Weak ferromagnetism and short range polar order in NaMnF₃ thin films

There is much interest in multiferroic (MF) materials that show the coexistence of two or more long-range orders such as ferroelectricity, (anti-) ferromagnetism, and/or ferroelasticity. Magnetoelectric (ME) coupling can allow for the control of magnetization M (electric polarization P) using an electric field E (a magnetic field H), which can be used for potential applications in data storage, sensors, and spintronic devices.^1–5 As a result, there has been much effort expended towards synthesizing MF insulating materials with strong ME coupling at room temperature.^6–9 

Many MF materials are oxides, but multiferroicity can also be found in fluorides. For example, in the orthorhombic BaMF₄ family, the M=Co and Ni compounds in bulk form are ferroelectric (FE) and antiferromagnetic.¹⁰ While in oxide perovskites ABO₃ the simultaneous presence of FE and magnetic orderings is limited by conflicting requirements for the dⁿ electronic configuration of the transition metal in the B-site,¹¹ ferroelectricity in BaMnF₄ is mainly due to geometric (topological) reasons.¹² Recent experiments showed that BaCoF₄ thin films are weakly ferromagnetic at low temperatures due to strain.¹³ 

NaMnF₃ (NMF) is another fluoride compound that is possibly MF with ME coupling. The NMF crystal structure is described by the orthorhombic space group Pnma. At room temperature, the lattice constants are a_o = 5.751 Å, b_o = 8.008 Å, and c_o = 5.548 Å (Fig. 1(a)).¹⁴ In the pseudo-cubic unit cell, the corresponding lattice parameters are $apc=cpc=ao2+co2/2=3.995 Å$ and b_pc = b_o/2 = 4.004 Å. NMF exhibits G-type antiferromagnetism of the Mn magnetic moments centered within tilted MnF₆ octahedra, with additional weak ferromagnetic canting.^14–16 The Néel temperature has been reported to be T_N = 66 K.^15–20 Recent computational work predicted a ferroelectric instability in NMF originating from Na displacements (the A-site perovskite cation in the ABF₃ structure). Calculations assuming that a cubic substrate forces the a and c lattice parameters to be equal to each other have shown that the soft polar mode B_2u freezes, resulting in a transformation to the polar Pna2₁ space group which causes a polarization of P = 6 μC/cm² along the long axis b. Calculations also indicate that P can be enhanced by negative or positive strains (12 μC cm² at +5% strain 9 μC/cm² at −5% strain).^21,22 A weak ferromagnetic phase and an amplification of ME coupling are also expected in the strained films, with a ME response comparable to Cr₂O₃.²² Therefore, NMF is an attractive ME material because its ferroelectricity can be modulated by strain and is not subject to the dⁿ rule because there is no displacement of the Mn ion from the center of the MnF₆ octahedra.

Here, we report on the growth of NMF thin films on cubic SrTiO₃ (STO) substrates and on SrRuO₃ (SRO) layers, their weak ferromagnetism (FM) below the T_N, and their short range dipolar order. NMF thin films approximately 50 nm thick were grown on pre-polished single crystal (001) STO substrates (a = 3.905 Å) by molecular beam epitaxy (MBE) by co-depositing NaF and MnF₂ in an ultra-high vacuum chamber with growth pressure P ≈ 5.0 × 10⁻⁹ Torr. Atomically flat surface and single termination of STO substrates was achieved by the combination of two thermal annealing steps and a de-ionized (DI) water treatment.²³ For additional details of the growth process, see supplementary material. The crystallography of the substrate and film surfaces was monitored in-situ using reflection high energy electron diffraction (RHEED). Further crystallographic characterization was carried out ex-situ using X-ray diffraction (XRD) and X-ray reflectivity (XRR) techniques using a rotating anode source with a graphite bent crystal monochromator optimized for Cu K_α radiation. The surface morphology was studied by the atomic force microscopy (AFM). Magnetic measurements were performed using a superconducting quantum interference device (SQUID) magnetometer with the substrate signal subtracted by performing identical measurements on a blank substrate.

Dielectric measurements of NMF films were performed by first depositing a 30 nm SRO film on the STO substrate as a bottom electrical contact via pulsed laser deposition (PLD) in a separate vacuum chamber. The SRO was grown at a substrate temperature of 600 °C in 100 mTorr O₂ partial pressure. A 150 nm NMF thin film was then deposited on the SRO using the conditions described above. The dielectric constant was measured using a lock-in amplifier in the frequency range of 100 Hz to 100 kHz from 10 K to room temperature using a cryostat and an ac voltage amplitude of V_ac = 100 mV. Similar results were obtained for V_ac = 10 mV.

The best film quality was achieved at T_s = 250 and 300 °C as determined from XRD and RHEED. Figure 1(a) shows the θ – 2θ XRD scan for a film grown at T_s = 300 °C. The out-of-plane lattice parameters were 3.989 Å and 3.982 Å, and the rocking curve full width at half maximum (FWHM) values were 1.32° and 2.23° for NMF/STO and NMF/SRO films, respectively (see supplementary material). The positions of the SRO peaks were consistent with data in the literature.²⁴ XRR data (supplementary material) were fitted using GenX software for the NMF/STO film²⁵ to obtain a NMF thickness of 51.76 nm and a NMF surface roughness of 0.4 nm. Streaky RHEED patterns (Fig. 1(a) inset i) confirmed that the NMF/STO film was smooth and crystalline while the NMF/SRO sample also had streaky patterns with a somewhat fuzzy streaky background (Fig. 1(a) inset ii).

Using the Pnma NMF structure, reflections associated with b_o -axis pointing out-of-plane [NMF(010)/STO(001)] as well as in-plane [NMF(101)/STO(001)] domains were identified via XRD $ϕ$-scans and reciprocal space maps (RSMs) (the corresponding (010) and (101) lattice parameters are virtually identical; see supplementary material Fig. S4 for RSM data). These results indicate the absence of a_o = c_o distortion in the NMF/STO film for both of these domains. The measured lattice parameters for NMF(010)/STO(001) and NMF(101)/STO(001) domains were a_o = 5.76 Å, b_o = 7.98 Å, and c_o = 5.57 Å and a_o = 5.86 Å, b_o = 8.04 Å, and c_o = 5.45 Å, respectively (see Table S1 in the supplementary material). Figure 1(b) shows the $ϕ$-scans of the (222)_o reflection of the NMF(101) and NMF(010) domains along with (012) reflections of the substrate. In both NMF domains, four peaks separated by 90° were observed, meaning that two NMF phases with orthogonal b_o axes were present. For NMF(010) domains, the orthorhombic a_o and c_o axes $∥$ STO [110]. For NMF(101) domains, the b_o axis commensurately aligns $∥$ STO [010] and $[1¯00]$ (labeled D₁ and D₂ in Fig. 1(d)). For the NMF(010) domains, each (222)_o reflection was additionally comprised of two peaks separated by 1.4° (Fig. 1(b), inset). Therefore, each NMF(010) domain was actually comprised of two in-plane twin domains rotated by 1.4° about the surface normal with respect to each other (labeled D₃ and D₄ in Figs. 1(b) inset and 1(d) top view). Similar epitaxies of orthorhombically distorted perovskite oxide films on STO have been reported elsewhere.^24,26–28

For the NMF/SRO sample, epitaxy could not be confirmed by $ϕ$-scans or RSM, despite the streaky RHEED pattern in Fig. 1(a). This indicates that the bulk of NMF was highly disordered, with small domains. Because of the similarity between STO and SRO, however, it is reasonable to assume that the strain in the NMF/SRO sample was not too different from the NMF/STO sample.

The temperature-dependent magnetization measurements on NMF/STO were carried out by first cooling the samples from room temperature to 5 K in H = 0, and then measuring the magnetic moment m as a function of T in H = 1 kOe applied in-plane along the STO [001] direction while warming up to 320 K (zero-field cooled, ZFC). Then the measurements were continued while cooling in the same H down to 5 K (field cooled, FC). Subsequently, the thermoremanent magnetization (TRM) was measured while warming in H = 0. ZFC and FC data in Fig. 2(a) show a Curie-Weiss-like increase with decreasing temperature until the ZFC data peak at T_N ≈ 66 K (see supplementary material). On the other hand, the FC m decreases sharply below T_N ≈ 66 K, in good agreement with the theory²¹ and previous experimental work.^15–19 This split between ZFC and FC magnetizations below 66 K is consistent with the weak FM below T_N, in agreement with previous studies of NMF nanoparticles and nanoplates.^19,20 The TRM data have a Brillouin-like increase below T_N, confirming the weak ferromagnetic nature of the transition. Fig. 2(b) shows the in-plane magnetization hysteresis loop M(H) measured at 5 K with the diamagnetic response from the substrate subtracted. The M(H) behavior is consistent with the weak FM, including an S-shaped loop with a lack of saturation in magnetic fields of up to 7 T and a remanent magnetization M(H = 0) ≈ 0.003 μ_B/f.u. in the low field region (Fig. 2(b), inset).

Figures 3(c) and 3(d) show the dielectric permittivity ε as a function of T for the 150 nm thick NMF/SRO film at frequencies from 1 to 78 kHz. The response at 0.1 kHz almost overlaps with that at 1 kHz so it is not shown. At large frequencies, the increase of the real part of the dielectric permittivity $ε′(T)$ at low T is superimposed with a broad maximum whose position shifts to lower T with decreasing frequency f. The imaginary part of dielectric permittivity, $ε″(T)$⁠, also shows a peak that shifts to lower T and becomes narrower with decreasing f. This behavior implies a strong relaxation of the dielectric permittivity and resembles properties of relaxor ferroelectrics,²⁹ but the frequency dependence of the position of the maximum of $ε′(T)$⁠, T_m, is much stronger than is typically observed in relaxors.³⁰ In our case, T_m shifts more than 200 K in only two frequency decades.

To gain more insight into relaxation dynamics, we analyzed the f dependence of ε. At low f, $ε″(f)$ follows a 1/f dependence which is due to a non-zero dc conductivity (Fig. 3(b)). Also, a typical relaxation maximum is observed for f > 3 kHz, whose position shifts to lower f as T decreases. The $ε′(f)$ behavior shows a step-like increase at frequencies where $ε″(f)$ is the maximum. As T decreases below 52 K, another relaxation peak seems to appear at a higher frequency. However, this peak was not within our experimental window down to the lowest T, and only the low-f wing is visible.

Assuming a Debye-like dielectric relaxation, the broad peaks (steps) in the $ε″(f) (ε′(f))$ data indicate the existence of a distribution of relaxation times. Using the Cole-Cole dispersion law³¹ 

where τ_eff is an effective characteristic relaxation time, ε_∞ is the dielectric permittivity at very high frequencies, Δε is the relaxation strength, and β (0 < β ≤ 1) indicates the width of the relaxation time distribution (β = 1 corresponds to a Debye relaxation with a single relaxation time). The values of τ_eff (T), extracted from the position of the maximum of $ε″(f), τeff=(2πfm)−1$⁠, are shown in Figs. 3(e) and 3(f). τ_eff (T) cannot be fitted well either by an Arrhenius law

or by a Vogel-Fulcher law applicable to the typical relaxor FE

where k is the Boltzmann's constant, and E_a is an activation energy, as seen in Fig. 3(e). In the case of Eq. (2), the quality of the fit is low, while the fit to Eq. (3) results in an unphysical negative value of the freezing temperature, T_f. On the other hand, τ_eff(T) is described well by the power law equation

The best fit (Fig. 3(f)) yields ν = −0.90 ± 0.07, τ₀ = (2.8 ± 1.5) × 10⁻⁵ s, and a transition temperature of T₀ = 8 K. The value of ν is close to ν = −1 that describes the divergence of the relaxation time for an order-disorder phase transition in the classical mean-field approximation.³² The actual transition temperature could be much lower than 8 K because only the behavior for T > T₀ was observed. Interestingly, in canonical ferroelectrics, the relaxation dynamics corresponding to such transition are typically observed in the GHz frequency range, indicating that relaxing dipolar entities are relatively large and slowly fluctuating microscopic regions spanning several unit cells.

Large dielectric dipolar regions could form because of the diffusion of a FE soft mode^21,22 as a result of the disorder in the NMF/SRO film. Although the NMF/STO film has only a small strain, local measurements like piezoelectric force microscopy (PFM) need to be performed to verify ferroelectricity because of the lack of an electrically conductive bottom layer. In addition, further research is required to obtain a tetragonal a_o = c_o lattice distortion in NaMnF₃ thin films to verify the large ferroelectric polarization predicted by theory.^21,22

See supplementary material for details of the growth procedure and structural characteristics of the films, as well as the magnetic susceptibility data.

This work was supported by the National Science Foundation (Grant No. 1434897) and the WVU Shared Research Facilities. We thank A. H. Romero and C. Chen for helpful discussions.