3. 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
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
2CoMnO
6 series shows slightly larger lattice parameters than the A
2NiMnO
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 polycrystalline bulk samples of both double-perovskite series [
5,
6,
7], the pseudocubic out-of-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
2CoMnO
6 (LCMO) and ε = −0.87% for La
2NiMnO
6 (LNMO) to ε = −3.76% for Gd
2CoMnO
6 (GCMO) and ε = −3.99% for Gd
2NiMnO
6 (GNMO).
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
2CoMnO
6 and A
2NiMnO
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
2NiMnO
6 series compared to the A
2CoMnO
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
2BMnO
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
2CoMnO
6 and A
2NiMnO
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.
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
2NiMnO
6 series is shifted by ~20–30 cm
−1 to higher wavenumbers compared to those in the corresponding A
2CoMnO
6 series, due to the larger degree of covalence of the A–O and Ni/Mn–O bonds in the A
2NiMnO
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 double-perovskite 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
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
2max 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
2BB’O
6 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 M
max is the saturation magnetization at T = 0 K. In
Figure 4a–e and
Figure 5a–e we present the softening
of the A
g breathing mode plotted against squared normalized magnetization
M2(T)/M2max for the Co- and Ni-film series, respectively. An excellent agreement between the
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.
The shift
of the A
g breathing mode, plotted against the magnetization
M2(T)/M2max, can be described by a linear approximation—as shown for both film series in
Figure 4f and
Figure 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
2CoMnO
6 and A
2NiMnO
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
2CoMnO
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
2NiMnO
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
2CoMnO
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
2NiMnO
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
2NiMnO
6/STO(111) films (see
Figure 6).
4. 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
2CoMnO
6 and the A
2NiMnO
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
2CoMnO
6 and the A
2NiMnO
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
2CoMnO
6 film series shows slightly larger λ values (the only exception being A = La) than the A
2NiMnO
6 film series, although T
C values are significantly higher for the A
2NiMnO
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
2CoMnO
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
2NiMnO
6 series, no comparative values are available. Recent studies on polycrystalline A
2NiMnO
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
2NiMnO
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
2CoMnO
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
2O
3(0001) and on (LaAlO
3)
0.3(Sr
2TaAlO
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
2BMnO
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
2O
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.