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Predicting New Iron Garnet Thin Films with Perpendicular Magnetic Anisotropy

Perpendicular magnetic anisotropy (PMA) is a necessary condition for many spintronic applications like spin-orbit torques switching, logic and memory devices. An important class of magnetic insulators with low Gilbert damping at room temperature are iron garnets, which only have a few PMA types such as terbium and samarium iron garnet. More and stable PMA garnet options are necessary for researchers to be able to investigate new spintronic phenomena. In this study, we predict 20 new substrate/magnetic iron garnet film pairs with stable PMA at room temperature. The effective anisotropy energies of 10 different garnet films that are lattice-matched to 5 different commercially available garnet substrates have been calculated using shape, magnetoelastic and magnetocrystalline anisotropy terms. Strain type, tensile or compressive depending on substrate choice, as well as the sign and the magnitude of the magnetostriction constants of garnets determine if a garnet film may possess PMA. We show the conditions in which Samarium, Gadolinium, Terbium, Holmium, Dysprosium and Thulium garnets may possess PMA on the investigated garnet substrate types. Guidelines for obtaining garnet films with low damping are presented. New PMA garnet films with tunable saturation moment and field may improve spinorbit torque memory and compensated magnonic thin film devices.


Introduction
With the development of sputtering and pulsed laser deposition of high-quality iron garnet thin films with ultralow Gilbert damping, researchers have been able to investigate a wide variety of magnetization switching and spin wave phenomena [1][2][3] . The key enabler in many of these studies has been Yttrium iron garnet (Y3Fe5O12, YIG) 4 which has a very low Gilbert damping allowing spin waves to propagate over multiple millimeters across chip. YIG thin films are useful for spin wave device applications, but since their easy axes lie along film plane, their utility cannot be extended to different mechanisms such as spin-orbit torques, Rashba-Edelstein effect, logic devices, forward volume magnetostatic spin waves 1 . At the same time, to have reliable and fast response using low current densities as in spin-orbit torque switching, magnetization orientation needs to be perpendicular to the surface plane 5 . The possibility of having Dzyaloshinskii-Moriya interaction (DMI) in TmIG/GGG may enable stabilizing skyrmions and help drive skyrmion motion with pure spin currents 6 .
There is a number of studies on tuning anisotropy or obtaining perpendicular magnetic anisotropy in insulator thin films [7][8][9][10][11][12] . Among the materials studied, insulating magnetic garnets whose magnetic properties can be tuned have been a matter of interest over the past decades [13][14][15] due to their low damping and high magnetooptical Faraday rotation. In order to obtain perpendicular magnetic anisotropy in magnetic garnets, one needs to engineer the anisotropy terms that give rise to out-of-plane easy axis. Angular dependence of total magnetization energy density is called magnetic anisotropy energy and consists of contributions from shape anisotropy, strain-induced (magnetoelastic) and magnetocrystalline anisotropy. A magnetic material preferentially relaxes its magnetization vector towards its easy axis, which is the least energy axis, when there is no external field bias. Such energy minimization process drives magnetic switching rates as well as the stability of total magnetization vector. Controlling magnetic anisotropy in thin film garnets not only offers researchers different testbeds for experimenting new PMA-based switching phenomena, but also allows the investigation of anisotropy-driven ultrafast dynamic magnetic response in thin films and nanostructures.
The most extensively studied garnet thin film is Yttrium Iron Garnet (YIG). YIG films display inplane easy axis because of their large shape anisotropy and negligible magnetocrystalline anisotropy 3 . Although PMA of ultrathin epitaxial YIG films has been reported 16,17 , the tolerance for fabrication condition variations for PMA YIG is very limited and strain effects were found to change magnetocrystalline anisotropy in YIG. Strain-controlled anisotropy has been observed in polycrystalline ultrathin YIG films 17,18 . In case of YIG thin film grown on Gadolinium Gallium Garnet (GGG), only partial anisotropy control has been possible through significant change in oxygen stoichiometry 19 , which increases damping. Since the fabrication of high-quality and highly PMA YIG films is not easy for practical thicknesses on gadolinium gallium garnet substrates (GGG), researchers have explored tuning magnetic anisotropy by substituting Yttrium sites with other rare earth elements 20,21 . New garnet thin films that can exhibit PMA with different coercivities, saturation fields, compensation points and tunable Gilbert damping values must be developed in order to evaluate the effect of these parameters on optimized spintronic insulator devices.
Since the dominant anisotropy energy term is shape anisotropy in thin film YIG, some studies focus on reducing the shape anisotropy contribution by micro and nanopatterning [22][23][24] . Continuous YIG films were etched to form rectangular nanostrips with nanometer-scale thicknesses, as schematically shown on Fig. 1(a). Thus, least magnetic saturation field is needed along the longest dimension of YIG nanostrips. By growing ultrathin YIG, magnetoelastic strain contributions lead to a negative anisotropy field and thus PMA in YIG films 24 . As the length-to-thickness ratio decreases, the effect of shape anisotropy is reduced and in-plane easy axis is converted to PMA 17 .
While reducing the effect of shape anisotropy is necessary, one also needs to use magnetoelastic anisotropy contribution to reorient magnetic easy axis towards out of film plane, as schematically shown on Fig. 1(b). Strain-induced perpendicular magnetic anisotropy in rare earth (RE) iron garnets, especially in YIG, has been demonstrated to overcome shape anisotropy 16,17,25,26 . If magnetoelastic anisotropy term induced by crystal lattice mismatch is larger than shape anisotropy and has opposite sign, then magnetoelastic anisotropy overcomes shape. Thus, the easy axis of the film becomes perpendicular to the film plane and the hysteresis loop becomes square-shaped with low saturation field 8 . One can also achieve PMA in other RE magnetic iron garnets due to their lattice parameter mismatch with their substrates. PMA has previously been achieved using Substituted Gadolinium Gallium Garnet (SGGG) as substrate and a Samarium Gallium Garnet (SmGG) ultrathin film as buffer layer under (and on) YIG 16 . In case of thicker YIG films (40nm), the magnetic easy axis becomes in-plane again. An important case shown by Kubota et.al 19 indicates that increasing in-plane strain (ε||) or anisotropy field (Ha) helps achieve perpendicular magnetic anisotropy. In ref. 8,19 , they reported that if magnetostriction coefficient (λ111) is negative and large enough to overcome shape anisotropy, and tensile strain is introduced to the thin film sample (ε||>0), the easy axis becomes perpendicular to the sample plane as in the case of Thulium iron garnet (Tm3Fe5O12, TmIG).
A different form of magnetoelastic anisotropy effect can be induced by using porosity in garnet thin films. Mesoporous Holmium Iron Garnet (Ho3Fe5O12, HoIG) thin film on Si (001) 27 exhibits PMA due to reduced shape anisotropy, increased magnetostrictive and growth-induced anisotropy effects. Such combined effects lead to PMA in HoIG. In this porous thin film, the PMA was found to be independent of the substrate used, because the mechanical stress does not result from a lattice or thermal expansion mismatch between the substrate and the film. Instead, the pore-solid architecture itself imposes an intrinsic strain on the solution processed garnet film. This example indicates that the film structure can be engineered in addition to the substrate choices in order to overcome shape anisotropy in thin film iron garnets.
Another key method for controlling anisotropy is strain doping through substitutional elements and using their growth-induced anisotropy effects, as schematically shown on Fig. 1(c). Bi-doped yttrium iron garnet (Bi:YIG and Bi:GdIG) has been reported to possess perpendicular magnetic anisotropy due to the chemical composition change as the result of increased annealing temperature 21,28 . Another reason for PMA in these thin films is strain from GGG substrate 29 .
Doping of oxides by Helium implantation was shown to reversibly and locally tune magnetic anisotropy 30 . For TbxY3-xFe5O12 (x=2.5, 2.0, 1.0, 0.37) samples grown by spontaneous nucleation technique 31 , magnetic easy axis was found to change from [111] to [100] direction as Tb concentration was decreased. The first-order anisotropy constant K1 undergoes a change of sign near 190K. Another temperature-dependent lattice distortion effect that caused anisotropy change was also reported for YIG films 32 . These results indicate that temperature also plays an important role in both magnetic compensation, lattice distortion and change in anisotropy.
Out of these 20 cases, 7 film/substrate pairs were experimentally tested and shown to exhibit characteristics originating from PMA. The remaining 13 pairs, to the best of our knowledge, have not been tested for PMA experimentally. We indicate through systematic anisotropy calculations that large strain-induced magnetic anisotropy terms may overcome shape when the films are highly strained. We use only the room temperature values of λ111 33 and only report predictions for room temperature (300K). Throughout the rest of this study, the films are labelled as XIG (X = Y, Tm, Dy, Ho, Er, Yb, Tb, Gd, Sm, Eu), i.e. TbIG (Terbium iron garnet) or SmIG (Samarium iron garnet) etc. to distinguish them based on the rare earth element. Our model could accurately predict the magnetic easy axis in almost all experimentally tested garnet film/substrate cases provided that the actual film properties are entered in the model and that the experimental film properties satisfy cubic lattice matching condition to the substrate. (a) Micro/nano-patterning reduces shape anisotropy and magnetoelastic anisotropies overcome shape. (b) Large strain-induced anisotropy must overcome shape anisotropy to yield out-of-plane easy axis. (c) Substitutional doping in garnets overcome shape anisotropy by enhancing magnetocrystalline, growth-induced or magnetoelastic terms.

Anisotropy energy density calculations
Total anisotropy energy density contains three main contributions; according the Equation 1, shape anisotropy (K shape ), first order cubic magnetocrystalline anisotropy (K 1 ), and strain-induced (magnetoelastic) anisotropy (K indu ) parameters determine the total effective anisotropy energy density 16 .
K eff = K indu + K shape + K 1 (1) In case of garnet film magnetized along [111] direction (i.e. on a 111 substrate), the magneto-elastic anisotropy energy density, resulting from magneto-elastic coupling is calculated by Equation 2: where λ 111 is magnetostriction constant along [111] direction and it is usually negative at room temperature 34 . In Eqn. 2, σ || is the in-plane stress induced in the material as a result of lattice mismatch between film and the substrate, and the in-plane stress is calculated from Equation 3 35 : where Y is elastic modulus, and ν is Poisson's ratio 36 .
Assuming the lattice parameter of the thin film matches with that of the substrate, the lattice constant of substrate can be used as the lattice constant of thin film for calculation of strain in Equation 5 37 : The lattice constants used for the films and substrates examined for this study are presented on Table 1.
Shape anisotropy energy density depends on the geometry and the intrinsic saturation magnetic moment of the iron garnet material. Shape anisotropy has a demagnetizing effect on the total anisotropy energy density. These significant anisotropy effects can be observed in magnetic hysteresis loops and FMR measurements 24 .
The most common anisotropies in magnetic materials are shape anisotropy and magneto-crystalline anisotropy 38,39 . Considering that the film is continuous, the shape anisotropy is calculated as 16 By obtaining the values of Ms for rare earth iron garnets as a function of temperature 40,41 , the value for shape anisotropy energy density have been calculated using Equation 6.
Intrinsic magnetic anisotropy 42 , so called magnetocrystalline anisotropy, has the weakest contribution to anisotropy energy density compared to shape, and strain-induced anisotropies 9,11,16,19 . The values for the first order magnetocrystalline anisotropy is calculated and reported previously for rare earth iron garnets at different temperatures 43 . A key consideration in magnetic thin films is saturation field. In anisotropic magnetic thin films, the anisotropy fields have also been calculated using equation 7 as a measure of how much field the films need for magnetic saturation along the easy axis: respectively, based on ref. 36 . We also assume that the saturation magnetization, used for calculation of shape anisotropy, does not change with the film thickness. The saturation magnetization (Ms) values and shape anisotropy for iron garnet films are presented in the third and fourth columns, respectively. The stress values for fully lattice-matched films σ calculated using equation 3 and magnetostriction constants of the films, λ111, are presented on columns 6 and 7. Magnetoelastic anisotropy Kindu, magnetocrystalline anisotropy energy density K1, and the total magnetic anisotropy energy density Keff are calculated and listed on columns 8, 9 and 10, respectively. HA on column 11 is the anisotropy field (the fields required to saturate the films). In this study, we take the same sign convention as in ref. 16  spontaneously minimize its energy and reorient its magnetic moment along the easy axis. As a result, we use here Keff < 0 for out-of-plane easy axis. Due to the thermodynamic arguments mentioned above, we suggest researchers to use Keff < 0 definition for PMA.

Effect of Substrate on Anisotropy Energy Density
Changing the substrate alters the strain in the film, which also changes strain-induced anisotropy in the film. Figure 2 shows the calculated anisotropy energy density of rare earth iron garnet thin films grown on five commercially available different substrates: Gadolinium Gallium Garnet  Consequently, the anisotropy energy density is negative for GdIG when grown on SGGG due to the influence of magnetocrystalline anisotropy energy density.
One other candidate for a PMA rare earth iron garnet on SGGG substrate is SmIG. Since the film lattice parameter is greater than that of the substrate, compressive strain (-3.99 ×10 -3 ) is induced in the film such that the resulting anisotropy energy density possesses a negative value of an order of magnitude (-1.45×10 5 erg·cm -3 ) comparable to the shape anisotropy energy density (1.23×10 5 erg·cm -3 ). With its relatively large magnetocrystalline anisotropy energy density (-1.74×10 4 erg·cm -3 ), SmIG has a perpendicular magnetic anisotropy due to negative value for effective anisotropy energy density (-4.80×10 5 erg·cm -3 ). TbIG film on SGGG substrate is a PMA candidate with positive strain and this film was also recently experimentally demonstrated to have PMA 47 .
Since TbIG's lattice constant is smaller than that of the substrate, the film becomes subject to tensile strain (+1.61 ×10 -3 ). Since TbIG also has a positive λ111 (in contrast to that of SmIG), the film's magnetoelastic anisotropy term becomes large and negative and overcomes the shape anisotropy.
In case of TbIG, magnetocrystalline anisotropy alone overcomes shape and renders the film PMA on SGGG. With the additional magnetoelastic anisotropy contribution (-8.14×10 4 erg·cm -3 ), significant stability of PMA can be achieved.     In case of TmIG, as shown in Fig. 3

Sensitivity of Anisotropy to Variations in Saturation Magnetic Moment and Film Relaxation
The films with predicted effective anisotropy energy and field may come out differently when fabricated due to unintentional variability in fabrication process conditions, film stoichiometry (rare earth ion to iron ratio and oxygen deficiency) as well as process-induced non-equilibrium phases in the garnet films. These changes may partially or completely relax the films or increase strain further due to secondary crystalline phases. Practical minor changes in strain may dramatically alter both the sign and the magnitude of magnetoelastic anisotropy energy and may cause a film predicted as PMA to come out with in-plane easy axis. On the other hand, offstoichiometry may cause reduction in saturation magnetic moment. Reduction in saturation magnetic moment decreases shape anisotropy term quadratically (Kshape = 2πMs 2 ), which implies that a 10% reduction in Ms leads to a 19% decrease in Kshape and anisotropy field may increase (HA = 2Keff/Ms). Therefore, sample fabrication issues and the consequent changes in anisotropy terms may weaken or completely eliminate the PMA of a film/substrate pair and alter anisotropy field.
While these effects may arise unintentionally, one can also use these effects deliberately for engineering garnet films for devices. Therefore, the sensitivity of anisotropy properties of garnet thin films such as anisotropy field and effective anisotropy energy density needs to be evaluated with respect to changes in film strain and saturation magnetic moment. K eff = K indu + K shape + K 1 (8) Relaxing each fully strained and lattice-matched thin film towards unstrained state (ε → 0 or moving from left to right on each plot in Fig. 5 causes the magnetoelastic anisotropy energy term to decrease in magnitude and gradually vanish. The total anisotropy energy decreases in intensity for decreasing strain and constant Ms. When Ms increases, shape anisotropy term also increases and overcomes magnetoelastic anisotropy term. As a result, higher Ms for relaxed films (i.e. relatively thick and iron-rich garnets) may lose PMA. Therefore, one needs to optimize the film stoichiometry and deposition process conditions, especially growth temperature, oxygen partial pressure and film thickness, to ensure that the films are strained and stoichiometric. Since strains are less than 1% in Fig. 5(a), 5(c)-(e), these samples are predicted to be experimentally more reproducible. For YIG on YAG, as shown in Fig. 5(b), the strains are around 3%, which may be challenging to reproduce. The cases presented in Fig. 5 50 , the lattice matching in case of the thick samples becomes challenging to sustain, and the strain relaxes inside the film. Thus, the decrease in the anisotropy field is one consequences of the lower strain state, which is an advantage for magnonics or spin-orbit torque devices. Below a critical thickness, HoIG grown on GGG has PMA. However, as the film reaches this critical thickness, the 40% or more strain relaxation is expected and the easy axis becomes in-plane. So thinner films are preferred to be grown in integrated device applications. SmIG on NGG are presented when strain relaxation and magnetic saturation moments change independently. Film strain may vary from a completely lattice-matched state to the substrate to a relaxed state or a highly strained state due to microparticle nucleation. Strain variability alters magnetoelastic anisotropy and cause a PMA film become in-plane easy axis. On the other hand, magnetic saturation moments may deviate from the tabulated values because of process-induced off-stoichiometry in the films (i.e. rare earth ion to iron ratio or iron deficiency or excess, oxygen deficiency). Relaxing the films reduces the magnetoelastic anisotropy term and diminishes PMA. Increasing Ms strengthens shape anisotropy and eliminates PMA for low enough strains for all five cases presented.
Minimizing Gilbert damping coefficient in garnet thin films is also an important goal for spintronic device applications. First principles predictions of physical origins of Gilbert damping 51 indicate that magnetic materials with lower Ms tend to have lower damping. Based on this prediction, DyIG, HoIG and GdIG films are predicted to have lower Gilbert damping with respect to the others. Since the compensation temperatures of these films could be engineered near room temperature, one may optimize their damping for wide bandwidths all the way up to terahertz (THz) 52 spin waves or magnons. The first principles predictions also indicate that higher magnetic susceptibility (χm) in the films helps reduce damping (i.e. lower saturation field). Therefore, the PMA garnet films with lower anisotropy fields are estimated to have lower Gilbert damping parameters with respect to PMA garnets with higher anisotropy fields.

Conclusion
Shape, magnetoelastic and magnetocrystalline magnetic anisotropy energy terms have been calculated for ten different garnet thin films epitaxially grown on five different garnet substrates.
Negative Keff (effective magnetic anisotropy energy) corresponds to perpendicular magnetic anisotropy in the convention used here. By choosing a substrate with a lattice parameter smaller than that of the film, one can induce compressive strain in the films to the extent that one can always overcome shape anisotropy and achieve PMA for large and negative λ111. Among the PMA films predicted, SmIG possesses a high anisotropy energy density and this film is estimated to be a robust PMA when grown on all five different substrates.
In order to obtain PMA, magnetoelastic anisotropy term must be large enough to overcome shape anisotropy. Magnetoelastic anisotropy overcomes shape anisotropy when the strain type (compressive or tensile) and magnetoelastic anisotropy constants λ111 of the garnet film have the correct signs (not necessarily opposite or same) and the magnetoelastic anisotropy term has a magnitude larger than shape anisotropy. Both compressive and tensile-strained films can, in principle, become PMA as long as shape anisotropy can be overcome with large magnetoelastic strain effects. Here, in almost all cases that yield PMA on the given substrates, PMA iron garnets form under compressive lattice strain, except TbIG on SGGG and TbIG on NGG. These two cases have tensile strain and relatively large magnetocrystalline anisotropy, which could already overcome shape anisotropy without strain. Experiments are therefore suggested to target mainly compressive lattice strain.
20 different garnet film/substrate pairs have been predicted to exhibit PMA and their properties are listed on Table 2. For 7 of these 20 potential PMA cases, we could find unambiguous experimental demonstration of PMA. Among the 20 PMA cases, HoIG/GGG, YIG/YAG, SmIG/SGGG, HoIG/TGG and SmIG/NGG cases have been found to be sensitive to fabrication process or stoiochiometry-induced variations in Ms and strain. In order to control effective anisotropy in rare earth iron garnets (RIGs), shape anisotropy could be tuned by doping garnet films with Ce 53 , Tb 31 and Bi 54 or by micro/nano-patterning. Saturation magnetization could also be increased significantly by doping, which results in increasing the shape anisotropy in the magnetic thin films.
Among the cases predicted to possess PMA, anisotropy fields ranging from 310 Oe (0.31 T) to 12.6 T have been calculated. Such a wide anisotropy field range could be spanned and engineered through strain state, stoichiometry as well as substrate choice. For integrated magnonic devices and circuits, garnets with low Ms and lower anisotropy fields (HA < 0.5 T) would require less energy for switching and would be more appropriate due to their lower estimated Gilbert damping.

Methods
Calculation of anisotropy energy density. We used Keff = Kindu + Kshape + K1 equation to calculate the total anisotropy energy density for each thin film rare earth iron garnet/substrate pair. Each anisotropy term consist of the following parameters: K eff = −