Abstract
Whilst it has long been known that disorder profoundly affects transport properties, recent measurements on a series of solid solution 3dtransition 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 meanfreepath can vary from ~10 Å (strong scattering limit) to ~10^{3} Å (weak scattering limit). Here, we delineate the underlying electron scattering mechanisms responsible for this disparate behavior. While sitediagonal, spin dependent, potential scattering is always dominant, for alloys containing only Fe, Co, and Ni the majorityspin 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.
Introduction
Electrical resistivity is one of the most fundamental properties of materials. At the coarsest level, it distinguishes between metals, semiconductors and insulators. As such, it provides a window into the properties of the electron glue responsible for cohesion. In metals and alloys the electrical resistivity is directly related to the mean free path, λ_{e}[ε_{F}], (alternatively the lifetime, τ_{e}[ε_{F}]) of electrons at the Fermi energy. In a pure crystalline metal at absolute zero of temperature (T = 0 K), Bloch states are eigenstates of the system, λ_{e}[ε_{F}] and τ_{e}[ε_{F}] are infinite, and the resistivity vanishes. In disordered solid solution alloys, the chemical disorder that results from the random distribution of the alloying elements on the underlying crystalline lattice induces electron scattering and finite λ_{e}[ε_{F}] and τ_{e}[ε_{F}] even at absolute zero. As a result, the T = 0 K resistivity, or residual resistivity ρ_{0}, is finite and its precise value provides a direct measure of the disorder induced changes in the underlying electronic structure.
In a general Ncomponent solid solution alloy the chemical disorder, as measured by the ideal entropy of mixing, is maximal at equiatomic composition and increases with the number of components. Equiatomic high entropy alloys (HEA), are exemplars of such maximally disordered alloys in that they are comprised of N ≥ 5 components yet unexpectedly form highly stable, singlephase, disordered solidsolutions on a simple crystal lattice. The first singlephase HEA, NiFeCoCrMn, was synthesized by Cantor et al.^{1,2} in 2004. Since then HEAs have become a subject of intense scientific and technological interest^{3,4,5,6}. In 2014, Wu et al.^{7} showed that alloying the elements of Cantor’s alloy (supplemented with Pd) yields a series of 2, 3, 4component equiatomic fcc solidsolutions: NiPd, NiCo, NiFe, NiFeCo, NiCoCr, NiCoMn, NiCrCoMn, NiFeCoMn, and NiFeCoCr. This set of alloys combined with NiFeCoCrMn and NiFeCoCrPd (here collectively referred to as CantorWu alloys), constitute a rich playground for comprehensive studies of the role of maximal disorder on the properties of multicomponent alloys by controlling both the number (increasing configurational entropy) and types (chemical specificity) of alloying elements^{4,5,8}.
Of interest here are the results of recent residual resistivity measurements^{5,8} of a subset of CantorWu alloys that show, rather than increasing monotonically with increasing numbers of components, values of ρ_{0} break into two subgroups of low (ρ_{0} < 10 μΩ·cm) and high (ρ_{0} > 75 μΩ·cm) resistivity alloys. In addition, two entropically identical alloys, NiCoFe (ρ_{0} = 1.7 μΩ·cm) and NiCoCr (ρ_{0} = 92.7 μΩ·cm), fall into different resistivity groupings. Remarkably, the least and most resistive alloys differ by almost two orders of magnitude, ρ_{0}(NiCo) = 1.3 μΩ·cm; ρ_{0}(NiFeCoCrPd) = 124.8 μΩ·cm. Interestingly, the low resistivity group have ρ_{0} values typical of dilute weak scattering alloys in which there are clearly defined host (solvent) and impurity (solute) elements. In such alloys, ρ_{0} arises from the scattering of a low Fermi energy DOS of nearlyfreeelectron spstates with large λ_{e}[ε_{F}] and ρ_{0} generally obeys both Nordheim’s relation (ρ_{0} ∝ c((1−c); where c is impurity concentration)^{9} and Linde’s “law” (ρ_{0} ∝ (ΔZ)^{2}; where ΔZ is the valence difference between host and impurity atoms)^{10} (see ref. ^{11} for a discussion). This, despite the fact that, in equiatomic alloys, the concept of host and impurity elements is lost and the Fermi energy falls in the high density of state (DOS) dbands^{5}. At the other extreme, highρ_{0} NiFeCoCrPd is close to the MottIoffeRegel (MIR) limit^{12,13}, which is characterized by a λ_{e}[ε_{F}] value comparable to the lattice spacing^{12,14,15}. Combined, these observations suggest that, although the CantorWu alloys are highly crystalline and have uniformly high Fermi energy carrier densities, λ_{e}[ε_{F}] can be controlled, from ~10 Å to ~10^{3} Å, by the specifics of the number and types of alloying elements. Furthermore, distinct from many other metallic materials with high resistivity, the disorderinduced short λ_{e}[ε_{F}] of highly resistive CantorWu alloys does not require strong electron correlation as in incoherent metals^{16,17,18}, large atomic displacements associated with very high temperatures, or complete loss of translational symmetry as in quasicrystals^{19}. As such the CantorWu alloys provide a unique opportunity for uncovering the underlying scattering mechanisms that give such disparate and nonmonotonic behavior in 3dtransition metal alloys that form on welldefined, in this case fcc, crystalline lattices.
Here, we report the first calculations of the residual resistivity of the full set of CantorWu alloys using stateoftheart ab initio transport theory for disordered alloys. Consistent with experiment, we find that the calculated ρ_{0} break into highρ_{0} (alloys involving Mn/Cr elements) and lowρ_{0} (the others) sets. We show that it is the magnitude of the spindependent sitediagonal potential scattering that makes the dominant contribution to ρ_{0} and gives rise to this remarkable difference in ρ_{0} between the two sets. We explicitly evaluate effects of disorder that go beyond those captured by conventional CPA—local lattice displacements, the distribution of both the magnitude and orientation of the local magnetic moments. Surprisingly, we find that the scattering from local lattice distortions as well as the sitetosite variations in local moment magnitude and orientation are relatively weak in most alloys. This despite the fact that one of these, lattice displacements, has been shown to strongly correlate with such a seemingly unrelated property as yield strength.
Results and Discussion
In solid solution alloys, all electron scattering ultimately results from the disorderinduced sitetosite potential fluctuations. However, to understand the fundamentals of the scattering mechanisms, it is useful to divide the total scattering according to a number of distinguishable submechanisms. Singlesite electron scattering can be thought of as resulting from the sitetosite variation (δ) in the local potential due to the random distribution of elements. In the presence of magnetism, conduction electrons experience an additional inhomogeneous exchange field (Δ_{Exch}), which further increases the sitedisorder and is different in separate spin channels. In the following, we shall refer to sitediagonal disorder as being the combined effects of [δ, Δ_{Exch}]. The singlesite picture is further modified by including the effects of displacement scattering caused by relaxation of the atoms away from their ideal lattice sites due to the fact that every atom is surrounded by a different configuration of other atoms. Moreover, additional magnetic scattering can arise from fluctuations about the speciesdependent singlesite average in both the size of the local moments and how they couple amongst themselves—ferromagnetic, antiferromagnetic, mixed ferro/antiferro, noncollinear, noncoplanar.
Sitediagonal disorder
Using the conventional spinpolarized KKRCPA method, we explore the effect of sitediagonal disorder, i.e., [δ_{σ}, Δ_{Exch}], on the electronic structure and ρ_{0} in CantorWu alloys. Figure 1 compares the calculated ρ_{0} with the measured values^{5,8}. From the figure, there are three clear conclusions. Firstly, consistent with the experiments, the calculated values of ρ_{0} separate into two groups: lowρ_{0} alloys (NiPd, NiCo, NiFe, NiFeCo), having ρ_{0} < 10 μΩ·cm and highρ_{0} alloys (the others). This finding is independent of the particular exchangecorrelation functional used. Secondly, while the calculated value of ρ_{0} including only sitediagonal disorder, underestimates ρ_{0}, the contribution from sitediagonal disorder is dominant across all CantorWu alloys. Thirdly, the magnitude of ρ_{0} correlates with the types of alloying elements. In particular, for alloys containing only the Ni, Fe, Co, that have nearlyfilled 3dbands, ρ_{0} is low. While for alloys containing both Ni, Fe, Co, and Cr, Mn, whose d bands are approximately halffilled, ρ_{0} is large. Notably, the latter set of alloys are also characterized by mixed exchange coupling between the local moments of Ni, Fe, Co (ferromagnetic) and Cr, Mn (antiferromagnetic) while the former exhibit only ferromagnetic coupling.
The underlying reason for the breakdown into two distinct resistivity groups can be understood in terms of disorder smearing of the Fermi surface. Figure 2a shows the spinresolved Fermi surfaces of four selected CantorWu alloys—two each from the lowρ_{0} and highρ_{0} group. While the minorityspin Fermi surfaces exhibit large disorder smearing for all of the alloys, the majorityspin channels are very different in the two classes. In particular, the majorityspin Fermi surfaces for NiCo and NiFeCo remain very sharp which corresponds to a long λ_{e}[ε_{F}]. As a result, the majorityspin channel acts as a short circuit for electron conduction resulting in an overall low resistivity. On the contrary, the majorityspin Fermi surfaces of NiFeCoCr and NiFeCoCrMn alloys are washed out with the consequence that the λ_{e}[ε_{F}] in both spin channels is very short and thus ρ_{0} is high. In the absence of a direct calculation of the residual resistivities, it has been previously noted that the transport properties of the CantorWu alloys qualitatively reflect the large differences in disorder smearing of the Fermi energy Bloch spectral functions^{5,8}, that are driven by differences in magnetic (FM versus mixed FM/AFM) coupling—an conclusion that turns out to be inadequate and even misleading. Notably, NiCoCr also has a very smeared Femi energy Bloch spectral function, and correspondingly high ρ_{0}, despite being robustly nonmagnetic^{20}.
Figure 2b shows a cartoon of the underlying spinresolved partial DOS of the alloying elements Ni, Co, Fe, Cr that illustrates why the spinresolved Fermi surface smearing is so different in the two alloy groups. Within the KKRCPA, the strength of the disorder scattering can be characterized by the ratio of two important energy scales: the energy separation (δ) between the band centers of different species a.k.a “band center mismatch” and the overall band width (W)^{21}. In transition metals, the most relevant band center is simply the dscattering resonance (ε_{d}) of the singlesite potential, while W encapsulates the spread of the dbands due to hybridization. These energy scales are illustrated in Fig. 2b. If δ/W << 1, the disorder scattering is weak, and the electron bands are welldefined. However, if δ/W ~ 1, disorder scattering is strong, leading to large disorder broadening (smearing) of the energy bands. For magnetic alloys, the electrons propagate and are scattered in two separate and independent spin channels^{22}—neglecting the spinmixing contribution. The spinmixing arising from spin–orbit coupling however affects ρ_{0} and brings about the anisotropic magnetoresistance as shown by Banhart et al.^{23}. As a result, the above argument applies to each spin channel independently, distinguished by subscript σ = ↑ and ↓ for spinup and spindown.
For alloys containing only Fe, Co, and Ni, the majorityspin 3d band centers are aligned due to minimization of the kinetic energy. As a result, δ_{↑} between all atom pairs is small and thus δ_{↑}/W_{↑} is always in the weak scattering regime. Because, different local moments form on different species and they couple ferromagnetically, the additional exchange splitting (Δ_{Exch}) which is proportional to the size of the local moment—with proportionality constant ~1 eV/μ_{B}, leads to a large band center mismatch (δ_{↓}) in the minorityspin channel and consequent large disorder scattering (δ_{↓}/W_{↓} ~ 1) (see Fig. 2b). A similar argument has been applied in Ni_{35}Fe_{65}^{24}, and NiCo^{25}. As a result, while the majorityspin Fermi surface is welldefined and λ_{e}[ε_{F}] is long, minorityspin channel electron transport is “blocked” by the strong disorder smearing of the Fermi surface. On the other hand, when alloying with lower band filling Cr, the band center in both channels is shifted towards the Fermi energy in order to realize charge neutrality. As a consequence, δ is large in both spin channels (large disorder scattering), thereby washing out the Fermi surface. In addition, the moments on Cr can couple either ferromagnetically or antiferromagnetically, further modifying δ. However, this does not substantially diminish δ, and thus δ/W remains in the strong scattering regime. Similar arguments also apply to Mn. It is worth noting that ρ_{0} is high only if strong disorder scattering is present at the Fermi energy. For example, strong disorder scattering in NiPd notwithstanding, ρ_{0} is low (ρ_{0} = 2.19 μΩ·cm) because the Fermi surface is mainly of sp character and large disorder scattering of d electrons does not pollute the Fermi surface in either spin channel (see the Fermi surface in the Supplementary Information, Section 5(D), Fig. S7). Whilst, these kinds of arguments are well known^{24,26,27,28}, the way they operate in this class of alloys is particularly startling.
As noted previously, while the calculated ρ_{0} of NiCo, NiFeCo, NiFeCoCr agree quantitatively with the experiments^{5,8} (see Fig. 1), ρ_{0} is still underestimated by a substantial fraction, particularly in NiFe, NiPd and also in some highρ_{0} alloys such as NiCoCr, NiFeCoCrMn, and NiFeCoCrPd. To shed light on this underestimation of ρ_{0}, we explore the effects of other scattering mechanisms—displacement scattering, magnetic scattering—beyond the singlesite approximation. The supercell method is employed to explore the effect of sitetosite atomic displacement and the complex magnetic effect (see the Method section). Figure 3a illustrates the supercell of NiFeCoCrMn alloy with different species randomly distributed. The direction and length of the black arrows on each site indicate the orientation and magnitude of each local moment, respectively.
Displacement scattering
Recently, atomic displacements have been shown to correlate with yield strength^{29}, suggesting the root mean square (rms) atomic displacements as a descriptor of the mechanical properties of CantorWu alloys. Given that all materials properties ultimately originate from the electronic structure, studying the effect of displacement scattering on the electronic transport provides a window into their importance as a scattering mechanism—albeit only at the Fermi energy.
Based on fully relaxed supercell calculations, the statistics of the magnitude of the atomic displacements—resolved by species—are shown in Fig. 3c for selected CantorWu alloys (see the Supplementary Information, Section 3, Fig. S2 for other alloys). As seen in the NiFeCo ternary alloy, atomic displacements (Δu) on all species are small with Δu(Ni) < Δu(Co) < Δu(Fe). This is consistent with the elements having similar atomic size and electronegativities. On the other hand, for alloys involving Cr and/or Mn, the atomic displacements on Cr and Mn are much larger. Again, consistent with expected larger size mismatch and charge transfer effects. As such, speciesdependent displacements become more pronounced towards the left side of the 3dtransition metal elements in the periodic table. Moreover, if alloying with Pd, as in NiPd and NiFeCoCrPd alloys, atomic displacements on all species are large due to sizemismatch between 3d and 4d elements (see Review article^{30} for the sizemismatch effect). Notably, the statistics of the angular dependence of the displacements appears to be random (see Fig. 3b for an illustration in NiFe alloy).
Assuming that the sitetosite variation of both magnitude and the orientation of the atomic displacements are uncorrelated, their effects on the electronic structure and ρ_{0} can be assessed by the alloy analogy model (AAM)^{31}. The results are shown in Fig. 4a. In most alloys the enhancement of ρ_{0} due to local displacements is rather small because the atomic displacements (of most alloys) are only a small fraction of the interatomic spacing. The exceptions are NiPd and NiFeCoCrPd alloys whose atomic displacements are large for all component species (see Fig. 3c). For NiPd, the resulting displacementenhanced resistivity is in good agreement with experiment. While for NiFeCoCrPd the inclusion of displacement scattering increases ρ_{0} by ~12%, the actual ρ_{0} is still underestimated. Therefore, the overall effect of displacement scattering in most alloys is small. Thus, the reasons for the general underestimation of ρ_{0} by singlesite theory alone must be sought elsewhere. Furthermore, this finding makes the strong correlation between the rms displacements and yield strength all the more interesting; perhaps, suggesting the existence of a more fundamental descriptor, rooted in the (common) underlying electronic structure.
Magnetism beyond the singlesite approximation
Unlike in the KKRCPA treatment of magnetism which deals with speciesdependent singlesite averaged magnetic moment, in the real alloys, the local moments of each species take on a distribution of values and can possibly point along arbitrary directions. In Fig. 3d we show that speciesresolved local moment distributions for selected CantorWu alloys, obtained using the supercell method, with moments constrained to be colinear. For the local moment distribution in other alloys, see Supplementary Information, Section 4(A), Fig. S3. It is noteworthy that, although supercell calculations yield a distribution of local moments, the speciesdependent averaged local moments turn out to be consistent with those obtained from KKRCPA (see the Supplementary Information, Section 4(B), Tab. S2).
For alloys considered here, several features of the local moment patterns can be found. Firstly, the local moments in lowρ_{0} alloys are ferromagnetically coupled and display only a small variation in the size of the moments. In highρ_{0} alloys with Cr, the magnitudes of Cr moments vary widely from negative (antiferromagnetically aligned) to positive (ferromagnetically aligned). In contrast to Cr, the magnitudes of Mn moments fall into two welldefined groups, large positive and large negative. This suggests that large onsite Hund's exchange promotes the formation of local moments on Mn, while the interatomic exchange interaction between Mn atoms is antiferromagnetic. The antiferromagnetic coupling associated with Cr and Mn can be attributed to the approximately halffilled d bands^{32,33,34}. Clearly, the complicated magnetic configurations just described have the potential to induce significant additional electron scattering beyond that included in the KKRCPA^{22,35,36,37}.
Using NiFe as an example, where ρ_{0} is underestimated (ρ^{Calc}_{0} = 3.3 versus ρ^{Exp}_{0} = 10.3 μΩ·cm), we first evaluate the effect of having a distribution of the local moment sizes. Based on supercell calculations, we find a Fe moment distribution that has a 0.2 μ_{B} broadening. To mimic these moment fluctuations using CPA, we discretize the Fe moment distribution into three types, having moments m_{0} and m_{0} ± 0.1 μ_{B} (where m_{0} is the average Fe moment), through scaling of the spin part of the exchangecorrelation functional. Unsurprisingly, ρ_{0} only increases by 0.3 μΩ·cm, indicating the ρ_{0} is insensitive to such longitudinal spin fluctuations.
Taking NiCoMn and NiFeCoCrMn alloys as examples, it is noted that supercell calculations find two wellseparated Mn moment distributions with opposite orientations and roughly similar amounts of Mn_{↑} (~65–35%) and Mn_{↓} (~35–65%) moments as the ground state. Due to the very localized nature of the Mn local moments such states can also be described within the KKRCPA approach in a manner analogous to disordered local moment state (DLM)^{38,39} previously used to describe paramagnetic Fe. Rather than investigate all possible ratios of Mn_{↑} and Mn_{↓,} we focus the system having equal concentrations as an approximated representative of the state having maximal “Mnmoment” disorder—we denote this state as DLMMn. Similarly, another collinear magnetic state with 100% Mn_{↓}, and one fully DLM state (DLM on all species) are also found to be stable solutions. Table 1 lists the CPA total energies for the three states. As expected, the DLMMn state of NiCoMn is the ground state. Moreover, the DLMMn state of NiCoMn gives ±2.2 μ_{B} for the local moment of Mn_{↑} and Mn_{↓}, which is consistent with the averaged Mn_{↑} and Mn_{↓} moments (~±2.3 μ_{B}) obtained from the supercell calculation. However, the Mn moment within the AFM state is only −0.7 μ_{B}, further casting suspicion on 100% Mn_{↓} as representative of Mn containing CantorWu alloys. Further justification of the efficacy of the DLMMn state can be obtained by comparing the speciesresolved DOS with the supercell and studying stability of the Heisenberg interactions (unpublished data). For ρ_{0}, we find its value depends sensitively on the assumed magnetic state—ρ_{0} is lowest in the AFM state and increases by ~50% in the DLMMn ground state. Unfortunately, the experimental value has not yet been measured. The reason for the different ρ_{0} behavior can be easily traced to the underlying electronic structure: the AFM state exhibits a relatively sharp Fermi surface in the minorityspin channel while the Fermi surface in both spin channels smears out for the DLMMn state (see the Supplementary Information, Section 5(B), Fig. S6). In contrast to NiCoMn, it turns out the AFM and DLMMn states in NiFeCoCrMn alloy are not only close in energy but their ρ_{0} are insensitive to which state is considered because the electron scattering by magneticdriven disorder is already almost saturated.
So far, ρ_{0} has been calculated assuming collinear spin configurations. However, spin noncollinearity is also possible, particularly in Mn and/or Crcontaining alloys due to the geometric frustration of antiferromagnetism on a triangular lattice (as the (111) plane of the fcc lattice)^{40,41,42}, oscillating exchange interactions as a function of distance^{28,43}, and spin–orbit interaction. For example, NiFeCoCrMn alloy is found to have a spin glass state both experimentally^{44} and theoretically (unpublished data).
In principle, the spin noncollinearity can be dealt with straightforwardly from the spin noncollinear calculations based on the supercell method. However, such calculations for supercell size sufficient to gain good statistics of distributions of spin orientations are extremely demanding and remain a research project in their own right. Such calculations are made particularly difficult by the need not only treat the spin noncollinearity but also include the spin–orbit interactions—particularly for Pdcontaining CantorWu alloys, making them beyond the scope of this paper.
Without such a sophisticated evaluation of the spin noncollinearity at zero temperature (probability distribution of the spin orientations for each species), here we calculate the maximum contribution to the residual resistivity (ρ_{0}) that can arise from spin disorder. To assess this, we again employ the AAM by using a discrete set of random and uncorrelated spin moments that are distributed uniformly in space, where the magnitude of the speciesdependent local moments are obtained from CPA ground state^{31}. Notably, spin noncollinearity in lowρ_{0} alloys is negligibly small, as verified by fully relativistic supercell calculations. Therefore, we only explore the effect of full spin disorder in the highρ_{0} alloys. The resulting resistivities, which can be viewed as the maximum effect of spin disorder on ρ_{0}, are shown in Fig. 4b. A sizable ρ_{0} enhancement, as large as 10 μΩ·cm, is observed in NiFeCoCr, NiFeCoMn, and NiFeCoCrPd. Therefore, the full spin disorder produces a modest increase of ρ_{0} in highρ_{0} alloys.
Notwithstanding the overall improved agreement with the measurement resulting from inclusion of the additional scattering mechanisms discussed above, the remaining moderate underestimation of ρ_{0} in some alloys (NiFe, NiCoCr, NiFeCoCrMn, NiFeCoCrPd) requires consideration of other possible theoretical shortcomings. The three most obvious being: going beyond the singlesite treatment of disorder; inclusion of any possible (but currently mostly unknown except for NiFe^{45}) short range order (SRO); consideration of additional electron correlation effects, beyond LDA. For binary alloys where the first two effects have been considered their impact on ρ_{0} has been found to be small^{46,47} (with specific exceptions, e.g., in Kstate alloys^{48}). However, it is not clear whether the impacts of the first two effects in the CantorWu alloys are also small. Specifically, the specific heat measurements of Crcontaining CantorWu alloys^{49} show a Kstate transition at 800–1000 K, which is usually attributed to the orderdisorder transition^{50,51,52}. Although the effects of SRO on ρ_{0} are clearly worthy investigating, accounting for them requires treatments of multisite scattering processes that go beyond the singlesite approximation—e.g., nonlocal CPA (see Review article^{53} and references therein)—and are beyond current capabilities. In principle the investigation of additional Coulomb correlation effects on ρ_{0} could be addressed by a combination of KKRCPA and Dynamical Mean Field Theory^{54}, however, it is beyond the scope of the present work.
In conclusion, we have demonstrated that the abnormal and disparate electronic transport in CantorWu alloys at zero temperature is dominated by electron scattering arising from sitetosite potential disorder. In particular, it is found that Cr and/or Mn produce strong disorder scattering as a result of the proximity of the dscattering resonance to the Fermi energy which results from their reduced band filling. Additionally, other electron scattering mechanisms are explored explicitly and shown to be small in most alloys with the exception of NiPd and NiFeCoCrPd, where the effect of displacement scattering is large; NiCoMn, where the DLMMn ground state significantly raises ρ_{0}; and NiFeCoCrMn/NiFeCoCrPd, where the effect of spinnoncollinearity is large. A profound understanding of the electronic transport in disordered alloys not only provides insights on the Fermi surface, but also on the overall effects of disorder throughout the occupied dbands. As such it sheds light on the bonding mechanisms responsible for many of the exotic properties of HEAs, such as mechanical^{4,55}, and radiation response^{5} properties. Here it is notable that while the effect of displacement scattering on ρ_{0} is small, the highly unusual solid solution strength in several of CantorWu alloys has been correlated with the magnitude of displacement fluctuations^{29}. Resolving this apparent dichotomy is clearly a challenge worthy further investigation.
Methods
Our calculations employ the ab initio spinpolarized fully relativistic KorringaKohnRostoker Green’s function method^{56,57}, combined with the Coherent Potential Approximation^{58} (hereafter referred to as KKRCPA)—as implemented in the Munich SPRKKR package^{59,60}—to calculate the effect of disorder on the electronic structure and residual resistivity. The CPA method is a selfconsistent theory of the effects of substitutional disorder on the configurationally averaged singlesite properties of alloys. As such it includes, the effect of [δ, Δ_{Exch}] on electronic band structure and quasiparticle lifetime on the same ab initio footing. The conductivity tensor is calculated by using the linear response KuboGreenwood formula^{61,62} for the configurational averaged state, described by the CPA^{63}:
where \(j_\alpha ^\mu\) denotes the μcomponent of the current density operator j for species α with concentration x_{α} and G^{+}(ε_{F}) is the retarded Green’s function at the Fermi energy. Within the KKRCPA calculations, the local density approximation (LDA)^{64} is employed for exchange and correlation. The sensitivity of the results to the exchangecorrelation functional is discussed in the Supplementary Information, Section 2, Fig. S1. To study the effect of displacement scattering and spin disorder on ρ_{0}, we used the socalled alloy analogy model (AAM) to perform the configurational average over a discrete set of speciesresolved atomic displacements and local moment orientations^{31}. To obtain the statistics of the atomic displacements and local moments, we performed standard supercell calculations for a 256atom special quasirandom structure (SQS)^{65} using the projector augmented wave method (PAW)^{66} as implemented in the Vienna ab initio simulation package (VASP)^{67} (see the Supplementary Information, Section 1).
Data availability
The authors declare that the data supporting this study are available from the corresponding author upon request.
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Acknowledgements
This work was supported by the Energy Dissipation and Defect Evolution (EDDE), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences under contract number DEAC0500OR22725. This research used resources of Oak Ridge National Laboratory’s Computer and Data Environment for Science (CADES) and the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DEAC0500OR22725. S.M. acknowledges fruitful discussions with K.D. Belashchenko, B.C. Sales, and K. Jin. S.W., S.M., and H.E. would like to thank the DFG (Deutsche Forschungsgemeinschaft) for financial support within the priority program SPP 1538 and the collaborative research centers 689 and 1277.
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G.M.S. conceived this research; S.M. carried out the firstprinciples calculations and analyzed the results with G.M.S., G.D.S., and S.W.; S.M. and G.M.S. wrote the paper, and all authors participated in the discussions and contributed to finalize the paper.
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Correspondence to Sai Mu or George M. Stocks.
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Mu, S., Samolyuk, G.D., Wimmer, S. et al. Uncovering electron scattering mechanisms in NiFeCoCrMn derived concentrated solid solution and high entropy alloys. npj Comput Mater 5, 1 (2019) doi:10.1038/s415240180138z
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