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Uncovering electron scattering mechanisms in NiFeCoCrMn derived concentrated solid solution and high entropy alloys

Uncovering electron scattering mechanisms in NiFeCoCrMn derived concentrated solid solution and... www.nature.com/npjcompumats ARTICLE OPEN Uncovering electron scattering mechanisms in NiFeCoCrMn derived concentrated solid solution and high entropy alloys 1 1 2 1 1 2 2 Sai Mu , German D. Samolyuk , Sebastian Wimmer , Maria C. Troparevsky , Suffian N. Khan , Sergiy Mankovsky , Hubert Ebert and George M. Stocks Whilst it has long been known that disorder profoundly affects transport properties, recent measurements on a series of solid solution 3d-transition metal alloys reveal two orders of magnitude variations in the residual resistivity. Using ab initio methods, we demonstrate that, while the carrier density of all alloys is as high as in normal metals, the electron mean-free-path can vary from ~10 Å (strong scattering limit) to ~10 Å (weak scattering limit). Here, we delineate the underlying electron scattering mechanisms responsible for this disparate behavior. While site-diagonal, spin dependent, potential scattering is always dominant, for alloys containing only Fe, Co, and Ni the majority-spin channel experiences negligible disorder scattering, thereby providing a short circuit, while for Cr/Mn containing alloys both spin channels experience strong disorder scattering due to an electron filling effect. Somewhat surprisingly, other scattering mechanisms—including displacement, or size effect, scattering which has been shown to strongly correlate with such diverse properties as yield strength—are found to be relatively weak in most cases. npj Computational Materials (2019) 5:1 ; https://doi.org/10.1038/s41524-018-0138-z INTRODUCTION Cantor-Wu alloys), constitute a rich playground for comprehensive studies of the role of maximal disorder on the properties of multi- Electrical resistivity is one of the most fundamental properties of component alloys by controlling both the number (increasing materials. At the coarsest level, it distinguishes between metals, configurational entropy) and types (chemical specificity) of semi-conductors and insulators. As such, it provides a window into 4,5,8 alloying elements . the properties of the electron glue responsible for cohesion. In Of interest here are the results of recent residual resistivity metals and alloys the electrical resistivity is directly related to the 5,8 measurements of a subset of Cantor-Wu alloys that show, rather mean free path, λ [ε ], (alternatively the lifetime, τ [ε ]) of e F e F than increasing monotonically with increasing numbers of electrons at the Fermi energy. In a pure crystalline metal at components, values of ρ break into two subgroups of low (ρ < absolute zero of temperature (T = 0 K), Bloch states are eigen- 0 0 10 μΩ·cm) and high (ρ >75 μΩ·cm) resistivity alloys. In addition, states of the system, λ [ε ] and τ [ε ] are infinite, and the resistivity 0 e F e F two entropically identical alloys, NiCoFe (ρ = 1.7 μΩ·cm) and vanishes. In disordered solid solution alloys, the chemical disorder 0 NiCoCr (ρ = 92.7 μΩ·cm), fall into different resistivity groupings. that results from the random distribution of the alloying elements 0 Remarkably, the least and most resistive alloys differ by almost on the underlying crystalline lattice induces electron scattering two orders of magnitude, ρ (NiCo) = 1.3 μΩ·cm; ρ (NiFeCoCrPd) = and finite λ [ε ] and τ [ε ] even at absolute zero. As a result, the T 0 0 e F e F 124.8 μΩ·cm. Interestingly, the low resistivity group have ρ values = 0 K resistivity, or residual resistivity ρ ,is finite and its precise 0 typical of dilute weak scattering alloys in which there are clearly value provides a direct measure of the disorder induced changes defined host (solvent) and impurity (solute) elements. In such in the underlying electronic structure. alloys, ρ arises from the scattering of a low Fermi energy DOS of In a general N-component solid solution alloy the chemical nearly-free-electron sp-states with large λ [ε ] and ρ generally disorder, as measured by the ideal entropy of mixing, is maximal e F 0 obeys both Nordheim’s relation (ρ ∝ c((1−c); where c is impurity at equiatomic composition and increases with the number of 9 2 components. Equiatomic high entropy alloys (HEA), are exemplars concentration) and Linde’s “law” (ρ ∝ (ΔZ) ; where ΔZ is the 10 11 of such maximally disordered alloys in that they are comprised of valence difference between host and impurity atoms) (see ref. N ≥ 5 components yet unexpectedly form highly stable, single- for a discussion). This, despite the fact that, in equiatomic alloys, phase, disordered solid-solutions on a simple crystal lattice. The the concept of host and impurity elements is lost and the Fermi first single-phase HEA, NiFeCoCrMn, was synthesized by Cantor energy falls in the high density of state (DOS) d-bands . At the 1,2 et al. in 2004. Since then HEAs have become a subject of intense other extreme, high-ρ NiFeCoCrPd is close to the Mott-Ioffe-Regel 12,13 3–6 7 (MIR) limit , which is characterized by a λ [ε ] value comparable scientific and technological interest . In 2014, Wu et al. showed e F 12,14,15 to the lattice spacing . Combined, these observations that alloying the elements of Cantor’s alloy (supplemented with suggest that, although the Cantor-Wu alloys are highly crystalline Pd) yields a series of 2-, 3-, 4-component equiatomic fcc solid- and have uniformly high Fermi energy carrier densities, λ [ε ] can solutions: NiPd, NiCo, NiFe, NiFeCo, NiCoCr, NiCoMn, NiCrCoMn, e F NiFeCoMn, and NiFeCoCr. This set of alloys combined with be controlled, from ~10 Å to ~10 Å, by the specifics of the NiFeCoCrMn and NiFeCoCrPd (here collectively referred to as number and types of alloying elements. Furthermore, distinct from 1 2 Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA and Department of Chemistry, Ludwig-Maximilians-Universität, München D- 81377, Germany Correspondence: Sai Mu ([email protected]) or George M. Stocks ([email protected]) Received: 16 July 2018 Accepted: 6 December 2018 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences S. Mu et al. additional magnetic scattering can arise from fluctuations about the species-dependent single-site average in both the size of the local moments and how they couple amongst themselves— ferromagnetic, antiferromagnetic, mixed ferro/antiferro, noncol- linear, non-coplanar. Site-diagonal disorder Using the conventional spin-polarized KKR-CPA method, we explore the effect of site-diagonal disorder, i.e., [δ , Δ ], on σ Exch the electronic structure and ρ in Cantor-Wu alloys. Figure 1 5,8 compares the calculated ρ with the measured values . From the figure, there are three clear conclusions. Firstly, consistent with the experiments, the calculated values of ρ separate into two groups: low-ρ alloys (NiPd, NiCo, NiFe, NiFeCo), having ρ <10 μΩ·cm and 0 0 high-ρ alloys (the others). This finding is independent of the particular exchange-correlation functional used. Secondly, while the calculated value of ρ including only site-diagonal disorder, underestimates ρ , the contribution from site-diagonal disorder is dominant across all Cantor-Wu alloys. Thirdly, the magnitude of ρ Fig. 1 Residual resistivity of Cantor-Wu alloys due to site-diagonal disorder (SDD) (blue bars), compared with experimental data (red correlates with the types of alloying elements. In particular, for bars) alloys containing only the Ni, Fe, Co, that have nearly-filled 3d- bands, ρ is low. While for alloys containing both Ni, Fe, Co, and Cr, many other metallic materials with high resistivity, the disorder- Mn, whose d bands are approximately half-filled, ρ is large. induced short λ [ε ] of highly resistive Cantor-Wu alloys does not e F Notably, the latter set of alloys are also characterized by mixed 16–18 require strong electron correlation as in incoherent metals , exchange coupling between the local moments of Ni, Fe, Co large atomic displacements associated with very high tempera- (ferromagnetic) and Cr, Mn (antiferromagnetic) while the former tures, or complete loss of translational symmetry as in quasicrys- exhibit only ferromagnetic coupling. tals . As such the Cantor-Wu alloys provide a unique opportunity The underlying reason for the breakdown into two distinct for uncovering the underlying scattering mechanisms that give resistivity groups can be understood in terms of disorder smearing such disparate and non-monotonic behavior in 3d-transition metal of the Fermi surface. Figure 2a shows the spin-resolved Fermi alloys that form on well-defined, in this case fcc, crystalline lattices. surfaces of four selected Cantor-Wu alloys—two each from the Here, we report the first calculations of the residual resistivity of low-ρ and high-ρ group. While the minority-spin Fermi surfaces 0 0 the full set of Cantor-Wu alloys using state-of-the-art ab initio exhibit large disorder smearing for all of the alloys, the majority- transport theory for disordered alloys. Consistent with experiment, spin channels are very different in the two classes. In particular, we find that the calculated ρ break into high-ρ (alloys involving 0 0 the majority-spin Fermi surfaces for NiCo and NiFeCo remain very Mn/Cr elements) and low-ρ (the others) sets. We show that it is 0 sharp which corresponds to a long λ [ε ]. As a result, the majority- e F the magnitude of the spin-dependent site-diagonal potential spin channel acts as a short circuit for electron conduction scattering that makes the dominant contribution to ρ and gives 0 resulting in an overall low resistivity. On the contrary, the majority- rise to this remarkable difference in ρ between the two sets. We 0 spin Fermi surfaces of NiFeCoCr and NiFeCoCrMn alloys are explicitly evaluate effects of disorder that go beyond those washed out with the consequence that the λ [ε ] in both spin e F captured by conventional CPA—local lattice displacements, the channels is very short and thus ρ is high. In the absence of a distribution of both the magnitude and orientation of the local direct calculation of the residual resistivities, it has been previously magnetic moments. Surprisingly, we find that the scattering from noted that the transport properties of the Cantor-Wu alloys local lattice distortions as well as the site-to-site variations in local qualitatively reflect the large differences in disorder smearing of 5,8 moment magnitude and orientation are relatively weak in most the Fermi energy Bloch spectral functions , that are driven by alloys. This despite the fact that one of these, lattice displace- differences in magnetic (FM versus mixed FM/AFM) coupling—an ments, has been shown to strongly correlate with such a conclusion that turns out to be inadequate and even misleading. seemingly unrelated property as yield strength. Notably, NiCoCr also has a very smeared Femi energy Bloch spectral function, and correspondingly high ρ , despite being robustly nonmagnetic . RESULTS AND DISCUSSION Figure 2b shows a cartoon of the underlying spin-resolved In solid solution alloys, all electron scattering ultimately results partial DOS of the alloying elements Ni, Co, Fe, Cr that illustrates from the disorder-induced site-to-site potential fluctuations. why the spin-resolved Fermi surface smearing is so different in the However, to understand the fundamentals of the scattering two alloy groups. Within the KKR-CPA, the strength of the disorder mechanisms, it is useful to divide the total scattering according to scattering can be characterized by the ratio of two important a number of distinguishable sub-mechanisms. Single-site electron energy scales: the energy separation (δ) between the band centers scattering can be thought of as resulting from the site-to-site of different species a.k.a “band center mismatch” and the overall variation (δ) in the local potential due to the random distribution band width (W) . In transition metals, the most relevant band of elements. In the presence of magnetism, conduction electrons center is simply the d-scattering resonance (ε ) of the single-site experience an additional inhomogeneous exchange field (Δ ), potential, while W encapsulates the spread of the d-bands due to Exch which further increases the site-disorder and is different in hybridization. These energy scales are illustrated in Fig. 2b. If δ/W separate spin channels. In the following, we shall refer to site- << 1, the disorder scattering is weak, and the electron bands are diagonal disorder as being the combined effects of [δ, Δ ]. The well-defined. However, if δ/W ~ 1, disorder scattering is strong, Exch single-site picture is further modified by including the effects of leading to large disorder broadening (smearing) of the energy displacement scattering caused by relaxation of the atoms away bands. For magnetic alloys, the electrons propagate and are from their ideal lattice sites due to the fact that every atom is scattered in two separate and independent spin channels — surrounded by a different configuration of other atoms. Moreover, neglecting the spin-mixing contribution. The spin-mixing arising npj Computational Materials (2019) 1 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; S. Mu et al. Fig. 2 Fermi surface and energy scale of disorder scattering. a Fermi surface (Γ–X–W plane) in NiCo, NiFeCo, NiFeCoCr, and NiFeCoCrMn alloys (given in arbitrary units). Majority- and minority-spin channels are distinguished using ↑ and ↓; b Magnetic origin of the energy scales for disorder scattering: Simplified depiction of the species-resolved DOS in NiFeCoCr, in which ε denotes the 3d band centers for spin σ (σ = ↑ or σ σ ↓). The band center mismatch δ is defined as max{ε (i)}−min{ε (i)} with i denoting different species. The exchange splitting is denoted as σ d d ↓ ↑ Δ : = ε −ε Exch d d from spin–orbit coupling however affects ρ and brings about the scattering regime. Similar arguments also apply to Mn. It is worth anisotropic magnetoresistance as shown by Banhart et al. .As a noting that ρ is high only if strong disorder scattering is present result, the above argument applies to each spin channel at the Fermi energy. For example, strong disorder scattering in independently, distinguished by subscript σ = ↑ and ↓ for spin- NiPd notwithstanding, ρ is low (ρ = 2.19 μΩ·cm) because the 0 0 up and spin-down. Fermi surface is mainly of sp character and large disorder For alloys containing only Fe, Co, and Ni, the majority-spin 3d scattering of d electrons does not pollute the Fermi surface in band centers are aligned due to minimization of the kinetic either spin channel (see the Fermi surface in the Supplementary energy. As a result, δ between all atom pairs is small and thus δ / ↑ ↑ Information, Section 5(D), Fig. S7). Whilst, these kinds of W is always in the weak scattering regime. Because, different local 24,26–28 arguments are well known , the way they operate in this moments form on different species and they couple ferromagne- class of alloys is particularly startling. tically, the additional exchange splitting (Δ ) which is propor- Exch As noted previously, while the calculated ρ of NiCo, NiFeCo, tional to the size of the local moment—with proportionality 5,8 NiFeCoCr agree quantitatively with the experiments (see Fig. 1), constant ~1 eV/μ , leads to a large band center mismatch (δ )in B ↓ ρ is still underestimated by a substantial fraction, particularly in the minority-spin channel and consequent large disorder scatter- NiFe, NiPd and also in some high-ρ alloys such as NiCoCr, ing (δ /W ~ 1) (see Fig. 2b). A similar argument has been applied ↓ ↓ 24 25 NiFeCoCrMn, and NiFeCoCrPd. To shed light on this under- in Ni Fe , and NiCo . As a result, while the majority-spin Fermi 35 65 estimation of ρ , we explore the effects of other scattering surface is well-defined and λ [ε ] is long, minority-spin channel 0 e F mechanisms—displacement scattering, magnetic scattering— electron transport is “blocked” by the strong disorder smearing of beyond the single-site approximation. The supercell method is the Fermi surface. On the other hand, when alloying with lower band filling Cr, the band center in both channels is shifted towards employed to explore the effect of site-to-site atomic displacement the Fermi energy in order to realize charge neutrality. As a and the complex magnetic effect (see the Method section). Figure consequence, δ is large in both spin channels (large disorder 3a illustrates the supercell of NiFeCoCrMn alloy with different scattering), thereby washing out the Fermi surface. In addition, the species randomly distributed. The direction and length of the moments on Cr can couple either ferromagnetically or antiferro- black arrows on each site indicate the orientation and magnitude magnetically, further modifying δ. However, this does not of each local moment, respectively. substantially diminish δ, and thus δ/W remains in the strong Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 1 S. Mu et al. Fig. 3 Displacements and local moments distributions in selected alloys. a A 256-atom supercell of NiFeCoCrMn with the arrows denoting the local moments. b The atomic displacements—with arrows denoting displacement vector away from the ideal positions–on both Ni (green sphere) and Fe (blue sphere) species in a 256-atom supercell of NiFe. Displacement c and local moment d distributions of each species in NiFeCo, NiCoMn, NiFeCoCrMn, and NiFeCoCrPd, calculated from a 256-atom supercell. The arrows in the moment statistics of NiCoMn and NiFeCoCrMn label the local moment for each species in different states Fig. 4 Residual resistivity including other scattering mechanisms. a Effect of displacement scattering (DS) on ρ in Cantor-Wu alloys. b Effect of additional spin disorder (SD) on ρ of high-ρ alloys 0 0 Displacement scattering displacement scattering on the electronic transport provides a window into their importance as a scattering mechanism—albeit Recently, atomic displacements have been shown to correlate only at the Fermi energy. with yield strength , suggesting the root mean square (rms) Based on fully relaxed supercell calculations, the statistics of the atomic displacements as a descriptor of the mechanical properties magnitude of the atomic displacements—resolved by species— of Cantor-Wu alloys. Given that all materials properties ultimately are shown in Fig. 3c for selected Cantor-Wu alloys (see the originate from the electronic structure, studying the effect of Supplementary Information, Section 3, Fig. S2 for other alloys). As npj Computational Materials (2019) 1 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences S. Mu et al. seen in the NiFeCo ternary alloy, atomic displacements (Δu) on all having a distribution of the local moment sizes. Based on supercell species are small with Δu(Ni) < Δu(Co) < Δu(Fe). This is consistent calculations, we find a Fe moment distribution that has a 0.2 μ with the elements having similar atomic size and electronegativ- broadening. To mimic these moment fluctuations using CPA, we ities. On the other hand, for alloys involving Cr and/or Mn, the discretize the Fe moment distribution into three types, having atomic displacements on Cr and Mn are much larger. Again, moments m and m ± 0.1 μ (where m is the average Fe 0 0 B 0 consistent with expected larger size mismatch and charge transfer moment), through scaling of the spin part of the exchange- effects. As such, species-dependent displacements become more correlation functional. Unsurprisingly, ρ only increases by pronounced towards the left side of the 3d-transition metal 0.3 μΩ·cm, indicating the ρ is insensitive to such longitudinal elements in the periodic table. Moreover, if alloying with Pd, as in spin fluctuations. NiPd and NiFeCoCrPd alloys, atomic displacements on all species Taking NiCoMn and NiFeCoCrMn alloys as examples, it is noted are large due to size-mismatch between 3d and 4d elements (see that supercell calculations find two well-separated Mn moment Review article for the size-mismatch effect). Notably, the distributions with opposite orientations and roughly similar statistics of the angular dependence of the displacements appears amounts of Mn (~65–35%) and Mn (~35–65%) moments as ↑ ↓ to be random (see Fig. 3b for an illustration in NiFe alloy). the ground state. Due to the very localized nature of the Mn local Assuming that the site-to-site variation of both magnitude and moments such states can also be described within the KKR-CPA the orientation of the atomic displacements are uncorrelated, their approach in a manner analogous to disordered local moment 38,39 effects on the electronic structure and ρ can be assessed by the state (DLM) previously used to describe paramagnetic Fe. alloy analogy model (AAM) . The results are shown in Fig. 4a. In Rather than investigate all possible ratios of Mn and Mn we ↑ ↓, most alloys the enhancement of ρ due to local displacements is focus the system having equal concentrations as an approximated rather small because the atomic displacements (of most alloys) are representative of the state having maximal “Mn-moment” disorder only a small fraction of the interatomic spacing. The exceptions —we denote this state as DLM-Mn. Similarly, another collinear are NiPd and NiFeCoCrPd alloys whose atomic displacements are magnetic state with 100% Mn , and one fully DLM state (DLM on large for all component species (see Fig. 3c). For NiPd, the all species) are also found to be stable solutions. Table 1 lists the resulting displacement-enhanced resistivity is in good agreement CPA total energies for the three states. As expected, the DLM-Mn with experiment. While for NiFeCoCrPd the inclusion of displace- state of NiCoMn is the ground state. Moreover, the DLM-Mn state ment scattering increases ρ by ~12%, the actual ρ is still 0 0 of NiCoMn gives ±2.2 μ for the local moment of Mn and Mn , B ↑ ↓ underestimated. Therefore, the overall effect of displacement which is consistent with the averaged Mn and Mn moments (~ ↑ ↓ scattering in most alloys is small. Thus, the reasons for the general ±2.3 μ ) obtained from the supercell calculation. However, the Mn underestimation of ρ by single-site theory alone must be sought moment within the AFM state is only −0.7 μ , further casting elsewhere. Furthermore, this finding makes the strong correlation suspicion on 100% Mn as representative of Mn containing between the rms displacements and yield strength all the more Cantor-Wu alloys. Further justification of the efficacy of the DLM- interesting; perhaps, suggesting the existence of a more funda- Mn state can be obtained by comparing the species-resolved DOS mental descriptor, rooted in the (common) underlying electronic with the supercell and studying stability of the Heisenberg structure. interactions (unpublished data). For ρ ,we find its value depends sensitively on the assumed magnetic state—ρ is lowest in the Magnetism beyond the single-site approximation AFM state and increases by ~50% in the DLM-Mn ground state. Unlike in the KKR-CPA treatment of magnetism which deals with Unfortunately, the experimental value has not yet been measured. species-dependent single-site averaged magnetic moment, in the The reason for the different ρ behavior can be easily traced to the real alloys, the local moments of each species take on a underlying electronic structure: the AFM state exhibits a relatively distribution of values and can possibly point along arbitrary sharp Fermi surface in the minority-spin channel while the Fermi directions. In Fig. 3d we show that species-resolved local moment surface in both spin channels smears out for the DLM-Mn state distributions for selected Cantor-Wu alloys, obtained using the (see the Supplementary Information, Section 5(B), Fig. S6). In supercell method, with moments constrained to be co-linear. For contrast to NiCoMn, it turns out the AFM and DLM-Mn states in the local moment distribution in other alloys, see Supplementary NiFeCoCrMn alloy are not only close in energy but their ρ are Information, Section 4(A), Fig. S3. It is noteworthy that, although insensitive to which state is considered because the electron supercell calculations yield a distribution of local moments, the scattering by magnetic-driven disorder is already almost species-dependent averaged local moments turn out to be saturated. consistent with those obtained from KKR-CPA (see the Supple- So far, ρ has been calculated assuming collinear spin mentary Information, Section 4(B), Tab. S2). configurations. However, spin noncollinearity is also possible, For alloys considered here, several features of the local moment particularly in Mn- and/or Cr-containing alloys due to the patterns can be found. Firstly, the local moments in low-ρ alloys geometric frustration of antiferromagnetism on a triangular lattice are ferromagnetically coupled and display only a small variation in 40–42 (as the (111) plane of the fcc lattice) , oscillating exchange the size of the moments. In high-ρ alloys with Cr, the magnitudes 0 28,43 interactions as a function of distance , and spin–orbit of Cr moments vary widely from negative (antiferromagnetically interaction. For example, NiFeCoCrMn alloy is found to have a aligned) to positive (ferromagnetically aligned). In contrast to Cr, the magnitudes of Mn moments fall into two well-defined groups, large positive and large negative. This suggests that large on-site Table 1. The total energies (E , meV/site) and residual resistivity (ρ , tot 0 Hund's exchange promotes the formation of local moments on μΩ·cm) for the multiple magnetic states in NiCoMn and NiFeCoCrMn Mn, while the interatomic exchange interaction between Mn from KKR-CPA atoms is antiferromagnetic. The antiferromagnetic coupling Alloys NiCoMn NiFeCoCrMn associated with Cr and Mn can be attributed to the approximately 32–34 half-filled d bands . Clearly, the complicated magnetic config- States E ρ E ρ tot 0 tot 0 urations just described have the potential to induce significant additional electron scattering beyond that included in the KKR- AFM-Mn 22.5 46.7 0 61.0 22,35–37 CPA . DLM 15.7 66.7 4.2 62.1 Calc Using NiFe as an example, where ρ is underestimated (ρ = 0 0 DLM-Mn 0 72.0 0.12 65.7 Exp 3.3 versus ρ = 10.3 μΩ·cm), we first evaluate the effect of Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 1 S. Mu et al. spin glass state both experimentally and theoretically (unpub- but also on the overall effects of disorder throughout the lished data). occupied d-bands. As such it sheds light on the bonding In principle, the spin noncollinearity can be dealt with mechanisms responsible for many of the exotic properties of 4,55 5 straightforwardly from the spin noncollinear calculations based HEAs, such as mechanical , and radiation response properties. on the supercell method. However, such calculations for supercell Here it is notable that while the effect of displacement scattering size sufficient to gain good statistics of distributions of spin on ρ is small, the highly unusual solid solution strength in several orientations are extremely demanding and remain a research of Cantor-Wu alloys has been correlated with the magnitude of project in their own right. Such calculations are made particularly displacement fluctuations . Resolving this apparent dichotomy is difficult by the need not only treat the spin noncollinearity but clearly a challenge worthy further investigation. also include the spin–orbit interactions—particularly for Pd- containing Cantor-Wu alloys, making them beyond the scope of METHODS this paper. Our calculations employ the ab initio spin-polarized fully relativistic Without such a sophisticated evaluation of the spin noncolli- 56,57 Korringa-Kohn-Rostoker Green’s function method , combined with the nearity at zero temperature (probability distribution of the spin Coherent Potential Approximation (hereafter referred to as KKR-CPA)—as orientations for each species), here we calculate the maximum 59,60 implemented in the Munich SPR-KKR package —to calculate the effect contribution to the residual resistivity (ρ ) that can arise from spin of disorder on the electronic structure and residual resistivity. The CPA disorder. To assess this, we again employ the AAM by using a method is a self-consistent theory of the effects of substitutional disorder discrete set of random and uncorrelated spin moments that are on the configurationally averaged single-site properties of alloys. As such it distributed uniformly in space, where the magnitude of the includes, the effect of [δ, Δ ] on electronic band structure and Exch species-dependent local moments are obtained from CPA ground quasiparticle lifetime on the same ab initio footing. The conductivity tensor is calculated by using the linear response Kubo-Greenwood state . Notably, spin noncollinearity in low-ρ alloys is negligibly 61,62 63 formula for the configurational averaged state, described by the CPA : small, as verified by fully relativistic supercell calculations. There- no fore, we only explore the effect of full spin disorder in the high-ρ μ þ ν þ σ ¼ Tr <j ImG ðÞ ε j ImG ðÞ ε > ; μν F F α β alloys. The resulting resistivities, which can be viewed as the π NΩ maximum effect of spin disorder on ρ , are shown in Fig. 4b. A μ where j denotes the μ-component of the current density operator j for sizable ρ enhancement, as large as 10 μΩ·cm, is observed in 0 species α with concentration x and G (ε ) is the retarded Green’s function α F NiFeCoCr, NiFeCoMn, and NiFeCoCrPd. Therefore, the full spin at the Fermi energy. Within the KKR-CPA calculations, the local density approximation (LDA) is employed for exchange and correlation. The disorder produces a modest increase of ρ in high-ρ alloys. 0 0 sensitivity of the results to the exchange-correlation functional is discussed Notwithstanding the overall improved agreement with the in the Supplementary Information, Section 2, Fig. S1. To study the effect of measurement resulting from inclusion of the additional scattering displacement scattering and spin disorder on ρ , we used the so-called mechanisms discussed above, the remaining moderate under- alloy analogy model (AAM) to perform the configurational average over a estimation of ρ in some alloys (NiFe, NiCoCr, NiFeCoCrMn, discrete set of species-resolved atomic displacements and local moment NiFeCoCrPd) requires consideration of other possible theoretical 31 orientations . To obtain the statistics of the atomic displacements and shortcomings. The three most obvious being: going beyond the local moments, we performed standard supercell calculations for a 256- single-site treatment of disorder; inclusion of any possible (but atom special quasi-random structure (SQS) using the projector augmen- currently mostly unknown except for NiFe ) short range order ted wave method (PAW) as implemented in the Vienna ab initio simulation package (VASP) (see the Supplementary Information, Section (SRO); consideration of additional electron correlation effects, 1). beyond LDA. For binary alloys where the first two effects have been considered their impact on ρ has been found to be 46,47 48 small (with specific exceptions, e.g., in K-state alloys ). DATA AVAILABILITY However, it is not clear whether the impacts of the first two The authors declare that the data supporting this study are available from the effects in the Cantor-Wu alloys are also small. Specifically, the 49 corresponding author upon request. specific heat measurements of Cr-containing Cantor-Wu alloys show a K-state transition at 800–1000 K, which is usually attributed 50–52 to the order-disorder transition . Although the effects of SRO ACKNOWLEDGEMENTS on ρ are clearly worthy investigating, accounting for them This work was supported by the Energy Dissipation and Defect Evolution (EDDE), an requires treatments of multisite scattering processes that go Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), beyond the single-site approximation—e.g., nonlocal CPA (see Office of Science, Basic Energy Sciences under contract number DE-AC05-00OR22725. Review article and references therein)—and are beyond current This research used resources of Oak Ridge National Laboratory’s Computer and Data capabilities. In principle the investigation of additional Coulomb Environment for Science (CADES) and the Oak Ridge Leadership Computing Facility, correlation effects on ρ could be addressed by a combination of which is a DOE Office of Science User Facility supported under Contract DE-AC05- 00OR22725. S.M. acknowledges fruitful discussions with K.D. Belashchenko, B.C. Sales, KKR-CPA and Dynamical Mean Field Theory , however, it is and K. Jin. S.W., S.M., and H.E. would like to thank the DFG (Deutsche beyond the scope of the present work. Forschungsgemeinschaft) for financial support within the priority program SPP In conclusion, we have demonstrated that the abnormal and 1538 and the collaborative research centers 689 and 1277. disparate electronic transport in Cantor-Wu alloys at zero temperature is dominated by electron scattering arising from site-to-site potential disorder. In particular, it is found that Cr and/ AUTHOR CONTRIBUTIONS or Mn produce strong disorder scattering as a result of the G.M.S. conceived this research; S.M. carried out the first-principles calculations and proximity of the d-scattering resonance to the Fermi energy which analyzed the results with G.M.S., G.D.S., and S.W.; S.M. and G.M.S. wrote the paper, results from their reduced band filling. Additionally, other electron and all authors participated in the discussions and contributed to finalize the paper. scattering mechanisms are explored explicitly and shown to be small in most alloys with the exception of NiPd and NiFeCoCrPd, ADDITIONAL INFORMATION where the effect of displacement scattering is large; NiCoMn, where the DLM-Mn ground state significantly raises ρ ; and Supplementary information accompanies the paper on the npj Computational Materials website (https://doi.org/10.1038/s41524-018-0138-z). NiFeCoCrMn/NiFeCoCrPd, where the effect of spin-noncollinearity is large. 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Uncovering electron scattering mechanisms in NiFeCoCrMn derived concentrated solid solution and high entropy alloys

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www.nature.com/npjcompumats ARTICLE OPEN Uncovering electron scattering mechanisms in NiFeCoCrMn derived concentrated solid solution and high entropy alloys 1 1 2 1 1 2 2 Sai Mu , German D. Samolyuk , Sebastian Wimmer , Maria C. Troparevsky , Suffian N. Khan , Sergiy Mankovsky , Hubert Ebert and George M. Stocks Whilst it has long been known that disorder profoundly affects transport properties, recent measurements on a series of solid solution 3d-transition metal alloys reveal two orders of magnitude variations in the residual resistivity. Using ab initio methods, we demonstrate that, while the carrier density of all alloys is as high as in normal metals, the electron mean-free-path can vary from ~10 Å (strong scattering limit) to ~10 Å (weak scattering limit). Here, we delineate the underlying electron scattering mechanisms responsible for this disparate behavior. While site-diagonal, spin dependent, potential scattering is always dominant, for alloys containing only Fe, Co, and Ni the majority-spin channel experiences negligible disorder scattering, thereby providing a short circuit, while for Cr/Mn containing alloys both spin channels experience strong disorder scattering due to an electron filling effect. Somewhat surprisingly, other scattering mechanisms—including displacement, or size effect, scattering which has been shown to strongly correlate with such diverse properties as yield strength—are found to be relatively weak in most cases. npj Computational Materials (2019) 5:1 ; https://doi.org/10.1038/s41524-018-0138-z INTRODUCTION Cantor-Wu alloys), constitute a rich playground for comprehensive studies of the role of maximal disorder on the properties of multi- Electrical resistivity is one of the most fundamental properties of component alloys by controlling both the number (increasing materials. At the coarsest level, it distinguishes between metals, configurational entropy) and types (chemical specificity) of semi-conductors and insulators. As such, it provides a window into 4,5,8 alloying elements . the properties of the electron glue responsible for cohesion. In Of interest here are the results of recent residual resistivity metals and alloys the electrical resistivity is directly related to the 5,8 measurements of a subset of Cantor-Wu alloys that show, rather mean free path, λ [ε ], (alternatively the lifetime, τ [ε ]) of e F e F than increasing monotonically with increasing numbers of electrons at the Fermi energy. In a pure crystalline metal at components, values of ρ break into two subgroups of low (ρ < absolute zero of temperature (T = 0 K), Bloch states are eigen- 0 0 10 μΩ·cm) and high (ρ >75 μΩ·cm) resistivity alloys. In addition, states of the system, λ [ε ] and τ [ε ] are infinite, and the resistivity 0 e F e F two entropically identical alloys, NiCoFe (ρ = 1.7 μΩ·cm) and vanishes. In disordered solid solution alloys, the chemical disorder 0 NiCoCr (ρ = 92.7 μΩ·cm), fall into different resistivity groupings. that results from the random distribution of the alloying elements 0 Remarkably, the least and most resistive alloys differ by almost on the underlying crystalline lattice induces electron scattering two orders of magnitude, ρ (NiCo) = 1.3 μΩ·cm; ρ (NiFeCoCrPd) = and finite λ [ε ] and τ [ε ] even at absolute zero. As a result, the T 0 0 e F e F 124.8 μΩ·cm. Interestingly, the low resistivity group have ρ values = 0 K resistivity, or residual resistivity ρ ,is finite and its precise 0 typical of dilute weak scattering alloys in which there are clearly value provides a direct measure of the disorder induced changes defined host (solvent) and impurity (solute) elements. In such in the underlying electronic structure. alloys, ρ arises from the scattering of a low Fermi energy DOS of In a general N-component solid solution alloy the chemical nearly-free-electron sp-states with large λ [ε ] and ρ generally disorder, as measured by the ideal entropy of mixing, is maximal e F 0 obeys both Nordheim’s relation (ρ ∝ c((1−c); where c is impurity at equiatomic composition and increases with the number of 9 2 components. Equiatomic high entropy alloys (HEA), are exemplars concentration) and Linde’s “law” (ρ ∝ (ΔZ) ; where ΔZ is the 10 11 of such maximally disordered alloys in that they are comprised of valence difference between host and impurity atoms) (see ref. N ≥ 5 components yet unexpectedly form highly stable, single- for a discussion). This, despite the fact that, in equiatomic alloys, phase, disordered solid-solutions on a simple crystal lattice. The the concept of host and impurity elements is lost and the Fermi first single-phase HEA, NiFeCoCrMn, was synthesized by Cantor energy falls in the high density of state (DOS) d-bands . At the 1,2 et al. in 2004. Since then HEAs have become a subject of intense other extreme, high-ρ NiFeCoCrPd is close to the Mott-Ioffe-Regel 12,13 3–6 7 (MIR) limit , which is characterized by a λ [ε ] value comparable scientific and technological interest . In 2014, Wu et al. showed e F 12,14,15 to the lattice spacing . Combined, these observations that alloying the elements of Cantor’s alloy (supplemented with suggest that, although the Cantor-Wu alloys are highly crystalline Pd) yields a series of 2-, 3-, 4-component equiatomic fcc solid- and have uniformly high Fermi energy carrier densities, λ [ε ] can solutions: NiPd, NiCo, NiFe, NiFeCo, NiCoCr, NiCoMn, NiCrCoMn, e F NiFeCoMn, and NiFeCoCr. This set of alloys combined with be controlled, from ~10 Å to ~10 Å, by the specifics of the NiFeCoCrMn and NiFeCoCrPd (here collectively referred to as number and types of alloying elements. Furthermore, distinct from 1 2 Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA and Department of Chemistry, Ludwig-Maximilians-Universität, München D- 81377, Germany Correspondence: Sai Mu ([email protected]) or George M. Stocks ([email protected]) Received: 16 July 2018 Accepted: 6 December 2018 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences S. Mu et al. additional magnetic scattering can arise from fluctuations about the species-dependent single-site average in both the size of the local moments and how they couple amongst themselves— ferromagnetic, antiferromagnetic, mixed ferro/antiferro, noncol- linear, non-coplanar. Site-diagonal disorder Using the conventional spin-polarized KKR-CPA method, we explore the effect of site-diagonal disorder, i.e., [δ , Δ ], on σ Exch the electronic structure and ρ in Cantor-Wu alloys. Figure 1 5,8 compares the calculated ρ with the measured values . From the figure, there are three clear conclusions. Firstly, consistent with the experiments, the calculated values of ρ separate into two groups: low-ρ alloys (NiPd, NiCo, NiFe, NiFeCo), having ρ <10 μΩ·cm and 0 0 high-ρ alloys (the others). This finding is independent of the particular exchange-correlation functional used. Secondly, while the calculated value of ρ including only site-diagonal disorder, underestimates ρ , the contribution from site-diagonal disorder is dominant across all Cantor-Wu alloys. Thirdly, the magnitude of ρ Fig. 1 Residual resistivity of Cantor-Wu alloys due to site-diagonal disorder (SDD) (blue bars), compared with experimental data (red correlates with the types of alloying elements. In particular, for bars) alloys containing only the Ni, Fe, Co, that have nearly-filled 3d- bands, ρ is low. While for alloys containing both Ni, Fe, Co, and Cr, many other metallic materials with high resistivity, the disorder- Mn, whose d bands are approximately half-filled, ρ is large. induced short λ [ε ] of highly resistive Cantor-Wu alloys does not e F Notably, the latter set of alloys are also characterized by mixed 16–18 require strong electron correlation as in incoherent metals , exchange coupling between the local moments of Ni, Fe, Co large atomic displacements associated with very high tempera- (ferromagnetic) and Cr, Mn (antiferromagnetic) while the former tures, or complete loss of translational symmetry as in quasicrys- exhibit only ferromagnetic coupling. tals . As such the Cantor-Wu alloys provide a unique opportunity The underlying reason for the breakdown into two distinct for uncovering the underlying scattering mechanisms that give resistivity groups can be understood in terms of disorder smearing such disparate and non-monotonic behavior in 3d-transition metal of the Fermi surface. Figure 2a shows the spin-resolved Fermi alloys that form on well-defined, in this case fcc, crystalline lattices. surfaces of four selected Cantor-Wu alloys—two each from the Here, we report the first calculations of the residual resistivity of low-ρ and high-ρ group. While the minority-spin Fermi surfaces 0 0 the full set of Cantor-Wu alloys using state-of-the-art ab initio exhibit large disorder smearing for all of the alloys, the majority- transport theory for disordered alloys. Consistent with experiment, spin channels are very different in the two classes. In particular, we find that the calculated ρ break into high-ρ (alloys involving 0 0 the majority-spin Fermi surfaces for NiCo and NiFeCo remain very Mn/Cr elements) and low-ρ (the others) sets. We show that it is 0 sharp which corresponds to a long λ [ε ]. As a result, the majority- e F the magnitude of the spin-dependent site-diagonal potential spin channel acts as a short circuit for electron conduction scattering that makes the dominant contribution to ρ and gives 0 resulting in an overall low resistivity. On the contrary, the majority- rise to this remarkable difference in ρ between the two sets. We 0 spin Fermi surfaces of NiFeCoCr and NiFeCoCrMn alloys are explicitly evaluate effects of disorder that go beyond those washed out with the consequence that the λ [ε ] in both spin e F captured by conventional CPA—local lattice displacements, the channels is very short and thus ρ is high. In the absence of a distribution of both the magnitude and orientation of the local direct calculation of the residual resistivities, it has been previously magnetic moments. Surprisingly, we find that the scattering from noted that the transport properties of the Cantor-Wu alloys local lattice distortions as well as the site-to-site variations in local qualitatively reflect the large differences in disorder smearing of 5,8 moment magnitude and orientation are relatively weak in most the Fermi energy Bloch spectral functions , that are driven by alloys. This despite the fact that one of these, lattice displace- differences in magnetic (FM versus mixed FM/AFM) coupling—an ments, has been shown to strongly correlate with such a conclusion that turns out to be inadequate and even misleading. seemingly unrelated property as yield strength. Notably, NiCoCr also has a very smeared Femi energy Bloch spectral function, and correspondingly high ρ , despite being robustly nonmagnetic . RESULTS AND DISCUSSION Figure 2b shows a cartoon of the underlying spin-resolved In solid solution alloys, all electron scattering ultimately results partial DOS of the alloying elements Ni, Co, Fe, Cr that illustrates from the disorder-induced site-to-site potential fluctuations. why the spin-resolved Fermi surface smearing is so different in the However, to understand the fundamentals of the scattering two alloy groups. Within the KKR-CPA, the strength of the disorder mechanisms, it is useful to divide the total scattering according to scattering can be characterized by the ratio of two important a number of distinguishable sub-mechanisms. Single-site electron energy scales: the energy separation (δ) between the band centers scattering can be thought of as resulting from the site-to-site of different species a.k.a “band center mismatch” and the overall variation (δ) in the local potential due to the random distribution band width (W) . In transition metals, the most relevant band of elements. In the presence of magnetism, conduction electrons center is simply the d-scattering resonance (ε ) of the single-site experience an additional inhomogeneous exchange field (Δ ), potential, while W encapsulates the spread of the d-bands due to Exch which further increases the site-disorder and is different in hybridization. These energy scales are illustrated in Fig. 2b. If δ/W separate spin channels. In the following, we shall refer to site- << 1, the disorder scattering is weak, and the electron bands are diagonal disorder as being the combined effects of [δ, Δ ]. The well-defined. However, if δ/W ~ 1, disorder scattering is strong, Exch single-site picture is further modified by including the effects of leading to large disorder broadening (smearing) of the energy displacement scattering caused by relaxation of the atoms away bands. For magnetic alloys, the electrons propagate and are from their ideal lattice sites due to the fact that every atom is scattered in two separate and independent spin channels — surrounded by a different configuration of other atoms. Moreover, neglecting the spin-mixing contribution. The spin-mixing arising npj Computational Materials (2019) 1 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences 1234567890():,; S. Mu et al. Fig. 2 Fermi surface and energy scale of disorder scattering. a Fermi surface (Γ–X–W plane) in NiCo, NiFeCo, NiFeCoCr, and NiFeCoCrMn alloys (given in arbitrary units). Majority- and minority-spin channels are distinguished using ↑ and ↓; b Magnetic origin of the energy scales for disorder scattering: Simplified depiction of the species-resolved DOS in NiFeCoCr, in which ε denotes the 3d band centers for spin σ (σ = ↑ or σ σ ↓). The band center mismatch δ is defined as max{ε (i)}−min{ε (i)} with i denoting different species. The exchange splitting is denoted as σ d d ↓ ↑ Δ : = ε −ε Exch d d from spin–orbit coupling however affects ρ and brings about the scattering regime. Similar arguments also apply to Mn. It is worth anisotropic magnetoresistance as shown by Banhart et al. .As a noting that ρ is high only if strong disorder scattering is present result, the above argument applies to each spin channel at the Fermi energy. For example, strong disorder scattering in independently, distinguished by subscript σ = ↑ and ↓ for spin- NiPd notwithstanding, ρ is low (ρ = 2.19 μΩ·cm) because the 0 0 up and spin-down. Fermi surface is mainly of sp character and large disorder For alloys containing only Fe, Co, and Ni, the majority-spin 3d scattering of d electrons does not pollute the Fermi surface in band centers are aligned due to minimization of the kinetic either spin channel (see the Fermi surface in the Supplementary energy. As a result, δ between all atom pairs is small and thus δ / ↑ ↑ Information, Section 5(D), Fig. S7). Whilst, these kinds of W is always in the weak scattering regime. Because, different local 24,26–28 arguments are well known , the way they operate in this moments form on different species and they couple ferromagne- class of alloys is particularly startling. tically, the additional exchange splitting (Δ ) which is propor- Exch As noted previously, while the calculated ρ of NiCo, NiFeCo, tional to the size of the local moment—with proportionality 5,8 NiFeCoCr agree quantitatively with the experiments (see Fig. 1), constant ~1 eV/μ , leads to a large band center mismatch (δ )in B ↓ ρ is still underestimated by a substantial fraction, particularly in the minority-spin channel and consequent large disorder scatter- NiFe, NiPd and also in some high-ρ alloys such as NiCoCr, ing (δ /W ~ 1) (see Fig. 2b). A similar argument has been applied ↓ ↓ 24 25 NiFeCoCrMn, and NiFeCoCrPd. To shed light on this under- in Ni Fe , and NiCo . As a result, while the majority-spin Fermi 35 65 estimation of ρ , we explore the effects of other scattering surface is well-defined and λ [ε ] is long, minority-spin channel 0 e F mechanisms—displacement scattering, magnetic scattering— electron transport is “blocked” by the strong disorder smearing of beyond the single-site approximation. The supercell method is the Fermi surface. On the other hand, when alloying with lower band filling Cr, the band center in both channels is shifted towards employed to explore the effect of site-to-site atomic displacement the Fermi energy in order to realize charge neutrality. As a and the complex magnetic effect (see the Method section). Figure consequence, δ is large in both spin channels (large disorder 3a illustrates the supercell of NiFeCoCrMn alloy with different scattering), thereby washing out the Fermi surface. In addition, the species randomly distributed. The direction and length of the moments on Cr can couple either ferromagnetically or antiferro- black arrows on each site indicate the orientation and magnitude magnetically, further modifying δ. However, this does not of each local moment, respectively. substantially diminish δ, and thus δ/W remains in the strong Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 1 S. Mu et al. Fig. 3 Displacements and local moments distributions in selected alloys. a A 256-atom supercell of NiFeCoCrMn with the arrows denoting the local moments. b The atomic displacements—with arrows denoting displacement vector away from the ideal positions–on both Ni (green sphere) and Fe (blue sphere) species in a 256-atom supercell of NiFe. Displacement c and local moment d distributions of each species in NiFeCo, NiCoMn, NiFeCoCrMn, and NiFeCoCrPd, calculated from a 256-atom supercell. The arrows in the moment statistics of NiCoMn and NiFeCoCrMn label the local moment for each species in different states Fig. 4 Residual resistivity including other scattering mechanisms. a Effect of displacement scattering (DS) on ρ in Cantor-Wu alloys. b Effect of additional spin disorder (SD) on ρ of high-ρ alloys 0 0 Displacement scattering displacement scattering on the electronic transport provides a window into their importance as a scattering mechanism—albeit Recently, atomic displacements have been shown to correlate only at the Fermi energy. with yield strength , suggesting the root mean square (rms) Based on fully relaxed supercell calculations, the statistics of the atomic displacements as a descriptor of the mechanical properties magnitude of the atomic displacements—resolved by species— of Cantor-Wu alloys. Given that all materials properties ultimately are shown in Fig. 3c for selected Cantor-Wu alloys (see the originate from the electronic structure, studying the effect of Supplementary Information, Section 3, Fig. S2 for other alloys). As npj Computational Materials (2019) 1 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences S. Mu et al. seen in the NiFeCo ternary alloy, atomic displacements (Δu) on all having a distribution of the local moment sizes. Based on supercell species are small with Δu(Ni) < Δu(Co) < Δu(Fe). This is consistent calculations, we find a Fe moment distribution that has a 0.2 μ with the elements having similar atomic size and electronegativ- broadening. To mimic these moment fluctuations using CPA, we ities. On the other hand, for alloys involving Cr and/or Mn, the discretize the Fe moment distribution into three types, having atomic displacements on Cr and Mn are much larger. Again, moments m and m ± 0.1 μ (where m is the average Fe 0 0 B 0 consistent with expected larger size mismatch and charge transfer moment), through scaling of the spin part of the exchange- effects. As such, species-dependent displacements become more correlation functional. Unsurprisingly, ρ only increases by pronounced towards the left side of the 3d-transition metal 0.3 μΩ·cm, indicating the ρ is insensitive to such longitudinal elements in the periodic table. Moreover, if alloying with Pd, as in spin fluctuations. NiPd and NiFeCoCrPd alloys, atomic displacements on all species Taking NiCoMn and NiFeCoCrMn alloys as examples, it is noted are large due to size-mismatch between 3d and 4d elements (see that supercell calculations find two well-separated Mn moment Review article for the size-mismatch effect). Notably, the distributions with opposite orientations and roughly similar statistics of the angular dependence of the displacements appears amounts of Mn (~65–35%) and Mn (~35–65%) moments as ↑ ↓ to be random (see Fig. 3b for an illustration in NiFe alloy). the ground state. Due to the very localized nature of the Mn local Assuming that the site-to-site variation of both magnitude and moments such states can also be described within the KKR-CPA the orientation of the atomic displacements are uncorrelated, their approach in a manner analogous to disordered local moment 38,39 effects on the electronic structure and ρ can be assessed by the state (DLM) previously used to describe paramagnetic Fe. alloy analogy model (AAM) . The results are shown in Fig. 4a. In Rather than investigate all possible ratios of Mn and Mn we ↑ ↓, most alloys the enhancement of ρ due to local displacements is focus the system having equal concentrations as an approximated rather small because the atomic displacements (of most alloys) are representative of the state having maximal “Mn-moment” disorder only a small fraction of the interatomic spacing. The exceptions —we denote this state as DLM-Mn. Similarly, another collinear are NiPd and NiFeCoCrPd alloys whose atomic displacements are magnetic state with 100% Mn , and one fully DLM state (DLM on large for all component species (see Fig. 3c). For NiPd, the all species) are also found to be stable solutions. Table 1 lists the resulting displacement-enhanced resistivity is in good agreement CPA total energies for the three states. As expected, the DLM-Mn with experiment. While for NiFeCoCrPd the inclusion of displace- state of NiCoMn is the ground state. Moreover, the DLM-Mn state ment scattering increases ρ by ~12%, the actual ρ is still 0 0 of NiCoMn gives ±2.2 μ for the local moment of Mn and Mn , B ↑ ↓ underestimated. Therefore, the overall effect of displacement which is consistent with the averaged Mn and Mn moments (~ ↑ ↓ scattering in most alloys is small. Thus, the reasons for the general ±2.3 μ ) obtained from the supercell calculation. However, the Mn underestimation of ρ by single-site theory alone must be sought moment within the AFM state is only −0.7 μ , further casting elsewhere. Furthermore, this finding makes the strong correlation suspicion on 100% Mn as representative of Mn containing between the rms displacements and yield strength all the more Cantor-Wu alloys. Further justification of the efficacy of the DLM- interesting; perhaps, suggesting the existence of a more funda- Mn state can be obtained by comparing the species-resolved DOS mental descriptor, rooted in the (common) underlying electronic with the supercell and studying stability of the Heisenberg structure. interactions (unpublished data). For ρ ,we find its value depends sensitively on the assumed magnetic state—ρ is lowest in the Magnetism beyond the single-site approximation AFM state and increases by ~50% in the DLM-Mn ground state. Unlike in the KKR-CPA treatment of magnetism which deals with Unfortunately, the experimental value has not yet been measured. species-dependent single-site averaged magnetic moment, in the The reason for the different ρ behavior can be easily traced to the real alloys, the local moments of each species take on a underlying electronic structure: the AFM state exhibits a relatively distribution of values and can possibly point along arbitrary sharp Fermi surface in the minority-spin channel while the Fermi directions. In Fig. 3d we show that species-resolved local moment surface in both spin channels smears out for the DLM-Mn state distributions for selected Cantor-Wu alloys, obtained using the (see the Supplementary Information, Section 5(B), Fig. S6). In supercell method, with moments constrained to be co-linear. For contrast to NiCoMn, it turns out the AFM and DLM-Mn states in the local moment distribution in other alloys, see Supplementary NiFeCoCrMn alloy are not only close in energy but their ρ are Information, Section 4(A), Fig. S3. It is noteworthy that, although insensitive to which state is considered because the electron supercell calculations yield a distribution of local moments, the scattering by magnetic-driven disorder is already almost species-dependent averaged local moments turn out to be saturated. consistent with those obtained from KKR-CPA (see the Supple- So far, ρ has been calculated assuming collinear spin mentary Information, Section 4(B), Tab. S2). configurations. However, spin noncollinearity is also possible, For alloys considered here, several features of the local moment particularly in Mn- and/or Cr-containing alloys due to the patterns can be found. Firstly, the local moments in low-ρ alloys geometric frustration of antiferromagnetism on a triangular lattice are ferromagnetically coupled and display only a small variation in 40–42 (as the (111) plane of the fcc lattice) , oscillating exchange the size of the moments. In high-ρ alloys with Cr, the magnitudes 0 28,43 interactions as a function of distance , and spin–orbit of Cr moments vary widely from negative (antiferromagnetically interaction. For example, NiFeCoCrMn alloy is found to have a aligned) to positive (ferromagnetically aligned). In contrast to Cr, the magnitudes of Mn moments fall into two well-defined groups, large positive and large negative. This suggests that large on-site Table 1. The total energies (E , meV/site) and residual resistivity (ρ , tot 0 Hund's exchange promotes the formation of local moments on μΩ·cm) for the multiple magnetic states in NiCoMn and NiFeCoCrMn Mn, while the interatomic exchange interaction between Mn from KKR-CPA atoms is antiferromagnetic. The antiferromagnetic coupling Alloys NiCoMn NiFeCoCrMn associated with Cr and Mn can be attributed to the approximately 32–34 half-filled d bands . Clearly, the complicated magnetic config- States E ρ E ρ tot 0 tot 0 urations just described have the potential to induce significant additional electron scattering beyond that included in the KKR- AFM-Mn 22.5 46.7 0 61.0 22,35–37 CPA . DLM 15.7 66.7 4.2 62.1 Calc Using NiFe as an example, where ρ is underestimated (ρ = 0 0 DLM-Mn 0 72.0 0.12 65.7 Exp 3.3 versus ρ = 10.3 μΩ·cm), we first evaluate the effect of Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2019) 1 S. Mu et al. spin glass state both experimentally and theoretically (unpub- but also on the overall effects of disorder throughout the lished data). occupied d-bands. As such it sheds light on the bonding In principle, the spin noncollinearity can be dealt with mechanisms responsible for many of the exotic properties of 4,55 5 straightforwardly from the spin noncollinear calculations based HEAs, such as mechanical , and radiation response properties. on the supercell method. However, such calculations for supercell Here it is notable that while the effect of displacement scattering size sufficient to gain good statistics of distributions of spin on ρ is small, the highly unusual solid solution strength in several orientations are extremely demanding and remain a research of Cantor-Wu alloys has been correlated with the magnitude of project in their own right. Such calculations are made particularly displacement fluctuations . Resolving this apparent dichotomy is difficult by the need not only treat the spin noncollinearity but clearly a challenge worthy further investigation. also include the spin–orbit interactions—particularly for Pd- containing Cantor-Wu alloys, making them beyond the scope of METHODS this paper. Our calculations employ the ab initio spin-polarized fully relativistic Without such a sophisticated evaluation of the spin noncolli- 56,57 Korringa-Kohn-Rostoker Green’s function method , combined with the nearity at zero temperature (probability distribution of the spin Coherent Potential Approximation (hereafter referred to as KKR-CPA)—as orientations for each species), here we calculate the maximum 59,60 implemented in the Munich SPR-KKR package —to calculate the effect contribution to the residual resistivity (ρ ) that can arise from spin of disorder on the electronic structure and residual resistivity. The CPA disorder. To assess this, we again employ the AAM by using a method is a self-consistent theory of the effects of substitutional disorder discrete set of random and uncorrelated spin moments that are on the configurationally averaged single-site properties of alloys. As such it distributed uniformly in space, where the magnitude of the includes, the effect of [δ, Δ ] on electronic band structure and Exch species-dependent local moments are obtained from CPA ground quasiparticle lifetime on the same ab initio footing. The conductivity tensor is calculated by using the linear response Kubo-Greenwood state . Notably, spin noncollinearity in low-ρ alloys is negligibly 61,62 63 formula for the configurational averaged state, described by the CPA : small, as verified by fully relativistic supercell calculations. There- no fore, we only explore the effect of full spin disorder in the high-ρ μ þ ν þ σ ¼ Tr <j ImG ðÞ ε j ImG ðÞ ε > ; μν F F α β alloys. The resulting resistivities, which can be viewed as the π NΩ maximum effect of spin disorder on ρ , are shown in Fig. 4b. A μ where j denotes the μ-component of the current density operator j for sizable ρ enhancement, as large as 10 μΩ·cm, is observed in 0 species α with concentration x and G (ε ) is the retarded Green’s function α F NiFeCoCr, NiFeCoMn, and NiFeCoCrPd. Therefore, the full spin at the Fermi energy. Within the KKR-CPA calculations, the local density approximation (LDA) is employed for exchange and correlation. The disorder produces a modest increase of ρ in high-ρ alloys. 0 0 sensitivity of the results to the exchange-correlation functional is discussed Notwithstanding the overall improved agreement with the in the Supplementary Information, Section 2, Fig. S1. To study the effect of measurement resulting from inclusion of the additional scattering displacement scattering and spin disorder on ρ , we used the so-called mechanisms discussed above, the remaining moderate under- alloy analogy model (AAM) to perform the configurational average over a estimation of ρ in some alloys (NiFe, NiCoCr, NiFeCoCrMn, discrete set of species-resolved atomic displacements and local moment NiFeCoCrPd) requires consideration of other possible theoretical 31 orientations . To obtain the statistics of the atomic displacements and shortcomings. The three most obvious being: going beyond the local moments, we performed standard supercell calculations for a 256- single-site treatment of disorder; inclusion of any possible (but atom special quasi-random structure (SQS) using the projector augmen- currently mostly unknown except for NiFe ) short range order ted wave method (PAW) as implemented in the Vienna ab initio simulation package (VASP) (see the Supplementary Information, Section (SRO); consideration of additional electron correlation effects, 1). beyond LDA. For binary alloys where the first two effects have been considered their impact on ρ has been found to be 46,47 48 small (with specific exceptions, e.g., in K-state alloys ). DATA AVAILABILITY However, it is not clear whether the impacts of the first two The authors declare that the data supporting this study are available from the effects in the Cantor-Wu alloys are also small. Specifically, the 49 corresponding author upon request. specific heat measurements of Cr-containing Cantor-Wu alloys show a K-state transition at 800–1000 K, which is usually attributed 50–52 to the order-disorder transition . Although the effects of SRO ACKNOWLEDGEMENTS on ρ are clearly worthy investigating, accounting for them This work was supported by the Energy Dissipation and Defect Evolution (EDDE), an requires treatments of multisite scattering processes that go Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), beyond the single-site approximation—e.g., nonlocal CPA (see Office of Science, Basic Energy Sciences under contract number DE-AC05-00OR22725. Review article and references therein)—and are beyond current This research used resources of Oak Ridge National Laboratory’s Computer and Data capabilities. In principle the investigation of additional Coulomb Environment for Science (CADES) and the Oak Ridge Leadership Computing Facility, correlation effects on ρ could be addressed by a combination of which is a DOE Office of Science User Facility supported under Contract DE-AC05- 00OR22725. S.M. acknowledges fruitful discussions with K.D. Belashchenko, B.C. Sales, KKR-CPA and Dynamical Mean Field Theory , however, it is and K. Jin. S.W., S.M., and H.E. would like to thank the DFG (Deutsche beyond the scope of the present work. Forschungsgemeinschaft) for financial support within the priority program SPP In conclusion, we have demonstrated that the abnormal and 1538 and the collaborative research centers 689 and 1277. disparate electronic transport in Cantor-Wu alloys at zero temperature is dominated by electron scattering arising from site-to-site potential disorder. In particular, it is found that Cr and/ AUTHOR CONTRIBUTIONS or Mn produce strong disorder scattering as a result of the G.M.S. conceived this research; S.M. carried out the first-principles calculations and proximity of the d-scattering resonance to the Fermi energy which analyzed the results with G.M.S., G.D.S., and S.W.; S.M. and G.M.S. wrote the paper, results from their reduced band filling. Additionally, other electron and all authors participated in the discussions and contributed to finalize the paper. scattering mechanisms are explored explicitly and shown to be small in most alloys with the exception of NiPd and NiFeCoCrPd, ADDITIONAL INFORMATION where the effect of displacement scattering is large; NiCoMn, where the DLM-Mn ground state significantly raises ρ ; and Supplementary information accompanies the paper on the npj Computational Materials website (https://doi.org/10.1038/s41524-018-0138-z). NiFeCoCrMn/NiFeCoCrPd, where the effect of spin-noncollinearity is large. A profound understanding of the electronic transport in Competing interests: The authors declare no competing interests. disordered alloys not only provides insights on the Fermi surface, npj Computational Materials (2019) 1 Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences S. Mu et al. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims 32. Moriya, T. Ferro-and antiferromagnetism of transition metals and alloys. Prog. in published maps and institutional affiliations. Theor. Phys. 33, 157–183 (1965). 33. Hamada, N. Calculation of a local-environment effect on the atomic magnetic moment in ferromagnetic Ni–Cu alloys. J. Phys. Soc. Jpn. 50,77–82 (1981). 34. Heine, V. & Samson, J. H. Magnetic, chemical and structural ordering in transition REFERENCES metals. J. Phys. F. Met. Phys. 13, 2155–2168 (1983). 1. Cantor, B., Chang, I. T. H., Knight, P. & Vincent, A. J. B. Microstructural develop- 35. 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