Coupled weak ferromagnetic-paramagnetic excitations in (Gd0.5Eu0.5)2CuO4

The electron paramagnetic resonance spectra of Gd3+ in single crystals of (Gd0.5Eu0.5)2CuO4 show an anomalous anisotropy for temperatures below the magnetic ordering of the Cu ions (T approximately=280 K). The authors have performed a detailed experimental study of this feature, and they discuss their results in terms of a model that assumes a weak Heisenberg interaction between Cu and Gd moments. The authors show that that anisotropy is a consequence of the dynamic coupling of the Gd3+ paramagnetic mode with a weak ferromagnetic mode originating in the Cu magnetic moments.


Introduction
The rare-earth cuprates, R2Cu04, are parent compounds of the electron-doped high T, superconductors, R,-,(Ce, Th), Cu04 [l]. These cuprates form in a tetragonal ( T ' ) crystal structure with Cu02 planes in which oxygen atoms are square planar coordinated about the Cu atoms [2].
Three different types of magnetic interactions have to be considered in the rareearth cuprates, namely the Cu-Cu, Cu-R and R-R interactions; since their relative magnitude differ considerably, their effects can be separated by measurements in different temperature ranges [3]. For temperatures above 50 K the magnetic properties are mainly due to the Cu-Cu interaction, leading to magnetic ordering of the Cu lattice around 280K. Below 50K the effects of the R-Cu and R-R interactions become increasingly important and must be taken into account in order to describe the magnetic behaviour of the system, including the spin reorientation transition around 15 K and the magnetic ordering of the R lattice for some of the rare-earths at lower temperatures [3, The WF component in (Gd,,,Euu,5)2Cu0, has been interpreted as being due to a combinationof both a cantingofthe Cu moments away from perfect antiferromagnetism, and a polarization of the paramagnetic rare-earth ions, which are coupled to the CuO, planes through an effective internal field of about 1 kGauss [3,4]. The Dzyaloshinski-Moriya (DM) antisymmetric exchange interaction was suggested to be responsible for the canting of the Cu moments [4,8].
This WF also introduces specific features in the microwave magneto-absorption spectra. In addition to the Gd3+ electron-spin resonance (EPR) lines, intense signals have been measured at low fields [3]. These low-field spectral lines were successfully described as ferromagnetic modes associated with the wF component of the Cu lattice, with a model that included the DM interaction [9]. On the other hand, a very unusual anisotropy of the EPR signal that appears below the ordering temperature of the Cu magnetic moments, first reported in [3]. remains to be understood.
In order to understand the observed behaviour, we have performed a detailed experimental study of the Gd resonance in single crystals of (Gdu.5Eu&Cu04. In section 2 we present these data. In section 3 we discuss these results in relation to previous data on the DC magnetization and the weak ferromagnetic resonance; we show that a static coupling between the Cu and Gd systems is not enough to understand the observed anisotropyoftheGd3+ EPR. Insection4 we introduceamodel forthismagnetic system. Themodelisbasedon theindependentexcitationsoftheCuweakferromagnetic and the Gd paramagnetic lattices, and a coupling between the Cu and Gd moments is included through a weak scalar exchange interaction. We show that coupled weak ferromagnetic-paramagnetic resonance modes can be derived from it and we give a quantitative account of the data in terms of these mixed excitations.

Experimental results
Single crystals of (Gd,,,Eu&CuO, were grown from PbO and CuO fluxes as described in [lo]. The crystals were thin platelets of typical size 3 X 3 x 0.1 mm3 with the c-axis (perpendicular to the Cu02 planes) parallel to the thinnest dimension.
The EPR experiments were performed in a 9.5 GHz spectrometer operating in the conventional derivative absorption mode. The experimental set-up was precisely mounted so that the external field (H,) could rotate from a plane direction ([ 1 lo]) to the c-axis. The crystals were mounted in the resonant cavity so that the time dependent microwaveexcitation fieldwaskept parallel to theCuOzplanes,andperpendiculartothe external magnetic field. These conditions are required for thesimultaneous excitationof the two resonance modes, i.e. the WF resonance and the Gd3+ EPR [9].
The microwave absorption spectrum of (G&.sEuu.s)~Cu04 at room temperatures, shows one single broad line (AHp,=2 kGauss) at g=2.0 (resonant field = 3350 Gauss) that can be assigned to trivalent gadolinium. For temperatures below 280 K. there is an extra strong absorption at H,.wF-100 Gauss. In figure 1 we show a typical spectrum, measured at 180 K; the low field absorption associated with a wFmodeand the paramagnetic resonancedue to the Gd'+ ionsareobserved. DPPH isused asamarkerk = 2.0). We willonlypresent heredatareferred to theGd3+ resonance.The properties of the low-field absorption have been extensively discussed in [3] and [9]. and we will only give a brief summary of the main features in section 3.
In figure 2 we show the temperature dependence of the resonant field assigned to the EPR of Gd", for two directions of the applied external magnetic field H,, one parallel to the CuO, planes ([llo]), and one nearly perpendicular to them (1" from the c-axis). Hr.Gd was observed to be isotropic in the Cu02 planes. There is a shift for Ho applied parallel to the Cu02 planes, that starts at T =_ 280 K and saturates at a field (450 ? 20) Gauss lower than the g = 2.0 value. A quite different behaviour is observed for H , perpendicular to the CuO, planes, where a rapid change is developed, also at T 3 280 K, shifting the resonant field to extreme high values as the temperature is lowered. This behaviour can be analysed from the angular dependence of the resonant magnetic field as Ho rotates from [110] to the c-axis at a fixed temperature. In figure 3 we show spectra taken at 184 K for different external magnetic field angles 0 (0 = 0 for the [llO]direction). Both magneticresonancesareshown, the wFmodeand the Gd3+signal; it can be seen how both spectral lines shift to higher magnetic fields as the direction of H , approaches the c-axis, merging together in a single broad absorption for 0 close to 90". compared with HDppH (the resonant field corresponding to the DPPH marker) as a function of the angle 0 for two temperatures, namely 344 K and 184 K. As shown in figure 2, Hr.Gd is nearly independent of angle for T > 280K, whereas for T < 280 K an extreme out-of-plane anisotropy is observed. As previously mentioned, is shifted 450 Gauss lower than In figure 4 we show the shift of the resonant magnetic field 1225L K v: 9.5 G Hz ~~~~ ~~~ ~~ ~~~ ~~~ . HDppH for B = 0, it reaches the value of HDPPH for 0 75" and continues increasing as B approaches 90". Within a few degrees from the c-axis (k2') this behaviour changes and theresonant field drops, defining an asymmetric peaked structure (see inset in figure 4).
As seen in figure 3, another signature of the Gd'+ EPR is the strong variation of the intensityofthespectralline with themagneticfieldangle B.Infigure5weshowtheangle dependence of the integrated intensity, measured at 184 K. An increase is observed as 0 approaches 90". except for a very narrow range around the c-axis where the intensity drops in a similar way as observed for the resonant magnetic field (see the inset in figure   5). Again, an asymmetric peaked structure is defined around the c-axis.
In figure 6 we show the temperature dependence of the spectral line intensity, for B = 0" and 0 = 89". Whereas for Ho parallel to the Cu02 planes there are no important changes in the intensity as the temperature is lowered, for Hu near the c-axis there is an abrupt increase starting at T 1 280 K. This shows that for temperatures above 280 K the intensity is isotropic, while angular dependence, as shown in figure 5, appears as the temperature passes through 280 K.

Analysis of the data
In order to discuss the features of the Gd3+ magnetic resonance, we will first briefly review previous data on the magnetic properties of (Gd, .xEu,)2Cu0, solid solutions, namely the DC magnetization and the weak ferromagnetic resonance.
Below T = 280 K weak ferromagnetism appears: MDc presents a non-linear behaviour as a function of the applied magnetic field, Ho. Its component parallel to the CuO, planes, MI,, shows an initially fast increase as a function of field and, above a relative We assume that the presence of Eu has no determinant effect on the observed behaviour of the Gd3' EPR. In fact, similar experiments in Gd,CuO, show exactly the same features. We have chosen (Gdo,sEuo.5)2Cu04 for the presentation of the characteristics of the Gd3' EPR spectra because rhe spectral lines are narrower than in Gd2Cu0, and so the features we want to stress are more clear in order to compare them with the model in section 4.
The angular dependence of MDc in single crystal measurements suggests that the internal field HI,,,, is constant and orientated within the CuOl planes. Besides, from the angular dependence of H,, it can be deduced that only the component parallel to the CuO, planes of the applied magnetic field is effective in the process of developing the measured weak ferromagnetic component.

Weak ferromagnetic resonant mode
The main featuresof this resonant mode are [3,9] asfollows. This resonance was described in [9] as a weak ferromagnetic mode, associated with the Cu system and characterized by the following resonance equation, first derived by where y is the gyromagnetic factor appropriate for the Cu moments, and HDM is the socalled Dzyaloshinski-Moriya field, that phenomenologically describes the effect of the antisymmetric DM interaction. The cos(0) factor in (2) arises from a strong anisotropy energy term favouring the orientation of the Cu moments within the Cu02 planes, and is the origin of the l/cos(0) dependence in Hr.wF.
In order to obtain coincident values of HDM from DC magnetization and the analysis of the WF mode, it is necessary to introduce the coupling to the Gd lattice through an effective field 'seen' by the Cu moments. HDM was estimated to be of the order of lo5 Gauss [9].
Based on the above results we will now discuss the characteristics of the Gd3+ EPR; we will analyse separately three features of the data, i.e. the resonant field H,.Gd, the integrated intensity of the spectral line, and the near c-axis region defined in the angular variations.

The resonant magneticfield
Above 280 K, the resonant field is nearly isotropic and corresponds tog = 2, in agreement with the magnetization data. Below 280 K, specific featuresappear associated with the weak ferromagnetism. DC magnetization measurements show that an internal field Hi.Gd, due tothecusystem, isactingalong theCu0,planesat eachGdsite. Thisinternal field is expected to produce a shift of the resonant field. The observed shift of at 0 = 0 (450 Gauss) is in the right direction but is significantly smaller than the 800 Gauss obtained from DC magnetization measurements. On the other hand, it can be expected that the value H, -HoppH would go to zero at 0 = 90"; instead, a divergent behaviour is observed as the direction of Ho approaches the c-axis. This divergence can be represented by a l/cos(b') behaviour near the c-axis, this feature being a signature of the WF mode. It should be noted that the hypothesis that leads to the angular dependence of Hr,WF in (2), namely the anisotropy that favours the orientation of the Cu magnetic moments parallel to the CuO, planes, cannot be used here since the Gd3+ ions are in a paramagnetic state and follow the magnetic field.
It is clear that the anisotropy of the Gd3+ resonance has to be related to the magnetic ordering of the Cu moments, but a staticcoupling, represented by the internal field Hi,Gd obtained from the DC magnetization, does not explain the observed features.
Angular dependence of the transition probabilities can appear in systems with a high crystalline anisotropy, but this can be ruled out in our case since, above the temperature of the magnetic ordering of the Cu moments, the integrated intensity is isotropic.
The number of Gd3+ ions is constant and their magnitude depends on temperature with a Curie-Weiss law; hence it would be expected that there is a l/Tdependence of the intensity, as in fact is observed when Ho is applied parallel to the CuO, planes, but not the strong anisotropy that appears below 280 K, shown in figures 5 and 6.
An internal field would only shift the resonance, but w'ould not produce an angular dependence of the intensity; hence, another mechanism associated with a coupling between the Gd and Cu magnetic lattices must be considered.

The central region
In the angular dependence of H r .~d (see figure 4) and the angular dependence of the integrated intensity (see figure 5) below 280 K, a centre region is defined around the caxis (6 = 90" t 2') where both Hr,cd and the intensity change their overall behaviour.
Two features related to this centre region must be analysed, namely the drop for 6 3 90", and the asymmetry of the formed peaked structure. Althoughnot evident, themechanism that givesrise to theobservedanisotropymust be related totheappearanceofweakferromagnetismat280 K.Hence,thedropobserved when the direction of H o is nearly parallel to the c-axis, can be interpreted as a consequence of there no longer being a large enough component of the field applied in the plane to 'set' fully the weak ferromagnetism.
We do not have a complete explanation for the origin of the asymmetry, but at least two mechanisms have to be considered: (i) the non-perfect alignment between the external magnetic field Ho and the c-axis, as 8 passes through 90"; and (ii) the angular modulation of the absorption. Angular modulation arises when Hoand the modulation field used for lock-in signal detection are not aligned. It can be important in very anisotropicresonancesI 131andsinceit dependson theangularderivativeofthe resonant field, aH,(@)/ae, its contribution to the total intensity of the signal must change sign as the c-axis is crossed by Hu. In order to evaluate the importance of this experimental feature, we have repeated the same measurements changing the angle q between Ho and the modulation field. We observed that, although the general characteristics of the angular dependence of H,,,, and the intensity do not change, the detail of the central part, i.e. the relative magnitude of the peaks. is sensitive to changes assmall as loin v.

Theoretical model
The anomalous behaviour described above can be explained in terms of a coupled mode of the Gd and Cu magnetic lattices. Mode mixing results in new excitations analogous to polaritons or magneto-elastic modes [14,15]. Figure 8 shows the expected angular dependence of the individual Gd and Cu excitations and the mixed modes derived from them. The resonant fields for the individual modes, and hence their resonant frequencies in a given field, are well separated when 6 is close to 0" or 180" but become degenerate at some angle close to 90' . We may therefore expect the modes to preserve their independent character when the field lies inor near toa [llO]direction,whereasstrongcouplingwilloccur whenthe fielddirection is close to the c-axis.
In what follows, we will show that this resonant coupling of modes can be derived on the basisof WeaklyinteractingGdand Cumagneticlattices,coupled throughanisotropic Heisenberg Hamiltonian.
We will consider not the individual magnetic moments but the macroscopic magnetization fields. of the Gd paramagnetic lattice (MGd), and of the Cu WF lattice (Mcu). The macroscopic field theory is specially appropriate for magnetic resonance [14], where. since the excitation field H I ( [ ) = H , eIW'is uniform in space. only k = 0 magnons can be excited.
We propose a zero-order energy that describes the system, where &, i s the equilibrium energy, ct 10) is a state of one k = 0 W F magnon with energy given by (2) and g' 10) is a state of one paramagnetic excitation with energy wpM = yHO (we take fi = 1). For the coupling between the Gd and Cu magneticmoments we assume We define the mean internal fields Hi,Gd = AMc, as due to the copper moments acting on theGdsites, andH,,,. = M G d o f the Gdmagneticmomentsactingon thecumagnetic moments. Hi,Gd and MGd can be obtained from DC magnetization measurements. The constant A can be related to the exchange constant Jc-.Gd on a molecular field basis. If we assume that Gd ions interact mainly with the four near neighbour copper ions, and since M, , = 2 x 10-3pB/Cu-atom and Hi,Gd 3 800 Gauss [3], we can estimate JCu.Gd/ Equation (4) can be written in terms of creation and annihilation operators using the Holstein-Primakoff transformation [14]. In that case, each magnetization must be referred to a different frame of axes: we assume that disregarding small perturbations Mcu is parallel to the CuOz planes, while MGd follows the external field as it moves away from the planes to a direction perpendicular to them (see figure 9). The total energy %e = X 0 + Xiot can then be written as the sum of the following terms, k~ 30 K. Although, in principle, it is possible to obtain the eigenvalues of the entire energy X, more insight on the physical meaning of the solution can be obtained by separately analysing each term in (5).
The diagonal part Xdd includes the energies of the independent modes shifted by the internal fields, as expected from a mean field static approach.
The second term X i with non-diagonal terms cg+ and c+g, corresponds to the annihilation of one type of excitation with the simultaneous creation of an excitation of the other system. These terms are typical of resonantly coupled modes; Xdd + X , t . can be diagonalized with the transformation (Y' = ug' + uc+ and /3' = ug' -uc+ with U and U real numbers, and where n'10)and/3+lO)correspond to mixed states. The energies of these mixed excitations are w e = (WGF + w&4)/2 2 { [ ( U & -wbM)/2]' + U:}"* (6) where the U * are the independent frequencies shifted by the internal fields, and wI = (.Ay/2)(MCuMGd)'!*(1 + cos (6)) is the interaction term that couples both modes. The value of wI is always non-zero since 6 vanes only from 0" to 90". Even for small values of w , it cannot be considered just as a perturbation since, as can be seen from (6). wI has to be compared with the difference of both diagonal frequencies, Am*. In fact, when U & = U&, a gap opens with an energy difference of 204. In this way, two separate branches are obtained as shown in the schematic diagram in figure 8.
This resonant coupling can also be seen analysing the eigenvectors; the constants U and U that diagonalize Xdd + X i , are If the frequency difference Aw* is much larger than U' then A 5 0 and there are two solutions for each (Y or /3: U = 0 and U = 1 or U = 1 and U = 0, i.e. the eigenstates correspond essentially to the independent excitations. If Aw* = 0 then u2 = U ' = 1/2 and the eigenstates LY and /3 correspond to a mixing of the paramagnetic and weak ferromagnetic modes, both with the same weight. The lower branch in figure 8 is weak ferromagnetic-like when 6 is close to 0" or NO", and becomes paramagnetic-like for 6 90", while the upper branch, that we associate with the Gd3+ EPR. is paramagneticlike when the field is near to a [110] direction, but becomes weak ferromagnetic in character when the field direction is close to the c-axis. The third term of (4). X? , , with non-diagonal operators cg and c+g' corresponds to the simultaneous creation or annihilation of both types of excitations. It is zero for 0 = 0. corresponding to the ferromagnetic alignment of both magnetic systems, but similarly to the case of antiferromagnetic magnons [14] they show that for a nonferromagnetic situation the Nee1 state is not the exact ground state and point-zero oscillationsappear. It can be diagonalized with using theBogoliubov transformation a = uc -ug+ and / 3 = ug -uc+. The diagonal energies in this case are with wI, = (Ay/2)(McuMGd)'~2(1cos(@)). Thecase now . isdifferent from the preceding w * = -wFM)/2 + { ( ( W G F + w;M)/2]2 +w?,}'" (8) one; wII is zero for Ho in the plane and it can be considered as a perturbation since it has to be compared with the sum of the independent excitation energies. In fact, when wII Q + U$, (8) can be expanded to give Equation 9 shows that the two energies are lowered by the same amount, and that the shift can be considered as a perturbation. The eigenvectors can be analysed calculating the numbers U and U ; it can be shown that the new states (Y and / 3 differ from the independent excitations c and g also in order O ,~/ ( W ; ,~ + w ;~) .
The last term of ( 5 ) generates only a shift in the constant part of the total energy H and does not need to be considered in order to describe the excitations of the system.
Since we have shown that H I , can be taken asa perturbation, and that the reduced energy H = Hdd + H I 1 contains the necessary elements to describe the coupling of modes, in what follows we will only consider these tn'o terms for a more quantitative approach.
We first consider the case for Hu parallel to the C u 0 2 planes, i.e. B = 0". If, in (6).
we assume that Aw* 9 wlr the frequency of the paramagnetic mode corrected by the interaction can be shown to be

@/Y HO + H L G d -Hi,CuHi,Gd/([HO(HO + HD)]"' + H~. C U -H U -~~. G~) . (10)
Equation (10) shows that the resonant field will be shifted not only by Hi,cd as expected from a static approach, but that there is also a shift that makes H,,,, go to higher values and is originated in the dynamic coupling of the two modes. If we assume that this is the origin of the difference between the measured shift of Hr,Gd for B = 0 (450 Gauss) and Hi,Gd (800 Gauss), we can obtain from this model a value for HDM since all other terms in (6) or (10) are already known. From our data at T = 184 K we get H,, 2 lo5 Gauss, whichisinagreement with the valueestimatedfromocmagnetization and the weak ferromagnetic resonance.
With this value for HDM, the resonant field of the signal assigned to the Gd3+ EPR can be calculated for any external field angle B. from (6), with no free parameters. In figure   4 we show the experimental results for the angular dependence of this resonance at T = 184 K, with the theoretical curve deduced from this model (full curves); the agreement is remarkable, reproducing the observed divergent behaviour. A quantitative estimation of the angular dependence of the integrated intensity of the spectral line can also be obtained from our model. Since the eigenstate we are exciting is defined by ol+lO) = ug+/O) + uc'lo), with U and U given in (7). the intensity of the absorption due to the coupled mode can be obtained as z,(q = u2rGd(e) + u2rc,(e) where IGd(0) and IC"((?) are the intensities of the absorptions expected for the independent modes.
Following the discussion in section 3, we assume that IGd(B) = fGd(0), with IGd(0) the value of the intensity assigned to the Gd'+ EPR at 6 ' = On, where @+lo) On the other hand, we do not have a model to describe Ic,,(B). Nevertheless, a phenomenological argument can be given. based on the observed angle dependence of the WF resonance [9]. In fact, when this resonance is clearly separated from the Gd" EPR line, its peak to peak linewidth varies with Bas AHpp(B) = AHpp(0)/cos(B), while its amplitude, k , is nearly constant. The integrated intensity is proportional to hAHZp; g' 10).
In figure 5 we show the experimental results for the angular dependence of the intensity of the spectral line assigned to Gd3+ at 184 K, with the theoretical curve calculated this way (full curves). Although this model is admittedly oversimplified, the overall behaviour is correctly reproduced. It is clear that a more detailed description of lcu(0) is needed for a precise determination of the expected intensity for the coupled modes. Moreover, the angular variation of the measured intensity due to angular modulation of the absorption, as mentioned in section 3, was not taken into account. Experimental errors in the determination of I J O ) which originated in the superposition of the two observed modes must also be considered.

Conclusions
We have presented a detailed experimental investigation of the anomalous behaviour observed on the magnetic resonance assigned to Gd3' in (Gdo,5Eu,,,)LCu0,.
We introduced a model for this magnetic system, that describes these anomalies in terms of a new type of magnetic excitation, i.e. coupled paramagnetic-weak ferromagnetic modes. All of the presented unusual featurescan be explained considering the following.
(i) The difference bet-zeen the resonant field with Ha parallel to the CuOz planes and the expected value including the static internal field H i ,~d is interpreted as being due to a dynamic coupling of the modes. From this analysis we obtained H,, 3 lo5 Gauss, which is in agreement with previousresults. In fact, it could have been taken as an input to the model.
(ii) The l/cos(O) behaviour of the resonant field Hr.Gd is a signature of the weak ferromagneticcharacterofthe upper branchof the mixedexcitations when O approaches 90".
(iii) The anomalous angular dependence of the intensity of the absorption is a consequence of this mixing. Since the excitations are linear combinations of the independent modes, not only the individual transition probabilities must be considered but also the weight with which each mode enters the mixed state.
(iv) The decrease of H,.td and of the intensity, for O = go", can also be understood with this model: since for H , parallel to the c-axis there is no weak ferromagnetic magnetization, the independent Gd3+ excitation must be recovered.
A final test for the model would be the observation of the complete lower branch of figure 8, that is, the WF mode saturating at a resonant field near the g s 2 value. This is not possible in (Gdo,5Euo,5)LCu0,, nor in Gd,CuO,: broadeningof the spectral line and superposition with the Gd3* EPR prevents a precise determination of the resonant field for angles nearer than 10" from the c-axis, and hence critical effects due to the dynamic coupling of modes are not evident.