Revisited Stress-Dependent Curie-Temperature in BCC Iron: LSDA+*U* Cleansing of Magnon Ambiguities

Published Date : 2019-08-23

DOI : https://doi.org/10.12982/CMUJNS.2017.0017

Journal Issues : Number 3 , July-September 2017

**ABSTRACT**

This study proposes that the strong correlation of the 3d electrons in Fe is an important key to understanding the stress dependence behavior of Curie temperature (*T _{c}*). We proved our proposed hypothesis using density functional theory (DFT) within an LSDA+U (local spin density approximation +U) framework. Applying LSDA+U correction increased both the magnetic moment and magnon energy. The increased magnon energy directly contributed to the higher magnitude of the calculated

**Keywords:** Stress dependence, Curie temperature, Body-centered cubic iron, Frozen spin spiral, Magnon, LSDA+U

**INTRODUCTION**

Curie temperature (*T _{c}*) is one of the most important and studied factors in magnetic-based applications. The most common material for studying magnetic properties is iron (Fe), because of its high

Theoretically, Morán et al. (2003) and Körmann et al. (2009) used nearly identical methodologies, but obtained different results. Both calculated *T _{c}* by integrating magnon energy along a high symmetry path in the Brillouin zone (the so-called ‘frozen magnon’ approach) (Halilov et al., 1998). However, Körmann used an extremely high fixed cone angle (

In addition to the choice of *θ*, a strong coulomb interaction should be included in the calculation when using materials composed of Fe (or other 3d transition metals), in particular, iron oxides. While the insertion of on-site coulomb interaction via *U *parameter may be excessive in the case of pure Fe, we hypothesize that this may be the key factor in the dependence of *T _{c}* on stress. Therefore, this study first determines the proper

**MATERIALS AND METHODS**

**Theory and computational methods**

The magnon calculation strategy in this study is based on the frozen magnon approach, in which the magnetic interactions are mapped to Heisenberg Hamiltonian of the form:

where *J _{ij}* is the exchange interaction and

This study takes into account the generalized Bloch’s theorem to perform self-consistent spin spiral calculations at different volumes, *V*. The spin spiral state of the system for a single magnetic atom per unit cell can be characterized via the moment:

with ** q** the position in reciprocal space,

**Figure 1.** (a) The linear fitting between Δ*E _{q,θ}* and sin

The total energy difference between the perturbed spin spirals and the ferromagnetic states of the system containing the single magnetic atom per unit cell, Δ*E _{q,θ} *can be attributed to the magnon energy (ωq(

where *M(V) *is the volume dependent magnetic moment.

To choose *θ*, we first investigated the relation between Δ*E _{q,θ}* and sin

We employed the mean-field approximation (MFA) derived from the Heisenberg model to obtain the Curie temperature (Pajda et al., 2001):

The dependence of *T _{c}* on stress σ,

We obtained the total energies by the full-potential linearized augmented plane-wave method (FP-LAPW) as implemented within ELK package. We used 16×16×16 of k-point mesh in the irreducible Brillouin zone. The potential and density were expanded within a plane wave cut-off of |G| = 14/*a _{0}* (

The calculations were performed using the LSDA and the LSDA+*U*. We used the parameters from Perdew-Wang (1992) for the LSDA function. For the LSDA+*U* calculation, the *d-d* interactions were counted into the localized part by adding the Hubbard term or the screened Coulomb and exchange parameters, i.e., *U* and *J*. The non-localized part was calculated in the same way as in LSDA. While many choices of *U* and *J* for bcc Fe exist, we chose the one calculated from the linear response calculation (Cococcioni and Gironcoli, 2005): *U*_{eff }~ 2.2 eV. *U _{eff}* is the effective Hubbard parameter, including

**RESULTS**

Before performing the first principles calculation under the non-collinear spin spiral treatment of our system, we optimized the computational accuracy by considering Fe as a collinear magnetic system. The magnetic moment acquired from the collinear calculation (2.18 μ_{B}) agreed well with our experimental results of 2.20 μ_{B} (at 120 K) (Billas et al., 1993). We then performed the non-collinear calculations. At a fixed experimental volume, the non-collinear LSDA gave a magnetic moment of about 2.3 μ_{B}, while the non-collinear LSDA+*U* increased the magnetic moment (up to 15%); this was consistent with the theoretical work of Rollmann et al. (2004). The magnetic moments as a function of cell volume from both the LSDA and LSDA+_{U} calculations are shown in Figure 1(b).

The approximation in equation 3 allows us to characterize the magnon behavior of the given system. Our calculated magnon spectrum at fixed volume is shown in Figure 2(a). The results agreed quite well with Morán et al. (2003), in both magnitude and spectrum shape. From the LSDA results, Curie temperature calculated via magnon dispersion was consistent with our experimental results, although of a slightly lower magnitude. Our LSDA calculation predicted *T _{c}* of 1,030 K for bcc iron; the experimental results were 1,044 K. However, since LSDA+

To investigate the effect of stress on *T _{c}*, we calculated the magnon spectra at different volumes, ranging from 11.20 - 12.51 Å

**Figure 2. **(a) Magnon spectrum from LSDA and LSDA+*U *at different unit cell volume and (b) the dependence of Curie temperature (*T _{c}*) on unit cell volume.

The relation between *T _{c}* and

**DISCUSSION**

The use of non-collinear spin spiral treatment (either with or without +*U* correction) moderately increased the magnitude of the calculated magnetic moment. This enhancement was induced by a small external magnetic field applied to constrain spin spiral cone angle. Although the use of a small enough external magnetic field led to a calculated magnetic moment with the same magnitude as the collinear one (and the experimental one), it may not be a sufficiently large fixed cone angle, and resulted in the instability of the SCF calculation. The Fe magnetic moments calculated using +*U* correction (either with collinear or non-collinear treatment) were higher than the experimental values (and higher than bare LSDA). This is because the Hubbard +*U* usually increases the spin splitting between majority and minority spin states (Tao et al., 2014). Thus, the magnetic moment, defined as the difference of electron density in majority and minority spin channels, is enhanced. This behavior of Hubbard +*U* correction has been shown to exist in 3d transition metals, as well as 4f transition metals, such as Gd (Tao et al., 2014).

Our LSDA results (at equilibrium volume) produced slightly lower *T _{c }*values than the experiment values. This was not reasonable, because we used mean-field approximation (MFA) (Pajda et al., 2001) that generally overestimates

In contrast, combining LSDA+*U* with MFA produced *T _{c}* of about 1,400 K with a more acceptable consistency; these results agreed well with the GGA results of Körmann et al. (2009). Their GGA results analyzed with a more accurate scheme, i.e., random phase approximation (RPA), accurately predicted

The magnitude of *T _{c}* as a function of cell volume (Figure 2(b)), with both LSDA and LSDA+

**CONCLUSION**

We aimed to determine whether adding strong electron correlation via on-site coulomb interaction (*U* parameter) would clarify the *T _{c}* dependence on stress. We found that the relationship between

**ACKNOWLEDGEMENTS**

The authors wish to acknowledge financial support from CMU Post-Doctoral Fellowship Program 2015, Chiang Mai University.

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Kanokwan Kanchiang^{1,2}, Sittichain Pramchu^{1}, Atchara Punya Jareonjittichai^{1} and Yongyut Laosiritaworn^{1*}

1 Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

2 Program of Material Science, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand

*Corresponding author. E-mail: yongyut_laosiritaworn@yahoo.com

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