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Magnetic Dynamics of Phase Separation Domains in GdMn2O5 and Gd0.8Ce0.2Mn2O5 Multiferroics

Specific features of the magnetic properties and magnetic dynamics of isolated phase separation domains in GdMn2O5 and Gd0.8Ce0.2Mn2O5 have been investigated. These domains represent 1D superlattices consisting of dielectric and conducting layers with the ferromagnetic orientation of their spins. A set of ferromagnetic resonances of separate superlattice layers has been studied. The properties of the 1D superlattices in GdMn2O5 and Gd0.8Ce0.2Mn2O5 are compared with the properties of the previously investigated RMn2O5 (R = Eu, Tb, Er, and Bi) series. The similarity of the properties for all the RMn2O5 compounds with different R ion types is established. Based on the concepts of the magnetic dynamics of ferromagnetic multilayers and properties of semiconductor superlattices, a 1D model of the superlattices in RMn2O5 is built.


INTRODUCTION.
RMn2O5 manganites, where R is the rare-earth ions, Y and Bi, belong to the class of type-II multiferroics, in which the ferroelectric ordering (the Curie temperature is TC≈ 30-35 K) is induced by the magnetic order (the Neel temperature is TN≈ 35-45 K) lowering the crystal symmetry to noncentral. Due to the similarity of the TC and TN values, the RMn2O5 manganites exhibit the pronounced magnetoelectric effect, which evokes great interest of researchers in these materials [1,2]. Until recently, it has been commonly accepted that at T< TC the RMn2O5 manganites are characterized by the centrosymmetric sp. gr. Pbam. When these crystals contain the same number of ions of different valences (Mn3+ and Mn4+), it becomes energetically more favorable to establish a certain order (charge order) in their distribution. Mn3+ and Mn4+ ions characterized by alternating ferro-and antiferromagnetic spin orientations at temperatures of T ≤ TN are distributed pairwise along the b axis. The difference between the ferromagnetic double exchange (the exchange integral is JDE ≈ 300 meV [3,4]) and indirect antiferromagnetic exchange (JAF≈ 10 meV) values leads to the exchange striction, which breaks the lattice symmetry, and to the occurrence of ferroelectric ordering along the b axis at T ≤ TC [5].
In the recently published structural work of a series of RMn2O5 (R = Pr, Nd, Gd, Tb, and Dy) multiferroics, the room-temperature synchrotron resonance scattering study revealed, along with high intensit reflections of the sp. gr. Pbam, the weak-intensity reflections, which could not be described by the central symmetry [6]. Based only on the structural data, the authors of [6] failed to give preference to one of the two possible monoclinic space groups, i.e., P2, which allows polarization along the c axis, and Pm with the polarization in the ab plane. The authors believed that the noncentral groups belong to the entire crystal, in which the uniform polarization can only be established along one crystal axis. Since at low temperatures the striction polarization is oriented along the b axis, the authors chose the sp. gr. Pm. It followed from [6] that at room temperature the RMn2O5 man-ganites should be characterized by the uniform electric polarization along the b axis, the origin of which is different from that of the low-temperature polarization. The electric polarization in RMn2O5 (R = Gd and Bi) in the temperature range of 5-330 K along all the crystal axes was studied in [7,8]. At temperatures above TC≈ 30-35 K, the electric polarization uniform over the entire crystal was not observed. However, over the entire temperature range, there were local polar domains of phase separation. These regions formed the frozen superparaelectric state, which existed from the lowest temperatures to certain temperatures in the paramagnetic temperature range depending on the crystal axes directions. In these states, the response to the applied electric field had the form of hysteresis loops. This situation was theoretically investigated in [9], but not experimentally observed. The noncentrosymmetric reflections observed in [6] apparently correspond to the polar phase separation domains and the main crystal matrix has the sp. gr. Pbam in the paramagnetic temperature range [7,8].
The RMn2O5 unit cell contains one Mn 3+ ion and one Mn 4+ ion. Mn 4+ ions are located in the nearly undistorted oxygen octahedra (MnO6) and occupy the positions with z= 0.25c and z= (1 -0.25)c.
Their 3d shells contain three localized t2g electrons. Mn 3+ ions are located in the noncentral pentagonal pyramids (MnO5) in the positions with z= 0.5c and contain three localized t2g electrons and one delocalized eg electron on the degenerate orbital doublet. R 3+ ions are located in the positions with z= 0c and have the environment similar to that of Mn 3+ ions [10]. Thus, the RMn2O5 manganite is characterized by the layer-by-layer ion distribution perpendicular to the c axis. Transport of eg electrons between Mn 3+ and Mn 4+ ion pairs in the neighboring lattice layers leads to the phase separation, which affects the properties of RMn2O5 in both the low-temperature multiferroic state and paramagnetic state. This phase separation is analogous to that observed in the doped LaAMnO3 (A=Sr, Ba, and Ca) manganite perovskites, which are magnetic semiconductors with the colossal magneto-resistance [4,11] RMn2O5 tend to be accumulated in the ferromagnetic conducting phase separation domains inside the dielectric antiferromagnetic (paramagnetic) crystal matrix due to the strong interactions between them and the crystal matrix [4,11]. These interactions are the double exchange caused by transport of the eg electron between the ion pairs Mn 3+ -Mn 4+ with the ferromagnetic spin orientation of these ions. As was mentioned above, the transfer integral at the double exchange is t≈ 300 meV [3,4]. The double exchange leads to accumulation of eg electrons in the phase separation domains. This is facilitated also by local structural distortions near the Jahn-Teller Mn 3+ ions in octahedra (EJT≈ 700 meV). The Coulomb repulsion (EQ≈ 1 meV) prevents charge accumulation in these regions. This results in the formation of the dynamical equilibrium localized ferromagnetic conducting phase separation domains. Since the phase separation is induced by the strong interactions, it exists from the lowest temperatures to the temperatures above room temperature. In [4,11], these regions in LaAMnO3 were considered as nanoscale (~30 nm) spheres.
In the later works, it was shown that the phase separation domains arise also in other crystals containing ions of different valences. The shape of these regions depends on the ratio between values of the interactions inducing them and can be 2D plane or 1D [12][13][14][15][16].
The states induced by the phase separation and self-organization of eg electrons were thoroughly studied in EuMn2O5 multiferroic manganites and doped Eu0.8Ce0.2Mn2O5 manganites in [15,16], where their dielectric and magnetic properties were compared, the specific heat were investigated, and X-ray diffraction and Raman scattering examinations were carried out. The phase separation was observed in the initial crystals, but doping of EuMn2O5with Ce 4+ ions replacing Eu 3+ ions led to its significant enhancement via forming an additional doping channel. Indeed, in Eu0.8Ce0.2Mn2O5 in the z= 0 plane, electrons were induced by the reaction Eu 3+ = Ce 4+ + e, which transformed Mn 4+ ions to Mn 3+ ions in the planes z= 0.25c and 1 -z= 0.75c. The phase separation led to the sharp growth of the permittivity and local conductivity at temperatures above 185 K [15]. Study of the fine structure of Bragg reflections by the high-sensitivity three-crystal X-ray diffraction technique revealed the layered superstructure in EuMn2O5 and Eu0.8Ce0.2Mn2O5 along the c axis at room temperature, which consists of the alternating initial EuMn2O5 dielectric crystal layers and layers with excess Mn 3+ ions. The layer widths in the 2D superstructure were ~900 Å for EuMn2O5 and ~700 Å for Eu0.8Ce0.2Mn2O5 [15]. At low temperatures, the 2D phase separation layers transformed to the limited 1D superlattices consisting of the alternating conducting and dielectric layers with the ferromagnetically oriented spins. Study of EuMn2O5 and Eu0.8Ce0.2Mn2O5 specific heats showed that at low temperatures they undergo the same set of phase transitions [15]. It means that the 1D superlattices occupy a small crystal volume both in the initial and doped states. In the superlattice layers, a set of ferromagnetic resonances (FMRs) was observed, the intensity of which sharply dropped at T ≤30 K [17,18]. As was shown in [16], at temperatures above 40-50 K, carrier hoppings from the 1D superlattices are intensified and, as the temperature increases, the 2D layered structure forms, which acquires the final shape at 185-320 K [15].
The aim of this study was to compare the magnetic properties of GdMn2O5 and Gd0.8Ce0.2Mn2O5 and magnetic dynamics of the 1D superlattices (a set of FMRs of their separate layers) at temperatures of 5-90 K, which still have been understudied. The especially interesting RMn2O5 multiferroic is GdMn2O5. Gd 3+ ions (the ground state is 8 S7/2) have the maximum spin in the series of R ions and are not coupled to the lattice, in contrast to most magnetic R ions. As was shown in [19][20][21], in GdMn2O5 at temperatures of T≤30 K = TC, the homogeneous antiferromagnetic state with a wave vector of q= (1/2, 0, 0) is established. In this case, the electric polarization appeared higher than that observed usually in RMn2O5 with other R ions by an order of magnitude, which is related to the polarization enhancement by the strong homogeneous Gd-Mn exchange [19]. In this study, we investigate specific features of the magnetic state and lowtemperature phase separation in GdMn2O5 and Gd0.8Ce0.2Mn2O5, compare their properties with those of the previously investigated RMn2O5 manganites, and establish the nature and main peculiarities of the low-temperature phase separation domains in RMn2O5, which were found to be weakly dependent on the R ion type.

EXPERIMENTAL
The GdMn2O5 and Gd0.8Ce0.2Mn2O5 single crystals were grown by spontaneous crystallization and had the form of plates with a thickness of 1-3 mm and an area of 3-5 mm 2 . The magnetic measurements were performed on a Quantum Design PPMS magnetometer. The microwave magnetic dynamics was studied on a transmission magnetic resonance spectrometer in the frequency range of 28-40 GHz with the low-frequency magnetic modulation [17,18] The RMn2O5 compounds with different R ions usually have a complex magnetic structure with the wave vector q= (1/2 + δx, 0, 1/4 + δz) and undergo a great number of magnetic phase transitions at temperatures of T ≤ TN. During these phase transitions, commensurability parameters δx and δz change step-wise [10]. As the temperature near TN decreases, the incommensurable magnetic phase is observed. With a further decrease in temperature, this incommensurable phase transforms to the commensurable one. Then, at lower temperatures, the incommensurable phase forms again.
The ferroelectric ordering in RMn2O5 is observed, as a rule, in the intermediate commensurable phase. The GdMn2O5 magnetic structure differs from the structure observed usually in RMn2O5.
As was mentioned above, the commensurable homogeneous collinear antiferromagnetic structure with the wave vec-tor q= (1/2, 0, 0) is observed in the temperature range of 0-30 K (TN= 35 K) [19][20][21]. Figure 1 shows temperature dependences of the magnetization of GdMn2O5 in a magnetic field of H= 5 kOe oriented along different crystal axes. It is noteworthy that the magnetization of GdMn2O5 is much higher than in many RMn2O5 compounds, which is reasonable to attribute to the contribution of Gd 3+ ions with the spin S= 7/2. The anomaly caused by the antiferromagnetic transition of Mn ions near TN is barely noticeable against the background of large contribution of Gd ions to the total magnetization (see the left inset in Fig. 1). For Gd 3+ ions, the magnetization maximum is observed along the a axis near 13 K, which evidences for the intrinsic ordering of these ions. The right inset in Fig. 1 shows the temperature dependence of the inverse magnetization of GdMn2O5, which made it possible to determine Curie-Weiss temperatures TCW along different axes. It was found that these temperatures are almost identical to TN. This means that the magnetic state of GdMn2O5 is not frustrated and the antiferromagnetic transition is sharp. Usually, the

Ferromagnetic Resonances of the 1D Layers of the Superlattices: Phase Separation Domains at Low Temperatures
As was mentioned above, the authors of [17,18] found the 1D superlattices of the alternating conducting and dielectric layers with the ferromagnetic orientation of their spins at low temperatures in a wide series of the RMn2O5 compounds. Figure 3 shows sets of FMRs detected from separate superlattice layers for these RMn2O5 compounds. The magnetic dynamics was most thoroughly studied for the phase separation domains in the EuMn2O5 and Eu0.8Ce0.2Mn2O5 multiferroics [17]. It appeared that the low-temperature phase separation in GdMn2O5 and Gd0.8Ce0.2Mn2O5 is also a set of isolated 1D superlattices located in the initial multiferroic matrix, which occupies a small matrix volume.     To describe the spin-wave excitations in the superlattice layers, we can use the dispersion equation for spin waves of isotropic magnetic films in multilayers [22] ω 2 =(ωH +ηk 2 ) (ωH +ηk 2 +ωMsin θk 2 ) Here ωH= 4πγH, ωM= 4πγM0, η is the inhomogeneous exchange constant, θk is the angle between directions of wave vector k of the spin wave and magnetic field H, and M0 is the saturation magnetization. In the case of longitudinal magnetization (θk= 0) or at ωH ≫ ωM, at the transverse magnetization implemented in our case, the dispersion law for spin waves is ω= ωH+ ηk 2 . In addition, we should take into account the boundary conditions caused by spin pinning at the boundaries of separate superlattice layers. The pinning parameter ξ is a ratio between the energy of surface anisotropy KS of the layer and the inhomogeneous exchange energy (qM0) 2 in it, i.e., ξ= 2KS/(qM0) 2 . The pinning parameters ξ1 and ξ2 on the opposite surfaces determine wave vector k of the standing spin wave excited in a layer with thickness d.
This relation is described by the equation [22] coth (kd) = ( In our case, the dynamical equilibrium states of the superlattice are established and maintained by permanent hoppings of eg electrons between Mn 3+ and Mn 4+ ions both inside the layers and between separate layers. As was mentioned above, such hoppings can occur only when Mn ion spins are oriented ferromagnetically in all the neighboring layers, which is caused by the double exchange. As a result, the condition ξ1= ξ2= 0 should be met for all the inner layers of the superlattices. In this case, as follows from Eq. (2), the spin-wave excitations with the wave vectors km= (m-1)π/d (m= 1, 3, 5…) can occur in the inner layers of the superlattices. Then, the fundamental modes with m=1 are the homogeneous spin-wave excitations with k=0, i.e., homogeneous ferromagnetic resonances. The higher-order modes with m= 3, 5, …, etc., are the standing spin waves with k≠ 0. At a fixed frequency, such modes, which exhibit decreasing intensity with increasing m, should be excited in the decreasing magnetic fields due to the increasing contribution of the inhomogeneous exchange ηk 2 . In the 1D superlattices, we apparently only observe the fundamental modes of the homogeneous FMR in all superlattice layers. The rigid spin pinning can occur in the investigated superlattices only on the outer interfaces between the edge layers and the antiferromagnetic crystal matrix. Near such interfaces, the inhomogeneous internal magnetic field is induced, which changes the wave vector k of the spin wave. The outer interface is a rotation surface (Fig. 9), on which wave vector k of the spin wave changes its sign and passes through zero. It is positive inside the superlattice and negative outside it. This means that the homogeneous spin excitation with alternating magnetic moment m propagates inside the superlattice layer, but decays exponentially in the antiferromagnetic matrix. As a result, the spin excitation with k= 0 (Fig. 9) occurs also on the rotation surface [22]. Thus, the homogeneous spin-wave excitations with k= 0 (FMR) are induced in all the superlattice layers and on their rotation surfaces. The presence of the rotation surfaces near the superlattice edges makes the ferromagnetic superlattices independent of the main antiferromagnetic matrix and their spin moments are oriented by only external field H. Fig. 9. Inhomogeneity of field H0 and wave vector k 2 of the spin wave and propagation of the spin wave with variable magnetization m near the rotation surface (at the boundary surface of the superlattice layer with coordinate z=0) [22].
The aforementioned features of the spin excitations in the investigated superlattices are caused by the fact that they are formed in the bulk of multiferroic single crystals by selforganization. The condition ξ1= ξ2=0 cannot be met in artificial ferromagnetic superlattices due to the inevitable defect formation at the interfaces between different layers. In such multilayers, the higher-order inhomogeneous modes with k≠0 are usually excited. As was mentioned above, such modes with decreasing intensity are localized in decreasing magnetic fields from the most intense mode. Only the surface modes in artificial ferromagnetic multilayers are excited in strong magnetic fields with respect to the fundamental line, but usually close to it. In this case, the surface mode intensities can exceed the intensity of the fundamental mode in the film layer [22].  The symmetry of the L and R lines relative to the LO line is caused by the requirement for electroneutrality of the superlattice in the presence of different charges of Mn ions and electron and hole carriers in the right and left superlattice layers (Fig. 10). Fig. 11. Splitting of the energy band E(kz) of the crystal with lattice constant a into the minibands Ef(kz) by the potential of a superlattice with period d. The number of minizones is d/a [27].
In terms of the semiconductor band diagrams, the superlattices in multiferroics are similar to ordinary semiconductor superlattices and represent a set of split minibands (Fig. 11) [27] with gaps between them. Figure 12 shows the frequency dependence of the observed FMR intensity (the L0 lines) in GdMn2O5 at T= 5 K, which reveals minibands of the 1D superlattices and gaps between them. Fig.12. The frequency dependence of FMR intensity (the L0 lines) in GdMn2O5 at T= 5 K.
As the temperature increases, upon breaking of the antiferromagnetic ordering and simultaneous growth of the hopping conductivity, the 1D superlattices change their shape and state. With an increase in the kinetic energy of electrons, the layers with deeper barriers (the R layers) are populated in a greater degree. The number of superlattice layers decreases and the shape of phase separation domains changes, transforming to the 2D structure observed by us at room temperature in GdMn2O5 [7,8] and Gd0.8Ce0.2Mn2O5 [28].

CONCLUSIONS
Thus, the dynamical equilibrium 1D superlattices occur at low temperatures in a wide series of