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ISRN Condensed Matter Physics

Volume 2013 (2013), Article ID 497073, 43 pages

http://dx.doi.org/10.1155/2013/497073

## Hexagonal Manganites—(RMnO_{3}): Class (I) Multiferroics with Strong Coupling of Magnetism and Ferroelectricity

Texas Center for Superconductivity and Department of Physics, University of Houston, Houston, TX 77204-5002, USA

Received 15 October 2012; Accepted 2 November 2012

Academic Editors: I. Galanakis, M. Higuchi, and V. Kochereshko

Copyright © 2013 Bernd Lorenz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Hexagonal manganites belong to an exciting class of materials exhibiting strong interactions between a highly frustrated magnetic system, the ferroelectric polarization, and the lattice. The existence and mutual interaction of different magnetic ions (Mn and rare earth) results in complex magnetic phase diagrams and novel physical phenomena. A summary and discussion of the various properties, underlying physical mechanisms, the role of the rare earth ions, and the complex interactions in multiferroic hexagonal manganites, are presented in this paper.

#### 1. Introduction and Brief History

Since the discovery of the intimate relations between electric and magnetic phenomena by Oersted, Ampère, Faraday, and others, which ultimately led James Clerk Maxwell to formulate the unified theory of electromagnetism, the mutual interaction between magnetic (electric) properties and electric (magnetic) fields has been in the focus of interest for more than a century. Wilhelm Conrad Röntgen discovered in 1888 that a dielectric became magnetized when moving in a uniform electric field [1]. His study was motivated by the following reasoning: when a dielectric sheet, polarized in the external electric field, is moved in the direction perpendicular to the field lines, the motion of negative and positive charges, separated through the induced polarization, becomes equivalent to two electrical currents moving into opposite directions on either side of the sheet. Those currents generate a magnetic field, that is, the dielectric becomes magnetized. The experimental setup to prove the magnetoelectric effect did involve a fast rotating dielectric disc between two horizontal capacitor plates. In the same communication, Röntgen also conjectured that the inverse effect should exist, namely the change of the polarization of a moving dielectric induced by an external magnetic field, which was indeed experimentally shown by Wilson in 1905 [2].

The possibility of a magnetoelectric effect in (non-moving) materials was later discussed by P. Curie from a symmetry point of view [3]. It took, however, many more years to understand the importance of the violation of time reversal symmetry (either by moving the dielectric, by external magnetic fields, or by magnetic orders) for the magnetoelectric coupling to become effective. The realization of the magnetoelectric effect in the antiferromagnetic phase of Cr_{2}O_{3}, predicted by Dzyaloshinskii [4] on symmetry grounds, was experimentally verified through the demonstration that an electric field did induce a magnetization [5–7] as well as the reverse effect, the magnetic field induced electrical polarization [8]. It should be noted that the observed magnetoelectric effect in Cr_{2}O_{3} is linear, that is, the induced magnetization (polarization) increases linearly with the electric (magnetic) field, in agreement with the lowest order of the magnetoelectric part of the thermodynamic potential, derived by Dzyaloshinskii [4]:
and are the diagonal elements of the linear magnetoelectric tensor parallel and perpendicular to the -axis, respectively. In an attempt to search for the microscopic origin of the magnetoelectric effect in Cr_{2}O_{3}, Rado linked the linear magnetoelectric coefficient to the sublattice magnetization of the antiferromagnetic order and emphasized already on the important role of the spin orbit interaction [9].

In the following years, many more magnetoelectric materials have been discovered. In 1972, Hornreich listed about twenty magnetoelectric materials [10]. A more recent list of compounds showing a linear magnetoelectric effect can be found in the article by Schmid [11]. It should be noted that the linear magnetoelectric coefficient is relatively small because it is limited by the dielectric () and magnetic () susceptibilities according to the relation [12–14].

The largest coefficient was found in the antiferromagnetic state of TbPO_{4} at K, ps/m [11, 15, 16]. This coefficient is about 180 times larger than of Cr_{2}O_{3}, however, the field-induced electrical polarization of 58 *μ*C/m^{2} measured at a field of 1 kOe [16] (or 585 *μ*C/m^{2} if extrapolated to a field of 10 kOe) is relatively small. Furthermore, the linear magnetoelectric effect is often forbidden by symmetry, or it breaks down at larger field values. For example, in TbPO_{4} the antiferromagnetic order and the magnetoelectric effect are destroyed in fields above 8 kOe [17]. The allowed tensor elements of in different magnetic point groups are listed and discussed in [11, 18].

Under the limitations for the linear magnetoelectric effect discussed above, the higher order coupling terms in the free energy may be important and of interest, particularly at larger magnetic fields. They are derived from the expansion of the free energy density with respect to electric and magnetic fields (in SI units) [19, 20]:

We use the convention of summation over pairs of equal indices in (3). The first term, , is the free energy in the absence of magnetic or electric fields, the next two terms represent the energy gain due to the interaction of a spontaneous polarization (magnetization) with the electric (magnetic) field, for example, in a ferroelectric (ferromagnetic) state. The following two terms in (3) are due to the interaction of the field-induced polarization (magnetization) with the corresponding electric (magnetic) fields. () and () are the absolute and relative dielectric permitivities (magnetic permeabilities), respectively. and are symmetric second rank tensors. All terms in (3) represent sums over the three respective indices , , and , representing the three spatial directions.

The mixed field terms in line two of (3) start with the linear magnetoelectric energy with the second rank magnetoelectric tensor , followed by the next higher order terms coupling electric and magnetic fields. The third rank tensors and are also called the coefficients of the bilinear (quadratic) magnetoelectric effect [11]. This are the terms that may result in a magnetic field-induced polarization (or an electric field-induced magnetization), even if the linear magnetoelectric effect is forbidden by symmetry (). From (3) the magnetic field-induced polarization change in the absence of an electric field can be calculated:

A similar expression is obtained for the change in magnetization due to external electric fields in the absence of a magnetic field:

Bilinear magnetoelectric effects have indeed been measured in different compounds in the 1960’s [21, 22] and the superposition of contributions from both, the linear and the bilinear magnetoelectric effects have been reported for a member of the boracite family of compounds [23] and very recently in the antiferromagnetic state of LiFeP_{2}O_{7} [24]. In the boracites, the bilinear magnetoelectric effect turned out to reduce the field-induced polarization from the linear coupling and only moderate values of less than 2 *μ*C/ have been achieved in fields up to 10 kOe. In the search for materials with large magnetoelectric effect, the focus was recently directed toward the rare earth iron borate compounds, RFe_{3}(BO_{3})_{4} (R: rare earth) (for a review see [25]). Field-induced polarization values of up to 500 *μ*C/ have been reported in SmFe_{3}(BO_{3})_{4} at 10 kOe [26] and NdFe_{3}(BO_{3})_{4} [27]. Replacing the transition metal iron by nonmagnetic aluminium, even larger bilinear magnetoelectric effects have been discovered recently in the rare earth aluminum borate, RAl_{3}(BO_{3})_{4}, with polarization values up to 3600 *μ*C/m^{2} in HoAl_{3}(BO_{3})_{4} at the maximum field of 70 kOe [28]. This seems to be the largest magnetic field-induced polarization change reported so far. It was further shown that the polarization in fields up to 10 kOe scaled perfectly with , that is, these compounds exhibit exclusively the bilinear magnetoelectric coupling [29].

The thermodynamic condition (2), limiting the linear magnetoelectric effect, has guided researchers to study materials with large dielectric and magnetic susceptibilities. and are large in materials exhibiting ferroic (ferroelectric or ferromagnetic) long range orders, at least near the respective transition temperatures. It has been argued that materials showing ferroelectric and ferromagnetic orders in one and the same phase are rare because the d-electrons of transition metals favoring magnetism are detrimental to the off-center distortions needed to sustain ferroelectricity [30]. However, as early as 1966, E. Ascher et al. [31] reported the discovery of ferroelectricity arising simultaneously with weak ferromagnetism at 64 K in Ni_{3}B_{7}O_{13}I. A sizable magnetoelectric effect and the reversal of the polarization in an external magnetic field was also observed in the coexistence region of both ferroic orders. This work was preceded by the report of the coexistence of ferroelectricity and antiferromagnetism in the perovskite Pb(Fe_{2/3}W_{1/3})O_{3} [32, 33] with significantly different transition temperatures for the two ordered states.

The coexistence and mutual interaction of magnetic (including antiferromagnetic) and ferroelectric orders in matter have attracted increasing attention since then. Many additional materials, originally called ferroelectromagnets, have been studied in the following years, predominantly by groups in the former Soviet Union. An early review by Smolenskii and Chupis (published in 1982, [34]) lists more than 50 compounds showing the coexistence of ferroelectricity and magnetic order in some range of temperature. Among them are more members of the boracite family of compounds, M_{3}B_{7}O_{13}X (M = Cr, Mn, Fe, Co, Ni, Cu and X = Cl, Br, I), the hexagonal manganites, RMnO_{3} (R = Sc, Y, Dy, Ho, Er, Tm, Yb, Lu), compounds with perovskite structure (e.g., BiFeO_{3}), members of the polar BaMF_{4} (M: transition metal) family, and others.

The term “multiferroic”, describing materials in which at least two of three ferroic properties (ferroelectricity, ferromagnetism, or ferroelasticity) occur in the same phase, was introduced by Schmid in 1994 [35]. The definition was later extended to include also antiferroic orders, like antiferromagnetism. A significant number of multiferroic materials known today combine ferroelectricity with antiferromagnetic orders. The continuously increasing interest in the study of multiferroic materials is reflected in the number of scientific papers published annually. Figure 1 shows the annual number of publications with “multiferroic” or “magnetoelectric” in their topic (an extension of a graph originally shown by Manfred Fiebig in a review of the magnetoelectric effect [36]). The graph covers the last 15 years of research, from 1996 to 2011, and it proves the yet growing interest in the field. The number of publications did reach nearly 1000 in 2011 and the tendency is still up. It is also remarkable that a strong increase of research activities is visible past the year 2003, after the magnetic order induced ferroelectric state was discovered in the rare earth manganite TbMnO_{3} [37].

In general terms, multiferroic compounds can be separated into two classes depending on whether the magnetic order and the ferroelectric state arise independently (and at very different temperatures) or the magnetic order breaks the inversion symmetry and induces a ferroelectric state (due to electronic or ionic displacements). In the first class of multiferroics, the ferroelectricity is usually established at much higher temperature (often above room temperature) and the magnetic order appears at low temperatures. In the second class, the transition into the magnetic and ferroelectric states happen at the same temperature. Examples for class one (I) multiferroics are the rare earth manganites RMnO_{3} (R: rare earth, Y, Sc) with hexagonal structure [38–41]. While the ferroelectric transition temperature, , is of the order of 900 K [42], the antiferromagnetic order sets in at typically below 100 K. The first signature of a coupling between both orders had been observed in YMnO_{3} [43]. Another compound family of interest are polar crystals with strongly interacting magnetic ions and a magnetically ordered phase at low-temperature. Although those crystals may not exhibit a ferroelectric phase transition below their melting point, the fact that they crystallize in a structure with a finite lattice polarization makes them an interesting candidate to study the coupling of the magnetic order parameter below with the existing polarization, as shown very recently in LiFeP_{2}O_{7} [24].

The second class (II) of multiferroics includes all materials with a magnetically ordered structure that breaks the spatial inversion symmetry and induces simultaneously a ferroelectric state. Different types of magnetic structures which can stabilize a ferroelectric state have been revealed within the last decade of research. Among them are the transverse spin spiral (realized, e.g., in TbMnO_{3} [37], Ni_{3}V_{2}O_{8} [44], MnWO_{4} [45–47], LiCu_{2}O_{2} [48], CuFeO_{2} [49], CoCr_{2}O_{4} [50], and others), the exchange striction mediated displacement of spins and the associated charges [51] (found in [52] and in the RMn_{2}O_{5} compounds [53, 54]), and the E-type magnetic structures in orthorhombic RMnO_{3} (R = Ho to Lu) [55, 56]. The origin of the aforementioned inversion symmetry breaking magnetic orders was found in strong frustration of the magnetic system due to geometric constraints or competing magnetic exchange interactions. Therefore, different magnetic orders are close in energy and compete for the ground state. This circumstance explains the extreme sensitivity of class (II) multiferroics with respect to small perturbations in form of external magnetic or electric fields [46, 57–60], physical pressure [61–64], and ionic substitutions [65–69]. This sensitivity is an essential ingredient for the development of prospective applications of multiferroic materials as magnetoelectric sensor or a new type of memory elements. Several good reviews of class (II) multiferroics can be found in a number of recent publications [36, 51, 70–74]. It should also be mentioned that ferroelectricity can be induced by charge order. A recent review was devoted to multiferroics with different types of charge order [75].

In the following sections, we will discuss novel phenomena, complex magnetic and multiferroic phase diagrams, and the effects of magnetic fields on the multiferroic properties of class (I) multiferroics. The physics of hexagonal manganites, as representatives of class (I) multiferroics, will be reviewed in detail in Section 2 and a brief summary is presented in Section 3.

#### 2. The Hexagonal Manganites: Class (I) Multiferroics

##### 2.1. Structure, Ferroelectricity, and Magnetism

Ferroelectricity in hexagonal manganites RMnO_{3}, with R = Y, Dy, Ho, Er, Tm, Yb, Lu, or Sc, was discovered as early as 1963 by Bertaut et al. [38]. The structure refinement at ambient temperature revealed the crystal structure described by the polar space group (no. 185) with an unusual five- and sevenfold coordination polyhedra about the Mn and R ions, respectively [39]. A sketch of the structure is shown in Figure 2. The origin of ferroelectricity in hexagonal RMnO_{3} has been a matter of discussion. It should be noted that the manganites do not fulfil the condition of “d^{0}-ness” which leads to an off center displacement in common perovskite ferroelectrics (like BaTiO_{3}) due to the hybridization of empty transition metal orbitals with the oxygen 2p states and the associated second-order Jahn-Teller effect [76]. They also do not possess a “lone pair” (s^{2}) set of electrons which may cause the loss of inversion symmetry through a mixing with an excited (s^{1})(p^{1}) state [77] as, for example, realized in BiMnO_{3} [78]. The lattice distortion in RMnO_{3} resulting in a noncentrosymmetric and polar structure has to be sought in other physical mechanisms.

Extensive studies of the structure over a large temperature range and the ferroelectric properties have been conducted for YMnO_{3}, some of the results will be discussed in the following. Based on first principle calculations, Van Aken et al. [79] have concluded that the ferroelectricity in YMnO_{3} is due to electrostatic and size effects. The structural distortion involves mainly the rotation of the MnO_{5} bipyramids which displaces the oxygen atoms from their centrosymmetric positions and a displacement of Y atoms along the -axis forming a buckled triangular lattice in the ferroelectric state. The resulting huge Y-O displacements along the -axis create large local electric dipoles which are antiparallel (but of different magnitude) for the two inequivalent yttrium ions of the structure. Therefore, the polar state of YMnO_{3} (and all hexagonal RMnO_{3}) is ferrielectric.

The centrosymmetric high temperature phase of YMnO_{3} is stable above K [80–84]. The space group was determined as (no. 195). The transition temperature into the ferroelectric phase was reported to be about 300 K lower and the possible existence of an intermediate phase (space group ) between and was suggested [81, 85]. The symmetry analysis of different modes explaining the possible distortions which lead from the high-temperature to the low-temperature structure are discussed for example, in [86]. However, recent neutron scattering experiments have shown that the best refinement below could only be achieved for the space group, indicating a direct transition from to with a tripling of the unit cell and no intermediate symmetry phase [84]. Anomalies observed by different authors near 900 K were attributed to an isosymmetric transition within the same space group, .

The magnetism in hexagonal RMnO_{3} and the details of the long range order of Mn spins have been studied in the early 1960’s by Bertaut and Mercier [87] and Koehler et al. [88] All hexagonal manganites show antiferromagnetic (AFM) order of the Mn^{3+} spins below their respective Néel temperatures, . Table 1 shows the lattice constants and the Néel temperatures of nine hexagonal RMnO_{3} (R = In, Sc, Y, Dy to Lu). The AFM transition happens below 100 K (except for ScMnO_{3} and InMnO_{3}).

The understanding of the magnetic order compatible with the hexagonal symmetry requires a closer inspection of the lattice structure, mainly the sublattice of the magnetic Mn^{3+} ions. Figure 3 shows a projection of the Mn sublattice along the hexagonal -axis. The Mn^{3+} ions form layers of triangular structure stacked along . Two subsequent layers of Mn are distinguished by color in Figure 3. The R ions (not shown in the figure) are located in the open spaces between the Mn layers. The magnetic interactions of the Mn spins are antiferromagnetic (AFM) super exchange interactions. The spins on a triangular lattice with AFM interactions between nearest neighbors are highly frustrated and the corresponding magnetic structure, compatible with the hexagonal symmetry, is characterized by a non collinear spin arrangement where the three spins on a triangle form an angle of 120^{∘} with one another.

There are two possibilities to arrange the relative spin orientation between neighboring Mn layers, depending on the relative orientation of the pairs of spins (belonging to two layers) along the edges of the magnetic unit cell (and the pair of two spins in the center): in the model the spins of a so defined pair are parallel (shown in Figure 3(a)) whereas they are antiparallel in the model (Figure 3(b)). The angle of the spins with the hexagonal axis determines the details of the magnetic space group. Two preferred values of , and , define the magnetic orders corresponding to the four one-dimensional irreducible representations, to , of the little group () which is identical to the crystal’s space group . A complete group theoretical analysis of the little group was presented by Muñoz et al. [95]. Two of the possible magnetic symmetries (assigned to the and models) had been considered for YMnO_{3} by Bertaut and Mercier based on the results of powder neutron scattering experiments [87], however, a definite conclusion of whether the or model describes the magnetic order best could not be drawn.

The four magnetic space groups associated with the to irreducible representations are listed in Table 2 and their characteristic spin order is visualized in Figure 4. Besides the one-dimensional representations to , there exist two more two-dimensional irreducible representations, and , as discussed in [95]. According to the irreducible representations listed in Table 2, the original models proposed for YMnO_{3} in [87] are described by the ( model) and ( model) representations. Muñoz et al. have conducted extensive powder neutron studies of YMnO_{3} and concluded that the favored magnetic symmetry is described by the representation ( model, ) [95]. However, more recent powder neutron scattering experiments could not distinguish between the originally suggested and representations [96]. It should also be noted that combinations of different irreducible representations are possible and they describe a magnetic structure with an angle between and . A total of four intermediate () magnetic structures can be visualized if the spins on the two sublattices are allowed to rotate by an angle either in the same or in opposite directions, leading to transitions within or between the and models. Their magnetic space groups are , , , and [93]. The four intermediate structures, which may be important to understand spin rotation transitions and new phases induced by magnetic fields, are included in Table 2.

The problem with powder neutron scattering experiments in determining complex magnetic orders is the limited resolution in fitting different magnetic structures to the scattering spectra. Therefore, alternative methods, preferentially working with single crystals, need to be used for a precise magnetic structure determination. Due to the non centrosymmetric structure of hexagonal manganites, nonlinear optical methods can be employed [97, 98]. It was shown that the second harmonic generation (SHG) provides a very sensitive probe of magnetic symmetries since the second order susceptibility tensor obeys selection rules that are characteristic for different magnetic structures. This method was first used by Fröhlich et al. to study the magnetic and ferroelectric orders in YMnO_{3} [99]. Fiebig et al. developed the SHG method further and applied it to the extensive study of the magnetic symmetry of hexagonal manganites [100]. The non-linear susceptibility tensors describing the second harmonic generation experiments have been derived for the hexagonal manganites in [101] and it was shown that they can be used to distinguish different magnetic structures. The results for most of the hexagonal RMnO_{3} are shown in Figure 5. The symmetry of the ordered phase of YMnO_{3} has been uniquely identified as (). It is interesting that ScMnO_{3} and LuMnO_{3} appear to show the coexistence of different magnetic symmetries in some temperature range and HoMnO_{3} experiences a sudden spin rotation from () to () upon decreasing temperature (the spins rotate by an angle of ). The spin rotation in HoMnO_{3} was already proposed by Koehler et al. in their early neutron scattering experiments of hexagonal RMnO_{3} compounds [88].

The above discussion of the magnetic orders in the RMnO_{3} system focusses only on the Mn^{3+} spins; however, most rare earth ions carry their own magnetic moment oriented along the -axis due to a strong uniaxial anisotropy. The interaction of the rare earth moments may cause another magnetic phase transition at low temperature involving mainly the rare earth moments with a possible effect on the Mn^{3+} spins. The rare earth moment order was indeed observed in ErMnO_{3} below K, in YbMnO_{3} below 4 K, and in HoMnO_{3} below 5.4 K [41, 94, 102, 103]. The complex phase diagrams of various RMnO_{3} will be discussed in the following sections.

##### 2.2. Magnetoelectric Coupling and the Complex Phase Diagram of Hexagonal HoMnO_{3}

The coexistence of ferroelectric and magnetic orders below the Néel temperature raises the question of how the two order parameters mutually interact with one another and how different physical properties might be affected by their coupling. Symmetry does not allow for a linear coupling of the AFM order parameter and the -axis ferroelectric polarization. However, higher order couplings mediated through the strong spin-lattice interaction may result in sizable anomalies, for example, of dielectric quantities at the magnetic phase transitions.

The first signature of a coupling between the magnetic order and dielectric properties in hexagonal manganites was observed in YMnO_{3} [43]. A distinct anomaly of the dielectric constant and the loss factor at the Néel temperature provides clear evidence for a strong correlation between magnetic and ferroelectric orders in this class of compounds, as shown in Figure 6. A magnetic field of 5 T caused a small shift of the anomaly to higher temperature which could be an effect of the field on . While the early measurements had been conducted on polycrystalline samples of YMnO_{3}, later studies on single crystals of various RMnO_{3} have confirmed the existence of a clear, kink-like dielectric anomaly at the magnetic transition temperature [90, 104, 105].

###### 2.2.1. Magnetic Order and Dielectric and Thermodynamic Properties of HoMnO_{3} in the Absence of Magnetic Fields

The most pronounced anomalies of the dielectric constant at magnetic phase transitions had been reported in hexagonal HoMnO_{3}. Upon decreasing temperature, three distinct and sharp anomalies could be observed at K, K, and K (Figure 7). The kink of at is the typical signature of the onset of frustrated magnetic order in the hexagonal manganites. The sharp peak at , however, is unusual and has only be seen this sharp in HoMnO_{3}. The transition at was attributed to the rotation of the Mn^{3+} spins by [88]. With regard to the exact magnetic symmetries above and below , contradicting results have been reported. Muñoz et al. [106] proposed a transition from () to () upon decreasing temperature. However, subsequent neutron scattering [103, 107, 108] and optical experiments [109, 110] have shown that the transition at is most likely from () to () and the magnetic structure () is only realized after the second transition at below 5 K. The two magnetic orders of the Mn^{3+} spins above and below are included as and structures, respectively, in Figure 4. A partial magnetic polarization of the Ho^{3+} moments below , with moments aligned antiferromagnetically along the hexagonal -axis, was detected in neutron [103] and magnetic X-ray scattering experiments [111]. Based on symmetry arguments and a theoretical calculation, the origin of the magnetic coupling between Mn^{3+} and Ho^{3+} and the magnetic polarization of Ho moments below was attributed to a trigonal anisotropy term [112]. The increasing magnetic fluctuations of the subsystem of Ho^{3+} moments upon decreasing temperature and their coupling to the Mn spins may be considered as the possible origin of the Mn^{3+} spin rotation.

Notably, at much lower temperatures, there is a sharp increase of indicating another change in the magnetic structure of the Mn^{3+} and Ho^{3+} moments (see inset in Figure 7). This transition at was identified as a second Mn spin rotation transition to symmetry ( model, in Figure 4) with a significant increase of both, the Mn- and Ho-sublattice magnetizations [103, 108, 111]. Whereas below the Ho^{3+} moments in both Wyckoff positions occupied by Ho (2a and 4b) show AFM order, the symmetry does not allow magnetic order of the Ho in 2a position. It should be noted that all three phase transitions observed in HoMnO_{3} are extremely sharp when high quality single crystals are studied. From the dielectric data of Figure 7 the transition widths are all found close to the experimental resolution. The slope change (kink) of at happens within less than 0.7 K, the peak of the dielectric constant at is 0.5 K wide, and the sudden increase of at has a width of 0.07 K [113, 114]. The sharp dielectric anomalies of HoMnO_{3} have been confirmed by different research groups [115–120].

The three magnetic transitions in HoMnO_{3} are also reflected in sharp anomalies of other physical quantities [121]. The magnetization in the hexagonal plane and along the -axis is shown in Figure 8. The magnetization is moderately anisotropic with the in-plane magnetization slightly larger than the -axis values. Similar data have been obtained by [122]. The Curie-Weiss fit to the high temperature part of yields an effective magnetic moment of 11.43 which is in reasonably good agreement with the value of 12.14 which is expected for the sum of the free ion values of (5.92 ) and (10.6 ). The deviation of from the Curie-Weiss line sets in below 130 K indicating the onset of magnetic fluctuations and short range correlations between the Mn-spins. The Néel transition at 76 K is barely detectable in the magnetization data since the large Ho moment dominates the magnetic response. However, a clear magnetization drop at indicates the Mn spin rotation transition and the onset of AFM Ho moment order with orientation along the -axis. At , the magnetization shows a sudden increase followed by a continuous decrease toward lower temperatures due to the increasing Ho moment order.

All transitions are also accompanied by distinct anomalies of the heat capacity, as shown in Figure 9 [121]. The three sharp anomalies in have been verified by different groups [117, 119, 123–125]. Near the Néel temperature, exhibits a -shaped peak characteristic for a second order phase transition. The critical magnetic fluctuations at have been studied recently [126] and the critical exponents were found close to those of the three dimensional Heisenberg universality class. The critical scaling properties prove the second order nature of the transition.

The spin rotation at is revealed by a small, but sharp peak of (lower inset in Figure 9). The narrow peak suggests that this transition is first order in nature. The entropy change across the spin rotation transition, as estimated from the integral over the peak area, is relatively small, J/(mol K). A careful entropy analysis showed that the value of is consistent with the changes of volume and magnetization across the transition and the Clausius-Clapeyron equation, valid at first order transitions, is fulfilled [127]. It appears conceivable that the rotation of the Mn-spin system by does not contribute significantly to the entropy change at but the onset of AFM moment order is the main cause of .

At the second spin rotation transition temperature, , the specific heat shows a sharp increase followed by a decrease towards lower temperatures. A closer inspection reveals a two-peak structure of at this transition (upper inset in Figure 9) which could indicate that the spin rotation and the low-temperature order of the moments happen at slightly different temperatures. It should be noted that the two transitions near 5.3 K and 4.9 K, respectively, are also resolved in the dielectric constant data (a small kink is clearly visible in the inset of Figure 7) and in the -axis magnetization ( increases sharply at 5.3 K and drops suddenly at 4.9 K). In magnetic fields (see below), both anomalies actually split apart and define different phases in the phase diagram.

The sharp anomalies of the dielectric constant (Figure 7) at all three magnetic phase transitions reveal a sizable magnetoelectric effect mediated by strong interaction of the magnetic moments with the lattice. The experimental evidence for strong spin-lattice coupling was first found in distinct anomalies of the thermal expansion coefficients [127]. The hexagonal -axis shrinks with decreasing temperature and exhibits a clear anomaly at the onset temperature of magnetic order, . In contrast, the -axis expands and shows a similar anomaly at , but with opposite sign. The strong 2D spin fluctuations in the frustrated magnetic system of spins on the triangular lattice are responsible for the anisotropic response of the lattice. The gain of magnetic exchange energy apparently causes the in-plane contraction, particularly near , and the expansion of the -axis is mediated through the elastic forces of the lattice. The thermal expansivities, , are shown in the inset of Figure 10. The -type anomalies of and reflect the second order nature of the Néel transition in agreement with the specific heat anomaly (Figure 9). It is worth noting that the spin rotation transition at is also accompanied by a sudden change of the lattice constants, as shown by the sharp peaks of and (lower inset in Figure 10). The thermal expansion measurements and the observed anomalies at the magnetic transitions provide unambiguous evidence for extraordinarily strong spin-lattice interaction in the hexagonal rare earth manganites.

Further signatures for the strong coupling of the magnetic order to the lattice or phonons was derived from Raman experiments showing an enhancement of the phonon frequencies of two modes, which modulate the Mn-Mn interaction, below the Néel temperature of HoMnO_{3} [128]. A similar phonon enhancement had also been reported for hexagonal LuMnO_{3} near [129]. The thermal conductivity of HoMnO_{3} and other hexagonal RMnO_{3} is suppressed in the paramagnetic state, but it experiences a sudden increase with the onset of magnetic ordering, suggestive of a strong dynamic coupling between acoustic phonons and low-energy spin fluctuations [115]. High resolution neutron scattering experiments have revealed the atomic displacements of the Mn^{3+} ions in passing from the paramagnetic to different magnetically ordered states and their effect on the spin wave excitations [130]. The local structure and the Mn-Mn, Mn-Ho, and the Mn-O bond distances have been studied through X-ray absorption spectroscopy [131]. The results show that different displacements of ion pairs happen at the three magnetic phase transaitions of HoMnO_{3}. The structural distortion at is dominated by the change of in-plane distances between Mn ions, the first spin rotation transition causes a change of Mn-Mn and nearest neighbor Mn-Ho distances, and the low-temperature transition involves a structural distortion of all ions, including Ho-Ho pairs. The enhanced magnetoelastic effects in HoMnO_{3} have finally be studied with ultrasonic techniques [132]. The sudden change of the elastic moduli at and are further indications for the importance of the spin-lattice coupling in hexagonal manganites.

The structural changes at the magnetic transitions of HoMnO_{3} and the distinct dielectric anomalies observed raise the question about the response of the ferroelectric polarization to the magnetic order. It is conceivable to expect significant changes of , the -axis polarization, due to the strong coupling of the magnetic system to the lattice. The polarization changes below 100 K have recently been investigated through pyroelectric measurements [120]. Figure 11 shows as function of temperature. At , a minute kink of indicates the entrance into the magnetically ordered phase. The sharp drop near and the increase of at correlate well with the anomalies of the dielectric constant (as shown in Figure 7).

###### 2.2.2. The Magnetic Phase Diagram of HoMnO_{3}: Dielectric Measurements

Based on second harmonic generation optical experiments, Fiebig et al. have proposed a complex field-temperature phase diagram for HoMnO_{3} [93, 109]. With the external field oriented along the -axis, four different magnetic structures according to the 1D irreducible representations ( to , Table 2) had been proposed. Besides the three phases at zero field, a high-field phase of symmetry was detected, stable above 20 kOe and below 10 K.

The sharp anomalies shown in the dielectric constant at all magnetic transitions, which are strongly correlated with the magnetic and thermodynamic anomalies, and the high sensitivity of state of the art capacitance meters makes the dielectric constant an extremely sensitive probe of phase transitions in the RMnO_{3} compound system. Therefore, the dielectric measurements shown in Figure 7 (at zero magnetic field) have been extended to study the complete magnetic phase diagram of HoMnO_{3} in external fields along the hexagonal -axis [113, 114]. It turns out that the magnetic phase diagram is far more complex than originally suggested [109].

The signature of the spin rotation () and the low temperature () transitions are sharp peaks of the dielectric constant at zero field (Figure 7). With applied -axis magnetic field, the peak at broadens into a plateau-like structure while moving to lower temperatures. At the same time, the sharp increase of at moves higher in , forms a second plateau which finally merges with the high-temperature plateau at about 34 kOe. Between 34 kOe and 40 kOe only one plateau-like enhancement of , centered at about 18 K, exists. Above 40 kOe, no anomaly of the dielectric constant exists between 6 K and . This is shown in detail in Figure 12(a). Additional anomalies like sharp steps and peaks of at very low temperatures will be discussed separately below.

The data shown in Figure 12(a) suggest that the plateau-like enhancements of , evolving with magnetic fields from and , are the signature of one and the same phase. The phase diagram, constructed from the rapid increase/decrease of at the edges of the high- and low-temperature plateaus, is shown in Figure 13 (note that the low-temperature part of the phase diagram will be discussed later). In contrast to earlier studies [109], the transition between the to magnetic structures in higher magnetic fields is not instantaneous (as in zero field), but it passes through a well defined intermediate phase, characterized by an enhanced dielectric constant. The magnetic symmetry group of this phase is most likely , that is, the angle of the spins with the -axis is intermediate between () and (). The two transition temperatures are tentatively labeled () and (). The plateau-like enhancement of in the phase is further confirmed by isothermal measurements shown in Figure 12(b).

It appears conceivable that is continuously changing between the to limiting values and that it has a well defined value at any given temperature and field. This makes the intermediate phase a true thermodynamic phase, in contrast to a possible coexistence of the bordering and magnetic phases. However, since the low-frequency dielectric measurements could be sensitive to domain boundaries, possibly formed by a coexistence of the two major magnetic structures ( and ), further evidence is needed to prove the existence of a new phase. The increased value of the dielectric susceptibility in the phase could be an expression of the softness of the magnetic system and the strong coupling to the lattice, as discussed below. An alternative explanation of the plateau like increase of was given by Lottermoser and Fiebig [133]. In a careful study of the magnetic domain structure in the phase it was concluded that domain walls separating two magnetic domains with spin rotation angles of and , respectively, may result in a lowering of the local symmetry within the domain walls and a magnetoelectric effect which modifies (enhances) the dielectric function. The strong coupling between magnetic and ferroelectric domain walls was also proposed from the study of local magnetism and magnetoelectricity by muon-spin relaxation measurements [134].

The magnetic susceptibility is another bulk property and its changes in the different magnetic phases of HoMnO_{3} are discussed in the following. The -axis magnetic susceptibility was measured at different magnetic fields between 100 Oe and 45 kOe. The low-field data of the inverse susceptibility shown in Figure 8(b) indicate that is strictly linear between and with a kink-like anomaly at . The derivative suits best to visualize the changes of the inverse magnetization in the intermediate phase. The data at different fields are shown in Figure 14 (note that data in Figure 14(a) are vertically offset for better clarity). It is obvious that the derivative plotted in Figure 14 clearly deviates from the constant in the phase. This deviation is negative above 20 K and positive below.

According to the phase diagram (Figure 13), at 33 kOe the sequence of phases upon decreasing temperature is , that is, the system passes twice through the intermediate phase. The corresponding transitions and critical temperatures and are indicated in Figure 14(a). At slightly higher fields, 35 kOe and 38 kOe, the magnetic system does not pass into the phase and the sequence is , as indicated by the continuous and linear increase of between the two phase transitions. Above 40 kOe no transition could be observed above 5 K, in agreement with the phase diagram of Figure 13.

If all data of Figure 14(a) are plotted without any offset on the same scale (Figure 14(b)), it becomes obvious that the values of within the phase are well defined and follow the dashed line in Figure 14(b). This can be considered as further evidence that the intermediate phase is not a mixture of coexisting and phases, but instead it is a uniform phase determined by the spin angle in the magnetic structure. It should also be noted that no signature of a large temperature or field hysteresis had been observed at or in all measurements, although small hysteretic effects cannot be completely excluded.

At , the two phase boundaries with transition temperatures and merge into a single transition at . The temperature range of stability of the phase shrinks to zero and the transition at () is characterized by an instantaneous rotation of the spins by from the to the magnetic structure. It is interesting, that the dielectric constant of the phase (the envelop to the plateaus of , as shown by the dashed line in Figure 12(a)) appears to diverge in approaching the critical point , .

On the low temperature side, K, the phase diagram of HoMnO_{3} exhibits an unprecedented complexity [114]. Starting from a detailed analysis of the temperature and field dependence of the dielectric constant, a series of distinct anomalies define the extension of the phase boundaries at and , as well as new phase transitions. The low temperature data of are summarized in Figure 15. The two phase boundaries separating the three high temperature phases, and , can be traced to lower temperature. The inner phase boundary of the phase, , is characterized by a sharp increase of , as indicated in Figures 15(a), 15(b), and 15(c). It can be traced to the zero field , as discussed above. The outer phase boundary of the phase , however, decreases to zero near a critical field of about 20 kOe, as shown in Figure 15(c). The derived phase diagram below 9 K is shown in Figure 16. Additional sharp anomalies of are labeled to . At about 6 kOe, the dielectric constant develops a sharp step (Figure 15(a)) at low temperature (denoted ). increases first up to 3.3 K with increasing and decreases again to zero at 20 kOe. The corresponding phase boundary is shown by the red squares in Figure 16. Above 20 kOe, another step-like increase of develops at from low temperatures. With increasing field, rises to 4.5 K and drops to zero at about 80 kOe, as shown in Figures 15(c) and 15(d). This phase boundary is included in Figure 16 as blue triangles. At 12 kOe, another anomaly of splits off from the step at (c.f. 12 kOe data in Figure 15(b)). This anomaly, labeled in Figures 15 and 16, shifts to lower temperature and merges with the phase boundary, as shown by the pink squares in the phase diagram of Figure 16.

The assignment of the different magnetic structures was made based on the results from second harmonic generation optical experiments [93, 109, 110] and neutron scattering measurements [103, 106, 108, 130]. The phase boundary apparently separates the low-field phase from the high-field phase. It should be noted that a significant thermal hysteresis is observed in crossing (see Figure 15(b)) which shows the strong first order character of this transition. In addition to the sharp phase boundaries to , other subtle anomalies of are observed and indicated by and in Figure 15. Of particular interest is the sharp peak of developing at below 5 K and 22 kOe, with a maximum height near 2.8 K and 19 kOe. It is remarkable that this low-temperature peak of appears near a point in the phase diagram where three phase boundaries, , , and , come very close and, possibly, form a multicritical point. The -peak indicates a softness of the dielectric system in approaching the critical point. The magnetic structures of the two low-temperature phases labeled LT1 and LT2 in Figure 16 have yet to be explored.

The multitude of phase boundaries as revealed by the temperature dependent dielectric measurements at low temperatures is shown in more detail in isothermal measurements of as a function of field in Figure 17. At 9 K (Figure 17(a)), a sharp increase followed by a drop of defines the range of the phase, that is, and . can be traced to lower temperature until it ends at = 0 as defined in the zero field measurements (Figure 7). At about 7 K, a peak of develops at and it sharpens significantly to lower temperatures. This peak and can be traced to 2.6 K when it meets the phase boundary of the LT1 phase (Figures 17(a), 17(b), and 17(c)).

The phase boundary between the and phases, , splits off from between 5.5 and 6 K. It’s characteristics is a small step and a sizable field hysteresis, as shown in Figure 17(a) (5.5 K data). The decrease of towards the boundary of the LT1 dome is demonstrated in Figure 17(b). Several phase boundaries merge smoothly in the phase diagram of Figure 16. Those deserve a more careful study. and approach one another upon decreasing temperature (Figure 17(b)) and they finally merge at about 2 K (Figure 17(c)). The phase boundary defines the second low-temperature phase, LT2. The LT2 phase emerges below 4.7 K and it is characterized by another valley of (shown in Figures 17(d) and 17(f)), similar to the LT1 phase. The two low-temperature phases, LT1 and LT2, appear to share one phase boundary at very low temperature. As shown in Figure 17(e), the two corresponding phase boundaries and merge at 1.4 K. The dielectric data at 1.4 K (Figures 17(e) and 17(f)) reveal a direct transition from LT1 to LT2 without any signature of an intermediate phase with enhanced . The step of at 1.4 K and 76 kOe (Figure 17(f)) designates the approximate upper field limit of the LT2 phase. No further phase transition could be detected above 80 kOe, as demonstrated by the -data shown in Figure 17(e) up to 130 kOe.

###### 2.2.3. Magnetic Properties and Heat Capacity in the Phase Diagram of HoMnO_{3}

The subtle anomalies of the dielectric constant have to be related to corresponding changes of the magnetic order and should be reflected also in anomalies of the magnetic properties of HoMnO_{3}. This is best verified by measuring the ac magnetic susceptibility as function of a dc bias field along the hexagonal -axis. Figure 18 shows the results in two different temperature ranges, (a) and (b). In the high-temperature range (Figure 18(a)), the real part shows a significant enhancement in the stability range of the intermediate phase. This increased susceptibility indicates a softness of the magnetic system with respect to the external field. The softness results from the fact that the angle of the spins is not locked in at or as in the and phases, respectively. The two boundary angles apparently increase the stiffness of the magnetic system resulting in a reduced susceptibility with respect to the external magnetic field. The dielectric constant is also enhanced in the phase due to the strong spin-lattice interaction, as shown in Figure 12(b). The change of the ferroelectric polarization was measured by Hur et al. [120] and they found a sizable increase of in the phase with increasing .

In the low-temperature region (Figure 18(b)), several peaks and anomalies of can be distinguished. The sharp drop at the high-field end corresponds to the phase boundary. On the low-field side, for temperatures below 3.2 K, a sharp peak of develops between 5 and 10 kOe. This peak coincides with the sharp drop of the dielectric constant (Figure 17(c)) and determines the phase boundary between the and the LT1 phases. Most remarkably, a broad maximum of below 5 K becomes a sharp peak near 3 K and 20 kOe before it is cut off by the entrance into the LT3 phase by crossing the phase boundary. The sharp peak of in this temperature-field range and the equivalent peak of the dielectric constant (Figures 17(b) and 17(c), marked as in the phase diagram of Figure 16) may suggest that the magnetic system is heading towards an instability at zero temperature. The increase of the peak maximum of both, the magnetic () and dielectric () susceptibilities are shown in Figure 19. The singular increase is interrupted by the phase boundary, indicated by the vertical dotted line. One may speculate about the possible existence of a quantum critical point which is hidden in the emerging LT1 phase. The linear extrapolation of the peak position to zero temperature would locate this critical point near kOe.

The complex phase diagram of Figure 16 was quantitatively confirmed by other groups reporting the results of a microwave study of HoMnO_{3} single crystals [135] and an investigation of the magnetoelastic coupling through the temperature and field dependence of elastic moduli [132]. Neutron scattering experiments have revealed characteristic changes of the peak intensities measured along different scattering geometries [108, 136]. While the observed anomalies are consistent with the phase boundaries drawn in Figure 16, the results of the neutron study had been interpreted differently in the low-temperature section ( K) of the phase diagram. Vajk et al. [108] derived from their data an extension of the phase boundary into the dome-shaped LT1 phase and the corresponding phase diagram is shown in Figure 20. The major difference are the phase boundaries drawn below 5 K, merging in a pentacritical point near 6 K and 13 kOe. A slightly revised low-temperature phase diagram of HoMnO_{3} was presented recently based on dielectric constant and polarization measurements [120].

Comparing the low-temperature part of the phase diagram of [108, 120, 136] with Figure 16 and those proposed in [132, 135], there arises the question whether the LT1 phase forms the dome shaped stability region defined by the phase boundary. Heat capacity () experiments have been chosen to define the thermodynamic phase boundaries of HoMnO_{3} [121]. The data shown in Figure 21 clearly define the phase boundary of the LT1 phase by a sharp peak of starting at kOe. The peak rises in magnitude when the field increases to 12 kOe at K and it traces back to lower temperature with further increasing field, eventually disappearing at the upper limiting field of about 20 kOe. This smooth and continuous development and shift of the -peak supports the interpretation of the dielectric and magnetic data and it defines unambiguously the dome shaped LT1 phase and its boundary, as shown in Figure 16. Whether or not the transition line extends into the LT1 phase to zero temperature, as suggested in [108, 120, 136], is not clear. No anomaly of the dielectric constant or the ac magnetic susceptibility, that could be associated with such extension of , was detected in temperature or field dependent data discussed above.

The phase boundary is defined by a sharp increase of upon decreasing temperature. and are highlighted by the dashed lines in Figure 21. At magnetic fields above 20 kOe, the heat capacity develops another sharp peak at low temperature, indicating the transition into the LT2 phase across , as shown in Figure 22. At the high temperature end, the heat capacity peak at the spin rotation transition temperature shifts to lower temperature and broadens with increasing (Figure 23). Above about 40 kOe, the heat capacity peak cannot be discerned anymore indicating that the only magnetic phase transition below is the transition into the LT2 phase at about 4 K, in agreement with the phase diagram of Figures 13 and 16. The heat capacity data thus confirm the major phase transitions in HoMnO_{3} in the magnetic field oriented along the -axis.

###### 2.2.4. Magnetoelastic Effects as Evidence for Strong Spin-Lattice Coupling in HoMnO_{3}

The large effects of the magnetic order on the ferroelectricity, as revealed in the pyroelectric measurements by Hur et al. [120] (Figure 7), can only be explained by the presence of strong spin-lattice interactions. Direct evidence for magnetoelastic effects was found in sizable anomalies of the lattice constants at the magnetic phase transitions. Figure 10 shows the temperature dependence of the - and -axis lattice constants of HoMnO_{3} as determined through thermal expansion measurements [127]. Interestingly, the -axis displays a negative expansion coefficient over the whole temperature range, that is, is expanding upon decreasing temperature. With the onset of magnetic order at , both axes experience a sharp anomaly, the -axis expanding more and the -axis shrinking to lower temperatures.

The thermal expansivities, shown in the left inset of Figure 10, exhibit a pronounced -shaped peak anomaly, similar to the heat capacity (Figure 9), as expected at a second order phase transition. Based on a systematic study of the heat capacity, Oleaga et al. [126] concluded that the critical properties of the hexagonal manganites near fit best to the 3D-Heisenberg universality class. At the spin rotation transition, the thermal expansivities show a narrow peak with opposite sign for the - and -axes. This peak is enlarged in the lower right inset to Figure 10. The sharpness of this peak reveals a sudden step-like change of (decreasing) and (increasing) resulting in a decrease of the relative volume upon cooling by . Other relevant physical quantities showing step-like changes at are the magnetization and the entropy. The sudden changes suggest that the spin rotation transition in HoMnO_{3} is of first order. This could be proven by calculating the entropy change at which can be expressed by the Clausius-Clapeyron equation:
and denote the external pressure and magnetic field, respectively. and are the measured changes of the volume and the magnetization across the transition and the entropy change can be derived by integrating the peak of the heat capacity at . The pressure and field dependencies of had been determined and (6) was shown to be valid, proving the first order nature of the transition [127]. This is also consistent with the sudden flop of the spins and the symmetry change from to at .

The negative thermal expansion of the -axis over a large temperature range appears unusual at first and it could be associated with strong magnetic correlations among the quasi-2D manganese spin system, resulting in an enhanced contraction of the in-plane distances and an expansion of the -axis through elastic forces upon decreasing temperature. A similar negative expansivity was also reported for the related compound, YMnO_{3}, below room temperature [96, 137, 138]. High-temperature studies of the structure of YMnO_{3} have shown that the expansivity of the -axis becomes negative below 1260 K, the transition temperature from the centrosymmetric structure to the polar phase [82–84, 91]. Therefore, the negative -axis expansivity appears to be more related to the properties of the ferroelectric phase.

Longitudinal magnetostriction measurements along the hexagonal -axis of HoMnO_{3} confirm the strong magnetoelastic effects suggested by the thermal expansion anomalies at zero magnetic field [41]. In the high-temperature range, the -axis length increases nonlinearly in the phase with increasing magnetic field, as shown in Figure 24(a). At the phase boundary, suddenly changes slope and increases at a steeper pace in the intermediate phase, until experiences another sudden decrease of slope with the entrance into the magnetic structure at . In the high-field phase appears to change linearly with further increasing field. The sharper increase of with in the intermediate phase reflects a remarkably strong response of the lattice to the spin rotation from to , resulting in an increased expansion of the -axis.

At lower temperature, several of the phase boundaries shown in Figure 16 are reflected in distinct magnetostriction anomalies of the -axis. The low-temperature data are summarized in Figure 24(b). At 5 K, the only transition that is clearly resolved in is from to across the phase boundary. The sharp increase of just below this transition coincides with the peaks of the dielectric (Figure 17) and magnetic (Figure 18) susceptibilities, again proving the softness of the magnetic, dielectric, and elastic properties. At lower temperature, the entrance into and exit from the LT1 phase by crossing the phase boundary twice is accompanied by a sudden increase and decrease of the -axis, respectively. This is clearly seen in the first two peaks of the derivative of the 3 K data, shown in the inset to Figure 24(b). The next sharp peak is associated with the -line and the last peak above 20 kOe signals the entrance into the LT2 phase (note that the and phase boundaries merge at this point, see Figure 16).

At the lowest temperature of this study (1.43 K), only two sharp anomalies of can be distinguished. Both result in a step-like increase of in crossing from the phase into the LT1 phase and eventually into the LT2 phase. It is interesting to note that no magnetostrictive anomaly of the -axis was detected at the phase boundary although magnetic as well as dielectric properties exhibit distinct anomalies with a strong field and temperature hysteresis.

The thermal expansion and magnetostriction measurements discussed above have revealed the macroscopic length changes of for example, the -axis with an extraordinary resolution that can only be obtained by the high-precision capacitance dilatometer employed for the studies. This makes the measurements significantly more sensitive than any structural characterization using scattering methods. However, the dilatometry cannot resolve the microscopic distortions within one unit cell which should be studied for a more fundamental understanding of the magnetoelectric coupling in multiferroics.

Recent high-resolution neutron diffraction experiments on different RMnO_{3} have proposed a correlation between the magnetic structure according to Table 2 and the position of the ion in the primitive cell [130]. In the structure, the relative position of the Mn ion is defined by one free parameter which varies between different rare earth ions in the RMnO_{3} compounds and also with temperature and magnetic orders. In HoMnO_{3}, increases from 0.325 above to 0.335 below . Since the super exchange interactions between the spins within the -planes as well as along the -axis depend on the crucial parameter , it was proposed that the different magnetic orders realized in HoMnO_{3} and in other hexagonal manganites (e.g., ScMnO_{3}, YbMnO_{3}) are strictly correlated with [130].

The local structure of HoMnO_{3} was studied using X-ray absorption spectroscopy [131]. The bond distances between different Mn- and Ho-ions were extracted from the atomic distribution functions and it was shown that different interatomic distances experience sudden changes at the three magnetic phase transitions. The onset of magnetic order at did mainly affect the inplane Mn-Mn bond distances. At the spin rotation transition temperature, , the Mn-Mn as well as Ho-Mn bonds show distinct anomalies. At the lowest transition temperature, , all bonds are distorted, including the nearest and next nearest neighbor Ho-Mn and the Ho-Ho distances. A model calculation based on the local spin density approximation was carried out to study the correlated spin-lattice system and the role of different magnetic exchange interactions (Mn-Mn, Ho-Mn, Ho-Ho) in driving the three observed magnetic phase transitions was determined [131].

The important role of the Ho-Mn magnetic interactions was also derived from an investigation of the magnetoelastic coupling in HoMnO_{3} through measurements of the elastic moduli [132]. An elastic softening was observed over a wide temperature range, with pronounced anomalies at and , and it was attributed to spin fluctuations induced by the Ho-Mn interactions. It should also be noted that magnetic ordering effects below result in a sizable hardening of phonon modes that modulate the Mn-Mn interactions which is further evidence of the strong spin-lattice coupling in [128].

###### 2.2.5. The Role of the Ho^{3+} Magnetic Moment in HoMnO_{3}

As already discussed in the previous section, the involvement of the magnetic moment is essential to understand the complex sequence of magnetic phase transitions in HoMnO_{3}. The moments interact with the spins and, at lower temperature, with one another. They also participate in the magnetic ordering in certain temperature and field ranges. There are two crystallographically inequivalent sites for the Ho ions: Ho(1) and Ho(2) occupying the 2a and 4b positions, respectively, and six formula units per primitive cell, including two Ho(1) and four Ho(2). The complete group theoretical representation of the structure was given by Muñoz et al. [106]. Above the spin rotation transition temperature, the magnetic symmetry would only allow the Ho moments to order, however, neutron scattering experiments have not found any indication of a moment order above the spin rotation transition temperature . A gradual increase of the Ho sublattice magnetization below about 30 K is an indication of a polarization and the onset of the Ho magnetic order at [106].

Below and above , the moments of both Ho(1) and Ho(2) are allowed to order antiferromagnetically. Thereby, the moments of Ho(1) within one hexagonal -plane form ferromagnetically aligned planes which are antiferromagnetically oriented between to neighboring planes. The Ho(2) moments form similar ferromagnetic planes but they are opposite in direction to the Ho(1) within each plane, as shown in Figure 25(a). The magnetic moments of Ho(1) and Ho(2) within one plane do not completely compensate so that a ferrimagnetic moment remains in each plane forming an antiferromagnetic alignment along the hexagonal -axis. Various experimental data are consistent with this order [103, 107, 111, 133].

Below , only the antiferromagnetic order of the Ho(2) moments is allowed by the () magnetic space group and the Ho(1) are paramagnetic. The transition into the phase is accompanied by a large increase of the Ho(2) sublattice magnetization and a sudden jump of the Mn ordered magnetic moment. The AFM order of the Ho(2) below is schematically shown in Figure 25(b). It should be noted that, although the phase transition at is well established through magnetic, dielectric, and thermodynamic measurements (see previous sections), and neutron scattering [103, 108, 130], second harmonic generation [93, 133], and X-ray resonant magnetic scattering experiments [111] support the magnetic symmetry below , a recent report did not find a change of the magnetic order of the Ho moments below [107]. The origin of the discrepancy is not clear at this point.

In magnetic fields along the axis, the Ho moments can be expected to become systematically aligned with the field (note that the Mn spins are in the plane and are less susceptible to the field ). A ferromagnetic alignment of the Ho moments along the axis is allowed by symmetry, for example, in the magnetic space group, corresponding to the irreducible representation (Figure 4). It was suggested that this phase is realized in HoMnO_{3} at low-temperatures and high magnetic fields ( kOe) [93, 139]. However, according to the phase diagram of Figure 16, there are at least to successive phase transitions as a function of increasing field below 3 K, the first transition into the LT1 phase at 5 kOe and a second transition into the LT2 phase at 20 kOe. It is likely that at high enough magnetic fields the magnetic structure turns into the symmetry, but the details of the the magnetic orders in the LT1 and LT2 phases remain a puzzle. The symmetry change with increasing field from to allows all Ho moments to order and it is accompanied by another rotation of the Mn spins resulting in the magnetic structure shown in Figure 4.

The symmetry allowed magnetic orders of the Ho(1) and Ho(2) ions in the unit cell and the linear magnetoelectric effect are summarized in Table 1 of [139]. It is particularly interesting that the linear magnetoelectric coupling along the axis is allowed only in symmetry, that is, the system can gain magnetoelectric energy through a coupling of the electrical polarization with the magnetization of the Ho ions, ( is the longitudinal magnetoelectric tensor element along the axis, is the ferroelectric polarization, and is the axis component of the Ho magnetization). This coupling may result in an electric field effect on the magnetization of the Ho moments. Lottermoser et al. [139] indeed found that an applied electric field did quench the second harmonic generation signal at all temperatures below indicating a major change of the Mn spin magnetic order. The response of the Ho sublattice moment was derived from a change of the Faraday rotation in electric fields applied with different polarities. The observed results indicate that the electric field did induce a macroscopic magnetization attributed to the ferromagnetic order of the Ho moments. The results of the optical study of [139] are shown in Figure 26.

The thermodynamic origin of the electric field effects on the magnetic order in , as discussed in [110], is found in the gain of magnetoelectric energy through the linear magnetoelectric coupling of magnetization and polarization. Since the linear magnetoelectric effect with the -axis electrical polarization is only allowed in the magnetic structure, the external electric field stabilizes this symmetry through the control of the electrical polarization with the consequence of a ferromagnetic alignment of the Ho moments and a rotation of the Mn spins. The magnetoelectric energy gain competes with the change of the magnetic anisotropy and superexchange energy. Fiebig et al. [110] have shown that the different ferroelectric displacements of the Ho(1) and Ho(2) ions as well as the finite sublattice magnetization of the Mn spins are essential for a macroscopic magnetoelectric effect in the case of symmetry.

Other groups have searched for the electric-field induced magnetic moment in HoMnO_{3} and the symmetry allowed linear magnetoelectric effect at low temperatures and high fields. In field-dependent polarization measurements below 3 K and magnetic fields above 30 kOe, Hur et al. could not detect a linear magnetoelectric effect within their experimental resolution [120]. Using X-ray resonant magnetic scattering, Nandi et al. have elucidated the role of ions and studied the magnetic order of the Ho moments [111]. The results confirm the onset of Ho moment order at and a change of the order at . However, measurements in high electric fields were found to be identical to the results of zero field experiments and the suggested change of the Ho moment order in electric fields could not be confirmed. Small angle neutron scattering experiments conducted on single crystals of HoMnO_{3} in magnetic and electric fields have indicated that the ferromagnetic moment in electric fields possibly arises from uncompensated spins in the antiferromagnetic domain walls rather then from bulk magnetism of the Ho moments [140]. This issue is still not decided and more studies have to be conducted.

##### 2.3. Magnetoelectric Effects in Hexagonal Manganites without Rare Earth Moments

###### 2.3.1. Ferroelectricity, Magnetic Order, and Magnetoelectric Coupling in YMnO_{3}

YMnO_{3} was in the focus of interest because of the absence of the rare earth moment which dominates the magnetic response in other hexagonal manganites. Structurally, YMnO_{3} is quite similar to HoMnO_{3} because of the very similar size of the and ions, however, the magnetic system appears to be simpler and the only magnetic ion is the . Therefore, YMnO_{3} has been studied extensively and compared to HoMnO_{3} and other hexagonal manganites. While there is no complete consensus about the high-temperature structures and phase transitions in YMnO_{3} [82–85], most studies confirm the onset of ferroelectricity near 1200 K with the change of structure to the phase. After the discovery of ferroelectricity in YMnO_{3} [38], Smolenskii and Bokov revealed the coexistence of the ferroelectric state with antiferromagnetism at low temperatures [42].

The origin of ferroelectricity was attributed to the buckling of the MnO_{5} polyhedra which is accompanied by displacements of the and ions along the -axis, away from a centrosymmetric position. The distortions in the ferroelectric phase are schematically shown in Figure 27. The ions in different Wyckoff positions are displaced along the -axis in opposite directions creating local dipolar moments of different sign. The local moments are not completely compensated and, in combination with the displacements of the oxygen ions, generate the macroscopic polarization. Theoretical calculations of the bonding and the Born effective charges () come to different conclusions. Van Aken et al. found the values for all ions close to the formal ionic charges suggesting that there is no significant rehybridization and charge transfer with the entrance into the ferroelectric phase [79]. The macroscopic polarization arises mainly from the Y-O_{p} displacements along the -axis. The origin of the instability is apparently a strong coupling of a zone boundary instability with the polarization resulting in an improper ferroelectric state [141, 142]. However, recent polarization dependent X-ray absorption measurements have suggested the existence of strong anisotropic hybridization of the Y 4-O 2 bonds which creates the off-center displacements of and the resulting large anomalies in the Born effective charges as the possible cause of the ferroelectric instability [143].

The details of the magnetic order below K in YMnO_{3} has been explored through bulk magnetization, neutron scattering, and second harmonic generation optical experiments. Unlike in HoMnO_{3}, where the large Ho moment dominates the magnetic susceptibility and makes the Néel transition of the Mn spins almost invisible (see Figure 8(a)), a clear change of slope of the -dependent magnetic susceptibility signals the onset of magnetic order in YMnO_{3}, as shown in Figure 28 for a floating zone grown single crystal. The sharp slope change is consistent with earlier reports for single crystals [144–146]. However, the magnetic signature at may be less clear in powder samples, possibly due to sample quality or grain size problems [43, 90, 95, 147–150].

The first neutron study of YMnO_{3} by Bertaut and Mercier [87] proposed two possible magnetic structures, (, -model) or (, -model), according to Table 2. Later, the same authors favored the magnetic symmetry [151]. Subsequent neutron studies found it difficult to distinguish between the two magnetic symmetries ( and ) since both did describe the spectra equally well [96, 152], but other data have been interpreted in favor of the () symmetry [95, 153, 154]. Recent neutron diffraction and polarimetric studies have suggested the lower magnetic symmetry with a spin tilt angle of = with reference to the structure [107]. Second harmonic genaration optical spectra are sensitive to the details of the magnetic order in hexagonal manganites [99]. For , the results of SHG experiments suggest the () magnetic symmetry, the same symmetry as in HoMnO_{3} below the spin rotation transition [99, 100].

The discussion about the magnetic symmetry of YMnO_{3} below the Néel temperature is even more complicated after signatures of diffuse scattering as well as unconventional short range spin fluctuations have been reported above as well as in the ordered phase [147, 148, 155–157]. The results seem to be consistent with a quasi two-dimensional frustrated magnetic system with weak interplane coupling and a spin liquid phase above which extends into the magnetically ordered phase below . Thermal conductivity measurements show an unusual suppression in a large temperature range, stretching from to nearly room temperature, which was attributed to strong spin fluctuations as a result of the 2D character of the Mn magnetic sublattice and geometric frustration [115]. Furthermore, a large deviation of the low-temperature () heat capacity from the phonon contribution seems to indicate the presence of a residual magnetic , possibly due to a magnetic glassy state coexisting with the ordered antiferromagnetism in some RMnO_{3} (R = Y, Lu, Sc) [90, 144]. In contrast, a recent study of the heat capacity of YMnO_{3} and the critical scaling near the Néel transition has argued that the deviation of the low-temperature from the Debye law, previously attributed to an abnormal magnetic contribution, can be accommodated by an additional Einstein contribution and the critical exponents derived are well within the range of a 3D Heisenberg model, thus not supporting 2D or chiral models for the magnetic system [126, 158]. It appears that the true nature of the magnetic order and the spin fluctuations below and above has yet to be revealed.

The magnetoelectric coupling between the ferroelectric polarization (oriented along the hexagonal -axis) and the spin order in the plane was first reported in form of a distinct anomaly of the dielectric constant, as shown in Figure 6, by Huang et al. [43]. These results were confirmed for YMnO_{3}, LuMnO_{3}, and ScMnO_{3} by Tomuta et al. [90]. While those measurements were conducted on polycrystalline ceramic samples, a clearer picture is obtained from single crystals. Katsufuji et al. [144] reported a sudden drop of the dielectric constant measured with the electric field orientation within the -plane in YMnO_{3} and LuMnO_{3}.

The dielectric response to the magnetic order at suggests a very strong coupling of the spins to the lattice. The spin-lattice interaction in YMnO_{3} is not complicated by the presence of a rare earth moment with a strong uniaxial anisotropy, as for example in HoMnO_{3} discussed above. First evidence for the existence of magnetoelastic effects was derived from subtle anomalies of the lattice parameters detected in neutron scattering experiments [96], although the observed anomalies are almost within the resolution of the scattering experiment. Much higher resolution of length measurements can be achieved if a capacitance dilatometer is used. Thermal expansion data for YMnO_{3} are shown in Figure 29. Similar to HoMnO_{3}, the -axis shows a negative expansivity at all temperatures below ambient whereas the -axis contracts with decreasing . Those results are consistent with data from neutron scattering experiments [95]. At , both axes respond to the onset of magnetic order but with opposite sign, the -axis expands faster and the inplane distances experience an enhanced contraction. The -shaped sharp anomalies of the expansivities (Figure 29(b)) are consistent with the second order nature of the magnetic phase transition and the existence of critical spin fluctuations near . The anomalous -dependence of the lattice parameters was confirmed by neutron scattering experiments [138, 159].

The details of the ionic distortions in the structure of YMnO_{3} and LuMnO_{3} have been investigated through high resolution X-ray diffraction and neutron scattering experiments only very recently [137]. This work mapped out the details of the structural changes at on an atomic scale and showed that all atoms in the unit cell exhibit giant displacements, two orders of magnitude larger than in typical magnetic materials. The coupling between the ferroelectric polarization and the antiferromagnetic order (the internal magnetoelectric effect) was explained by the magnitude and nature of the atomic displacements. The driving mechanism for the huge magnetoelastic effect was proposed to be the displacement of the ions from their ideal symmetric position () which results in a coupling to the electric dipole moments [159]. It is interesting to note that the deviation from the position for the is of opposite sign in YMnO_{3} and LuMnO_{3}, resulting in different magnetic structures below . The possible effects of the displacements on the magnetic exchange coupling parameters and the resulting magnetic orders was also discussed by Fabrèges et al. [130]. Similar to the internal magnetic order, external magnetic fields have a significant influence on the atomic position in the unit cell. This was demonstrated in neutron scattering experiments in magnetic fields up to 50 kOe [150].

The strong magnetoelastic effects observed in YMnO_{3} and other hexagonal RMnO_{3} leave their imprint also on other physical quantities as, for example, the elastic moduli studied in ultrasound measurements [160]. The study of magnetic and lattice excitations using inelastic neutron scattering experiments provided evidence for a strong coupling between magnons and phonons which appear to hybridize in one mixed elementary excitation [154], similar to the electromagnons studied in multiferroic orthorhombic manganites [161, 162]. Raman and infrared absorption investigations of the temperature dependent phonon spectrum of YMnO_{3} and LuMnO_{3} have found a kink at and an abnormal hardening in the magnetically ordered phase, indicating strong spin-phonon coupling [163–165]. Similar phonon hardening was also found in compounds with rare earth magnetic moments, HoMnO_{3} and ErMnO_{3} [128, 166].

###### 2.3.2. Ferroelectric and Magnetic Domains in YMnO_{3}

Ferroelectric and magnetic domains and the associated domain walls play an essential role in multiferroic compounds. Novel techniques of visualizing and distinguishing those domains had to be developed in order to study the domain physics in detail. One of the more sensitive methods is the second harmonic generation (SHG) spectroscopy and it’s power in making domains of magnetic materials (e.g., antiferromagnetic Cr_{2}O_{3}) visible has been demonstrated [167, 168]. The SHG technique was later refined to investigate the multiferroic domain structure of hexagonal manganites [169]. Most investigations have focused on YMnO_{3} to avoid the influence of the rare earth magnetic moments. The principle of the second harmonic generation technique is the sensitivity of the third rank SHG susceptibility tensor with respect to magnetic symmetry and polar structure.

It is the beauty of the nonlinear optical process (SHG) that it allows to distinguish between electric and magnetic domains within one experiment. This technique was successfully applied to the study of ferroelectric and magnetic domains in YMnO_{3} [170]. The results shown in Figure 30 clearly show different domain pattern of the same area of the crystal’s surface, depending on the polarization geometry of the incoming and SHG light. By choosing the appropriate geometry, different domains (ferroelectric, magnetic, or a combination of both) can be made visible through an optical interference technique with an external reference signal [171, 172]. Examples are shown in Figure 30, left panel. The top-left image reveals the ferroelectric domains only, the dark and bright areas corresponding to and , respectively. The antiferromagnetic domains are made visible in the bottom image, dark and bright areas denoting and , respectively. The top right image of Figure 30 was taken in a geometry that distinguishes the sign of the product of and , the dark area corresponds to and the bright area denotes . From combining all images, a complete domain picture can be constructed, as shown in the right panel of Figure 30 [170]. The most interesting result of this study was the observation that ferroelectric domain walls are not forming independently of the antiferromagnetic domains. Instead a “clamping” property was shown to exist between ferroelectric and magnetic domain walls, as illustrated in Figure 30 (right panel). In contrast, some magnetic domain walls do exist within one ferroelectric domain and are not necessarily attached to ferroelectric domain walls. This behavior indicates the strong interaction of ferroelectric and magnetic orders and novel physical phenomena happening in between the different domains.

After the discovery of the clamping feature of ferroelectric and magnetic domain walls in YMnO_{3}, the possible microscopic origin was discussed by Hanamura et al. [173]. The authors considered the isotropic as well as the antisymmetry (Dzyaloshinskii-Moriya) exchange interactions between the spins within the -plane of YMnO_{3} and calculated the energy of the ferroelectric and antiferromagnetic domain boundaries within a continuum approximation. It was shown that the Dzyaloshinskii-Moriya interaction did stabilize the ferroelectric-antiferromagnetic domain boundary and a single ferroelectric domain wall was not a stable solution in the calculations. These calculations highlight the importance of the antisymmetric Dzyaloshinskii-Moriya interactions in hexagonal manganites, at least in the boundaries between different domains.

An alternative explanation of the clamping property of ferroelectric and antiferromagnetic domain walls was proposed by Goltsev et al. [174, 175] based on a microscopic model that includes the in-plane and out-of-plane exchange interactions of the spins, their anisotropy, the local distortion at the ferroelectric domain wall, and the piezomagnetic coupling between the local strain and the local magnetic moment. The energy gain achieved through this piezomagnetic effect favors the clamping of magnetic to ferroelectric domain walls at . Since the ferroelectric domains do exist above , the first magnetic domain walls formed in the ordered state coincide with the ferroelectric walls, however, additional magnetic domains may form within one ferroelectric domain. It should be noted that the proposed mechanism does not involve the antisymmetric Dzyaloshinskii-Moriya exchange interaction.

###### 2.3.3. Multiferroic Properties of LuMnO_{3} and ScMnO_{3}

LuMnO_{3} and ScMnO_{3} are of interest since and both do not carry their own magnetic moment, similar to , but they are significantly smaller resulting in a compression of the lattice (see Table 1) and stronger magnetic exchange interactions within the -plane as well as along the -axis. As a consequence, the Néel temperatures increase from YMnO_{3} (72 K) to LuMnO_{3} (90 K) and to ScMnO_{3} (130 K). In addition, the in-plane magnetic anisotropy decreases and is quite small for ScMnO_{3} making the magnetic orders with the preferred Mn spin angles and (see Table 2) less favorable.

The crystal structure at high temperatures ( K) had been studied and compared with data for YMnO_{3} [91]. The -axis lattice parameters show a similar change with temperature, however, there are significant changes with respect to the thermal expansion of the -axis. While YMnO_{3} exhibits a strongly negative expansivity, similar to HoMnO_{3} (see also Figures 10 and 29), the -axis is almost temperature independent for LuMnO_{3} and it shows a “normal” behavior (positive expansivity) in the case of ScMnO_{3}. The differences in the thermal expansion property of the -axis and the negative -axis expansivity of YMnO_{3} has been attributed to differences of the buckling angle of the MnO_{5} bipyramids and it’s specific temperature dependence. The -axis length scales with the buckling angle and, for YMnO_{3}, the decrease of this angle with increasing temperature results in the contraction of the -axis, as observed experimentally [91]. For ScMnO_{3}, the buckling angle slightly increases with temperature explaining the normal expansion property of the -axis in this compound.

The transition into the magnetically ordered state of LuMnO_{3} at 90 K is well documented in anomalies of the bulk magnetic susceptibility and the heat capacity [90, 144, 176]. Measurements of polycrystalline samples show a kink in the magnetic susceptibility which continues to increase below the magnetic transition temperature [90, 177]. However, high-quality single crystals reveal a clear peak of the magnetic susceptibility at , as expected for an antiferromagnetic phase transition [122, 144, 176]. This is similar to data obtained for polycrystalline samples of YMnO_{3} [90] as compared to single crystals (see Figure 28). The magnetic contribution to the heat capacity of LuMnO_{3} and ScMnO_{3} was extracted and it was shown that the entropy change between zero temperature and corresponds to the expected value for spin 2, , with a significant part of the entropy change shifted to lower temperatures [90].

The magnetic susceptibility of ScMnO_{3} exhibits a different anomaly at with a field-dependent irreversible behavior (difference between field-cooled and zero field-cooled data) below and an obvious magnetization hysteresis loop at the lowest temperatures [90, 95, 178, 179]. This has been interpreted as a weak ferromagnetic moment associated with the magnetic order below . However, it has been pointed out by Fiebig et al. that the magnetic structure of the spins ( or ), as derived from neutron scattering [179] and SHG experiments [180], is not compatible with a ferromagnetic moment. There arises the question of whether or not the polycrystalline nature of the samples or possible impurity phases (not detectable in X-ray measurements) could have been the source of the irreversibility in the bulk magnetization. A recent study of single crystals and ceramic samples of HoMnO_{3} has shown magnetic irreversibility in the polycrystalline material [181].

The detailed structure of LuMnO_{3} at low temperature had been studied by high resolution neutron scattering. Interestingly, the -axis of LuMnO_{3} experiences a sizable increase with decreasing temperature below 200 K, signaling the onset of magnetic fluctuations [137, 176, 182]. This increase and the related anomaly in YMnO_{3} (also in HoMnO_{3}) show that the buckling of the MnO_{5} units and the related ferroelectric distortion discussed above are not the only mechanism affecting the thermal expansion behavior of the -axis. The origin of the structural distortions in the magnetically ordered phase was suggested to lie in the displacements of the ions from their ideal position in the unit cell. Interestingly, the deviation from is opposite for YMnO_{3} ( increases below ) and LuMnO_{3} ( decreases below ) [137, 182]. As pointed out by Fabrèges et al. [130], the deviation from determines the sign of the magnetic coupling between spins in adjacent planes and the magnetic structure for different RMnO_{3}.

The magnetic structure of LuMnO_{3} was discussed based on neutron scattering experiments. The proposed models for the magnetic order vary significantly among different publications. Koehler et al. [88] determined an intermediate angle of within the model (Figure 3) with the magnetic symmetry . In contrast, Katsufuji et al. [91] and Kozlenko et al. [152] suggested two possible spin configurations for LuMnO_{3}, ( model, ) or ( model, ). The latter model is supported by the neutron study of Bieringer et al. [183]. The model magnetic order in LuMnO_{3} was also derived from SHG optical experiments [100] and it was concluded that LuMnO_{3} apparently exhibits a spin reorientation from to magnetic symmetry with an intermediate phase () coexisting with the two other phases below 60 K. It is not clear why this spin reorientation has not been observed in neutron scattering experiments. The spatial coexistence of phases with different magnetic symmetry can make the correct determination of the spin order below quite difficult.

Unlike in LuMnO_{3}, the spin reorientation in ScMnO_{3} was detected in early neutron scattering experiments [179] and later confirmed by other neutron studies [95, 130]. While all neutron investigations agree about a smooth spin reorientation starting between 60 and 80 K and slowly progressing toward lower temperature, there is no consensus about the specifics of the magnetic order ( or model). Bieringer et al. [179] assigned the magnetic symmetry with a spin angle of (close to ) to the high-temperature phase and suggested a spin rotation within the model to a low-temperature angle of (close to ), the same type of spin reorientation as reported for HoMnO_{3}. Muñoz et al., however, described the magnetic order within the model with the spin angle ( symmetry) at high temperatures ( K) and continuously increasing from at 75 K to at 1.8 K. In this model, the symmetry of the low-temperature phase would be . The SHG optical experiments support the model with below coexisting with below 60 K, and all three phases (including ) coexisting at the lowest temperatures within different regions of the same crystal [93, 100, 184].

It is interesting to note that the spin reorientation in ScMnO_{3} is realized in a system of exclusively spins and no rare earth moment is involved, unlike in the sister compound HoMnO_{3}. Therefore, the change in the in-plane magnetic anisotropy has to be considered as the origin of those phenomena. In ScMnO_{3}, the magnetic anisotropy is less pronounced making this system more susceptible to spin reorientations and phases with intermediate spin angles . The strong spin lattice interaction causes the structural distortions and the change of the Mn position in the unit cell, as reported in recent high resolution neutron studies [130]. The “softness” of ScMnO_{3} with respect to an in-plane spin rotation is also reflected in the photoinduced control of the average spin angle at low temperature [180].

##### 2.4. Multiferroic Properties and Phase Diagrams of ErMnO_{3}, TmMnO_{3}, and YbMnO_{3}

The multiferroic phase diagrams of ErMnO_{3}, TmMnO_{3}, and YbMnO_{3} in -axis magnetic fields and the associated magnetic orders were first proposed by Fiebig et al. [93, 102], as shown in Figure 31, based on SHG and Faraday rotation optical experiments. It is interesting that in the case of ErMnO_{3}, TmMnO_{3}, and YbMnO_{3}, the high-field phase assumes the symmetry which can be understood by the energy gain through the coupling of the external magnetic field to the -axis magnetization [102]. In between the pure and magnetic phases there is a region (grey area in Figure 31) showing hysteretic and irreversible effects resulting in a history dependence of the final state within a certain mode of field and temperature change [93]. The Néel temperatures of ErMnO_{3}, TmMnO_{3}, and YbMnO_{3} slightly decrease with the external field up to about 40 kOe and than trace back to low temperatures and low fields resulting in the typical “nose” shape of the phase boundaries to the high-field phase (Figure 31).

In contrast to the results of Figure 31, bulk measurements of the magnetization and dielectric constant have shown that the phase boundary to the high-field phase extends to much higher magnetic fields, for example, above 140 kOe for YbMnO_{3} [41]. A more detailed discussion of the anomalies of bulk physical properties of all three compounds, ErMnO_{3}, TmMnO_{3}, and YbMnO_{3}, is therefore warranted. Bulk measurements such as dc or ac magnetization and heat capacity can be utilized to detect the thermodynamic nature of the phase transitions. The dielectric constant is extremely sensitive to the magnetic transitions because of the strong spin lattice coupling in all hexagonal manganites. This was clearly demonstrated in Section 2.2.2 for HoMnO_{3}. The measurement of , in combination with other physical quantities, is therefore an ideal tool to study the high-field phase diagrams of other RMnO_{3}.

The zero-field heat capacity of reveals two distinct peaks at K and K, as shown in Figure 32(a). The dc magnetization measured at 200 Oe is shown in Figure 32(b). A subtle slope change of at (right inset) and a strong rise at indicate the onset of antiferromagnetic spin order and another low-temperature change of the magnetic orders of Mn/Er moments, respectively. The anomaly of the heat capacity at , similar to the one shown in Figure 32(a), was also reported earlier [185]. Sugie et al. [105] found a magnetic hysteresis loop at low temperatures indicative of a possible ferro- or ferrimagnetic moment, however, no such irreversibility could be detected in our *M*-*H* measurements. The dielectric constant of ErMnO_{3}, shown in Figure 32(c), displays the kink-like anomaly at , as reported earlier [105].

The phase diagram of ErMnO_{3} in external -axis magnetic fields can be derived by tracing the anomalies of magnetization and dielectric constant as functions of temperature and field. The low-temperature region can be studied through magnetization measurements as function of the field (shown in Figure 34(a)). increases sharply at very low fields and experiences a sudden change of slope at 450 Oe (1.85 K) indicating a metamagnetic phase transition. With increasing temperature the critical field for this transition decreases to 270 Oe at 2.3 K, as shown in the inset of Figure 34(a). No transition is observed in the low field range above the critical temperature of 2.4 K (). At higher magnetic fields, a second steep increase of signals a second phase transition (see Figure 34(a)). The critical field of this transition increases quickly with temperature and is already above the maximum field of 50 kOe at about 14 K. Note the curves of shown in Figure 34(a) include data collected with increasing as well as decreasing field and no magnetic hysteresis is observed, unlike in the previous publication by Sugie et al. The absence of any hysteretic behavior upon increasing or decreasing temperature or field was also confirmed in a recent study of the magnetization of ErMnO_{3} to temperatures as low as 80 mK [186]. Furthermore, the first transition at fields below 1000 Oe (inset of Figure 34(a)) was not reported in [105].

The second phase transition in the high-field region of the phase diagram ( 7 kOe) is also manifested in temperature dependent magnetic susceptibility data, shown on a logarithmic scale in Figure 34(b). The Néel temperature , indicated by the dashed vertical line, decreases only very little at higher fields. At low temperatures, however, a clear kink of identifies a second phase boundary, labeled in Figure 34(b). The anomaly at develops at magnetic fields above 7 kOe and moves quickly to higher temperature with increasing field . The field and temperature dependent data of the magnetic susceptibility recently published [186] are in good agreement with our results shown in Figure 34.

The high-field phase diagram of ErMnO_{3} is revealed through dielectric constant measurements. Since exhibits a sharp kink at , it is expected that this anomaly can be traced to higher magnetic fields (note that the magnetization measurements in Figure 34 are limited to 50 kOe). Upon closer inspection of , a second anomaly in form of a sharp peak is revealed above a magnetic field of 9 kOe. This peak, shown in the data of Figure 34(a), shifts to higher temperature with increasing field and it’s position coincides with the anomalies of the magnetization (Figure 34). It therefore is the signature of the dielectric constant in crossing the phase boundary which can be traced to much higher fields. At lower magnetic fields, is also defined by the peak of shown in the inset of Figure 34(b). At high magnetic fields, 70 kOe, the two transition temperatures and are detected as small kink-like anomalies of in the main panel of Figure 34(b). With increasing field, decreases while increases and both temperatures eventually merge above 125 kOe.

The resulting phase diagram is shown in Figure 35. The magnetically ordered phase below exhibits a remarkable stability with respect to the field , at least in the intermediate temperature range, extending up to 130 kOe (unlike earlier reports [102]). The phase boundaries given in Figure 35 have been confirmed in recent studies up to 40 kOe [186]. The low-field and low-temperature section of the phase diagram is dominated by two transitions separating three different magnetic phases and the details are shown on an enlarged scale in the inset of Figure 35. Meier et al. [186] suggested another phase boundary based on the inflection point observed in the temperature dependence of the dc susceptibility . This phase boundary is shown as the red dotted line in the inset of Figure 35. While the magnetization data shown in Figure 33(b) indeed show the inflection point of the susceptibility near 4 K, it is not clear whether this feature provides clear evidence for an additional phase transition since the inflection point of (minimum of *dM/dT*) extends to even higher fields at about the same temperature of 4 to 5 K (see 40 kOe and 50 kOe data of *dM/dT* in the inset of Figure 33(b)). The minimum of *dM/dT*, as obtained from our magnetization measurements, is shown as the blue dotted line in the high-field region ( kOe). Dielectric constant data measured at different magnetic fields between 3 kOe and 40 kOe do not indicate any anomaly at the temperature defined by the inflection point of . Therefore, it is not clear if the phase boundary proposed by Meier et al. [186], shown as the red dotted line in Figure 35, indicates a real phase transition associated with distinct changes of physical quantities or rather a cross-over with a smooth change of the magnetic structure.

The magnetic orders and symmetries in the various phases of ErMnO_{3} are still a matter of discussion. Koehler et al. [88] in their early work determined the magnetic structure of the spins within the model with a spin angle to the hexagonal -axis as and independent of temperature corresponding to the magnetic symmetry (close to ). Park et al. [187] found two possible solutions fitting their neutron spectra equally well, ( model) or ( model). Fiebig et al., using optical SHG methods, identified the magnetic space group of ErMnO_{3} as at all temperatures below [93, 100].

In a recent work [186], the discussion of magnetic symmetries was extended to include the low-temperature phase (below ) and the phases induced by -axis magnetic fields. Based on bulk magnetic measurements, SHG optical, and neutron scattering studies, the following development of magnetic structures with temperature and field was proposed: The magnetic order in zero magnetic field sets in below with the magnetic space group , similar to other . The interaction with the -moments results in the partial AFM order of the Er spins on crystallographic 4b sites whereas the Er moments on 2a sites remain paramagnetic. This is consistent with the magnetic symmetry of the Mn spins. At low temperatures, the 4f exchange interaction of on 2a sites becomes stronger, resulting in their magnetic order according to the magnetic symmetry, triggering a simultaneous spin reorientation of the spin system. This could explain the phase transition at into the phase. The phase is ferrimagnetic since the magnetic moments of the on 2a and 4b sites are counter aligned along the -axis but they do not completely compensate each other. The transition from the phase to the phase with increasing magnetic fields is considered to be a change from a multi domain () to a single domain state (). At higher fields (crossing the phase boundary ) all moments are ferromagnetically aligned along the -axis (FM phase) without any change of the spin arrangement. Note that the magnetic symmetry is compatible with a ferromagnetic moment along the -axis.

The phase diagram of ErMnO_{3} at low temperatures is nearly as complex as that of HoMnO_{3} due to the magnetic exchange between rare earth moments and manganese spins which also have been shown to affect the magnon spectra and crystal field transitions in optical experiments [188]. On the other hand, there is also evidence for strong spin lattice interactions as, for example, shown in the phonon hardening below [166]. This could explain the anomalies of the dielectric constant at the various phase boundaries, similar to other rare earth manganites.

TmMnO_{3} was less intensively studied in the past. The magnetic order of spins sets in at K and the symmetry was determined as [93, 100]. The gradual polarization of the moments on 4b sites was detected below in Mössbauer experiments [189]. The measured temperature dependence of the hyperfine field at the 4b site could quantitatively well fitted by a crystal field model with the lowest level split by the molecular field arising from the Mn-Tm exchange interaction. The Néel temperature obtained is consistent with magnetic susceptibility and dielectric measurements [41, 177, 190]. There is no evidence so far of an additional phase transition which could be associated with a possible moment order on 2a sites and a spin rotation, as observed in HoMnO_{3}, ErMnO_{3}, and YbMnO_{3}. The dielectric constant at zero magnetic field shows the kink at , but no further anomaly at lower temperatures (Figure 36(a)).

With increasing magnetic field oriented along the -axis, the kink anomaly at shifts minutely to lower temperature. Above 42 kOe, a second step of at low temperatures indicates another phase boundary, similar to the high-field transition in ErMnO_{3} (Figure 36(a)). The critical temperature, , increases with the field and merges with just below 110 kOe and about 65 K. The low-temperature anomaly of is clearly seen as a sharp step in the field-dependent data of Figure 36(b). The phase diagram, derived from the dielectric data is shown in Figure 37. Similar to ErMnO_{3}, the stability of the magnetic order below extends to much higher fields as earlier proposed [93]. The fact that there is only one transition at the lowest temperatures with increasing magnetic field indicates that the magnetic structure with the antiferromagnetic order of Tm(4b) moments is very stable and the order of the Tm(2a) moments with the reorientation of the Mn spin system requires significantly higher fields (>42 kOe) than in ErMnO_{3}.

The bulk magnetic and dielectric properties of YbMnO_{3} were studied by Sugie et al. [105]. Besides the onset of magnetic order at K, a second phase transition was found below 5 K and at magnetic fields above 30 kOe a metamagnetic transition was detected in field-dependent magnetization data. The details of the magnetic phase diagram, however, have only been revealed through high-field magnetic and dielectric measurements [41]. The temperature dependence of at low field (100 Oe) indeed shows a sharp increase at 4 K (see inset in Figure 38(a)) indicating a possible ferrimagnetic moment arising from the ytterbium sublattice order, in analogy to ErMnO_{3}. This transition is marked by a sharp peak of the heat capacity, in addition to the -shaped peak at (Figure 39). The isothermal magnetization as function of the -axis field, shown in Figure 38(a), is very similar to the results obtained for ErMnO_{3}. The steep increase of the magnetization for small fields is typical for the phase discussed in [186] for ErMnO_{3}.

The sudden decrease of the -slope defines the first phase boundary, . The step-like increase of above 30 kOe could be understood as the transition to the complete ferromagnetic alignment of the moments. The corresponding phase boundary is labeled . The two low-temperature transitions are also reflected in distinct anomalies of the magnetization, , shown in Figure 38(b). In the low-field range ( kOe), the sharp increase of near 4 K signals the onset of the ferrimagnetic order of the moments. At magnetic fields betyween 1 and 30 kOe, however, no sharp anomaly is detected in . A broad maximum and a smooth drop to lower temperature is the characteristic feature of . At higher field, another sharp increase of indicates the transition into the ferromagnetic phase with all moments aligned with the field. Comparing the magnetic moments of the 40 kOe and 50 kOe data in Figure 38(b), it becomes obvious that saturation is reached at the lowest temperatures. This is consistent with the proposed ferromagnetic state. The fact that no significant difference between field-cooled and zero field-cooled data was observed in this high-field range indicates that the magnetic field is strong enough to create a single domain state. Some of the results for the temperature and field dependence of the magnetization discussed above have been confirmed in recent studies [94, 124, 191].

The complete phase diagram of YbMnO_{3} is derived from thermodynamic, magnetic, and dielectric measurements in a similar way as for ErMnO_{3} and TmMnO_{3} above and it is shown in Figure 40. The similarity to ErMnO_{3} is obvious, however, the antiferromagnetic phase extends to much higher fields, beyond the limit of 140 kOe of this investigation. The phase assignment could be similar to ErMnO_{3}. SHG optical measurements favor the magnetic symmetry below . The low-temperature transition almost certainly does involve the order of moments. Following the same scenario proposed for ErMnO_{3} [186], we can assume that the *f*-moments of the Yb on 4b sites are systematically polarized according to the antiferromagnetic structure shown in Figure 25(d). The phase transition at involves the order of the moments on 2a sites antiparallel to the 4b moments and a reorientation of the spins resulting in the ferrimagnetic order of the *f*-moments according to the magnetic symmetry. At higher magnetic fields, the flop of the moments at 2a positions results in the alignment of all f moments with the external field at .

The physical picture of the magnetic orders of spins and moments on 4b and 2a sites was first proposed by Fabrèges et al. based on a comprehensive study combining magnetization measurements, Mössbauer, and neutron scattering experiments. The ordered magnetic moments of Mn, Yb(4b), and Yb(2a) could be separately determined and fit to a mean field model. According to the model, the Yb(4b) moment order below is driven by the molecular field generated by the Mn spins and the Yb(2a) order below results from the Yb-Yb exchange interactions. The order of Mn spins and rare earth moments at 4b and 2a sites has been studied independently by Salama et al., arriving at similar conclusions [192, 193].

##### 2.5. Metastable Hexagonal DyMnO_{3}

The hexagonal structure of DyMnO_{3} is metastable and it can be synthesized using special chemical techniques [92, 194]. In the hexagonal form, DyMnO_{3} is ferroelectric, similar to all other hexagonal manganites. The ion is the largest in the series, resulting in the most expanded structure with lattice constants and exceeding those of all other hexagonal RMnO_{3}. A reduction of the magnetic exchange couplings between the different ions can be expected and it is not surprising that the Néel temperature of DyMnO_{3} ( K) is the lowest among all RMnO_{3} [195]. The magnetic properties of DyMnO_{3} were first studied by Ivanov et al. [194]. The most notable feature was a sharp peak in the ac magnetic susceptibility observed at K and a remanent magnetization below this temperature, indicating a possibly ferrimagnetic state at low . The low-temperature ferrimagnetic phase was found to extend to higher temperature in external magnetic fields oriented along the -axis.

Subsequent studies of the magnetic properties are in qualitative agreement with the original data [111, 195], however, the values for and scatter among different investigations. Harikrishnan et al. [195] derived K and K from magnetic and heat capacity data. Nandi et al. [111] found K and K. The differences may be due to small variations in the chemical composition of the samples used by different groups. The magnetic order of the moments was studied using X-ray resonant magnetic scattering [111]. The magnetic symmetry in the temperature range between and was determined as (), with the moments all parallel to the -axis. It is interesting that this symmetry is different from the magnetic symmetry of ErMnO_{3}, TmMnO_{3}, and YbMnO_{3}, but it coincides with the symmetry of below the spin reorientation temperature, . For the low-temperature phase, the () magnetic symmetry which allows for a ferromagnetic moment was proposed. It should be noted that the X-ray resonant magnetic scattering experiment only determines the magnetic symmetry of the ordered Dy sublattices, with no information about the magnetic structure of the spins.

The magnetic symmetry of the Mn sublattice was studied through SHG optical measurements as well as neutron scattering experiments and compared with the magnetic symmetry of the Dy sublattice in the temperature-field phase diagram of DyMnO_{3} [196]. Interestingly, different magnetic symmetries have been found for Mn and Dy moments in the temperature range between and in zero magnetic field. The magnetic space group for the spins was determined as while the space group for the moment order was , confirming the earlier studies [111]. This indicates that the 3d-4f coupling between the order parameters describing the spin order and the moments is weaker than assumed. To understand the coexistence of two order parameters with different symmetry, Wehrenfennig et al. [196] proposed the coupling between 3d spins and 4f moments to be of the biquadratic form. This biquadratic interaction can trigger the simultaneous order of the spin and moment systems with different magnetic symmetries. The proposed mechanism would introduce novel physics into the basic understanding of multiferroic hexagonal manganites.

In the low-temperature phase () and in magnetic fields applied along the -axis the order of the spins was found to follow the same magnetic symmetry as the moments, namely . The high-field magnetic symmetry appears reasonable in view of the fact that allows for a ferromagnetic moment along the -axis which is certainly stabilized by the external field.

##### 2.6. InMnO_{3}: A Paraelectric Hexagonal Manganite

InMnO_{3} was synthesized in 1992 [197] and the structure was determined to be hexagonal with space group , similar to other RMnO_{3} discussed above [89]. First magnetic measurements have shown a clear anomaly at K. The 120 K transition was identified as the onset of magnetic order through neutron scattering experiments. Two more anomalies of the magnetic susceptibility at 40 K, and 15 K, observed at low fields, did disappear in moderate magnetic fields [89, 198, 199]. Based on the temperature dependence of different neutron scattering peaks and the additional anomalies of the susceptibility at lower temperatures, Greedan et al. suggested the possible existence of spin reorientation transitions. However, the intrinsic nature of those low-temperature anomalies has been questioned and attributed to possible impurity phases since high-pressure synthesized samples of InMnO_{3} did not provide support for additional phase transitions below [200]. A weak ferromagnetic-like hysteresis was reported in field-dependent magnetization measurements at low temperature. The phase transition at 120 K is clearly supported by heat capacity measurements showing a distinct peak at [200, 201]. The low-frequency dielectric constant exhibits a kink at , similar to other hexagonal manganites, which is an indication of the strong interaction of the spins with the lattice.

The existence of ferroelectricity in InMnO_{3} is a matter of dispute. Based on the structural data derived from neutron scattering experiments [89], Abrahams estimated the ferroelectric transition temperature as K [202]. Dielectric and *P*-*E* hysteresis measurements have been interpreted as evidence for a ferroelectric transition at 465 K [199]. The origin of the suggested ferroelectric phase in was proposed to be an intra-atomic orbital mixing of In and a covalent bonding (4(In)-2(O)) along the -axis [203]. However, Belik et al. [200, 201] have not found any changes of the dielectric permittivity, the number of phonon excitations, or other quantities sensitive to structural changes at high temperatures up to 900 K, questioning the possible existence of a structural (ferroelectric) phase transition. The relaxor-like property of the *P*-*E* hysteresis loop was reproduced and attributed to extrinsic, nonferroelectric effects like high-loss dielectric behavior caused by defect-induced conductivity.

The lattice parameters of InMnO_{3}, as determined from X-ray and neutron scattering experiments [89, 200, 204], deviate significantly from other hexagonal RMnO_{3}. While the lattice parameter of all hexagonal RMnO_{3}, including ScMnO_{3}, are a linear function of the ionic radius, and of InMnO_{3} deviate significantly with and found to be clearly smaller and larger, respectively, as expected from the linear dependence [89]. The large -axis length suggests a reduced magnetic interplane exchange coupling. This effect is further enhanced by frustration effects due to the almost ideal position of the ion in the unit cell (). The two possible exchange paths between ions in neighboring planes are therefore almost identical, increasing the magnetic frustration between spins of neighboring planes.

In contrast to other hexagonal RMnO_{3}, the magnetic order below causes a doubling of the unit cell along the -axis [89]. High resolution neutron scattering experiments are consistent with the doubling of the magnetic unit cell, the onset of magnetic order at 120 K, and the absence of any further spin reorientation at lower temperatures [204]. The magnetic symmetry of the ordered spins was determined to be either or (Figure 4), with the spins of two subsequent unit cells along the -axis reversed by . Above , a strong diffuse scattering indicates the presence of a significant amount of two-dimensional spin fluctuations which supports the conjecture that InMnO_{3} appears to be the most frustrated system among the hexagonal RMnO_{3} compounds. The 2D short range correlations of the spins arise at much higher temperatures, however, 3D long range order is stabilized only at lower temperature by the weak interplane magnetic interactions. The nature of these interactions was discussed by Fabrèges et al. [204] and they were attributed to long-range pseudodipolar interactions stabilizing the magnetic order along the -axis with the modulation vector .

The possible absence of ferroelectricity in InMnO_{3} seems to be incompatible with the polar space group which was used to fit numerous X-ray diffraction and neutron scattering data. However, Rusakov et al. [201] pointed out that the centrosymmetric, trigonal space group (No. 165) could also be used to fit their X-ray spectra with almost the same values as the polar group . This motivated Kumagai et al. [205] to revisit and study the structure and polarization property of InMnO_{3} combining X-ray diffraction, piezoelectric force microscopy, and optical SHG measurements with density functional calculations. Based on the calculated lower energy of the structure, the authors concluded that the actual lattice symmetry of InMnO_{3} is not but rather . Furthermore, no evidence for ferroelectricity or ferroelectric domains was found in SHG and piezoelectric force measurements. The two structures under discussion are shown in Figure 41 (Figure 1 in [205]). The main difference between the and structures is the way the MnO_{5} bipyramids are tilted from their ideal position and the related displacements of the ions. In symmetry, the ions on different crystallographic sites (2a and 4b) are displaced along the -axis in opposite directions. The generated dipolar momenta are not compensated yielding a macroscopic polarization (see left column in Figure 41(a)). In the structure, however, one ion remains in the centrosymmetric 2b position without any displacement along . The structure has an inversion center and no ferroelectric polarization is allowed. This is shown in the right column of Figure 41(a). The corresponding spin orders are shown in Figure 41(b).

With all experimental evidence for a missing polarization it can be concluded that InMnO_{3} is so far the only paraelectric hexagonal manganite. The possible origin of this “anomaly” was discussed by Kumagai et al. [205] based on the results of band structure calculations. It was concluded that the key to understand the different lattice structures of InMnO_{3} and other RMnO_{3} is the degree of covalency of the R-O bond. As a result of the competition between electrostatic energy and R-O covalency, either the structure is in favor (for lower R-O covalency) or, for higher covalency, the structure becomes the stable ground state.

##### 2.7. Symmetry Analysis and Landau Theory

To understand the complex magnetic phase diagrams, phase transitions between different magnetic symmetries, and partially coexisting phases, a group theoretical symmetry analysis was conducted by different authors [95, 106, 206]. The point group has four one-dimensional (labeled , , , and ) and two two-dimensional irreducible representations ( and ). The magnetic space groups in the ordered phases corresponding to the 1D representations are: (), (), (), and (). The compatible magnetic structures have been discussed above and are shown in Figures 4 and 25. The 2D representations can be disregarded since there is no experimental evidence for any ordered magnetic structure corresponding to or in the phase diagrams of all hexagonal RMnO_{3}.

The spin configurations of the , (2a), and (4b) can be classified according to the four 1D irreducible representations. The different magnetic structures are discussed in detail in [95] and the corresponding spin configurations of rare earth and manganese moments are listed in Table 1 of [206]. The most general Landau free energy expansion with respect to all order parameters of the spins and moments on two crystallographic sites can be very extensive with too many parameters involved. It is also not clear how the different order parameters of -spins and -moments couple to one another, as demonstrated in the case of DyMnO_{3} above. Therefore, Munawar and Curnoe limited the free energy expansion to two competing order parameters, defined by the and representations [206]. This choice is consistent with experimental results for ErMnO_{3}, TmMnO_{3}, and YbMnO_{3}. The free energy expression was derived in [206]:

and are the order parameters corresponding to the () and () irreducible representations, respectively. , , , and are the expansion parameters. The first four terms represent the energy of the ordered state for both symmetries, the next term describes the coupling between the two order parameters, and the last term is the energy gain due to the coupling of the magnetic order parameters to the external magnetic field. Minimizing the free energy (7) with respect to the order parameters and assuming proper sets of the expansion coefficients [206] could derive temperature-field phase diagrams showing the main features as observed in experiments [93], for example, the transition from the phase to the phase upon increasing magnetic field and the existence of a two-phase region in the phase diagram. With the expansion (7), however, it was assumed that the order of spins and moments in a single phase follow the same irreducible representation, unlike the special case of DyMnO_{3} discussed in the previous section.

The same Landau free energy expression (7) was used to describe the phase diagram of YbMnO_{3} and the critical scaling of the in plane dielectric constant as a function of temperature and -axis field [191]. The observed scaling of the magneto-dielectric effect could only be explained by assuming a competition of an antiferromagnetic () with a ferromagnetic () state. It was shown that any small magnetic field did induce a ferromagnetic order parameter of symmetry and the high-field transition is actually from a phase mixture into a phase with sole magnetic symmetry. This model did describe the observed scaling of the dielectric constant. The transition between the mixed () phase and the ferromagnetic phase is second order in the high-temperature range, but it becomes a first order transition at lower temperatures (critical point), close to the maximum field along the phase boundary. The two sections of the phase boundary correspond to the two critical temperatures labeled (2nd order transition) and (1st order), respectively, in Section 2.2.4.

#### 3. Summary

Hexagonal manganites RMnO_{3} belong to an exceptionally interesting family of type (I) multiferroics with high ferroelectric transition temperatures and frustrated magnetic orders existing at much lower temperature. The complex magnetic system of Mn spins and rare earth moments and the inherent frustration due to the triangular geometry of the magnetic sublattices results in a wealth of physical phenomena, phase transitions between different magnetic orders, and magnetoelectric as well as magnetoelastic effects. The series of compounds includes all RMnO_{3} with rare earth elements (R) from Dy to Lu and InMnO_{3}, ScMnO_{3}, and YMnO_{3}.

HoMnO_{3} and YMnO_{3} have been studied most extensively. HoMnO_{3} possesses the most complex phase diagram in -axis magnetic fields with a multitude of different phases clearly separated by the properties of magnetic, thermodynamic, and dielectric properties, although the precise magnetic orders and symmetries of all phases have yet to be explored. The strong interaction between the Mn spins and Ho moments, the Mn-Ho exchange, and the coupling to the lattice and the ferroelectric order parameter, together with the frustration of the magnetic exchange pathways, makes this compound a highly correlated system, the study of which appears to be quite challenging. Bulk magnetic, dielectric, heat capacity, thermal expansion, and other measurements have contributed to unravel the complex phase diagram and to identify the various phase boundaries. Neutron scattering and second harmonic generation optical measurements have determined the magnetic orders of Mn spins as well as Ho moments in different phases. Combining all studies has helped to arrive at a more complete understanding of the complex interactions of highly frustrated magnetic systems and the origin of the magnetoelectric effects in this class of multiferroics.

YMnO_{3} appeared interesting because of the missing rare earth magnetic moment which simplifies the magnetic subsystem and should make it easier to study the magnetoelectric interaction of the Mn spin order with the ferroelectricity. Nevertheless, new phenomena have been discovered in the ferroelectric and magnetic domain walls, such as a clamping property which causes a ferroelectric domain boundary always to coincide with a magnetic domain wall. These observations show that magnetoelectric interactions can be particularly strong in domain boundaries of multiferroic compounds.

Among the other hexagonal manganites, several family members exhibit further interesting properties and phenomena. ScMnO_{3} undergoes an Mn spin reorientation transition well below it’s Néel temperature due to the weak in-plane magnetic anisotropy of the Mn spins. ErMnO_{3}, YbMnO_{3}, and DyMnO_{3}, similar to HoMnO_{3}, show low-temperature phases which are determined by a ferromagnetic order of their respective moments on the two inequivalent rare earth sublattices. The strong *f*-*f* exchange interaction is assumed to be the origin of the *f*-moment alignment and the exchange with the manganese spins than forces the Mn spin system to rotate to be compatible with the same magnetic symmetry. However, hexagonal DyMnO_{3} was found to adopt different magnetic symmetries below for the Mn spins and Dy moments, respectively, which was not observed in other RMnO_{3}. InMnO_{3} seems to be the only paramagnetic hexagonal manganite since the originally reported ferroelectricity in this compound was disputed and no evidence of a sizable polarization was found in more detailed experiments.

The wealth of interesting phenomena and the unraveling of novel physics in multiferroic hexagonal manganites is closely related to the simultaneous presence of geometric frustration, particularly of the magnetic system, the coupling and mutual interaction of different magnetic ions, the important role of magnetic anisotropy, mainly controlled by the rare earth ions, and the strong interaction of moments with the lattice and the ferroelectric order parameter. This makes the compound family unique and a perfect subject for further studies. The results are expected to also provide fundamental insight into other complex multiferroics or highly frustrated magnetic systems.

#### Acknowledgments

This work is supported in part by the US Air Force Office of Scientific Research, the T.L.L. Temple Foundation, the J. J. and R. Moores Endowment, and the State of Texas through the Texas Center for Superconductivity at the University of Houston.

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