Longitudinal fluctuations of Co spin moments and their impact on the Curie temperature of the Heusler alloy Co 2 FeSi

The magnetism of the full Heusler alloy Co 2 FeSi with its high magnetic ordering temperature is studied on a first-principles basis employing the disordered local moment approximation, the magnetic force theorem and single-site spin fluctuation theory as formulated recently in the framework of the Local Spin Density Approximation. We find that the magnetic moments of Fe and Co in Co 2 FeSi exhibit a quite distinctive behavior at high temperatures. The Fe moments are well localized and keep their magnitude unchanged with temperature, whereas the Co moments are itinerant and show a temperature dependence. We find that the effects of magnetic disorder strongly renormalize the Fe-Co inter-atomic exchange interactions. Our results suggest a deficiency of the classical Heisenberg model with rigid localized atomic spin moments for the description of magnetism in Co 2 FeSi. An accurate estimation of the Curie temperature is obtained by taking into the account the thermal longitudinal fluctuations the Co moments.


Introduction
The Heusler alloys have been in the focus of intensive experimental and theoretical research for more than two decades [1,2].A root of such an interest is the possibility of tuning their magnetic and electronic properties by varying the chemical composition in a very wide range keeping the lattice structure fixed.As a result, several of the magnetic Heusler alloys have been developed with physical properties that are highly attractive for applications [1,3,4].One of the most intriguing of these properties is half-metallic behavior that has been predicted for a few of them [5,6].These ferromagnetic half-metallic materials, i.e. the materials that have a band gap only in the minority spin-channel attract huge theoretical [7] and experimental interest during the last few decades [8].The technological prospectus of these materials relies on the possibility of their application as a source of a spin-polarized current in spintronics devices [9].The most studied class of the full Heusler alloys with potentially half-metallic behavior were perhaps [10] Co-based compounds with formula Co 2 XY, where X = Fe, Mn and Y = Al, Si.A very high value of spin polarization in Co 2 MnSi has been found [11] experimentally.Another representative of this class is ferromagnetic Co 2 FeSi.Although experimental evidence of the half-metallic state in this material is still debated [12] it attracts a huge interest [13,14] due to its large magnetization (~6 µ B /f.u.) and a very high Curie temperature (1100 K).Thin films, various quasi-quaternary and offstoichiometric alloys based on Co 2 FeSi [15,16] have been widely studied for the potential applications in magnetic tunnel junctions [17,18,19].Thus, it is not surprising that the Co 2 FeSi alloy has attracted a considerable theoretical attention also from first-principles perspectives.Since the very first studies it has become clear [12] that the conventional ab-initio band structure methods based on the Local-Spin Density Approximation (LSDA) have a problem to describe the ground state magnetic properties of this system.The LSDA predicts a significantly smaller magnetic moment with respect to experiment [20] even in the framework of a General Gradient Approximation (GGA).This is a signal for the importance of correlation effects.Indeed, the LDA + U methodology with properly chosen U parameter gives correct magnetic moments and put the Fermi level in the pseudo-gap of the minority spin channel [20,21,22].However, it was also shown that the LDA + U method worsen the spectral properties of the Co 2 FeSi compare to the experiment [23].Meinert et al. [24] have shown that one can solve the problem by considering the correlation effects in the framework of the ab-initio GW approximation.
Another problem is an estimation of the magnetic ordering temperature of Co 2 FeSi alloy from first principles.The inter-atomic exchange constants of the Heisenberg Hamiltonian have been calculated using the magnetic force theorem in the framework of Density Functional Theory in several works [25,26,27].In all these investigations, the interactions have been calculated in the reference of the ferromagnetic ground state.It was shown that the dominating interaction appeared to be Co-Fe nearest neighbor one.However, the simulated Curie temperature (T c ) appears to be smaller than in experiment with use of LSDA (750 K in Ref. [26] and 650 K in Ref. [27]) and GGA (800 K) [27] as well.The same also holds for the results of mean-field results presented in Ref. [25] (T c = 1100 K) since the mean-field approximation for the Heisenberg model notoriously overestimates the value for the critical temperature.It has been commonly agreed [26,27] that both problems, the Curie temperature and the underestimation of the ground state magnetic moment in LSDA/GGA, are connected.This proposition became even more obvious when Chico et al. [27] have shown that the LDA + U method, with a parameterization that yields the correct value of the ground state magnetic moment in Co 2 FeSi, also leads to a larger Heisenberg exchange constant and thus a Curie temperature in close agreement with experiment.However, in the present work we will argue that the two abovementioned problems might be of different origin.The ground state moment underestimation in LSDA is indeed related to correlations effects.However, the source of the problems in the high temperature regime is the itinerant character of the Co moments that change their magnitude with temperature.We also find a strong dependence of the Fe-Co exchange interactions on the state of magnetic disorder.Thus, the exchange interactions calculated in the ferromagnetic ground state are not relevant to the discussion of the Curie temperature.A similar problem with the application of exchange constants derived for the ground state to estimate the magnetic ordering temperature has been reported previously for various metals and alloys [28,29].We will argue here that Co 2 FeSi is a nice example of a metallic system where the application of the conventional Heisenberg model, that assumes fixed local moments on the atomic sites and temperature independent interactions, are not sufficient for the description of high temperature properties.
Our paper is organized as follows: after the description of the applied methodology in the next section, in section III we perform an analysis of the exchange interactions calculated for the ferromagnetic state.We obtain similar results as in a couple the previous works based on the magnetic force theorem and first-principles calculations, where LSDA and GGA essentially underestimate the magnetization and the Curie temperature.However, we also show there that by fixing the total magnetization magnitude to the experimental value and calculating the interactions for such FM state, one can get the same estimate of the T c as was derived previously [27] within the LDA + U approximation.In section IV, we investigate the high temperature paramagnetic state using in the ab-initio framework developed as described in Refs.[30,31,32].We find a strong itinerant character of the Co moments and derive an accurate estimation of the Curie temperature by taking into the account the temperature induced longitudinal spin fluctuations.

Methodology
We use gthe bulk Korringa-Kohn-Rostoker (KKR) method in the atomic sphere approximation (ASA) [33,34] in the framework of the LSDA33 and the GGA34 method to calculate an electronic structure of Co 2 FeSi for the experimental lattice geometry [12].The partial waves have been expanded up to l max = 3 (spdf basis) inside the atomic spheres that were set equal for all nonequivalent atomic sites.
After the derivation of the self-consistent electronic structure for a selected magnetic configuration (reference magnetic state), the exchange interaction parameters of the classical Heisenberg Hamiltonian: J ij e i e j (1) where e ⇀ i is a directional unit vector of the magnetic spin moment at the i th lattice site, has been estimated using the first-principles magnetic force theorem (MFT) based on the Green function formalism [35] implemented in KKR ASA [36].A Monte-Carlo simulation with the Hamiltonian (1) and the calculated J ij constants is performed to obtain the corresponding magnetic ordering temperature.Let's note than in metals the exchange constant might depends on the choice of the reference magnetic state, even if atomic spin moments are very rigid/ localized, since the structure of the electronic bands are very sensitive to the long-range magnetic configurations [37].In the next section we use the ferromagnetic ground state as a reference as a reference to earlier computational investigations on finite temperature magnetism in Co 2 FeSi.In Section IV we will demonstrate the itinerant character of the Co moment and the necessity of considering the thermally induced longitudinal fluctuation of the Cobalt atomic moment.There we will deal with a high temperature paramagnetic state and use the Disordered Local Moment [38] (DLM) state as a reference for calculations of the inter-atomic exchange interaction.The DLM formalism is used for modelling the direction thermal magnetic disorder above the Curie temperature and its influence on the electronic structure (see details in Ref. [30,39,40]).Since the Cobalt moment has a strongly fluctuating longitudinal component in the paramagnetic regime at high temperature (>1000 K) instead of the Hamiltonian (1) we use the extended version allowing for these fluctuationsthe Longitudinal Spin Fluctuation (LSF) Hamiltonian [30] where ⃒ is a length of the atomic spin-moment at the i th lattice site.
The first term is the moment dependent on-site energy and the second term is a Heisenberg interaction, similar to those in Eq. ( 1) with exchange constants depending on the size of the atomic moments on neighboring sites.The procedure applied here for estimating the magnetic ordering temperature using Eq.(2) has been described previously in detail [31,32].The onsite term in Eq. ( 2) is approximated as DLM total energy, E DLM (m), calculated with a fixed atomic spin moment m.The temperature dependence of the atomic moments in the high temperature paramagnetic state has been calculated as a statistical average: where g(m) is the longitudinal integration measure in classical spin space [32,39].We determine the ordering temperature as a crossing point between a < m> T curve calculated from equ.(3) and T ord (<m > ) calculated by MC simulations with the Hamiltonian (1) where the J ij are calculated for each fixed value of m.We use the common choice [41,40] of a vector space measure g(m) = m 2 , which we found here to perform better for Co 2 FeSi (as well as for pure hcp Co -see Ref. [41]) than alternative linear form, [32] A notable difference of the application of the described procedure to Co 2 FeSi with respect to previous works cited above is the presence of two different magnetic sub-lattices, Fe and Co.In general this would require a full 2D ab-intio mapping of the Hamiltonian (2) with respect of Fe and Co fluctuating moments and 2D integration in Eq. (3).To avoid this mapping further approximations should be used (see for example Ref. [42]).The huge simplification for the present case is due to the fact that Fe moments are very well localized, so one can consider LSF only on Co sites allowing the Fe moments to converge to the self-consistent values.In section IV we will provide a full justification of this simplification.
S. Khmelevskyi and P. Mohn

Ferromagnetic ground state and exchange interactions
The self-consistent magnetic moments calculated with LSDA and GGA exchange-correlation potentials for the ferromagnetic (FM) ground state of Co 2 FeSi are given in the Table 1.In full agreement with earlier similar calculations [26,27], the total magnetic moments in both cases are smaller than the ideal half-metallic value of 6μ B /f.u.found in experiment at low temperatures.The calculated interatomic exchange interactions (Fig. 1) suggest a dominating first nearest neighbor's (NN) Co-Fe coupling with a smaller contribution from 1NN Co-Co coupling and almost vanishing interactions with the more distant shells.The results of the Monte-Carlo simulations with Hamiltonian (1) using the calculated interactions from Fig. 1 yields estimates for the magnetic ordering temperature (see Table 1) similar to those obtained earlier by Chico et al. [27] The considerable underestimation of T c compared to the experimental value of 1060 K, has been ascribed to the underestimation of the magnetic moments [26,27] from LSDA/GGA and the presence of correlation effects.Indeed, it was shown [27] that the use of LDA + U with a proper choice for the U-parameter gives a half-metallic state and an improved value of T c .Thus, the following logic might be applied: the underestimation of the correlation effects leads to the underestimation of the magnetic moments and as results to the underestimation of interatomic Co-Co and Co-Fe exchange interactions in the FM state and consequently a too low value of T c in LDA/GGA.
In order to verify the statement that the underestimation of the moment is a key issue in evaluating T c we performed calculations of the exchange interactions for the self-consistently derived LDA/GGA FM state with Co and Fe atomic moments fixed to the ideal "half-metallic" values.Indeed, the results of the Monte-Carlo simulations with J ij obtained in this way gives a T c value (Table 1) very close to the experiment and the LDA + U result.It thus appears, that our results completely confirm the above mentioned earlier conclusions.However, the main idea of the present work is to challenge this interpretation.
Due to the metallicity of the system the exchange interactions calculated in the magnetic ground state can be and must be used for simulations and the explanation of the magnetic properties at temperatures lower than the magnetic ordering temperature, when the thermal magnetic directional disorder is very small.Normally the average electronic structure is affected by the magnetic thermal disorder at high temperatures near the T c and thus the exchange interactions are altered.Only in a small number of cases (dominant direct NN exchange, i.e. the electronic structure averaged over random magnetic configuration is similar to the ground state one) or by chance, if some random compensation effects occur, the exchange interactions in the ground state and in the high temperature paramagnetic state might produce nearly the same value of T c .For itinerant metallic system such cases should be very exceptional.In addition, the longitudinal fluctuations of the magnetic moments in the paramagnetic state of a metallic system (although being largely frozen in the magnetically ordered state) bring in another complication.In the next section we show that such an exceptional situation might occur in the Co 2 FeSi compound.

High temperature paramagnetic state
Among the first ones who pointed out a possible importance of longitudinal spin fluctuations in Co 2 FeSi was Kübler [43].He noted that in the spin fluctuation approximation (SFA) and for the spherical model a reasonable value of T c in Co 2 FeSi can be derived from calculated exchange constants estimated from spin-spiral LSDA total energies within the Random Phase Approximation (RPA).Although not too many details were given on LSF magnitude of Co 2 FeSi it was noted [43]] a "somewhat surprising [43] successes of the SFA method in the framework of bare LSDA approximation without special local treatment of the correlation effects.
Here we look at the problem from another perspective.We investigate a high temperature paramagnetic state (PM) of Co2FeSi using the DLM formalism.A self-consistent DLM calculation converges to the DLM state with a magnitude of the spin moment on the Fe site of 2.75μ B /Fe being almost exactly the same as in the FM ground state (2.74μ B /Fe).However, the atomic Co moment vanishes completely in the DLM state.This suggests that Cobalt in Co 2 FeSi does not fulfill the Andersson criterium for the formation of local moments in a paramagnetic state [39] and the formation of the spin moments on the Cobalt sites in the PM state is entirely due to thermal longitudinal spin fluctuations [41,44,45].On the opposite, the Fe sites develop a well localized "robust" moment in the PM state.To illustrate the role of the temperature effects on the local moment formation we show the total energies of DLM states calculated with fixed spin moment constrained (Fig. 2).The results presented on both panels of the figure were calculated by fixing a respective spin moment on the given atomic sites (Co/Fe) allowing the other magnetic sites (Fe/Co) to converge freely.The energies given in the figure are in Kelvin/atom units to make a connection the temperature scale of LSF.One can see that thermal excitations of the order of 1000 K (=about the experimental T c ) can induce a rather significant thermally averaged spin moment on the Co cites of about ~1 μ B , departing strongly from the minimal energy zero value.In contrast, on the Fe sites thermal fluctuations on the same temperature scale lead only to rather insignificant fluctuations of the moment around its equilibrium value, giving on average the value close to the minimum of the total energy.Thus, one

Table 1
Calculated and experimental total (m tot ) and atomic (m Co/Fe ) magnetic moments on Co and Fe sites in the ferromagnetic state, first nearest neighbor Co-Fe (J Co-Fe ) and Co-Co (J Co-Co ) exchange interactions, and Curie temperatures (T c ) for Co 2 FeSi.The experimental (exp.)values are taken from Ref. [12].The three sets of the calculated values correspond to the self-consistent LSDA, GGA and fixedspin moment calculations (FSM).In the latter case, the self-consistent ferromagnetic solution was derived by fixing the Co and Fe moment to the ideal values for half-metallic state.We add also the results of the full potential LDA + U calculations for the atomic moments from Ref. [13] and their experimental estimation of the T c .Note that direct correspondence between FSM KKR-ASA and full potential LDA + U atomic moments is subject of some uncertainty since inevitably different choice of the muffin tin spheres.3), using the results presented in the upper panel of Fig. 2, we calculate the dependence of <m Co >(T).The interception of the two curves gives the value of the physical Curie temperature.The visualization and details of this procedure for different transition metal magnetic materials can be found elsewhere [31,41].
For Co 2 FeSi we thus obtain T c = 1100 K in GGA, and 1070 K in LDA approximations (within the applied vector space integration measure).Both results are in the fair agreement with the experimental value of 1040 K.Moreover, our LDA result are in full agreement with Kübler's SFA model (1058 K).
Thus, two essentially different ways of calculations: i) the LSF model presented here or Kübler's SFA model, and ii) a straightforward application of the classical Heisenberg model with calculated exchanges in FM state obtained either in correlated LDA + U calculations [27] or just by fixing the atomic moments to correct "half-metallic" values (previous section), provide equally good ab-initio values of T c .This result is rather intriguing, since both approaches rely on completely different physical points of view.
To understand this coincidence further, in Fig. 3, we plot the calculated dependence of the interatomic exchange interactions on the value of the Co moment in the DLM state and the corresponding values derived in the FM state (see section III).We only show two NN interactions, which give the dominating contribution to the ordering temperature.One can see that in the DLM state the interactions are greatly enhanced as compared to the FM state with respective to the Co moment.The converged Fe moment is essentially the same in both calculations.This enhancement might be understood taking into the account the fact that in the PM state the half-metallic pseudo-gap (or the real gap in the true half-metallic state) in the spectral function vanishes when both spin-channels, up and down, become equally populated (see e,g, Ref. [46]).This provides an additional channel for indirect exchange between atomic sites.The values of the exchange interactions in true half-metallic states (stars in the figure) becomes approximately the same as the exchange in DLM states with values of the Co moments close to their average values around 1000 K.We illustrate this statement by the dashed lines in Fig. 3.It thus appears that the similar values of T c obtained from the both methods described above is thus a pure random coincidence, peculiar for a given compound.

Conclusions
The main conclusion of the given work is that for an itinerant electron magnetic system a simple comparison of the calculated magnetic ordering temperatures with experiment cannot be used as a judgment for the validity if the proposed ab-initio model.The exchange interactions calculated in the magnetically ordered state and the application of the straightforward Heisenberg model might give an excellent result purely by coincidence.However, just considering one particular system it might be difficult to resolve the issue.The Heusler alloy Co 2 FeSi is such a peculiar example.Here we have shown that the itinerant character of the Co moments in Co 2 FeSi and dominant role of the longitudinal spin fluctuations together with an enhancement of the inter-atomic exchange interactions in the magnetically disordered PM state, is a requirement for the formation of very high Curie temperature.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence  The Co moments was fixed to the corresponding values using FSM method.The asteriks denote the exchange interactions calculated in the FM state.The asteriks are given at the corresponding self-consistently calculated values of the Co moments in the FM state.For an explanation of the dashed lines, see text.HMe states for half-metallci state modelled by FSM method.

Fig. 1 .
Fig. 1.Calculated interatomic exchange interactions in the ferromagnetic ground state of Co 2 FeSi.The exchanges in half-metallic state (HMe) are calculated using FSM method described in the text.

Fig. 2 .
Fig. 2. Calculated dependence of the total energies in the Disordered Local Moment State of Co 2 FeSi with the atomic moment of Co constrained (upper panel) and atomic Fe moment constrained (lower panel).Circles: LSDA-, squares: GGA-calculations.

Fig. 3 .
Fig. 3. Nearest neighbour inter-atomic exchange interactions: 1NN Co-Co (closed squares) and 2NN Co-Fe (open circles) calculated in the DLM state.The Co moments was fixed to the corresponding values using FSM method.The asteriks denote the exchange interactions calculated in the FM state.The asteriks are given at the corresponding self-consistently calculated values of the Co moments in the FM state.For an explanation of the dashed lines, see text.HMe states for half-metallci state modelled by FSM method.