Exchange Interactions in Paramagnetic Amorphous and Disordered Crystalline CrN-based Systems
A. Lindmaa, R. Lizárraga, E. Holmström, I. A. Abrikosov, B. Alling
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J un Exchange Interactions in Paramagnetic Amorphous and Disordered CrystallineCrN-based Systems
A. Lindmaa, ∗ R. Lizárraga, E. Holmström, I. A. Abrikosov, and B. Alling Department of Physics, Chemistry and Biology (IFM),Linköping University, SE-581 83 Linköping, Sweden Instituto de Ciencias Físicas y Matemáticas, Facultad de Ciencias,Universidad Austral de Chile, Casilla 567, Valdivia, Chile (Dated: December 27, 2017)We present a first principles supercell methodology for the calculation of exchange interactionsof magnetic materials with arbitrary degrees of structural and chemical disorder in their high tem-perature paramagnetic state. It is based on a projection of the total magnetic energy of the systemonto local pair clusters, allowing the interactions to vary independently as a response to their localenvironments. We demonstrate our method by deriving the distance dependent exchange interac-tions in vibrating crystalline CrN, a Ti . Cr . N solid solution as well as in amorphous CrN. Ourmethod reveals strong local environment effects in all three systems. In the amorphous case we usethe full set of exchange interactions in a search for the non-collinear magnetic ground state.
The understanding and accurate modeling of magneticmaterials at finite temperatures is a grand challenge insolid state physics [1]. Its importance is highlighted bydemands for new iron-based materials in the steel in-dustry, improvements in materials for electrical motorsand generators, and the outlook of the full utilizationof the spin degree of freedom in electronics [2, 3]. Thedifficulties in providing an accurate description of high-temperature magnetic materials stem from the complex-ity of the quantum excitations that must be included inthe models and simulations. Such excitations are of elec-tronic, magnetic, vibrational, and structural nature withunknown individual impact.Particularly, the treatment of amorphous magnets isproblematic, as the lack of crystal symmetry hampersboth experimental characterization as well as drasti-cally increases computational costs in the theoretical ap-proaches [4, 5]. Furthermore, the topological disorder islikely to induce unintuitive non-collinear magnetic con-figurations and methodological development is needed tounderstand their excitations.In crystalline cases, one key approach to get aroundthe quantum complexity, has been the mapping of themagnetic configurational degree of freedom onto a semi-classical model Hamiltonian such as the Heisenbergmodel known for its applicability on systems with robustlocal moments. This model Hamiltonian, H = − X i = j J ij e i , e j , (1)where the J ij ’s are the magnetic exchange interactions(MEI) between pairs of magnetic atoms ( i, j ) and each e i is a unit vector directed along the local atomic momentat site i , can subsequently be used in Monte Carlo [6] orspin dynamics simulations [7, 8] to obtain critical tem-peratures for ordering (T C ), or to find ground state con-figurations. This Heisenberg Hamiltonian, with only bi-linear pairinteractions gives only an approximate description of thereal complex magnetism in solid state systems. Its in-teractions correspond to the first terms of the completeexpansion series of the magnetic configurational energyaround the value of the fully disordered magnet. It ne-glects interactions corresponding to multi-site clusters,bi-quadratic terms and other higher order terms in theexpansion. Nevertheless it has proven to be valuable inpractice and is known to give an accurate description ofthe magnetic energies of several crystalline systems. Cer-tainly, limitations of this magnetic Hamiltonian shouldbe kept in mind.Several mapping procedures based on first principleselectronic structure calculations have been successfullyemployed to obtain the MEI. The perturbative magneticforce theorem proposed by Liechtenstein et al. [9, 10]has played an important role and has been implementedtogether with the disordered local moments (DLM) [11]treatment of paramagnetism in the generalized perturba-tion method (GPM) [12]. Supercell approaches includethe frozen magnon approach [13, 14] and the structureinversion methodology [15]. Despite their success, allthose methods suffer from difficulties to treat systemswith arbitrary disorder as they, this far, have relied onthe existence of an underlying lattice geometry. Hence,in our opinion, there is no established approach to obtainparamagnetic MEIs in systems with topological disorder,or even in configurationally disordered crystalline alloyswith large local lattice relaxations. Moreover, the impactof temperature induced vibrations, the need to deriveMEI from a disordered magnetic reference state [12, 16],and local environment effects are often neglected.In this work we propose a magnetic direct cluster av-eraging (MDCA) method, to calculate paramagnetic ex-change interactions in a system with arbitrary geometry.The method is based on a conventional first principle su-percell approach treating crystalline and non-crystallinematerials on an equal footing. We illustrate the methodfor crystalline disordered rock-salt structure CrN andTi . Cr . N as well as amorphous CrN. The studied ma-terials have attained substantial attention due to cou-plings between magnetic, electronic, and structural pa-rameters with significant implications for technologicalapplications [17–21]. We also demonstrate that a benefitfrom the MDCA method is that it can be used as a base-line to ensure convergence within the Connolly-Williams(CW) [22] structure inversion method.The Hamiltonian in Eq. 1 describes both collinear andnon-collinear magnetic configurations. Under the as-sumption that there are no important non-linear higherorder terms present, such as the ones discussed inRef [23], the J ij can be obtained from calculations re-stricted to collinear configurations. This observationis especially valid when the interactions in the hightemperature paramagnetic state are desired, where thespin correlation functions are small and one can followthe philosophy of the DLM approach [11, 24]. Thus,to obtain the exchange interactions we consider a sys-tem with a collinear magnetic configuration specified as σ = { σ , σ , . . . , σ n } , where σ i ∈ {± } are spin vari-ables for each site with a magnetic atom. This picture isequivalent to that of the configurational aspects of a bi-nary alloy for which a general mathematical frameworkwere developed by Sanchez et al. [25]. Following thisanalogy the magnetic interactions may be viewed as ef-fective pair interactions in a cluster expansion procedure.However, since our aim is to calculate magnetic interac-tions in disordered systems, where all pairs of atoms areunique , we can not rely directly on the traditional struc-ture inversion approach, as it requires the presence of anunderlying lattice symmetry to reduce the number of freefitting parameters. Instead we start with the very defi-nition of the pair interactions as projections of the totalenergy, E , onto the cluster basis function of each individ-ual pair of atoms. Accordingly, these projections definethe exchange interactions, J ∗ ij , as J ∗ ij = 1 N σ ′ X σ ′ (cid:20) − X σ i ,σ j = ± E (cid:0) { σ i , σ j } ; σ ′ (cid:1) Y k = i,j σ k (cid:21) , (2)where the summation is performed over all possible con-figurations σ of N magnetic atoms divided in a summa-tion over the configuration within the cluster { σ i , σ j } andin the rest of the cluster denoted σ ′ . N σ ′ = 2 ( N − is thetotal number of collinear magnetic configurations of theremaining ( N − sites. This procedure does have a coun-terpart in alloy theory that was suggested by Berera [26]and is referred to as direct cluster averaging (DCA). Here,we define the expression within the brackets of Eq. 2 astwo-site magnetic potentials, according to J σ ′ ij = − X σ i ,σ j = ± E (cid:0) { σ i , σ j } ; σ ′ (cid:1) Y k = i,j σ k (3) = − (cid:20) E (cid:0) {↑ , ↑} ; σ ′ (cid:1) + E (cid:0) {↓ , ↓} ; σ ′ (cid:1) − E (cid:0) {↑ , ↓} ; σ ′ (cid:1) − E (cid:0) {↓ , ↑} ; σ ′ (cid:1)(cid:21) . We note that by including all the four energy terms inEq. 3 the pair interaction between the moments on sites i and j is singled out as the effect of any other pair in-teraction between one of the moments within the clusterand any moment outside it is cancelled. The averagingover the configurations σ ′ in Eq 2 removes any possibleeffect of multisite interactions. In practical calculations,restrictions have to be imposed on the number of config-urations that are considered, introducing an uncertaintyin the values of J ∗ ij . However if we choose a subset of N ζ ′ configurations { ζ ′ } ⊂ { σ ′ } randomly, we simultaneouslyobtain paramagnetic-like configurations and allow for atreatment of the J σ ′ ij as stochastic variables. Thus wecan estimate the MEI J ∗ ij ≈ J ij = 1 N ζ ′ X σ ′ ∈{ ζ ′ } J σ ′ ij , (4)with a confidence interval of degree − α with respect tothe statistical sampling of magnetic configurations out-side the pair: J ij − t α/ ( f ) d < J ∗ ij < J ij + t α/ ( f ) d , where t α/ ( f ) is a t − distribution with f = N ζ ′ − degrees offreedom and d = s/ p N ζ ′ where s is the estimated stan-dard deviation of the J ij . In this work we have used 95%confidence intervals plotted as error bars in the figures.The inset of Fig. 1 shows the individual two-site poten-tials (Eq. 3) and the accumulated estimate of J ∗ ij (Eq. 4)for the case of nearest neighbor MEI in ideal CrN. Inthis case the set of configurations { ζ ′ } were based on 30randomly generated supercells with an average spin cor-relation function on the first coordination shell as smallas -0.006.The electronic structure problem was solved usingdensity functional theory (DFT) [27, 28] and the pro-jector augmented wave (PAW) method [29], as im-plemented in the Vienna ab-initio simulation package(VASP) [30, 31]. To accurately describe strongly cor-related Cr 3d-electrons, the local spin density approxi-mation with an additional Hubbard U-term (LDA+U)[32, 33] of eV was employed as discussed in details inRef. [24]. 64 atom supercells were used for the deriva-tion of NN interactions in CrN while 96 atoms cells wereused to obtain next-NN interactions in CrN and the in-teractions in Ti . Cr . N. 250 atoms were used in thesimulations of the amorphous CrN structure. The size ofour calculation cells were carefully checked so that effectsfrom interactions between periodic images were negligi-ble.The first application of the method is to investigate theinfluence of lattice vibrations on the magnetic exchangeinteractions. Fig. 1 shows the calculated (Eq. 3 and 4)nearest neighbor (NN) interactions in a realistic finitetemperature geometry obtained in Ref. [34] by the disor-dered local moment molecular dynamics (DLM-MD) at300 K. The figure shows interactions derived between thesame pair of atoms at 16 different MD-time steps, thuswith different inter-atomic distances. To visualize thepure distance effect in J ij ( | r i − r j | ) we also calculatedthe values of the nearest and next-nearest neighbor in-teractions, when only the interaction pair were distortedfrom ideal lattice positions.We observe a distinct dependency of the MEI on thedistances between the atoms, even for slight deviations.The NN as well as the next-NN interactions are anti-ferromagnetic at zero-deviation (-7.5 meV and -6.8 meVrespectively) with a sharp increase in antiferromagnetic(AFM) strength at smaller distances. However, whenpositioned far away, but still at realistic distance as illus-trated with presence in the MD-data, the NN interactionactually does become positive. This qualitative differ-ence between magnetic interactions induced by K vi-brations opens perspectives of phonon induced dynamicmagnetic short range correlations in the paramagneticstate, and possibly, the opposite coupling depending onthe magnetic and vibrational timescales. | r i − r j |/R α -0.03-0.02-0.010 E x c h a ng e i n t e r ac ti on s J ij ( e V ) NN interaction from MD supercellNN interaction idealized displacement2nd NN interaction idealized displacement NN i n t e r ac ti on ( e V ) J σ′ ij Eq. 3 J ij Eq. 4
FIG. 1: (Color online) The calculated magnetic exchange in-teractions on the first two Cr-Cr coordination shells of cubicCrN as a function of the interatomic distance. The values ob-tained from geometries derived with DLM-MD as well as ide-alized displacements of the atoms are shown. R α is the equi-librium pair distance of the α :th coordination shell ( α = 1 , .The inset illustrates the application of Equations 3 and 4. The second application of the method is to study ef-fects from substitutional chemical disorder in solid so-lutions. In such systems, different local chemical envi- r i − r j | (Å)-0.0100.010.02 J ij ( e V ) ≤
10 Ti NN>10 Ti NNpure CrN
Number of Ti NN to the pair -0.0100.010.02 (a)(b)
FIG. 2: (Color online) The magnetic exchange interactionsbetween Cr-Cr pairs in a Ti . Cr . N alloy as a function of theinteratomic distances (top panel) and number of Ti nearestneighbors to the Cr-Cr pairs (lower panel). In the top panelthe Ti rich environments are highlighted by open symbols andthe values in pure CrN are included for comparison. In thelower panel, a linear regression trend line is provided as aguide for the eye. C r- C r R D F ( a . u . ) a-CrN 300 Kc-CrN 300 K | r i − r j | (Å) -0.12-0.1-0.08-0.06-0.04-0.0200.02 J ij ( e V ) Magnetic CWMagnetic DCA -0.06-0.04-0.02
FIG. 3: (Color online) The Cr-Cr radial distribution func-tion for the amorphous CrN model (a-CrN) compared to crys-talline cubic CrN (c-CrN) (top panel) and the magnetic ex-change interactions (lower panel) in the amorphous cell ob-tained with MDCA and MCW approaches respectively. Theinset in the lower panel is a zoom in of the short distanceinteractions comparing the MDCA results with the same in-teractions obtained with MCW. ronments give rise to lattice relaxations and differencesin the electronic structure of the magnetic components,and thus possibly differences in the exchange interac-tions. Such effects are for instance crucial to understandthe INVAR-effect in FeNi alloys [35, 36].Shown in Fig. 2, are the exchange interactions be-tween a selection of Cr-Cr NN pairs on the metal fcc sub-lattice in a supercell model for the alloy Ti . Cr . N [37],as a function of the interatomic distance in Å(top panel)and as a function of the number of Ti atoms that are NNto the pair (lower panel). The exchange interactions aregenerally ferromagnetic, in line with experiments [38, 39]but in sharp contrast to AFM CrN. Cr-Cr pairs withmore than 10 NN Ti atoms seem to exclusively inter-act ferromagnetically. Thus, the increasing presence ofTi atoms in the immediate surrounding of a Cr-Cr pairstrongly influences their exchange coupling. This studyreveals the importance of local environment effects onmagnetic exchange interactions in these alloys, addingfurther complexity to the concentration dependence ob-served previously in this material [37].The third application of our method is to study an ex-treme case of disorder in the form of topologically disor-dered amorphous CrN. We obtained our structural modelfor amorphous CrN by means of the stochastic quenchmethod [40, 41] using 250 atoms in the supercell. To in-clude the effect of temperature induced vibrations on thestructure we first run 10 000 time steps of 1 fs by meansof a standard quantum molecular dynamics method. Themagnetic state was kept fixed and the temperature was700 K, a typical growth temperature for amorphous ni-tride thin films [42]. The QMD simulations are carriedout using a canonical ensemble (NVT), neglecting ther-mal expansion. In order to maintain the temperatureand avoid artificial energy drift we use the standard Noséthermostat [43] implemented in VASP, with the Nosé-mass corresponding to a 40 time step period. Then werun 1400 time steps of 1 fs of DLM-MD [34] at 300 Kto average out any possible memory effects of specificmagnetic orientations in the geometry.The top panel of figure 3 shows the obtained Cr-Crradial distribution functions (RDF) for our amorphousmodel as well as for our crystalline cubic CrN model afterthe MD simulation, both convoluted with a 0.2 Å gaus-sian. The first peak of the a-CrN Cr-Cr RDF is at slightlylower distances as compared to c-CrN as several of theCr-atoms in the a-CrN case are positioned in a N-poorlocal environment allowing for smaller Cr-Cr distances.However, the volume of a-CrN is as expected larger thanc-CrN.In our supercell approach, we first investigated therange of the magnetic interactions. This was achievedby using the MDCA approach to calculate a subset of20 chosen interactions out of the more than 4000 uniqueCr-Cr pairs in our cell that has distances less than half ofthe supercell periodicity. We found that Cr-Cr pairs withlarger separations than 5 Å have negligible interactionstrengths of at most 1 meV, a value which is within ourstatistical error bars. This distance corresponds roughlyto that of the 3rd metal coordination shell in a CrN crys-tal. Thus, we could focus on the 1456 Cr-Cr pairs withinteratomic distances less than 5Å, in our supercell ofsize 13.603 Å. In the lower panel of Fig. 3 we show 127 exchangeinteractions obtained with our MDCA method. Our ap-proach gives swift access to any chosen individual pairinteraction but if all the thousands of interactions in theamorphous supercell are desired, the method is compu-tationally cumbersome. However, the initial MDCA sur-vey of the J ij s has revealed both the relevant cut-off andprovided a substantial subset of reliable interaction val-ues. This translates into both a necessary limitation ofthe number of free parameters and a reliable convergencecriteria to attempt a brute force CW structure inversion.Thus, we performed a structure inversion using the 1456 J ij s within the 5 Å cut-off as independent free fittingparameters and gradually increased the included numberof first-principles calculations of the energies of cells withrandom generated configurations σ in the procedure. Wefound that when the number of considered configurationswas 4000, the values of the MEI had converged to theMDCA obtained values, with a mean absolute deviationof only 1.6 meV. The good agreement between the twomethods is illustrated in the inset in Fig. 3. The MEIin amorphous CrN are predominantly AFM, especiallyat short distances, but at larger distances a fraction areferromagnetic (FM). For all pair distances the spread inMEI is huge, underlying the need to treat all pair in-teractions as independent in any quantitative modelingscheme of amorphous magnets.The obtained full set of exchange interactions up to | r i − r j | = 5 Å were then used in the non-collinear Heisen-berg Hamiltonian in Eq. 1. We minimized the magneticenergy with respect to non-collinear configurations usinga simulated annealing procedure with a Metropolis-typeMonte Carlo simulation [44]. The energy was minimizedin a 4x4x4 supercell of our amorphous structure model,including 8000 Cr magnetic moments coupled by the cal-culated exchange interactions. The system was initiatedin a disordered non-collinear spin configuration followedby a gradual decrease in temperature with 100000 ran-dom trial spin rotations each temperature step. The pro-cedure was initiated at several different starting temper-atures in the interval 1000 - 100 K. The lowest energystate found was a non-collinear AFM configuration witha ferromagnetic component no larger than 0.01 µ B perCr-atom. It was obtained for runs with starting temper-atures in the range 200 - 100 K.In conclusion we demonstrate the vital importance oflocal environment effects on the magnetic exchange in-teractions in disordered systems, including amorphousmagnets. For this purpose we have suggested a super-cell technique inspired by the direct cluster averagingmethod from alloy theory, which could be fruitfully com-bined with a structure inversion approach. Our workopens up for theoretical predictions of ground state or-dering and finite temperature magnetism in amorphoussystems, as well as simulations of vibrational and configu-rational local environment effects on magnetism in alloysand compounds.Financial support from the Swedish Research Coun-cil (VR) grants no. 621-2011-4417 and 621-2011-4426and from the Swedish Foundation for Strategic Research(SSF) program SRLL10-0026 are gratefully acknowl-edged. EH and RL thank FONDECYT projects 1110602and 1120334. Calculations were performed utilizing su-percomputer resources supplied by the Swedish NationalInfrastructure for Computing (SNIC) at the PDC andNSC centers. ∗ Electronic address: [email protected][1] T. Moriya,
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