# Engineering the Exchange Spin Waves in Graded Thin Ferromagnetic Films

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{1−x}Fe

_{x}alloy (0.02 < x < 0.11) with different distributions of the magnetic properties across the thickness are presented. Films with linear and stepwise, as well as more complex Lorentzian, sine and cosine profiles of iron concentration in the alloy, and thicknesses from 20 to 400 nm are considered. A crucial influence of the magnetic properties profile on the spectrum of spin wave resonances is demonstrated. A capability of engineering the standing spin waves in graded ferromagnetic films for applications in magnonics is discussed.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of Samples and Research Methods

- Films with a fixed iron concentration gradient from 2 at.% to 10 at.%—linear, with thicknesses in the range of 50–400 nm;
- Bilayer and trilayer structures with different iron concentrations in each layer and a total thickness in the range of 20–220 nm. Each individual layer was epitaxial and uniform in composition. The layers were deposited on top of each other in a single run. Several concentrations of iron were chosen in the bilayer and trilayer samples: 4 at.% and 8 at.% in the bilayer and 2 at.% and 10 at.% in the trilayer ones;
- Films with non-linear composition distributions across the thickness—Lorentzian, sine and cosine, with thicknesses in the range of 200–400 nm.

^{−10}mbar) in the molecular beam epitaxy (MBE) chamber manufactured by SPECS GmbH (SPECS, Berlin, Germany). The iron concentration profile in the palladium matrix was realized by a controllable variation of the iron evaporation cell temperature with a fixed palladium cell temperature. The temperature variation cyclogram was loaded in the Eurotherm 3504 temperature controller of the iron effusion cell. Immediately after synthesis, all films were high-temperature annealed at 873 K under ultrahigh vacuum conditions for their structural optimization. A detailed description of the stages of film synthesis and methods for controlling their structural perfection and chemical composition, as well as their thickness, were described in detail in our previous works [20,21,22]. The magnetic characteristics of the films were studied by vibration sample magnetometry (VSM) with the Quantum Design (San Diego, California, USA) PPMS-9 system and by magnetic resonance with the Bruker ESP300 continuous wave X-band spectrometer in a wide field range of 0–1.4 T and a temperature range of 20–300 K [19,21,22].

#### 2.2. Simulations of Thermomagnetic Curves and Spin Waves

_{1−x}Fe

_{x}alloys, the Curie temperature T

_{C}and the saturation magnetization ${M}_{s}$ depend on the iron concentration [22]. Therefore, in Pd–Fe films with inhomogeneous iron distribution across the thickness, each sufficiently thin layer has its local ${T}_{C}(z)$ and ${M}_{s}(z)$. The profile of iron concentration in the film will manifest itself in the experimental dependence of the integral saturation magnetization on temperature $\overline{{M}_{s}}(T)$. Then the simulation of $\overline{{M}_{s}}(T)$ can be used as an indicator of whether the real iron concentration profile corresponds to its target shape. In the calculation of the dependence of saturation magnetization on temperature, we used the earlier established fact that the dependence ${M}_{s}(T/{T}_{C})/{M}_{s}(0)$ for all homogeneous epitaxial Pd

_{1−x}Fe

_{x}films (0 < x < 0.08) can be described by a general expression [23]:

_{1−x}Fe

_{x}[22]. Based on the predefined dependence of the iron concentration on thickness ${c}_{\mathrm{Fe}}(z)$, as well as the concentration dependence of the saturation magnetization ${M}_{s}({c}_{\mathrm{Fe}})$ and the Curie temperature ${T}_{C}({c}_{\mathrm{Fe}})$, and introducing ${M}_{s}(0,z)={M}_{s}(T=0K,{c}_{\mathrm{Fe}}(z))$, $\sigma (T,z)=\sigma (T,{T}_{C}({c}_{\mathrm{Fe}}(z)))$, the $\overline{{M}_{s}}(T)$ dependence for each sample can be calculated using the expression:

## 3. Results and Discussion

#### 3.1. Temperature Dependences

#### 3.2. SSW Resonance Spectra of Linear-Profile Samples Lin50–400

^{2}, ${\sigma}_{f}=170$ nm and ${\beta}_{k}=1.35$ that have not been varied for the rest of the samples. The value of ${\beta}_{k}>1$ indicates the hard direction along the normal to the film, which is verified by the shape of the magnetization curves in the out-of-plane geometry (not shown here, see Ref. [19]). A small value of ${\alpha}_{s}$ testifies insignificant pinning of the spins at the film surface. The inaccuracy of the x(z) profile and the film thickness measurement could be at the origin of the n = 3 mode discrepancy in Figure 3a.

#### 3.3. SSW in Bilayer Samples BiL20–200

#### 3.4. SSW in the Trilayer Structure and Films with Non-Linear Magnetization Profiles

## 4. Conclusions

^{2}, the surface pinning coefficient ${\alpha}_{s}=-\text{}0.05$, the skin layer depth ${\sigma}_{f}=170$ nm and the ratio of the effective magnetization to saturation magnetization ${M}_{eff}/{M}_{s}={\beta}_{}=1.35$. The obtained results indicate that the approach suggested and realized in this work can be applied to engineer the spin waves dispersion in graded ferromagnetic films.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Dependences of the saturation magnetization ${M}_{s}({c}_{\mathrm{Fe}})$ (left axis) and the Curie temperature (right axis) on iron concentration ${c}_{\mathrm{Fe}}$ in Pd–Fe films used in the calculations (lines) and corresponding experimental data for homogeneous thin films (symbols). (

**b**) A sketch for a simulation of the standing spin waves in a film.

**Figure 2.**Measured $\overline{{M}_{s}}(T)$ dependences (symbols) of the samples listed in Table 1 with the magnetic field of μ

_{0}H = 5 mT applied along the easy direction in the film plane (

**a**–

**e**). Lines—the calculated $\overline{{M}_{s}}(T)$ dependences following Equation (2); in the insets to panels (

**a**–

**e**), the iron concentration profiles are presented. (

**f**)—temperature evolution of the magnetic resonance spectra of sample Lin200 in the out-of-plane arrangement.

**Figure 3.**Measured (black lines) at T = 20 K and simulated (orange lines) SSW resonance spectra of the Lin50–400 samples (

**a**,

**c**,

**e**,

**g**). In the right-hand panels, the spatial distributions of the magnetization precession amplitude ${m}_{n}(z)$ are shown for each mode in its resonance field $(-{H}_{n}^{r})$ marked by the horizontal lines (

**b**,

**d**,

**f**,

**h**). The red dashed line is the potential well profile V(z).

**Figure 4.**Experimental (black lines) and simulated (orange lines) spectra of SSW resonances at T = 20 K of the bilayer BiL20–200 samples (

**a**,

**c**,

**e**,

**g**). In the right-hand panels, the spatial distribution of the magnetization precession amplitude ${m}_{n}(z)$ for each mode is displayed (

**b**,

**d**,

**f**,

**h**).

**Figure 5.**Experimental (black lines) and simulated (orange lines) SSW resonance spectra of the TriL220 (

**a**) and Lor400 (

**c**) samples at T = 20 K. In the right-hand panels, the spatial distribution of the magnetization precession amplitude ${m}_{n}(z)$ for each mode is shown (

**b**,

**d**); the dashed violet lines indicate the antisymmetric modes.

**Figure 6.**Experimental (black lines) and simulated (orange lines) SSW resonance spectra of the Sin200 (

**a**) and Cos200 (

**c**) samples at T = 20 K. In the right-hand panels, the spatial distribution of the magnetization precession amplitude ${m}_{n}(z)$ for each mode is shown (

**b**,

**d**).

**Figure 7.**Experimental resonance field difference between the 1st and n-th SSWs as a function of the mode index for TriL220, Lin400 and Lor400 (

**a**), and for Sin200 and Cos200 (

**b**), samples. The lines present the best fit results with $({H}_{1}^{r}-{H}_{n}^{r})\propto {n}^{p}$ function (

**a**).

Profile Type | Label | Thickness, nm | c_{1} | c_{2} |
---|---|---|---|---|

Linear | Lin50 | 53 | 3 ^{a} | 9 ^{b} |

Lin120 | 116 | 2.4 ^{a} | 9 ^{b} | |

Lin200 | 202 | 2 ^{a} | 9.6 ^{b} | |

Lin400 | 397 | 2 ^{a} | 10 ^{b} | |

Bilayer | BiL20 | 10/10 | 3.9 | 7.8 |

BiL40 | 20/20 | 4.1 | 8.1 | |

BiL60 | 30/30 | 3.7 | 7.9 | |

BiL200 | 100/100 | 4 | 8 | |

Trilayer | TriL220 | 10/200/10 | 2 ^{c} | 9.8 |

Lorentzian | Lor400 | 390/100 ^{d} | 2 ^{a} | 11 ^{b} |

Sine | Sin200 | 193 ^{e} | 2 ^{a} | 10 ^{b} |

Cosine | Cos200 | 190 ^{e} | 2 ^{a} | 10 ^{b} |

^{a}Minimal concentration.

^{b}Maximal concentration.

^{c}Concentration in the first and third layers.

^{d}Width at half-maximum.

^{e}The period of Sin and Cos fits the thickness of the film.

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**MDPI and ACS Style**

Yanilkin, I.; Gumarov, A.; Golovchanskiy, I.; Gabbasov, B.; Yusupov, R.; Tagirov, L.
Engineering the Exchange Spin Waves in Graded Thin Ferromagnetic Films. *Nanomaterials* **2022**, *12*, 4361.
https://doi.org/10.3390/nano12244361

**AMA Style**

Yanilkin I, Gumarov A, Golovchanskiy I, Gabbasov B, Yusupov R, Tagirov L.
Engineering the Exchange Spin Waves in Graded Thin Ferromagnetic Films. *Nanomaterials*. 2022; 12(24):4361.
https://doi.org/10.3390/nano12244361

**Chicago/Turabian Style**

Yanilkin, Igor, Amir Gumarov, Igor Golovchanskiy, Bulat Gabbasov, Roman Yusupov, and Lenar Tagirov.
2022. "Engineering the Exchange Spin Waves in Graded Thin Ferromagnetic Films" *Nanomaterials* 12, no. 24: 4361.
https://doi.org/10.3390/nano12244361