advances.sciencemag.org/cgi/content/full/6/32/eabb1724/DC1
Supplementary Materials for
Nonreciprocal surface acoustic wave propagation via magneto-rotation coupling
Mingran Xu, Kei Yamamoto, Jorge Puebla*, Korbinian Baumgaertl, Bivas Rana, Katsuya Miura,
Hiromasa Takahashi, Dirk Grundler, Sadamichi Maekawa, Yoshichika Otani*
*Corresponding author. Email: [email protected] (J.P.); [email protected] (Y.O.)
Published 7 August 2020, Sci. Adv. 6, eabb1724 (2020)
DOI: 10.1126/sciadv.abb1724
This PDF file includes:
Sections S1 to S5 Figs. S1 to S8 Tables S1 and S2 References
Supplementary Materials Section S1. Theoretical modelling We present a theoretical description of surface acoustic wave absorption by a ferromagnetic thin film. We model the experimental setup by an isotropic elastic material filling the half-space ๐ง๐ง < ๐๐/2 with its top layer of thickness ๐๐ assumed to be ferromagnetic with magnetization ๐ด๐ด. Although the elastic properties could be anisotropic and differ between the magnet and the substrate, this simplified model turns out to be sufficient for our purposes. Let ๐ข๐ข๐ฅ๐ฅ,๐ฆ๐ฆ,๐ง๐ง be the Cartesian components of the displacement vector field. The acoustic waves in isotropic media are characterized by Lamรฉ constants ๐๐, ๐๐ through the elastic free energy
๐น๐น = โซ ๐๐3๐๐ ๏ฟฝ๐๐2๏ฟฝ๐๐๐ฅ๐ฅ๐ฅ๐ฅ + ๐๐๐ฆ๐ฆ๐ฆ๐ฆ + ๐๐๐ง๐ง๐ง๐ง๏ฟฝ
2+ ๐๐๏ฟฝ๐๐๐ฅ๐ฅ๐ฅ๐ฅ2 + ๐๐๐ฆ๐ฆ๐ฆ๐ฆ2 + ๐๐๐ง๐ง๐ง๐ง2 + 2๐๐๐ฅ๐ฅ๐ฆ๐ฆ2 + 2ฮต๐ฆ๐ฆ๐ง๐ง2 + 2๐๐๐ง๐ง๐ฅ๐ฅ2 ๏ฟฝ๏ฟฝ (๐๐1)
where ๐๐๐๐๐๐ = ๏ฟฝ๐๐๐๐๐ข๐ข๐๐ + ๐๐๐๐๐ข๐ข๐๐๏ฟฝ/2 are the components of the strain tensor. The dynamics of ๐๐ with the isotropic free energy is studied in any textbook on continuum mechanics. In particular, the longitudinal and transverse bulk acoustic waves travel at respective speeds of sound ๐๐๐ฟ๐ฟ = ๏ฟฝ(๐๐ + 2๐๐)/๐๐, ๐๐๐๐ = ๏ฟฝ๐๐/๐๐ where ๐๐ is the mass density, and the surface acoustic waves propagating in positive and negative ๐ฅ๐ฅ directions with respective wavenumbers ยฑ๐๐,๐๐ > 0 along the surface ๐ง๐ง = ๐๐/2 are described by the solution
๏ฟฝ๐ข๐ข๐ฅ๐ฅยฑ
๐ข๐ข๐ง๐งยฑ๏ฟฝ = ๐ถ๐ถRe ๏ฟฝ๏ฟฝ
(1 โ ๐๐๐๐2/2)๏ฟฝ2๐๐๐ ๐ ๐ฟ๐ฟ(๐ง๐งโ๐๐/2) โ (2 โ ๐๐๐๐2)๐๐๐ ๐ ๐๐(๐ง๐งโ๐๐/2)๏ฟฝ
โ๐๐๏ฟฝ1 โ ๐๐๐ฟ๐ฟ2๏ฟฝ(2 โ ๐๐๐๐2)๐๐๐ ๐ ๐ฟ๐ฟ(๐ง๐งโ๐๐/2) โ 2๐๐๐ ๐ ๐๐(๐ง๐งโ๐๐/2)๏ฟฝ๏ฟฝ ๐๐โ๐๐(๐๐๐๐โ๐๐๐ฅ๐ฅ)๏ฟฝ (๐๐2)
where ๐ถ๐ถ is an arbitrary real constant, and the parameters ๐ ๐ ๐ฟ๐ฟ,๐๐ , ๐๐๐ฟ๐ฟ,๐๐ are related to the bulk speeds of sound ๐๐๐ฟ๐ฟ,๐๐ and the surface speed of sound ๐๐๐๐ by
๐ ๐ ๐ฟ๐ฟ,๐๐ = ๏ฟฝ๐๐2 โ๐ฅ๐ฅ2
๐๐๐ฟ๐ฟ,๐๐2 , ๐๐๐ฟ๐ฟ,๐๐ =
๐๐๐๐๐๐๐ฟ๐ฟ,๐๐
. (๐๐3)
Note that the value of ๐๐๐๐ depends implicitly on ๐๐๐ฟ๐ฟ,๐๐ through the algebraic equation
๐๐๐๐6 โ 8๐๐๐๐4 + 8๏ฟฝ3 โ2๐๐๐๐2
๐๐๐ฟ๐ฟ2๏ฟฝ ๐๐๐๐2 โ 16 ๏ฟฝ1 โ
๐๐๐๐2
๐๐๐ฟ๐ฟ2๏ฟฝ = 0. (๐๐4)
The โspinโ-momentum locking of surface acoustic waves manifests itself in the phase difference between ๐ข๐ข๐ฅ๐ฅยฑ and ๐ข๐ข๐ง๐งยฑ by โ๐๐ = ๐๐โ๐๐๐๐/2, which changes the sign under ๐๐ โ โ๐๐.
We regard the surface acoustic wave solution (S2) with fixed ๐ฅ๐ฅ and ๐๐ being given as an effective rf field and study how the magnet responds. Back reactions of magnetization dynamics onto acoustic waves are neglected. The purely magnetic part of the free energy is assumed to include an external magnetic field, cubic crystalline and interface-induced uniaxial magnetic anisotropies, and exchange, dipole-dipole and Dzyaloshinskii-Moriya interactions:
๐๐ = โซ ๐๐3๐๐ ๏ฟฝโ๐๐0๐๐๐๐๐ฏ๐ฏ โ ๐๐ โ๐ด๐ด2๐๐ โ ๐ป๐ป2๐๐ +
๐๐0๐๐๐๐2
8๐๐โซ ๐๐3๐๐โฒ(๐๐ โ ๐ป๐ป)(๐๐โฒ โ ๐ป๐ปโฒ)
1|๐๐ โ ๐๐โฒ|
+ ๐พ๐พ๐๐๏ฟฝ๐๐๐ฅ๐ฅ2๐๐๐ฆ๐ฆ2 + ๐๐๐ฆ๐ฆ2๐๐๐ง๐ง2 + ๐๐๐ง๐ง2๐๐๐ฅ๐ฅ2๏ฟฝ โ ๐พ๐พโฅ๐๐๐ง๐ง2 โ ๐พ๐พโฅ๐๐๐ฅ๐ฅ2 + ๐ท๐ท๐๐
โ ๏ฟฝ๏ฟฝ๐๐^๐๐๐ฆ๐ฆ โ ๐๐
^๐๐๐ฅ๐ฅ๏ฟฝ ร ๐๐๏ฟฝ๏ฟฝ , (๐๐5)
where ๐๐๐๐ is the saturation magnetization, ๐ง๐ง = ๐๐/๐๐๐๐, ๐ง๐งโฒ is the value of ๐ง๐ง evaluated at ๐ซ๐ซโฒ, ๐ป๐ปโฒ is the spatial derivative with respect to ๐ซ๐ซโฒ and ๐ฑ๐ฑ๏ฟฝ, ๐ฒ๐ฒ๏ฟฝ are unit vectors in ๐ฅ๐ฅ and ๐ฆ๐ฆ directions. Note that ๐พ๐พโฅ arises from the interface inversion symmetry breaking while ๐พ๐พโฅ is present due to the crystallographic ๐๐-axis alignment of the single crystalline LiNbO3 substrate. Taking the external field to be spatially homogeneous and in-plane ๐๐ =๐ป๐ป(cos๐๐,sin๐๐, 0), the ground state magnetization is also in the plane ๐ง๐ง = (cos๐๐,sin๐๐, 0) and ๐๐ = ๐๐ for sufficiently strong ๐๐ if ๐พ๐พ๐๐ = ๐พ๐พโฅ = 0 . The spin waves excited by the surface acoustic waves ๐ฎ๐ฎยฑ have wavevectors ยฑ๐๐๐ฑ๐ฑ๏ฟฝ respectively. In the thin film limit ๐๐๐๐ โช 1, linear perturbation around the ground state yields the dispersion relation
๐ฅ๐ฅ๐๐ = |๐พ๐พ|๐๐0๏ฟฝ{๐ป๐ป cos(๐๐ โ ๐๐) + ๐ป๐ป๐ฃ๐ฃ(๐๐)}{๐ป๐ป cos(๐๐ โ ๐๐) + ๐ป๐ป๐ง๐ง(๐๐)} + ๐พ๐พ๐๐0๐ป๐ปDMI(๐๐), (๐๐6)
where ๐พ๐พ < 0 is the gyromagnetic ratio and
๐ป๐ป๐ฃ๐ฃ(๐๐) = ๐ด๐ด๐๐2
๐๐0๐๐๐ ๐ + ๐๐๐๐ ๏ฟฝ1 โ 1โ๐๐โ๐๐๐๐
๐๐๐๐๏ฟฝ sin2 ๐๐ + 2๐พ๐พโฅ cos2๐๐โ๐พ๐พ๐๐๏ฟฝ1โ3 cos2 2๐๐๏ฟฝ
๐๐0๐๐๐ ๐ (๐๐7)
๐ป๐ป๐ง๐ง(๐๐) = ๐ด๐ด๐๐2
๐๐0๐๐๐ ๐ + ๐๐๐๐
1โ๐๐โ๐๐๐๐
๐๐๐๐โ 2๐พ๐พโฅโ2๐พ๐พโฅ cos2 ๐๐โ๐พ๐พ๐๐๏ฟฝ1+cos2 2๐๐๏ฟฝ
๐๐0๐๐๐ ๐ (๐๐8)
๐ป๐ปDMI(๐๐) = ยฑ2๐ท๐ท๐๐ sin ๐๐๐๐0๐๐๐๐
(๐๐9)
Note that we throughout use the convention to take frequencies to be positive. In the main text, we chose not to discuss ๐พ๐พโฅ and ๐พ๐พ๐๐, which we shall see are small, and denoted the total perpendicular anisotropy by ๐พ๐พ๐ข๐ข = ๐พ๐พโฅ โ ๐๐0๐๐๐๐
2/2. For a given driving frequency ๐ฅ๐ฅ, the resonance field is determined by
๐ป๐ปres cos(๐๐ โ ๐๐) =โ(๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง) + ๏ฟฝ(๐ป๐ป๐ฃ๐ฃ โ ๐ป๐ป๐ง๐ง)2 + 4(๐ป๐ปDMI โ ๐ฅ๐ฅ/๐พ๐พ๐๐0)2
2 . (๐๐10)
and expanding the square root to linear order in ๐ป๐ปDMI (and setting ๐๐ = ๐๐) yields Eq. (2) in the main text.
There are a variety of ways in which ๐ด๐ด interacts with acoustic waves. For cubic crystals, one usually includes the conventional magnetoelastic coupling in the free energy
๐ผ๐ผ1 = โซ ๐๐3๐๐๏ฟฝ๐๐1๏ฟฝ๐๐๐ฅ๐ฅ2๐๐๐ฅ๐ฅ๐ฅ๐ฅ + ๐๐๐ฆ๐ฆ2๐๐๐ฆ๐ฆ๐ฆ๐ฆ + ๐๐๐ง๐ง2๐๐๐ง๐ง๐ง๐ง๏ฟฝ + 2๐๐2๏ฟฝ๐๐๐ฅ๐ฅ๐๐๐ฆ๐ฆ๐๐๐ฅ๐ฅ๐ฆ๐ฆ + ๐๐๐ฆ๐ฆ๐๐๐ง๐ง๐๐๐ง๐ง๐ฅ๐ฅ + ๐๐๐ง๐ง๐๐๐ฅ๐ฅ๐๐๐ง๐ง๐ฅ๐ฅ๏ฟฝ๏ฟฝ. (๐๐11)
It turns out to be insufficient, however, for explaining the large non-reciprocal response seen in our experiment. For this purpose, we consider free energy terms that describe interactions between ๐ด๐ด and the rotation of elastic deformations ๐ฅ๐ฅ๐๐๐๐ = (๐๐๐๐๐ข๐ข๐๐ โ ๐๐๐๐๐ข๐ข๐๐)/2 . There are many possible mechanisms for magneto-rotation coupling, and here we discuss those that are directly related to the purely magnetic free energy ๐๐. As first pointed out by Maekawa and Tachiki(12), magnetic anisotropy fields induce magneto-rotation couplings through reorientations of crystalline axes, which for uniaxial and cubic anisotropies read
๐ผ๐ผ2 = 2๐พ๐พโฅโซ ๐๐3๐๐๏ฟฝ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅ๐๐๐ฅ๐ฅ + ๐ฅ๐ฅ๐ง๐ง๐ฆ๐ฆ๐๐๐ฆ๐ฆ๏ฟฝ๐๐๐ง๐ง + 2๐พ๐พโฅโซ ๐๐3๐๐๏ฟฝ๐ฅ๐ฅ๐ฅ๐ฅ๐ฆ๐ฆ๐๐๐ฆ๐ฆ + ๐ฅ๐ฅ๐ฅ๐ฅ๐ง๐ง๐๐๐ง๐ง๏ฟฝ๐๐๐ฅ๐ฅ+ 2๐พ๐พ๐๐โซ ๐๐3๐๐๏ฟฝ๐๐๐ฅ๐ฅ๐๐๐ข๐ข๏ฟฝ๐๐๐ฅ๐ฅ2 โ ๐๐๐ฆ๐ฆ2๏ฟฝ๐ฅ๐ฅ๐ฅ๐ฅ๐ฆ๐ฆ + ๐๐๐ฆ๐ฆ๐๐๐ง๐ง๏ฟฝ๐๐๐ฆ๐ฆ2 โ ๐๐๐ง๐ง2๏ฟฝ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐ง+ ๐๐๐ง๐ง๐๐๐ฅ๐ฅ(๐๐๐ง๐ง2 โ ๐๐๐ฅ๐ฅ2)๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅ๏ฟฝ, (๐๐12)
Similarly, the dipolar shape anisotropy results in a magneto-rotation coupling via change of the surface normal directions induced by SAWs. Based on the model of coupling between magnons and surface deformations given in (21), one derives the interaction energy
๐ผ๐ผ3 =๐๐0๐๐๐๐
2
8๐๐โซ ๐๐3๐๐โซ ๐๐3๐๐โฒ ๏ฟฝ๏ฟฝ๐ฟ๐ฟ ๏ฟฝ๐ง๐ง โ
๐๐2๏ฟฝ โ ๐ฟ๐ฟ ๏ฟฝ๐ง๐ง +
๐๐2๏ฟฝ๏ฟฝ ๐ข๐ข๐ง๐ง(๐ซ๐ซ)
+ ๏ฟฝ๐ฟ๐ฟ ๏ฟฝ๐ง๐งโฒ โ๐๐2๏ฟฝ โ ๐ฟ๐ฟ ๏ฟฝ๐ง๐งโฒ +
๐๐2๏ฟฝ๏ฟฝ ๐ข๐ข๐ง๐ง(๐ซ๐ซโฒ)๏ฟฝ (๐ง๐ง โ ๐ป๐ป)(๐ง๐งโฒ โ ๐ป๐ปโฒ)
1|๐ซ๐ซ โ ๐ซ๐ซโฒ|
. (๐๐13)
Although this contains both strain and rotation, we shall see shortly that for in-plane magnetization and surface acoustic waves, the strain can be neglected. The Dzyaloshinkii-Moriya interactions are also affected by the crystalline orientation and consequently generate couplings between magnet and lattice deformations;
๐ผ๐ผ4 = ๐ท๐ทโซ ๐๐3๐๐๐ง๐ง โ ๏ฟฝ๏ฟฝ ๏ฟฝ (๐๐๐ฅ๐ฅ๐ฅ๐ฅ โ ๐ฅ๐ฅ๐ฅ๐ฅ๐ฅ๐ฅ)๐ฅ๐ฅ=๐ฆ๐ฆ,๐ง๐ง
๐ซ๐ซ๏ฟฝ๐ฅ๐ฅ๐๐๐ฆ๐ฆ โ ๏ฟฝ ๏ฟฝ๐๐๐ฆ๐ฆ๐ฅ๐ฅ โ ๐ฅ๐ฅ๐ฆ๐ฆ๐ฅ๐ฅ๏ฟฝ๐ฅ๐ฅ=๐ฅ๐ฅ,๐ง๐ง
๐ซ๐ซ๏ฟฝ๐ฅ๐ฅ๐๐๐ฅ๐ฅ
+ ๐ฑ๐ฑ๏ฟฝ ๏ฟฝ ๏ฟฝ๐๐๐ฆ๐ฆ๐ฅ๐ฅ โ ๐ฅ๐ฅ๐ฆ๐ฆ๐ฅ๐ฅ๏ฟฝ๐ฅ๐ฅ=๐ง๐ง,๐ฅ๐ฅ
๐๐๐ฅ๐ฅ โ ๐ฒ๐ฒ๏ฟฝ ๏ฟฝ (๐๐๐ฅ๐ฅ๐ฅ๐ฅ โ ๐ฅ๐ฅ๐ฅ๐ฅ๐ฅ๐ฅ)๐ฅ๐ฅ=๐ฆ๐ฆ,๐ง๐ง
๐๐๐ฅ๐ฅ๏ฟฝ ร ๐ง๐ง๏ฟฝ , (๐๐14)
where ๐ซ๐ซ๏ฟฝ๐ฅ๐ฅ = ๐ฑ๐ฑ๏ฟฝ, ๐ซ๐ซ๏ฟฝ๐ฆ๐ฆ = ๐ฒ๐ฒ๏ฟฝ, ๐ซ๐ซ๏ฟฝ๐ง๐ง = ๐ณ๐ณ๏ฟฝ Again, the strain couplings are discarded later. Finally, when the microscopic magnetic moments may be considered fixed on individual atomic sites and adiabatically following the motion of the lattice, there will be an analogue of Coriolis force called spin-rotation coupling (22) given by
๐ผ๐ผ5 =โ๐๐๐๐โซ ๐๐3๐๐๏ฟฝ๐๐๐ฅ๐ฅ๐๐๐๐๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐ง + ๐๐๐ฆ๐ฆ๐๐๐๐๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅ + ๐๐๐ง๐ง๐๐๐๐๐ฅ๐ฅ๐ฅ๐ฅ๐ฆ๐ฆ๏ฟฝ, (๐๐15)
where ๐๐/๐๐ is the effective length of spin per unit cell.
Each of the interaction terms introduces an effective rf field ๐๐๐ฅ๐ฅ acting on the magnetization ๐ด๐ด where ๐๐๐ฅ๐ฅ = โ๐ฟ๐ฟ๐ผ๐ผ๐ฅ๐ฅ/๐ฟ๐ฟ(๐๐0๐๐๐๐๐๐),๐๐ = 1,โฏ ,5. Evaluating these fields for the ground state configuration of ๐ด๐ด yields
๐ก๐ก1 =๐๐1๏ฟฝ๐๐๐ฅ๐ฅ๐ฅ๐ฅ โ ๐๐๐ฆ๐ฆ๐ฆ๐ฆ๏ฟฝsin2๐๐ โ 2๐๐2๐๐๐ฅ๐ฅ๐ฆ๐ฆcos2๐๐
๐๐0๐๐๐๐๐ฏ๐ฏ๏ฟฝ โ
2๐๐2๏ฟฝ๐๐๐ง๐ง๐ฅ๐ฅcos๐๐ + ๐๐๐ง๐ง๐ฆ๐ฆsin๐๐๏ฟฝ๐๐0๐๐๐๐
๐ณ๐ณ๏ฟฝ, (๐๐16)
๐ก๐ก2= โ
2๐พ๐พโฅcos2๐๐ โ 2๐พ๐พ๐๐cos4๐๐๐๐0๐๐๐๐
๐ฅ๐ฅ๐ฅ๐ฅ๐ฆ๐ฆ๐ฏ๐ฏ๏ฟฝ
โ2๐พ๐พโฅ๏ฟฝ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅcos๐๐ + ๐ฅ๐ฅ๐ง๐ง๐ฆ๐ฆsin๐๐๏ฟฝ โ 2๐พ๐พโฅ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅcos๐๐ โ 2๐พ๐พ๐๐๏ฟฝ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅcos3๐๐ + ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐งsin3๐๐๏ฟฝ
๐๐0๐๐๐๐๐ณ๐ณ๏ฟฝ, (๐๐17)
๐ก๐ก3 =๐๐๐๐
4๐๐{๐ฏ๐ฏ๏ฟฝ(๐ฏ๐ฏ๏ฟฝ โ ๐ป๐ป) + ๐ณ๐ณ๏ฟฝ๐๐๐ง๐ง}
ร โซ ๐๐3๐๐โฒ ๏ฟฝ๐ฟ๐ฟ ๏ฟฝ๐ง๐งโฒ โ๐๐2๏ฟฝ โ ๐ฟ๐ฟ ๏ฟฝ๐ง๐งโฒ +
๐๐2๏ฟฝ๏ฟฝ
1|๐ซ๐ซ โ ๐ซ๐ซโฒ|
๏ฟฝcos๐๐๐๐๐ฅ๐ฅโฒ
+ sin๐๐๐๐๐ฆ๐ฆโฒ๏ฟฝ๐ข๐ข๐ง๐ง(๐ซ๐ซโฒ), (๐๐18)
๐ก๐ก4 =๐ท๐ท
๐๐0๐๐๐๐๏ฟฝ๏ฟฝ๐๐๐ฆ๐ฆ(๐๐๐ง๐ง๐ฅ๐ฅ + ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅ) โ ๐๐๐ฅ๐ฅ๏ฟฝ๐๐๐ฆ๐ฆ๐ง๐ง โ ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐ง๏ฟฝ๏ฟฝ๐ฏ๐ฏ๏ฟฝ
+ ๐๐๐ง๐ง๏ฟฝ(๐๐๐ง๐ง๐ฅ๐ฅ + ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅ)cos๐๐ + ๏ฟฝ๐๐๐ฆ๐ฆ๐ง๐ง โ ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐ง๏ฟฝsin๐๐๏ฟฝ๐ณ๐ณ๏ฟฝ๏ฟฝ, (๐๐19)
๐ก๐ก5 = โโ๐๐
๐๐0๐๐๐๐๐๐๐๐๐๐๏ฟฝ๏ฟฝ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅcos๐๐ โ ๐ฅ๐ฅ๐ฆ๐ฆ๐ง๐งsin๐๐๏ฟฝ๐ฏ๐ฏ๏ฟฝ + ๐ฅ๐ฅ๐ฅ๐ฅ๐ฆ๐ฆ๐ณ๐ณ๏ฟฝ๏ฟฝ, (๐๐20)
where ๐ฏ๐ฏ๏ฟฝ = โ๐ฑ๐ฑ๏ฟฝsin๐๐ + ๐ฒ๐ฒ๏ฟฝcos๐๐ โฅ ๐ง๐ง0 and the in-plane components have been projected onto this axis. To obtain Eq. (1) of the main text, we set ๐พ๐พโฅ = ๐พ๐พ๐๐ = 0 and discarded ๐๐4 and ๐๐5, which is justified in the next section. The main contribution of ๐๐3 is to replace ๐พ๐พโฅ in ๐๐2 by ๐พ๐พ๐ข๐ข . For the surface acoustic waves ๐๐ยฑ propagating in the positive and negative ๐ฅ๐ฅ directions, their non-vanishing components are given as follows:
๐๐๐ฅ๐ฅ๐ฅ๐ฅยฑ = ยฑ๐๐๐ถ๐ถ ๏ฟฝ1 โ๐๐๐๐2
2๏ฟฝ ๏ฟฝ๐๐๐ ๐ ๐ฟ๐ฟ๏ฟฝ๐ง๐งโ
๐๐2๏ฟฝ โ ๏ฟฝ1 โ
๐๐๐๐2
2๏ฟฝ ๐๐๐ ๐ ๐๐๏ฟฝ๐ง๐งโ
๐๐2๏ฟฝ๏ฟฝ , (๐๐21)
๐๐๐ง๐ง๐ง๐งยฑ = โ๐๐๐ถ๐ถ ๏ฟฝ1 โ๐๐๐๐2
2๏ฟฝ ๏ฟฝ(1 โ ๐๐๐ฟ๐ฟ2)๐๐๐ ๐ ๐ฟ๐ฟ๏ฟฝ๐ง๐งโ
๐๐2๏ฟฝ โ ๏ฟฝ1 โ
๐๐๐๐2
2๏ฟฝ ๐๐๐ ๐ ๐๐๏ฟฝ๐ง๐งโ
๐๐2๏ฟฝ๏ฟฝ , (๐๐22)
๐๐๐ฅ๐ฅ๐ง๐งยฑ = ๐ถ๐ถ ๏ฟฝ1 โ๐๐๐๐2
2๏ฟฝ๏ฟฝ1 โ ๐๐๐ฟ๐ฟ2 ๏ฟฝ๐๐
๐ ๐ ๐ฟ๐ฟ๏ฟฝ๐ง๐งโ๐๐2๏ฟฝ โ ๐๐๐ ๐ ๐๐๏ฟฝ๐ง๐งโ
๐๐2๏ฟฝ๏ฟฝ , (๐๐23)
๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅยฑ = โ๐ถ๐ถ๐๐๐๐2
2๏ฟฝ1 โ ๐๐๐ฟ๐ฟ2๐๐
๐ ๐ ๐๐๏ฟฝ๐ง๐งโ๐๐2๏ฟฝ. (๐๐24)
Note that we now omit the operation of taking real parts. Due to the free surface boundary conditions used to derive the solution, ๐๐๐ฅ๐ฅ๐ง๐ง vanishes at the surface and is smaller by a factor of ๐๐๐๐ in the magnetic region than ๐๐๐ฅ๐ฅ๐ฅ๐ฅ,๐ง๐ง๐ง๐ง and ๐ฅ๐ฅ๐ง๐ง๐ฅ๐ฅ. Thus in the thin film limit, one can ignore ๐๐๐ฅ๐ฅ๐ง๐ง in ๐๐1,๐๐3 and ๐๐4.
The linearized Landau-Lifshitz-Gilbert equation with Gilbert damping ๐ผ๐ผ , after Fourier transforms in time and space, reads
๏ฟฝ๐๐๐ฃ๐ฃ๐๐๐ง๐ง๏ฟฝ =
1(๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)(๐ป๐ป + ๐ป๐ป๐ง๐ง) โ ๐ผ๐ผ2๐ป๐ป๐๐2 + ๐๐๐ผ๐ผ๐ป๐ป๐๐(2๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง) โ (๐ป๐ป๐๐ โ ๐ป๐ป๐ท๐ท๐๐๐ท๐ท)2
ร ๏ฟฝ๐ป๐ป + ๐ป๐ป๐ง๐ง + ๐๐๐ผ๐ผ๐ป๐ป๐๐ โ๐๐(๐ป๐ป๐๐ โ ๐ป๐ป๐ท๐ท๐๐๐ท๐ท)๐๐(๐ป๐ป๐๐ โ ๐ป๐ป๐ท๐ท๐๐๐ท๐ท) ๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ + ๐๐๐ผ๐ผ๐ป๐ป๐๐
๏ฟฝ ๏ฟฝโ๐ฃ๐ฃโ๐ง๐ง๏ฟฝ , (๐๐25)
where we set ๐๐ = ๐๐ for simplicity (i.e. ignoring the effect of cubic anisotropy on the ground state), introduced ๐ป๐ป๐๐ = ๐ฅ๐ฅ/๐พ๐พ๐๐0, and the components of the total effective field ๐๐ยฑ from SAW with wavenumber ยฑ๐๐ are given by
โ๐ฃ๐ฃยฑ = ๐๐๐ถ๐ถcos๐๐๐๐0๐๐๐๐
๐๐๐๐2
2๏ฟฝยฑ๐๐1(2 โ ๐๐๐๐2)sin๐๐ +
โ๐ฅ๐ฅ๐๐๐๐
๏ฟฝ1 โ ๐๐๐ฟ๐ฟ2๏ฟฝ , (๐๐26)
โ๐ง๐งยฑ =๐ถ๐ถcos๐๐๐๐0๐๐๐๐
๏ฟฝ๏ฟฝ๐พ๐พโฅ โ๐๐0๐๐๐๐
2
2โ ๐พ๐พโฅ โ ๐พ๐พ๐๐cos2๐๐๏ฟฝ ๐๐๐๐2๏ฟฝ1 โ ๐๐๐ฟ๐ฟ2 โ ๐ท๐ท๐๐๐๐๐ฟ๐ฟ2 ๏ฟฝ1 โ
๐๐๐๐2
2๏ฟฝ๏ฟฝ . (๐๐27)
The constant ๐ถ๐ถ is irrelevant as it multiplies the overall magnitude of the excited spin waves. The energy absorbed from SAW into spin waves per unit time is expected to be proportional to the power ๐๐ยฑ๐๐ dissipated by the spin waves, which is equal to the amount of work done by ๐๐ยฑ
๐๐ = ๐๐0๐๐๐๐ ๏ฟฝRe ๏ฟฝ๐๐๐๐๐๐๐๐๏ฟฝ โ Re[โยฑ]๏ฟฝ , (๐๐28)
where the angled bracket denotes averaging over one period of SAW. Substituting Eqs. ,(๐๐25), (๐๐26) and (๐๐27), one derives the formula for SAW absorption
๐๐ยฑ๐๐ =๐ถ๐ถ2๐ผ๐ผ๐ฅ๐ฅ2๐๐๐๐4 cos2 ๐๐ /2|๐พ๐พ|๐๐0๐๐๐๐
{(๐ป๐ป๐๐ โ ๐ป๐ปDMI)2 โ (๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)(๐ป๐ป + ๐ป๐ป๐ง๐ง) + ๐ผ๐ผ2๐ป๐ป๐๐2 }2 + ๐ผ๐ผ2๐ป๐ป๐๐2 (2๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง)2ร [{(๐ป๐ป + ๐ป๐ป๐ง๐ง)2 + (๐ป๐ป๐๐ โ ๐ป๐ปDMI)2 + ๐ผ๐ผ2๐ป๐ป๐๐2 }(ยฑ๐๐ME sin๐๐ + ๐๐SR)2]+ 2(๐ป๐ป๐๐ โ ๐ป๐ปDMI)(2๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง)(ยฑ๐๐ME sin๐๐ + ๐๐SR)๐๐MR+ {(๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)2 + (๐ป๐ป๐๐ โ ๐ป๐ปDMI)2 + ๐ผ๐ผ2๐บ๐บ2}๐๐MR
2 , (๐๐29)
The parameters ๐๐ME,SR,MR measure, in unit of energy density, the contributions from magneto-elastic, spin-rotation and magneto-rotation couplings respectively, defined by
๐๐ME=๐๐1 ๏ฟฝ1 โ ๐๐๐๐2
2๏ฟฝ ,๐๐SR = โ๐๐๐๐
2๐๐๏ฟฝ1 โ ๐๐๐ฟ๐ฟ2, (๐๐30)
๐๐MR = ๏ฟฝ๐พ๐พโฅ โ๐๐0๐๐๐ ๐
2
2โ ๐พ๐พโฅ โ ๐พ๐พ๐๐ cos2 ๐๐๏ฟฝ๏ฟฝ1 โ ๐๐๐ฟ๐ฟ2 โ
๐ท๐ท๐๐๐๐๐ฟ๐ฟ2
2๐๐๐๐2 ๏ฟฝ1 โ ๐๐๐๐
2
2๏ฟฝ , (๐๐31)
It can be clearly observed that nonreciprocity, i.e. dependence of ๐๐ยฑ๐๐ on the sign ยฑ, comes from cross terms between ๐๐ME and ๐๐SR or ๐๐ME and ๐๐MR . Any ยฑ dependence is accompanied by a factor sin ๐๐, which is dictated by the time-reversal symmetry. Note that in Eq. (๐๐29), the dependence of the absorption on the strength of external magnetic field has been all explicitly spelt out so that it may be directly used to fit the SAW transmission data. Section S2. Details of the attenuation fitting
Although the expression (๐๐29) can, in principle, be directly compared with the measured attenuation of SAWs, fitting of the data in practice is not entirely straightforward due to the relatively large number of fitting parameters. It should also be mentioned that the modeling is oversimplified in some aspects so that some discrepancies between the data and the theory are inevitable due to the neglected factors including crystalline anisotropy of LiNbO3 , roughness of the interfaces, and the difference in elastic properties between the different materials among other things. To extract the salient features of the relevant physics under discussion in this article, we split the fitting procedure into several steps, which is guided by the underlying physics. Throughout, fitting functions are denoted by block capitals and the corresponding script letters are used for the experimental data to be fitted. Step 1: Lorentzian fitting.
Let ๐ซ๐ซยฑ๐๐(๐ป๐ป,๐๐) be the attenuation data as a function of the external field ๐ป๐ป for a fixed angle ๐๐ and wavenumber ยฑ๐๐. We fit each data set ๐ซ๐ซยฑ๐๐(๐ป๐ป,๐๐) by a function ๐๐ยฑ๐๐(๐๐) with three parameters ๐๐ยฑ๐๐(๐๐), โยฑ๐๐
๐๐๐๐๐๐(๐๐), ๐ฅ๐ฅโยฑ๐๐(๐๐):
๐๐ยฑ๐๐(๐๐) =1
2๐๐๐๐ยฑ๐๐(๐๐) ๐ฅ๐ฅโยฑ๐๐(๐๐)
๏ฟฝ๐ป๐ป โโยฑ๐๐๐๐๐๐๐๐(๐๐)๏ฟฝ
2+ ๐ฅ๐ฅโยฑ๐๐(๐๐)2/4
(๐๐32)
The best fit values for the parameters are now fed into the next steps as the data points. Step 2: Resonance field fitting.
We determine five parameters ๐ด๐ด,๐พ๐พโฅ,๐พ๐พโฅ,๐พ๐พ๐๐,๐ท๐ท by fitting the data โยฑ๐๐res(๐๐)as a function of
๐๐ by the function
๐ป๐ปยฑ๐๐res(๐๐) =
โ๐ป๐ป๐ฃ๐ฃ(๐๐) โ ๐ป๐ป๐ง๐ง(๐๐) + ๏ฟฝ{๐ป๐ป๐ฃ๐ฃ(๐๐) โ ๐ป๐ป๐ง๐ง(๐๐)}2 + 4{๐ฅ๐ฅ/๐พ๐พ๐๐0 โ ๐ป๐ปDMI(๐๐)}2
2 (๐๐33)
where ๐ป๐ป๐ฃ๐ฃ,๐ง๐ง,DMI(๐๐) are given by Eqs. (๐๐7) - (๐๐9). We use the value|๐พ๐พ| = 29.4 quoted from Ref. and measure ๐๐๐๐ independently by MPMS. The best fit values for the parameters
are given in TABLE. S1 and plotted in Fig. S1. The DMI constant value presented in the main text was obtained at this step. Table S1. Summary of resonance field fitting.
๐ด๐ด(๐ฝ๐ฝ/m) ๐พ๐พโฅ(๐ฝ๐ฝ/๐๐3) ๐พ๐พโฅ(๐ฝ๐ฝ/๐๐3) ๐พ๐พ๐๐(๐ฝ๐ฝ/๐๐3) ๐ท๐ท(๐ฝ๐ฝ/๐๐2)
Best fit value
9.138ร 10โ11
6.134 ร 105 7.212 ร 103 โ9.023ร 103
8.9 ร 10โ5
Fitting error 1.112ร 10โ12
2.059ร 10โ15
6.30ร 10โ16
9.341ร10โ16 1.1 ร 10โ5
Note that the value of ๐ด๐ด is presumably overestimated due to the unquantifiable spatial variation across the film thickness costing significant exchange energy.
Fig. S1. Fittings for โ+๐๐๐๐๐๐๐๐(ฯ), โโ๐๐
๐๐๐๐๐๐(๐๐) and โ+๐๐๐๐๐๐๐๐(ฯ) โโโ๐๐
๐๐๐๐๐๐(๐๐) , respectively
Step 3: Linewidth fitting.
In our simple phenomenology, the linewidth is independent of the angle and given by ๐ฅ๐ฅ๐ป๐ป =๐ผ๐ผ|๐ฅ๐ฅ/2๐พ๐พ๐๐0|. We determine the value of Gilbert damping constant ๐ผ๐ผ by equating ๐ฅ๐ฅ๐ป๐ป to the ๐๐- and ยฑ๐๐-average of โยฑ๐๐(๐๐). The estimated value is ๐ผ๐ผ = 0.059873, and the fitting results are plotted in Fig. S2.
Fig. S2. Fitting for ๐ฅ๐ฅ๐ป๐ป+๐๐ and ๐ฅ๐ฅ๐ป๐ป+๐๐ respectively.
Step 4: Amplitude fitting.
Finally, we carry out the fitting of the amplitude data ๐๐ยฑ๐๐(๐๐) by the function
๐๐ยฑ๐๐(๐๐) =๏ฟฝฬ๏ฟฝ๐ถ2 cos2 ฯ
๏ฟฝ2๐ป๐ปยฑ๐๐๐๐๐๐๐๐(ฯ) + ๐ป๐ป๐ฃ๐ฃ(ฯ) + ๐ป๐ป๐ง๐ง(ฯ)๏ฟฝ
2
ร ๏ฟฝ๏ฟฝ๏ฟฝ๐ป๐ปยฑ๐๐๐๐๐๐๐๐(ฯ) + ๐ป๐ป๐ง๐ง(ฯ)๏ฟฝ
2+ ๏ฟฝ
ฯฮณฮผ0
โ ๐ป๐ป๐ท๐ท๐๐๐ท๐ท(ฯ)๏ฟฝ2๏ฟฝ (๐๐2 ยฑ sinฯ)2
+ 2 ๏ฟฝฯฮณฮผ0
โ ๐ป๐ป๐ท๐ท๐๐๐ท๐ท(ฯ)๏ฟฝ ๏ฟฝ2๐ป๐ปยฑ๐๐๐๐๐๐๐๐(ฯ) + ๐ป๐ป๐ฃ๐ฃ(ฯ) + ๐ป๐ป๐ง๐ง(ฯ)๏ฟฝ(๐๐2
ยฑ sinฯ)๐๐1+๏ฟฝ๐ป๐ปยฑ๐๐res (๐๐) + ๐ป๐ป๐ฃ๐ฃ(๐๐)๏ฟฝ
2+ ๏ฟฝ
๐ฅ๐ฅ๐พ๐พ๐๐0
โ ๐ป๐ป๐ท๐ท๐๐๐ท๐ท(๐๐)๏ฟฝ2๐๐12๏ฟฝ (๐๐34)
The functions and parameters appearing in the above equation have all been obtained in the previous steps except for those to be fitted, i.e. ๏ฟฝฬ๏ฟฝ๐ถ, ๐๐1 and ๐๐2. The overall constant ๏ฟฝฬ๏ฟฝ๐ถ does not contain any meaningful information. The other two ๐๐1,2 are the central objects of interest in this study, which respectively measure the ratio of the magneto-rotation and spin-rotation coupling energies ๐๐MR , ๐๐SR to the magnetoelastic coupling energy ๐๐ME : ๐๐1 =๐๐MR/๐๐ME, ๐๐2 = ๐๐SR/๐๐ME (c.f. Eqs. (๐๐30) and (๐๐31)). The nonreciprocity arises from the terms that are linear in ๐๐1,2. It turns out that these two parameters are highly degenerate: both of them can fit the data equally well on their own and when being fitted at the same time, the error bars tend to be much greater than when only one of them is fitted. The results of the fitting are given in TABLE. S2 and plotted in Fig. S3. In the end, we convert ๐ด๐ดยฑ๐๐(๐๐) into ๐๐ยฑ๐๐(๐๐) and plot ๐๐ยฑ๐๐(๐๐) and rectifier ratio [๐๐+๐๐(๐๐) โ ๐๐+๐๐(๐๐)]/[๐๐+๐๐(๐๐) + ๐๐+๐๐(๐๐] in Fig. S4. Table S2. Summary of amplitude fitting.
๐ถ๐ถ ๐ถ๐ถerr ๐๐1 ๐๐1err ๐๐2 ๐๐2err
Fitting with ๐๐2 = 0 192.340259 1.8981 0.187254 0.013432 N/A N/A
Fitting with ๐๐1 = 0 192.229955 1.88400 N/A N/A -0.08537 0.006042
3-parameter fitting 1.717393 -5.190078 0.976959 -2.441063 0.442856 1.717393
Fig. S3. Fitting for ๐ด๐ด+๐๐(๐๐) , ๐ด๐ด+๐๐(๐๐) and [๐ด๐ด+๐๐(๐๐) โ ๐ด๐ด+๐๐(๐๐)]/[๐ด๐ด+๐๐(๐๐) + ๐ด๐ด+๐๐(๐๐]) under conditions ๐๐2 = 0, ๐๐1 = 0 and ๐๐1, ๐๐2 โ 0, respectively.
Fig. S4. Fitting for ๐๐+๐๐(๐๐), ๐๐+๐๐(๐๐) and [๐๐+๐๐(๐๐) โ ๐๐+๐๐(๐๐)]/[๐๐+๐๐(๐๐) + ๐๐+๐๐(๐๐]) under the condition ๐๐2 = 0. Even though the nonreciprocity data alone is insufficient to decide which of the magneto-rotation and spin-rotation couplings is the dominant mechanism, we can argue in favor of the former by considering how plausible the best fit values of ๐๐1,2 are. First of all, we note that the value of ๐๐MR is completely known from the resonance field fitting and estimated to be ๐๐MR โผ โ106 J/m3. In order to estimate ๐๐SR, one would need to know the effective spin density ๐๐/๐๐ for CoFeB thin films. Although it cannot be precisely determined due to the uncertainties in the microscopic magnetic structure, one could safely assume ๐๐
๐๐< 1030 m3
since ๐๐ โผ ๐๐(1) and the unit cell size cannot be smaller than 1 ร . Thus for ๐ฅ๐ฅ =2๐๐ ร 6.1 GHz , one obtains ๐๐SR < 6๐๐ ร 105 J/m3 . Therefore, |๐๐2| would be at best comparable to |๐๐1| even in the most optimistic scenario. While we do not know the value of ๐๐1 for our sample, typical values for transition metals are of order 107 J/m3 (23) so that the best fit value ๐๐1 โ 0.2 is very reasonable while achieving ๐๐2 โ โ0.1 would require a significantly lower magnetostriction for CoFeB than Co or Fe alone. These estimates also suggest that the shear strain mechanism should be far less effective than the magneto-rotation coupling for our thin films as ๐๐2 should be of the same order as ๐๐1 and ๐๐๐๐ โผ1/500. Therefore, we conclude that the observed giant nonreciprocity is mainly due to the
magneto-rotation coupling induced by the uniaxial anisotropy field ๐พ๐พ๐ข๐ข = ๐พ๐พโฅ โ ๐๐0๐๐๐๐2/2.
Although we are unable to exclude a contribution from the spin-rotation coupling, it would be at most of similar order of magnitude to the contribution from the magneto-rotation coupling. Section S3. Influence of the substrate anisotropy on the angular dependence of the SAW attenuation
The theoretical model used for fitting the data above assumes an isotropic elastic medium. However, LiNbO3 is trigonal and its elastic properties are anisotropic. Here we consider possible corrections to the SAW attenuation signal arising from the substrate anisotropy. The power dissipated by spin waves is given by
๐๐ = โฮฑ๐๐๐๐ฯ2
2ฮณ[{(๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)(๐ป๐ป + ๐ป๐ป๐ง๐ง) โ (๐ป๐ปฯ โ๐ป๐ปDMI)2}2 + ฮฑ2(2๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง)2๐ป๐ปฯ2 ]โ1
ร [{(๐ป๐ป + ๐ป๐ป๐ง๐ง)2 + (๐ป๐ปฯ โ ๐ป๐ปDMI)2}|โ๐ฃ๐ฃ|2+ {(๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)2 + (๐ป๐ปฯ โ ๐ป๐ปDMI)2}|โ๐ง๐ง|2 +2(2๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง)(๐ป๐ปฯโ ๐ป๐ปDMI)โ๏ฟฝโ๐ง๐งโ๐ฃ๐ฃ๏ฟฝ๏ฟฝ
= โฮฑ๐๐๐๐ฯ2
2ฮณ[{(๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)(๐ป๐ป + ๐ป๐ป๐ง๐ง) โ (๐ป๐ปฯ โ ๐ป๐ปDMI)2}2 + ฮฑ2(2๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง)2๐ป๐ปฯ2 ]โ1
ร {|(๐ป๐ป + ๐ป๐ป๐ง๐ง)โ๐ฃ๐ฃ + ๐๐(๐ป๐ปฯ โ ๐ป๐ปDMI)โ๐ง๐ง|2+ |(๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)โ๐ง๐ง โ ๐๐(๐ป๐ปฯ โ ๐ป๐ปDMI)โ๐ฃ๐ฃ|2} , (๐๐35)
This is essentially Eq. (S29), written in terms of hv,z instead of ฯME,MR ,SR via Eqs. (S26) and (S27). Assuming near resonance H~Hres, we separate it into the Lorentzian and the residual amplitude
๐๐ โฮฑ|๐ป๐ปฯ|๐ด๐ด/ฯ
(๐ป๐ป โ๐ป๐ปres)2 + ฮฑ2๐ป๐ปฯ2, (๐๐36)
๐ด๐ด =ฯ2
ฮผ0๐๐๐๐|ฯ|(2๐ป๐ปres + ๐ป๐ป๐ฃ๐ฃ + ๐ป๐ป๐ง๐ง)2
ร {|(๐ป๐ป + ๐ป๐ป๐ง๐ง)โ๐ฃ๐ฃ + ๐๐(๐ป๐ปฯ โ ๐ป๐ปDMI)โ๐ง๐ง|2+ |(๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)โ๐ง๐ง โ ๐๐(๐ป๐ปฯ โ ๐ป๐ปDMI)โ๐ฃ๐ฃ|2} , (๐๐37)
We are doing this splitting because this model evidently fails to fit the observed anisotropic linewidth data (which is down to the back reaction of spin waves onto SAWs) so that the amplitude part should be isolated in comparing the theory with the data. If the magnetic resonance is isotropic, i.e. ๐ป๐ป๐ง๐ง = ๐ป๐ป๐ฃ๐ฃ, one also has (๐ป๐ป + ๐ป๐ป๐ฃ๐ฃ)2 = (๐ป๐ป + ๐ป๐ป๐ง๐ง)2 = (๐ป๐ปฯ โ๐ป๐ปDMI)2 at the resonance. Although this approximation is not very good in the present setup where the dipolar shape anisotropy is clearly visible, for simplicity we take it here. Conventionally choosing ๐ป๐ปฯ > 0, one obtains at the resonance
๐ด๐ด =ฯ4ฮผ0๐๐๐๐|ฯ||โ๐ฃ๐ฃ + ๐๐โ๐ง๐ง|2 , (๐๐38)
With the cubic magneto-elastic coupling and out-of-plane uniaxial anisotropy, the effective magnetic field generated by acoustic waves is given by
โ๐ฃ๐ฃ =1
ฮผ0๐๐๐๐๏ฟฝ๐๐1๏ฟฝฯต๐ฅ๐ฅ๐ฅ๐ฅ โ ฯต๐ฆ๐ฆ๐ฆ๐ฆ๏ฟฝ sin 2ฯ โ 2๐๐2ฯต๐ฅ๐ฅ๐ฆ๐ฆ cos 2ฯ๏ฟฝ , (๐๐39)
โ๐ง๐ง = โ2
ฮผ0๐๐๐๐๏ฟฝ(๐๐2ฯต๐ง๐ง๐ฅ๐ฅ + ๐พ๐พโฅฯ๐ง๐ง๐ฅ๐ฅ) cosฯ + ๏ฟฝ๐๐2ฯต๐ง๐ง๐ฆ๐ฆ + ๐พ๐พโฅฯ๐ง๐ง๐ฆ๐ฆ๏ฟฝ sinฯ๏ฟฝ, (๐๐40)
Suppose that the surface acoustic wave propagates in the ๐ฅ๐ฅ direction, but still has a nonzero ๐ฆ๐ฆ component of the deformation. In our original analysis, we did not include this component since it is absent for SAWs in isotropic media. The boundary conditions force ฯต๐ง๐ง๐ฅ๐ฅ = ฯต๐ง๐ง๐ฆ๐ฆ at the boundary, and the effective magnetic field reduces to
โ๐ฃ๐ฃ =1
ฮผ0๐๐๐๐๏ฟฝ๐๐1ฯต๐ฅ๐ฅ๐ฅ๐ฅ sin 2ฯโ 2๐๐2ฯต๐ฅ๐ฅ๐ฆ๐ฆ cos 2ฯ๏ฟฝ, (๐๐41)
โ๐ง๐ง = โ2๐พ๐พโฅฮผ0๐๐๐๐
๏ฟฝฯ๐ง๐ง๐ฅ๐ฅ cosฯ + ฯ๐ง๐ง๐ฆ๐ฆ sinฯ๏ฟฝ , (๐๐42)
We cannot derive analytical expressions for the strain and vorticity tensor components in general anisotropic media, but here the purpose is to capture the qualitative trend. First of all, let us assume ฯต๐ฅ๐ฅ๐ฅ๐ฅ,ฯ๐ง๐ง๐ฅ๐ฅ are given by those of the SAWs in isotropic media, meaning they are of a similar order of magnitude and have a phase difference of ยฑฯ/2 for ยฑk respectively. Next, ฯต๐ฅ๐ฅ๐ฆ๐ฆ = โ๐ฅ๐ฅ๐ข๐ข๐ฆ๐ฆ/2,ฯ๐ง๐ง๐ฆ๐ฆ = โ๐ง๐ง๐ข๐ข๐ฆ๐ฆ/2 arise from the anisotropy correction so that they are expected to be smaller than ฯต๐ฅ๐ฅ๐ฅ๐ฅ,ฯ๐ง๐ง๐ฅ๐ฅ. For surface localized waves, one expects โ๐ฅ๐ฅ โผ ๐๐๐๐, โ๐ง๐ง โผ ฮบ > 0, where ๐๐, ๐ ๐ are real so that it is reasonable to assume the relative phase between ฯต๐ฅ๐ฅ๐ฆ๐ฆ and ฯ๐ง๐ง๐ฅ๐ฅ is also ยฑฯ/2. Hence we introduce the following parameterisation:
๐๐1ฯต๐ฅ๐ฅ๐ฅ๐ฅ = ๐๐๐๐, 2๐พ๐พโฅฯ๐ง๐ง๐ฅ๐ฅ = ๐๐, 2๐๐2ฯต๐ฅ๐ฅ๐ฆ๐ฆ = ๐๐๐๐๐๐๐๐ฮด, 2๐พ๐พโฅฯ๐ง๐ง๐ฆ๐ฆ = ๐๐๐๐๐๐ฮด , (๐๐43)
where ๐๐, ๐๐, ๐๐,๐๐, ๐ฟ๐ฟ can be taken to be real. The experimental data already suggested |๐๐/๐๐| โผ 0.35 and ๐๐,๐๐ represent the anisotropy correction so that|๐๐|, |๐๐| โช |๐๐|.๐๐ and ๐๐ are even and odd with respect to ยฑ๐๐ respectively, while the behavior under +๐๐ โ โ๐๐ is not known for ๐๐,๐๐. However, given ฯต๐ฅ๐ฅ๐ฆ๐ฆ โผ ๐๐๐๐๐ข๐ข๐ฆ๐ฆ/2,ฯ๐ง๐ง๐ฆ๐ฆ โผ ฮบ๐ข๐ข๐ฆ๐ฆ/2, it is expected that one is odd and the other is even. One obtains
๐ด๐ด =ฯ4
|ฯ|ฮผ0๐๐๐๐
๏ฟฝ๐๐ sin 2ฯ โ ๐๐ cosฯ โ (๐๐ cos 2ฯ + ๐๐ sinฯ)๐๐๐๐ฮด๏ฟฝ2
, (๐๐44)
where the cross term between ๐๐ sin 2๐๐ and ๐๐ cos๐๐ gives the main nonreciprocal term in the amplitude, while the anisotropy corrections may have angular dependencies that do not appear from the isotropic part, i.e. a term proportional to sin 4๐๐. These terms can explain at least parts of the features in Fig. 3 that are not accounted for by our fitting curve. Section S4. Details of 100% nonreciprocity
In the angular dependence spectrum near ๐๐ =180o, there is an abrupt change of nonreciprocity (Fig. 3.). It is also the region where the nonreciprocity reaches its maxima. According to the theory, the maxima are indeed 100% as we shall demonstrate now. Defining nonreciprocity by (๐ด๐ด+ โ ๐ด๐ดโ)/(๐ด๐ด+ + ๐ด๐ดโ) where ๐ด๐ดยฑ corresponds to ๐ด๐ด in Eq. (S44) evaluated for ยฑ๐๐ , i.e. ยฑ๐๐ respectively, it is expected that 100% nonreciprocity may be achieved at angles where either ๐ด๐ด+ or ๐ด๐ดโ is equal to zero. By considering the isotropic case with ๐๐ = ๐๐ = 0, this angle can be determined by the condition
2๐๐ sin๐๐ + ๐๐ = 0 . (๐๐45) Since |๐๐/2๐๐| < 0 in our sample, this always has a solution near ฯ = 0,๐๐ and if ๐๐ >0 one gets ๐ด๐ด+ = 0 at a ๐๐ < 0 for instance. And obviously ๐ด๐ด+ = 0 implies (๐ด๐ด+ โ ๐ด๐ดโ)/(๐ด๐ด+ + ๐ด๐ดโ) = โ1, i.e. 100% non-reciprocity.
Fig. S5. The nonreciprocity ratio of the absorption amplitude ๐ด๐ด (๐ธ๐ธ๐ธ๐ธ. (๐๐45)) when the SAW is assumed isotropic, i.e. ๐๐ = ๐๐ = 0 . We set ๐๐ = 1, ๐๐ = ยฑ0.35. The non-reciprocity reaches 100 % at an angle very close to ฯ = 0.
In the experiment, we rotated the magnetic field angle from ๐๐ = 172o to ๐๐ = 188o , tracking the variation in the nonreciprocity (Fig. S6). From the spectra, we confirmed the rapid change of nonreciprocity amplitude and sign. Also, interestingly, when ๐๐ =184o , we observed a total flat line for SAW(-k), i.e. a vanishing ๐ด๐ดโ, while maintaining SAW(+k) with a robust peak, namely 100% nonreciprocity ratio in accordance with the theory.
Fig. S6. (A- I) Absorption spectra at ๐๐ = 172o ,174o ,176o ,178o ,180o ,182o ,184o , 186o , 188o, respectively. Section S5. Characterization of Dzyaloshinskii-Moriya interaction via Brillouin light scattering spectroscopy Dzyaloshinskii-Moriya interaction (DMI) is the antisymmetric exchange coupling, which favours the canting alignment of the neighboring magnetic spins ๐๐๐๐ and ๐๐๐๐. In recent years, due to its intriguing application in stabilizing magnetic skyrmions and chiral domain walls, DMI has attracted intensive research. In the magnetic heterostructure, DMI appears as a consequence of the broken structural inversion symmetry in the magnet. Among the experimental methods for investigating DMI, Brillouin light scattering (BLS) spectroscopy has been most widely used due to its high sensitivity. In the presence of the DMI, because of the different canting arrangement, spin waves with wavenumbers ยฑ๐๐ give opposite contributions to the total energy, which results in an asymmetric spin wave dispersion relation. And this asymmetry in ยฑ๐๐ leads to Eq. (๐๐45) (10, 14, 24, 25) for estimating DMI constant ๐ท๐ท:
๐ฅ๐ฅ๐ฅ๐ฅ =๐ฅ๐ฅ(โ๐๐) โ ๐ฅ๐ฅ(๐๐)
2๐๐=
2๐ท๐ท๐๐|๐พ๐พ|๐๐๐๐๐๐
(๐๐46)
๐๐ = sgn(๐๐๐ฅ๐ฅ) 4๐๐sinฮฮปlaser
(๐๐47)
where we take gyromagnetic ratio |๐พ๐พ| = 29.4 GHz/T (26), saturation magnetization ๐๐0๐๐๐๐ = 1.5 T , wavelength of the laser ฮปlaser = 473 nm and sgn(๐๐๐ฅ๐ฅ) is the polarity of the ๐ฅ๐ฅ component of static magnetization, ฮ the angle between incident light and sample plane.
Fig. S7. Characterization of Dzyaloshinskii-Moriya interaction via Brillouin light scattering spectroscopy. (A) Schematics of Brillouin light scattering geometry, with scattering plane (in blue), and Cartesian coordinates. (B) Brillouin spectra of the Ta/CoFeB(1.6nm)/MgO film measured at incident angle ฮ = 65โ . Red and blue dots represent spectra measured under applied field ๐๐0๐ป๐ป= 50 mT along respective +๐ฅ๐ฅ and โ๐ฅ๐ฅ directions. Solid lines represent Lorentzian fitting of spectra. ๐๐๐๐ is the magnitude of wavenumber ๐๐. Stokes and anti-Stokes peaks were normalized to a peak amplitude of 1, respectively.
Fig. S8. Frequency difference ๐ฅ๐ฅ๐ฅ๐ฅ of ยฑ๐๐ spin waves as a function of wavevector ๐๐. Purple circles and solid line denote measured data and fitting by Eq. (๐๐46)
In order to estimate ๐ท๐ท, we performed BLS measurement on the Ta/CoFeB(1.6nm)/MgO thin film in Damon-Eshbach geometry (as depicted in Fig. S7A). Figure Fig. S7B shows measurement of BLS spectra at ฮ =65โ while applying magnetic field ๐๐0๐ป๐ป of ยฑ50mT. Owing to in-plane momentum conservation of the light scattering process, spin waves travelling with the wavenumber ยฑ๐๐ appear as anti-Stokes and Stokes peaks, respectively. The difference of spectra center frequency ๐ฅ๐ฅ๐ฅ๐ฅ in ยฑ๐๐ are plotted in Fig. S8. By fitting with Eq. (๐๐46), we obtain DMI constant ๐ท๐ท = 0.063ยฑ 0.0023 mJ/m2 , which is in a good agreement with the estimation from acoustic ferromagnetic resonance ๐ท๐ทa-FMR = 0.089 ยฑ0.011 mJ/m2.
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