Observation of mode splitting in artificial spin ice
Sergi Lendinez, Mojtaba Taghipour Kaffash, M. Benjamin Jungfleisch
OObservation of mode splitting in artificial spin ice
Sergi Lendinez, Mojtaba Taghipour Kaffash, and M. Benjamin Jungfleisch Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716,United States a) We report the dependence of the magnetization dynamics in a square artificial spin-ice lattice on the in-plane magneticfield angle. Using two complementary measurement techniques – broadband ferromagnetic resonance and micro-focused Brillouin light scattering spectroscopy – we systematically study the evolution of the lattice dynamics, bothfor a coherent radiofrequency excitation and an incoherent thermal excitation of spin dynamics. We observe a splittingof modes facilitated by inter-element interactions that can be controlled by the external field angle and magnitude.Detailed time-dependent micromagnetic simulations reveal that the split modes are localized in different regions of thesquare network. This observation suggests that it is possible to disentangle modes with different spatial profiles bytuning the external field configuration.Propagating spin waves (or their elementary quanta –magnons) with wavelengths at the sub-micrometer lengthscale can carry and transport spin information in magneticmaterials with low losses . Therefore, they have been dis-cussed as data carriers in next-generation information tech-nologies. Essential in this regard is magnonic crystals, whichare artificially-designed periodic lattices in which the spin-wave band structure is engineered for optimized spin-waveproperties. Hence, they are promising for applications indata processing, information technologies, and microwavedevices . In recent years, artificial spin ices (ASIs) havebeen proposed as potential magnonic crystals, as they al-low unprecedented reconfigurability, precise control of theirground state, and tuning of the magnetization dynamics .Micromagnetic modeling , microwavespectroscopy , and Brillouin light scattering (BLS)spectroscopy have been used to gain a better under-standing of the complex dynamics in ASI. Gliga et al. firstproposed square ASI as a potential platform for spin-waveconduits . First experimental efforts followed shortlyafter that and mainly relied on ferromagnetic resonance(FMR) spectroscopy . Due to their high sensitivity,microwave-based techniques are well suited for detectingsmall signal levels such as those found in ASI. There hasbeen an increased interest in the effects of dynamic modehybridization in magnetic materials and nanostructures and, recently, high-resolution anticrossing spectra havebeen reported even in interacting square artificial spin ice .However, the high sensitivity comes at the cost of losinginformation about the spatial extent of the dynamics.Furthermore, by relying on a microwave antenna of finitesize, one is limited to probing only a limited wavevectorrange. Additionally, microwave antennas couple efficientlyonly to odd spin-wave modes. Hence, they cannot be usedto excite/detect even spin-wave modes reliably. As an al-ternative, micro-focused BLS has been used more recentlyto investigate the spatial distribution of microwave-drivenspin dynamics in ASI . On the other hand, wavevector-resolved BLS revealed the spin-wave dispersion of thermallyexcited ASI . However, how the thermal spin-wave spec-trum without any resonant microwave excitation compares a) Electronic mail: [email protected] to the traditionally-used microwave-spectroscopy techniquessuch as FMR, and even more importantly, how the dynamicsand inter-element coupling depend on the in-plane magneticfield angle, have been open questions until now.Here, we study the detailed angular-dependent dynamicproperties of a square artificial spin-ice lattice by three com-plementary techniques: we present a comparison of angular-dependent inductive FMR measurements with micro-focusedBLS characterizations of thermally-excited spin dynamics.We observe a spin-wave mode splitting facilitated by the inter-element interactions that depends on both the in-plane fieldangle and magnitude. The experimentally-acquired spectraare interpreted using time-dependent micromagnetic simula-tions. Furthermore, the two-dimensional micromagnetic mod-eling results reveal that the split modes observed in the ex-periments reside in different regions of a single ASI vertex,suggesting that it is possible to control which portion of thenetwork oscillates.Nanomagnetic islands with a size of 260 nm ×
80 nm ar-
200 nm260 nm80 nm (a)(b) (c) M i c r o w a v e B L S l a s e r bea m SEM
FIG. 1. Illustration of the experimental setup and angular-dependentmeasurement configuration for (a) ferromagnetic resonance and (b)Brillouin light scattering measurement. The square artificial spin iceis patterned directly on the signal line (gray area). The bias magneticfield µ H is applied in-plane at an angle θ , where θ = ◦ meansalong the x -axis, while θ = ◦ corresponds to a field along the y -axis. (c) Scanning electron micrograph of the studied artificial spinice with the dimensions of the nanomagnets. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b ranged on a square lattice with a lattice constant of 341 nm anda minimum gap of 34 nm have been fabricated. We patternedthe lattice with electron beam lithography, deposited 20 nm ofNi Fe using electron beam evaporation, followed by lift-off. The ASI was grown directly on top of the signal line ofa coplanar waveguide that had previously been fabricated us-ing optical lithography and electron beam evaporation of 150nm of Au, see Figs. 1(a,b). The signal line width is 20 µ m,and the gap between the signal and ground lines is 10 µ m.This geometry results in an ASI lattice composed of hundredsof thousands of elements, which we expect to behave as aninfinitely-extended lattice. A scanning electron microscopyimage of a section of the lattice with its dimensions is shownin Fig. 1(c).Two separate sets of measurements have been carried out:in-plane magnetic field dependent FMR and thermally-excitedmicro-focused BLS. For the FMR measurements, we usea vector network analyzer FMR (VNA-FMR) approach, inwhich a microwave signal is sent through the coplanar wave-guide, and the transmitted signal is detected by the VNA(Keysight N5225A) measuring the transmission parameterS . A higher absorption at a given field and frequency in-dicates that the system magnetization is on resonance. Sincethere is an oscillating driving, the measurement is more sensi-tive when the magnetization is perpendicular to the oscillationdirection (parallel to the signal line). At a given in-plane fieldangle, we sweep the magnetic field from negative to positivevalues, and we record the frequency-dependent S parameterat each field step. To vary the in-plane field angle, the direc-tion of the magnetic field is rotated with respect to the signalline.The inductive FMR measurements are compared to micro-focused-BLS studies, which reveal the thermally excited spec-tra at various in-plane field angles. For this purpose, a contin-uous single-mode 532-nm wavelength laser (Spectra PhysicsExcelsior) is focused on the surface of the same ASI lattice asstudied by FMR. A sketch of the measurement configurationis shown in Fig. 1(b). The BLS process can be understoodby the inelastic scattering of laser photons with magnons.The BLS process is energy and momentum conserving, andthus the scattered photons carry information about the probedmagnon . In our micro-focused system, a high-numericalaperture (NA = 0.75) objective lens is used. Therefore, thescattered and reflected light is collimated within a large coneangle with respect to the sample surface normal resulting inthe detection of an extensive wavevector range between 0 –17.8 rad/ µ m . In our setup, we can rotate the sample withrespect to the external magnetic field to perform angular-dependent measurements. Since the magnetization is ther-mally excited, all magnetization directions are equally sen-sitive regardless of their orientation with respect to the signalline, which is in stark contrast to the typically-employed FMRtechnique. Additionally, unlike FMR, which relies on a cou-pling of the microwave drive with spin dynamics, both oddand even spin-wave modes can be probed with equal sensitiv-ity in thermal-BLS measurements.The experimental data are compared to micromagnetic sim-ulations performed using MuMax3 . The ASI is simulated (a)(b) m z , b y FF T i n t . FIG. 2. Illustration of the dynamic micromagnetic simulation ap-proach. (a) Temporal evolution of the exciting magnetic field sinc-pulse b y (orange) and the resulting onset of magnetization dynamics,shown here: m z (blue). The insets show m z and b y on a magnifiedscale in the instance of time when b y is applied and long after thatwhen a steady-state oscillation of m z is observed. (b) CorrespondingFourier transform shows the eigenexcitations of the square lattice atan external magnetic field of B = θ = ◦ .The inset shows the Fourier transform of the sinc pulse shown in (a)that features a precise cut-off frequency of 50 GHz. using a single vertex and periodic boundary conditions, withthe same geometry and dimensions as the nanofabricated sam-ple. The simulation space is divided in 230 × × . × . ×
15 nm . The lateral cell sizeis chosen to be smaller than 5 nm, corresponding to the ex-change length of Ni Fe . Starting from an equilibrium state,the magnetization dynamics are simulated by sending an ex-citation sinc field pulse in the y -axis, with an amplitude of b y = . z -componentin Fig. 2(a) for an external field of B = z -component of the magnetization. The FFT intensity is com-puted as the square power of the FFT amplitude, and the peaksin intensity correspond to the dominant oscillation frequen-cies, as shown in Fig. 2(b) for the averaged z -component. Thesinc pulse is chosen since its transformed spectrum is flat up tothe cut-off frequency, at which point its amplitude decreasesto almost 0, as is evidenced in the inset of Fig. 2(b); this re-duces a spurious contribution of characteristic frequencies inthe FFT intensity spectrum associated with the pulse. Spa- (a) (b)
100 0 100Field (mT)1020 F r e q u e n c y ( G H z ) S (arb. units)maxmin 100 0 100Field (mT)1020 F r e q u e n c y ( G H z ) S (arb. units)maxmin FIG. 3. Experimentally observed broadband ferromagnetic reso-nance spectra at magnetic fields applied at (a) θ = ◦ and (b) θ = ◦ .The color-coded maps show the strongest microwave absorption cor-responding to a minimum scattering parameter S in dark colors. tial profiles can be obtained similarly by performing the FFTof the time series of each cell. The FFT intensity spectra arehence obtained by simulating the time evolution in the fieldrange of −
300 mT to 300 mT.In the following, we discuss the experimental results ob-tained by broadband FMR. Figure 3 shows the experimen-tal FMR data at applied magnetic field angles of θ = ◦ and θ = ◦ . The dark lines in the FMR spectra (Fig. 3) show adecrease in the transmission parameter S indicating a reso-nant excitation of spin dynamics in the ASI. For an applied-field angle of θ = ◦ [Fig. 3(a)], the most intense mode ischaracterized by a monotonous increase in frequency as themagnitude of the external field increases and is produced bythe islands aligned along the signal line (in the following werefer to those islands as horizontal islands). Fainter absorp-tion lines running parallel to the main absorption lines but at a ∼ . At even lower frequencies, weobserve a characteristic W-shaped line, with minima around −
175 mT and 175 mT, and a maximum of ≈
10 GHz at 0mT. This absorption line originates from the vertical islands(aligned perpendicular to the signal line), and, hence, it be-comes weaker as the magnetic field approaches 0 mT and themagnetization is perpendicular to the signal line. In this sit-uation, the microwave field is parallel to the magnetizationleading to a reduced torque.In order to tune the inter-element interactions, we changethe magnetic field angle. Figure 3(b) shows the FMR spectraat applied magnetic fields of θ = ◦ . As is apparent from thefigure, the most intense mode (predominantly associated withthe dynamics in the horizontal islands) is lowered in frequencyand is mostly flat; i.e., ∂ f / ∂ H ≈ ◦ [Fig. 3(a)], where the slope of the most in-tense curve ∂ f / ∂ H monotonically decreases as the field isswept from negative to positive fields until the horizontal is-lands switch at around +
80 mT. Moreover, we find indicationsthat modes cross as we change the field angle.To gain further insights, we employ a second experimen-tal technique – micro-focused BLS on thermally-excited spinwaves. Figures 4(a,b) show the thermally-excited BLS spec-tra for 0 ◦ and 75 ◦ , respectively. While the same general trendof the main modes is observed, we find a distinct differencein the BLS data. Significantly more modes are detectable in (a) (b)(c) (d)
300 150 0 150 300Field (mT)102030 F r e q u e n c y ( G H z ) Norm. BLS int. (arb. units)maxmin300 150 0 150 300Field (mT)102030 F r e q u e n c y ( G H z ) Norm. BLS int. (arb. units)maxmin300 150 0 150 300Field (mT)102030 F r e q u e n c y ( G H z ) FFT int. (arb. units)maxmin 300 150 0 150 300Field (mT)102030 F r e q u e n c y ( G H z ) FFT int. (arb. units)maxmin
FIG. 4. (a) Brillouin light scattering data for applied magnetic fieldsat θ = ◦ and (b) θ = ◦ . In this case, dark color in the color-codedmaps represents higher intensity. (c) Corresponding micromagneticsimulations at θ = ◦ and (d) θ = ◦ . the thermal BLS spectra. Furthermore, the most prominentresonant lines – coming from horizontal and vertical elements– have comparable intensities. This can be understood fromthe fact that Figs. 4(a,b) show the thermal BLS signal; i.e., theresonances are driven by thermal activation, which is inde-pendent of the relative orientation of the signal line (and thusthe driving microwave magnetic field in FMR experiments)and the direction of the magnetization. Therefore, the torqueon the vertical islands’ net magnetization is not reduced asthe field is approaching 0 mT. Moreover, additional modes,barely visible in the FMR data, can be detected by BLS. Wewill discuss this interesting observation below.We now compare the experimental data with micromag-netic simulations, shown in Figs. 4(c,d). As is evident froma comparison to the BLS data [Figs. 4(a,b)], the agreementbetween the logarithmic color-coded micromagnetic simula-tion results and the micro-BLS data is remarkable. It high-lights that micro-BLS is well suited for detecting most eigen-modes in the ASI network, even those inaccessible by stan-dard broadband FMR.Inspecting the spectra obtained at an in-plane field angle of θ = ◦ more closely, we can observe several modes that arecrossing. For example, in the BLS [Fig. 4(a)] and the cor-responding simulated data [Fig. 4(c)], we find such a modecrossing at a magnetic field of approximately −
100 mT and afrequency of 8 GHz. This crossing is also visible in the FMRresults [Fig. 3(a)]; however, the FMR signal of the intersect-ing modes is much weaker. As discussed above, the FMRdata shows a change in the lineshape and crossings as the in-plane field angle is changed. With the BLS technique, we arebetter equipped to understand the dynamics since we can de-tect more modes and are equally sensitive to both sublattices.By changing the direction of the external field to θ = ◦ [Figs. 4(b,d)], the field dependence of the main resonant linechanges and, as a result, the crossing behavior is altered. Atmagnetic fields around −
100 mT, we find that the main reso- N o r m . FF T i n t . FIG. 5. Simulated frequency response of the modes observed at θ = ◦ for −
150 mT (red, bottom curve) and −
100 mT (orange,top curve). The spatial profiles of the observed modes are shownas insets as color coded images: red represents a high intensity andblack a small intensity. nance splits in different modes. In particular, in the BLS data,we observe that the resonance splits into at least two modeswith opposite slopes [Fig. 4(b)], while in the micromagneticsimulated spectra, a mode splitting into four lines is found[Fig. 4(d)]. The corresponding frequency response obtainedin simulations is shown in Fig. 5 for two different field magni-tudes of −
150 mT (red curve) and −
100 mT (orange curve) at75 ◦ : at a higher field magnitude, there is a high-intensity mode(red dot) and a very small-intensity mode (red triangle), whileat the lower field magnitude the same modes appear to havesplit into four separated peaks (indicated by an orange dot,star, cross, and triangle). In the micromagnetic spectra, themode splitting appears to be accompanied by a hybridization-like behavior indicated by avoided crossings . However, wecannot observe clear evidence of such a mode hybridizationfacilitated by inter-element coupling in the experiments – nei-ther in the BLS nor in FMR results.We conduct spatially-resolved micromagnetic simulationsand determine the exact location of the excitations in the lat-tice, see insets in Fig. 5. While not done here, we note thatmicro-focused BLS could potentially serve as an experimen-tal method to record the spatially-resolved dynamics . Be-fore the splitting occurs, at a magnetic field of −
150 mT(red curve), the main excitation mode, centered at 9.2 GHz(marked with a red dot), is located on the horizontal islandsand the edges of the vertical islands. Since the external fieldis applied at an angle of θ = ◦ , the mode in the horizon-tal islands is tilted, and the regions oscillating with a higheramplitude are pushed to the edge of the islands. A higher-order mode can also be observed at a higher frequency of11.15 GHz, but with a much smaller intensity (marked witha red triangle). When the magnetic field is reduced to − ACKNOWLEDGMENTS
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