants. All data for the Fe-Pt system are shifted upwards by 100 meV. Some data points for the model with fixed force constants corresponding to strongly relaxing structures are beyond the field of the figure.

Figure 4

16-atom structure X competing with $\text{L}{1}_0$ in the CuAu system.

Figure 5

Phonon spectra calculated using a $8\times 8\times 8$ cubic cell containing 2048 atoms and the fitted force constants. Solid lines: Random alloy. Dashed lines: Fully ordered alloy ($\text{L}{1}_0$ for 50% and $\text{L}{1}_2$ for 1:3 compositions). Black lines: Total phonon DOS; red and blue lines: partial DOS for $3d$ and $5d$ elements, respectively. Green circles: Neutron-weighted phonon DOS for disordered Cu-Au alloys from Fig. 1 of Ref. [39]. Green dot-dashed line for ${\mathrm{Cu}}_3\text{Au}$: Phonon DOS for the $\text{L}{1}_2$ phase corrected for neutron weighting from Ref. [39]. Green dot-dashed lines for ${\mathrm{FePt}}_3$ and ${\mathrm{Fe}}_3\text{Pt}$: Phonon DOS for the $\text{L}{1}_2$ phases calculated in Ref. [40] from the force constants fitted to experimental phonon dispersions. The frequency scale is in meV; DOS is in arbitrary units.

Figure 6

Accuracy of the real-space cluster expansions of the CLDM relaxation energy for (a) Cu-Au and (b) Fe-Pt alloys. Each structure is represented by a data point. Data sets for different concentrations are shifted along the $y$ axis.

Figure 7

Effective cluster interactions (ECI) representing the strain-induced interaction in Cu-Au (left panel) and Fe-Pt (right panel) alloys. Red circles: Pair ECI; blue diamonds: many-body ECI. Large empty symbols: These ECIs were divided by 5 to fit in the figure. The ECIs are multiplied by the multiplicity factor giving the number of the corresponding clusters per lattice site. OV: Optimized volume.

Figure 8

Effective pair interaction $J_{\mathrm{eff}}\left(\mathbf{k}\right)$ (meV units) in Cu-Au and Fe-Pt alloys for small deviations from the random alloy. “OV” stands for optimized volume. Dashed green: Chemical contribution; solid blue: strain-induced contribution; dotted red: homogeneous strain contribution (only for CuAu); thick black lines: total effective interaction. The dotted segments of the lines near the $\Gamma$ point show the region where the strain-induced contribution is not correctly captured by the auxiliary cluster expansion.