Phase stability, ordering tendencies, and magnetism in single-phase fcc Au-Fe nanoalloys

Bulk Au-Fe alloys separate into Au-based fcc and Fe-based bcc phases, but ${\mathrm{L}1}_0$ and ${\mathrm{L}1}_2$ orderings were reported in single-phase Au-Fe nanoparticles. Motivated by these observations, we study the structural and ordering energetics in this alloy by combining density functional theory (DFT) calculations with effective Hamiltonian techniques: a cluster expansion with structural filters, and the configuration-dependent lattice deformation model. The phase separation tendency in Au-Fe persists even if the fcc-bcc decomposition is suppressed. The relative stability of disordered bcc and fcc phases observed in nanoparticles is reproduced, but the fully ordered ${\mathrm{L}1}_0$ AuFe, ${\mathrm{L}1}_2{\mathrm{Au}}_3\mathrm{Fe}$, and ${\mathrm{L}1}_2{\mathrm{AuFe}}_3$ structures are unstable in DFT. However, a tendency to form concentration waves at the corresponding [001] ordering vector is revealed in nearly random alloys in a certain range of concentrations. This incipient ordering requires enrichment by Fe relative to the equiatomic composition, which may occur in the core of a nanoparticle due to the segregation of Au to the surface. Effects of magnetism on the chemical ordering are also discussed.

Figure 1

Formation enthalpies of ferromagnetic, fully relaxed fcc (black) and bcc (red) ordered Au-Fe structures (only those with up to five atoms per cell are shown). Open symbols: DFT results; crosses or pluses: CE predictions; lines: CE predictions for random alloys (see legend). Diamonds (purple): unmappable structures (see text). Dotted lines: asymptotic tangents estimated for dilute alloys (see Appendix pp5).

Figure 2

Phase diagram of bulk Au-Fe based on the experimental data from Ref. [3]. Shaded areas: single-phase regions (blue: bcc; gray: fcc; yellow: liquid). Point A: predicted intersection of the bcc and fcc random-alloy formation enthalpies from Fig. 1. Point C: eutectoid point. CDE (green): two-phase ($\alpha +\gamma$) region. Point B lies directly above point A on the continuation of the DC line. The martensitic transformation line $T_{\text{mart}}\left(x\right)$, which shows equilibrium between random single-phase fcc and bcc alloys, is predicted to lie inside the hashed (red) area ABCDEA. Labels (“fcc” and “bcc”) indicate the regions of relative stability of these random alloys.

Figure 3

Final scaled ratio of fcc/bcc scores after geometry relaxation for structures initially at (a) ideal fcc and (b) ideal bcc positions. The shaded areas highlight structures that relax to a different type of lattice geometry.

Figure 4

Total energy [(a)–(c)] and stress tensor components [(d)–(f)] as a function of the $c/a$ ratio plotted along the volume-conserving Bain path of $\mathrm{L}{1}_0$ AuFe with FM [(a) and (d)], C-type AFM [(b) and (e)], and G-type AFM [(c) and (f)] spin orderings. The volume of the equilibrium FM phase is used throughout, and it is also taken as the energy reference. The stress components are ${\sigma }_{xx}={\sigma }_{yy}$ (light blue dots) and ${\sigma }_{zz}$ (black dots). Insets in (a)–(c) show structural and magnetic order at the bcc and fcc values of the $c/a$ ratios. Dashed lines are guides for the eye.

Figure 5

Energy as a function of the $c/a$ ratio, along the volume-conserving Bain path, for the paramagnetic $\mathrm{L}{1}_0$ AuFe. Models 1 (gray line) and 2 (black line) average over different magnetic states (see text). The volume of the equilibrium FM phase is used throughout, and it is also taken as the energy reference.

Figure 6

The low-energy ordered Au-Fe structures, as determined by different approaches (cf. Table 1). The conventional cells are shown in most cases, except W2 is displayed within a nonperiodic $2\times 2\times 1$ fcc unit for clarity.

Figure 7

Total effective potential $J_{\mathrm{eff}}\left(\mathbf{k}\right)$ (black) and the individual chemical (green) and relaxation-induced (blue) contributions, plotted along the special directions in the Brillouin zone, at the following concentrations of ${\mathrm{Au}}_{1-x}{\mathrm{Fe}}_x$: (a) $x=0.75$, (b) 0.666, and (c) 0.5. Thick green line: $J_{\mathrm{chem}}\left(\mathbf{k}\right)$. Dashed blue line: $J_{\mathrm{SI}}\left(\mathbf{k}\right)$ from the CLDM-based CE (Appendix pp2). Solid blue line: $J_{\mathrm{SI}}\left(\mathbf{k}\right)$ from S-CLDM [first term in Eq. (C1)]. Dotted red line: $J_{\mathrm{res}}\left(\mathbf{k}\right)$ from Eq. (C1). Dashed black and solid thick black lines: total $J_{\mathrm{eff}}\left(\mathbf{k}\right)$ with $J_{\mathrm{SI}}\left(\mathbf{k}\right)$ from the CLDM-based CE and from S-CLDM, respectively.

Figure 8

Total effective potential $J_{\mathrm{eff}}\left(\mathbf{k}\right)$ in AuFe from S-CLDM [Eq. (C1)] at different volumes. Black, red, green, and blue lines (labeled 1, 2, 3, 4): $a=3.953$ Å (equilibrium at zero pressure); 3.901; 3.8; and 3.7 Å.

Figure 9

Solid lines: formation enthalpy of the random ${\mathrm{Au}}_{1-x}{\mathrm{Fe}}_x$ alloy in the ferromagnetic (FM) and paramagnetic (PM) states from the coherent potential approximation (CPA) calculations [with disordered local moments (DLM) for the PM state]. Dashed line: the mean-field Curie temperature (right axis).

Figure 10

Relaxation energy (per atom) from S-CLDM vs its input value from the full CLDM, for three concentrations. The set of structures is the same as the one used in the CE for $\mathrm{\Delta }{H}_{\mathrm{rel}}$, see Table 5.

Figure 11

Formation enthalpies of substitutional defects (diamonds: Fe in fcc Au matrix, squares: Au in bcc Fe matrix) vs linear supercell size $L$. Filled symbols: uncorrected values; open symbols: including the elastic corrections (only for $L>2$). Multiple symbols for the same $L$ correspond to different (but reasonably dense) $k$-point meshes.