APPLIED PHYSICS LETTERS 93, 262501 共2008兲

Spin-wave confinement in rolled-up ferromagnetic tubes Stefan Mendach,a兲 Jan Podbielski, Jesco Topp, Wolfgang Hansen, and Detlef Heitmann Institut für Angewandte Physik und Mikrostrukturforschungszentrum, Universität Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany

共Received 12 November 2008; accepted 7 December 2008; published online 29 December 2008兲 We investigate the spin-wave dispersion in rolled-up Permalloy microtubes based on self-rolling strained semiconductor layers. Using microwave absorption spectroscopy we find that these structures exhibit a characteristic spin-wave mode spectrum. The magnetization and spin-wave resonance at zero external magnetic field is determined by curvature induced dynamic demagnetization fields. At high magnetic fields transverse to the tube axis, the three-dimensional shape anisotropy of the tube results in spin-wave confinement in well-defined regions along the tube perimeter. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3058764兴 The spin-wave mode spectrum of micro- and nanostructured ferromagnets is a field of ongoing research. Most commonly the subjects of investigation are either laterally patterned thin fims,1–6 sequences of thin ferromagnetic and nonferromagnetic layers, or both.7 Recently, atomic layer deposition was shown to be a powerful tool to create threedimensional ferromagnetic objects on the nanometer scale, e.g., ferromagnetic tubes.8 In this letter, we propose and demonstrate that strained semiconductor layers9,10 can be used as templates to transfer planar Permalloy films into three-dimensional rolled-up Permalloy tubes 共RUPTs兲 as sketched in Fig. 1共a兲. By means of broadband microwave spectroscopy supported by micromagnetic simulations, we investigate the characteristic spin-wave ជ 兲 of these RUPTs. In analogy to Permalloy dispersion f共H 11,12 we find that the dispersion splits into two branches rings ជ applied transverse to the tube axis. at high magnetic fields H Our simulations reveal that the splitting results from a characteristic magnetostatic configuration with two domain walls at the poles of the tube 关cf. Fig. 3共b兲兴, which leads to an azimuthal spin-wave confinement. This spin-wave confinement is induced solely by the three-dimensional shape anisotropy of the RUPTs and not connected to lithographically defined film edges as in planar structures.13 Furthermore, we find that the curvature of the Permalloy film induces dynamic demagnetization fields. These demagnetization fields determine the resonance frequency of the RUPTs as well as the magnetization configuration at zero external magnetic field. The RUPTs are prepared as follows. Initially a GaAs buffer layer 共100 nm兲 is grown on a GaAs substrate using molecular beam epitaxy, followed by an AlAs sacrificial layer 共40 nm兲, a strained In18Al20Ga62As layer 共5 nm兲, and an Al24Ga76As layer 共5 nm兲. Subsequently a Ni80Fe20 layer 共15 nm兲 is deposited on the semiconductor by thermal evaporation. The so-called starting edges are processed into the sample by scratching with sandpaper, as described in detail in Ref. 14. The strained In18Al20Ga62As/ Al24Ga76As/ Ni80Fe20 layer system is then released from the substrate at these starting edges by selectively etching away the AlAs sacrificial layer. Once released, it minimizes its strain energy by rolling up. Due to the statistical distribution a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected].

0003-6951/2008/93共26兲/262501/3/$23.00

of the scratched starting edges, we obtain RUPTs of various lengths and numbers of rotations. For the measurements described in this paper, we concentrate on a RUPT with a length of 40 ␮m and n = 1.3 rotations. In good agreement with continuum strain theory,15 the bending radius of the RUPT is rtube = 500 nm 关Fig. 1共a兲兴. The spin-wave eigenmode spectrum of the RUPT was determined by means of broadband microwave spectroscopy. For this purpose the RUPT was removed from the GaAs substrate and placed on the 2.4 ␮m wide signal line 共S兲 of a coplanar waveguide 共CPW兲 using a piezocontrolled manipulation needle 关Fig. 1共c兲兴. As shown in Fig. 1共c兲 the RUPT was aligned in the z direction parallel to the signal line 共S兲. ជ was applied in the plane of the An external magnetic field H CPW 共xz plane兲 with varying angle ␣ relative to the RUPT axis 关Fig. 1共d兲兴. A vector network analyzer was used to measure the microwave transmission through the CPW in depen-

FIG. 1. 共Color online兲 共a兲 Sketch of the RUPT shown in 共c兲 and 共d兲. We ជ in the xz plane and label E1/E2 and P1/P2 as the equators and assume H poles of the tube, respectively. 共b兲 Layer sequence of the tube walls. The strained semiconductor layers serve as rolling templates for the Permalloy layer. 共c兲 Top-view optical micrograph of the CPW consisting of a signal line 共S兲 and two grounds 共G兲. The manipulation needle is used to deposit the RUPT on the signal line. 共d兲 Zoom-in taken with a scanning electron microscope after manipulation. The outer rolling edge of the tube is indicated by ជ in the the black arrow. The orientation of the external magnetic field H xz-plane is described by the angle ␣. The coordinate systems of 共a兲 and 共d兲 refer to each other.

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© 2008 American Institute of Physics

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FIG. 3. 共Color online兲 关共a兲 and 共b兲兴 Sketch of the magnetization in the ជ parallel 共␣ = 0°兲 and tilted 共␣ ⬎ 0°兲 with respect to the tube axis. RUPT for H 共c兲 In a RUPT spin precession tangential to the curvature induces dynamic demagnetization fields. For ␣ = 0° these demagnetization fields can be accounted for by reducing the width of the wire used to model the RUPT.

FIG. 2. 共Color online兲 Spin-wave dispersion of the RUPT shown in Fig. 1 with external magnetic field parallel 共␣ = 0°兲 and tilted 共␣ = 75°兲 with respect to the tube axis. 关共a兲 and 共b兲兴 Experimental data obtained with microwave absorption spectroscopy. In 共a兲 all frequency sweeps are normalized to the same value. In 共b兲 every frequency sweep is normalized individually to its maximum absorption value for a better visibility 共blue, white, and red, respectively, mean low, medium, and high absorption兲. 关共c兲 and 共d兲兴 Corresponding simulated data obtained within the OOMMF framework. The RUPT is modeled with a wire of the width wsim = 2␲rtuben = 4.1 ␮m in a modulated magnetic field. The dotted curve in 共a兲 is calculated with Kittel’s formula 共Ref. 18兲 for a wire with a reduced width w = 1 ␮m ⬍ wsim. 共e兲 Calculated ជ 兩 = 74 mT兲 and mode B 共5.5 absolute amplitude of mode A 共7.4 GHz at 兩H ជ GHz at 兩H兩 = 74 mT兲 for ␣ = 75° plotted in dependence of the azimuthal position in the RUPT. Mode B is located at the poles. Mode A exhibits a more complex pattern with the highest amplitudes at the equators.

dence of the microwave frequency for varying external magnetic fields. The amplitude of the microwave induced rf magnetic field hជ rf was chosen smaller than 0.5 mT to avoid nonlinear effects.16 Resonant magnetic excitations in the RUPT result in a decreased microwave transmission due to absorption of power from the CPW at the corresponding exជ . The micitation frequency f and external magnetic field H crowave absorption spectra were normalized to reference ជ 兩 = 90 mT and ␣ = 90°. Prior to every spectra taken with 兩H frequency sweep the RUPT was magnetized along the rolling ជ 兩 = 90 mT and ␣ = 0°. For a detailed description axis with 兩H of our techniques, see Ref. 12. Figure 2 shows microwave absorption spectra of the RUPT for ␣ = 0° and ␣ = 75° measured as described above 关Figs. 2共a兲 and 2共b兲兴 together with corresponding micromagnetic simulations obtained using the OOMMF framework17 关Figs. 2共c兲 and 2共d兲兴. In the micromagnetic simulation the rolled-up Permalloy film in a homogeneous external magnetic field is modeled with a flat Permalloy wire in a sinusoidally modulated external magnetic field. The modulation runs along the width 共wsim = 2␲rtuben = 4.1 ␮m兲 of the wire. For ␣ = 0 the magnetization is oriented along the tube axis, i.e., along the z direction 关cf. Fig. 3共a兲兴. In this configuration we observe one pronounced dispersion branch in both

the simulated and the measured data with a good agreement at high magnetic fields. In the low field regime near zero, however, we find deviations between measurement and simulation. The measurements exhibit a stronger absorption than the simulations in the low field regime. Furthermore, the zero field resonance in the measurements f expt共0兲 = 2.85 GHz is considerably increased in comparison to the simulated value of f sim共0兲 = 1.4 GHz. We attribute this difference to the curvature of the RUPT, which is not included in our micromagnetic simulations. In a homogenously magnetized ferromagnet without any intrinsic anisotropy, the natural resonance frequency f共0兲 is determined by the dynamic demagnetization fields of the precession. In the case of a thin film the lack of such fields in the plane of the film results in f film共0兲 = 0. This situation changes if the film is bent, as sketched in Fig. 3共c兲, and the spin precession tangential to the curvature of the film induces dynamic demagnetization fields. This is most easily understood if large angle precessions are considered. For ␣ = 0 these additional demagnetization fields can be accounted for by decreasing the width of the wire used in the simulation, i.e., by increasing the dynamic demagnetization fields due to the lateral wire edges. In this simple configuration without modulation of the external magnetic field, Kittel’s formula18 for ferromagnetic resonances in a thin wire, f=

␥␮0 2␲

冑冋 冉 冊 册冋 冉 冊 册 H+ 1−

2t Ms ␲w

H+

2t Ms ␲w

共1兲

can be used to fit our data. M s is the saturation magnetization of Permalloy, t = 15 nm is the film thickness, and w is the width of the wire. We find a good agreement with the measurement for a wire with a reduced width of w = 1 ␮m. The corresponding calculated dispersion curve is indicated by the dots in Fig. 2共a兲. For ␣ ⬎ 0° the three-dimensional shape of the RUPT leads to the formation of two Néel-type domain walls at the poles of the tube 关cf. Fig. 3共b兲兴. This is an immediate consequence of the different magnitudes of the demagnetization field Hdemag created by magnetizing a thin film either in plane 共small Hdemag兲 or out of plane 共large Hdemag兲. In this configuration the high field magnetization pattern in the RUPT re-

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sembles onionlike states in thin-film rings, which are well understood.11,12,19 Like in rings the internal field varies along the circumference of the tube. In the equatorial regions, ជ and the external magnetic field H ជ are where magnetization M parallel, Hint is large. At the poles, where the shape anisoជ and H ជ , H is tropy prohibits a parallel alignment of M int smaller. This results in the occurrence of two prominent spinwave resonances. Mode A is localized in the equatorial region and exhibits a higher frequency because of the larger internal field. Mode B, which is localized around the poles, features a lower frequency because of the smaller internal field. As shown in Fig. 2共b兲 we indeed find a pronounced dispersion splitting for ␣ = 75° in the measurements. The ជ 兩 = 40 mT and splitting occurs above magnetic fields of 兩H ជ 兩 = 74 mT. Note that the full increases to ⌬f = 1.75 GHz at 兩H width at half maximum value of the absorption is comparable for ␣ = 75° and ␣ = 0° and the seemingly broadening of the dispersion in Fig. 2共b兲 compared to Fig. 2共a兲 is due to a different normalization of the spectra. In the corresponding simulations 关Fig. 2共d兲兴 the splitting of the dispersion is well reproduced. However, the excitation of mode A is much weaker in the simulations, and higher orders of mode B appear, which are not present in the measurements. We attribute these deviations to the different dynamic excitations used in the simulations and the experiment. In Fig. 2共e兲 we use the simulated data to plot the absolute amplitude of mode A and mode B in dependence of the azimuthal position in the RUPT. As expected from the above discussion we find that mode A is located at the equator of the RUPT and mode B is located at the poles of the RUPT. Furthermore, we find that mode A exhibits 共i兲 a more complex mode pattern than mode B and 共ii兲 a less pronounced confinement. In conclusion, we report on spin-wave excitation in three-dimensional rolled-up ferromagnetic tubes. We used an established self-rolling technique based on strained semiconductor layers to roll up Ni80Fe20 films into tubes with radii on the submicron scale. By combining microwave absorption experiments with numerical and analytical calculations, we demonstrate that these ferromagnetic microstructures with a rolled-up-carpet-like shape exhibit a characteristic spin-wave mode spectrum. At zero magnetic field the magnetization configuration and the resonance frequency are determined by dynamic demagnetization fields resulting from the curvature

of the rolled-up film. In a homogenous external magnetic field transverse to the tube axis, spin waves are confined at the equators and the poles. The fact that the spin-wave confinement is solely due to their shape anisotropy and not connected to lithographically defined edges makes them promising candidates for low-loss spin-wave guiding. We would like to thank Markus Bröll and Stephan Schwaiger for help during the tube micromanipulation process. Dirk Grundler, Bernhard Botters, Andreas Krohn, and Felix Balhorn are acknowledged for fruitful discussions. This work was supported by the DFG via Grant Nos. SFB 668, SFB 508, and GrK 1286. 1

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