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

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 {\AA} (strong scattering limit) to ~10$^3$ {\AA} (weak scattering limit). Here, we delineate the underlying electron scattering mechanisms responsible for this disparate behavior. While spin dependent site-diagonal disorder 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. Unexpectedly, other scattering mechanisms (e.g. displacement scattering) 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, semi-conductors 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 r0, is finite and its precise value provides a direct measure of the disorder induced changes in the underlying electronic structure.
In a general N-component 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, single-phase, disordered solid-solutions on a simple crystal lattice. The first single-phase 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  Of interest here are the results of recent residual resistivity measurements 5,8

of a subset of
Cantor-Wu alloys that show, rather than increasing monotonically with increasing numbers of components, values of r0 break into two subgroups of low (r0 <10 µW•cm) and high (r0 >75 µW•cm) resistivity alloys. In addition, two entropically identical alloys, NiCoFe (r0 = 1.7µW•cm) and NiCoCr (r0 =92.7 µW•cm), fall into different resistivity groupings. Remarkably, the least and most resistive alloys differ by almost two orders of magnitude, r0(NiCo)=1.3µW•cm; r0(NiFeCoCrPd)=124.8µW•cm. Interestingly, the low resistivity group have r0 values typical of dilute weak scattering alloys in which there are clearly defined host (solvent) and impurity (solute) elements. In such alloys, r0 arises from the scattering of a low Fermi energy DOS of nearly-freeelectron sp-states with large λ e [ε F ] and r0 generally obeys both Nordheim's relation (r0∝c ((1-c); where c is impurity concentration) 9 and Linde's "law" (r0 ∝ (DZ) 2 ; where DZ 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) d-bands 5 . At the other extreme, high-r0 NiFeCoCrPd is 5 close to the Mott-Ioffe-Regel (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 Cantor-Wu 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 Cantor-Wu 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 Cantor-Wu alloys provide a unique opportunity for uncovering the underlying scattering mechanisms that give such disparate and non-monotonic behavior in 3d-transition 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 Cantor-Wu alloys using state-of-the-art ab initio transport theory for disordered alloys. Consistent with experiment, we find that the calculated r0 break into high-r0 (alloys involving Mn/Cr elements) and low-r0 (the others) sets. We show that it is the magnitude of the spin-dependent site-diagonal potential scattering that makes the dominant contribution to r0 and gives rise to this remarkable difference in r0 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 site-to-site variations in local moment magnitude and 6 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 Discussions
In solid solution alloys, all electron scattering ultimately results from the disorder-induced siteto-site 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. Single-site electron scattering can be thought of as resulting from the site-to-site variation (d) in the local potential due to the random distribution of elements. In the presence of magnetism, conduction electrons experience an additional inhomogeneous exchange field (DExch), which further increases the site-disorder and is different in separate spin channels. In the following,  Figure 1 compares the calculated r0 with the measured values 5,8 . From the figure, there are three clear conclusions. Firstly, consistent with the experiments, the calculated values of r0 separate into two groups: low-r0 alloys (NiPd, NiCo, NiFe, NiFeCo), having r0 < 10 µW•cm and high-r0 alloys (the others). This finding is independent of the particular exchange correlation functional used. Secondly, while the calculated value of r0 including only sitediagonal disorder, underestimates r0, the contribution from site-diagonal disorder is dominant across all Cantor-Wu alloys. Thirdly, the magnitude of r0 correlates with the types of alloying elements. In particular, for alloys containing only the Ni, Fe, Co, that have nearly-filled 3dbands, r0 is low. While for alloys containing both Ni, Fe, Co, and Cr, Mn, whose d bands are approximately half-filled, r0 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 2 (a) shows the spinresolved Fermi surfaces of four selected Cantor-Wu alloys -two each from the low-r0 and high-r0 group. While the minority-spin Fermi surfaces exhibit large disorder smearing for all of the alloys, the majority-spin channels are very different in the two classes. In particular, the majority-spin Fermi surfaces for NiCo and NiFeCo remain very sharp which corresponds to a long λ e [ε F ]. As a result, the majority spin channel acts as a short circuit for electron conduction resulting in an overall low resistivity. On the contrary, the majority-spin Fermi surfaces of NiFeCoCr and NiFeCoCrMn alloys are washed out with the consequence that the λ e [ε F ] in both 8 spin channels is very short and thus r0 is high. In the absence of a direct calculation of the residual resistivities, it has been previously noted that the transport properties of the Cantor-Wu 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 r0, despite being robustly nonmagnetic 20 . 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 d-scattering resonance (ed) of the single-site potential, while W encapsulates the spread of the d-bands due to hybridization. These energy scales are illustrated in Fig. 2 (b). If δ/W << 1, the disorder scattering is weak, and the electron bands are well-defined. 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 spin-mixing contribution. The spin-mixing arising from spin-orbit coupling however affects r0 and brings about the anisotropic magnetoresistance as shown by 9 Banhart et al. 23 . As a result, the above argument applies to each spin channel independently, distinguished by subscript s = and ¯ for spin-up and spin-down.
For alloys containing only Fe, Co and Ni, the majority-spin 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 (DExch) which is proportional to the size of the local moment -with proportionality constant ~ 1eV/µB, leads to a large band center mismatch (δ ↓ ) in the minority-spin channel and consequent large disorder scattering (δ¯/W¯~1) (see Fig. 2(b)). A similar argument has been applied in Ni35Fe65 24 , and NiCo 25 . As a result, while the majority-spin Fermi surface is well-defined and λ e [ε F ] is long, minority-spin 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 r0 is high only if strong disorder scattering is present at the Fermi energy.  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 r0 of NiCo, NiFeCo, NiFeCoCr agree quantitatively with the experiments 5,8 (see Fig. 1), r0 is still underestimated by a substantial fraction, particularly in NiFe, NiPd and also in some high-r0 alloys such as NiCoCr, NiFeCoCrPd alloys, atomic displacements on all species are large due to size-mismatch between 3d and 4d elements (see Review article 30 for the size-mismatch effect). Notably, the statistics of the angular dependence of the displacements appears to be random (see Fig. 3

(b) for an illustration in NiFe alloy).
Assuming that the site-to-site variation of both magnitude and the orientation of the atomic displacements are uncorrelated, their effects on the electronic structure and r0 can be assessed by the alloy analogy model (AAM) 31 . The results are shown in Figure 4 (a). In most alloys the enhancement of r0 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. 3 (c)). For NiPd, the resulting displacement-enhanced resistivity is in good agreement with experiment. While for NiFeCoCrPd the inclusion of displacement scattering increases r0 by ~12%, the actual r0 is still underestimated. Therefore, the overall effect of displacement scattering in most alloys is small. Thus, the reasons for the general underestimation of r0 by single-site 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.  Fig. S6). In contrast to NiCoMn, it turns out the AFM and DLM-Mn states in NiFeCoCrMn alloy are not only close in energy but their r0 are insensitive to which state is considered because the electron scattering by magnetic-driven disorder is already almost saturated.

Magnetism beyond the single-site approximation
So far, r0 has been calculated assuming collinear spin configurations. However, spin noncollinearity is also possible, particularly in Mn-and/or Cr-containing alloys due to the geometric frustration of antiferromagnetism on a triangular lattice (as the (111) plane of the fcc lattice) [41][42][43] , oscillating exchange interactions as a function of distance 28,44 , and spin-orbit interaction. For example, NiFeCoCrMn alloy is found to have a spin glass state both experimentally 45 and theoretically 46 .
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 spinorbit interactions -particularly for Pd-containing Cantor-Wu 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 (r0) 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 species-dependent local moments are obtained from CPA ground state 31 . Notably, spin noncollinearity in low-r0 alloys is negligibly small, as verified by fully relativistic supercell calculations. Therefore, we only explore the effect of full spin disorder in the high-r0 alloys. The resulting resistivities, which can be viewed as the maximum effect of spin disorder on r0, are shown in Fig. 4 (b). A sizable r0 enhancement, as large as 10 µW•cm, is observed in NiFeCoCr, NiFeCoMn, and NiFeCoCrPd. Therefore, the full spin disorder produces a modest increase of r0 in high-r0 alloys.
Notwithstanding the overall improved agreement with the measurement resulting from inclusion of the additional scattering mechanisms discussed above, the remaining moderate 16 underestimation of r0 in some alloys (NiFe, NiCoCr, NiFeCoCrMn, NiFeCoCrPd) requires consideration of other possible theoretical shortcomings. The three most obvious being: going beyond the single-site treatment of disorder; inclusion of any possible (but currently mostly unknown except for NiFe 47 ) 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 r0 has been found to be small 48,49 (with specific exceptions, e.g., in Kstate alloys 50 ). However, it is not clear whether the impacts of the first two effects in the Cantor-Wu alloys are also small. Specifically, the specific heat measurements of Cr-containing Cantor-Wu alloys 51 show a K-state transition at 800-1000 K, which is usually attributed to the orderdisorder transition [52][53][54] . Although the effects of SRO on r0 are clearly worthy investigating, accounting for them requires treatments of multisite scattering processes that go beyond the single-site approximation -e.g. nonlocal CPA (see Review article 55 and references therein) -and are beyond current capabilities. In principle the investigation of additional Coulomb correlation effects on r0 could be addressed by a combination of KKR-CPA and Dynamical Mean Field Theory 56 , however, it is beyond the scope of the present work.
In conclusion, we have demonstrated that the abnormal and 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/or Mn produce strong disorder scattering

Data availability
The authors declare that the data supporting this study are available from the corresponding author upon request. Figure 1 Residual resistivity of Cantor-Wu alloys due to site-diagonal disorder (SDD) (blue bars), compared with experimental data (red bars).

Figures
28 Figure 2 (a) Fermi surface (G-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 speciesresolved DOS in NiFeCoCr, in which e s d denotes the 3d band centers for spin s(s = or ¯). The 29 band center mismatch ds is defined as max{e s d (i)}-min{e s d (i)} with i denoting different species.
The exchange splitting is denoted as DExch:= e¯d -e d.  Tables: Table 1 The total energies (Etot, meV/site) and residual resistivity (r0, µW•cm) for the multiple magnetic states in NiCoMn and NiFeCoCrMn from KKR-CPA.