SSnell’s Law for Spin Waves
J. Stigloher, M. Decker, H.S. Körner, K. Tanabe, T. Moriyama, T. Taniguchi, H.Hata, M. Madami, G. Gubbiotti, K. Kobayashi, T. Ono, and C.H. Back Department of Physics, Regensburg University, 93053 Regensburg, Germany Department of Physics, Nagoya University, Nagoya, Aichi 464-8602, Japan Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Dipartimento di Fisica e Geologia,Universita di Perugia, I-06123 Perugia, Italy Istituto Officina dei Materiali del Consiglio Nazionale delle Ricerche (IOM-CNR),Sede di Perugia, c/o Dipartimento di Fisica e Geologia,Via A. Pascoli, I-06123 Perugia, Italy Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan (Dated: June 10, 2016)
Abstract
We report the experimental observation of Snell’s law for magneto-static spin waves in thinferromagnetic Permalloy films by imaging incident, refracted and reflected waves. We use a thicknessstep as the interface between two media with different dispersion relation. Since the dispersionrelation for magneto-static waves in thin ferromagnetic films is anisotropic, deviations from theisotropic Snell’s law known in optics are observed for incidence angles larger than 25° with respectto the interface normal between the two magnetic media. Furthermore, we can show that thethickness step modifies the wavelength and the amplitude of the incident waves. Our findings openup a new way of spin wave steering for magnonic applications. a r X i v : . [ c ond - m a t . m e s - h a ll ] J un nell’s law describes the refraction of waves at the transition between two media withdifferent indices of refraction. In optics the dispersion relation of light is isotropic and thusthe relation between the incident and refracted angles is solely determined by the ratio of therefractive indices. In contrast, for spin waves in thin films with in-plane magnetization thedispersion relation is inherently anisotropic and thus deviations from Snell’s law in opticsare expected [1–6] but have so far not been reported directly.In the emerging field of magnonics, it is foreseen that spin waves can be used as carrierstransmitting information from one medium to another. Thus, it is important to study theirrefraction and reflection at the interface between two magnetic media. Furthermore, theirefficient manipulation and steering is one of the fundamental problems that needs to besolved before spin waves or magnons can be used in magnonic devices [7, 8]. Attempts insteering range from using artificially designed magnonic crystals [9–11] to spin wave guidingin nanostructures [12, 13].Most magnonic devices realized so far are rather large [14], although they could poten-tially be scaled down to the nanometer range. One reason is that spin waves are typicallygenerated by lithographically defined microwave antennas that limit the experimentally ac-cessible wavelengths to a few hundred nanometers. There is major interest in overcomingthis limit and different schemes have been proposed [15, 16].In the experiments presented here, we use Snell’s law for spin waves in the dipolar regimeas an efficient means of spin wave steering and as a way to reach lower wavelengths. We use athickness step to realize the transition between two magnetic media with different dispersionrelations for propagating spin waves. Spin waves are excited in a thick Permalloy film andsubsequently propagate into a film with lower thickness, see Fig. 1 a). This idea [17] has onlyrecently been put into the context of magnonics [18, 19]. We show refraction and reflectionof the waves and find deviations from Snell’s law in optics for incidence angles larger than25° with respect to the interface normal. Furthermore we can show experimentally that thespin wave amplitudes are enhanced in the vicinity of the transition region counteractinglosses on lenght scales of a few micrometers.To explain our findings, we have to incorporate the anisotropic dispersion relation forspin waves in thin films into Snell’s law. Let us first consider the case of a dipolar spin waveimpinging onto an arbitrary interface between two isotropic magnetic media. The continuityof the tangential component of the wave vector k of any wave when experiencing reflection2r when being transmitted to a different medium can be regarded as Snell’s law [20–22],namely sin( θ ) = k , k sin( θ , ) , (1)with θ i the angles with respect to the interface normal. The indices 1-3 denote incoming,refracted and reflected waves, compare Fig. 1 b) . In optics, this reduces to the well knownSnell’s law for refracted waves, where k , can be substituted by the respective refractiveindices due to isotropic and linear dispersion relations in most materials. For the samereasons, it simply follows θ = θ for a reflected wave since it remains in the same medium.In contrast, the wave vector of spin waves in thin films depends on the angle ϕ between thepropagation direction with respect to the direction of the externally applied field H . Thisfollows directly from the dispersion relation [23] (cid:18) ωµ γ (cid:19) = (cid:18) H + Jk + M − Mkd (cid:19) (cid:18) H + Jk + Mkd · sin ( ϕ ) (cid:19) (2) with ω the angular frequency of the wave, γ the gyromagnetic ratio, M the saturationmagnetization, d the thickness of the film, µ the vacuum permeability and J = Aµ M withthe exchange stiffness constant A . For the wave propagation discussed experimentally inthis paper, it is safe to neglect exchange interactions (i.e. A = 0 ), since we are limited torather small wave vectors around k = 1 µm − . In this range of k , the propagation is mainlygoverned by the dynamic dipolar forces originating from the precessing magnetization [23]. ϕ can be identified as ( ϕ + θ − θ ) (see Fig. 1 b)) and Eq. (2), can be rewritten in thefollowing form k = (cid:32) − (cid:114)(cid:0) ( H + M ) sin ( ϕ ) + H (cid:1) − (cid:16) ϕ ) ωµ γ (cid:17) + ( H + M ) sin ( ϕ ) − H (cid:33) d M · sin ( ϕ ) . (3) This expression for the k -vector can be inserted into Eq. (1) to obtain Snell’s law forspin waves. Besides the known material and experimental parameters, the resulting implicitequation only depends on θ and θ and can therefore be used to predict refraction anglesfor spin waves. Similarly, the angle of reflection can be determined by identifying ϕ =180 ° − ( ϕ + θ + θ ) . Feeding the calculated angles back into Eq. (3) allows calculating thewave vector amplitudes. The formalism is not limited to our experiments; it can also beused for interfaces consisting of different magnetic materials.3 ) b)interface normal60 nm100 µm interface C P W
30 nm
H H
10 µm
Figure 1. a) Sketch of the sample with the z -axis not drawn to scale. The red arrow indicatesthe direction of the externally applied magnetic field which is aligned parallel to the coplanar waveguide (yellow). The latter is used to excite spin waves which propagate perpendicular to it. b) Topview of a) with exemplary data acquired by TRMOKE. The green arrows show the wave vectors k , k and k relevant for the analysis. ϕ − denote the angles of the wave vectors with respect tothe external field, while θ − denote the angles with respect to the interface normal. The indices1–3 correspond to the incident, refracted and reflected wave, respectively. In the trivial case of spin waves impinging at normal incidence, i.e. ϕ = ϕ = 90 °, ontoa step interface between two media with thicknesses d i , it is straightforward to define anangle-independent relative refractive index: Since k ∝ d , k k in Eq. (1) reduces to d d = c . Inthe experimental case discussed below c = 2 . This case corresponds to Snell’s law in optics.Experimentally, we use time resolved scanning Kerr microscopy (TRMOKE) and micro-focused Brillouin light scattering (µ-BLS) to verify Snell’s law for spin waves.For the TRMOKE experiments, a 800 nm wavelength Ti:Sapphire laser is focused to aspot of 450 nm at normal incidence onto the sample. Upon reflection, the rotation of thepolarization vector of the incident light is detected which is directly proportional to theout-of-plane component of the magnetization. In order to reach time-resolution, the laserpulses are phase locked to the microwave excitation frequency.Using a x -, y -, z -piezo stage, the sample can be scanned enabling direct access to thecharacteristics of the spin waves, namely wave vector, phase and relative amplitude. Simul-4aneously, the reflectivity of the sample is recorded which is used to identify the thick andthin parts of the sample. Typical dimensions of the images are × µm. We use a stepsize of nm.µ-BLS measurements are performed by focusing about mW of monochromatic lightfrom a Diode-Pumped-Solid-State laser operating at nm onto the sample. All featuresof the experimental apparatus are described in detail elsewhere [24]. Conventional BLSmeasurements are only sensitive to the spin wave intensity, not to its phase. In order tomeasure the propagation direction of spin waves it is necessary to extract the required phaseinformation. This can be realized with the so called phase-sensitive micro-focused BLS whichrelies on the interference between the inelastically scattered light and a reference beam ofconstant phase [25]. Two-dimensional µ-BLS maps are acquired by scanning the laser spotover an area of about . × . µm² with nm step size [26].By sputter deposition and standard lithography techniques, we fabricate a 100 µm wideferromagnetic thin film sample out of Permalloy (Py) which features a well-defined thicknessstep of ∆ z = 30 nm, see Fig 1 a). Spin waves are excited in the 60 nm thick part by ashorted coplanar wave guide (CPW) deposited on top of the films. The spin waves thenpropagate away from the CPW in the Damon-Eshbach (DE) geometry i.e. with k -vector (cid:126)k perpendicular to the direction of both the CPW and the applied magnetic field. Atsome distance from the CPW the spin waves reach a thickness step and are refracted intoa medium with lower thickness, in the present case 30 nm. In thin films, a change in thethickness of the magnetic material causes a drastic change of the dispersion relation whichis therefore used in the experiments to model a transition to a different medium. In fact,in Fig. 1 b) we can clearly observe a refracted wave with altered k -vector (cid:126)k . Similarly, areflected wave can also be observed in the upper part of the thick region. The angle and k -vector definitions are drawn on top of the experimental data obtained by TRMOKE.In total, twelve different samples with varying angle of incidence θ between 0° and 60°in steps of 5° were measured at a fixed excitation frequency of ω = 2 π · GHz in TRMOKEexperiments and ω = 2 π · . GHz in µ-BLS experiments. Examples of the raw TRMOKEdata are shown in Fig. 2 a) and b). In the data, we notice the incoming wave in medium1 (left of the grey line) and the refracted wave in medium 2 (right of the grey line). Whenclosely analyzing the dynamic magnetic contrast in medium 1, also a reflected wave can beobserved. To emphasize the reflected waves we show linescans along the wavefronts of the5ncoming waves in Fig. 2 e) and f). The crest (trough) of the incoming wave leads to apositive (negative) offset in the Kerr signal.As clearly seen in the images, the k -vector of the spin waves is significantly enhancedbehind the thickness step. This means that the natural limit for short wave length spinwave generation given by the geometrical constraint of the CPW can be elegantly overcome.Furthermore, near the interface the signal in the thin part of the Permalloy film is sub-stantially larger than in the thick part. This is counter-intuitive at first, since the refractedwave is induced by the incoming wave. However, the combined action of exchange and dipo-lar interaction leads to an increased excursion angle. To avoid dynamic magnetic charges,purely dipolar coupling would lead to a doubling of the excursion angle (since the thick-ness ratio of the two media is 2:1). At the same time, exchange prefers reducing the tiltangle between the precessing magnetic moments in both media. As a result, the Kerr sig-nal increases by a factor slightly less than two. Note that also an increased in-plane shapeanisotropy might contribute to the deviation from the factor of two. The enhancement ofthe amplitude is an important point and means that we can in fact boost the signal somedistance from the excitation, thus counteracting natural attenuation by damping. This is alocal effect, since the attenuation length in the thin part becomes shorter mainly due to thereduction of the group velocity that scales linearily with thickness. The attenuation lengthis further reduced since the k -vector increases and since the propagation direction tilts awayfrom the Damon Eshbach geometry [27]. However, a net boost of the signal is clearly seensome micrometers from the interface.To further analyze the experiments, we fit the data in the thin part to a 2D plane wavefor the refracted waves to obtain the quantities of interest, namely wave vector amplitudes k and the angles of refraction θ . Additionally, amplitude, phase and attenuation lengthare included in this model. The thick part is fitted with a superposition of incoming andreflected wave yielding k and θ . The fits are displayed in Fig. 2 c) and d). The results arealso used to characterize the sample, as described in the Supplementary Material [26].For the fitting procedure, we avoid regions where the wave is disturbed by sample defectsor where additional reflected waves or static demagnetizing effects near the edges of thePermalloy film alter the plane wave. This is especially important near the interface: Sincecomponents of the external magnetic field point along the interface normal, demagnetizingeffects arise in the thick as well as in the thin region near the thickness step. We expect a6 y ( µ m ) a) b) x (µm )010203040 y ( µ m ) c) x (µm ) d) m z (arb. u.)0 5 10 15 20 25 30distance (µm )0.500.751.00 m z ( a r b . u . ) e) f) Figure 2. Experimental results for two samples. In a) the incoming wave has an angle of θ =20 ◦ with respect to the interface normal and in b) the angle is θ = 40 ◦ . c) and d) show thecorresponding plane wave fits. The x - and y -axis give the dimensions of the images while the color-code provides a scale for the dynamic magnetization component in arbitrary units. The gray linemarks the step between thick (on the left) and thin (on the right) Permalloy films and the whiteboxes indicate the area of the fit. The images are recorded at a fixed frequency of ω = 2 π · GHzand an external field of µ H = 54 mT along the wave fronts of the incoming wave. The color-scale iscropped in order to enhance the contrast in the areas with lower signal. To emphasize the reflectedwaves e) and f) show line scans along the blue lines in a) and b). The blue dots are interpolatedfrom the data; the red lines are fits extracted from c) and d), see main text. "distance" indicatesthe distance from the lower left to the upper right of the blue lines. θ > ◦ in the case of refraction and θ > ◦ in the case of reflection. One of theimportant results that we conclude from our experiments is that the wave vector can be veryefficiently enhanced for incidence angles θ > ◦ . We observe that while the refracted anglestarts decreasing again for θ > ◦ , k keeps increasing due to the anisotropic dispersionrelation (in the case of reflection a decrease is observed for θ > ◦ ) . Essentially, tomatch the condition of Snell’s law and the dispersion relation at the same time, the k -vectorneeds to increase considerably for dipolar spin waves: on an iso-frequency curve, DamonEshbach spin waves have the lowest k -vector. This allows reducing the magnon wavelengthefficiently. In contrast, in an isotropic system — where the wave vector is solely determinedby the refractive index which is generally not angle dependent — it would stay constant.One should note, that the results depend crucially on the orientation of the externalmagnetic field (which is aligned parallel to the antenna in all measurements), while itsmagnitude is negligible for the angular dependence of the refracted wave. In contrast, thewave vector amplitude is influenced substantially by the magnitude of the external field.This can be observed in Fig. 3 a) and b). In the µ-BLS experiments we use µ H = 41 mTas external magnetic field at a frequency of 8.1 GHz while in TRMOKE we use µ H =54 mT at a frequency of 8.0 GHz. Since increasing the external magnetic field shifts thedispersion relation upwards, we detect a k -vector smaller by about a factor of two in theTRMOKE experiments (the slight frequency difference is negligible). Note that surprisinglythe refracted angles remain unaffected.We conclude that Snell’s law for spin waves in the dipolar regime can be predicted withhigh accuracy. Our experiments can be fully reproduced by incorporating the anisotropicdispersion relation. We observe efficient spin wave steering due to the step interface while8 ( / μ m )
54 mT/8.0 GHz 41 mT/8.1 GHz θ ( deg r ee ) TRMOKEμ-BLSSnell opticsSnell spin waves b) d) c)a) θ ( deg r ee ) θ (degree) k ( / μ m ) θ (degree)0 10 20 30 40 50 60 70 Figure 3. a) Refracted angle θ b) refracted wave vector k c) reflected angle θ and d) reflectedwave vector k , all shown versus incident angle θ . In all graphs, the blue dots are experimentalvalues measured with TRMOKE, while red dots are measured with µ-BLS. The orange line showsSnell’s law for an isotropic dispersion relation and the green and purple curves Snell’s law for spinwaves. The latter are calculated with the help of the anisotropic dispersion relation, eq. (2),reflecting the different experimental conditions: The µ-BLS data are measured at an external fieldof µ H = 41 mT and an excitation frequency of 8.1 GHz, while TRMOKE data was recorded atan external field of µ H = 54 mT and an excitation frequency of 8.0 GHz. The purple curve is notshown in a), since it overlaps with the green curve. The errors are the result from least squarefitting.
9t the same time the wave length of the spin waves can be reduced. In the vicinity of theinterface a signal boost is observed that we attribute to dynamic dipolar coupling. Ourfindings should be important in the field of magnonics where efficient spin wave steeringremains a serious problem to be solved. For example, it can be envisaged that a seriesof stepped interfaces results in an increased refracted angle while at the same time shortwave length spin waves can be generated. Note that Snell’s law in the form presented here,should also hold for hetero-interfaces composed of different magnetic materials. In this casethe material parameters (e.g. saturation magnetization and gyro-magnetic ratio) of thedifferent regions have to be inserted in Eq. (3).We gratefully acknowledge funding from the following sources: JSPS KAKENHI GrantNumbers 15H05702, 25103003, 26103002, 26220711, 26870300, 26870304, CollaborativeResearch Program of the Institute for Chemical Research, Kyoto University, DeutscheForschungsgemeinschaft via SFB 689. M. M. and G. G. thank the MIUR under PRINProject No. 2010ECA8P3 "DyNanoMag". K.T. acknowledges a Grant-in-Aid for YoungScientists (B) (No.15K17436). [1] Gruszecki, P. et al.
Influence of magnetic surface anisotropy on spin wave reflection from theedge of ferromagnetic film.
Phys. Rev. B , 054427 (2015).[2] Dadoenkova, Y. S. et al. Huge Goos-Hänchen effect for spin waves: A promising tool for studymagnetic properties at interfaces.
Appl. Phys. Lett. , 042404 (2012).[3] Yasumoto, K. & Oishi, Y. A new evaluation of the Goos-Hänchen shift and associated timedelay.
J. Appl. Phys. , 2170–2176 (1983).[4] Gieniusz, R., Bessonov, V. D., Guzowska, U., Stognii, A. I. & Maziewski, A. An antidot arrayas an edge for total non-reflection of spin waves in yttrium iron garnet films. Appl. Phys. Lett. , 082412 (2014).[5] Gorobets, Y. I. & Reshetnyak, S. A. Reflection and refraction of spin waves in uniaxial magnetsin the geometrical-optics approximation.
Technical Physics , 188–191 (1998).[6] Jeong, D.-E., Han, D.-S. & Kim, S.-K. Refractive index and snell’s law for dipole-exchangespin waves in restricted geometry. Spin , 27–31 (2011).[7] Kruglyak, V. V., Demokritov, S. O. & Grundler, D. Magnonics. J. Phys. D: Appl. Phys. , et al. Realization of a spin-wave multiplexer.
Nat. Commun. , 3727 (2014).[9] Chumak, A. V. et al. All-linear time reversal by a dynamic artificial crystal.
Nat. Commun. , 141 (2010).[10] Duerr, G. et al. Spatial control of spin-wave modes in ni80fe20 antidot lattices by embeddedco nanodisks.
Appl. Phys. Lett. , 202502 (2011).[11] Haldar, A., Kumar, D. & Adeyeye, A. O. A reconfigurable waveguide for energy-efficienttransmission and local manipulation of information in a nanomagnetic device. Nat Nano et al.
Spin waves turning a corner.
Appl. Phys. Lett. , 042410 (2012).[13] Wagner, K. et al.
Magnetic domain walls as reconfigurable spin-wave nanochannels.
Nat Nano
339 (2016).[14] Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics.
NatPhys , 453–461 (2015).[15] Yu, H. et al. Omnidirectional spin-wave nanograting coupler.
Nat. Commun. , 2702 (2013).[16] Demidov, V. E. et al. Excitation of short-wavelength spin waves in magnonic waveguides.
Applied Physics Letters (2011).[17] Vashkovskii, A., Stal’makhov, A. & Shakhnazaryan, D. Formation, reflection, and refractionof magnetostatic wave beams. Sov. Phys. J. , 908–915 (1988).[18] Tanabe, K. et al. Real-time observation of snells law for spin waves in thin ferromagnetic films.
Appl. Phys. Exp. , 053001 (2014).[19] Hata, H. et al. Micromagnetic simulation of spin wave propagation in a ferromagnetic filmwith different thicknesses.
J. Magn. Soc. of Japan , 151–155 (2015).[20] Hecht, E. Optics (Addison-Wesley, 2002), 4 edn.[21] Reshetnyak, S. Refraction of surface spin waves in spatially inhomogeneous ferrodielectricswith biaxial magnetic anisotropy.
Phys. Solid State , 1061–1067 (2004).[22] Kim, S.-K. et al. Negative refraction of dipole-exchange spin waves through a magnetic twininterface in restricted geometry.
Appl. Phys. Lett. , 212501 (2008).[23] Kalinikos, B. A. & Slavin, A. N. Theory of dipole-exchange spin wave spectrum for ferromag-netic films with mixed exchange boundary conditions. J. Phys. C: Sol. St. Phys. , 7013(1986).
24] Madami, M., Gubbiotti, G., Tacchi, S. & Carlotti, G. Application of microfocused brillouinlight scattering to the study of spin waves in low-dimensional magnetic systems.
Sol. StatePhys. , 79–150 (2012).[25] Serga, A. A., Schneider, T., Hillebrands, B., Demokritov, S. O. & Kostylev, M. P. Phase-sensitive brillouin light scattering spectroscopy from spin-wave packets. Appl. Phys. Lett. ,063506 (2006).[26] See Supplemental Material which inlcudes Refs. [1, 28, 29] .[27] Kabos, P. & Stalmachov, V. S. Magnetostatic Waves and Their Application (Chapman & Hall,1994).[28] Davies, C. S. et al.
Towards graded-index magnonics: Steering spin waves in magnonic net-works.
Phys. Rev. B , 020408 (2015).[29] Vansteenkiste, A. et al. The design and verification of mumax3.
AIP Adv. , 107133 (2014)., 107133 (2014).