Antiferromagnets provide a flexible platform for the exploration of magnetically driven phase transitions. Examples include the classic spin flop [1][2][3][4][5][6], magnetic saturation transitions [7][8][9], and those with a sequence of metamagnetic, helical, and nanopantograph spin states [10][11][12]. The emergent class of tunable multiferroics with chemical formula [(CH 3 ) 2 NH 2 ]M(HCOO) 3 (M = Mn 2+ ,F e 2+ ,N i 2+ ,C o 2+ ) provides a number of exciting opportunities in this regard. The Mn analog [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 is ferroelectric below T C = 185 K with polarization P = 1.5 µC/cm 2 .I t is also a weakly canted antiferromagnet below T N = 8.5K with an 0.08 • canting angle [13,14]. Cross coupling between the electric and magnetic orders has been observed [15][16][17][18], and low-temperature phase separation leads to quantum tunneling of the magnetization [19]. Figure 1 displays the crystal structure of this compound and summarizes the aforementioned energy scales. The Mn 2+ centers are connected with formate superexchange ligands with J =-0.64 K [20]. Dimethylammonium resides inside the anion pocket and is disordered (ordered) above (below) T C . Surprisingly and counterintuitively, polarization is enhanced in magnetic field at temperatures well above T N [16]. This dovetails with a recent report of sizable magnetoelectric coupling at elevated temperatures [21]. These findings suggest that there may be a magnetically driven transition at higher fields-beyond the phase space that has been thus far explored [18,22]. Other perovskitelike materials including the rare earth manganites also sport exotic phase diagrams [23], providing additional motivation for measurements of [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 under external stimuli [24].
In order to explore the spin behavior of a model metalorganic multiferroic, we measured the magnetization of [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 in pulsed fields up to 65 T and temperatures down to 0.37 K. The magnetic field-temperature (B-T ) phase diagram developed from these data reveals a spin flop at 0.31 T and a transition to the fully saturated state at 15.3 T. The overall character of the B-T phase diagram and the in-phase → out-of-phase nature of the calculated spin-density pattern that accompanies the transition to the fully polarized magnetic state is reminiscent of other moleculebased quantum magnets like Mn[N(CN) 2 ] 2 and MnCl 2 (urea) 2 [25,26], although in this system, we have not only an adaptable framework but also a mechanism for multiferroic behavior. In contrast to the antiferromagnetic rare earth manganites, the energy scales are smaller which makes the high-field phase accessible with laboratory-based magnets. This facilitates development of the full phase diagram and opens the door to deeper exploration of this system. Residual magnetization above the 8.5 K magnetic ordering temperature suggests the importance of short-range interactions, findings that we discuss in terms of magnetoelectric coupling above T N [16,21].
Polycrystalline [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 was prepared by solvothermal reaction techniques as described previously [13,27]. Magnetization measurements were performed using a 65 T short-pulse magnet at the National High Magnetic Field Laboratory in Los Alamos using a 1.5 mm bore, 1.5 mm long, 1500 turn compensated-coil susceptometer, constructed from 50 gauge high-purity copper wire. The sample was loaded into a 1.3 mm diameter ampoule that moves in and out of the coil. The signal voltage V when the sample is in the coil is proportional to dM/dt, where t is time. M is evaluated via numerical integration. Accurate values of M were obtained by subtracting empty coil data from data measured when the sample is present, under identical conditions. Temperature was controlled with a 3 He system obtaining temperatures down to 0.37 K. An induced voltage in a ten-turn coil, calibrated by observing the de Haas-van Alphen oscillations of the belly orbits of the copper coils of the susceptometer, is integrated to obtain the measured field B. In addition to full field pulses, we carried out 4 and 25 T shots to better resolve the spin flop and saturation. The critical fields (B SF and B Sat )w e r e determined using first derivative techniques. The pulsed field magnetization values were calibrated against measurements on a transparent single crystal with cuboidal shape in a 14 T dc magnet using a vibrating sample magnetometer.
Electronic structures for the antiferromagnetic and ferromagnetic states of [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 were calculated on the basis of density functional theory calculations by employing the frozen-core projector augmented wave method [28,29] encoded in the Vienna Ab initio Simulation Package [30] with the generalized-gradient approximation for the exchange-correlation functional [31], a plane-wave cutoff energy of 520 eV, and 76 k points for the irreducible Brillouin zone. To describe the strong electron correlation at the magnetic site, the on-site repulsion U eff = U -J = 4eV was added on Mn 2+ [32]. Spin-density distributions were calculated for the antiferromagnetic and ferromagnetic states of [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 using the XCrySDen program [33]. [25,26]. The linear character of M(B) between B SF and B Sat is in line with expectations for a three-dimensional system [7].
In our analysis of the magnetization and exchange interactions, we employ the experimentally determined saturation field (B Sat = 15.3 T) and the spin Hamiltonian
where Z is the coordination number of the Mn 2+ center, S = 5/2, g = 2.0, and µ B is the Bohr magneton. We extract J =-0.69 K [34]. This value compares well with that estimated from the zero-field susceptibility (-0.64 K) [13], although the absence of adjustable fit parameters makes it much less ambiguous [35]. The overall low energy scale of the exchange interactions explains why the spins can be saturated in experimentally realizable fields. Above T N , magnetization increases gradually toward saturation. The general shape of M(B) is evolving, but M(B) does not turn into a Brillouin function above T N . The spin-flop transition is no longer apparent, and the sharp knee denoting B Sat is rounded due to the loss of long-range order. The gradual rise indicates that there are important short-range interactions above T N .Thisis discussed in detail below. Figure 2(b) displays the calculated spin density of [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 in the antiferromagnetic and ferromagnetic states. In the zero-field (antiferromagnetic) state, spin density resides primarily on the Mn 2+ centers and to a lesser extent on the formate ligand. The carbon center in each formate group is the node between the up-and down-spin states. In the B B Sat (ferromagnetic) state, the spin-density pattern on Mn 2+ and formate is similar although with an overall in-phase (rather than out-of-phase) arrangement. The node still resides on the carbon center in each formate group. This reflects participation of the nonbonding π level of HCOO - in the magnetic orbitals of the high-spin Mn 2+ , which has no contribution from the carbon atom.
We carried out pulsed field magnetization shots at numerous temperatures in order to track the behavior of the 0.31 T spin flop and 15.3 T saturation transitions. Plots of (∂M/∂B) T vs B reveal inflection points that define these phase boundaries. In the former case, the peak locations define the position of B SF whereas in the latter case, the region of maximum slope [36] defines B Sat . Figure 3 summarizes the trends, and the arrows denote these positions. The signature of the spin-flop transition is subtle in the absolute magnetization [Fig. 2(a), inset], but it becomes apparent in the derivative plots. The strength of the feature (B SF ) diminishes with increasing temperature until it is no longer evident above T N . The transition to the fully polarized state (B Sat ) is sharp at 0.37 K, and with increasing temperature, it broadens and shifts to lower fields. Trends below and above T N represent the contributions of long-range magnetic order vs short-range ordering, respectively. The latter manifests as a broad derivativelike structure rather than a sharp jump. Recent work on other magnetic materials reveals that short-range correlations can survive well above the ordering temperature-sometimes to a surprising extent [37][38][39][40]. In [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 , this suggests that the magnetic aspects of multiferroicity can persist to much higher temperatures than previously supposed and certainly much higher than T N = 8.5K [21].
Bringing these results together, we generated the magnetic field-temperature phase diagram of [(CH 3 ) 2 NH 2 ] Mn(HCOO) 3 (Fig. 4). As expected from the derivative analysis of the magnetization, there are two main phase boundary lines. The spin-flop transition is relatively stable at B SF = 0.31 T, disappearing completely at T N = 8.5 K. Increasing FIG. 4. B-T phase diagram [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 obtained from an analysis of pulsed field magnetization. PM = paramagnetic, cAFM = canted antiferromagnet. Closed circles correspond to boundaries involving long-range order. Open circles denote the region of short-range magnetic order. The 8.5-K Néel temperature is takenfromRef. [13]. Pulses up to 65 T reveal no additional features. field suppresses the quantum fluctuations of the canted antiferromagnetic state and drives into the fully saturated magnetic state. The limiting low-temperature value of this phase boundary is B Sat = 15.3 T. Whether this is a true magnetic quantum phase transition or not depends upon the behavior as T → 0K [ 41]. In any case, this magnetically driven transition is strongly temperature dependent: at T N /2 = 4.2K,B Sat drops by 11%, and at 7 K, B Sat falls by 32%. Even so, there is a strong remnant of the long-range-ordered state (in the form of short-range interactions) above T N .T h i si s evidenced by the broad, sweeping magnetization curve that also saturates (above 55 T at 10.00 K) but without the sharp knee that characterizes the critical field. These effects are diagramed by the open circles in Fig. 4. Similar short-range effects appear in other antiferromagnets including CrSiTe 3 , SrMnO 3 , TbInO 3 , and Cu(pyz) 2 HF 2 BF 4 [37][38][39][40].
The phase diagram of this molecule-based multiferroic is analogous to that of other quantum magnets like Mn[N(CN) 2 ] 2 and MnCl 2 (urea) 2 [25,26,42] in that it contains only spin-flop and saturation fields, although in this system, we have not only an adaptable framework but also a mechanism for multiferroic behavior. Thus, as magnetic field drives across the spin-flop and saturation transitions, changes in electronic properties may become apparent. The simplicity of the phase diagram and the overall accessibility of the high-field phase makes [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 amenable to in-depth exploration [43] compared to the rare earth manganites. The B-T phase diagrams of the latter tend to be much more complicated, with numerous competing phases, noncollinear spin structures, and eventual saturation at higher fields [23].
Finally, the unexpected trends in the high-temperature polarization of [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 can be linked to the structure of the B-T phase diagram and in particular the 15.3 T saturation field. The heart of this puzzle is that dielectric polarization is enhanced by 22% in a 13 T magnetic field between 150 and 190 K [16]. These temperatures are well above the magnetic ordering transition. At the same time, fits to the susceptibility in the vicinity of the ferroelectric transition [16], our magnetization data above T N which extends to 12 K (Fig. 4), and the sizable magnetoelectric coupling at elevated temperatures (up to 40 K) [21] indicate that short-range interactions are important, although the temperature range is more limited. Nuclear magnetic resonance measurements on the Zn analog, on the other hand, reveal that T C itself is insensitive to magnetic field and that there is no ordering of the amine-at least up to 5 T [44]. That spin resides on the metal-organic framework is consistent with our spin-density calculations in Fig. 2(b) as well as recent magnetoinfrared and neutron-scattering work [43,45]. Taken together, these findings suggest that [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 is sensitive to magnetic field above T N due to a remnant of the saturation field. In the absence of thermal fluctuations (well into the canted antiferromagnetic state), it is likely that dielectric polarization will be greatly enhanced under magnetic field. High-field polarization measurements across B Sat will be useful to test this supposition.
In summary, we investigated the magnetically driven phase transitions in [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 , a building block for a flexible new family of multiferroic materials. Using pulsed field magnetization, we unveil two features: a spin flop at 0.31 T and a transition to the fully polarized magnetic state at 15.3 T. The latter is in excellent agreement with mean-field theory, and first-principles spin-density calculations reveal the microscopic character of the phases. Tracking these critical fields with temperature, we generate a magnetic field-temperature phase diagram, which we discuss in terms of emerging properties in the high-field phase and its overall simplicity compared to the rare earth manganites. Importantly, we can explore the complete phase diagram of [(CH 3 ) 2 NH 2 ]Mn(HCOO) 3 at laboratory magnetic fields due to the overall low energy scales in this system. This work opens the door to exploration of the high-field properties, provides a likely mechanism for emerging properties, and motivates the study of polarization in high magnetic fields.