Thin film rare earth iron garnets with perpendicular magnetic anisotropy for spintronic applications

Perpendicular magnetic anisotropy (PMA) in garnet thin films is important for achieving numerous spintronic applications including spin-orbit switching. In this study, we computationally investigated how to control PMA by tuning substrate strain in Holmium Iron Garnet (HoIG) films grown on five different (111) single crystal garnet substrates of Gadolinium Gallium Garnet (GGG, Gd3Ga5O12), Yttrium Aluminum Garnet (YAG, Y3Al5O12), Terbium Gallium Garnet (TGG, Tb3Ga5O12), Substituted Gadolinium Gallium Garnet (sGGG, Gd3Sc2Ga3O12), and Neodymium Gallium Garnet (NGG, Nd3Ga5O12). The negative sign of effective anisotropy energy density, Keff < 0, and anisotropy field, Ha < 0, determines the easy magnetization axis of the film to be perpendicular to the film surface. Here, we show that magnetoelastic anisotropy energy density determines the sign of the total anisotropy and it can be manipulated by altering the lattice parameter mismatch of the film and its substrate. Based on this study, HoIG is predicted to have PMA when grown on GGG, TGG and YAG among all five substrates mentioned. Moreover, the saturation field magnitude is calculated as an order of several hundreds of Oersteds, which is feasible in practical applications to saturate rare earth iron garnets with perpendicular magnetic anisotropy.


INTRODUCTION
Magnetization is required to be perpendicular (out-of-plane) for spin-orbit switching using low current densities in order to have reliable and fast response. 1 Insulating magnetic garnets have been a matter of interest in the past decades because of their low Gilbert damping and their tunable magnetic properties. [2][3][4] Yttrium Iron Garnet (Y 3 Fe5O 12 , YIG) is a common major material for such spintronic applications and spin wave phenomena. 5 YIG's magnetic properties could be controlled by substituting its Y sites with other rare earth elements. 6 YIG films display in-plane easy axis because of their large shape anisotropy and their negligible magnetocrystalline anisotropy, which is a direction-dependent description of magnetic energy due to bond structure. Since the fabrication of high quality YIG film with out-of-plane magnetization is challenging, new iron garnet thin films with perpendicular magnetic anisotropy (PMA, perpendicular easy axis) and different saturation fields are needed.
The magnetic properties of any magnetic thin film can be described using anisotropy energy density, 7 which is the angular dependence of magnetic energy of a magnetic material. When no external magnetic field is applied; by second law of thermodynamics, entropy is maximized and energy is minimized within the magnetic material by aligning magnetization vector along its axis of minimum total magnetic energy, 8 which is known as the magnetic easy axis. The anisotropy energy density of a magnetic thin film contains three terms 9 that may alter the easy axis; shape, magnetoelastic and magnetocrystalline anisotropy energy density. Each of these terms are described more rigorously in the next section. By engineering these terms, PMA can be induced. Increasing in-plane strain (ε || ) ARTICLE scitation.org/journal/adv contributes to magnetoelastic anisotropy and may lead to PMA. 10 Strain-dependence of magnetic anisotropy in rare earth iron garnets, especially in YIG, has been studied previously. [11][12][13][14] If the anisotropy induced by lattice parameter mismatch is strong enough to overcome shape term, the hysteresis loop becomes square-shaped with low saturation field when magnetized out of plane. 15 PMA has been reported in YIG using substituted Gadolinium Gallium Garnet (sGGG) as substrate. 16 The ability to fabricate new iron garnet thin films with PMA and with different saturation fields and damping values may enable researchers to experimentally evaluate the effects of changing saturation fields, damping and rare earth ions on spin switching, magnetooptical switching and other spin-based phenomena. 1,17,18 In this study, we systematically calculate the anisotropy energy density terms to understand how one may achieve PMA for Holmium iron garnet (Ho 3 Fe5O 12 , HoIG) films when they are epitaxially grown on five different (111) single crystal garnet substrates: Gadolinium Gallium Garnet (GGG, Gd 3 Ga5O 12 ), Yttrium Aluminum Garnet (YAG, Y 3 Al5O 12 ), Terbium Gallium Garnet (TGG, Tb 3 Ga5O 12 ), Substituted-Gadolinium Gallium Garnet (sGGG, Gd 3 Sc 2 Ga 3 O 12 ) and Neodymium Gallium Garnet (NGG, Nd 3 Ga5O 12 ). We show that by using different substrates, one can alter anisotropy terms and change the magnetic easy axis of the films. Changing the anisotropy energy also helps tune the field required for saturating magnetic moment in the HoIG films (Ha: anisotropy field). Finally, we evaluate the sensitivity of the anisotropy energy terms on the experimental variability of film strain.
Holmium iron garnet has been chosen because it was not extensively explored previously for spintronics, yet it has many promising applications, which include magnetooptical response, 19 multiferroicity, 20 low temperature compensation point near 120 K for cryogenic ultrafast quantum spintronic experiments, 21 cancer radiotherapy and chemotherapy 22 as well as tunable magnetic saturation fields. HoIG, compared with YIG, is interesting because its stoichiometric film has been experimentally shown to exhibit PMA in a previous study. 23 Suchomski et al. 24 reported a different method to induce PMA in HoIG by introducing a mesoporous structure in thin film grown on Si (001). In this case, PMA is achieved as the result of combined effect of reducing shape, increasing magnetostrictive and growth-induced anisotropy. PMA is independent of the lattice mismatch between the substrate and the film. The intrinsic strain leading to PMA in HoIG is the result of the porous structure. Therefore, film strain can be manipulated not only by substrate choice, but also by processing and synthesis method.

CALCULATION METHODS FOR ANISOTROPY ENERGY DENSITY
The effective anisotropy energy density main contributions are shape anisotropy (K shape ), magnetoelastic anisotropy (K indu ) and magnetocrystalline anisotropy (K 1 ) as in equation 1; 16 Shape anisotropy, as one of the most common anisotropy in the magnetic materials, 25,26 varies as the geometry and the intrinsic saturation magnetization (Ms) of the iron garnet material changes.
It has a demagnetizing effect on the total anisotropy energy density. For thin films, the shape anisotropy energy is calculated as K shape = 2πM 2 s . 16 For calculation of K shape , we assume that the Ms does not change with the thickness, but it is temperature dependent 27,28 and constant temperature (300 K) is assumed in this study. Magnetoelastic coupling (or magnetoelastic anisotropy energy density) emerges because epitaxially grown films are strained 29 and this forces magnetic moments to realign along vertical axis depending on the sign of the film lattice strain (ε || ) and the magnetoelastic constants of the films (λ 111 ). λ 111 quantifies the coupling of magnetic properties with elastic strain. Magnetoelastic (strain-induced) anisotropy energy density is calculated by 16 assuming that the garnet film is magnetized along [111] direction on a (111) substrate, 30 and σ ∥ the in-plane stress induced in the material from lattice mismatch between the lattice parameters of the film and the substrate, σ ∥ = Y 1−ν ε ∥ . 31 The Young's modulus of HoIG, Y, and the poison ratio ν are constants with values 2.00×10 12 dyne⋅cm -2 and 0.29, respectively, based on Ref. 32. The film strain is calculated using ε ∥ = a s −a f a f , where as and a f represent the lattice parameters of the substrate and the film, respectively. 29 The weakest contribution for the total effective anisotropy energy comes from magnetocrystalline anisotropy (K 1 ), which is a stoichiometry-dependent intrinsic effect that originates from the directional bonding structure of the magnetic material. 33 HoIG's first order magnetocrystalline anisotropy (K 1 ) value reported for room temperature (300K) was used. 34

ARTICLE scitation.org/journal/adv
After calculating the effective anisotropy energy density, one can obtain the anisotropy field, Ha, as given in equation 2. 16 Ha is the magnetic field required to saturate the garnet films.
(2) Figure 1 shows that lattice strain may force magnetic easy axis of HoIG from in-plane to out-of-plane. The lattice of the film is strained due to the lattice parameter difference between the film (a f ) and the substrate (as). It is illustrated that if the lattice parameter of the film is larger than the lattice parameter of the substrate (as-a f = ∆a < 0), compressive in-plane strain is induced inside the film (ε || <0), so the induced anisotropy energy density will be negative. As K indu is a dominant term (at least one order of magnitude) in equation 1, negative strain results in K eff < 0 and Ha < 0. On the other hand, ∆a>0 dictates tensile strain inside the film which results in positive value for both effective anisotropy energy density and anisotropy field. The unit cell of the film (a f ) and the unit cell of the substrate (as) are shown schematically in Fig. 1.

RESULTS AND DISCUSSION
The individual parameters affecting each anisotropy term in the total magnetic anisotropy energy density and the calculated anisotropy energy density values at 300 K are presented on Table I for HoIG on five different commercially available substrates (GGG, YAG, TGG, sGGG and NGG). The lattice parameters of the film and the substrates have been extracted from the tabulated data in Ref. 35. The Ms for HoIG, shape anisotropy, lattice parameters for each substrate are presented in the second, third and fourth columns, respectively. The columns 5 and 6 include the calculated strain values for a fully lattice-matched film ε || (using ε ∥ = a s −a f a f ) and film magnetoelastic constants, λ 111 . Magnetoelastic anisotropy K indu , first-order magnetocrystalline anisotropy energy density, K 1 , and the effective magnetic anisotropy energy density, K eff , are calculated and shown on seventh, eighth, and ninth columns, respectively. The anisotropy field required to saturate the magnetic thin film, Ha is presented on column 10.
The sign convention for Ha in this paper has been chosen based on Ref. 16, in which the K eff < 0 (consequently Ha < 0) is accepted to have PMA magnetic thin film. To obtain this, K indu in the equation 1 should be negative and large enough to overcome K shape . Since the value of magnetoelastic constant, λ 111 , is negative for HoIG at room temperature, it is necessary for the in-plane strain to be compressive (ε || < 0) to yield K indu < 0.

SUBSTRATE INFLUENCE ON EFFECTIVE ANISOTROPY FIELD
The effect of substrate on the total anisotropy field becomes more evident mainly in the value of magnetoelastic energy density. The sign of the strain due to substrate lattice mismatch determines whether the magnetoelastic anisotropy is positive or negative. In case of HoIG (lattice parameter of 12.4 Å) grown epitaxially and lattice-matched on GGG (12.383 Å), YAG (12.005 Å), and TGG (12.355 Å) substrates, the strain is negative because the lattice parameters of these substrates are smaller than that of HoIG film. The strain is even larger when YAG is used as a substrate as the result of nearly 3% of in-plane strain induced in HoIG. Consequently, the large and negative K indu overcomes both shape and magnetocrystalline anisotropy terms in equation 1 which results in Ha <0 and PMA in HoIG thin film. Considering the data on the last column of Table I, the field needed to saturate the HoIG film is an experimentally feasible amount for integrated magnonics applications, especially for the case of GGG (-311 Oe). Both sGGG (12.48 Å) and NGG (12.509 Å) induce tensile strain (ε || > 0) because their lattice parameter is larger than lattice constant of the HoIG film. Consequently, the total anisotropy fields become positive, which make the latter two substrates not suitable candidates for HoIG with PMA. Figure 2 illustrates the effect of five different substrates on effective anisotropy field of HoIG. This plot presents that GGG, YAG and TGG are suitable substrates to induce negative strain anisotropy inside HoIG film, because negative anisotropy field in HoIG can be induced in these film/substrate combinations. In Fig. 2(a), we compare the anisotropy fields and strains of HoIG films on each substrate. Since films with strains above 1-2% may relax spontaneously in experiments, the sensitivity of anisotropy for partial strain relaxation is presented in Fig. 2(b).
As shown on Fig. 2(a), the calculations highlight the necessity of substrate lattice parameters to be smaller than that of HoIG thin film to yield ε || < 0 and to achieve K indu < 0. On the other hand, the tensile stress induced in the film from sGGG and NGG substrates results in Ha > 0, and thus in-plane anisotropy in HoIG film. In experimental studies, one may observe variability of film strain due to partial relaxation during fabrication process or the tabulated mechanical properties of the HoIG film may change for various growth conditions. As a result, one may observe variability in the film strain with respect to the ideal film/substrate mismatch-based magnetoelastic anisotropy model. In Fig. 2(b), the calculated effective anisotropy field values for HoIG films on the five substrates are presented when film is partially relaxed as much as 40% (i.e. the film is near or thicker than its critical thickness) or additionally strained by as much as 40% (i.e. due to growth-induced microstructure evolution). This partial relaxation may occur because of growing films thicker than their critical thicknesses or small shifts in stoichiometry during growth or oxygen vacancies. As a result, we investigate the effect of partial strain relaxation on anisotropy change on Fig. 2(b). Except on GGG, the film's anisotropy orientation remains the same although the magnitude of effective field increases with increasing film strain. If HoIG is grown thick enough on GGG such that it relaxes itself 40% or more than the lattice-matched case, the film's easy axis shifts from out-of-plane to in-plane.

CONCLUSIONS
In this study, we investigated the effect of lattice strain on the effective anisotropy field (Ha) needed to saturate the HoIG film. The effect of lattice strain induced by five different substrates (GGG, YAG, TGG, sGGG, NGG) has been studied by calculating the strains induced by the substrates in HoIG thin film as the result of lattice parameter mismatch between the film and the substrate. The calculations indicate that if the film lattice parameter is larger than that of the substrate, compressive strain is induced in the film and thus K indu < 0. Based on the calculated Ha values, growing HoIG films on GGG, YAG and TGG are estimated to result in PMA (compressive strain in the film yields K eff < 0, and Ha < 0). When the substrate strain variations are considered, the film is PMA on GGG, YAG and TGG and in-plane easy axis for sGGG and NGG. Thicker films would relax as lattice matching to the substrate cannot be sustained for all thicknesses. As a result, anisotropy fields decrease with thicker films (lower strain state) and the fields might become more feasible magnonic or spin-orbit devices. When grown on GGG, HoIG is PMA below a few nanometer critical thickness limit. At this critical thickness limit, the film undergoes 40% or more relaxation and becomes in-plane easy axis. Therefore, for device applications, HoIG must be grown thinner than the critical thickness on GGG for PMA.
The predictions presented in this study could be tested experimentally. HoIG has smaller (about one third) saturation magnetization compared with YIG. This difference helps HoIG have more robust PMA than YIG. Smaller saturation magnetization leads to lower shape anisotropy energy barrier against achieving PMA. HoIG films with PMA and saturation fields tunable with thickness may enable on-chip polarization control for magnon modes and spin waves.