Magnetic and magnetoelectric properties of $A$FeF$_5$ ($A$ = Ca, Sr) spin-chain compounds

TL;DR

This study investigates the magnetic and magnetoelectric properties of CaFeF$_5$ and SrFeF$_5$ spin-chain compounds, revealing SrFeF$_5$ as a multiferroic material through theoretical calculations.

Contribution

It provides a theoretical analysis of the magnetic and magnetoelectric properties of CaFeF$_5$ and SrFeF$_5$, identifying SrFeF$_5$ as a multiferroic compound.

Findings

CaFeF$_5$ does not exhibit magnetoelectric properties.

SrFeF$_5$ is identified as a multiferroic.

Magnetic exchange constants were determined using density functional theory.

Abstract

Fluorides in general are characterized by big variety of crystal structures, whereas those containing transition metals also often show sizable magnetic properties. The tendency of fluorine to form linear chain structures in many cases results in low-dimensional magnetism. Despite the plethora of magnetic phenomena in fluorides, their magnetoelectric properties are less studied than those of oxides. In the present work we theoretically study the magnetic and magnetoelectric properties of spin-chain compounds CaFeF$_5$ and SrFeF$_5$. The density functional theory is employed for determination of magnetic exchange constants, which are then used in Monte Carlo calculations. The symmetry analysis reveals that CaFeF$_5$ does not show magnetoelectric properties, whereas SrFeF$_5$ is a multiferroic.

Figures

Tables

Table 1. Table 1: Calculated magnetic exchange constants for CaFeF 5 and respective Fe – Fe distances.

	$J_1$	$J_2$	$J_3$	$J_4$	$J_5$	$J_6$
Fe – Fe, Å	3.800	5.428	5.484	5.492	5.506	5.559
J, meV	21.9	-0.037	0.076	0.18	0.086	0.24
	$J_7$	$J_8$	$J_9$	$J_{10}$	$J_{11}$
Fe – Fe, Å	6.437	6.549	7.506	7.540	7.599
J, meV	0.78	-0.2	0.14	0.01	0.25

Table 2. Table 2: Calculated magnetic exchange constants for SrFeF 5 and respective Fe – Fe distances.

	$J_1$	$J_2$	$J_3$	$J_4$	$J_5$	$J_6$
Fe – Fe, Å	3.666	3.891	4.486	4.692	5.044	5.094
J, meV	15.5	21.4	0.05	0.38	0.34	-0.91
	$J_7$	$J_8$	$J_9$	$J_{10}$	$J_{11}$	$J_{12}$
Fe – Fe, Å	5.114	5.840	5.983	6.092	6.329	6.336
J, meV	1.35	-0.89	0.16	2.06	-1.01	1.53

Table 3. Table 1: Structural parameters of CaFeF 5 obtained experimentally ( a 𝑎 a =5.492 Å, b 𝑏 b =10.076 Å, c 𝑐 c =7.599 Å, β 𝛽 \beta =110.02 ∘ , sp. gr. P2 1 /c) [ 14 ] and by DFT calculations in this work ( a 𝑎 a =5.571 Å, b 𝑏 b =10.218 Å, c 𝑐 c =7.694 Å, β 𝛽 \beta =109.9 ∘ )

Atom		$x$	$y$	$z$
Fe	exp.	0.10314	0.24644	0.3996
	DFT	0.10542	0.24653	0.4001
Ca	exp.	0.52690	-0.01783	0.2540
	DFT	0.52914	0.98047	0.2548
F1	exp.	0.38640	0.11840	0.4448
	DFT	0.38841	0.11776	0.4443
F2	exp.	0.35480	0.38740	0.4924
	DFT	0.35781	0.38768	0.4945
F3	exp.	0.12140	0.60360	0.1878
	DFT	0.12073	0.60462	0.1880
F4	exp.	0.17220	0.86320	0.1413
	DFT	0.16907	0.86377	0.1405
F5	exp.	0.14990	0.29350	0.1621
	DFT	0.15270	0.29490	0.1629

Table 4. Table 2: Structural parameters of SrFeF 5 obtained experimentally ( a 𝑎 a =7.062 Å, b 𝑏 b =7.289 Å, c 𝑐 c =14.704 Å, β 𝛽 \beta =95.4 ∘ , sp. gr. P2 1 /c) [ 13 ] and by DFT calculations in this work ( a 𝑎 a =7.127 Å, b 𝑏 b =7.395 Å, c 𝑐 c =14.891 Å, β 𝛽 \beta =95.2 ∘ )

Atom		$x$	$y$	$z$
Fe1	exp.	0.5062	0.6185	0.1614
	DFT	0.5070	0.6170	0.1611
Fe2	exp.	0.8107	0.6832	0.3750
	DFT	0.8118	0.6846	0.3753
Sr1	exp.	0.6896	0.1536	0.0873
	DFT	0.6847	0.1559	0.0870
Sr2	exp.	0.0015	0.6529	0.1374
	DFT	0.0003	0.6587	0.1385
F1	exp.	0.6977	0.4951	0.0986
	DFT	0.7001	0.4964	0.0978
F2	exp.	0.3190	0.6978	0.0678
	DFT	0.3190	0.6920	0.0676
F3	exp.	0.3145	0.6658	0.2414
	DFT	0.3130	0.6586	0.2417
F4	exp.	0.6797	0.5565	0.2715
	DFT	0.6831	0.5543	0.2714
F5	exp.	0.6548	0.8301	0.1503
	DFT	0.6526	0.8293	0.1488
F6	exp.	0.9935	0.4878	0.3911
	DFT	0.9997	0.4935	0.3890
F7	exp.	0.9680	0.8067	0.2936
	DFT	0.9669	0.8129	0.2941
F8	exp.	0.6113	0.8693	0.3540
	DFT	0.6036	0.8617	0.3560
F9	exp.	0.6531	0.5802	0.4594
	DFT	0.6545	0.5734	0.4584
F10	exp.	0.9559	0.8206	0.4657
	DFT	0.9561	0.8205	0.4677

Equations

$$\mathcal{H}=\sum _{ij}{J}_{ij}{\vec{S}}_i\cdot {\vec{S}}_j+\sum _i\left(D_x{S}_{ix}^2+D_y{S}_{iy}^2+D_z{S}_{iz}^2\right),$$

$$a_1\frac{\partial a_2}{\partial x}-a_2\frac{\partial a_1}{\partial x},a_1\frac{\partial a_2}{\partial z}-a_2\frac{\partial a_1}{\partial z},$$

$$b_1\frac{\partial b_2}{\partial x}-b_2\frac{\partial b_1}{\partial x},b_1\frac{\partial b_2}{\partial z}-b_2\frac{\partial b_1}{\partial z},$$

$$(a_1{b}_1+a_2{b}_2)P_y,$$

$$(a_1{a}_2^{\prime }-a_2{a}_1^{\prime })P_x,(a_1{a}_2^{\prime }-a_2{a}_1^{\prime })P_z,$$

$$(b_1{b}_2^{\prime }-b_2{b}_1^{\prime })P_x,(b_1{b}_2^{\prime }-b_2{b}_1^{\prime })P_z,$$

Full text

††journal: Journal of Magnetism and Magnetic Materials

1 Introduction

Multiferroics and specifically magnetoelectrics continue to be one of the focal points in condensed matter physics due to promising practical applications. Combination of magnetic and electric properties in a single material opens up new opportunities for creating various sensors, logical elements, or spin electronic devices [1]. However, interacting magnetic and electric subsystems are also of fundamental interest. In magnetism low-dimensional systems continue to serve as playground to explore exotic quantum phenomena such as spin liquids, Bose-Einstein condensation, or spin-Peierls transition [2]. Such quantum phenomena can be easily tuned, changed, or completely destroyed by small external influences. Vast experimental data on magnetoelectric compounds allows concluding that interaction of magnetic and electric subsystems is frequent for both simple or more complex magnetic orderings. Therefore, exploiting the electric subsystem as a means for tuning the magnetic subsystems via, e.g., application of external electric field, can significantly extend the plethora of magnetic phenomena in low-dimensional systems or potentially result in new effects.

Fluoride compounds offer unique opportunities for low-dimensional magnetic systems, because fluorine exhibits strong preference for linear bridging modes, which can provide the necessary topology. A great number of fluoride inorganic compounds have been synthesized and studied to date, ranging from 0D to 3D connectivity of $M$Fx polyhedra surrounding the metal $M$ [3, 4]. Even when the magnetic network has three-dimensional character, the magnetic interactions are often anisotropic leading to low-dimensional magnetism.

Many fluoride compounds have already been shown to exhibit multiferroic properties and the available experimental data were summarized in recent reviews [5, 6], however most of the compounds are the, so-called, type-I multiferroics with magnetic and electric subsystems ordering independently [7]. Therefore, from our point of view searching for new fluoride type-II multiferroic compounds, which have stronger magnetoelectric coupling than type-I compounds, can be fruitful because (i) fluorides have diverse compositions and crystal structures, (ii) in many cases the exchange coupling in them is high, and (iii) low-dimensional magnetism in combination with magnetoelectric phenomena can potentially result in new physics.

In this work we theoretically study the magnetic and possible magnetoelectric phenomena in spin-chain compounds

CaFeF5 and SrFeF5. We find the magnetic exchange constants using the density functional theory, which are then used in Monte Carlo calculations. Supplemented with the symmetry analysis, the study reveals the magnetic and magnetoelectric properties of these fluorides.

2 Methods

We performed density functional theory (DFT) calculations using the Vienna Ab-initio Simulation Package (VASP) [8] and the projected augmented wave method [9]. The generalized gradient approximation (GGA) of exchange correlation corrected by means of the GGA+U formalism for the Fe atoms with $U_{\rm eff}=U-J=3$ eV was used within the Dudarev approach [10]. The energy cutoff was 500 eV, whereas the Brillouin zone integration was done using the Monkhorst-Pack scheme.

The crystal structure relaxation was done using the stopping criterion for absolute values of forces on atoms of 10*-3* eV/Å. For the determination of exchange constants spin polarized collinear calculations were used and the Hamiltonian was fitted to relative total energies of different collinear magnetic structures.

The obtained exchange constants were used in classical

Monte Carlo (MC) simulations of the Hamiltonian

[TABLE]

where $\vec{S}$ are classical vectors of unit length and $D_{\alpha}$ ($\alpha=x,y,z$) are anisotropy constants. In the Hamiltonian Eq. (1) we account only for the anisotropy terms $D_{\alpha}\neq 0$, which we consider being the same for all spins, pursuing a minimal model to characterize the magnetic structure. The Metropolis scheme was used for MC simulations and the size of the simulation box was chosen large enough as to not influence significantly the results (typically larger than $15\times 15\times 15$ unit cells). After each change of temperature the system was allowed to relax for $5\cdot 10^{3}$ Monte Carlo steps per spin (MCS), whereas the statistical information was subsequently gathered over the next $10^{4}$ MCS. The crystal and magnetic structures in this work were visualized using the software VESTA [11].

3 Crystal structure and magnetic exchange interactions

The compounds with the general chemical formula

$A^{2+}M^{3+}$F5 ($A$ = Ca, Sr, Ba; $M$ = transition metal) present many examples of 1D chain structures, which contain isolated ($M$F5)n chains of octahedra. The arrangement of infinite chains depends on the ionic radii of both $A^{2+}$ and $M^{3+}$ and several structural types have been found [12]. CaFeF5 and CaCrF5 present two structural types, in which the $M$F6 octahedra are linked via opposite (trans-)corners and the main difference is in the positions of Ca2+ ions. Several other fluorides, e.g., CaVF5 and CaTiF5, crystallize in the structure of CaCrF5, whereas the structure of CaFeF5 is also shared by, e.g., CdCrF5 and CdGaF5 [12].

In contrast to the CaFeF5 and CaCrF5 structures, in which the chains of Fe3+ ions are almost or completely linear (177.8∘ and 180∘, respectively), the SrFeF5-type structure contains helicoidal (FeF5)n chains and the octahedra are connected by corners in the cis-position [13]. Two octahedral chains with opposite rotations are present in the structure. To this structural type belong also SrVF5, SrCoF5, and SrTiF5.

The crystal structures of CaFeF5 and SrFeF5 are shown in Figs. 1(a,d). Tables 1 and 2 present the lattice parameters and atomic positions obtained by DFT calculations in comparison with the experimental results taken from literature. Overall we obtain good correspondence with the experimental results with lattice parameters difference of not more than 1.5% [13, 14].

We restrict the DFT calculations of magnetic exchange constants $J_{i}$ of CaFeF5 to eleventh nearest neighbor (NN). The exchange paths in CaFeF5 are shown in Figs. 1(b,c). Thus, the largest distance between the Fe3+ ions, for which the exchange constant is determined corresponds to next nearest neighbors (NNN) within the chains. The obtained values of $J_{i}$ ordered with respect to increasing distance between the spins are given in Table 1. It can be found that $J_{1}$ and $J_{11}$ are the only intrachain interactions representing NN and NNN exchanges within the chains, respectively, whereas all other interactions are between different chains. The dominant exchange interaction is NN within a chain ($J_{1}$), while the NNN interaction $J_{11}$ is almost two orders of magnitude smaller. The strongest interchain interaction $J_{7}$ is roughly 30 times smaller than $J_{1}$. Moreover, given a strong tendency for $\ldots$-up-down-up-down-$\ldots$ ordering within the chains, some interchain interactions introduce frustration in the system, e.g., the interactions $J_{7}$ and $J_{9}$ cannot be simultaneously satisfied. Overall, CaFeF5 can be characterized as having a strong NN antiferromagnetic intrachain interaction, weak NNN antiferromagnetic exchange within the chains, and weak interchain magnetic coupling.

The magnetic exchange paths of SrFeF5, shown in

Figs. 1(e,f), are determined up to the twelfth nearest neighbor. The calculated values of $J_{i}$ are given in Table 2. It can be found that the chains are composed of alternating Fe1 and Fe2 octahedra with alternating NN exchange constants $J_{1}$ and $J_{2}$, whose values are comparable to that in CaFeF5, whereas $J_{2}/J_{1}\approx 1.38$. The NNN intrachain interaction is $J_{3}$, whereas interaction $J_{4}$ can be considered as next next nearest neighbor along the chain. The exchange interactions $J_{i}$ with $i$ = 5 – 12 are interchain couplings. Similar to the case of CaFeF5 the dominant exchange interactions in SrFeF5 are NN intrachain couplings, whereas further neighbor exchanges within chains are significantly smaller. However, in contrast to CaFeF5, interchain interactions in SrFeF5 are relatively stronger.

4 Magnetic properties

The magnetic exchange constants obtained with the help of DFT calculations were used in Monte Carlo simulations. We have not calculated the magnetic anisotropy terms in Eq. 1 and used tentative values of $D_{\alpha}$ ($\alpha=x$, $y$, $z$) in our MC studies. In the following we assume that the axes $x$ and $y$ are parallel to the axes $a$ and $b$, respectively, and, the axis $z$ is perpendicular to both the $a$ and $b$ axes. In our MC calculations of CaFeF5 we used $D_{x}=-0.1$, $D_{y}=0.1$, and $D_{z}=0$ meV. Figure 2(a) shows the temperature dependencies of magnetic susceptibility $\chi$ of CaFeF5 studied along various directions. It can be found that $\chi$ experiences a broad plateau characteristic of 1D magnetic chains structure. The fitting of $\chi$ in the range of temperatures from 500 to 1000 K gives the Curie-Weiss temperature $\theta_{\rm CW}$=-219 K, whereas from the inset in Fig. 2(a) it follows that CaFeF5 experiences an antiferromagnetic phase transition at $T_{N}=23$ K. Overall the results of calculations are in good agreement with the experimental data on magnetic properties of CaFeF5, which give the Curie-Weiss and Neel temperatures of -202 K and 21 K, respectively [15]. From our point of view the presence of frustrating interchain interactions contribute to the frustration parameter $f=|\theta_{\rm CW}|/T_{N}\approx 9.5$, without which it would have been smaller. The analysis of magnetic structure resulting from MC calculations shown in Fig. 2(b) below $T_{N}$ reveals that it is characterized by the wave vector $\vec{k}=(1/2,0,0)$. Thus, the magnetic cell is twice the crystallographic one (i.e. doubled along the $a$ axis) and the spins are directed along the easy axis.

Figure 3 shows the magnetic susceptibility of SrFeF5 obtained from MC calculations, in which we used the anisotropy constants $D_{x}=-0.05$, $D_{y}=0.05$, and $D_{z}=0$ meV. Similar to CaFeF5 the magnetic susceptibility shows a broad maximum characteristic of a 1D magnetism. The Curie-Weiss temperature obtained by fitting the data in the temperature range 500 – 1000 K is $\theta_{\rm CW}$=-166 K. In the inset of Fig. 3 one can discern two magnetic phase transitions at $T_{N1}=35$ and $T_{N2}=28$ K, corresponding to anomalies in the temperature dependence of $\chi_{x}$ and $\chi_{z}$. In SrFeF5, similar to the case of CaFeF5, given a strong tendency for $\ldots$-up-down-up-down-$\ldots$ ordering within the chains, several exchange interactions cannot be simultaneously satisfied (e.g. the pairs $J_{5}$ and $J_{7}$, as well as $J_{6}$ and $J_{11}$), which contribute to both the frustration parameter and to the appearance of incommensurate modulation as described below.

The analysis of magnetic structure reveals that the first phase transition at $T_{N1}$ is connected with the appearance of sinusoidally modulated structure with spins along the easy axis ($x$ axis in our case), whereas at the second phase transitions at $T_{N2}$ additional component along the $z$ axis appears resulting in cycloidal magnetic structure. The modulation wave vector is $\vec{k}_{inc}\approx(0.22,0,0.32)$. The plane of rotation of spins in the low-temperature magnetic structure is governed by the magnetic anisotropy, which in our model is determined by the tentative choice of the single-ion anisotropy constants $D_{\alpha}$. Moreover, in our simplified model we assumed the same single-ion anisotropies for both crystallographically different iron positions Fe1 and Fe2, which may differ in general. Thus, the magnetic order in SrFeF5 can in principle be more complex than a cycloidal rotation of spins.

5 Possible magnetoelectric properties and discussion

As found in the previous section, the magnetic structure of CaFeF5 below the Neel temperature is characterized by the wave vector $\vec{k}=(1/2,0,0)$, which is the Y-point of the Brillouin zone of the space group P21/c. The analysis of possible magnetic structures induced by a single exchange multiplet in this point [16], shows that they cannot induce electric polarization, because they do not break inversion symmetry. This conclusion is also in accord with the absence of modulated magnetic ordering because the irreducible representations (IR) in the Y-point of the Brillouin zone satisfy the Lifshitz criterion [17].

Monte Carlo calculations reveal that the magnetic ordering wave vector in SrFeF5 is $\vec{k}_{inc}\approx(0.22,0,0.32)$. This modulation vector is close to the commensurate vector $\vec{k}_{c}=(1/4,0,1/3)$, which is the F-point of the Brillouin zone and which is realized in our MC calculations assuming stronger single-ion anisotropy constants $D_{\alpha}$. The phenomenological treatment of magnetic phase transitions in SrFeF5 can be performed based on the instability with the wave vector $\vec{k}_{c}$, whereas the incommensurate modulation can be described by Lifshitz invariants. Indeed, the F-point lies in the interior of the Brillouin zone and the two two-dimensional IR’s F1 and F2 in this point do not satisfy the Lifshitz criterion. Thus, the crystal symmetry permits the invariants responsible for long-wavelength modulation of magnetic order in the $ac$ plane

[TABLE]

[TABLE]

where $(a_{1},a_{2})$ and $(b_{1},b_{2})$ are magnetic order parameters transforming according to IR’s F1 and F2, respectively. Depending on the orientation of spins in the low-temperature magnetic structure of SrFeF5 (e.g. cycloidal, proper screw, or more complex magnetic ordering) at least two magnetic order parameters describe the magnetic structure, which transform either by the same single IR F1 or F2 or by two different IR’s F1 and F2. In both cases the magnetoelectric interactions

[TABLE]

[TABLE]

[TABLE]

where the order parameters $(a_{1},a_{2})$ and $(a^{\prime}_{1},a^{\prime}_{2})$ transform according to F1, while $(b_{1},b_{2})$ and $(b^{\prime}_{1},b^{\prime}_{2})$ according to F2, should result in appearance of electric polarization $P_{\alpha}$. Thus, SrFeF5 is predicted to be multiferroic below the magnetic ordering temperature $T_{N2}$, where two magnetic order parameters condense. This picture of magnetically induced electric polarization is similar to that in many other multiferroics [18, 19, 20]. Given the absence of inversion center at the positions of Fe3+ ions in SrFeF5 the single-spin effects should contribute to magnetoelectric coupling  [21]. These single-ion effects together with the nearest neighbor two-spin couplings along the spin chains are arguably the strongest magnetoelectric interactions in SrFeF5.

Both studied compounds, CaFeF5 and SrFeF5, contain spin chains with strong NN intrachain interactions, weaker NNN intrachain couplings and relatively smaller interchain interactions. In SrFeF5 the interchain couplings are relatively stronger compared with intrachain one, than in CaFeF5, which is probably determined by the helicoidal arrangements of spin chains in SrFeF5 and, thus, shorter interchain distances. In both cases, within the chains the spins order in $\ldots$-up-down-up-down-$\ldots$ fashion, whereas relative ordering of chains is determined by a big number of interchain exchange constants. Our calculations show that in CaFeF5 this results in magnetic ordering with

$\vec{k}=(1/2,0,0)$, which does not permit magnetoelectric properties, whereas in SrFeF5 a modulated magnetic structure is realized resulting in multiferroic behavior. In both cases the magnetic wave vector is a result of combined action of many exchange parameters. Given possible inaccuracy in determination of exchange constants by DFT calculations the real magnetic structures in CaFeF5 and SrFeF5 can be characterized by wave vectors other, then those found in the present work. However, since the space group P21/c is non-symmorphic, analysis shows that only the magnetic structures induced by the Y-point instability should not give rise to magnetoelectric properties, whereas all other Brillouin zone boundary points (A, B, C, and Z), which are compatible with the $\ldots$-up-down-up-down-$\ldots$ spin ordering of the chains, contain IR’s not satisfying the Lifshitz criterion and, thus, inducing improper electric polarization [17]. It has to be noted, that in CaFeF5 the energy of the obtained magnetic structure with $\vec{k}=(1/2,0,0)$ is very close to that of magnetic structure with $\vec{k}=0$, in which the spins of of Fe3+ ions located at positions $(x,y,z)$, $(1-x,1-y,1-z)$, $(1-x,1/2+y,1/2-z)$, and $(x,1/2-y,1/2+z)$ are ordered in $(+-+-)$ fashion. Such magnetic ordering breaks inversion symmetry and results in appearance of linear magnetoelectric effect below $T_{N}$. Neutron diffraction experiments are necessary to clarify the magnetic structures in the studied fluorides.

6 Conclusions

We have studied the spin-chain fluorides CaFeF5 and SrFeF5 by means of density functional theory and Monte Carlo calculations, as well as by symmetry analysis. The chains in both compounds are characterized by strong NN exchange couplings and weak NNN interactions. The interchain interactions are much weaker than the NN intrachain couplings, which gives rise to low-dimensional magnetic behavior. Our results indicate that the magnetic structure of CaFeF5 does not result in magnetoelectric effects, whereas SrFeF5 is multiferroic in the low-temperature incommensurate magnetically ordered phase.

Acknowledgements

V. O. acknowledges partial financial support from the Science Committee of the Ministry of Education and Science of Armenia (grant SFU-02), ANSEF project condmatth-5212 as well as the support from the HORIZON 2020 RISE ”CoExAN” project (GA644076). He also expresses his gratitude to the Institute of Physics, Southern Federal University, Rostov-on-Don for warm hospitality during his visit in 2017. N.V.T. acknowledges financial support by the Russian Foundation for Basic Research grant No. 18-52-80028 (BRICS STI Framework Programme).