^{1}, M. Krawczyk

^{1}, V. K. Sakharov

^{2,3}, Yu. V. Khivintsev

^{2,3}, Yu. A. Filimonov

^{2,3}and S. A. Nikitov

^{3,4}

### Abstract

The features of standing spin waves (SWs) excited during ferromagnetic resonance in three different one-dimensional magnonic crystals (MC) are intensively studied. The investigated magnonic crystals were: an array of air-spaced cobalt stripes, an array of air-spaced permalloy (Py) stripes, and a bi-component MC composed of alternating Co and Py stripes. All MC structures were made by etching technique from Co and Py thin films deposited onto Si substrates. Two configurations are considered with the in-plane external magnetic field applied parallel or perpendicular to the stripes. The supporting calculations are performed by the finite element method in the frequency domain. A number of intensive SW modes occurred in periodic structures under ferromagnetic resonance conditions as a consequence of standing spin waves excitation. These modes were analyzed theoretically in order to explain the origins of SW excitations. With the support of numerical calculations, we analyze also the possible scenarios for the occurrence of standing SWs in the investigated structures. It is demonstrated that the SW propagation length is an important factor conditioning the standing SW formation in MCs.

We acknowledge the financial support from the European Community's Seventh Framework Programme (FP7/2007-2013) under Grant Agreement No. 247556 (People) for NoWaPhen Project and under Grant Agreement No. 228673 for Magnonics Project.

This study was also supported by a Grant from the Government of the Russian Federation for the Support of Scientific Research in the Russian Universities under the Guidance of Leading Scientists, Project No. 11.G34.31.0030 and by RFBR Grant Nos. 12-07-31155 and 12-07-00203.

I. INTRODUCTION

II. THEORETICAL MODEL

III. SPIN WAVE PROFILES AND RELATIVE INTENSITIES OF THE ABSORPTION PEAKS

IV. EXPERIMENT

V. FORMATION OF STANDING SPIN WAVES

VI. RESULTS

A. DE spin waves geometry

B. BV spin waves geometry

VII. CONCLUSIONS

### Key Topics

- Monte Carlo methods
- 42.0
- Magnetic fields
- 39.0
- Spin waves
- 33.0
- Demagnetization
- 20.0
- Magnetic films
- 16.0

##### H01F10/00

## Figures

Magnonic crystals investigated in this study. Left: periodic lattice of 6.6 *μ*m wide Co stripes with a lattice constant of 10 *μ*m. Center: periodic lattice of 3.4 *μ*m wide Py stripes with a lattice constant of 10 *μ*m. Right: bi-component MC of alternate 3.4 *μ*m wide Py stripes and 6.6 *μ*m Co stripes in direct contact. Two geometries are considered: the magnetostatic surface wave geometry with the bias magnetic field oriented along the stripes (*z*-axis), and the backward volume magnetostatic wave geometry with the field oriented along the direction of periodicity (*x*-axis). In both geometries, the spin waves propagate along the direction of periodicity.

Magnonic crystals investigated in this study. Left: periodic lattice of 6.6 *μ*m wide Co stripes with a lattice constant of 10 *μ*m. Center: periodic lattice of 3.4 *μ*m wide Py stripes with a lattice constant of 10 *μ*m. Right: bi-component MC of alternate 3.4 *μ*m wide Py stripes and 6.6 *μ*m Co stripes in direct contact. Two geometries are considered: the magnetostatic surface wave geometry with the bias magnetic field oriented along the stripes (*z*-axis), and the backward volume magnetostatic wave geometry with the field oriented along the direction of periodicity (*x*-axis). In both geometries, the spin waves propagate along the direction of periodicity.

AFM image of the fabricated bi-component structure.

AFM image of the fabricated bi-component structure.

Resonance magnetic field versus wavevector in a 50 nm thick uniform film of Py (solid lines) and Co (dashed lines) at a frequency of 9.8 GHz. The blue lines on the left refer to the DE geometry, and the red lines on the right to the BV geometry. Points *i* and *ii* correspond to the wavevector values *k* indicated by vertical grid lines in Figs. 13(d)–13(f) .

Resonance magnetic field versus wavevector in a 50 nm thick uniform film of Py (solid lines) and Co (dashed lines) at a frequency of 9.8 GHz. The blue lines on the left refer to the DE geometry, and the red lines on the right to the BV geometry. Points *i* and *ii* correspond to the wavevector values *k* indicated by vertical grid lines in Figs. 13(d)–13(f) .

(a), (b) Group velocity and (c), (d) propagation length of uniform (*k* = 0) SWs in plain Py (solid lines) and Co (dashed lines) thin films of thickness *d* = 50 nm versus bias magnetic field in two geometries: (a), (c) DE and (b), (d) BV. Highlighted points correspond to uniform film excitations at a frequency of 9.8 GHz.

(a), (b) Group velocity and (c), (d) propagation length of uniform (*k* = 0) SWs in plain Py (solid lines) and Co (dashed lines) thin films of thickness *d* = 50 nm versus bias magnetic field in two geometries: (a), (c) DE and (b), (d) BV. Highlighted points correspond to uniform film excitations at a frequency of 9.8 GHz.

Dispersion relation, i.e., plot of frequency versus wavevector in the first Brillouin zone for the bi-component MC composed of alternate 3.4 *μ*m wide Py stripes and 6.6 *μ*m wide Co stripes (with a periodicity *a* = 10 *μ*m) in magnetic field . The field is oriented along the stripes; the spin waves propagate in the plane of the MC orthogonally to the bias field, i.e., in the DE geometry.

Dispersion relation, i.e., plot of frequency versus wavevector in the first Brillouin zone for the bi-component MC composed of alternate 3.4 *μ*m wide Py stripes and 6.6 *μ*m wide Co stripes (with a periodicity *a* = 10 *μ*m) in magnetic field . The field is oriented along the stripes; the spin waves propagate in the plane of the MC orthogonally to the bias field, i.e., in the DE geometry.

Profiles of two lowest-frequency SW modes in the bi-component MC in the DE geometry. (a) and (b) Color map of the amplitude of the *x* component *m _{x} * of the dynamic magnetization vector in the plane defined by the thickness and the periodicity direction. (c) and (d) The amplitude of

*m*in the unit cell along the

_{x}*x*-axis at the mid-thickness of the MC. The profiles are calculated for and

*k*= 0, i.e., in the center of the BZ in Fig. 5 .

Profiles of two lowest-frequency SW modes in the bi-component MC in the DE geometry. (a) and (b) Color map of the amplitude of the *x* component *m _{x} * of the dynamic magnetization vector in the plane defined by the thickness and the periodicity direction. (c) and (d) The amplitude of

*m*in the unit cell along the

_{x}*x*-axis at the mid-thickness of the MC. The profiles are calculated for and

*k*= 0, i.e., in the center of the BZ in Fig. 5 .

Spin wave frequency versus bias magnetic field for: (a) an array of Co stripes (6.6 *μ*m wide), (b) an array of Py stripes (3.4 *μ*m wide), and (c) a bi-component Co/Py MCs (with a period of 10 *μ*m) in the DE geometry for *k* = 0. All the three structures have equal thickness *d* = 50 nm and periodicity *a* = 10 *μ*m. The horizontal solid line marks the resonance frequency of 9.8 GHz. The colored vertical grid lines indicate the position of the most intensive modes, the profiles of which are plotted in Fig. 8 with the corresponding numbers and colors.

Spin wave frequency versus bias magnetic field for: (a) an array of Co stripes (6.6 *μ*m wide), (b) an array of Py stripes (3.4 *μ*m wide), and (c) a bi-component Co/Py MCs (with a period of 10 *μ*m) in the DE geometry for *k* = 0. All the three structures have equal thickness *d* = 50 nm and periodicity *a* = 10 *μ*m. The horizontal solid line marks the resonance frequency of 9.8 GHz. The colored vertical grid lines indicate the position of the most intensive modes, the profiles of which are plotted in Fig. 8 with the corresponding numbers and colors.

Results of numerical calculations and experimental data in the DE geometry for: (a), (d), and (g) the array of Co stripes; (b), (e), and (h) the array of Py stripes; (c), (f), and (i) the bi-component Co/Py MC. (a)-(c) Amplitudes of relative absorption calculated numerically from Eq. (14) . (d)-(f) FMR absorption spectra obtained by numerical integration of the experimental FMR signal (solid lines) and from calculations (dashed lines). (g)-(i) Experimental FMR signal (solid lines) and numerical curves (dashed lines). Colored vertical lines in (a)-(i) indicate the position of the most intensive modes, the profiles of which are plotted in Figs. 9(a)–9(d) with the corresponding labels (I–IV) and colors (black, red, and blue).

Results of numerical calculations and experimental data in the DE geometry for: (a), (d), and (g) the array of Co stripes; (b), (e), and (h) the array of Py stripes; (c), (f), and (i) the bi-component Co/Py MC. (a)-(c) Amplitudes of relative absorption calculated numerically from Eq. (14) . (d)-(f) FMR absorption spectra obtained by numerical integration of the experimental FMR signal (solid lines) and from calculations (dashed lines). (g)-(i) Experimental FMR signal (solid lines) and numerical curves (dashed lines). Colored vertical lines in (a)-(i) indicate the position of the most intensive modes, the profiles of which are plotted in Figs. 9(a)–9(d) with the corresponding labels (I–IV) and colors (black, red, and blue).

Amplitude of the *x* component of the magnetization vector for the most pronounced absorption peaks in: (a) the array of Co stripes; (b) the array of Py stripes; (c) and (d) the bi-component Co/Py MC. The position of the peaks is indicated by colored vertical lines in Figs. 8(a)–8(i) with corresponding labels (I–IV) and colors (black, red, and blue).

Amplitude of the *x* component of the magnetization vector for the most pronounced absorption peaks in: (a) the array of Co stripes; (b) the array of Py stripes; (c) and (d) the bi-component Co/Py MC. The position of the peaks is indicated by colored vertical lines in Figs. 8(a)–8(i) with corresponding labels (I–IV) and colors (black, red, and blue).

Numerical results and measurement data for the considered structures in the BV geometry: (a), (d), and (g) the array of Co stripes; (b), (e), and (h) the array of Py stripes; (c), (f), and (i) the bi-component Co/Py MC. (a)-(c) Amplitudes of relative absorption calculated numerically from Eq. (14) . (d)-(f) FMR absorption spectra (solid lines) obtained by numerical integration of the experimental FMR signal and the absorption spectra obtained from calculations (dashed lines). (g)-(i) Experimental FMR signal (solid lines) and numerical curves (dashed lines). Colored vertical lines in (a)-(i) indicate the position of the most intensive modes, the profiles of which are plotted in Figs. 11(a)–11(d) with corresponding labels (I–VI) and colors (black, red, and blue).

Numerical results and measurement data for the considered structures in the BV geometry: (a), (d), and (g) the array of Co stripes; (b), (e), and (h) the array of Py stripes; (c), (f), and (i) the bi-component Co/Py MC. (a)-(c) Amplitudes of relative absorption calculated numerically from Eq. (14) . (d)-(f) FMR absorption spectra (solid lines) obtained by numerical integration of the experimental FMR signal and the absorption spectra obtained from calculations (dashed lines). (g)-(i) Experimental FMR signal (solid lines) and numerical curves (dashed lines). Colored vertical lines in (a)-(i) indicate the position of the most intensive modes, the profiles of which are plotted in Figs. 11(a)–11(d) with corresponding labels (I–VI) and colors (black, red, and blue).

Amplitude of the *x* component of the dynamic magnetization vector for the most pronounced absorption peaks for: (a) the array of Co stripes; (b) the array of Py stripes; (c) and (d) the bi-component Co/Py MC. The position of the peaks is indicated in Figs. 10(a)–10(i) by colored vertical lines with corresponding labels (I–VI) and colors (black, red, and blue).

Amplitude of the *x* component of the dynamic magnetization vector for the most pronounced absorption peaks for: (a) the array of Co stripes; (b) the array of Py stripes; (c) and (d) the bi-component Co/Py MC. The position of the peaks is indicated in Figs. 10(a)–10(i) by colored vertical lines with corresponding labels (I–VI) and colors (black, red, and blue).

Internal magnetic field in (a) an array of 6.6 *μ*m wide Co stripes with a 3.4 *μ*m air spacing; (b) an array of 3.4 *μ*m wide Py stripes with a 6.6 *μ*m air spacing; (c) a bi-component MC composed of alternate 3.4 *μ*m wide Py stripes and 6.6 *μ*m wide Co stripes. All the stripes are 50 nm thick and are fully saturated along the *x*-axis. Calculations were performed for .

Internal magnetic field in (a) an array of 6.6 *μ*m wide Co stripes with a 3.4 *μ*m air spacing; (b) an array of 3.4 *μ*m wide Py stripes with a 6.6 *μ*m air spacing; (c) a bi-component MC composed of alternate 3.4 *μ*m wide Py stripes and 6.6 *μ*m wide Co stripes. All the stripes are 50 nm thick and are fully saturated along the *x*-axis. Calculations were performed for .

Profiles of the *x* component of the dynamic magnetization in different regions of the bi-component MC: (a) in Co stripes, without the demagnetizing field, (b) in Py stripes, without the demagnetizing field, (c) in Py stripes with the demagnetizing field taken into account. Plots (d)-(f) show the spectra obtained from the Fourier analysis of the respective profiles (a)-(c). Vertical grid lines indicate the value of the wavevector in plain films of Co or Py in the exchange region of the dispersion; see Fig. 3 , points *i* and *ii* for Co and Py, respectively. The corresponding frequency in the plain film is equal to the frequency of the uniform SW.

Profiles of the *x* component of the dynamic magnetization in different regions of the bi-component MC: (a) in Co stripes, without the demagnetizing field, (b) in Py stripes, without the demagnetizing field, (c) in Py stripes with the demagnetizing field taken into account. Plots (d)-(f) show the spectra obtained from the Fourier analysis of the respective profiles (a)-(c). Vertical grid lines indicate the value of the wavevector in plain films of Co or Py in the exchange region of the dispersion; see Fig. 3 , points *i* and *ii* for Co and Py, respectively. The corresponding frequency in the plain film is equal to the frequency of the uniform SW.

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