Spin-Phonon Coupling in A 2 BMnO 6 (A = La, Pr, Nd, Sm, Gd; B = Co, Ni) Double-Perovskite Thin Films: Impact of the A-Site Cation Radius

: Two series of B-site ordered, double-perovskite A 2 CoMnO 6 and A 2 NiMnO 6 (A = La, Pr, Nd, Sm, Gd) epitaxial ﬁlms with thickness d ~ 100 nm were grown on SrTiO 3 (111) substrates via metalorganic aerosol deposition. Polarization and temperature-dependent Raman spectroscopy were carried out in order to determine the spin-phonon coupling constant, λ , and the impact of the A-site cation radius on the phonon properties. The reduction of the A-site cation radius from La 3+ down to Gd 3+ systematically shifts the Raman modes to lower wavenumbers, and decreases the magnetization-induced softening of the Ag breathing mode, described by the spin-phonon coupling constant, λ , which changes from λ = 1.42 cm − 1 (La 2 CoMnO 6 ) and λ = 1.53 cm − 1 (La 2 NiMnO 6 ) down to λ = 0.58 cm − 1 (Gd 2 CoMnO 6 ) and λ = 0.44 cm − 1 (Gd 2 NiMnO 6 ). A similar effect of the A-cation radius was established for the c-lattice parameter and Curie temperature, TC, in this series of double-perovskite ﬁlms. Our observations directly demonstrate a strong impact of the lattice structure on the ferromagnetic superexchange interaction in double perovskites. Moreover, the A 2 CoMnO 6 and A 2 NiMnO 6 series exhibit very similar behavior of spin-phonon coupling due to the only moderate difference of Co 2+ and Ni 2+ cation size.

Here, we report the impact of the A-site cation size on the structural, magnetic, and phonon properties of A 2 CoMnO 6 and A 2 NiMnO 6 (A = La, Pr, Nd, Sm, Gd) epitaxial films grown on (111)-oriented SrTiO 3 (STO) substrates via a metalorganic aerosol deposition (MAD) technique [23]. The easy stoichiometry control within MAD is very useful for the growth of high-quality epitaxial films with different compositions. Polarization and temperature-dependent Raman spectroscopy carried out for temperatures T = 80-300 K allow us to relate the characteristic changes in the Raman spectra, induced by the A-cation size, to the changes in structure and magnetism. We observed the systematic reduction of the spin-phonon coupling by decreasing the A-site cation radius, and discussed it in the framework of variation of the internal chemical pressure induced by A-site cations.

Experimental Details
Double-perovskite A 2 CoMnO 6 and A 2 NiMnO 6 (A = La, Pr, Nd, Sm, Gd; B = Co, Ni)further referred to as Co-and Ni-series, respectively-epitaxial thin films with thickness d~100 nm were grown on SrTiO 3 (STO) substrates with (111) orientation by means of MAD [23], using a solution of acetylacetonates for the A-and B-site cations in dimethylformamide (DMF) under ambient oxygen partial pressure (pO 2~0 .2 bar), with growth rates of r~6 nm/min, substrate temperature of T sub~9 00-950 • C, and a heating/cooling rate of~50 K/min before and after deposition [10,11,24]. X-ray reflection (XRR) and X-ray diffraction (XRD, Cu-K α radiation, Bruker AXS D8 Advance) were used to determine the thickness, the quality, and the out-of-plane lattice parameters of the films. Temperaturedependent magnetization was measured using a SQUID magnetometer (MPMS 3, Quantum Design) at temperatures of T = 5-400 K and under an external magnetic field of H = 1000 Oe. Raman spectra were recorded using a LabRAM HR Evolution (HORIBA Jobin Yvon) spectrometer in the backscattered top-illumination geometry. A Linkam THMS350EV thermal vacuum stage, cooled by liquid nitrogen, was used for temperature-dependent measurements in the temperature range T = 80-400 K. The sample illumination and the collection of backscattered light was conducted using a 50× magnification long working distance objective (Olympus, NA = 0.5). The excitation was carried out using a linear polarized Nd:YAG laser (λ = 532 nm, Laser Quantum torus 532, second harmonic generation) with an effective incident power of P~2 mW at the film surface and a laser spot size of 1 µm 2 laterally. Polarized Raman spectra were recorded in a sample reference system with X 112 parallel to the incident laser polarization, Y 110 perpendicular to it, and Z [111] in the direction of light propagation. The scattered light polarization was filtered by an analyzer with two configurations (x and y) in front of the detector. Consequently, XX implies parallel scattering with the analyzer aligned parallel to the incident laser polarization, while XY is a cross-scattering polarization configuration with the analyzer aligned perpendicular to it. Transmission electron microscopy was performed using a FEI Titan 80-300 G2 environmental transmission electron microscope (TEM) operated at an acceleration voltage of 300 kV.

Results
To characterize the impact of the A-site cation size on the structure, we performed the XRD measurements on all of the grown films (see Supplementary Materials (SM), Figure S1 for details). XRD and TEM (SM, Figure S2) evidence an epitaxial growth for both series of samples, in good agreement with previously reported MAD-grown, double-perovskite films [10,11,18]. In Figure 1a we present the evaluated pseudocubic out-of-plane lattice parameters c pc = c(111)/ √ 3 of the Co-and Ni-series on STO(111) as a function of the A-site cation radius, taken from Ref. [25] for coordination number CN = 8. One can see that c pc systematically deceases with the decrease in the A-site cation radius, and the data for both film series can be quite well described by linear fits. Note that the A 2 CoMnO 6 series shows slightly larger lattice parameters than the A 2 NiMnO 6 series, which can be attributed to the slightly larger B-site cation radius of Co 2+ compared to the Ni 2+ cation-i.e., 74.5 pm for high-spin (HS)-Co 2+ vs. 69 pm for Ni 2+ ; CN = 6, Ref. [26]. Related to the corresponding Crystals 2021, 11, 747 3 of 12 polycrystalline bulk samples of both double-perovskite series [5][6][7], the pseudocubic outof-plane lattice parameters are slightly decreased due to an in-plane tensile epitaxy strain from the STO substrate, having a larger in-plane lattice parameter a STO = 0.3905 nm [25], and resulting in a lattice compression in the out-of-plane direction. As the lattice parameter shrinks further by reducing the A-site cation size, the out-of-plane compressive strain ε = (c pc − a sub )/a sub of the films systematically increases from ε = −0.77% for La 2 CoMnO 6 (LCMO) and ε = −0.87% for La 2 NiMnO 6 (LNMO) to ε = −3.76% for Gd 2 CoMnO 6 (GCMO) and ε = −3.99% for Gd 2 NiMnO 6 (GNMO). a systematic and almost linear reduction (solid lines for the linear approximation) of TC was observed here by decreasing the A-site cation size from Gd 3+ to Gd 3+ . A similar decrease of TC with decreasing rA was obtained in recent studies on polycrystalline A2CoMnO6 and A2NiMnO6 bulk samples [5][6][7], and was assigned to the successive reduction of the Co 2+ -O 2− -Mn 4+ and the Ni 2+ -O 2− -Mn 4+ bond angles; this reduces the overlap of Mn(eg) and O(2p) orbitals, and causes the apparent reduction of the ferromagnetic superexchange interaction between the two B-site cation species [5][6][7]. The higher Curie temperatures, TC, for the A2NiMnO6 series compared to the A2CoMnO6 films with the same A-site cation also indicate a stronger orbital overlap of the bonds between the (Ni/Mn)O6 octahedra compared to those of the (Co/Mn)O6 octahedra [6], although the differences in TC decrease by reducing the A-site cation size; this also displays the strong impact of the cations and the internal chemical pressure on the magnetic properties of the A2BMnO6 double-perovskite system. In addition to the optimally high values of TC in films of both Co and Ni series, the saturation magnetization measured at T = 5 K was found to be close to the theoretical values of 6 µB/f.u and 5 µB/f.u for the Co-and Mn-series, respectively. All of this indicates a high degree of B-site ordering in studied samples. In Figure 1b, the values of Curie temperatures, T C , for the Co-and Ni-series as a function of the A-site cation radius are presented; the corresponding magnetization M(T) curves are shown in Figures S3 and S4 in SM. T C values were calculated from the minimum of the temperature coefficient of magnetization, TCM = (1/M) (dM/dT). In addition, a systematic and almost linear reduction (solid lines for the linear approximation) of T C was observed here by decreasing the A-site cation size from Gd 3+ to Gd 3+ . A similar decrease of T C with decreasing r A was obtained in recent studies on polycrystalline A 2 CoMnO 6 and A 2 NiMnO 6 bulk samples [5][6][7], and was assigned to the successive reduction of the Co 2+ -O 2− -Mn 4+ and the Ni 2+ -O 2− -Mn 4+ bond angles; this reduces the overlap of Mn(e g ) and O(2p) orbitals, and causes the apparent reduction of the ferromagnetic superexchange interaction between the two B-site cation species [5][6][7]. The higher Curie temperatures, T C, for the A 2 NiMnO 6 series compared to the A 2 CoMnO 6 films with the same A-site cation also indicate a stronger orbital overlap of the bonds between the (Ni/Mn)O 6 octahedra compared to those of the (Co/Mn)O 6 octahedra [6], although the differences in T C decrease by reducing the A-site cation size; this also displays the strong impact of the cations and the internal chemical pressure on the magnetic properties of the A 2 BMnO 6 double-perovskite system. In addition to the optimally high values of T C in films of both Co and Ni series, the saturation magnetization measured at T = 5 K was found to be close to the theoretical values of 6 µ B /f.u and 5 µ B /f.u for the Co-and Mn-series, respectively. All of this indicates a high degree of B-site ordering in studied samples.
The Raman spectra recorded at T = 300 K of the Co-and Ni-series of double-perovskite thin films are shown in Figure 2a,b, respectively, in the parallel XX-scattering configuration, whereas Figure 2c,d show the corresponding Raman spectra in the crossed XY-scattering configuration. All films exhibit very similar Raman spectra, with two significant features: first, a strong and sharp mode at a Raman shift of~630-675 cm −1 , and second, a broader and less intensive band at around~470-530 cm −1 . Theoretical lattice dynamical calculations (LDCs) for LCMO [14] assigned the sharp mode to a symmetric stretching of the (Co/Mn)O 6 and (Ni/Mn)O 6 octahedra (breathing mode), whereas the broader band relates to a mixed type of antisymmetric stretching and bending of the octahedra (mixed mode). For B-site-ordered double perovskites of A 2 CoMnO 6 and A 2 NiMnO 6 with a monoclinic P12 1 /n1 structure, the breathing mode is predicted to have A g symmetry, which is allowed in the parallel XX-scattering and is forbidden in the crossed XY-scattering configurations; the mixed mode shows a B g symmetry with the opposite Raman selection rules [14,15,18,20]. From the polarized Raman spectra in Figure 2a-d, it is evident that all films follow this prediction for the monoclinic P12 1 /n1 structure. The additional observed modes around~1200-1350 cm −1 represent second-order overtones of the breathing mode and a breathing/mixed mode combination [18], confirming the high crystalline quality of the grown films.  In Figure 3a,b the temperature dependences of the Raman shift ω(T) of the Ag mo in the Raman spectra for the Co-and Ni-series, with different A-site cations-A = La, Nd, Sm, and Gd-are shown. In general, the temperature dependences of the positions the Raman modes result from the anharmonic contribution to the potential energy of atomic vibrations. By considering cubic anharmonic constants on the potential energy, scattering can be characterized by a three-phonon process, such that the Raman mo position as a function of the temperature can be described as [28,29]: Furthermore, a clear impact of the A-and B-site cations on the Raman spectra can be seen. Namely, the position of all Raman modes of the A 2 NiMnO 6 series is shifted bỹ 20-30 cm −1 to higher wavenumbers compared to those in the corresponding A 2 CoMnO 6 series, due to the larger degree of covalence of the A-O and Ni/Mn-O bonds in the A 2 NiMnO 6 system [6]. Moreover, the radius of the A-site cation has a significant impact on the position of the Raman modes as well. For both film series, a systematic softening of all Raman modes when decreasing the A-site cation radius was observed, indicating an increase in the Co(Ni)/Mn-O bond length [9,27]. For the A g breathing mode in the parallel XX-scattering spectra, the development for both series is illustrated in the insets of Figure 2a,b. A similarity of the phonon behavior with that of the c-lattice parameter (see Figure 1a) and T C (Figure 1b) as a function of the A-site cation radius evidences a strong interconnection of the structural, magnetic, and lattice properties.
In Figure 3a,b the temperature dependences of the Raman shift ω(T) of the A g mode in the Raman spectra for the Co-and Ni-series, with different A-site cations-A = La, Pr, Nd, Sm, and Gd-are shown. In general, the temperature dependences of the positions of the Raman modes result from the anharmonic contribution to the potential energy of the atomic vibrations. By considering cubic anharmonic constants on the potential energy, the scattering can be characterized by a three-phonon process, such that the Raman mode position as a function of the temperature can be described as [28,29]: where ω 0 is the position of a Raman mode at T = 0 K, and C is an anharmonic constant, both of which were taken as adjustable parameters to fit the temperature dependence of the experimentally determined position of the A g mode. One can see in Figure 3a,b that the anharmonic model (red curves) of the A g mode position ω(T) fits the experimental data (black data points) for temperatures T ≥ T C within the paramagnetic phase of the doubleperovskite thin films nicely. However, for all films of the Co-and Ni-series, a distinct deviation from the anharmonic model approximation occurs for 80 K < T < T C -namely, one can see a significant softening of the A g breathing mode, initiated by the transition into a ferromagnetic phase at T C and continuously increased by a subsequent cooling to lower temperatures. Comparing the position of the A g mode at the lowest temperature T = 80 K with that given by anharmonic approximation for different films, it is evident that the extent of the softening appears to be coupled to the radius of the A-site cations-i.e., the largest A-site cation Gd 3+ results in the strongest softening, which then decreases systematically by reducing the A-site radius to Gd 3+ . This is very similar to the above-shown effects of the A-site cation radius on the c-lattice parameter (Figure 1a), on the T C (Figure 1b), and on the softening of the breathing mode with decreasing r A , as shown in Figure 2a,b. Taken together, these data offer a clear indication of the connection between the mode softening, the lattice structure, and the magnetic properties. The softening of the A g breathing mode at T < T C originates from a phonon renormalization induced by ferromagnetic ordering, also observed for ferromagnetic manganites [29,30] and cobaltates [31], resulting in a coupling between magnetism and lattice known as a spin-phonon coupling. Considering the nearest-neighbor interaction, the phonon renormalization ∆ω(T) = ω(T) − ω anh. (T) is proportional to the spin-spin correlation function S i ·S j of the spins at the ith and jth sites, which is proportional in a molecular mean-field approximation to the normalized average magnetization M 2 (T)/M 2 max per magnetic ion at temperature T [14,[29][30][31]. As there are four nearest B'-site cation neighbors for each B-site cation in the A 2 BB'O 6 double perovskites with a rock-salt-type B-site ordering, the phonon renormalization ∆ω(T) can be written as [30][31][32]: Within this model, the spin-phonon coupling is quantified by the spin-phonon coupling constant, λ, whereas M max is the saturation magnetization at T = 0 K. In Figures 4a-e and 5a-e we present the softening ∆ω(T) = ω(T) − ω anh. (T) of the A g breathing mode plotted against squared normalized magnetization M 2 (T)/M 2 max for the Co-and Ni-film series, respectively. An excellent agreement between the ∆ω(T) and magnetization data for all films at temperatures of T < T C verifies that the A g mode softening arises from the spin-phonon coupling. It is evident that the softening, starting at T C , increases with increasing magnetization in the film by cooling; hence, a significant spin-phonon coupling is present for all of our films in their ferromagnetic phases.
Crystals 2021, 11, x FOR PEER REVIEW 6 of 1 shown in Figure 2a,b. Taken together, these data offer a clear indication of the connection between the mode softening, the lattice structure, and the magnetic properties. The softening of the Ag breathing mode at T < TC originates from a phonon renormal ization induced by ferromagnetic ordering, also observed for ferromagnetic manganite [29,30] and cobaltates [31], resulting in a coupling between magnetism and lattice known as a spin-phonon coupling. Considering the nearest-neighbor interaction, the phonon . Temperature dependence of the A g breathing mode position in the A 2 CoMnO 6 (a) and the A 2 NiMnO 6 (b) thin film series (thickness d~100 nm) with A = La, Pr, Nd, Sm, and Gd on STO(111) in the Raman spectra (unpolarized). The red solid lines denote the approximation from the anharmonic three-phonon model, which fits nicely for T > T C in the paramagnetic phase of the films.
The shift ∆ω(T) of the A g breathing mode, plotted against the magnetization M 2 (T)/M 2 max , can be described by a linear approximation-as shown for both film series in Figures 4f and 5f-and the spin-phonon coupling strength is determined by the corresponding slope S = 4λ. [10,16,19,31] One can see a systematic decrease in the slope and, hence, in the spin-phonon coupling constant when the A-site cation radius is reduced from the largest cation Gd 3+ to the smallest cation Gd 3+ within the A 2 CoMnO 6 and A 2 NiMnO 6 film series. The corresponding values for the spin-phonon coupling strength, λ, are presented for both series as a function of the A-site cation radius [26] in Figure 6. For the A 2 CoMnO 6 film series, the maximal value of λ = 1.42 cm −1 , obtained for LCMO, reduces systematically with the decreasing size of the A-cation, down to λ = 0.58 cm −1 (GCMO). For the A 2 NiMnO 6 films, the maximal value of λ = 1.53 cm −1 (LNMO) reduces to λ = 0.44 cm −1 (GNMO). Note that the λ values for GCMO and GNMO are likely underestimated, since these films show an enhanced saturation magnetization (see Table  S1 in the SM) due to the strong contribution of the magnetic moment of Gd. Hence, a systematic and similar decrease in the spin-phonon coupling strength, λ, by reducing the A-site cation radius is observed for both series. A similar behavior was observed recently for A 2 CoMnO 6 /LaAlO 3 (001) films (A = Pr, Nd, Sm) [19] and polycrystalline bulk samples (A = La, Pr, Nd) [16]. Remarkably, very different λ-values were determined in thin films and bulk samples: a PCMO film and a bulk sample show λ = 1.61 cm −1 and λ = 0.51 cm −1 , respectively. As for the A 2 NiMnO 6 series, no influence of the A-site cation radius on the spin-phonon coupling was detected in bulk samples for A = La, Pr, Nd, Gd, Y [17], in contradiction to our results for A 2 NiMnO 6 /STO(111) films (see Figure 6).
Crystals 2021, 11, x FOR PEER REVIEW 7 of 13 renormalization ∆ ( ) = ( ) − . ( ) is proportional to the spin-spin correlation function 〈 • 〉 of the spins at the ith and jth sites, which is proportional in a molecular mean-field approximation to the normalized average magnetization M 2 (T)/M 2 max per magnetic ion at temperature T [14,[29][30][31]. As there are four nearest B'-site cation neighbors for each B-site cation in the A2BB'O6 double perovskites with a rock-salt-type B-site ordering, the phonon renormalization ∆ ( ) can be written as [30][31][32]: Within this model, the spin-phonon coupling is quantified by the spin-phonon coupling constant, λ, whereas Mmax is the saturation magnetization at T = 0 K. In Figure 4a-e and Figure 5a-e we present the softening ∆ ( ) = ( ) − . ( ) of the Ag breathing mode plotted against squared normalized magnetization M 2 (T)/M 2 max for the Co-and Nifilm series, respectively. An excellent agreement between the ∆ ( ) and magnetization data for all films at temperatures of T < TC verifies that the Ag mode softening arises from the spin-phonon coupling. It is evident that the softening, starting at TC, increases with increasing magnetization in the film by cooling; hence, a significant spin-phonon coupling is present for all of our films in their ferromagnetic phases.   The shift ∆ ( ) of the Ag breathing mode, plotted against the magnetization M 2 (T)/M 2 max, can be described by a linear approximation-as shown for both film series in Figures 4f and 5f-and the spin-phonon coupling strength is determined by the corresponding slope S = 4λ. [10,16,19,31] One can see a systematic decrease in the slope and, hence, in the spin-phonon coupling constant when the A-site cation radius is reduced from the largest cation Gd 3+ to the smallest cation Gd 3+ within the A2CoMnO6 and A2NiMnO6 film series. The corresponding values for the spin-phonon coupling strength, λ, are presented for both series as a function of the A-site cation radius [26] in Figure 6. For the A2CoMnO6 film series, the maximal value of λ = 1.42 cm −1 , obtained for LCMO, reduces systematically with the decreasing size of the A-cation, down to λ = 0.58 cm −1  Table S1 in the SM) due to the strong contribution of the magnetic moment of Gd. Hence, a systematic and similar decrease in the spin-phonon coupling strength, λ, by reducing the A-site cation radius is observed for both series. A similar behavior was observed recently for A2CoMnO6/LaAlO3(001) films (A = Pr, Nd, Sm) [19] and polycrystalline bulk samples (A = La, Pr, Nd) [16]. Remarkably, very different λ-values were determined in thin films and bulk samples: a PCMO film and a bulk sample show λ = 1.61 cm −1 and λ = 0.51 cm −1 , respectively. As for the A2NiMnO6 series, no influence of the A-site cation radius on the spin-phonon coupling was detected in bulk samples for A = La, Pr, Nd, Gd, Y [17], in contradiction to our results for A2NiMnO6/STO(111) films (see Figure 6).

Discussion
To elucidate the origin of the A-site cation influence on the spin-phonon coupling in double-perovskite films, the intrinsic and extrinsic mechanisms should be considered. In general, the systematic decrease in the spin-phonon coupling strength, λ, for smaller Asite cations reflects the impact of chemical pressure on the structure-property (magnetic Figure 6. The spin-phonon coupling strength, λ, of the B-site-ordered A 2 CoMnO 6 and A 2 NiMNO 6 thin film series on STO(111) as a function of the A-site cation radius. The values were extracted from the phonon renormalization of the A g breathing mode in the ferromagnetic phase below T C .

Discussion
To elucidate the origin of the A-site cation influence on the spin-phonon coupling in double-perovskite films, the intrinsic and extrinsic mechanisms should be considered. In general, the systematic decrease in the spin-phonon coupling strength, λ, for smaller A-site cations reflects the impact of chemical pressure on the structure-property (magnetic and phononic) relationship. Initially, the orbital overlap and the apparent ferromagnetic superexchange depend on the Co 2+ -O 2− -Mn 4+ or Ni 2+ -O 2− -Mn 4+ bond angles between the B-site cations [5][6][7]. This is shown in Figure 1b for the A 2 CoMnO 6 and the A 2 NiMnO 6 series, for which a reduction in the A-site cation size from Gd 3+ to Gd 3+ leads to a systematic decrease in T C due to the chemical-pressure-induced decrease in the bond angle. Simultaneously, the reduction of the A-site cation radius modifies the internal bond lengths, yielding a decrease in the pseudocubic c-lattice parameter [5][6][7]33] (Figure 1a), an increase in the lattice mismatch and, finally, an enhancement of the epitaxy stress in films grown on the STO(111) substrates. As a result, the force constants of the perovskite lattice [30,32] also increase, and the lattice gets stiffer, as evidenced by the significant shift of the A g breathing mode in the Raman spectra towards larger wavenumbers (Figure 2a,b). As a consequence, the decrease in the A-site cation radius results in a systematic reduction of both the magnetic superexchange interaction and the spin-phonon coupling, as the latter connects magnetic and phonon properties. A similar behavior was seen also for the rare earth RMnO 3 manganites [30,32].
Comparing the A 2 CoMnO 6 and the A 2 NiMnO 6 double-perovskite film series, the spin-phonon coupling strength, λ, for the same A-site cation exhibits only small differences, indicating a weak impact of the Co 2+ and Ni 2+ cations on the spin-phonon coupling. In general, the A 2 CoMnO 6 film series shows slightly larger λ values (the only exception being A = La) than the A 2 NiMnO 6 film series, although T C values are significantly higher for the A 2 NiMnO 6 series ( Figure 1b). As the cation size of Co 2+ and Ni 2+ is comparable (74.5 pm for HS-Co 2+ vs. 69 pm for Ni 2+ ; CN = 6, Ref. [26]), and the lattice parameters are also similar (Figure 1a), the different spin contributions of HS-Co 2+ /Ni 2+ , and their impact on the bonds, force constants, and phonon properties (Figure 2a-d), must be involved. Obviously, the overall interplay results in a comparable spin-phonon coupling for both series. Recent studies on the A 2 CoMnO 6 series showed similar values for the coupling strength, with λ = 1.61 cm −1 for PCMO, λ = 1.20 cm −1 for NCMO, and λ = 1.16 cm −1 for SCMO for films on LAO(100) substrates [19], and λ~1.4-2.1 cm −1 , depending on the substrate, for LCMO thin films [10], whereas for polycrystalline bulk PCMO with λ = 0.51 cm −1 , a significantly lower coupling was obtained [16]. For the A 2 NiMnO 6 series, no comparative values are available. Recent studies on polycrystalline A 2 NiMnO 6 bulk samples showed no A-site cation influence, but from the A g breathing mode softening, the coupling strength can be estimated roughly to λ~0.6-0.7 cm −1 for the whole series [17], which is again significantly lower than for our A 2 NiMnO 6 thin films.
An explanation for this discrepancy can be given by the sample characteristics; whereas the epitaxial thin films exhibit a high crystalline quality and are in-plane strained due to the lattice misfit [10,12,19,24], the strain effects are absent for the polycrystalline bulk samples, but problems concerning the oxygen stoichiometry can occur and play a crucial role [8,9,16]. Furthermore, the degree of the B-site ordering of the double-perovskites must be included for a complete discussion [10,12,20,21]. Beginning with the strain, it is argued that for the case of A 2 CoMnO 6 films on LAO(100), the compressive biaxial strain has a significant influence on the magnetic interaction and the spin-phonon coupling, due to a reduction in oxygen vacancies and a higher degree of B-site ordering compared to bulk samples [16,19]. Our recent study on the B-site-ordered LCMO thin films with a (111) out-of-plane-orientation on various substrates also reveals a strong strain impact [10]. In this context, the largest spin-phonon coupling strength of λ~2.1 cm −1 was observed for LCMO(111) films on Al 2 O 3 (0001) and on (LaAlO 3 ) 0.3 (Sr 2 TaAlO 6 ) 0.7 (111)(LSAT) substrates, actuating a small, compressive, in-plane strain. This coupling constant is significantly larger than λ~1.4-1.5 cm −1 (see Ref. [10] and the present study), obtained on LCMO(111) films grown on substrates with a larger lattice mismatch, such as LAO(111) and STO(111).
On the other hand, the spin-phonon coupling of the double-perovskite A 2 BMnO 6 is affected by the degree of the B-site ordering, as it directly depends on structural and magnetic properties [8][9][10][11][12]. For B-site-ordered LCMO and LNMO films, this is reflected by a significant increase in the softening of the A g breathing mode in the ferromagnetic phase, and by the corresponding spin-phonon coupling for the B-site-ordered films compared to partially or fully disordered systems [10,20,21]. For LCMO thin films on c-orientated Al 2 O 3 (0001), this results in a spin-phonon coupling strength of λ~2.1 cm −1 in the case of B-site ordering, and a decrease to λ~1.7 cm −1 for a partially B-site-disordered film [10]. Additionally, the B-site ordering in the double-perovskite films can be stimulated and improved by a compressive in-plane strain, due to a different distortion of the B/B'O 6 octahedra [12]. In combination with the higher crystalline quality of the epitaxial thin films, a higher degree of B-site ordering-and, hence, a stronger mode softening and spin-phonon coupling compared to polycrystalline, double-perovskite bulk samples-is achieved. For partially B-site-disordered or polycrystalline bulk double perovskites, this also results in a decreased impact of the A-site cation on the coupling strength, as the crystal quality, ordering, and strain have a significantly stronger influence [10,16,17,[19][20][21]. Conversely, the spin-phonon coupling strength as an indicator of the crystal quality depicts a direct way to characterize and improve the suitability of a double-perovskite material for technological applications.

Conclusions
Epitaxial films of the B-site-ordered A 2 CoMnO 6 and A 2 NiMnO 6 (A = La, Pr, Nd, Sm, Gd) double-perovskite series with a monoclinic P12 1 /n1 structure were grown on SrTiO 3 (111) substrates using the MAD technique, in order to determine the impact of the A-site cation size on the structural, magnetic, and phonon properties. By carrying out polarization and temperature-dependent Raman measurements, the anomalous softening of the breathing mode in the ferromagnetic phase, and its dependence on the size of A-site cations, were revealed in both film series. The results were discussed within the influence of the A-site-induced variation of internal chemical pressure on the spin-lattice interaction, quantified by the corresponding spin-phonon coupling strength, λ. We observed a systematic reduction in λ by decreasing the size of the A-cation, which is very similar to the changes in the c-lattice parameters and Curie temperatures actuated by the A-cation size. Thus, the spin-phonon coupling constant-correlating structural, magnetic, and phonon properties together-displays the strong impact of the A-site cation size in the A 2 CoMnO 6 and A 2 NiMnO 6 double perovskites, and depicts a promising way to improve the material suitability of various double-perovskite systems for technological applications.