Temperature-dependent phonon spectra of magnetic random solid solutions

TL;DR

This paper introduces a first-principles method to compute phonon spectra in magnetic random solid solutions, accounting for thermal magnetic fluctuations, and explains unusual phonon hardening observed experimentally in Fe-Pd and Fe-Pt alloys.

Contribution

It develops a novel computational approach that incorporates magnetic and chemical disorder effects on phonons, explaining temperature-dependent phonon behavior in magnetic alloys.

Findings

Accurately predicts phonon hardening in Fe-Pd and Fe-Pt alloys with temperature.

Shows thermal magnetic fluctuations cause unusual phonon behavior.

Provides insights into phonon-magnetic interactions in disordered alloys.

Abstract

A first-principles-based method for computing phonons of magnetic random solid solutions including thermal magnetic fluctuations is developed. The method takes fluctuations of force constants (FCs) due to magnetic excitations as well as due to chemical disorder into account. The developed approach correctly predicts the experimentally observed unusual phonon hardening of a transverse acoustic mode in Fe-Pd an Fe-Pt Invar alloys with increasing temperature. This peculiar behavior, which cannot be explained within a conventional, harmonic picture, turns out to be a consequence of thermal magnetic fluctuations.

Equations

$${\Phi }_{\text{M–M′}}(T)$$

Full text

Magnetic random solid solutions represent a large and important class of crystalline materials ranging from structural materials such as steels Muñoz et al. (2011); Razumovskiy et al. (2011); Vitos et al. (2002, 2003), including Invar alloys Weiss (1963); van Schilfgaarde et al. (1999); Khmelevskyi et al. (2003); Yokoyama and Eguchi (2011, 2013), up to multi-component magnetic high-entropy alloys Zhang et al. (2015); Granberg et al. (2016); Li et al. (2016). The simultaneous presence of chemical disorder and thermal magnetic fluctuations as well as their couplings to lattice vibrations play pivotal roles in many of these alloys. Lattice vibrations largely dominate thermodynamic properties of materials Fultz (2010) and contribute to phase stability Fultz (2014), which is a key parameter for the computational design of new and innovative materials. A computational scheme that can simulate lattice vibrations in magnetic random solid solutions by properly taking into account both, magnetic fluctuations as well as chemical disorder (as sketched in Fig. 1), is therefore of genuine importance.

The delicate interactions between lattice vibrations, chemical disorder, and thermal magnetic fluctuations can cause extreme and unusual physical properties. A prominent example is the hardening of a transverse acoustic phonon mode and elastic constants with increasing temperature in Invar alloys Sato et al. (1982); Kästner et al. (1999). Since thermal expansion — usually dominating the temperature dependence of phonon modes — is in such alloys negligible Weiss (1963); van Schilfgaarde et al. (1999); Khmelevskyi et al. (2003); Yokoyama and Eguchi (2011, 2013), the inclusion of explicit temperature dependent excitations, such as magnetic fluctuations, is critical to resolve such peculiarities.

In the last few years significant progress has been made for the computation of lattice vibrations incorporating thermal magnetic fluctuations for pure elements such as Fe Leonov et al. (2012, 2014); Ruban and Razumovskiy (2012); *Ruban2014E; Körmann et al. (2012, 2014); Ikeda et al. (2014); Alling et al. (2016) and Ni Körmann et al. (2016) as well as for several ordered magnetic compounds Fennie and Rabe (2006); Shulumba et al. (2014); Zhou et al. (2014); Gruner et al. (2015); Dutta et al. (2016). While these methods clearly highlight the recent progress in first-principles-based thermodynamic approaches for magnetic systems, they are still limited to chemically ordered systems. To investigate magnetic random solid solutions, methods capable of including chemical disorder are required. For random solid solutions without thermal magnetic fluctuations, computational methods for lattice vibrations have also been advanced significantly in the last years. Among such methods, the itinerant coherent potential approximation (ICPA) Ghosh et al. (2002); *Ghosh2003E and the band unfolding Boykin et al. (2007); Allen et al. (2013a); *Allen2013E; Ikeda et al. (2017) have been used to successfully compute lattice vibrations for random solid solutions Dutta and Ghosh (2009a, b); Dutta et al. (2010); Dutta and Ghosh (2011); Grånäs et al. (2012); Boykin et al. (2014); Huang (2015); Zheng and Zhang (2016); Overy et al. (2016); Ikeda et al. (2017). However, a computational method for lattice vibrations considering both, thermal magnetic fluctuations and chemical disorder, is lacking so far.

We therefore propose a first-principles-based method to calculate lattice vibrations of magnetic random solid solutions, which addresses both, thermal magnetic fluctuations as well as chemical disorder on equal footing. This is achieved by extending and combining above-mentioned disjunct approaches into a methodological framework, which allows to predict temperature-dependent phonon spectra of magnetic random solid solutions. To achieve this goal, we apply a two-step procedure and adiabatically decouple the (fast) magnetic and the (slow) chemical degrees of freedom. To include thermal magnetic fluctuations, we utilize force constants (FCs) which implicitly depend on the magnetic temperature Körmann et al. (2014). Such FCs are obtained using the spin-space averaging (SSA) method Körmann et al. (2012) in combination with quantum Monte Carlo (QMC) simulations for an effective Heisenberg spin Hamiltonian. To account for chemical disorder, which induces variations of atomic masses and FCs among atomic sites, we employ the ICPA Ghosh et al. (2002); *Ghosh2003E and the band unfolding Boykin et al. (2007); Allen et al. (2013a); *Allen2013E; Ikeda et al. (2017).

In the first step, the FCs as functions of magnetic temperature are obtained as follows. First the FCs in the low-temperature “ideal” ferromagnetic (FM) state, where all the magnetic moments on atoms point to the same direction, and the FCs in the high-temperature “ideal” paramagnetic (PM) state, where the direction of the magnetic moments are fully disordered, are calculated. Particularly the FCs in the ideal PM state are obtained using the SSA method Körmann et al. (2012); Zhou et al. (2014), i.e., by a statistical average over a large set of randomly distributed collinear magnetic moments. Then the element-resolved FCs \Phi_{\textrm{M--M{}^{\prime}}}(T) for each pair of chemical elements M and M*′* at temperature $T$ are obtained by extending the recently developed magnon-phonon coupling formalism Körmann et al. (2014) to random solid solutions as

[TABLE]

where \Phi^{\textrm{FM}}_{\textrm{M--M{}^{\prime}}} and \Phi^{\textrm{PM}}_{\textrm{M--M{}^{\prime}}} denote the element-resolved FCs in the ideal FM and in the ideal PM states, respectively. The interpolation parameter $\alpha(T)$ is directly related to the magnetic energy Körmann et al. (2014) [see also Sec. B in Supplemental Material (SM)]. Spin quantization effects for the magnetic energy and hence for $\alpha(T)$, being critical below $T_{C}$ Körmann et al. (2010), are incorporated by performing numerically exact QMC simulations for an effective nearest-neighbor Heisenberg spin Hamiltonian Körmann et al. (2010, 2011, 2013, 2014). Here the solid solution is modeled by randomly distributing the chemical components having different spin values onto the magnetic sites.

In the second step, the thus obtained \Phi_{\textrm{M--M{}^{\prime}}}(T) is employed to derive the phonon spectrum of the magnetic random solid solution at any given $T$. In principle both, the ICPA Ghosh et al. (2002); *Ghosh2003E and the band unfolding Boykin et al. (2007); Allen et al. (2013a); *Allen2013E; Ikeda et al. (2017) can address the variations of atomic masses and FCs among atomic sites due to chemical disorder. The ICPA has the advantage that it analytically incorporates the variations of atomic masses and FCs; a fully random solid solution is rigorously modeled in the ICPA. This method, however, requires the numerical solution of a relatively complex set of equations Ghosh et al. (2002); *Ghosh2003E, making its extension to multicomponent systems challenging. In contrast, the band unfolding can be straightforwardly extended to such systems but is limited by the size of the supercell model; undesired periodicity due to the limited size of the model may cause spurious features in the computed phonon spectra. The possible influences of a limited supercell size can be, however, straightforwardly eliminated by constructing a much larger, effective supercell model in which the \Phi_{\textrm{M--M{}^{\prime}}}(T) computed from the original (smaller) supercell model are assigned to the atomic sites (details of the mapping are given in the SM). Such an effective supercell model has longer periodicity than the original supercell model and therefore includes a larger number of distinct local configurations of chemical components.

We apply the developed approach to two experimentally well-studied magnetic alloys, namely to disordered face-centered cubic (fcc) Fe0.72Pd0.28 and Fe0.72Pt0.28 alloys. Both Invar alloys reveal the aforementioned characteristic phonon hardening of a $\langle 110\rangle$ (in fractional coordinates for the conventional fcc unit cell) transverse acoustic mode when heated up above the Curie temperature, $T_{C}$ Sato et al. (1982); Kawald et al. (1990). In the following we focus on (i) the impact of chemical disorder as well as the performance of the ICPA and the band unfolding to incorporate it, and, in particular, (ii) the impact of thermal magnetic fluctuations on the phonon spectra.

Chemical and magnetic (for the ideal PM state) disorder in these alloys were simulated by special quasirandom structures (SQSs) Zunger et al. (1990). The SQSs in this study were constructed on the 32-atom $2\times 2\times 2$ supercell of the conventional fcc unit cell (see also Sec. A in SM). Chemical compositions for the SQS supercell models were chosen to be Fe0.75Pd0.25 and Fe0.75Pt0.25, being close to the experimental ones. \Phi^{\textrm{FM}}_{\textrm{M--M{}^{\prime}}} and \Phi^{\textrm{PM}}_{\textrm{M--M{}^{\prime}}} were calculated for the SQS supercell models using the finite-displacement method. Among the computed FCs, those up to the fourth nearest neighbors were taken into account in the following steps. $\alpha(T)$ in Eq. (1) was determined using the effective Heisenberg spin Hamiltonian for a $7\times 7\times 7$ supercell including 1372 magnetic sites (see also Sec. B in SM). In the band-unfolding procedure, effective 864-atom $6\times 6\times 6$ supercell models, where \Phi_{\textrm{M--M{}^{\prime}}}(T) obtained from the $2\times 2\times 2$-SQS models were assigned among the atomic sites, were employed. Since thermal expansion is small for the chosen Invar alloys, fixed lattice constants were applied for the computer simulations, which were taken from experimental data at room temperature; 3.755 Å Sato et al. (1982) (disordered fcc Fe0.72Pd0.28) and 3.749 Å Sumiyama et al. (1976) (disordered fcc Fe0.72Pt0.28). Electronic structures were calculated in the framework of density-functional theory within the generalized gradient approximation of the Perdew-Burke-Ernzerhof from Perdew et al. (1996) using the plane-wave basis projector augmented wave method Blöchl (1994) as implemented in the vasp code Kresse (1995); Kresse and Furthmüller (1996); Kresse and Joubert (1999). A plane-wave energy cutoff of 350 eV was used. Internal atomic positions were fully relaxed while keeping the lattices of the SQS models fixed.

In order to separately discuss the impact of chemical disorder and thermal magnetic fluctuations, we first focus on the ideal FM state [$\alpha(T=0)=1$], i.e., in the absence of thermal magnetic fluctuations. Figure 2 shows the phonon spectra of Fe0.75Pd0.25 and Fe0.75Pt0.25 in the ideal FM state calculated using the band unfolding (top panels) and the ICPA (middle panels). The results obtained from the two different methods are very similar to each other for both alloys, although the formalisms of the ICPA and of the band unfolding look very different. From this analysis we conclude that both methods incorporate the impact of chemical disorder into the calculations of phonon spectra in a similar quality. It is also found that the peak positions of the computed phonon spectra are in good agreement with experimental phonon frequencies.

The spectra also show phonon broadening originating from the variations of atomic masses and FCs among atomic sites due to the chemical disorder. The broadening in the Fe0.75Pd0.25 is relatively small, while Fe0.75Pt0.25 shows large broadening in the frequency region around 3–5 THz. The increased broadening in Fe0.75Pt0.25 occurs probably because of the large differences of atomic masses and FCs among the chemical components. The atomic mass of Pt relative to Fe ($\approx 3.5$) is much larger than that of Pd ($\approx 1.9$). The FCs in Fe0.75Pt0.25 are also largely different among the distinct combinations of the chemical elements compared with those in Fe0.75Pd0.25 (see also Sec. D in SM).

To elucidate the impacts of the variations of atomic masses and FCs among atomic sites more clearly, the phonon dispersion relations are also calculated in the absence of FC and mass fluctuations using the concentration-weighted average atomic mass, $\bar{m}$, and the crystallographically-symmetrized FCs $\bar{\Phi}$. $\bar{\Phi}$ is obtained by first applying each symmetry operation of the fcc structure to the original FCs and then taking the average of the transformed FCs irrespective of chemical components. The results are shown in the bottom panels in Fig. 2. Note that by construction, no phonon broadening is obtained in this case. The phonon frequencies derived from $\bar{m}$ and $\bar{\Phi}$ agree reasonably well with experiment for Fe0.75Pd0.25. For Fe0.75Pt0.25 the deviations are, however, significant, in particular around the X point. This indicates the importance of taking the variations of atomic masses and FCs among atomic sites into account for accurate phonon computations of random solid solutions.

Having verified the importance of an appropriate treatment of chemical disorder in random solid solutions, we next analyze the impact of thermal magnetic fluctuations on the phonon spectra from the viewpoint of their temperature dependence. The temperature-dependent spectra are obtained using \Phi_{\textrm{M--M{}^{\prime}}}(T) from Eq. (1). Calculations are carried out at three representative characteristic temperatures, i.e., below, near, and above $T_{C}$ ($T_{C}$ is 575 K for Fe0.72Pd0.28 Sato et al. (1982) and 367 K for Fe0.72Pt0.28 Kawald et al. (1990)). We focus in particular on the lowest-frequency phonon branch along the $\langle 110\rangle$ direction, which shows the unusual hardening with increasing temperature. Note that within the harmonic approximation and in absence of explicit temperature dependent excitations, phonon spectra only implicitly depend on temperature via the volume expansion. However, as mentioned above, the considered Invar alloys reveal negligible expansion below $T_{C}$. A conventional harmonic approximation would therefore predict a temperature-independent (i.e. constant) phonon spectra and cannot explain the experimental data.

The results employing the band-unfolding method are shown in Figure 3 111Note that similarly to the discussed ideal FM state (Fig. 2), the ICPA results are again in excellent agreement with the band-unfolding results for the intermediate temperatures and therefore shown only in SM.. For comparison, the results for the ideal FM and the ideal PM states are also shown. The peak positions of the computed temperature-dependent phonon spectra are in agreement with experimental phonon frequencies at the corresponding temperatures, while the results for the ideal FM and the ideal PM states provide the lower and the upper bounds, respectively. This indicates that thermal magnetic fluctuations are crucial to reproduce the experimental data.

Our simulations further indicate the presence of considerable magnetic short-range ordering (SRO) (i.e., a finite value of $\alpha(T)$ in Fig. 3) even above $T_{C}$, similar as already observed for pure Fe Körmann et al. (2014). Without taking magnetic SRO into account, i.e., by limiting the simulations solely to the ideal PM state, the agreement with experiment is lost. Magnetic SRO hence plays a crucial role, not only for pure Fe, but also for Fe-based alloys. We also observe slight deviations between our simulations and the experiment above $T_{C}$. We attribute these deviations to the increase of lattice constants with temperature above $T_{C}$ (the Invar effect is lost), which is not yet included in our calculations.

From the simulated temperature dependence of the phonon modes, we can also deduce information on the martensitic transformation for the investigated alloys. In Fig. 3, the slope of the lowest-frequency branch along the $\langle 110\rangle$ direction around the $\Gamma$ point largely depends on temperature for both Fe0.75Pd0.25 and Fe0.75Pt0.25; the slope increases with increasing temperature. The slope of this branch is associated with the elastic stiffness constant $C^{\prime}\equiv(C_{11}-C_{12})/2$, which is related to the dynamical stability for the martensitic transformation along the Bain path Bain and Dunkirk (1924). The softening of this branch when lowering the temperature is considered as a precursor of the martensitic transformation. In experiments disordered fcc Fe1-xPdx and Fe1-xPtx ($x=$ 0.25–0.33) indeed show a martensitic transformation; for example, Fe0.72Pd0.28 and Fe0.73Pt0.27 show a martensitic transformation at 314.5 K Arabi-Hashemi et al. (2015) and approximately at 125 K Sumiyama et al. (1983), respectively. Thus, our computed result suggests that the martensitic transformation for these alloys originates from magnon-phonon coupling.

In summary, we propose a first-principles-based method to incorporate both, thermal magnetic fluctuations and chemical disorder into a unified computational framework of phonon calculations for magnetic random solid solutions. Chemical disorder, which leads to variations of atomic masses and FCs among atomic sites, is taken into account using the ICPA and the band unfolding in combination with SQSs. Thermal magnetic fluctuations are incorporated using the SSA in combination with QMC simulations for an effective Heisenberg spin Hamiltonian.

The proposed methodology is applied to Fe–Pd and Fe–Pt Invar alloys. Both, the ICPA and the band unfolding are found to be equally capable for computing phonon spectra of the chemically disordered alloys. Both methods also reveal phonon broadening due to the chemical disorder. Taking thermal magnetic fluctuations into account, the developed approach shows excellent agreement between the temperature-dependent peak positions of the computed phonon spectra and experimental phonon frequencies. In particular, the approach correctly reproduces the experimentally observed unusual hardening in iron-based Invar alloys at high temperatures and shows that magnetic fluctuations are responsible for it. Finite-temperature magnon-phonon contributions also cause the experimentally observed softening of the elastic constant $C^{\prime}$ and hence trigger the martensitic transformation in these alloys.

The proposed approach is general and can be straightforwardly employed to examine other complex magnetic random solid solutions such as magnetic high-entropy alloys, which are expected to reveal complex physical mechanisms caused by the interplay between thermal magnetic fluctuations, chemical disorder, and lattice vibrations.