Magnetically-Induced Ferro electricity Unraveled Through Spin-Phonon Coupling

Magnetically-Induced Ferroelectricity Unraveled Through Spin-Phonon Coupling Rui Miguel Abreu Vilarinho da Silva Thesis submitted to the Faculty of Sciences of the University of Porto in partial fulllment of the requirements for the degree of Master in Physics Supervisor: Prof. Joaquim Agostinho Moreira Department of Physics and Astronomy Faculty of Sciences of the University of Porto September 2013 . Aos meus pais . Acknowledgements Firstly, I would like to thank my supervisor, professor Joaquim Agostinho Moreira, for inviting me as his student and performing with excellency his orientation functions, with enormous dedication, always available at any day or time. I am thankful to professor Abílio Almeida, that constantly orientated me all along this work and helped me countless times. I thank professor Pedro Tavares, for the processing of every high quality samples used in this work, along with its initial characterization and quality guarantee through x-ray diraction, SEM imaging, x-ray photoelectron spectroscopy analysis and infrared spectroscopy. To professors Maria Armanda Sá and José Brochado Oliveira for the measurements of the specic heat, along with the supplying of the data analysis program. A special remark for my research colleges Daniel Mota and Yonny Romaguera Barcelay, who always presented themselves available for any kind of needed help. To the fellow IFIMUP-IN researchers, thanks for bringing me into all those stress-releasing football matches. Before reaching this closure, long years have passed within this faculty, and my thanks to every friend that I had the pleasure to make, particularly the ones I met in 2006, each one helping in its very own way. A special and profound thanks to the She-Who-Prefers-Not-To-Be-Named, without whom this work would never be as good. A true work inspiration and a very, very, very good (and patient..) listener, You-KnowWho! ¡ 5 . Abstract This work aims at studying the role of the spin-phonon coupling in the stabilization of magnetically-induced ferroelectric phases in magnetoelectric materials. To achieve this objective, we have characterize the thermodynamic, dielectric, magnetic, polar and magnetoelectric properties of Gd1−xYxMnO3 system, with 0≤x≤0.4 . The overall set of experimental results enable us to draw the ( x , T) phase diagram of this system and to elucidate the eect of an applied electric eld on the magnetic properties of the compounds. An induced electric polarization was observed in GdMnO3 , which is not associated with any kind of cooperative phenomena. For x≥0.1 , antiferromagnetic phases are stabilized at low temperatures, and some of them simultaneously exhibit polar properties. Although only the lowest temperature magnetic phase is polar, a magnetoelectric coupling was evidenced for higher temperatures. Moreover, the study of lattice dynamics through Raman spectroscopy reveals the existence of a spin-phonon coupling even in the paramagnetic phase. This result points for the existence of precursor eects of the low temperature magnetic phases. Clear deviations of the Raman modes renormalized frequency as a function of temperature relatively to the anharmonic classical behavior, is interpreted in the framework of spin-phonon coupling model, in which the balance between competitive ferro and antiferromagnetic interactions is considered. An association between the spin-spin correlation function and the renormalized frequency of the Raman modes is presented. 7 . Resumo Este trabalho destina-se ao estudo do papel do acoplamento spin-fonão na estabilização de fases ferroeléctricas magneticamente induzidas em materiais magnetoeléctricos. Para este propósito fez-se uma caracterização das propriedades termodinâmicas, dieléctricas, magnéticas, polares e magnetoeléctricas do sistema Gd1−xYxMnO3 , com 0≤x≤0.4 . O conjunto dos resultados experimentais permitiu-nos esboçar o diagrama de fases ( x , T) deste sistema e perceber o efeito de um campo eléctrico aplicado nas propriedades magnéticas destes compostos. Observou-se uma polarização eléctrica de natureza inductiva no GdMnO3 , não associada a nenhum fenómeno cooperativo. Para x≥0.1 , fases antiferromagnéticas são estabilizadas a baixas temperaturas, em que algumas simultâneamente exibem propriedades polares. Embora apenas a fase magnética de mais baixa temperatura seja polar, um acoplamento magnetoléctrico foi evidenciado para temperaturas superiores. Adicionalmente, o estudo da dinâmica de rede através da espectroscopia Raman revelou a existência dum acoplamento spinfonão ainda na fase paramagnética. Este resultado aponta para a existência de efeitos precursores das fases magnéticas de baixas temperaturas. Desvios signicativos da frequência renormalizada dos modos Raman em função da temperatura relativamente ao comportamento anarmónico clássico, são interpretados com base no modelo de spin-fonão, no qual são consideradas o balanço entre as interacções competitivas ferro e antiferromagnéticas. É apresentada uma relação entre a função de correlação entre spins e a frequência renormalizada dos modos Raman. 9 . Chapter 1 Introduction Magnetically-induced ferroelectricity arises from the coupling between magnetic and electric dipoles and opens new ways for technological applications. In magnetically-induced ferroelectrics, the distortions giving rise to electric polarization are a consequence of the coupling between spins and lattice, which can be detected through the anomalous behavior of the phonons. Thus, the physics of the magnetically-induced ferroelectricity could be enlightened through the study of the spin-phonon coupling. A large variety of materials exhibit spin-phonon coupling, namely superconductors, antiferromagnets, and magnetic semiconductors. Magnetoelectric materials are recent examples. In these materials, the coupling between magnetic and electric dipoles enables to change the electric polarization by applying a magnetic eld or to change the magnetization through an applied electric eld. Although scarce, several families of magnetoelectric materials, among them, the rare-earth manganites ( RMnO3 ), have drawn an enormous attention in the scientic community due to their rich phase diagrams, including magnetically-induced ferroelectric phases. The strong coupling between lattice, spin, charge and orbital degrees of freedom enable to tune the physical properties through well controlled distortions, which can be achieved by chemical substitution, temperature, strain, pressure and electric or magnetic elds. Up to now, the understanding of the microscopic mechanisms underlying the magnetoelectric coupling is still missing and, thus, their study is very challenging. In magnetoelectric rare-earth manganites, improper ferroelectricity has been attributed to spin-lattice interactions in a modulated non-centrosymmetric magnetic structure. It has been proposed that ferroelectricity can originate from a variety of spiral magnetic structures and can be explained in terms of the Dzyialowshinki-Moriya model. The study of spin-phonon coupling is particularly relevant in magnetoelectric materials, as this coupling is a necessary condition for the magnetoelectric eect to emerge. However, recent studies carried out in various systems have revealed that the magnetoelectric eect is weak, hindering their use in devices. Thus, studying spin-phonon coupling in those materials is a main issue both for understanding their fundamental aspects and underlying mechanisms for magnetically-induced ferroelectricity and to tailoring materials accordingly for possible applications. Raman and infrared spectroscopy studies of rare-earth manganites ( RMnO3 ) revealed a signicant spinphonon coupling for compounds with R=Pr to Sm near and below the Néel temperature. It is worth noting that the eect of the magnetic ordering on phonon frequencies becomes weaker or rather negligible in orthorhombic rare-earth manganites, whilst increasing the ionic rare-earth size, obtained going from Gd to Lu . Any comparative analysis of these compounds is rather complex, because both the variations of the rare-earth ionic radii and dierent values of magnetic moment in dierent R ions have to be taken into account. Very recently, a detailed analysis of the spin-phonon coupling in Eu1−xYxMnO3 revealed that the coupling strength could not be calculated from the direct analysis of the eigenfrequency shift of the normal temperature behavior described by the purely anharmonic model. In fact, more accurate models for the analysis of the spin-phonon coupling need to be considered. The frequency shift of a given phonon as a function of temperature, due to the spin-phonon coupling, is determined by the exchange integrals and spin-spin correlation function. Whenever ferro and antiferromagnetic competitive interactions are present, as is the case of some magnetoelectric rareearth manganites, the frequency deviation from the normal anharmonic temperature behavior depends on the relative strength of the exchange integrals, which depend on the ferro and antiferromagnetic interactions in the system. The model predicts a negative or positive shift to the normal frequency behavior of the eigenmode under examination, its magnitude being dependent from both symmetry and relative strength between the ferro and antiferromagnetic exchange interactions. Recently published results regarding the lattice deformations evidenced by the spectroscopic studies in Eu1−xYxMnO3 and their role in the polar 17 properties in these compounds show the correlation between the magnetic ordering and lattice deformations in these compounds. Although the magnetic structure and the modulation wavevector have been determined for some rare-earth manganites, the systematic study of the lattice deformation and magnetic structure is still missing for a large number of such compounds. Though previous experimental results are rather scarce and non-systematic, they provide clear evidence for the importance of the spin-phonon coupling underlying the physics of magnetoelectricity. However, a complete understanding of the interplay between spin-phonon coupling and other microscopic mechanisms, like orbital overlapping, orbital ordering, is still missing. This work is aimed at studying the macroscopic properties, the magnetoelectric eect and the spin-phonon coupling and its role in Gd1−xYxMnO3 , with 0≤x≤0.4 , underlying the cross-coupling between polar and magnetic orders. First, the crystal structure and the lattice dynamics will be analyzed in detail at room temperature. The structural distortions induced by the Gd -substitution will be studied, which have a major role in the physical properties of the rare-earth manganites at low temperatures. For this, x-ray diraction and Raman spectroscopy techniques will be carried out for all the samples, along with structure renement. The structural deformations will be compared with other rare-earth manganites through the tolerance factor scaling. A detailed characterization of the low temperature physical properties will be carried out, and a ( x , T) phase diagram is proposed for this system. To shed light on the spin-phonon coupling, we undertake a systematic investigation on the temperature dependencies of the optical phonons, across the magnetic and polar phase transitions. The magnetic and ferroelectric properties of all available samples, as well as the magnetoelectric coupling will be studied through the measurements of magnetic, ferroelectric and dielectric properties with and without applied electric eld. The obtained data will allow enlightening the nature of the magnetoelectric coupling. Raman scattering measurements will be carried out as a function of temperature, aiming at studying both spin and lattice excitations. Suitable models will be used for the spectra analysis and the frequency and linewidth will be calculated from the best t to the experimental data. Based on symmetry arguments, the temperature dependence of some selected optical modes will be thoroughly analyzed. The frequency deviation of a given phonon as a function of temperature due to the spin-phonon coupling is a function of the spin-spin correlation functions. The frequency shift depends on the relative strength of ferro and antiferromagnetic exchange integral values which may yield positive or negative frequency deviations. The linewidth of the optical phonons will also be the aim of detailed study. In fact, in the magnetically ordered phase an additional contribution to the phonon damping arises from the spin-phonon coupling, which vanishes in the paramagnetic phase. Taking into account the obtained data, we propose: (a) To scale the optical mode frequencies to the spin-spin correlation functions, in order to get information about the coupling constant and type of coupling occurring in each compound. Available models will be used to analyze the results; (b) To correlate the spin-phonon coupling with other mechanisms underlying the physical properties of each compound, as the magnetoelectric eect. Theoretical models based on relevant literature will be considered in order to build up a framework to understand and explain the spin-phonon coupling in the emergence of magnetically-induced ferroelectricity. The nearest-neighbor spin-spin correlation functions will be calculated from the measurements of magnetization and specic heat data. We will assume that the deviations from the aforementioned scaling emerge mainly from structural deformations, which induces changes on the exchange integrals. By using adequate models available in literature, the balance of the ferro and antiferromagnetic exchange integrals and the spin-phonon coupling parameter will be calculated. These constants will be useful in the analysis of the specic heat data and for discussing the Raman data. "Where will you go?", "No idea. But if I'm to discover the truth of life.. I had better get going." - Seta Sojiro 18 Chapter 2 General Considerations Magnetoelectric materials have attracted much attention of the scientic community due to their potential applications and the new physics that they bring. The coupling between ferroelectricity and magnetism opens the possibility to tune the polar/magnetic properties through magnetic/electric elds. The possibility of stabilize the remarkable property which reveals as a ferroelectric phase within the temperature range of stability of magnetic phases arouse the interest of understanding this coupling in the framework of the Landau theory or quantum models. In some microscopic models, the spin-phonon coupling has been considered as a key for the emergence of magnetically-induced ferroelectric phases. However, the spin-phonon coupling alone does not explain the main features of the magnetic and ferroelectric phases. Among the magnetoelectric materials, rare-earth manganites are the most known. Manganites refer to the manganese oxide compounds with a general formula of AMnO3 , which consist in many types of dierent materials. Within the manganites, a particular interest has been paid in the orthorhombic rare-earth manganese oxides, or simply, rare-earth manganites. These compounds present a perovskite-like structure in which, as a consequence of the size mismatch between the cavity formed by the oxygen-octahedra and the undersized cation in the A-site, the GdFeO3 -type distortion is presented [10]. Due to this distortion, the oxygen octahedra chains of the ideal cubic perovskite structure (see Figure 1) are tilted, which causes the reduction of the Acation coordination number from 12 to 8. In the particular case of manganites, the Mn3+ ion is Jahn-Teller active and, so, three dierent Mn −O bond lengths, between the center Mn3+ ion and the O2− ions in the octahedra corners, are exhibited. Octahedra tilting and cooperative Jahn-Teller distortion are the main distortions presented in the orthorhombic rare-earth manganites, which reduce the structural symmetry from Pm-3m to Pnma. These structural distortions are an important key underlying the ferroelectricity physics of these materials [11, 12]. Figure 1: Ideal cubic perovskite structure ( ABO3 ). 19 2.1 Multiferroic and Magnetoelectric Properties Multiferroicity denotes the co-existence of more than one primary ferroic order parameter simultaneously in a single material [13]. These ferroic orders can be ferromagnetism, ferroelectricity, ferroelasticity, and ferrotoroidicity [14]. Though the term multiferroic was rst used for materials which exhibit simultaneously ferroelectricity and ferromagnetism, up to date, this term has evolved and has been extended also to those materials which exhibit both ferroelectricity and antiferromagnetism [2, 15, 16]. Magnetoelectrics are materials in which the ferroelectric and magnetic order are coupled. Magnetoelectric multiferroics designated a class of materials in which ferroelectricity and ferromagnetism coexist and are coupled together [17]. The number of such materials is scarce and the few magnetoelectric multiferroics that have been identied to date, have no practical applications of magnetoelectric phenomena, mainly due to the small magnitude of the magnetoelectric or magnetocapacitive eects and to the low temperatures where both ferroelectric and magnetic order coexist. In most of the magnetoelectric multiferroics, the temperature range for ferroelectric order is much larger than for magnetic order [18]. This leads to only weak coupling between magnetism and ferroelectricity in these systems, which once blocked the development of multiferroicity in the last century. Till 2003, two milestone works, the discovery of magnetic-eld-controllable ferroelectric polarization in TbMnO3 crystals and a giant ferroelectric polarization in BiF eO3 lms, renewed the interest of research on magnetoelectricity [19, 20]. Then, multiferroics and magnetoelectrics have become a ourishing research area, with more and more multiferroic materials being discovered and the understanding of the underlying physical mechanisms has also been pushed forward gradually. More recently, gigantic magnetoelectric and magnetocapacitive eects have been found in certain rare-earth manganites having antiferromagnetic orders with long wavelengths as compared to their chemical unit cell [1]. These interactions provide an approach to engineering the couplings between magnetism and ferroelectricity. Recent theoretical and experimental studies of a series of rare-earth manganites with orthorhombically distorted perovskite structure revealed that the ferroelectricity originates from competing magnetic interactions which produce a long-wavelength antiferromagnetic spin order and accordingly lattice modulations with nonzero wavevector through magnetoelastic coupling [21]. This coupling between magnetic order and lattice distortions which produces ferroelectricity gives rise to strong magnetoelectric coupling and resultant gigantic magnetoelectric and magnetocapacitive eects [1]. 2.2 Phase Diagram of Unsubstituted Orthorhombic Rare-earth Manganites The paramagnetic phase of orthorhombic rare-earth manganites is also paraelectric, and the spontaneous ferroelectric polarization appears to be directly driven by a transition to a cycloidal modulated antiferromagnetic phase [1]. From a perspective based on symmetry and group theory, the idea of improper ferroelectricity driven by the condensation of a primary order parameter of a magnetic nature raises interesting questions [22]. In the case of the frustrated magnets, a key point is to know under which circumstances a modulated magnetic order parameter can induce ferroelectricity. According to this suggestion, the electric polarization in these compounds would result from the secondary lattice modulation, of a displacive nature, magnetoelasticallyinduced by the primary magnetic modulation [22]. In the case of orthorhombic RMnO3 , decreasing the ionic radius of the R ion ( rR ) changes the balance of the competition in magnetic interactions. As the A-site ionic radius decreases, the octahedra tilt distortion increases. By increasing the tilt angle, the Mn −O−Mn bond angle reduces, and so, the orbital overlap between adjacent Mn3+ and O2− ions is altered, which consequently modies the magnetic superexchange integrals. The balance between competitive ferro and antiferromagnetic interactions is crucial in the denition of the spin arrangement. This makes the rare-earth manganites very interesting compounds, as the magnetic 20 phase can be tailored in order to have the desired electric properties, due to direct coupling to the primary magnetic order parameter. Figure 2: Magnetoelectric phase diagram in temperature for RMnO3 compounds, as a function of Mn−O−Mn bond angle, dened by the rare-earth ionic radius [1]. Shadow area denotes the ferroelectric phase. Figure 3: Possible magnetic spin arrangements identied in RMnO3 compounds [2]. The phase diagram of RMnO3 as a function of the magnitude of the tilt angle or the ionic R -site radius 21 ( rR ) has been studied experimentally, and it is presented in Figure 2 [1]. For R=La to Sm , there is a single phase transition at TN , from the paramagnetic state to a canted A-type antiferromagnetic phase [1]. As rR decreases, for R=Eu and Gd , two phase transitions are now observed. One at TN , from the paramagnetic phase to a collinear-sinusoidal incommensurate antiferromagnetic one, and another at lower temperature to a canted A-type antiferromagnetic phase. As the A-site ionic radius further decreases, for R=Tb and Dy a commensurate antiferromagnetic phase below the collinear-sinusoidal incommensurate antiferromagnetic phase appears at Tlock , around 27 K and 20 K, respectively. This last magnetic phase is also ferroelectric. For the smallest A-site ionic radius within the orthorhombic rare-earth manganites, HoMnO3 , the rst phase transition remains unaltered, but the second phase transition is into a E-type antiferromagnetic phase. Figure 3 depicts the spin arrangements for each referred phase. Among the orthorhombic RMnO3 , it is well known that TbMnO3 and DyMnO3 are spontaneously ferroelectric at low temperatures, while GdMnO3 and EuMnO3 exhibit a ferroelectric phase under applied magnetic elds [5]. These compounds are the magnetoelectric ones. 2.3 Theory of Spin-phonon Coupling The vibrational modes of RMnO3 are not independent of the Mn3+ spins arrangement due to the magnetoelectric coupling. The Hamiltonian, when considering a simple ferromagnetic exchange and the lattice vibrations, is given by 3 terms, where the last one represents the coupling between spins and phonons [23] H=Hex +Hph +Hsp , (1) with Hex =−JX <i,j> Si.Sj, (2) Hph =X α hωα1 2+a| αaα, (3) where J is the magnetic exchange, ωα is the phonon frequency, a| α and aα are the operators that create and destroy, respectively, one lattice vibration excitation of energy hωα . The Raman-active modes involve displacements of non-magnetic ions only, while the magnetic ion Mn3+ remains stationary. With this, using a stationary perturbation theory with the exchange and phonon Hamiltonians, the spin-phonon coupling energy is determined as [23] Hsp =X <i,j> RiSi.Sj, (4) where Ri is the squared derivative of the magnetic exchange integrals to the normal coordinate for the i th atom. From Hsp one is able to calculate the renormalized frequency of the phonons, which is given by [23] ω=ω0+γ < Si.Sj> , (5) where < Si.Sj> is the spin-spin correlation function and, if a single ferromagnetic exchange integral is assumed, the spin-phonon coupling parameter γ is identied with R . The spin-spin correlation function plays a major role in the phonon frequency renormalization, as in a disordered phase, where < Si.Sj>= 0 , no renormalization is expected. So, in a more complex structure, where both ferro and antiferromagnetic interactions are present, such as the rare-earth manganites structure, when an ordered magnetic phase is achieved, if the ferromagnetic interactions are dominant ( Jeff >0 ) the phonons frequencies are expected to shift toward higher values, while on the other hand, if the antiferromagnetic interactions are dominant ( Jeff <0 ), the phonons frequencies are expected to shift toward lower values. 22 2.4 Magnetically Driven Ferroelectricity in Rare-earth Manganites The physical properties of RMnO3 compounds are mainly dened by the arrangement of the Mn3+ spins. Figure 2 can be determined from a classical Heisenberg model for the Mn3+ spins energies, when assuming they present s= 2 [2]. There is one theoretical model, the Mochizuki-Furukawa model, based on microscopic mechanisms, that is broadly assumed by the research community. The Hamiltonian consists in four major contributions [24], H=Hex +HSIA +HDM +Hcub, (6) with Hex =−Jac X <i,j> Si.Sj+J2X <i,j> Si.Sj+JbX <i,j> Si.Sj, (7) HSIA =DX i S2 ζi +EX i (−1)ix+iyS2 ξi −S2 ηi, (8) HDM =X <i,j> dα i,j (Si×Sj), (9) Hcub =a S(S+ 1) X iS4 xi +S4 yi +S4 zi. (10) The rst term Hex stands for the exchange and superexchange magnetic interactions between neighbors and nearest-neighbors Mn3+ spins, respectively. The Jac ferromagnetic exchange is orientated along the Mn−Mn bonds in the x and y axis. The J2 and Jb are antiferromagnetic exchanges, orientated on the in-plane diagonal between the Mn −Mn bonds, along the b -axis and on the Mn −Mn bonds along the c -axis, respectively. Figure 4 depicts the orientation of the exchanges in the crystallographic structure. It is very interesting that the strength of the next-nearest neighbor exchange J2 increases as the ionic radius of the rare-earth decreases. Conversely, the Jac and Jb exchanges are almost independent of the structural distortions. Within the Mochizuki-Furukawa model the Jac and Jb exchanges dependence with rR are neglected and only the rR - dependence of the antiferromagnetic exchange J2 is taken into account. So, as the octahedra tilting increases, the balance between competitive magnetic interactions favors the antiferromagnetic ones [24]. A Peierls-type spin-phonon coupling is here introduced, which consists in a further displacement of the O2− ion due to the magnetic ordering at low temperatures, when compared to the already displaced initial position at room temperature. This further displacement is reected in the eective magnetic exchange integral by Jij =Jac +J ´ acδi,j , (11) where J ´ ac =dJac dδ , and δi,j denotes the shift of the O2− ion between the i th and j th Mn3+ ions. The second term HSIA represents the single-ion anisotropies [24]. This anisotropy is dened by the valence electron of the Mn3+ ion, occupying the eg orbital, which aects mainly the environment inside the MnO6 octahedra. ζi denotes the local hard magnetization axis in the b -direction for every Mn3+ site, while ξi and ηi are alternately local hard magnetization axes in the ac -plane. The third term HDM stands for the interactions mediated by the Dzyialowshinki-Moriya mechanism [24]. The vector dα i,j is dened on the Mn(i)−O−Mn(j) bond, along the α direction, where α=x, y or z . This vector is antisymmetric, and so dα i,j =−dα j,i . The last term Hcub denotes the cubic anisotropy, arising from the nearly cubic symmetry of the distorted perovskite structure. In this term, xi , yi and zi are the coordinates of the i th Mn3+ ion with respect to the cubic x , y and z axes, while a is just a coupling constant. The contribution to this anisotropy coming from the lattice distortions is neglected, as it is expected to be very small. This theoretical model has proven to be suitable to reproduce many of the published experimental results for RMnO3 , from Sm to Ho, for Eu1−xYxMnO3 , with 0≤x≤0.5 and for Gd1−xTbxMnO3 , with 0≤x≤1 . 23 Figure 4: Orientation of the exchanges and superexchanges interactions Jac , J2 and Jb in the crystallographic structure. In RMnO3 compounds, the improper ferroelectric phases are magnetically driven. In the model referred to above, the emergence of ferroelectricity is explained by a microscopic spin-current model, derived from the Dzyialowshinki-Moriya mechanism [16]. This mechanism supposes a spin arrangement in a cycloidal plane, such as Figure 3(d) depicts, in which the overlap of the electronic orbitals of two adjacent ions, whose spins are mutually canted, is able to give origin to an electric polarization [25] pi=Aei,j ×(Si×Sj), (12) where A is a constant determined by the spin-exchange and spin-orbit interactions, ei,j is the vector that connects the i th ion with its nearest neighbor, and Si is the spin of the i th ion. So, the magnetic phase with a cycloidal spin order is expected to produce an electric polarization P , equal to the sum of each local polarization pi , orientated perpendicularly to the spiral propagation vector and the spin-helicity vector. Within the orthorhombic rare-earth manganites series, two cycloidal spin ordered phases can be stabilized, one with the cycloidal spin in the ac -plane and another in the ab -plane. The magnitude of the structural distortions, dependent of rR , denes which magnetic phase is most stable at xed external conditions, due to competitions between Single Ion anisotropy and Dzyialowshinki-Moriya interaction energies [24]. Consequently, the resulting electric polarization will be orientated in the b -axis, or in the c -axis, respectively. 2.5 Unknown Behavior of Gd1−xYxMnO3 The coexistence of ferroelectricity with antiferromagnetism is most interesting, and thus we will now focus on the frontier where it appears (cf. Figure 2). EuMnO3 exhibits no ferroelectricity for any temperature, in the absence magnetic eld [1]. It is known that ferroelectricity can be induced in EuMnO3 by an external magnetic eld of approximately 7 T [26]. In TbMnO3 the competing interactions give rise to a long wavelength antiferromagnetic phase, which produces ferroelectricity through magnetoelastically induced lattice modulations [3]. It exhibits a gigantic magnetoelectric eect, where electric polarization can be opped by applying a magnetic eld. GdMnO3 is located between these two compounds in the phase diagram of Figure 2. Its orthorhombic distortion is greater than EuMnO3 , but smaller than TbMnO3 . This localization in the phase diagram makes GdMnO3 an interesting compound to study. In fact some studies have been made, and although they evidence that GdMnO3 shows 24 no spontaneous ferroelectricity, although with some discussion about the existence of a polizarable ground state, it presents a magnetically-eld-induced ferroelectric transition, with an applied eld of 2 T [27]. As the distortions present in GdMnO3 locate it close to the boundary of the ferroelectric phase, it would be of great interest to be able to study them in detail, in order to understand how one can possibly alter them slightly and induce a ferroelectric phase with a much smaller, if not even none, external magnetic eld. In contrast to TbMnO3 and DyMnO3 , only very limited data on the magnetic ordering of GdMnO3 have been reported so far. Some of its magnetic phases are known by inductive logic using the experimental data for other RMnO3 , since Gd3+ has a too high cross-section for neutron scattering, which is the best experimental technique to obtain information regarding the magnetic structure. In GdMnO3 two phase transitions driven by the ordering of the Mn3+ spins are reported [28]. A rst phase transition occurs at TN= 42 K, from the paramagnetic to the collinear-sinusoidal incommensurate antiferromagnetic phase [27]. The other phase transition is located at TC≃20 K, to the canted A-type antiferromagnetic phase. This phase transition can be observed in the temperature dependence of the complex dielectric permittivity, by a peak-like anomaly, which occurs at 19 K [4], 21 K [29] and 23 K [21, 30] in single crystals, while for powder ceramic samples, this temperature is lower, around 14 K [28]. There are also other physical quantities whose temperature dependence marks this phase transition at TC , as the appearance of an electric polarization [4], a step-like anomaly in the lattice modulation vector [3] or the magnetic response [28]. Furthermore, through measurements with applied magnetic eld, the eld-cooled thermal expansion and the magnetostriction temperature and magnetic eld dependence the same temperature transition was reported at 18 K and 20 K, respectively [27]. Figure 5 presents some of the aforementioned reported anomalies. Figure 5: Reported anomalies at TC for GdMnO3 . Temperature dependence of (a) spin modulation vector [3], (b) complex dielectric permittivity [4], (c) and (d) complex dielectric permittivity with applied magnetic eld [5]. At around 5 K another phase transition is exhibited as the Gd3+ spins antiferromagnetically order [5, 30]. However, there is incomplete information in the unnished discussion whether ferroelectricity is a ground state or magnetically/electrically induced in GdMnO3 [1, 5, 27, 30]. Although rare-earth manganites give us a rich phase diagram from R = La to Dy , their location within it is well dened, as are their phase transitions at low temperatures. To overcome this discrete spacing between each compound, RMnO3 , the original compound can have a certain amount of the rare-earth R substituted by a smaller ion, R1−xXxMnO3 . Through it, the octahedra tilt angle can be tuned and a continuous change of the balance between competitive ferro and antiferromagnetic interactions obtained. Still, the possibility of distortion-induced ferroelectricity in GdMnO3 through substitution by a smaller ion is poorly explored, as no more than simply one work is published, where the studied system is Gd1−xYxMnO3 [6]. This published work 25 Figure 8: X-ray powder diraction spectra of Gd1−xYxMnO3 , with 0≤x≤1 , recorded at room temperature. 32 Figure 9: x -dependence of the molar percentage of orthorhombic and hexagonal phases. Figure 10: Pseudocubic lattice parameters as a function of x . (Error bars are smaller than the markers). 33 The slopes of the linear dependence of the pseudocubic lattice parameters as a function of Y -concentration are displayed in Table 1. The slope of apc(x) is the smallest, while the slope of cpc(x) is the largest one, slightly higher than the bpc(x) slope value. The dierence in the x -dependence of the lattice parameters evidence for an anisotropic volume reduction of the unit cell. The increase of the Y -amount becomes more asymmetric the unit cell dimensions, as the lattice parameters values become further apart. Pseudocubic Lattice Parameter Slope (Å) apc 0.013 ±0.001 bpc 0.039 ±0.003 cpc 0.041 ±0.002 Table 1: Slope of the linear relation between the pseudocubic lattice parameters and Y -concentration. Figure 11 shows the pseudocubic volume ( Vpc =V 4 ) as a function of x , where a linear decrease is evident. Figure 11: Pseudocubic cell volume as a function of x . (Error bars are smaller than the markers). Figure 12 depicts the representative unit cell of the Gd1−xYxMnO3 , with x≤0.4 , at room temperature. The crystal structure consists on a network of oxygen shared MnO6 octahedra, forming chains along the b - direction. The MnO6 octahedra are rotated about the pseudocubic [100]pc , [001]pc and [111]pc axes, described by the a+b−b− Glazer scheme [35]. The Gd3+ or Y3+ ions occupy the interstices between octahedra. Table 2 presents the Wycko positions occupied by each atom in the unit cell. Atom Wycko position Symmetry R = Gd, Y 4c .m. Mn 4b -1 O1 4c .m. O2 8d 1 Table 2: Wycko positions and symmetry. Due to the electronic distribution of the Mn3+ ions, Jahn-Teller distortion is present in the structure. So, 34 Figure 12: Structure of the representative unit cell of the Gd1−xYxMnO3 system, with x≤0.4 . Image adapted from [7]. three dierent Mn −O bond lengths are expected. The apical Mn −O1 bond lenghts present only one value, while two dierent values are obtained for the Mn −O2 bond lengths, in the equatorial plane of the MnO6 octahedra. Moreover, due to the MnO6 tilting, dierent Mn −O−Mn bond angles were measured. Figure 13 shows the Mn −O−Mn bond angles and Mn −O bond lengths as a function of Y -concentration. These quantities are known to be of great importantance, since they bear information about octahedra tilting and distortion, which are well known to tailor the low temperature behavior of rare-earth manganites [1]. As a consequence of their small x-ray atomic form factor, the positions of the oxygen atoms are not determined with high accuracy, which prevents the calculations of the Mn−O bond lengths and Mn−O−Mn bond angles with enough precision. Figure 13(a) shows the x -dependencies of the Mn −O−Mn bond angles values. The values obtained for the Mn−O1−Mn bond angle are in good agreement with those published for GdMnO3 [1]. The dispersion of the Mn−O−Mn bond angles values prevent the observation of variations less than 2 o , and, so, no clear dependencies on Y -amount can be discerned. Figure 13(b) shows the x -dependencies of the Mn −O bond lengths. We concluded from Figure 13(b) that the Mn −O1 bond length decreases very slowly as Y -concentration increases. The dierence between the long and the short Mn −O2 bond lengths is about 0.4 Å for GdMnO3 , and it decreases as Y-concentration increases. This is due to the decreament of the Mn −O2(long) bond length, and the increament of the Mn −O2(short) one, as x increases. This behavior evidences for a reduction of the octahedra asymmetry as Y-amount increases. Also, an important value to take into account when studying the structural changes in these materials is the orthorhombic distortion parameter ( e ), dened in Pnma as [38] e=2(a−c) (a+c), (16) where a and c are the lattice parameters. This parameter characterizes the orthorhombic distortion of the lattice, allowing the study of the deformation with respect to the ideal cubic perovskite structure induced by the smaller Y3+ ionic radius. From Figure 14, which shows the orthorhombic distortion parameter as a function of Y -concentration, a linear increase of the othorhombic distortion is observed. This result is a consequence of the dierente slopes of the linear relations of the lattice parameters, apc(x) and cpc(x) , as it was referred to above, and by the variation of tilt angles. 35 Figure 13: (a) Mn −O−Mn bond angles and (b) Mn −O bond lengths as a function of x . (a) The dashed lines are guide for the eyes, (b) the solid lines are the best linear t. 36 Figure 14: Orthorhombic distortion parameter x -dependence. (Error bars are smaller than symbols). Some years ago, M. A. Carpenter and C. J. Howard have analyzed the symmetry, order-parameter and strain/order-parameter relationships in ABX3 perovskites, exhibiting both Jahn-Teller distortion and octahedral tilting, in the framework of Landau theory [39]. According to the grounds of the model, for rare-earth manganites with symmetry Pnma, as it is our case, the Jahn-Teller ordering scheme is associated with the M+ 2 irreducible representation of the space group Pm-3m, while the tilting instability is associated with the M+ 3 and R+ 4 irreducible representations. The three irreducible representations are 3D. According to group theory, the non-vanishing components of the order-parameters are presented in Table 3 [39]. M+ 2M+ 3R+ 4 General ( q1JT , q2JT , q3JT ) ( q1 , q2 , q3 ) ( q4 , q5 , q6 ) Pnma (0, q2JT , 0) (0, q2 , 0) ( q4 , 0, q4 ) Table 3: Non-vanishing components of the order-parameters associated with the Jahn-Teller distortion ( M+ 2 ) and the octahedra tilting ( M+ 3 and R+ 4 ), for Pnma space group. The general equations derived from the symmetry rules and coupling between octahedra titing and JahnTeller distortion provide a basis for analyzing changes in lattice parameters, leading to the evolution of individual tilting and Jahn-Teller order-parameters. According to Carpenter et Howard [39], the shear strain e23 is directly associated with the tilt and Jahn-Teller distortions as follows |e23|=λ1.q2 4+λ2.q2JT .q2 c, (17) where λ1 , λ2 and c are phenomenological coupling parameters. The shear strain is experimentally determined through the strain tensor components as follows e23 =e22 −e33 =apc −a0 a0−cpc −a0 a0 , (18) where a0=3 √V0 is the cubic parameter. Figure 15 shows the shear strain as a function of x , where a linear increase of e23 with increasing x is apparent. In the following, we assume that the values of q2 and q4 are 37 proportional to the tilt angle, while q2JT to the Jahn-Teller distortion. As it was referred to above, as the Y -content increases, the dierence between the Mn −O2 bond lengths decreases, which we interpret as a decrease of the q2JT value. If so, the increase of e23 should be associated with the increase of q2 and/or q4 , meaning that the octahedra tilting must increase. We will address to this problem in the next section. Figure 15: Strain tensor e23 component as a function of x . (Error bars are smaller than the markers). 4.2 Lattice Dynamics and Structure The unpolarized Raman spectra of the Gd1−xYxMnO3 samples ( x= 0.0, 0.1, 0.2, 0.3 and 0.4), recorded at room temperature, are shown in Figure 16. The prole of the Raman spectra recorded for all compounds at room temperature is the typical of the orthorhombic rare-earth manganites [8]. In orthorhombic rare-earth manganites, the activation of the Raman modes is due to the distortions associated with symmetry reduction from the ideal perovskite structure. According to group theory, from the 60 normal modes at the Γ -point of the Brillouin zone, only 24 are Raman active, which have the following decomposition into irreducible representations of the mmm point group: ΓRaman = 7Ag+ 5B1g+ 7B2g+ 5B3g (19) Due to the polycrystalline nature of our samples, the Raman spectra presented in Figure 16 exhibit simultaneously the Raman modes of all symmetries. It is well-known that the Ag and B2g modes give rise to the most intense Raman bands [8]. In good agreement with reported spectra for GdMnO3 , our observed Ag and B2g modes are the most intense [40]. 38 Figure 16: Unpolarized Raman spectra of Gd1−xYxMnO3 , recorded at room temperature. In order to get detailed information regarding the dependence of the phonon parameter dependence on Y - concentration, we have simulated the experimental spectra by using a sum of damped oscillators, according to Equation 13. Figure 17 shows, as an example, the result of the simulation procedure of the Gd0.6Y0.4MnO3 Raman spectrum, along with a symmetry assignment of the observed bands, taking the information given in 39 [8]. Figure 17: Example of best t for the Gd0.6Y0.4MnO3 Raman spectrum. The presented symmetry assignement follows the notation of Iliev et al [8]. Table 4 summarizes the wavenumbers of the experimentally observed Raman bands at room temperature, the main atomic motions, symmetry and basic distortion giving rise to the Raman activity. We have adapted the symmetry assignment according to Iliev et al [8]. Detailed studies published by several authors in rare-earth manganites, both single crystal and ceramic forms [8, 40, 41], enable us to assign the most intense Raman bands in our spectra. In Gd1−xYxMnO3 , with 0≤x≤0.4 , the band at ∼610 cm−1 is assigned to the in-plane symmetric O2 stretching mode, with origin on the Jahn-Teller distortion of the MnO6 octahedra. This mode (with B2g symmetry) involves the stretching of the Mn −O2 bonds and it is known to be very sensitive to the magnetic arrangements [8]. The band at ∼520 cm−1 is assigned to an in-phase O2 scissor-like mode (symmetry B2g ), given rise by the tilt distortion. The band at ∼500 cm−1 and the one at ∼480 cm−1 (both with Ag symmetry) are assigned to a MnO6 octahedra bending mode and the in-plane O2 anti-stretching mode, respectively. These two bands, due to their similar frequency and same symmetry, are strongly mixed [8]. The band at ∼470 cm−1 is assigned to an out-of-phase MnO6 octahedra bending (symmetry B2g ). Finally, the band at ∼370 cm−1 is assigned to the out-of-phase MnO6 octahedra rotations, activated by the octahedra chains tilting. As it is a lattice mode, it is very sensitive to distortion of the crystal structure. Its frequency depends on the tilt angle. The relative variation of the frequency of the tilt mode is found to be proportional to the relative variation of the tilt angle θt [8] ∆ω ω∝∆θt θt . (20) So, an increase of the tilt angle causes the frequency of this mode to increase. On the basis of the presented modes assignment, we now correlate the x -dependencies of the frequency of these Raman bands with structural distortions induced by the Y -concentration. Figure 18 shows the wavenumber of several Raman modes as a function of Y -concentration. For these modes, the wavenumber increases with increasing x , although the rate of change is not the same for every mode. Table 5 presents the slope of the wavenumbers as a function of Y -amount, for the dierent analyzed modes. 40 Wavenumber ( cm−1) Denomination Symmetry (#) Main atomic motions Basic distortion 370 Tilt (T) Ag (4) out-of-phase MnO6 octahedra x -rotation Octahedra tilt 470 - B2g (3) out-of-phase MnO6 octahedra bending Octahedra tilt 480 Anti-Stretching (AS) Ag (1) equatorial O2 anti-stretching Jahn-Teller 500 Bending (B) Ag (3) MnO6 octahedra bending Octahedra tilt 520 Scissors (S) B2g (2) in-phase O2 scissors-like Octahedra tilt 610 Symmetric Stretching (SS) B2g (1) equatorial O2 symmetric stretching Jahn-Teller Table 4: Wavenumber, symmetry, atomic motions and basic distortions of the Raman modes observed for Gd1−xYxMnO3 at room temperature [8]. Raman mode Symmetry (#) Slope ( cm−1 ) Tilt Ag (4) 24 ±2 Anti-Stretching B2g (3) 9±3 Bending Ag (1) 14 ±2 Scissors Ag (3) 12 ±3 Symmetric Stretching B2g (2) 5±3 Table 5: Slope of the linear relation between Raman modes wavenumbers and Y -amount. 41 Figure 24: Mn −O bond lengths in rare-earth ferrites (open symbols) and manganites (lled symbols). Adapted from [9]. volume. For ferrites, which do not exhibit Jahn-Teller distortion, the volume reduction as the tolerance factor decreases is accommodated by bending the FeO6 octahedra network, as no signicant distortion of the FeO6 octahedra could be detected (cf. Figure 24). In both pure and substituted manganites, as the lattice parameters display a parallel t -dependence of ferrites, we can conclude that the volume reduction is also accommodated mainly by bending the MnO6 , which slightly alters the MnO6 octahedra distortion. The latter distortion is likely to have a small contribution for volume reduction, without much expression on the tolerance factor dependence of the lattice parameters. Figure 25 shows a more complete t -dependence of the Mn −O bond lengths of the unsubstituted rare-earth manganites and the substituted GdY and EuY , where we added our Mn −O bond lengths results of GdY and EuY (the latter obtained in a previous work [43]) to the Lufaso et al [9] results presented in Figure 24. Figure 25 evidences that the apical Mn −O1 bond length is weakly dependent on the A-site size, with a very small monotonous decrease from La to Dy . The equatorial Mn −O2 bonds lengths as a function of the tolerance factor, exhibit non-monotonous behaviors. Focusing on the Mn −O2(long) one, it clearly increases as the A-site size decreases from La to Gd , and then it decreases as the A-site size decreases from Gd to Dy . The Mn −O2(short) bond length has, qualitatively, a behavior which is symmetric to the Mn −O2(long) one. Note that the non-monotonous behavior of Mn −O2(long) with the tolerance factor is similar to the one presented by the lattice parameter a (cf. Figure 23). Since the Mn −O2(long) bond has a high projection on the a -direction, there is a suggestion that these results are associated. As it was referred to above, the determination of the oxygen positions by the renement of the x-ray diraction of the structure is not enough accurate. So, the calculated Mn −O−Mn bond angles are rather dispersed, preventing the determinations of their x -dependence. However, the T mode frequency dysplays a well dened dependence on Y -concentration, allowing to estimate the Mn −O−Mn bond angles variations with x . In the following, we will use the tilt angle, dened as 48 Figure 25: t -dependence of Mn −O bond lengths for RMnO3 , GdY and EuY systems. θt=180−(Mn −O1−Mn) 2. (23) Figure 26 shows the tilt angle of unsubstituted and substituted rare-earth manganites, as a function of the tolerance factor. It is clear that the tilt angle is perfectly scaled by the tolerance factor, exhibiting a linear increment as the the A-site size decreases. The θt obtained for the GdY system presents a wide dispersion, but when compared with the unsubstituted rare-earth manganites [1, 41] and with the GdTb results [29], it shows a clear agreement. Now we are able to state that the tilt angle in our GdY system increases as the Y -concentration increases (decreasing the tolerance factor), as in the Crystal Structure section one concluded it should, from the results of e23 . Moreover, the increment of the tilt angle is determined as 0.5 o , within the experimental error of our x-ray diraction data. The conrmation of the variation of the tilt angle is of great importance, as it suggest that the low temperature physical properties of our Gd1−xYxMnO3 system will be dependent of the Y -concentration, x , as the balance between ferro and antiferromagnetic exchanges alters through the tilt angle. It is possible from Figure 26 data to calculate a relation between the relative change of the tilt angle and of the tolerance factor, as follows ∆θt θt =−86.2∆t t. (24) This relation allows to determine the tilt angle of any rare-earth manganite from the tolerance factor (i.e., the A-site ionic radius). Since Raman modes frequency fully depend on structural parameters, it is reasonable to assume that they can similarly be normalized by the tolerance factor and therefore compared with other rare-earth manganites. The tilt mode wavenumber was plotted as a function of the tolerance factor, and the results are very concordant between dierent materials ( GdY , EuY , EuLu and every RMnO3 ), apart from a slight translation due to device calibrations, as shown in Figure 27. The tilt mode involves rotations of the MnO6 octahedra and thus, 49 Figure 26: Tilt angle t -dependence for GdY , GdTb and pure rare-earth manganites. probes the MnO6 network distortions induced by A-site size reduction. The t -dependence of the tilt mode wavenumber appears to be linear from Nd to Dy . In the inset we focus on the substituted series to show the concordance in detail. Figure 28 presents the plot of the Raman T mode frequency as a function of the tilt angle, combining the Raman spectroscopy data with the x-ray diraction one, for R=Nd to Dy , and for the GdY . From this combination a relation between the relative change of the Raman T mode frequency and of the tilt angle can be determined, resulting in ∆ωT ωT = 21.8∆θt θt . (25) This result is important, as it relates the frequency of the Raman T vibrational mode of the octahedra with the structural tilt angle of the octahedra chains. The SS mode, which involves the in-phase O2 stretching, is sensitive to the octahedra distortion. This mode is able to probe variations in the Mn −O bond lengths, which dene the octahedra distortion. Figure 29 shows the dependence of the SS mode wavenumber with tolerance factor for GdY , EuY and pure rare-earth manganites, from a previous work of ours and Iliev et al work. The SS mode wavenumber must increase as the Mn −O2(long) decreases. Note that as this mode depends only on atomic motions, it does not suer great variations as the tilt mode does. However, comparing with either the Mn−O2(long) bond length or the lattice parameter a , the Raman SS mode wavenumber conrms the behavior obtained through x-ray diraction. As the Mn −O2(long) bond length and the lattice parameter a increase from La to Sm , where they reach their maximum, the SS mode wavenumber decreases, reaching its minimum value in Sm . From Sm to Dy the lattice parameter a decreases, and the SS mode wavenumber increases as it should. Despite the counterintuitive non-monotonous behavior of the Mn −O2(long) bond length, the lattice parameter a and the SS mode wavenumber each conrms the other, presenting consisting evidences for the non-monotonous behavior of the MnO6 octahedra distortion with the A-site size. 50 Figure 27: Raman tilt mode wavenumber as a function of the tolerance factor. Inset presents a detail of the substituted manganites. Figure 28: Raman tilt mode wavenumber as a function of the tilt angle for R=Nd to Dy , and for the GdY . 51 Figure 29: Raman SS mode frequency dependence with the tolerance factor. Briey concluding, the Rietveld renement of the x-ray diraction spectra and the analysis of the Raman spectra reveal that the substitution of Gd3+ by the smaller Y3+ ion increases the orthorhombic distortion. As the volume is anisotropically reduced, its accomodation is mainly due to the increase of the tilt of the MnO6 octahedra chains, which is expected to have great importance in dening the physical properties of our system at low temperatures. Linear relations in both the unsubstituted and substituted rare-earth manganites have been determined for the tilt angle and for the frequency of the Raman T mode, through the results scaling by the tolerance factor. Moreover, our results for Gd1−xYxMnO3 , with 0≤x≤0.4 , perfectly t in these linear relations. 52 Chapter 5 Macroscopic characterization at low temperatures This chapter is addressed to ascertain the temperature dependence of the thermodynamic, dielectric and magnetic properties of the Gd1−xYxMnO3 , with 0≤x≤0.4 , in the 5 K to 300 K temperature range. 5.1 Specic Heat Figure 30 shows the temperature dependence of the specic heat divided by temperature, for the compositions x= 0.0, 0.1, 0.2, 0.3 and 0.4, in the 10 K to 170 K temperature range. The open symbols presented in Figure 30 was calculated from the best t of the lattice contribution for the specic heat between 100 K and 200 K, according to equation [44] CD= 9NhkBT θh D3ˆ θh D T 0 x3 ex−1dx + 9NlkBT θl D3ˆ θl D T 0 x3 ex−1dx, (26) where Nh and Nl stand for the number of heavy and light atoms, respectively. In our case, Nh= 2 and Nl= 3 . Furthermore, kB is the Boltzmann constant and, θh D and θl D are tting parameters, which mean the Debye temperature for heavy and light atoms, respectively. This model has been proven to be more accurate than the model with a single Debye temperature. Table 6 shows the values for the Debye temperatures for heavy and light atoms calculated from the tting procedure for each composition. While θh D takes values of the order of 300 K, θl D is of the order of 800 K. The values obtained in the Gd1−xYxMnO3 , with 0≤x≤0.4 , are in good agreement with those obtained in the Eu1−xYxMnO3 system, with 0≤x≤0.5 [34]. The similarity of the Debye temperatures for both systems reects the similarity of the crystal structure of the orthorhombic Gd1−xYxMnO3 and Eu1−xYxMnO3 . Composition θh D (K) θl D (K) x= 0.0 314 771 x= 0.1 326 750 x= 0.2 319 765 x= 0.3 315 818 x= 0.4 326 754 Table 6: Debye temperatures for heavy ( θh D ) and light ( θl D ) atoms, obtained from the best t of Equation 26 to experimental data, from 100 K to 200 K. Quantitatively, the C T curves presented in Figure 30 are in good agreement with the range obtained for GdMnO3 [28] and the Y -substituted EuMnO3 [45]. A well dened λ -like anomaly of the C T(T) curve is observed at around 42 K, being practically independent on Y -amount. For GdMnO3 , this anomaly is associated with the ordering of the Mn3+ spins from the paramagnetic to the collinear-sinusoidal incommensurate antiferromagnetic phase, as reported in the literature [28]. This phase transition has been also observed for TbMnO3 , DyMnO3 and Eu1−xYxMnO3 , with 0≤x≤0.5 . The critical temperature is not strongly dependent on the A-site size, and takes values around 42 K to 45 K [1]. As in Gd1−xYxMnO3 , x= 0.1 to 0.4, the temperature where the λ -like anomaly occurs in the C T(T) curve is nearly independent on Y -amount, we assume 53 that for these compositions a similar paramagnetic to a collinear-sinusoidal incommensurate antiferromagnetic phase transition also occurs at TN≃ 42 K. Table 7 presents the obtained values for TN . Composition TN (K) x= 0.0 41.9 x= 0.1 41.8 x= 0.2 41.7 x= 0.3 42.0 x= 0.4 41.7 Table 7: Experimental values of the Néel temperature for dierent x values. The area between the experimental curve and the extrapolated Debye behavior below 100 K refers to the magnetic contribution to the specic heat, associated with the magnetic phase transitions taking place at low temperatures. Figure 31 presents the magnetic contribution of the specic heat as a function of temperature. The phase transition temperature TN is marked by the dashed vertical line. At low temperatures and for low concentrations of Y , the specic heat data increases as temperature decreases. This increase is due to the magnetic phase transition associated with the antiferromagnetic ordering of the Gd3+ spins, as it is referred to in the literature [46]. It is worth stressing that this increase is clearly reduced as Gd -concentration decreases. However, this work is not devoted to study this phase transition. The entropy variation associated with the magnetic phase transitions at low temperatures is given by the integral of the magnetic contribution to the specic heat [44] ∆S= TM ˆ 10 Cmag TdT, (27) where TM is the maximum temperature registered. It is assumed that, for T > 10 K, the precursor eects of the ordering of the Gd3+ ions are small, and so, its contribution to the calculated entropy variation can be neglected. Table 8 shows the values of ∆S for the dierent compounds. Composition ∆SJK−1mol−1 x= 0.0 14.8±0.2 x= 0.1 13.7±0.2 x= 0.2 13.7±0.2 x= 0.3 13.7±0.2 x= 0.4 13.7±0.2 Table 8: Entropy variation associated with the Mn3+ spins ordering for each compound. The theoretical entropy variation associated with the magnetic transition as the Mn3+ spins order can be calculated by [44] ∆S=NkBln(2s+ 1), (28) where 4S is the maximum entropy variation obtained from Equation 27, N is the Avogadro number, kB is the Boltzmann constant and s is the total spin. Mn3+ ion is known to have s= 2 , in its high spin conguration [47], thus the expected entropy variation associated with the Mn3+ spin ordering is 13.4 J.K−1mol−1 . The experimental values for x= 0.1 up to 0.4 are in good agreement with the theoretical expected value, with a relative error of only 3%. The experimental entropy variation for the x= 0.0 composition is substantially 54 Figure 30: Temperature dependencies of the specic heat divided by temperature (closed symbols) and respective Debye lattice t (open symbols), for the compositions x= 0.0, 0.1, 0.2, 0.3 and 0.4. 55 Figure 31: Magnetic contribution to the specic heat for x= 0.0, 0.1, 0.2, 0.3 and 0.4 samples. Vertical dashed line marks the magnetic transition at TN . 56 higher, which suggests a contribution of the ordering of the Gd3+ ions at around 6 K. No other anomalous behavior in C T(T) curves could be detected. 5.2 Complex Dielectric Permittivity Figures 32 and 33 exhibit the real (ε0) and imaginary (ε00) parts of the complex dielectric permittivity of Gd1−xYxMnO3 , with x= 0.0, 0.1, 0.2, 0.3 and 0.4, as a function of temperature, measured at dierent xed frequencies in the 3 kHz to 1 MHz frequency range. Measurements were done both in heating and cooling runs, and no thermal hysteresis could be found. The range of experimentally values obtained for the real part of the complex dielectric permittivity in our ceramic powder samples are consistent with the ones found in literature for GdMnO3 and Gd0.9Y0.1MnO3 single crystals [4, 6, 48]. It is worthwhile to note that for the GdMnO3 and Gd0.9Y0.1MnO3 , the ε0(T) curve obtained in this work resemble the ε0(T) curves obtained along the a -direction of single crystals [6]. Two types of anomalies are evidenced in the dielectric permittivity curves. One of them, a peak-like anomaly in both ε0(T) and ε00(T) of every compound is observed at low temperatures, typically below 20 K, and it is frequency independent. These results are consistent with the only published work with ε0(T) curves for single crystals of Gd1−xYxMnO3 , with x= 0.0 and 0.1, where a peak-like anomaly at 18 K and 22 K for x= 0.0 and x= 0.1, respectively, is observed [6]. A similar anomaly at around 19 K is also reported by other works on GdMnO3 single crystal [4, 48]. As Y -concentration increases, the maximum of this anomaly shifts towards lower temperatures, in such a way, that for x= 0.3 and 0.4 it is no longer achieved in our temperatures range of operation. The other type consists on a step-like anomaly in ε0(T) and a broad peak in ε00(T) , strongly dependent on the measurement frequency. From the selected frequencies displayed in Figures 32 and 33, it is clear that the step of ε0(T) and the maximum of the broad peak in ε00(T) shift to lower temperatures as frequency decreases. This behavior is associated with a thermal-driven dielectric dipolar relaxation process and, like in other rareearth manganites, no correlation to any of the magnetic transitions exists, pointing out for a non-cooperative mechanism driver for this relaxation [49]. 5.2.1 Dielectric Relaxation In the following, we will discuss the main features of the dipolar relaxation process aforementioned. We start with the discussion of the model which better describes the relaxation mechanism. Figure 34 shows the imaginary part of the dielectric permittivity of Gd0.9Y0.1MnO3 , as a function of the real part, both measured at 30 K at dierent frequencies. For this case, ε00(ε0) lies on a semicircle with center on the ε0 -axis, following the equation ε002= [ε(0) −ε0].[ε0−ε(∞)] , (29) where ε(0) and ε(∞) are the intersection of the semicircle with the ε0 -axis. ε(0) stands for the lower frequency and ε(∞) for the upper frequency limits of the dielectric permittivity, associated with the relaxation process. The ε00(ε0) semicircle is typically observed in compounds exhibiting a relaxation process with a single relaxation time, described by the Debye model [50] ε0(ω) = ε(∞) + ε(0) −ε(∞) 1 + ω2τ2, (30) ε00(ω) = [ε(0) −ε(∞)] ωτ 1 + ω2τ2, (31) where τ is the relaxation time and ω is the angular frequency. 57 Figure 37: ε0(T) (closed symbols) and ε00(T) (open symbols) curves measured in heating runs at 500 kHz, for x= 0.0, 0.1, 0.2, 0.3 and 0.4. 64 of temperature occur at the boundaries of magnetic phases, pointing out for a magnetoelectric coupling in Gd1−xYxMnO3 , with 0≤x≤0.4 . 5.3 Magnetic Response 5.3.1 Paramagnetic phase Figure 38 shows the temperature dependence of H M , which corresponds to the inverse of the molar magnetic susceptibility in the paramagnetic phase, above TN . From the linear behavior of H M(T) above 60 K, all compounds closely follow the Curie-Weiss law χ=C T−θp⇔1 χ=−θp C+T C, (36) where χ is the magnetic susceptibility, θp is the Curie-Weiss temperature and C is the Curie-Weiss constant. Figure 38: Inverse of the M H ratio (open symbols) for x= 0.1, 0.2, 0.3 and 0.4. The solid line was calculated from the best t of Equation 36 to the experimental data, above 100 K. Fitting Equation 36 to the H M(T) data, above 80 K, the Curie-Weiss temperature (θp) and constant C were determined. The eective paramagnetic moment (µeff ) can be associated with the Curie-Weiss constant (C) by [51] µeff =p3kBC. (37) From Equation 37, the eective paramagnetic moment (µeff ) was calculated for all compositions. Table 12 shows the values of the Curie-Weiss temperature and the eective paramagnetic moment for the studied compositions. 65 Composition θp(K)µeff (µB) x= 0.0 -38 9.20 ±0.01 x= 0.1 -37 8.82 ±0.01 x= 0.2 -40 8.41 ±0.01 x= 0.3 -40 7.69 ±0.01 x= 0.4 -38 7.16 ±0.01 Table 12: Curie-Weiss temperature and eective paramagnetic moment for every compound. θp takes almost the same value, around -40 K, for every compound, and µeff decreases as Y -amount increases. This decreament was expected, as the Gd3+ ion is magnetic while the Y3+ one is not. Taking into account that s= 2 for the Mn3+ ions, which is corroborated by the analysis of the entropy variation referred to above, the contribution of the Mn3+ ions for the eective magnetic momentum is µMn = 4.80 µB [44]. For the particular composition of GdMnO3 , the eective magnetic momentum of the Gd3+ ions (µGd) can be calculated through µ2 eff =µ2 Gd +µ2 Mn , (38) where the obtained result is µGd = 7.85 µB . This result is concordant to others published in literature, which report µGd = 7.94 µB [44]. 5.3.2 Magnetic Ordering Figure 39 shows the temperature dependence of the molar magnetic response for x= 0.0, 0.1, 0.2, 0.3 and 0.4, measured in zero eld-cooling (ZFC) and eld-cooling (FC) conditions, under an applied eld DC magnetic eld of 40 Oe . First of all, we would like to stress that the increase of molar magnetic response at low temperatures is a consequence of the Gd3+ spin ordering, which takes place at TGd N∼ 5 K [46]. This interpretation is consistent with the decrease of the magnitude of magnetic response as Y -concentration increases. Moreover, the contribution coming from the Mn3+ spin system is superimposed and aects the magnitude of the magnetic response measured in ZFC and FC conditions. The contribution of the Mn3+ spins to the magnetic response is apparent in the dierence between the magnitude of the ZFC and FC magnetic response curves, as it will be explained in the following. The dierence between the FC and ZFC curves enable to arrange the compounds in two sets. One set consists of the compositions with x= 0.0 . This set is characterized by a much larger magnitude of the magnetic response and a dierence between the FC and ZFC curves. The increase of the magnetic response in FC conditions evidences for a weak ferromagnetism in the GdMnO3 compound, stable below 30 K. These dierences in the magnetic response emerge from the Mn3+ spin canting, which is characteristic of the weak ferromagnetism present in the canted A-type antiferromagnetic phase, and it is concordant with what has been reported for RMnO3 , with R= Eu to Dy [1]. Also, a small anomaly is detected in the M(T) curves around 18 K. This anomaly will be addressed later. The second set, comprising the compositions with x≥0.2 , shows a superposition of the ZFC and FC curves, and no enhancement of the magnetic response could be observed through the DC magnetic eld applied in the cooling run. The absence of an increase in the magnetic response under FC conditions points for an antiferromagnetic character at the aforementioned compositions. The absence of a typical antiferromagnetic M(T) curve, which is characterized by a gradually decrease of the magnetic response as temperature decreases below the critical temperature, is a consequence of the magnetic response of the system to the Gd3+ spins ordering at low temperature, that superimpose to the response of the Mn3+ spins. The interpretation of the M(T) curves for the case of x= 0.1 should be more prudent. The dierence between the FC and ZFC curves is non-vanishing, but it is far lesser than for the x= 0.0 composition. Also, 66 Figure 39: Temperature dependence of the molar magnetic response for Gd1−xYxMnO3 , with x= 0.0, 0.1, 0.2, 0.3 and 0.4, measured in ZFC (closed symbols) and FC (open symbols) conditions, under an applied magnetic eld H= 40 Oe . 67 the magnitude of the magnetic response is comparable with the obtained for the compositions with higher x . The dierence between the FC and ZFC curves could be attributed to a weak ferromagnetic character of this composition. However, an analysis of the M(H) relations leads us to another conclusion. Figures 40 to 43 show the magnetic hysteresis loop measurements, M(H), for x= 0.0, 0.1, 0.2 and 0.4, respectively. Figure 40: Magnetic hysteresis loops measurements for x= 0.0 at xed temperatures T = 5 K, 15 K and 25K. At 25 K, for the x= 0.0, 0.2 and 0.4 compositions, linear M(H) relations are observed in the Figures 40 to 43. In Figure 40, for GdMnO3 , a well dened ferromagnetic hysteresis loop appears at 15 K and persists down to 5 K. This result gives strong evidences of the weak ferromagnetic character of GdMnO3 , due to spin canting, which establishes below 20 K. Moreover, at this temperature, a small but clear anomaly in both the ZFC and FC curves is observed (cf. Figure 39). This temperature is in good agreement with the critical temperature reported for the collinear-sinusoidal incommensurate antiferromagnetic phase to canted A-type antiferromagnetic phase transition in GdMnO3 [1]. For the composition of x= 0.1 (Figure 41) instead of a typical ferromagnetic hysteresis loop, a linear M(H) relation in the low-strength magnetic eld range is observed at 5 K. A clear change of the M(H) relation is detected for magnetic elds stronger than 10 kOe , where double hysteresis loops appear, pointing for an antiferromagnetic response. The aforementioned result is in good agreement with the M(H) relation published by Ivanov et al [6], which evidences the antiferromagnetic character for the composition x= 0.1. The dierence between ZFC and FC M(T) curves (cf. Figure 39 ) could arise from the precursor eects of the Gd3+ ordering, which also can not be all discarded in the magnetic response of GdMnO3 . 68 Figure 41: Magnetic hysteresis loops measurements for x= 0.1 at xed temperatures T = 5 K and 15 K. Figure 42: Magnetic hysteresis loops measurements for x= 0.2 at xed temperatures T = 5 K, 15 K and 25K. 69 Figure 43: Magnetic hysteresis loops measurements for x= 0.4 at xed temperatures T = 5 K, 15 K and 25K. In Figures 42 and 43, for x= 0.2 and 0.4, respectively, the M(H) relations recorded at 5 K and 15 K do not exhibit any clear hysteresis, but only a non-linear behavior for strong magnetic elds. Moreover, no hint of double hysteresis were observed up to 50 kOe . We note that at 25 K, no hysteric M(H) relation could be detected for any composition. So, for x= 0.0, a weak ferromagnetism is observed, due to the Mn3+ spin canting, while for x= 0.1, an antiferromagnetic response is evident. The overall set of experimental results obtained for x= 0.2 and 0.4 are interpreted as evidence of an antiferromagnetic response. We will again touch this issue in the last chapter. If our interpretation is correct, we can conclude that the increase of Y -concentration favors the antiferromagnetic interactions against the ferromagnetic ones. 5.4 Critical Temperatures After characterizing the magnetic properties of the Gd1−xYxMnO3 , with 0≤x≤0.4 , it is interesting to correlate the M(T), ε0(T) and ε00(T) curves in order to get information regarding the anomalies which signalize the phase sequence in such materials. This information, along with the macroscopic characterization will enable us to draw the phase diagram of the system, which will be presented later in this work. Figures 44 and 45 show the ε0(T) and ε00(T) curves, measured at 500 kHz , along with the temperature derivative of the magnetic response curve measured in FC condition, and of the ε0(T) curve, for the compositions 70 x= 0.0, 0.1, 0.2, 0.3 and 0.4. Figure 44: ε0(T) , ε00(T) , dε0 dT (T) and dM dT (T) curves for x= 0.0 and 0.1. Vertical solid lines mark the critical temperatures. Let us focus on the Figure 44, referring to x= 0.0. The anomaly observed at around 18 K in the temperature dependence of the magnetic response is revealed as an evident anomaly in dM dT (T) . At this temperature, no anomaly could be observed in ε0(T) and ε00(T) curves nor in dε0 dT (T) . However, the anomaly of the ε0(T) and ε00(T) curves at TC1= 14 K must mark the same anomaly of the magnetic response. The dierent temperature rates in these measurements can account for the small temperature mismatch. The cases of x= 0.1 and 0.2 can be discussed together. The anomalies observed at TC1 and TC2 in both the ε0(T) and ε00(T) curves, also evident in the dε0 dT (T) curves, do not have counterparts in the dM dT (T) curves. For x= 0.3 and 0.4, a clear anomaly in the dM dT (T) curves is detected at TC2= 25 K and 28 K, respectively. At TC2 , the ε00(T) curve for x= 0.3 reveals a small, but well dened, step-like anomaly, while both the ε0(T) and the dε0 dT (T) curves could not display any anomalous behavior. For x= 0.4, no clear anomaly could be found in the ε0(T) , ε00(T) nor dε0 dT (T) curves. 71 Figure 45: ε0(T) , ε00(T) , dε0 dT (T) and dM dT (T) curves for x= 0.2, 0.3 and 0.4. Vertical solid lines mark the critical temperatures. 72 Chapter 6 Electric Polarization and Magnetoelectricity The polar properties are a matter of relevant importance in studying magnetoelectric systems. In magnetoelectric rare-earth manganites, the emergence of polar ordering is intrinsically associated with the magnetic structure, which implies the loss of the centro-symmetric structure, and it is consequence of the magnetoelectric coupling. Moreover, the eect of an applied electric eld on the magnetic response is another feature of the magnetoelectric coupling, very important for some technological applications, as the selective spin transport controlled by electric eld. This chapter is addressed to ascertain the polar properties of Gd1−xYxMnO3 system by studying the thermally stimulated depolarizing currents. The results obtained enable to sort out the eect of Y -concentration and external eld on the polar properties, giving relevant information concerning the driving mechanisms of the magnetoelectric coupling. This chapter ends with the study of the eect of an electric eld on the magnetic properties of this system. 6.1 Polar Properties Figure 46 shows the current density divided by the poling electric eld strength, measure in heating run at a rate of 5K.min−1 after poling the samples of Gd1−xYxMnO3 , with 0≤x≤0.4 , below 300 K. For all compounds, a broad anomaly above 60 K and a double peak located near 20 K are observed. In the following, the broad anomaly with be called higher temperature anomaly, and the other one as low temperature anomaly. In the parent Eu1−xYxMnO3 system, with 0≤x≤0.5 , such kind of anomalies were also observed [52]. In Y -substituted EuMnO3 compounds, the experimental study of the high temperature anomaly as a function of poling electric eld strength and temperature change rate allowed to conclude that the current density peak is associated with a dipolar relaxation mechanism, well described by the Vanderschueren-Gasiot model [53]. We have analyzed in detail the behavior of the high temperature anomaly under dierent poling electric eld strengths and temperature change rates. As a detailed study of this dipolar relaxation is out of the scope of this work, just for the sake of clarity we shall discuss the particular case of Gd0.9Y0.1MnO3 , in order to discuss the origin of the high temperature anomaly in Gd1−xYxMnO3 , with 0≤x≤0.4 . Figure 47(a) shows the current density measured at dierent xed poling electric eld strengths, for Gd0.9Y0.1MnO3 . As the poling electric eld strength increases, the amplitude of the high anomaly also increases. The maximum current density JM is a linear function of the poling electric eld strength, as it can be observed in Figure 47(b). Figure 48 shows the current density measured at dierent xed heating rates, after cooling the sample with a 137 V.cm−1 poling electric eld. The high temperature anomaly shifts to higher temperatures as the heating rate increases, while the low temperature anomaly does not shift. This result, along with those results referred to above, reveals the dipole relaxation character of the high temperature anomaly, and enable us to identify the displacement currents, measured above 50 K, as a consequence of the dipolar relaxation process. We assume that this is the mechanism underlying the high temperature anomaly observed in the current density for all studied compounds. As this dipolar relaxation is not associated with a cooperative polar phenomena neither a magnetic one, the study of the main characteristic of this dipolar relaxation process is beyond the scope of this work. Figures 49 and 50 show the current density as function of temperature, measured in heating run, at a rate of 5K.min−1 , after cooling the sample with a poling electric eld below 30 K, along with the electric polarization variation curve, determined by time integration of the current density, and the complex dielectric permittivity real and imaginary parts. By applying the poling electric eld below 30 K, we prevent the activation of the 73 ∆M(T) , for the compositions of x= 0.0, 0.1, 0.2 and 0.4. For the x= 0.3 sample no clear results were obtained. From Figure 52 the studied compounds can be divided in two sets. The rst set consists of GdMnO3 . For x= 0.0, it is clear that both M(E= 0, T) and M(E6= 0, T) curves coincide in such a way that above 28 K ∆M= 0 , and no eect of the applied electric eld on the magnetic structure can be detected. At T= 28 K, and down to T= 16 K, M(E6= 0, T) takes lower values than M(E= 0, T) , and ∆M is positive. This means that in between these temperatures the antiferromagnetic character is enhanced by the applied electric eld. Below 16 K and down to 4 K, M(E6= 0, T) becomes larger than M(E= 0, T) , and ∆M < 0 . The applied electric eld enhances the ferromagnetic interactions against the antiferromagnetic ones. At around TC , a well dened change of slope of the ∆M(T) function is observed, marking the low temperature magnetic phase. On further cooling, e ∆M(T) displays another change of slope at TGd N= 7 K, marking the magnetic ordering of the Gd3+ spins. The second set consists of the x= 0.1, 0.2 and 0.4 compositions. In Figure 52(b), for x= 0.1, both M(E= 0, T) and M(E6= 0, T) curves coincide until 25 K, close above TC2 . Below this temperature, M(E6= 0, T) takes lower values than M(E= 0, T) , and ∆M(T)>0 . This behavior of ∆M(T)>0 reaches a maximum around 19 K, and then decreases and reaches ∆M= 0 at 13.3 K, a little below TC1 . As temperature further decreases, ∆M becomes positive again, until it collapses at around 5 K. In these two well dened temperature regions, the antiferromagnetic interactions are enhanced by the applied electric eld. A similar behavior is found in Figure 52(b) for x= 0.2. For this composition, that temperatures where ∆M(T) becomes non-zero are closer to TC1 and TC2 . The magnitude of the higher temperature peak of ∆M(T) decreases rapidly as Y -concentration increases, no longer being detected for x= 0.4. As for the lower temperature peak of ∆M(T) decreases much slighter, still being clearly observed for x= 0.4. In the composition of x= 0.4, TC1 is concordant with the temperature where ∆M(T)>0 . In this second set, the applied electric eld clearly enhances the antiferromagnetic interactions below roughly TC1 , for x= 0.1, 0.2 and 0.4. Assuming this interpretation, the applied electric eld favors the stabilization of the ferroelectric phase below TC1 , as it should be expected. Several critical temperatures have been identied from the study of the temperature dependence of the specic heat, the complex dielectric permittivity, the magnetic response and the thermally stimulated depolarizing currents, along with the investigation of the magnetoelectric properties of the Gd1−xYxMnO3 system, with 0≤x≤0.4 . The studied compositions were already separated in two sets, where the rst one consists of GdMnO3 , where two transition temperatures were found: TC= 17 K and TN= 41.8 K, while the second one consists of the compositions of x= 0.1, 0.2, 0.3 and 0.4, which compositions present three transitions temperatures, TC1 , TC2 and TN . Finally, Table 13 summarizes all the critical temperatures for the latter set. Composition TC1 (K) TC2 (K) TN (K) x= 0.1 17.8 22 41.9 x= 0.2 14.8 22 41.7 x= 0.3 14.0 25 42.0 x= 0.4 13.5 29 41.7 Table 13: Phase transition temperatures for x= 0.1, 0.2, 0.3 and 0.4. 80 Figure 52: (a) Magnetic response at low temperatures, with and without applied electric eld for x= 0.0, 0.1 and 0.2 composition as a function of temperature. (b) Dierence of the above magnetic response curves plotted with temperature. Vertical dashed lines mark the critical temperatures. 81 . Chapter 7 Spin-Phonon Coupling In the Structural and Morphological Analysis at Room Temperature Chapter, we have presented a study of the lattice dynamics of Gd1−xYxMnO3 , with 0≤x≤0.4 , at room temperature. A detailed mode assignment and the main structural distortions induced by the substitution of the Gd3+ ion by the Y3+ one were discussed. In this chapter, we present the study of the lattice dynamics at low temperatures, which is aimed at correlating the anomalous behavior of the mode frequencies across the magnetic phases with the spin arrangement and the balance of the competitive ferro and antiferromagnetic interactions in these compounds. The results obtained will enlighten the coupling between phonons and spins and its role in the stabilization of the ferroelectric phase. Figure 53 shows the unpolarized Raman spectra recorded, for every compound, at several xed temperatures in the 9 K to 300 K temperature range. As it can be seen in Figure 53, the Raman spectra recorded below room temperature exhibit slight changes in their proles. As the temperature decreases, the Raman bands become narrower and better resolved, due to the decrease of disorder arising from thermal motions. Particularly, no new Raman bands were detected at low temperatures, even in the magnetic phases which allow ferroelectricity. A ferroelectric phase imposes a symmetry break, as the crystallographic structure must be non-centro-symmetric. The absence of new well-dened activated Raman bands and the prevalence of the high-temperature spectra prole point out for the weak polar character of these compounds, and for the improper nature of the ferroelectric phases, as it was stressed in the previous chapter. In fact, as the electric polarization of these compounds is very low (few pC/cm2 ), the intensity of the new Raman bands, which is proportional to the oscillator strengths of the infrared active modes, is very small and could not be detected with the available experimental conditions. In order to get deep insight in the structural changes induced by the magnetic phase transition, the Raman spectra were simulated according to Equation 13. Figure 54 shows an example of the quality of the simulation of the Raman spectrum of Gd0.6Y0.4MnO3 , recorded at 9 K, along with the mode assignment presented in Table 4. The magnetically induced structural changes reveal small changes in the spectral prole. Therefore, the mode parameters should be determined with high accuracy. So, we have chosen to analyze the Raman band assigned to the tilt mode, T (lattice mode), and to the symmetric stretching mode, SS (internal or molecular mode), as these bands are well dened and they are well separated from other Raman bands. Moreover, these modes are the best sensors of the structural distortions occurring at the magnetic phase transitions. The T mode, being a lattice mode involving the octahedra tilting, is suitable to probe lattice distortions associated with the long-range cooperative phenomena, like the stabilization of modulated structures. Conversely, the SS mode probes the relative octahedra distortion in the Mn −O2 plane, resulting from the oxygen displacement. According to the spin-phonon coupling model, a change in phonon frequency as entering the magnetic phases is expected, reecting the phonon renormalization, arising from the exchange integrals and spin-spin correlation function [23]. In order to calculate the temperature anomalous behavior of the frequency due to the magnetic interactions, we have described the purely anharmonic temperature dependence of the frequency by the model [54] ω(T) = ω(0) + C1−2 ex−1, (40) where C is a model constant and x is given by x=èω0 2kBT , where è is the reduced Planck constant, kB is the Boltzmann constant and T is temperature. Figures 55 and 56 show the temperature dependence of the wavenumber of the T and SS Raman bands, respectively obtained from the best t of Equation ?? to the 83 Figure 53: Raman spectra of Gd1−xYxMnO3 , with 0≤x≤0.4 , recorded at several xed temperatures in the 9 K to 300 K temperature range. 84 Figure 54: Example of the best t to the Raman spectrum of Gd0.6Y0.4MnO3 , recorded at 9 K. experimental spectra, recorded for the dierent compositions. Equation 40 was tted to the experimental curves ω(T) above 100 K, and the solid lines in Figures 55 and 56 were calculated from the best t, which was extrapolated for lower temperatures. For GdMnO3 and Gd0.9Y0.1MnO3 a downward deviation of the wavenumber as a function of temperature from the extrapolated anharmonic behavior is evident for both the T and SS modes, below 100 K and 75 K, respectively. This deviation starts to be observed well above TN . This is due to local ordering regarding the spins which is a sign of precursor eects of the magnetic phase transitions, as it is the case of EuMnO3 compound and its Y -substituted parents [43]. In contrast to this, for the compositions with x= 0.2 and 0.4 an upward deviation, starting near TN , is clearly observed. In the case of Gd0.7Y0.3MnO3 , the ω(T) curve of the SS mode exhibits only a hint of an upward deviation below 25 K. Note that the deviations from the purely anharmonic behavior are monotonous for x= 0.1, 0.2 and 0.3, but are not for x= 0.0 and 0.4. For the case of GdMnO3 , ω(T) of both the T the SS mode starts to increase below TC≈18 K For Gd0.6Y0.4MnO3 the ω(T) curve of the SS mode exhibits a maximum value at around 34 K, decreasing below TC2≈27 K, while the ω(T) curve of the T mode shows a downward deviation at around 21 K. These anomalies reect the occurrence of structural distortions in both lattice and MnO6 octahedra, involving the tilt angles and the Mn−O2 bond lengths, respectively, induced by the magnetic ordering taking place at those temperatures. Within the spin-phonon coupling model, the wavenumber deviation of the phonon as a function of temperature is determined by the spin-spin correlation function, according to Equation 5. When there exist competitive ferro and antiferromagnetic interactions, the last equation can be generalized to [23] ω(T)−ω0=−RF M < Si·Sj>+RAF M < Si·Sj> , (41) where RF M and RAF M are spin dependent force constants of the lattice vibrations, dened as the squared derivatives of the ferro and antiferromagnetic exchange integrals, respectively, with respect to the normal coordinate. The < Si·Sj> term is associated with the ferromagnetic nearest neighbor Mn3+ spins, while the < Si·Sj> one with the antiferromagnetic next-nearest neighbors spins. The magnetic properties depend on the balance between the ferro and antiferromagnetic interactions, which are characterized by the corresponding 85 Figure 55: Wavenumber of the (a) T and b) SS modes as a function of temperature, for the compositions x= 0.0, 0.1 and 0.2. 86 Figure 56: Wavenumber of the (a) T and b) SS modes as a function of temperature, for the compositions x= 0.3 and 0.4. 87 exchange integrals, and so, by the value of RF M and RAF M . Depending on the relative values between RF M and RAF M , positive or negative frequency deviations to the purely anharmonic behavior are predicted. Before continuing the analysis of the experimental data, considerations regarding the spin-phonon coupling model must be taken. In the following, we will assume that the spin-spin correlation function of the nearest neighbors and the next-nearest neighbors for the Mn3+ spins have the same temperature dependence. Moreover, for each normal mode, we also assume constant values for RF M and RAF M . Taking this in mind, Equation 41 is rewritten as follows ω(T)−ω0≈(RAF M −RF M )< Si.Sj> , (42) where from comparison with Equation ?? , the spin-phonon coupling parameter can be identied as follows, γ=RAF M −RF M . According to Equation 42, the frequency deviations observed in Figure 55 are interpreted by assuming the coexistence of competitive ferro and antiferromagnetic interactions, whose balance depends on the Y - concentration and spin-phonon coupling. For x≤0.1 , the Raman deviation is negative, and the dierence between ω(T) and the extrapolated anharmonic behavior decreases with increasing of Y -amount. This means that the dierence RAF M −RF M is negative and its absolute value decreases with increasing x up to 0.1, evidencing the reinforcement of the antiferromagnetic interactions as the Y -amount increases. This interpretation is not necessarily in contradition with the one presented in the Magnetic Response section. Despite being stated that the x= 0.1 composition has dominant antiferromagnetic interactions, presenting no weak ferromagnetism, the dierence RAF M −RF M can still be negative, as it results from the derivatives of the magnetic exchange integrals. So, it is worth to stress that the antiferromagnetic interactions can still dominate while RF M also does, indicating that the composition with 0.1 of Y -concentration has a transition nature. For x≥0.2 the positive frequency deviation points for a positive RAF M −RF M value, which means the reinforcement of the antiferromagnetic interactions against the ferromagnetic ones, in good agreement with the previous magnetic response data. Despite being always positive, for the x= 0.4 compound, the deviation behavior is not monotonous, having an inversion point at around 30 K. This means that, as temperature further decreases below this point, the dierence RAF M −RF M , although still positive, is decreasing, and so, the ferromagnetic interactions must be the ones being reinforced. Figure 57 presents the temperature dependence of the full width at half maximum (FWHM) of the SS Raman band for every compound. The linewidth is known to follow the anharmonic temperature behavior [54] Γ(T) = Γ(0) 1 + 2 ex−1, (43) where x=èω 2kBT . The t procedure of Equation 43 to the experimental data was done for temperatures above 100 K, and the obtained best t is also plotted in Figure 57, as the solid line. The magnetic ordering in the Gd1−xYxMnO3 , with 0≤x≤0.4 , are also revealed through the anomalies in the linewidth as a function of the temperature. In the case of the linewidth of the SS mode, whose temperature dependence is presented in Figure 57, deviations from the extrapolated anharmonic behavior below 100 K are apparent. The temperature dependence of the experimental data, above 100 K, is well described by Equation 43, for every compound. It is interesting to see that below around 75 K, above TN , a broad peak-like anomaly is evident in every Γ(T) curve. This anomaly is very clear in every compound. The anomaly in the x= 0.1 and 0.4 compounds exhibits a positive deviation to the anharmonic behavior close above TN . As it reaches TN the deviation shifts to a negative one, while for x= 0.2 and 0.3 the deviation is always negative. As temperature further decreases, for x= 0.3 and 0.4, the Γ(T) curve collapses into the extrapolated anharmonic behavior. We did not conclude any physical meaning from the positive or negative character of the anomaly, 88 Figure 57: Temperature dependence of the linewidth of the SS band for every compound. Solid line represents the best t of the anharmonic behavior. nevertheless, this anomaly is interpreted to be due to the magnetic transition at TN . The Γ(T) curve of the SS mode is able to sense the precursor eects of the magnetic ordering of the Mn3+ spins. For temperatures close above TN the system has lost its long-range correlation, but medium and short-range correlations still exist, modifying the system disorder. As Equation 41 shows, the mode frequency is dependent on the magnetic exchanges derivatives and also of the neighbors spin correlation function. Qualitatively, the frequencies deviations were interpreted by the dominant magnetic character, but one can further investigate the coupling between lattice and magnetic order quantitatively through the calculation of the spin-spin correlation function. The spin-spin correlation function can be directly calculated from the integral of the magnetic component of the specic heat, as follows [55] ˆ10 200 CmdT = 6J < Si·Sj> , (44) where Cm is the magnetic component of the specic heat, 6 is the number of nearest-neighbors and J is the magnetic exchange constant. In the Specic Heat section of the Temperature Characterization chapter, the curve of the magnetic component of the specic heat was presented. Figure 58 shows the temperature integration of the magnetic contribution to the specic heat, as a function of temperature, for the compositions x= 0.0, 0.1, 0.2, 0.3 and 0.4. The obtained result reveals that the high temperature phase is paramagnetic, as the spin-spin correlation function is zero, meaning no short-range magnetic order for any compound is exhibited. Note that, consistently with the previous section results of the vibrational modes frequency deviation in Fig.55, the spin correlation 89 . References [1] T. Goto, T. Kimura, G. Lawes, A. P. Ramirez and Y. Tokura, Ferroelectricity and Giant Magnetocapacitance in Perovskite Rare-Earth Manganites, Physical Review Letters , 92 , 257201 (2004). 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