# Spin Wave

Spin Wave | |
---|---|

관련코스 | 현대광학 |

소분류 | 물리 |

선행 키워드 | |

연관 키워드 |

In ferromagnetic or antiferromagnetic material, magnetic ordering by electron spin state is observed. According to LLG equation, external field makes a precession of magnetization. With interaction between electron spin such as exchange interaction or magnetic dipole interaction, magnetization perpendicular to static magnetization can make excitation modes. Such low lying collective excitation in magnetically ordered substances is spin wave. In analogy to phonon, the quantized excitation of spin wave is also called magnon. Spin wave has lots of difficulties in observation and steering, but is expected to be important role in understanding magnetic film and spintronic signal processing.

## Contents

## Theory

*Main article: Spin wave theory*

The magnetization of ferromagnetic material can be described by LLG equation. Only considering exchange and neglecting the damping, anisotropy and others, the equation becomes:

- [math] {\operatorname{d}\!\mathbf{M}\over\operatorname{d}\!t}=\gamma\mu_{0}\mathbf{M}\times\mathbf{H}_{eff} [/math]

The dispersion relation can be found:

- [math] \omega=\sqrt{(\omega_{0}+\omega_{M}\lambda_{ex}k^{2})(\omega_{0}+\omega_{M}(\lambda_{ex}k^{2}+\sin^{2}\theta))} [/math]

## Measurement

### Magneto-Optical Kerr Effect (MOKE)

*Main article: MOKE*

MOKE is named after John Kerr who reported it in 1875. It is similar to Faraday rotation, but it describes changes in polarization of light when it is reflected at a magnetized material. In general, magnetic experiments using magnetic thin films are used and the films with substrates are generally opaque. Also, it is less damaging for the samples than Faraday rotation.

### Brillouin Light Scattering (BLS) Spectroscopy

*Main article: BLS*

Brillouin scattering is named after Leon Birllouin. Brillouin predicted inelastic scattering of light with thermal vibrations(phonons). Brillouin scattering is discovered by Leonid Mandelstam. Since BLS occurs in GHz scale, magnon also experiences Brillouin scattering with photon. BLS spectroscopy is similar to CARS in that both are light scattering spectroscopy, but Raman scattering occurs in THz scale.

## Spin Wave Optics

There have been some tries to handle spin wave in analogy to light. The "Spin wave optics" is not officially established name, but in some studies such trial used to coined as it.

### Snell's law in spin wave

When a wave propagates through a transition between two media which as different indices of refraction, the wave refracts at the interface. Snell's law describes such action of wave. Snell's law is written as:

- [math] k_{i}\sin(\theta_{i})=k_{t,r}\sin(\theta_{t,r}) [/math]

When thickness of medium considered and the angle between external field H and propagation direction is [math]\phi[/math], the dispersion relation is:

- [math] (\frac{\omega}{\mu_{0}\gamma})^{2}=(H+Jk^{2}+M-\frac{Mkd}{2})\times(H+Jk^{2}+\frac{Mkd}{2}\sin^{2}(\phi)) [/math]

When spin wave propagate through a thickness step it experiences dispersion difference owing to thickness. So at the thickness step the wave refracts or reflects. Since the dispersion is not [math]\omega=kv[/math], in case of linear medium in optics. There is no total reflection and because of angle [math]\phi[/math] in dispersion relation, incident angle is not same with reflection angle.

### Spin wave fiber

Snell's law in spin wave depending on the thickness of film and angle between the external field and propagation direction was observed, but in that case no total reflection is possible. Researchers predicted a total reflection analogy to generalized Snell's law in optics.

#### Generalized Snell's law

A phase discontinuity is defined as [math]\Phi(\mathbf{r})[/math] at the interface between two materials. Consider two beam refraction path, if both paths are infinitesimally close to real light path, then the point is local extreme point:

- [math] [k_{0}n_{i}\sin(\theta_{i})dx+(\Phi+d\Phi)]-[k_{0}n_{t}\sin(\theta_{t})dx+\Phi]=0 [/math]

The equation leads to the generalized Snell's law:

- [math] \sin(\theta_{t})n_{t}-\sin(\theta_{i})n_{i}=\frac{\lambda_{0}}{2\pi}\frac{d\Phi}{dx} [/math]

From the generalized Snell's law, the total reflection is found when [math]n_{t} \lt n_{i}[/math] as:

- [math] \theta_{c}=\arcsin(\pm\frac{n_{t}}{n_{i}}-\frac{\lambda_{0}}{2\pi}\frac{d\Phi}{dx}) [/math]

#### Generalized Snell's law for spin wave

Similarly in magnetism, as spin wave goes through the domain wall, the spin wave experiences a phase discontinuity owing to asymmetry of Dzyaloshinskii-Moriya interaction(DMI):

- [math] k_{g}\sin\alpha+\delta=k_{g}\sin\beta-\delta [/math]

As in case of light, total reflection critical angle is found:

- [math] \theta_{c}=\arcsin[1-\frac{2\delta}{k_{g}(\omega)}] [/math]

Using the total reflection, propagation dependent spin wave optical fiber is projected. In contrast to the case of light, magnetic properties of the media are dependent on the propagation of the spin wave. In the spin wave fiber, if the spin wave experiences total reflection, spin wave propagating through the opposite direction are on opposite case. The spin wave feels like going through the cladding which is covered with core, so it will refract out the fiber and vanish vary fast.

### Polarization and Retardation

Waves, like electromagnetic wave or lattice vibration, has polarization. To steer the polarization, polarizer and waveplate is used.

Spin wave in the ferromagnetic material can have only one polarization, right-handed. But spin wave in antiferromagnetic material can have two polarization both right-handed and left-handed or, x-direction and y-direction. LLG equation for antiferromagnetic material in domain wall is:

- [math] \dot{m_{i}}(\mathbf{r},t)=-{\gamma}m_{i}(\mathbf{r},t)\times\mathbf{H}_{i}^{eff}+{\alpha}m_{i}(\mathbf{r},t)\times\dot{m_{i}}(\mathbf{r},t) [/math]

where i=1, 2 denote the two sublattices, [math]\gamma[/math] is the gyromagnetic ratio, [math]\alpha[/math] is the Gilbert damping constant. [math]\gamma\mathbf{H}_{i}^{eff}[/math] is:

- [math]\gamma\mathbf{H}_{i}^{eff}=Km_{i}^{z}\hat{\mathbf{z}}+A\nabla^{2}\mathbf{m}_{i}+D\nabla\times\mathbf{m}_{i}-J\mathbf{m}_{\bar{i}}[/math]

Here, [math]\bar{1}=2[/math] and [math]\bar{2}=1[/math], and [math]J[/math] is the exchange coupling constant between two sublattices. K is the easy-axis anisotropy along z direction. The spin wave dynamics is obtained by following decoupled linearized LLG equations:

- [math] \dot{m}^{\phi}_{\mp}=-[-A\frac{\partial^{2}}{{\partial}x^{2}}+V_{K}(x)+J{\mp}J]m^{\theta}_{\pm} [/math]

- [math] \dot{m}^{\theta}_{\pm}=[-A\frac{\partial^{2}}{{\partial}x^{2}}+V_{K}(x)+J{\pm}J+V_D(x)]m^{\phi}_{\mp} [/math]

where [math]m^{\phi,\theta}_{\pm}=m^{\phi}_{1}{\pm}m^{\phi}_{2}[/math], [math]V_{K}[/math] arises due to the easy-axis anisotropy and [math]V_{D}[/math] is due to the DMI and the inhomogeneous magnetic texture. Using WKB approximation, spin wave dispersion is obtained:

- in-plane: [math]\omega^{IP}(k,x)=\omega_{\theta}(k,x)[/math]

- [math] =\sqrt{[2J+Ak^{2}+V_{K}(x)+V_D(x)][Ak^{2}+V_{K}(x)]}, [/math]

- out-of-plane: [math]\omega^{OP}(k,x)=\omega_{\phi}(k,x)[/math]

- [math] =\sqrt{[2J+Ak^{2}+V_{K}(x)][Ak^{2}+V_{K}(x)+V_D(x)]} [/math]

From the dispersion relation, angular frequency should be at least [math]\omega_{min}=\sqrt{(2J+K)K}[/math]. In other angular frequency region, domain wall responses differently to x and y spin wave polarization. This can be calculated using Green's function method. At low frequencies([math]\omega_G \lt \omega \lt \omega_D[/math]), y-polarization experiences little or zero reflection but x-polarization is strongly reflected. At high frequency([math]\omega \gt \omega_D[/math]), both polarized spin waves almost perfectly transmit, but they undergo different phase delay. Especially when the frequency is about 8GHz or 16GHz, the domain wall makes phase difference [math]\frac{3\pi}{2}[/math] or [math]\frac{\pi}{2}[/math] respectively, and the it acts as quarter-waveplate(retarder).

## References

- Handbook of Spintronics. (2015). Xu, Y., Awschalom, D. D., & Nitta, J., Springer.
- Solid state physics. (1976). Ashcroft, N. W., and Mermin, N. D., Cengage Learning.
- Spin waves. (2008). Stancil, D. D., & Prabhakar, A., Springer.
- Lan, J., Yu, W. & Xiao, J. Antiferromagnetic domain wall as spin wave polarizer and retarder. Nat Commun 8, 178 (2017).
- Stigloher, J., Decker, M., Körner H. S., Tanabe, K., Moryama, T., Taniguchi, T., Hata, H., Madami, M., Gubbiotti, G., Kobayashi, K., Ono, T., and Back, C. H., Snell's law for spin waves, Phys. Rev. Lett. 117, 037204(2017)
- Yu, W., Lan, J., Wu, R. & Xiao, J. Magnetic Snell’s law and spin-wave fiber with Dzyaloshinskii-Moriya interaction. Phys. Rev. B 94, 140410 (2016).

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