Magnetoelectric multiferroic compounds are a subject of significant consideration because they are characterized simultaneously by different magnetic and electric ferroic orders that are coupled. The magnetoelectric effect resulting from such strong coupling is stemming for exceptional materials, which opens a new horizon for the next generation applications in functional devices. In recent years, the magnetically frustrated multiferroic RMn₂O₅ family of oxides have attracted a great deal of attention due to the strong coupling between the magnetic and electric ferroic orders. ^{[1, 2]} The magnetoelectric effect revealed in RMn₂O₅ is due to a variety of phenomena, among them are magnetically induced polarization ( ) in HoMn₂O₅, ^[1] magnetically reduced in ErMn₂O₅, ^[1] magnetically flipped in TbMn₂O₅ ^[2] and GdMn₂O₅, ^[3] and magnetically flopped in TmMn₂O₅ ^[4] and YbMn₂O₅, ^[5] all arising in the low temperature phase. A reversible magnetocaloric effect has also been observed in HoMn₂O₅. ^[6] Moreover, the magnetoelectric coupling effect has also been studied in thin films. ^[7–9]

The main structure of all manganese RMn₂O₅ compounds crystallizes in the isostructural orthorhombic symmetry with space group Pbam. ^{[10, 11]} All members of RMn₂O₅ are characterized as the geometrically frustrated system because a slight deviation from their mean structure has been detected. ^[12] In a unit cell, there exist two manganese ions with different valances and positions: 4h for Mn (t e ) forming Mn O₅ square pyramids and 4f for Mn (t e ) with corner sharing Mn O₆ octahedra. These manganese ions together with oxygen coordination form a loop of five Mn ions, three Mn and two Mn . In each loop, there are three inequivalent magnetic superexchange interactions in the ab plane between the Mn and Mn ions, which are usually referred to as magnetic interactions J ₃, J ₄, and J ₅. The J ₃ and J ₄ magnetic interactions correspond to the exchange interactions between the magnetic moments of Mn and Mn , and J ₅ corresponds to the magnetic interaction between two Mn magnetic moments. Considering the ac plane, one can easily outline two Mn –Mn magnetic interactions, namely, J ₁ and J ₂. The J ₁ is responsible for the magnetic interaction between adjacent Mn spins through oxygen at the layer and depends on the radius of the ion, which directly affects the z component (k _z ) of the propagation vector ( ). ^{[13, 14]} Blake et al. have studied the crystal and magnetic structures of RMn₂O₅ ( , Ho, Dy) as a function of temperature by using neutron diffraction. They found that the spin arrangement of Mn ions is identical within the plane of the commensurate phase for each system. It was also suggested that a shift of the Mn cations gives rise to a net polarization along the axis due to the canted antiferroelectric structure. ^[15] Moreover, the radius of is essentially insensitive in the ab plane, while it determines the exchange interaction with adjacent Mn spins in the plane.

The key issue in the family of RMn₂O₅ is to elucidate the influence of on the magnetoelectric properties and its effect on the Mn magnetic ordering through J ₁. It is important to note that RMn₂O₅ compounds with large size (La and Pr) are not ferroelectrics. ^[16] Interestingly, compounds with small size (Nd to Yb) all present a similar behavior, with the largest polarization in the commensurate (CM) phase, ^[2] except in the case of NdMn₂O₅. Very recently, it was observed that the multiferroic NdMn₂O₅ exhibiting the largest polarization along the b axis in the incommensurate (ICM) phase ^[17] differs from other RMn₂O₅. This makes NdMn₂O₅ an interesting compound to study because it lies between the nonferroelectric PrMn₂O₅ and ferroelectric SmMn₂O₅ and possesses a different magnetic origin for the ferroelectric polarization.

A spin–phonon interaction has been observed in the paramagnetic phase of RMn₂O₅ ( , Dy, and Bi) with the detection of Raman phonons ^[18] and infrared active phonons in DyMn₂O₅. ^[19] More interestingly, it was found that the phonon anomalies in the paramagnetic phase show a strong dependence on the ions. ^[18] Very recently, we observed an anomalous behavior in the dielectric constant slightly above the Néel temperature in GdMn₂O₅ ^[20] and DyMn₂O₅. ^[21] In this regard, we are motivated to conduct temperature dependent dielectric measurements in the paramagnetic phase of NdMn₂O₅ to reveal the existence of a dielectric response at high temperature in this compound. As Nd has a larger ionic size as compared to Gd and Dy , so we expect some interesting results that can shed light on the role of the ion in the family of RMn₂O₅.

Here, we report on the dielectric properties of polycrystalline NdMn₂O₅ in the temperature range T = 2–60 K with applied magnetic field H = 0–12 T. Based on our results, we provide a comprehensive explanation about the magnetoelectric transition in the NdMn₂O₅ system and also construct the magnetoelectric phase diagram. In addition, we will emphasize on the role of the ion in the paramagnetic phase.

A polycrystalline sample of NdMn₂O₅ was prepared by using the sol–gel method. ^{[20, 22]} X-ray diffraction (XRD) measurement was performed with a Rigaku powder x-ray diffractometer using Cu radiation in the 2θ range from 20° to 80° by step scanning at 0.02° per step. The Rietveld refinement of the XRD pattern was performed by using the JANA2006 program. The pseudo-Voigt function was used to fit the diffraction peaks in the space group Pbam. The complex dielectric constant was measured in the temperature range T = 2–60 K with applied magnetic field H = 0–12 T at a frequency of 10 kHz by using a physical property measurement system (PPMS). For the dielectric measurements, the powder was pressed into a pellet of diameter 13 mm under the pressure of 30 kN by using the Paul-Otto Weber hydraulic press. The silver paste was used for forming a capacitor.

The high quality single crystal of NdMn₂O₅ is difficult to grow by using the flux method. ^{[23, 24]} However, the polycrystalline NdMn₂O₅ has been synthesized by the chemical transport method under high oxygen pressure. ^[25] Here, we report the successful synthesis of polycrystalline NdMn₂O₅ by using the sol–gel method for the first time. Figure 1 presents the XRD pattern of polycrystalline NdMn₂O₅ at room temperature. The XRD pattern is analyzed by using the Rietveld refinement technique assuming an orthorhombic structure with Pbam space group (see inset of Fig. 1). A good agrement has been found between the refined and the observed XRD patterns as all XRD peaks are fitted, which indicates the single phase formation of the material and no trace of any impurity. The obtained unit cell parameters are Å, Å, and Å, which are consistent with the previously reported values. ^[12]

Fig. 1.

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	Fig. 1. (color online) Room temperature XRD pattern of polycrystalline NdMn2O5. Inset shows the Rietveld refinement of the XRD curve by using the JANA2006 program. The calculated crystallographic data are , , and .

The study of dielectric properties is an essential tool to identify the multiferroicity. Figure 2 shows the temperature variation of the dielectric constant during both heating and cooling with zero magnetic field. The dielectric constant increases sharply at ∼31 K, which is referred to as the Néel temperature. ^{[17, 26]} Just below , there is a sharp peak indicating a small thermal effect as observed previously. ^{[17, 26]} This sharp peak at K is attributed to the ferroelectric peak as observed in other RMn₂O₅. ^[27] It is worth mentioning that the thermal hysteresis of indicates the first order phase transition similar to the other ferroelectric members. A slight change in the dielectric curve observed at K may be associated with another weak dielectric anomaly.

Fig. 2.

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	Fig. 2. (color online) Dielectric constant ε of NdMn2O5 at zero magnetic field. The arrows indicate the transition temperatures between different magnetic phases.

If this anomalous behavior appeared as a sharp step, it might be attributed to excitation of electromagnons. ^{[28, 29]} Here, we suggest that the anomaly at K may not be related to any electromagnon excitation, rather it is related to a slight change in the magnetic propagation vector. ^{[17, 26]} The most interesting finding is the observation of the low temperature anomaly at ∼4 K, which is attributed to the independent Nd magnetic ordering as observed in other RMn₂O₅ ( Tb, ^[2] Gd, ^[20] Dy, ^[21] Er, ^[30] Tm, ^[31] and Yb ^[24]). Chattopadhyay et al. failed to observe the low temperature independent Nd ordering in their dielectric measurements. ^{[17, 26]} However, we have clearly observed Nd ordering in our dielectric measurements, which is consistent with the magnetic susceptibility and neutron diffraction measurements on NdMn₂O₅. ^{[17, 26]}

We have also measured the dielectric constant under a strong magnetic field in order to identify the magnetoelectric effect in NdMn₂O₅. Figure 3 demonstrates the variation in the dielectric constant under a magnetic field up to 12 T. Interestingly, the dielectric constant remains robust against the external magnetic field at the most prominent phase transitions ( , , and T _X ), however the anomaly at smoothly vanishes with increasing magnetic field. The anomaly at shows an interesting magnetoelectric behavior and is quite similar to that observed for other members of the RMn₂O₅ family. ^{[20, 21]} Astonishingly, no magnetoelectric excitation is observed in the paramagnetic phase (above ) as observed in the case of GdMn₂O₅ ^[20] and DyMn₂O₅ ^[21] that gradually moves to higher temperature with increasing field. This reflects the role of the ions in the paramagnetic phase of RMn₂O₅, which is an unusual result and the origin of this discrepancy will be discussed later.

Fig. 3.

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	Fig. 3. (color online) Temperature dependent dielectric constant ε for NdMn2O5 at various magnetic fields up to 12 T.

3.1. Magnetoelectric phase diagram

In order to elucidate the magnetic field effect on NdMn₂O₅, we have constructed the magnetoelectric phase diagram, as shown in Fig. 4. The phase boundaries defined by , , and T _X in the phase diagram remain robust under the external magnetic field. These phase boundaries correspond to the phase transitions PM/PE ICM1/PE at , ICM1/PE ICM1/FE at , and ICM1/FE ICM2/FE at T _X , respectively (we have used the conventions for different phases (i.e., ICM1, ICM2, PE, FE) from Ref. [17]). At the lowest temperatures ( K), an additional structure is seen due to an ordered state of the Nd spins, which corresponds to another phase transition ICM2/FE CM+ICM2/FE.

Fig. 4.

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	Fig. 4. (color online) Magnetoelectric phase diagram of NdMn2O5.

The phase diagram for H = 0 exhibits similar phase sequences as observed through neutron diffraction. ^[17] Therefore, we suggest that fundamental physics behind the different phases is the same as that in the NdMn₂O₅ phase diagram as described in Ref. [17]. Above 3 T, the CM+ICM2/FE phase becomes unstable and merges with ICM2/FE as the field increases up to 12 T. A possible explanation for this unstable phase may be a reorientation of the Nd sublattice under the external magnetic field.

We have synthesized polycrystalline NdMn₂O₅ and performed dielectric measurements under a strong magnetic field. Our results unveil an anomaly in the low temperature phase, attributed to the Nd magnetic ordering. To summarize our results, a magnetoelectric phase diagram was constructed. We found that the main transitions defined by , , and T _X in the phase diagram remain robust under the external magnetic field. However, a strong magnetoelectric effect is observed for the transition at that vanishes at fields higher than 3 T. Moreover, the absence of the dielectric anomaly in the paramagnetic phase may be due to the bigger ionic size of the Nd ion, which directly influences the J ₁ exchange interaction and results in a different magnetic correlation in the paramagnetic phase as compared to the small ionic radius of in the family of RMn₂O₅.