Magnetic phase diagram and ordered ground state of GdMn2O5 multiferroic studied by x-ray magnetic scattering

The magnetic structure of multiferroic GdMn2O5 is studied by x-ray magnetic scattering, both off resonance and in resonant conditions at the Gd LIII edge. Temperature dependence of magnetic Bragg reflections shows an initial incommensurate (ICM) ordering appearing at T~40K with magnetic propagation vector kICM ~(0.49 0 0.18) which condenses at T~33K in a commensurate (CM) wave vector kCM=(½ 0 0). In the CM phase Gd3+ ions appear to order spontaneously at the same temperature as Mn ions. Azimuthal scans allowed for the precise determination, from x-ray magnetic scattering only, of the orientation and magnitude of the Gd3+ magnetic moments. The Gd3+ magnetic isotropy allows for the Gd ions to mimic the Mn4+ magnetic sublattice via Gd-Mn-Gd superexchange interactions.


Introduction
Spin-driven ferroelectrics have attracted much interest due to the possibility of controlling their electric (magnetic) polarization by applying a magnetic (electric) field [1,2]. Over the last few years, the experimental and theoretical studies of very diverse materials allowed to identify several magnetoelectric coupling mechanisms at play, driven for example by symmetric and antisymmetric magnetic exchange terms or single-ion effects. In the so-called symmetric exchangestriction model, polar displacements are due to a structural relaxation induced by competing Heisenberg terms (J ij S i · S j ) in the magnetic Hamiltonian. This mechanism can lead to large electrical polarization (P) in type-II multiferroics (compared with that of proper ferroelectrics such as BaTiO 3 is still minuscule) and has been proposed in the case of the RMn 2 O 5 compounds (R=rare-earth, Y,Bi) [3], Ca 3 CoMnO6 [4] and the manganite E-phase [5]. Of importance here is that this model maximizes P for commensurate magnetic order and collinear spins.
The RMn 2 O 5 compounds have been extensively studied since the seminal work on the Tb analogue showing that P can be reversed by an applied magnetic field [6]. For this class of materials, it is now accepted, after much debate, that the leading mechanism is related to symmetric-exchange striction induced by magnetic ordering of the Mn. A universal feature of the magnetoelectric phase diagram (for R=lanthanide) is the complex sequence of phase transitions on cooling below ∼ 40K [7]: first a paraelectric phase with incommensurate magnetic order, followed by a ferroelectric phase (b polar-axis) with commensurate magnetic order (wavevector k=(0.5 0 0.25)), and a ferroelectric phase with incommensurate order where P is much reduced. Induced order of the R moments has been observed by scattering techniques in all phases, followed below ∼ 10K by a proper ordering that has however limited impact on the electrical properties.
In the present article, we report on the detailed magnetic structure of GdMn 2 O 5 and on the use of x-ray magnetic scattering to fully determine the magnetic structure in this compound. This work is also motivated by the absence of spin-orbit coupling at first-order for Gd 3+ (4f 7 electronic configuration) and its larger ionic radius with respect to known members (Tb, Ho, Er, Dy, Tm) and by the reported observation of ferroelectric behaviour together with a large spontaneous polarization (P b >3600 µC/m 2 ) [8,9] at low temperature. By combining x-ray magnetic scattering (XMS) data, collected off-resonance and at the Gd L III -edge, we derive the complete magnetic phase diagram. We show that an incommensurate phase with wave-vector k=(0.486,0,0.18) below T N 1 ∼40K uniquely collapses to a commensurate phase k=( 1 2 ,0,0) below T N 2 ∼33K, without any further change of wave-vector below T N 2 . Unlike any compounds of the same series, magnetic ordering of the Gd site follows a proper order parameter below T N 2 suggesting that this ion is actively involved in the radical change of k. From the quantitative analysis of the azimuthal dependence of the XMS signal, collected off-resonance and at the Gd L III -edge, we propose a model of the Gd and Mn ordering in the commensurate phase and are able to explain why this specific magnetic configuration will induce a much larger electric polarization than for any related compounds. In addition we report the investigation of the magnetic domain topology of GdMn 2 O 5 as function of electric field. We show maps of the crystal's surface in the ac plane on electric field switching along the b crystallographic direction. These maps evidence the evolution of the magneto-electric domains as function of the applied electric field.

Experimental
A high quality single crystal (1.2x0.9x0.5mm 3 ) of GdMn 2 O 5 grown using floating zone method [10] was used for the x-ray magnetic scattering experiment, performed at the I16 beamline (Diamond Light Source Ltd.) [11,12]. The sample was mounted on the beamline Kappadiffractometer in a closed circle refrigerator with base temperature of ∼ 5K. The crystal was oriented with the diffraction face perpendicular to the (110) direction (Fig. 1). The azimuthal angle reference (ψ=0) is defined when the (100) reflection is in the scattering plane and its projection onto the incoming beam parallel to it. The diffractometer was operated in the vertical-plane scattering mode with an azimuth setup to allow for a 360 • sample rotation about the scattering vectors. In this geometry the natural polarization of the incident beam is perpendicular to the scattering plane (σ). The integrated intensity of the reflections was measured using either a Dectris Pilatus 100K area detector and/or an avalanche photodiode point detector. In the latter case, the polarization of the reflected beam was linearly analyzed by rotating the scattering plane of a mosaic crystal chosen to have a Bragg angle close to 90 • at the energy of interest (Al (220) at 6.4 keV, and Au (222) at 7.2455 keV). The measured Bragg intensities have been corrected for self-absorption and Lorentz factors, multiplying them by the factor sin β(Q,ψ) ·sin 2θ(Q) where α(Q, ψ) and β(Q, ψ) are the incident and exit angles with respect to the crystal surface. To separate the contribution of the Gd from the Mn one, the photon energy was tuned close to the E res (Gd L III -edge)=7.2455 keV. In this condition a very large (factor ∼40) enhancement ( Fig. 2) of the magnetic scattering was observed which allowed to ascribe the observed signal entirely to the Gd magnetic moments. A different experimental configuration has been used to collect magnetic domain maps and apply an electric field along the b direction. The sample was rotated and mounted on a suitable sample holder with the ac plane in scattering conditions. Diamond phase plates have been used to select circular right and circular left light polarization of the incident x-ray beam and the beam size of 195x32 µm 2 has been reduced, using slits, to a ∼50x50 µm 2 beam footprint on the sample surface. A Cu220 crystal has been used to analyse the scattered light in off-resonant conditions at 6.4 keV.  Fig. 2, has been obtained by sampling a large area of reciprocal space using an area detector. Additional Bragg reflections appear below 40K (red triangles in fig. 2), and can all be indexed by the incommensurate propagation vector k ICM ∼(0.49 0 0.18). It is important to note that the observation of two peaks at (k x ,0,k z ) and (1-k x ,0,k z ) confirms that the structure is also incommensurate along x, albeit the deviation from 0.5 is small. Below T N 2 ∼33K, k locks at the commensurate value k CM =(1/2,0,0). In the commensurate phase in the vicinity of T N 2 , the temperature dependence of the magnetic peaks intensities, shown in Fig.2, are adequately fitted with a power law (critical exponent 2β since the intensity is proportional to the square of the magnetization). The critical exponents measured in non resonant and resonant conditions are identical within the experimental error, respectively β=0.26±0.02 and β =0.29±0.03 [13]. This critical behavior indicates a unique order parameter with contribution from both the Gd and Mn magnetizations. This is in contrast to the observed induced magnetic ordering (secondary coupled order parameter) of Ho, Tb and Er observed in other compounds of the same family [14,15]. The azimuthal dependence of five magnetic reflections measured in resonant conditions at T=5K ( fig. 3(a)) has been used to derive a model for the magnetic structure of the Gd sublattice. In the dipolar approximation of the resonant case, the scattering is observed only in the rotated channel (σπ ) and the scattering amplitude f for a magnetic Bragg peak at momentum transfer Q and azimuth ψ is given by [16]:

Results and Discussion
considering only the lowest order term, where k' is the scattered wave-vector and S Gd the Gd magnetic structure factor. The technique probes indirectly the magnetic order of the f-electrons through the polarized Gd d density of states, as the d-f hybridization is allowed by the absence  of inversion symmetry at the Gd site. The azimuthal scans present a two fold periodicity with maxima at positions close to ψ=0 and 180 • , which given the azimuthal reference in the (100) direction, indicates that the Gd moments are approximately aligned along the crystallographic a-axis. The Gd sublattice magnetic configuration and its magnetic symmetry were determined empirically by least-square refinements of all azimuthal scans considered simultaneously. There is a unique solution for which the Gd moments (labelled 1-4 in Fig. 5) are oriented in the ab-plane and related by the time-reversed two-fold rotation axis along b. 1 A further refinement imposing this symmetry restriction lead to excellent results as shown in Fig. 3(a). For the magnetic structure stabilization, given the order parameter in the special direction (a,0), two irreducible representations (X 1 ,X 2 ) are allowed. The full magnetic space group found, which corresponds to X 2 , is P a b2 1 a (P a ca2 1 in conventional International Tables for Crystallography settings). The proposed symmetry allowed magnetic mode (out of the six allowed ones spanning X 1 and X 2 ), is uniquely consistent with Gd moments in the ab-plane and a ferroelectric axis along b, as observed experimentally. With this symmetry, only the moments on site Gd1 and Gd2, on one hand, and Gd3 and Gd4, on the other, are related by the two-fold rotation, while the two sets are unrelated due to the loss of inversion symmetry. Finally, temperature dependent (T=5, 15, 25, 31K) energy scans of the resonant signal and of the azimuthal scans (not shown) suggested an unchanged Gd magnetic configuration. The complete magnetic structure, Mn and Gd magnetic ordering and the relative phase between Gd and Mn modulations, was probed using non-resonant magnetic scattering (NRMS). In this case, the azimuthal dependence of the scattering amplitudes [16,17,18] in the σσ and in the σπ channels used to fit the data are: where k is the incident wave-vector and S is the magnetic structure factor including Gd and Mn contributions. Due to the long collection time, only a single off-resonant azimuthal scan could be collected in the full azimuthal range, preventing to perform an unconstrained refinement including all Mn moments. Instead, the magnetic configuration of the Mn in the ab plane was fixed to that found for other commensurate structures of the RMn 2 O 5 compounds, the main difference being caused by the different magnetic periodicity along c. As shown in Fig.  5, Mn 4+ -Mn 3+ -Mn 3+ -Mn 4+ form zig-zag AFM chains along the a-axis, and the moments are tilted by ∼20 • with respect to the crystallographic a-direction, pointing along the axis of the pyramidal Mn 3+ site. Such configuration on its own does not account well for the observed off-resonance signal, in particular under-calculating the intensities in the σπ channel. However, by including the Gd contribution derived from the RMS work, the off-resonance scan can be adequately fitted, as shown in Fig. 3. Despite the fact that the relative orientation of the Mn moments with respect to the Gd moments could be inferred from the NRMS data since the total magnetic structure factor is sensitive to the relative phase, the lack of available data (only one scanned reflection) does not allow to uniquely discern between a ferromagnetic or antiferromagnetic Gd-Mn alignment. Preliminary neutron scattering performed on a recently isotropically substituted GdMn 2 O 5 crystal confirms, as reported in Fig. 5, the Gd moments are arranged almost antiferromagnetically relative to the neighboring Mn 3+ moments (pyramidal sites). By supposing that the Mn ordered moments are saturated at the spin-expected value of 3µ B (octahedral site, S=3/2) and 4µ B (pyramidal site, S=2), one obtains, from the NRMS data, an ordered moment for Gd 3+ at 5K of ∼ 5.14(4)µ B and ∼ 4.75(4)µ B for Gd1/Gd2 and Gd3/Gd4 respectively. The Gd ordered moment is much larger than that found for R=Tb or Colour maps representing inversion symmetry related magnetic domains (respectively ratio > 1 red, ratio < 1 blue). Pixels size is 50x50 µm 2 and the temperature is 5 K. Colours show the ratio between right and left circularly polarised light, corresponding to two different magnetic domain populations [19]. Ho.
In order to observe the magnetic domains and their evolution upon application of an external electric field, the sample was rotated and electrodes were applied close to the crystallographic b direction. The sample has been aligned with the [0.5, 0, 2] reflection in scattering conditions and it has been cooled to 5 K, with electric fields of + and -1100 V/mm. The ac surface have been mapped by using incident circularly polarised light and by analysing the polarization of the scattered signal [19]. The ratio between circular right and circular left light has been measured in non-resonant conditions and plotted as function of the position on the sample surface. The resulting magnetic domain maps are shown in figure 4(a) and 4(b)). The data show that the magnetic domain populations was partially switched with the reversal of electric field but not fully reversed. We believe the combination of ∼ 25 • tilt of the sample b axis with respect to the electric field direction and the maximum electric field generated by the Keithley electrometer used during this experiment wasn't enough to fully control the magnetic domain population in this compound. The domains size appear to be of the order of ∼ 100 microns and the larger ones (top left and centre right of fig 4, appear to act as pinning domains, blocking the domain population changes in the nearby regions.

Conclusions
The magnetic structure stable below T N 2 of GdMn 2 O 5 is the simplest of the series, and yet supports ferroelectricity along the b crystallographic axis. Although GdMn 2 O 5 shows similarities with respect to other RMn 2 O 5 compounds in the most polar phase, such as a locking of the propagation vector at half along the x direction, with the formation of AFM chains along the a-axis with constant moment amplitude, this is the only system for which k z =0. The direct consequence of this periodicity is the production of a FM stacking along c of the adjacent AFM planes. This effect can be explained by two phenomena: firstly, eight-coordinated Gd 3+ has a Since longer Mn-Mn inter-atomic distances promotes direct exchange (in manganese metals the crossover between AFM and FM is ∼2.83Å), the argument seems to hold for GdMn 2 O 5 , where the Mn-Mn distance is 2.89Å. There is however a notable exception for BiMn 2 O 5 which, despite an even larger ionic radius on the R site, displays antiferromagnetic stacking (k z = 1/2). Secondly, based on the absence of first-order orbital momentum on the Gd site (4f 7 electronic configuration), it can be assumed that magnetism on the Gd site is very isotropic, and that the Gd moment direction will simply align along the spin of its strongest interacting neighbor.
Since the Gd moments are nearly collinear with the first neighbor Mn 3+ spins (Gd-Mn distance ∼3.303Å), the latter explanation is in excellent agreement with experimental observations. The situation is very different in analogues with non-quenched orbital momentum displaying noncollinear arrangements of the R moments [10]. Due to this simple commensurate structure, unlike in any other ferroelectric RMn 2 O 5 , in GdMn 2 O 5 the rare earth exhibit a large ordered moment in every layer. This unique magnetic configuration has important consequences for the ferroelectric behavior: since the Gd spin configuration on its own breaks inversion symmetry, one expect a finite contribution to the polarization along b from the coupled polar ionic displacements allowed by symmetry on the Gd site and coordinated oxygens. In the presence of a large ordered moment on Gd, one can anticipate a large contribution to the total polarization, provided that all polar displacements (Mn,O,Gd) are in phase. In this scenario, the Gd-Mn symmetric exchange striction, together with the Mn-Mn exchange striction mechanism, could lead to very large ferroelectric polarizations, as experimentally observed [9]. Ab-initio calculations, that will be facilitated by the small magnetic unit-cell, should shed light on the respective contributions of different ions to the total electric polarization, and separate the part imputable to the Gd magnetic ordering.