Missing magnetism in Sr$_{4}$Ru$_{3}$O$_{10}$: Indication for Antisymmetric Exchange Interaction

TL;DR

This study investigates the magnetic behavior of Sr$_{4}$Ru$_{3}$O$_{10}$ near its metamagnetic transition, revealing the importance of antisymmetric exchange interactions like Dzyaloshinskii-Moriya in explaining observed phenomena.

Contribution

The paper demonstrates that Dzyaloshinskii-Moriya interactions are essential to account for the reduced magnetic moment and complex metamagnetic features in Sr$_{4}$Ru$_{3}$O$_{10}$, supported by experimental and numerical analysis.

Findings

Double-feature at $H_c$ is robust below 1.8 K.

Field rotation shifts the double anomaly to higher fields.

Dzyaloshinskii-Moriya interaction is crucial for explaining moment reduction.

Abstract

We report a detailed study of the magnetization modulus as a function of temperature and applied magnetic field under varying angle in Sr$_{4}$Ru$_{3}$O$_{10}$ close to the metamagnetic transition at $H_{c}\backsimeq 2.5\,$T for $H \perp c$. We confirm that the double-feature at $H_{c}$ is robust without further splitting for temperatures below 1.8 K down to 0.48 K. The metamagnetism in Sr$_{4}$Ru$_{3}$O$_{10}$ is accompanied by a reduction of the magnetic moment in the plane of rotation and large field-hysteretic behavior. The double anomaly shifts to higher fields by rotating the field from $H\,\perp \,c$ to $H\,\parallel\,c$. We compare our experimental findings with numerical simulations based on spin reorientation models caused by intrinsic magnetocrystalline anisotropy and Zeeman effect. Crystal anisotropy is able to explain a metamagnetic transition in the ferromagnetic ordered system Sr$_{4}$Ru$_{3}$O$_{10}$, but a Dzyaloshinskii-Moriya term is crucial to account for a reduction of the magnetic moment as discovered in the experiments.

Figures

Equations

$$F=F_0+K_1{\cos }^2\theta +(K_2+K_3\sin (2\phi )){\sin }^4\theta -F_Z.$$

$$F_Z=MH(\sin \theta \sin \psi (\cos \omega \cos \phi +\sin \omega \sin \phi )+\cos \theta \cos \psi ).$$

$$\Delta E=K\sum _{i,j}({\nu }_i-{\nu }_j)\cdot ({\vec{S}}_i\times {\vec{S}}_j)$$

$$F_{DM}=-D{\sin }^2(\theta )\sin (2\phi )$$

Full text

magnetization, metamagnetism, strontium ruthenate

I Introduction

Sr4Ru3O10 belongs to the Ruddlesden-Popper family of ruthenium oxide perovskites Srn+1RunO3n+1. This class of metallic compounds caught much attention in recent years due to its rich variety of ground states. Sr2RuO4  the $n=1$ member, is discussed as an example of rare $p$-wave superconductivity.Ishida et al. (1998) A quantum critical endpoint covered by a high entropy phase was found in the $n=2$ layer system Sr3Ru2O7.Grigera et al. (2001) The compound Sr4Ru3O10 ($n=3$) discussed here shows ferromagnetism below $T_{C}=105\,K$.Crawford et al. (2002) Neutron diffraction experiments in zero magnetic field reveal ordering of the Ru moments along the $c$-axis. No ferromagnetic (FM) or antiferromagnetic (AFM) correlations are observed in the $ab$-plane.Granata et al. (2013, 2016) Sr4Ru3O10 contains four inequivalent Ru sites with two different magnetic moments of 0.9 $\mu_{B}$ and 1.5 $\mu_{B}$ sitting on outer and inner RuO layers, respectively. The magnetic unit cell contains 8 of the smaller and 4 of the bigger magnetic moments averaging to 1.1 $\mu_{B}$ per Ru. The higher order ruthenate SrRuO3 with $n=\infty$ orders also FM at a Curie temperature of 165 K.Callaghan et al. (1966); Kanbayasi (1976). Srn+1RunO3n+1 are strongly 2-dimensional electron systems with the trend to become more isotropic for higher $n$, because of their layered structure. Two-dimensionality is reflected in anisotropic transport properties as seen for Sr4Ru3O10 in the ratio of the electrical resistivity $\rho_{c}/\rho_{ab}\simeq 400$Fobes et al. (2010) and confirmed by optical conductivity experiments.Mirri et al. (2012)

Metamagnetism is a phenomenon observed in magnetic materials, where hidden magnetism is suddenly uncovered by the application of an external field. The origin of metamagnetism ranges from spin flip transitions in antiferromagnets,Held et al. (1997); Eundeok Mun et al. (2016) to the reconstruction of the Fermi surface in metallic materials,Deppe et al. (2012) and in the vicinity of a quantum critical point.Weickert et al. (2010) In a general description, metamagnetism is a phase transition or crossover from a magnetically disordered or ordered state with small net magnetization to a field polarized (FP) or partially FP state. In the case of Sr4Ru3O10  the term metamagnetism refers to the sudden increase in the magnetization when the field applied in the ($ab$) in-plane of this layered compound exceeds  2 T. Magnetism is hidden only because the spontaneous moment is mainly aligned with the easy $c$-axis at smaller fields; we continue to refer to this as metamagnetic (MM) transition.

While metamagnetism of $H\perp c$ in Sr4Ru3O10 was discovered from early on in flux grown single crystals,Crawford et al. (2002) it took more than a decade to improve the crystal quality to a level to see a double step in the magnetization at the MM transition.Carleschi et al. (2014) This strong dependence of physical properties on the crystal purity is a characteristic signature of strontium ruthenates Srn+1RunO3n+1 and was also observed in the sister compound Sr3Ru2O7.Grigera et al. (2001); Perry et al. (2004); Grigera et al. (2004); Gegenwart et al. (2006) The MM transition in Sr4Ru3O10 develops below 68 K as a double-transition close to zero field and shifts gradually to $H_{c}\sim 2.5\,$T with temperatures down to 1.7 K.Carleschi et al. (2014) Carleschi et al.Carleschi et al. (2014) speculate that the double transition originates either in the ordering of Ru magnetic moments on two inequivalent crystallographic sites or in the presence of two van Hove singularities in the density of states close to the Fermi level. A transport study based on electrical resistivityMao et al. (2006); Fobes et al. (2007) reveals steps in the magnetoresistance at various critical fields around $H_{c}$ accompanied by pronounced hysteresis. Fobes et al.Fobes et al. (2007) interpret the transport data as domain movement of regions with high and low electronic spin polarization. Anomalous behavior at the MM transition was also observed in specific heat experiments up to 9 TCao et al. (2007) and in thermopower investigations.Xu et al. (2007) Neutron diffraction experiments up to 6 T reveal a change of lattice parameters at the critical field $H_{c}$.Granata et al. (2013) Field and pressure dependent Raman measurementsGupta et al. (2006) as well as a recent study of thermal expansion and magnetostrictionSchottenhamel et al. (2016) confirm strong magnetoelastic coupling in Sr4Ru3O10.

In this work we focus on magnetization measurements up to 7 T and down to lowest temperatures of 0.46 K and under rotational fields between the $c$-axis and the $ab$-plane as well as ($ab$) in-plane rotation. Our investigations include a detailed analysis of the behavior of the magnetization modulus $M$ at the MM transition and its individual components $M_{ab}$ and $M_{c}$ simultaneously. In the following, we analyze and interpret our data in a localized picture, meaning the magnetic moments are mainly confined on the Ru4+ sites in the crystal structure of Sr4Ru3O10. This scenario is supported by neutron diffraction experiments which have determined the spin and orbital momentum distribution in great detail.Granata et al. (2013, 2016) Our main discovery is the observation of a reduced measured moment at the MM transition caused by a spin component pointing out of the rotational plane which we assert can best be explained by significant anisotropic exchange interactions in Sr4Ru3O10.

II Results

At first, we focus on the magnetization measured for $H\perp c$ at temperatures below 2 K. Figure 1 shows $\chi_{AC}=\partial M/\partial H$ between 0.5 T and 3.5 T. We observe a clear double MM phase transition with a main anomaly at $H_{c1}=2.3$ T and a second anomaly at $H_{c2}=2.8$ T for increasing field as observed by Carleschi et al..Carleschi et al. (2014) Our new experimental data down to 0.46 K clarify that neither transition sharpens to lower temperatures nor is there splitting into more distinct anomalies. The inset in Fig. 1 shows the $H-T$ phase diagram with near-vertical phase boundaries for $T\rightarrow 0$ at the MM transition. Both anomalies are shifted by $-0.3$ T for measurements in decreasing magnetic field. The size of the hysteretic region, marked as striped pattern, remains similar for all temperatures below 1.8 K.

The operation mode of the SQUID magnetometer allows the simultaneous collection of longitudinal $M_{long}$ and transversal component $M_{trans}$ of the sample magnetization in respect to the applied magnetic field $\vec{H}$ as sketched in the inset of Fig. 2. The magnetization component $M_{perp}$ occurring perpendicular to the rotational plane is not recorded during the measurements. This geometry allows us to calculate the magnetization $M^{2}_{rot}=M^{2}_{long}+M^{2}_{trans}$ in the rotational plane. The knowledge of the rotation angle $\psi$ and relation $\tan(\psi-\theta)=M_{trans}/M_{long}$ enables the determination of the angle of the magnetization $\theta$ with respect to the magnetic easy axis $c$ in the rotational plane. We can now estimate $M_{ab}=M_{rot}\sin\theta$ and $M_{c}=M_{rot}\cos\psi$ magnetization, again occurring in the plane of rotation. Note, we follow closely the notation of angles used for magnetic anisotropic materials. Figure 2 illustrates the different components of the magnetization for one particular measurement with $\psi=81.6^{\circ}$ taken at 1.8 K. Prominent feature is the hysteresis loop around $\pm 1$ T, caused by FM domain dynamics. The longitudinal magnetization $M_{long}$ increases moderately in small fields and shows a sudden rise at $H_{c}\backsimeq 3$ T at the MM transition. $M_{trans}$ on the other hand consists mainly of the $M_{c}$ component with a sudden decrease of the magnetization at the same critical field $H_{c}$. $M_{trans}\neq 0$ above $H_{c}$ indicates incomplete field polarization meaning that $\vec{M}$ is not perfectly aligned with $\vec{H}$. This observation points to the presence of magnetic anisotropy. The calculated magnetization modulus $M_{rot}$ in the plane of rotation is depicted as black line in Fig. 2. We find a maximum moment of $1.5\,\mu_{B}$ slightly higher than obtained in neutron experiments,Granata et al. (2013, 2016) but in good agreement with previous magnetization studies.Cao et al. (2003); Zhou et al. (2005) Most peculiar is that $M_{rot}$ drops suddenly below 1.2 $\mu_{B}$ at the MM transition and only recovers partially to 1.2 $\mu_{B}$ up to maximum applied field of 7 T. This missing component of the magnetic moment in Sr4Ru3O10 was never recognized before. Furthermore, we observe strong hysteresis at the MM transition between up and down measurements as reported in previous investigations.Cao et al. (2003); Zhou et al. (2005); Fobes et al. (2007, 2010); Carleschi et al. (2014)

Geometrical effects can distort magnetic properties during magnetization experiments. Therefore, we plot in Fig. 3 $M_{ab}$ as a function of the field component in the $ab$-plane $H^{ab}=H\sin\psi$ to examine how $H_{c1,2}$ change with $\psi$. In contrast to previous results by Jo et al.Jo et al. (2007) obtained by torque magnetometry, we observe a clear simultaneous increase of both critical fields $H_{c1,2}$ to higher values while rotating from $H\perp c$ to $H\parallel c$. In fact, $H_{c1,2}$ move out of the observable field range of 7 T maximum for $\psi\lesssim 72^{\circ}$. Fig. 4 summarizes the critical fields $H_{c1,2}$ in the $H-\psi$ phase diagram for field up and down sweep measurements. The difference $H_{c1}-H_{c2}$ increases slightly with smaller $\psi$. As mentioned above, the double anomaly is accompanied by significant hysteresis. The inset of Fig. 4 shows the evolution of combined step size $\Delta M_{ab}$ of both MM transitions for decreasing $\psi$. It follows a quadratic fit function marked as solid line and extrapolates to zero at about $65^{\circ}$.

The magnetization modulus $M_{rot}$ recorded in the plane of rotation for $\psi$ between 85.3*∘* and 70.5*∘* is plotted in Fig. 5. We only show field-down sweep measurements for clarity. Striking is the occurrence of a drop from about 1.5 $\mu_{B}$ to below 1.2 $\mu_{B}$ at the critical field of the main anomaly $H_{c1}$ followed by a minimum and a small step at the second anomaly at $H_{c2}$. The described features are marked in Fig. 5 by arrows for the measurement at $\psi=85.3^{\circ}$. The MM anomaly broadens and moves to higher fields for decreasing angles $\psi$.

III Discussion

The ”loss” of magnetic moment in the rotational plane can be explained either by partial AFM alignment or by a moment $M_{perp}$ occurring perpendicular to the rotation (parallel to rotation axis of $\psi$). First scenario can be excluded based on neutron experiments where no short or long range AFM coupling neither in zero nor in magnetic fields $H>H_{c}$ was observed in the $ab$-plane.Granata et al. (2013, 2016) The second scenario is rather unexpected since magnetic moments tend to align with field and stay within the rotational plane, if no further coupling is present. We want to focus in our discussion on two mechanisms that potentially lead to a $M_{perp}$ component in the magnetization. First one is based on general magnetocrystalline anisotropy in tetragonal symmetry, with an easy $c$-axis and 4-fold in-plane anisotropy. The second mechanism is antisymmetric exchange between spins, also called Dzyaloshinskii-Moriya (DM) interaction, causing a canting of the spins $\vec{S_{i}}\times\vec{S_{j}}$.

III.1 General Anisotropy

We have to have a closer look at the crystal structure of Sr4Ru3O10 in order to understand and model its magnetic anisotropy caused by spin-orbit coupling. Sr4Ru3O10 crystals consist of three layers of corner sharing RuO6 octahedra separated by a double layer of Sr-O. Primary Bragg reflections in synchrotron experiments can be indexed assuming a tetragonal unit cell with space-group $I4/mmm$, but a more detailed analysis of secondary reflections reveals orthorhombic $Pbam$ symmetry.Crawford et al. (2002) The lower symmetry originates in $c$-axis rotation of the RuO6 octahedra that are correlated between different layers, meaning $+11.2^{\circ}$ clockwise rotation for inner and $-5.6^{\circ}$ counterclockwise rotation for outer layers.

The free energy $F$ accounting for magnetocrystalline anisotropy in a tetragonal lattice, can be modeledHorner and Varma (1968) by

[TABLE]

$F_{0}$ is a constant background contribution independent of $\vec{H}$ or $\vec{M}$. $\vec{M}$ is expressed in polar coordinates ($\theta,\varphi$), with $\theta$ = 0 along the crystal $c$-axis and $\varphi$ = 0 defining the in-plane hard axis for $K_{3}<0$. Here, $K_{1}<0$ defines the easy direction and the Zeeman term $F_{Z}$ can be written as

[TABLE]

The applied field has the polar coordinates ($\psi,\omega$), with $\omega=0$ corresponding to field rotation from the $c$- axis to the in-plane hard direction and $\omega=\pi/4$, field rotation to the easy direction.

We used numerical minimization of equation (1) to determine $\theta$ and $\varphi$ as functions of the applied field. In the uniaxial case ($K_{3}=0$), the MM behavior in $M_{ab}$ and $M_{c}$ at the critical field $H_{c}$ is reproduced by choosing correct parameters $K_{1}$ and $K_{2}$ (data not shown). However, uniaxial anisotropy is unable to reproduce any reduction of the magnetization in $M_{rot}$, since the moment always stays in the rotational plane.

Note, we do not know precisely the in-plane orientation of our sample. However, the rectangular shape suggests that $\psi$ rotation axis is parallel to one of the principal axes such as [100] or [110]. For 4-fold tetragonal symmetry either one of them would be the intermediate or hard axis, respectively. We consider in the following a projection of $\vec{H}$ onto the magnetic hard axis in the $ab$-plane with $\omega=0$, because tilting of $\vec{M}$ towards the hard axis forces the magnetic moments to align spontaneously toward either one of the intermediate axes, which are $45^{\circ}$ apart from the hard axis. This spontaneous alignment $\pm 45^{\circ}$ is energetically degenerated and could lead to domain formation with an overall smaller net magnetization as observed in our measurements.

Fig. 6 compares numerical results of $\psi=78^{\circ},81^{\circ},84^{\circ}$ based on equation (1) with experimental data of the magnetization modulus $M$ for $\psi=77.9^{\circ},81.6^{\circ},83.4^{\circ}$. We are able to reproduce i) a critical field value $H_{c}\backsimeq 2.5$ T that increases with smaller $\psi$, ii) a drop $\Delta M$ at $H_{c}$ that is comparable in size with the experimental data, and iii) a gradual slope $M(H)$ for $H>H_{c}$. The double feature at the MM transition is missing due to the simplicity of the model. We obtain anisotropy parameters $K_{1}=3.1$ K, $K_{2}=0.1$ K and $K_{3}=-2.2$ K in Kelvin energy scale which convert to the following values 300 kJ/m3, 10 kJ/m3, and -210 kJ/m3, respectively, in units widely used in magnetic anisotropy tables. The 4th order parameter $K_{2}$ being more than 10 times smaller than $K_{1}$ implies that it is irrelevant for the description of the anisotropy in Sr4Ru3O10. For comparison, the $3d$ FM metal cobalt has anisotropy constants of $K_{1}=450$ kJ/m3 and $K_{2}=150$ kJ/m3, which are of similar size as $K_{1}$ in Sr4Ru3O10.Cullity and Graham (2008)

Despite the reasonable agreement between experiment and model, it is necessary to check in a subsequent experiment our initial assumption of tilting the spins towards the magnetic hard axis in the $ab$-plane. Therefore, we rotate the sample by $\omega=45^{\circ}$ in the plane and measure again $M$ at three different angles $\psi$ as shown in Fig. 7. We anticipate the 45*∘* change would bring the intermediate anisotropy axis into the rotation plane. Specifically, the magnetization would rotate toward the intermediate axis with the total magnetization remaining in the rotation plane and therefore no ”loss” of magnetic moment effect. Surprisingly, the magnetization $M_{rot}$ shows exactly the same behavior as for the $\omega=0$ experiments within experimental uncertainty. Even if in both experiments $\omega=0$ and 45*∘, the plane of $\psi$-rotation would not include exactly the principal axes, we would at least expect the observation of a reduced anomaly in $M_{rot}$ at $H_{c}$. Based on our last finding, we exclude general magnetic anisotropy as sole cause for the reduction of moment at the MM transition in Sr4Ru3O10*.

III.2 Antisymmetric Exchange Coupling

Anisotropic exchange interactions caused by spin-orbit coupling under certain symmetry constraints were first considered by Dzyaloshinsky and MoriyaDzyaloshinsky (1958); Moriya (1960) to explain weak FM ordering inside an AFM phase in transition metal oxides. Recently, Bellaiche et al.Bellaiche et al. (2012) pointed out that tilting of oxygen octahedra in perovskites can be described by a pseudo-vector $\nu_{i}$ at spin $S_{i}$ position $i$ that leads to an energy reduction

[TABLE]

in analogy to DM antisymmetric exchange coupling. The summation is done over nearest-neighboring spins $\vec{S}_{i,j}$ and $K$ is a constant. Consequently, we approximate a DM interaction between the in-plane component of the Ru center magnetization and that of the two nearest-neighbor Ru atoms, whose spins are assumed to remain parallel. This gives rise to an effective energy term

[TABLE]

in replacement of the $K_{3}$ term of the magnetocrystalline anisotropy. Angle $\varphi$ is interpreted as the angle between in-plane magnetic moments sitting on one inner and two outer layers with $\varphi=0$ being the direction of in-plane magnetic field. The change in the dependence on angle $\varphi$ between the magnetization vector and the $c$-axis prevents an accurate minimization of the total energy. Nonetheless, the approximate solution shown in Fig. 8 for $\psi=82^{\circ}$ does produce a ”missing” portion of the total magnetization, with the minimum moving to higher field with decreasing $\psi$. The parameter $D$ is approximately 5.3 K and comparable to the energy scale of the magnetocrystalline anisotropy $K_{1}$.

We want to point out that the opening of $2\varphi$ between the inner and outer magnetic moments can be seen as AFM order if it occurs periodically along the $c$-axis. Neutron scattering experiments on holmium-yttrium superlatticesMoriya (1999) e.g., were able to distinguish between different types of AFM periodicity along $[00l]$ with increasing in-plane magnetic field, such as helical, helifan-shaped, fan-shaped and FM order. However, the particular crystal structure of Sr4Ru3O10 with three layers of RuO6 octahedra connected through a double layer of Sr-O doesn’t give rise to additional periodicity, even if the magnetic moments follow a repetitive pattern such as $-\varphi,+\varphi,-\varphi$ along $c$ inside the triple layer. Consistently, in-plane AFM ordering was excluded by neutron scattering studies.Granata et al. (2013)

IV Summary

In summary, the MM transition in Sr4Ru3O10 as a function of magnetic field and rotation angle between $H\perp c$ and $H\parallel c$ has been studied in great detail by magnetization measurements down to lowest temperatures in a SQUID magnetometer. Our experimental results reveal a reduced magnetic moment in the plane of rotation which was never recognized before. It is robust to ($ab$) in-plane rotation. We find furthermore that the double step at the MM transition is stable down to lowest temperatures of 0.46 K. Our experimental results are interpreted in a strict localized picture with magnetic moments of different sizes sitting on inner and outer RuO6 layers in the crystal structure. We completed our study with numerical calculations based on energy minimization including Zeeman effect, magnetocrystalline anisotropy and antisymmetric exchange and compared them with the experimental data. We conclude that all three contributions are essential ingredients to understand the magnetization behavior and establish that a Dzyaloshinsky and Moriya like component is essential to understand the occurrence of a reduced magnetic moment in Sr4Ru3O10.

V Methods

Sr4Ru3O10 single crystals were grown in an image furnace by a floating zone techniqueFittipaldi et al. (2007) and characterized by energy dispersive spectroscopy, scanning electron microscopy, electron backscattering diffraction and x-ray diffraction techniques.

Magnetization measurements in fields up to 7 T were carried out in a Quantum Design MPMS SQUID magnetometer equipped with a standard 4He setup for measurements between 1.7 K and 100 K and in an iQuantum 3He insert that fits inside the MPMS sample space for the temperature range 0.46 K to 2 K. Excellent agreement between the 3He and 4He data was observed in the temperature range of overlap. Angular dependent measurements at 1.8 K were obtained with a mechanical rotator mounted inside the MPMS magnetometer. Note that the MPMS operates with a pair of coils for signal detection mounted parallel (longitudinal coil) and perpendicular (transversal coil) to the applied magnetic field. The rotator is aligned with the rotation axis normal to the plane defined by both SQUID coil axes.

The measured single crystal has a rectangular shape of (1.87 x 2.99)mm2 in $ab$ and 0.54 mm along the crystallographic $c$-direction. We considered demagnetization effects by approximating the sample shape with an ellipsoidOsborn (1945) and estimated small correction fields of - 30 mT for $H\perp c$ and - 140 mT for $H\parallel c$.

Acknowledgements.

Work at Los Alamos was supported by the National Science Foundation, Division of Material Research under Grant NSF-DMR-1157490 and the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division.