Applicability of the strongly constrained and appropriately normed density functional to transition metal magnetism
AApplicability of the strongly constrained and appropriately normed density functionalto transition metal magnetism
Yuhao Fu and David J. Singh ∗ Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211-7010 USA (Dated: October 25, 2018)We find that the recently developed self consistent and appropriately normed (SCAN) meta-generalized gradient approximation, which has been found to provide highly accurate results formany materials, is, however, not able to describe the stability and properties of phases of Fe im-portant for steel. This is due to an overestimated tendency towards magnetism and exaggeration ofmagnetic energies, which we also find in other transition metals.
Density functional theory (DFT) calculations are acentral tool in condensed matter physics, chemistry, andmaterials science. This utility is the result of the avail-ability of sufficient accuracy in tractable approximatefunctionals. This enables predictive calculations of prop-erties of interest and elucidations of underlying mecha-nisms of physical behavior. Therefore the development ofnew practical functionals that improve the accuracy, andtherefore the range of behaviors and materials that canbe studied with DFT calculations, is of great interest.Steel is arguably the most important industrial ma-terial. Annual production exceeds 1.7 billion metrictonnes. Steels are complex materials whose proper-ties are controlled by microstructure. These microstruc-tures are what provides steel with desirable combina-tions of ductility, toughness and tensile strength. Thesemicrostructures come from balances between differentphases mainly in the Fe-C phase diagram. While theground state of Fe is body centered cubic (bcc), an anequilibrium face centered cubic (fcc) phase exists betweenbetween 1185 K and 1667 K. Carbon has a much highersolubility in this fcc phase (up to 2.14 wt% and 0.76wt% at the eutectoid) than in the bcc phase (maximumof 0.022 wt%), leading to an easily accessed eutectoidpoint in the phase diagram (at 1000 K and 0.77 wt%C). Cooling leads to nanoscale and microscale precipi-tation of cementite (Fe C, a very hard phase), in a bccFe matrix, as well as non-equilibrium austenite (fcc Fewith C) and sometimes other phases associated with al-loying elements, to form microstructures such as perlite,martensite and bainite. These microstructures, some-times modified by mechanical deformation steps, are keyto the properties of steel. First principles based under-standing of steel requires the ability to model these dif-ferent phases and their relationships, most importantlythe relationship between the ground state bcc structure(ferrite) and the fcc structure (austenite).This has posed ongoing challenges to density func-tional calculations. Early on it was found that the other-wise highly successful local (spin) density approximation(LDA), cannot describe Fe. In particular, it was shownthat the LDA predicts a non-magnetic fcc ground statefor Fe, with the ferromagnetic bcc structure lying higherin energy. The LDA does, however, provide an accuratevalue of the spin magnetization of Fe, when constrained to its experimental bcc structure.An important step was the development of generalizedgradient approximation (GGA) functionals, based onknowledge of the behavior of the exchange correlationhole in inhomogeneous electron gasses.
In addition tocorrectly predicting the bcc ground state and spin mag-netization of Fe, these GGA functionals greatly im-proved the energetics of a wide variety of molecules andsolids. This was a remarkable achievement, especiallyconsidering that these GGA functionals were based onconstraints and scaling for the electron gas and not fitsto known materials properties.Therefore, it is very reasonable to assume that func-tionals that incorporate additional known exact prop-erties of the inhomogeneous electron gas will at leaston average improve the description of atoms, moleculesand solids. A significant recent development along theselines was the construction of a strongly constrained andappropriately normed (SCAN) functional. This is asemi-local meta-GGA functional. Meta-GGA function-als are more convenient for calculations than hybridfunctionals, especially in extended systems.The SCAN functional satisfies exact constraints, in-cluding importantly the Lieb-Oxford lower bound for theexchange energy, also important for the construc-tion of the earlier GGA functionals, as well as scalingrelations. It is also designed to revert to the LDA forthe uniform electron gas (a norm) and also uses the hy-drogen atom as a norm for the exchange. This is impor-tant in regards to self-interaction errors. It is designed tobe accurate both for the slowly varying electron gas, andfor atoms, which is not possible in GGA functionals. Tests done to date generally confirm the expectationthat SCAN provides highly accurate results for manymaterials, as might be expected from the manyconstraints that it satisfies. However, there is at leastone indication that SCAN may not improve the alreadygenerally good description of magnetism in some metallicferromagnets. Isaacs and co-workers reported that themagnetization of Fe, Co and Ni are enhanced by 0.42 µ B , 0.13 µ B and 0.1 µ B , respectively, relative to thewidely used GGA functional of Perdew, Burke and Ernz-erhof (PBE). They observed that this degrades agree-ment with experiment for Fe and Ni. Ekholm and co-workers, also performed calculations for Fe, Co and Ni, a r X i v : . [ c ond - m a t . m t r l - s c i ] O c t and found that the moments were enhanced relative toexperiment, which they ascribed to a downshift of the 3dstates. We did calculations with the LDA, the PBE GGA andthe SCAN functional using two different methods, specif-ically the projector augmented wave (PAW) method as implemented in the VASP code, and the all elec-tron general potential linearized augmented planewave(LAPW) method, as implemented in the WIEN2kcode. The VASP code includes a self-consistent calcu-lation with the SCAN functional, except that that it re-lies on PAW potentials constructed for the PBE GGA,which is an approximation. The LAPW method as im-plemented in WIEN2k is an all electron method that doesnot rely on pseudopotentials. However, at present, SCANcalculations with this method must be done non-self-consistently, in particular, calculating the energy usingthe SCAN functional, but based on the density from asemi-local calculation. We used the PBE GGA with theconstrained DFT, specifically the fixed spin moment(FSM) procedure, to generate the spin densities forcalculating the SCAN total energies.This procedure involves solving the Kohn-Sham equa-tions with a constraint that the integrated spin den-sity (the spin-moment) equal a specified value. This isachieved by imposing the constraint via a difference inspin-up and spin-down Fermi levels, equivalent to a mag-netic field operating on spin only. We used dense gridsof discrete moments to obtain the plots shown here. Thisallows us also to calculate the total energy as a functionof the constrained moment for ferromagnetic materials,and provides insights into the problems in the treatmentof magnetic transition metals with SCAN. We carefullyconverged the calculations, using large basis sets, anddense convergence tested k-point grids for all materials.We compared the results from the two codes and find verysimilar results, which supports the different approxima-tions involved. We also did self consistent calculationsincluding spin orbit for the PBE and LDA functionalsto quantify the effect of spin orbit, which could not beapplied in FSM calculations for the SCAN functional.These show that the effects of spin orbit are small onthe scale of the differences between the functionals, andcannot resolve the discrepancies.The measured saturation magnetizations of Fe, Co andNi are 2.22 µ B , 1.72 µ B and 0.62 µ B , on a per atombasis. These include both spin and orbital contribu-tions. The orbital moments of Fe and Co from x-ray mag-netic circular dichroism (XMCD) experiments are 0.09 µ B and 0.15 µ B , per atom, while the experimental valuefor Ni is 0.05 µ B . Our spin ( m sp ) and orbital ( m orb )moments, at the experimental lattice parameters fromLAPW calculations with the PBE functional includingspin orbit are m sp =2.22 µ B and m orb =0.04 µ B for Fe, m sp =1.62 µ B and m orb =0.08 µ B for Co, and m sp =0.63 µ B and m orb =0.05 µ B for Ni, i.e. spin moments veryclose to the experimental values, and orbital momentsare small and underestimated for Fe and Co, as in prior m ( B /atom) E ( m e V / a t o m ) -165 156 593EXP fcc LDA fcc
SCAN fcc
PBE
LDAPBESCAN
FIG. 1. FSM energy for bcc Fe at the experimental latticeconstant of 2.86 ˚A, on a per atom basis. The dashed linesare the energies of non-spin-polarized fcc Fe, at the optimizedlattice parameter for the different functionals. The small dotsindicate the minimum energy points. calculations. In our calculations without spin orbit, thePBE spin moments are m sp =2.22 µ B , m sp =1.62 µ B , and m sp =0.63 µ B , for Fe, Co and Ni, respectively, which arethe same as those with spin orbit to the quoted precision.Thus spin orbit does not have a significant effect on thecalculated spin moments for these 3d ferromagnets. It isalso to be noted that any enhancement of the spin mo-ment over the PBE values will degrade agreement withexperiment, including the case of Co.Fig. 1 shows our results for the magnetic energy of bccFe at its experimental lattice parameter, in comparisonwith the energy of non-spin-polarized fcc Fe. Numericalvalues and magnetic moments are given in Table I. Asseen, the SCAN functional yields dramatically differentresults from the LDA and PBE functionals. Energy playsa central role in density functional theory. As mentioned,the LDA fails for Fe, predicting that the fcc structure haslower energy, in particular by 0.165 eV. The PBE func-tional yields the correct ordering, with an energy differ-ence of 0.156 eV, considering a non-magnetic fcc struc-ture. The SCAN functional predicts a much more stablebcc structure, with an overestimated spin moment of 2.63 µ B /atom and an fcc - bcc energy difference of 0.593 eV.This is due to a much larger magnetic energy. Self con-sistent calculations using VASP yield similar numbers,specifically a spin moment of 2.65 µ B and an energy dif-ference of 0.579 eV, for SCAN. While these numbers donot include the magnetic enthalpy of fcc Fe, it is clearthat SCAN predicts an overly stable ferromagnetic statefor bcc Fe. The experimental enthalpy difference between TABLE I. Calculated properties of Fe. a exp and a calc are theexperimental and calculated lattice parameters of bcc Fe, re-spectively. The fcc-bcc energy difference ∆ E fcc − bcc is as inFig. 1. ∆ E mag is the magnetic energy from the difference be-tween non-spin polarized and ferromagnetic states. Energiesare per atom. LDA PBE SCAN Expt. a (˚A) 2.76 2.84 2.85 2.86 m sp ( a exp ) ( µ B ) 2.21 2.21 2.63 2.13 m sp ( a calc ) ( µ B ) 2.00 2.16 2.60 - m orb ( a exp ) ( µ B ) 0.05 0.04 - 0.09∆ E mag ( a exp ) (meV) 448 566 1117 -∆ E mag ( a calc ) (meV) 317 529 1078 -∆ E fcc − bcc (meV) -165 156 593 60 aa estimate from extrapolated thermodynamic data (see text).FIG. 2. LDA, PBE and SCAN FSM energy in meV/atomfor bcc and fcc Fe as functions of lattice parameter and spinmoment. bcc and fcc Fe at 1185 K from assessed calorimetric mea-surements is 0.009 eV/atom, while the low temperatureenergy difference from thermodynamic models based onexperimental data is 0.06 eV/atom. Fig. 2 shows the FSM energy as functions of lattice pa-rameter and moment for bcc and fcc Fe. In accord witholder work, in addition to its failure to predict thecorrect ground state, the LDA strongly underestimatesthe lattice parameter of magnetic bcc Fe, while the PBEGGA give values in closer agreement with experimentaldata. The SCAN functional gives a lattice parametersimilar to PBE for the bcc structure. The SCAN func-tional predicts very different behavior for the fcc phase.When constrained to ferromagnetism, the LDA and PBEpredict either no magnetism or a low moment state. TheSCAN functional predicts a high moment state. Whilehigh moment ferromagnetism does not preclude a still
TABLE II. Magnetic data for Ni and Co from fixed spin mo-ment calculations at the experimental lattice parameters. Allquantities are per atom.LDA PBE SCAN Expt.Co m sp ( µ B ) 1.61 1.62 1.79 1.57 m orb ( µ B ) 0.08 0.08 - 0.15∆ E mag (meV) 199 255 574 -Ni m sp ( µ B ) 0.62 0.63 0.76 0.57 m orb ( µ B ) 0.05 0.05 - 0.05∆ E mag (meV) 50 61 129 - lower energy ground state with antiferromagnetism, it isincompatible with a weak low moment antiferromagneticstate, due to the large magnetic energy associated withthe high moment state.Experimental information on the magnetism of free fccFe is limited by the fact that it is not a stable low tem-perature phase. However, fcc Fe films grown epitaxiallyon Cu are paramagnetic at ambient temperature, and be-come antiferromagnetic at low temperature with T N ∼ similar to the behavior of small fcc Fe precipitatesin an fcc Cu matrix. According to neutron diffractionmeasurements these have a small moment of ∼ µ B per Fe. Based on this, on this, as well as the prop-erties of non-ferromagnetic austenitic steels, thermo-dynamic modeling, and extrapolation of alloy data it is thought that fcc Fe is an itinerant weak antifer-romagnet with a Neel temperature below 70 K, and arelatively small contribution of magnetism to the energy.DFT studies have indicted that there is an additionalhigh volume high spin ferromagnetic state with higherenergy, and this has been discussed in connection withthe stability of the fcc phase between 1185 K and 1667K. We also did self consistent calculations with VASP forthe energy and moments of a hypothetical antiferromag-netic bcc Fe, where the moments of the two Fe atomsin the conventional cubic cell are oppositely aligned. Wefind that with the PBE functional the moments as mea-sured by the spin density around Fe sites, is reduced from2.25 µ B in ferromagnetic case (note there is a small neg-ative interstitial spin moment of ∼ -0.03 µ B ) to 1.71 µ B .In contrast, the SCAN result for the antiferromagneticcase of 2.66 µ B is almost exactly the same as for theferromagnetic case, i.e. 2.65 µ B . PBE predicts interme-diate itinerant / local moment behavior for bcc Fe, whileSCAN predicts that Fe is in the local moment limit, ingeneral disagreement with experiment. Thus the known data is consistent with good agree-ment between the predictions of the PBE functional andexperiment. Importantly, it is inconsistent with the pre-dictions of the SCAN functional. Specifically, the resultspoint to severe problems in the SCAN predictions formagnetic energies and moments in Fe. It is notable thatthe differences in magnetic energies between SCAN andthe LDA and PBE functionals are much larger than thedifferences between predictions of those two functionals. E ( m e V / a t o m ) EXP Co LDAPBESCAN m ( B /atom) EXP
Ni0.0 0.3 0.61001020 Pd
FIG. 3. FSM calculations of the magnetic energy of transitionmetal elements with the LDA, PBE and SCAN functionals.
Considering the very different predictions of SCAN ascompared to standard functionals for the magnetic prop-erties of Fe, it is of interest to investigate whether thisis general problem, or if it is restricted to Fe. Accord-ingly, we performed fixed spin moment and self-consistentcalculations for other materials. We start with cemen-tite (Fe C), which is ferromagnetic and a key ingredi-ent in many steels. The calculated spin magnetizationper three iron atom formula unit is 5.75 µ B with thePBE functional and 6.87 µ B with SCAN (based on self-consistent VASP calculations at the experimental lat-tice parameters; very similar values were obtained fromLAPW FSM calculations). This compares with a totalroom temperature saturation magnetization of 5.3 µ B from experiment, indicating again a substantial errorwith SCAN.Fig. 3 and Table II give the results of FSM calcu-lations for other elements with the experimental struc- tures and lattice parameters. Hexagonal close packed(hcp) Co and fcc Ni are the other ferromagnetic 3d ele-ments. Ni is regarded as a prototypical itinerant ferro-magnet. SCAN gives very much larger magnetic energiesfor these two elements as compared with PBE and LDA.We also find enhanced spin moments with SCAN, andsimilar to Fe we find significant degradation with respectto experiment for both Ni and Co. The calculated spinmoments with the SCAN functional are 1.80 µ B for Coand 0.77 µ B for Ni. bcc V and fcc Pd are both paramag-netic metals down to 0 K according to experiment. Pd isvery close to ferromagnetism, and for this reason exhibitsstrong spin fluctuations that have been implicated in pre-venting a superconducting state in this element. Pdis a particularly interesting test for density functionals,since it is incorrectly predicted to be ferromagnetic bysome hybrid functionals, while showing borderline fer-romagnetic behavior with standard GGAs. V is notas close to ferromagnetism and is a superconductor at lowtemperature. Our PBE and LDA results are consistentwith these experimental facts. SCAN on the other handpredicts an effectively infinite susceptibility for V, anda low moment ferromagnetic state for Pd. Thus quali-tatively similar to Fe, SCAN strongly overestimates themagnetic tendencies of V, Co, Ni and Pd.The above results point to a surprising degradation ofthe predictions of SCAN relative to PBE in describingmagnetism in transition metals, and suggest caution inthe use of this functional for predicting magnetic prop-erties of materials. This may perhaps be due to thechallenge of obtaining the itinerant physics of systemslike Fe with multiple partially occupied d-orbitals, andat the same time reproducing correct physics of atoms,including cancellation of self-interaction. In any case, wehope that the above results may motivate further workto develop improved meta-GGA functionals, particularlyfunctionals that satisfy known constraints, from the inho-mogeneous electron gas, including the many constraintssatisfied by SCAN, and possibly additional constraints,and at the same time predict accurate magnetic proper-ties of metals.This work was supported by the U.S. Department ofEnergy, Office of Science, Basic Energy Sciences, AwardNumber de-sc0019114. We are grateful for helpful dis-cussions with Guangzong Xing. ∗ [email protected] W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965). B. L. Bramfitt, in
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