Impact of temperature-dependent local and global spin order in RMnO3 compounds for spin–phonon coupling and electromagnon activity

The orthorhombic rare-earth manganite compounds RMnO3 show a global magnetic order for T < T N , and several representatives are multiferroic with a cycloidal spin ground state order for T < T cycl < T N ≈ 40 K . We deduce from the temperature dependence of spin–phonon coupling in Raman spectroscopy for a series of RMnO3 compounds that their spin order locally persists up to about twice TN. Along the same line, our observation of the persistence of the electromagnon in GdMnO3 up to T ≈ 100 K is attributed to a local cycloidal spin order for T > T cycl , in contrast to the hitherto assumed incommensurate sinusoidal phase in the intermediate temperature range. The development of the magnetization pattern can be described in terms of an order–disorder transition at Tcycl within a pseudospin model of localized spin cycloids with opposite chirality.


Introduction
Since the discovery of the giant magnetoelectric effect [1][2][3][4], the study of multiferroics, where an electric field E can affect the magnetic properties and vice versa, has gained strong interest [5][6][7][8][9][10][11][12][13]. While a large number of compounds was shown to exhibit multiferroic behavior, the underlying mechanisms are often still not fully understood. Considering the extensively studied RMnO 3 family of orthorhombically distorted perovskites, the cycloid-type order of the Mn spin ground state is seen as the origin of ferroelectric polarization of the multiferroic members (R=Gd, Tb, Dy) [14][15][16]. As a prerequisite for a cycloidally ordered magnetic state, frustration needs to be introduced to the spin system to deviate from the conventional parallel or antiparallel ordering of spins. This can be achieved by geometrical frustration as, for example, for antiferromagnetically coupled spins on a triangular or Kagomé lattice, or by competing ferro-(FM) and antiferromagnetic (AFM) exchange interactions [9,10,17]. The latter is the case for the orthorhombically distorted RMnO 3 compounds, where the nearest-neighbor Mn-O-Mn exchange within the MnO 2 plane is FM and the next-nearest neighbor interaction along the perpendicular direction is AFM in nature [18]. By choosing R-ions with appropriate ionic radius, the orthorhombic distortion angle can be tuned. With decreasing R-ion radius and therefore increasing orthorhombic distortion angle, the FM in-plane interactions are weakened, while the AFM plane-to-plane interactions are enhanced. For the material series from LaMnO 3 to EuMnO 3 , this results in canted A-type antiferromagnetism with T N -values decreasing from »150 K for LaMnO 3 to »50 K for EuMnO 3 [18,19]. The angular range in which the competing FM and AFM contributions are comparable, resulting in a cycloidal spin structure, occurs between EuMnO 3 and HoMnO 3 and comprises the compounds GdMnO 3 , TbMnO 3 and DyMnO 3 . Besides by these stoichiometric RMnO 3 compounds, this angular range is also achievable by solid solutions of various combinations of R-site ions, which, moreover, allow a fine-tuning of the systems properties, e.g. by Dzyaloshinskii-Moriya (IDM) model, the non-collinear arrangement of spins induces an electric dipole moment [21] µ´ṕ e S S , 1 where e ij denotes the unit vector which connects the spins S i and S j . This was confirmed by the simultaneous flop of electric polarization, when the cycloid helicity is reversed by application of a magnetic field [22,23]. Another consequence of the magnetoelectric coupling in a cycloidal structure is the existence of an electricdipole-active magnon excitation, termed the electromagnon [24][25][26][27][28]. Additionally, especially in GdMnO 3 , the experimentally observed selection rules for the electromagnon deviate from the expected behavior: when the rotational plane of the spin cycloid is flipped, the selection rule for excitation should flip accordingly. In contrast, it was found that the largest part of the dipole activity was bound to the crystal lattice and not to the cycloidal plane [29], which implies contributions of magnetostrictive nature, according to [15] Thus, for electromagnons generally both IDM and magnetostriction mechanisms may be of relevance. Besides, the magnetic ordering of Mn 3+ spins may also influence the phonon mode dynamics. This effect is denoted as spin-phonon coupling (SPC). Its experimental analysis essentially occurs by Raman spectroscopy [20,[30][31][32][33], where it mostly results in a mode-specific frequency softening with decreasing temperature. This frequency shift is usually assumed to be proportional to the spin-correlations w D µ á ñ S S i j SPC · . Thus, the strongest SPC is expected in systems with full global spin order, but it might also occur to a lesser degree already for local spin order.
It has been argued that the magnetic structure of the various RMnO 3 compounds changes with increasing temperature from the cycloidal ground state below » T 28 ) before arriving at the paramagnetic state 5 ( > » T T 40 N K) [7]. A (static) sinusoidal order, however, does not agree with the fact, that Mn 3+ should exhibit a Heisenberg spin with constant magnitude of S=2. A more satisfying approach is to view the sinusoidal order as a time-averaged mixture of cycloidal phases, as suggested by model calculations [19] and by analysis of dielectric relaxation [34]. This raises the question about the underlying nature of the intermediate magnetic order above T cycl and how the transition from the lowtemperature cycloidal phase to the paramagnetic phase takes place.
We have investigated the temperature-dependent behavior of phonons and electromagnons on multiferroic RMnO 3 compounds by Raman and THz spectroscopy, respectively. Our results suggest that the cycloidal phase extends on a short-range scale towards much higher temperatures than T cycl , and local cycloidal order persists even distinctly above » T 40 N K. This is evidenced by the observation of the persistence of both the electromagnon and the SPC up to temperatures around 100 K. Furthermore, a characteristic activation energy of » E 100 A K for switching between cycloid chirality in the pseudospin model was found for DyMnO 3 by dielectric spectroscopy [34]. Additionally, it was reported by De et al [35] that the polarization of a poled sample is preserved up to about 90 K, which further indicates the presence of cycloids up to this temperature. Therefore we propose that the underlying order of the sinusoidal and even of the paramagnetic phase may be explained as a dynamical equilibrium of fluctuating cycloids with opposite chirality. Along the same line, we observe also for non-multiferroic RMnO 3 compounds a persistence of SPC up to about 100 K, which implies the occurrence of local spin order far above T N also in this case.

Experimental
The orthorhombically distorted RMnO 3 samples were grown by a floating zone method. Within the Pnma coordinate axes orientation (International tables orientation) the alternating MnO 2 -and RO-planes are denoted as (a, c)-planes, while the axis perpendicular to these planes is called the b-axis. The polarized Raman spectra for the SPC studies were recorded from (a, c) surfaces (b-cut samples) and from (b, c) surfaces (a-cut samples), using a Horiba LabRAM HR 800 spectrometer equipped with a notch filter, a Peltier-cooled CCD camera as detector, and a 632.8 nm He-Ne laser for excitation. Laser focusing as well as signal collection was performed using a microscope with a 50× ULWD objective. To obtain the temperature-dependent spectra (5 K   T 295 K), the samples were mounted inside a LHe-flow cryostat. The terahertz and far-infrared spectra of the electromagnons were obtained on thin plane-parallel samples in transmittance geometry with a Mach-Zehnder type interferometer, with backward-wave oscillators providing the monochromatic, linearly polarized radiation sources. For detection, either a Golay cell or a liquid-He-cooled bolometer was used. Further details of this technique are provided in [36].

Spin-phonon coupling
We have analyzed the temperature dependence of the SPC strength for various multiferroic and non-multiferroic RMnO 3 compounds: the stoichiometric EuMnO 3 , GdMnO 3 , and TbMnO 3 , as well as the doped (Eu, Y)MnO 3 and (Eu, Ho)MnO 3 with various compositions. Here, we present in detail the results on -Eu x 1 Ho x MnO 3 , which is multiferroic for > x 0.2 [37]. Details of the SPC results of the other compounds can be found elsewhere [31,38]. Figure 1 shows temperature-dependent polarized Raman spectra for Eu Ho 0.9 0.1 MnO 3 and Eu 0.7 Ho 0.3 MnO 3 as an example for SPC-induced frequency renormalization of the octahedron breathing mode B 1 g 2 ( ) (»610 cm −1 ). The former sample is in close vicinity to the multiferroic phase, the latter within the multiferroic region. Its orthorhombic distortion angle is located between those of the stoichiometric multiferroic compounds GdMnO 3 and TbMnO 3 . As shown in figure 1(e), the B 1 g 2 ( ) mode consists of breathing movements of the MnO 2 oxygen ions which are in-phase along the b-direction. Thus, this mode can be used to probe the in-plane FM magnetic correlations within the ac-plane. In contrast, within the B 1 g 3 ( ) mode the MnO 2 breathing movements are out-of-phase and, additionally, the apical oxygen ions move along the b-axis (figure 1(f)). As the interaction along the b-axis is AFM, this mode in conjunction with the B 1 g 2 ( ) mode allows to disentangle the FM from the AFM interaction strength as described in detail elsewhere [39]. Here, we will focus solely on the temperature-dependent frequency shift of the B 1 g 2 ( ) mode which is induced by the SPC, as the cycloidal order is oriented within the ac planes. The phonon peak positions in figure 1 are clearly temperature dependent. Upon cooling down from 300 K, the peak frequency is first increased due to the common reduced anharmonic decay, followed upon further cooling by a decrease, which can be attributed to SPC. This behavior is plotted quantitatively in figure 2, which shows the T-dependence of the B 1 g 2 ( ) eigenfrequency values for x=0.1 and x=0.3 in the insets of figures 2(b) and (c), respectively. For comparison, the corresponding data for stoichiometric EuMnO 3 , i.e., x=0, are plotted in the inset of figure 2(a). For the quantitative separation of the contribution of magnetic correlations to the temperature dependence from the intrinsic temperature-dependent phonon behavior due to the anharmonic decay of an optical phonon into two acoustic ones, we apply the anharmonic-decay-based formula [40,41]: also referred to as Klemens model. Here, )is the wavenumber of the phonon for  T 0 and C is a free parameter that describes the strength of the anharmonic decay. This formula is expected to fit to the experimental data in absence of the magnetic order. The deviation from this curve will be ascribed to SPC, i.e., a phonon frequency renormalization, caused by magnetic correlations. The fits according to the Klemens formula are shown in figure 2 as red curves. They describe very well the experimentally observed cooling-induced increasing eigenfrequencies data in the upper temperature range (see insets). However, for lower temperatures the model predicts a constant frequency, while the experimental data show upon further cooling a clear redshift.
The frequency renormalization strengths, i.e., relative frequency shifts of the B 1 Obviously, according to equation (4), SPC is not confined to a global cycloidal magnetic order. Therefore, it also occurs for non-multiferroics, such as EuMnO 3 (figure 2(a)) and Eu 0.9 Ho 0.1 MnO 3 ( figure 2(b)), which show A-type antiferromagnetic order [37]. It was shown before, that SPC is directly related to the distortion angle between adjacent octahedra and therefore reflects magnetic correlations at a local level [38].
A key result of our study is the observation of SPC as a clear deviation from equation (3) not only for the ordered phase below T N , but also far above the long-range-order temperature T N , up to about 100 K, indicating local spin correlations even at this elevated temperature. This is in contrast to an abrupt ordering of spins at T N or T cycl . There is no abrupt anomaly in the phonon behavior at both temperatures, which would correspond to a phase transition. We have obtained similar results for the multiferroics GdMnO 3 and TbMnO 3 , as well as for doped (Eu,Y)MnO 3 with various compositions [31,38]. The corresponding values of T N and the temperature ranges for the onset of SPC are listed in table 1. For all investigated compounds the temperature range of the SPC onset exceeds T N by far, for most of them by a factor of two or even more. Besides, we want to point out that the Raman data of several other research groups show SPC at  T T N generally for RMnO 3 compounds, e.g. for DyMnO 3 with = T 40 N K and » T 120 SPC K [32] and several others [32,42,43], but in none of these reports this observation is discussed in terms of Mn-spin order in this temperature range. In our opinion all existing results on the temperature dependence of SPC in RMnO 3 can be explained consistently by assuming a that this SPC is due to local, short-range order of the Mn spins, persisting far above the Néel temperature.

Electromagnon activity
The interpretation given above is supported by our THz data for electromagnons in GdMnO 3 as shown in figure 3. Interestingly, GdMnO 3 is a material with no clearly established cycloidal phase [24]. However, well defined electromagnons at Terahertz frequencies strongly suggest the existence of at least a disordered cycloid. In contrast to the similar compound DyMnO 3 , the disorder does not allow the observation of static electric polarization. In GdMnO 3 the electromagnons are observed up to 70 K as shown in figure 3(a). The intensity of the electromagnon modes has been estimated by fitting the data using a sum of Lorentzian modes and is given as the spectral weight of the modes, which is defined as where e 0 is the permittivity of vacuum, e D and w 0 are dielectric contribution and eigenfrequency of the Lorentzian, respectively. As demonstrated in figure 3(b), the extrapolation of the electromagnon spectral weight suggests nonzero spectral weight up to »100 K. This continuous decrease of the electromagnon intensity indicates the existence of spin cycloids until much higher temperatures than » T 40 N K.  3.3. Order-disorder limit: the pseudo-spin model A useful idea explaining the continuation of the magnetoelectric characteristics of multiferroic manganites into the paramagnetic phase is given by the pseudo-spin model [34]. In short, this model exactly reproduces the main results discussed above: (i) both the sinusoidal magnetic structure and the paramagnetic state below » T 100 K should be described as a mixture of the magnetic cycloids with opposite chiralities. (ii) The characteristic energy scale of the problem is not given by The pseudo-spin model [34] has been developed and applied for dielectric relaxation in DyMnO 3 and it is based on the relation between the spininduced electric polarization and the chirality of the magnetic cycloid [22,28]. Within this model, the polarization in the ferroelectric phase is proportional to the difference of opposite chiralities of Mn 3+ magnetic cycloids. Hence, the electric polarization can be considered as the primary order parameter of the system. In addition, an order-disorder type phase transition between paraelectric and ferroelectric states is suggested. The main idea of the pseudo-spin model is illustrated in figure 4. Here, (i) the electric dipole moments are associated with the displacement of the -O 2 ions due to inverse DM interaction [22,28], (ii) the direction of the electric dipoles depends on the chirality of the magnetic order, (iii) the two possible directions of the electric dipoles are separated by an energy barrier E A and are coupled. In the order-disorder limit ( > E k T 1 A B cycl [45,46]), the system can be described by the pseudospin formalism, the model Hamiltonian is essentially of Ising-type and may be written as: where Dx is the displacement of the -O 2 ion, ¢ J R R , is the coupling constant between -O 2 ions at position R and ¢ R , and s R is the pseudospin at position R with s = D x x R R , which can take the values +1 and −1 (figure 4), E A is the activation energy determining the characteristic temperature scale (the height of the energy barrier).
From the viewpoint of the order parameter dynamics the jumps between the minima of the potential wells in figure 4 are of main interest since this motion of ions implicates the flipping of pseudo-spins.
An analysis of the relaxation characteristics of DyMnO 3 near the magnetically induced ferroelectric phase transition which was performed according to the pseudo-spin model, confirmed an order-disorder type transition with an experimentally determined activation energy of » E 100 A K [34]. Thus, for temperatures < T E A , the ions reside in one or the other side of the double-well potential and a local cycloidal spin order is present due to IDM. On the other hand, for temperatures  T E A , the local potential does not force ions to stay on one side of the double-well potential. Instead, the ions vibrate around the averaged position, and the shape of the double-well potential has only a small effect to modify the phonon frequencies. Consequently, due to IDM interaction, the neighboring Mn 3+ spins must align collinearly (´= + S S 0 i i 1 ) and no local cycloidal spin order is present.

Summary
The analysis of the temperature dependence of the SPC strength in a series of various RMnO 3 compounds shows no abrupt disappearance at T N , but in contrast the persistence of local order of the Mn-spins up to » T 100 K. This is far above T N and even further above the phase transition at T cycl for the multiferroic representatives. This . Sequence showing the ordering of atoms in the order-disorder limit in a double-well potential model with a critical temperature T C . In our system, T C represents the global cycloid ordering temperature T cycl . At high temperatures (  T T C ) the positions of neighboring ions are not correlated in any significant way (magenta and yellow balls indicate ions in the left or in the right state of the double-well potential, respectively). Near the transition temperature a degree of short-range order is established. Below the transition temperature one side of the double-well potential is favored and a long-range order appears. Well below the transition temperature each ion occupies the same side of the double-well potential. observation is underscored by the temperature dependence of the spectral weight of electromagnon excitations in GdMnO 3 , which is explained by describing the evolution of the magnetic structure of a multiferroic compound with increasing temperature as a pseudo-spin system of two possible types of cycloids with opposite chirality, which locally persist up to around 100 K. Because of this local character, the temperature scale of 100 K is not to be understood as a phase transition, but rather as a gradual onset of local order. Thus, the phonons and the electromagnons serve as valuable quasi-local probes, revealing the development of the magnetic correlation between neighboring spins even in the absence of global magnetic order.