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High-finesse optical nanocavities coupled with nanomechanical torque sensors have enabled highly sensitive and broadband readout of magnetic torques, from timescales involving quasi-static hysteresis response to radio-frequency magnetic susceptibility [1-3]. The extension of torque magnetometry to higher mechanical frequencies will grant further access to spin dynamics, including mechanical investigations of spin-lattice relaxation times.

For nanomechanical torque magnetometry measurements into radio frequencies, the contribution of the Einstein-de Haas (EdH) effect can become comparable to, and even exceed, the conventional magnetic torque (cross-product of magnetic moment and applied field) signal [4]. Extending sensitive optomechanical detection across a ladder of higher-order mechanical modes is a natural way to extract further information, through examination of the relative scaling of EdH and cross-product torques.

Sufficiently high-order mechanical modes have application to co-resonant detection of magnetic resonances. Magnetic vortex resonances intersecting the mechanical resonance spectrum will be described, and allow for the observation of dynamic vortex core interactions with magnetic inhomogeneities. 

[1] M. Wu et al. Nat. Nanotechnol. 12, 127 (2017).
[2] G. Hajisalem et al. New J. Phys. 21, 095005 (2019).
[3] J. Losby et al. J. Phys D. 51, 483001 (2018).
[4] K. Mori et al. Phys. Rev. B. 102, 054415 (2020).

CAP Virtual Congress 2021

Multi-frequency torque magnetometry: contribution of the Einstein-de Haas effect, and direct detection of overlapping magnetic and mechanical resonance modes

spin resonance

nanomechanics

torque magnetometry

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poster

Recently, there has been an increasing interest in the study of optical/microwave cavity-confined photon-magnon interaction due to their potential applications in quantum information processing
and spintronics [1,2]. In those studies, a number of unique features have been observed at frequencies in the GHz range. On the other hand, less is known about the dynamics of antiferromagnets, which are characterized by frequencies in the THz range [3].


We report results from an analytical study of the coupling of an antiferromagnetic domain wall to optical photons via the inverse Faraday effect. We consider spin canting and complex spin textures in antiferromagnetic materials that can arise from Dzyaloshinskii-Moriya interactions (DMI) [4]. DMI is not easily measurable and is often inferred from other quantities. In this work, we find that the presence of DMI enables spin interactions with cavity photons in a geometry which otherwise allows no magneto-optical coupling. This result may be used to measure the DMI constant in optomagnonic experiments by measuring the interaction of antiferromagnetic resonances to optical modes in a cavity.

**Keywords**: optomagnonic, DMI, antiferromagnet, domain wall, spin texture

*This work was supported by NSERC, CFI-JELF, Research Manitoba and the University of Manitoba, Canada.*

**References**
1. Harder, M. & Hu, C. M. Cavity Spintronics: An Early Review of Recent Progress in the Study of
Magnon–Photon Level Repulsion. in Solid State Physics - Advances in Research and Applications (eds. Camley, R. E. & Stamps, R. L.) vol. 69 47–121 (Academic Press, 2018).
2. Bozhko, D. A., Vasyuchka, V. I., Chumak, A. V & Serga, A. A. Magnon-phonon interactions in
magnon spintronics (Review article). Low Temp. Phys. 46, 383–399 (2020).
3. Kampfrath, T. et al. Coherent terahertz control of antiferromagnetic spin waves. Nat. Photonics 5, 31–34 (2011).
4. Conzelmann, T., Selzer, S. & Nowak, U. Domain walls in antiferromagnets: The effect of Dzyaloshinskii-Moriya interactions. J. Appl. Phys. 127, 223908 (2020).

Collective dynamics of antiferromagnetic domain walls in an optical cavity

Geometrical frustration arises in systems whose structures support multiple degenerate ground states. Artificial spin ice (ASI), has been built in diverse configurations which allows us to control frustration by experimentally tuning suitable parameters[1]. We present computational results on ASI systems consisting
of planar arrays of nanosized elongated ferromagnetic islands where geometrical frustration takes place at the vertices due to dipolar interactions between elements. To date, most studies have employed two-dimensional geometries in order to study magnetic charge dynamics, phase transitions, vertex-based frustration, and other interesting effects[2]. Recently, there has been a surge of interest in realizing three-dimensional artificial spin ice[3]. The use of a third dimension allows increased configurability and optimization of the magnetostatic interaction between different elements of the system. The work mentioned here concerns non-
equilibrium dynamics of a bilayer artificial spin ice structure consisting of two identical arrays separated by some distance and rotated at an angle between the two array axes. Introducing an angle between the layers enables the observation of novel, emergent phenomena as it changes the nature of the interactions between
the spins. We consider pinwheel variants of a square ice geometry for each
array[4]. Using different Monte Carlo techniques, we examine ground state ordering as a function of the interaction strength between layers for different relative orientations. Avalanche processes are also investigated and contrasted with those found in single-layer arrays.

References:
[1] RF Wang, C Nisoli, RS Freitas, J Li, W McConville, BJ Cooley, MS Lund, N Samarth, C Leighton, VH Crespi, et al. Artificial `spin ice' in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature, 439(7074):303{306, 2006.
[2] Cristiano Nisoli, Vassilios Kapaklis, and Peter Schier. Deliberate exotic magnetism via frustration and topology. Nature Physics, 13(3):200{203, 2017.
[3] Gia-Wei Chern, Charles Reichhardt, and Cristiano Nisoli. Realizing three-dimensional artificial spin ice by stacking planar nano-arrays. Applied Physics Letters, 104(1):013101, 2014.
[4] R. Mac^edo, G. M. Macauley, F. S. Nascimento, and Robert L. Stamps. Apparent ferromagnetism in the pinwheel artificial spin ice. Physical Review B,
98(1):014437, 2018.

Investigation of the dynamics of twisted bilayer artificial spin ice structure

Majorana zero modes (MZM) have been a focal point of the condensed matter physics
community in recent times due in part to their potential applications in quantum computation.
Most notably, the exotic exchange statistics of MZMs form the basis of
topologically protected quantum gates. The physical exchange, or braiding, of MZMs is often modeled in networks of 1D topological superconducting wires. A key ingredient of these braiding protocols is the transport of MZMs across a superconductor. We consider a “piano key model” of transport, where segments of a superconductor are adiabatically tuned between the topologically trivial and the topologically non-trivial phases via electric gates. We examine the time scales over which the piano key tuning should be performed in order to minimize the likelihood of unwanted excitations of the ground state. We also examine the dynamical phase accrued by the ground state, which is of importance in the context of braiding.

Transport of Majorana zero modes in 1D topological superconductors

The discovery of time-reversal symmetry(TRS) breaking in elemental Re has ignited fresh interest in the Re based superconductors. The recent studies show that role of Re concentration and crystal symmetry is crucial in understanding the unconventional superconductivity of these compounds. Therefore, we studied two Re-based superconductors Re$_3$B (<var>TC</var> = 5.19 K) and Re$_7$B$_3$ 
(<var>TC</var> = 3.2 K) having a centrosymmetric and non-centrosymmetric crystal structure, respectively. In this talk, we will present a comprehensive study on superconducting properties of Re$_7$B$_3$ and Re$_3$B through specific heat, magnetic susceptibility, resistivity, and transverse and zero-field muon rotation/relaxation ($\mu$SR) experiments. This will include the temperature dependence of penetration depth using transverse-field $\mu$SR measurements, which will be used to determine the symmetry of the superconducting gap. To further explore these materials, we performed the zero-field $\mu$SR measurements, which show the evidence for TRS preserving superconducting state for both materials in the limits of our measurements.

$\mu$SR study of superconducting Re-B compounds

The Kitaev model on the honeycomb lattice recently attracted considerable attention. However, in candidate materials, other interactions are present, and 
among them the off-diagonal $\Gamma$ interaction has been particularly challenging. While several numerical studies on a minimal Kitaev-$\Gamma$ $(K\Gamma)$ model found multiple quantum disordered phases, definite conclusions on existence of quantum spin liquids remain elusive due to various numerical limitations. Here we present a classical phase diagram of the $K\Gamma$ model in the presence of the bond-anisotropy, which reveals intriguing competing phases and extended frustration. The stability of the classical phases to quantum fluctuations using a linear spin wave theory will be also discussed.

Extent of Frustration in the Classical Kitaev-$\Gamma$ Model

Dynamical backaction resulting from radiation pressure forces in optomechanical systems has proven to be a versatile tool for manipulating mechanical vibrations. Notably, dynamical backaction has resulted in the cooling of a mechanical resonator to its ground-state, driving phonon lasing, and observing the optical-spring effect. In certain magnetic materials, mechanical vibrations can interact with magnetic excitations (magnons) via the magnetostrictive interaction, resulting in an analogous magnon-induced dynamical backaction. This talk will discuss the direct observation of magnon-induced dynamical backaction and its effect on a spherical magnetic sample's mechanical vibrations. Moreover, dynamical backaction effects play a crucial role in many recent theoretical proposals; thus, our work provides the foundation for future experimental work perusing many of these novel theoretical proposals. foundation for future experimental work perusing many of these novel theoretical proposals.

Dynamical Backaction Magnomechanics

By engineering an anti-parity-time (anti-PT) symmetric cavity magnonics system with precise eigenspace controllability, we observe two different singularities in the same system. One type of singularity, the exceptional point (EP), is produced by tuning the magnon damping. Between two EPs, the maximal coherent superposition of photon and magnon states is robustly sustained by the preserved anti-PT symmetry. The other type of singularity, arising from the dissipative coupling of two antiresonances, is a zero-damping condition (ZDC). At the settings of ZDCs, the coupled system exhibits infinite discontinuities in the group delay. We find that both singularities coexist at the equator of the Bloch sphere, which reveals a unique hybrid state that simultaneously exhibits the maximal coherent superposition and slow light capability.

Unconventional singularity in anti-parity-time symmetric cavity magnonics

The classical Einstein-de Haas (EdH) effect [1] is a AC mechanical torque arising from a time rate of change of net magnetization, and represents the intrinsic relationship between magnetism and mechanical angular momentum. Nanofabrication of torque sensing devices is opening new avenues for exploration and applications of the EdH effect. The scale-up of resonance frequencies with continued miniaturization of mechanical torque sensing resonators enhances EdH torques, which increase linearly with drive frequency, relative to frequency-independent magnetic cross-product torques. Previously, miniaturized EdH experiments have been performed at the microscale using mechanical modes in the audio frequency range (13 kHz) [2]. The present work brings the measurements to radio frequencies. 

Single-crystal Yttrium Iron Garnet (YIG) disks with magnetic vortex ground states are mounted on nano-scale mechanical resonators with a torsional resonance mode close to 3 MHz. For DC bias fields below about 200 A/m the maximum cross-product torques remain smaller than the EdH torques [3]; no control of the field geometry to null cross-product torques is required. Quadrature lock-in measurements allow cross-product and EdH torque signals to be recorded simultaneously, owing to a 90$^{\circ}$ relative phase shift arising between the driving torques, and referenced to the phase of the driving field. The simultaneous measurement scheme enables determination of the magnetomechanical ratio, $g\prime$, without requiring separate experimental inputs. Results from measurements around the full vortex hysteresis loop, and comparisons with micromagnetic simulations of cross-product and EdH torques, also will be discussed.

[1] A. Einstein and W.J. de Haas, Proceedings - KNAW **18**, 696 (1915).

[2] T.M. Wallis, J. Moreland, and P. Kabos, Appl. Phys. Lett. **89**, 122502 (2006)

[3] K. Mori, M.G. Dunsmore, J.E. Losby, D.M. Jenson, M. Belov, and M.R. Freeman, Phys. Rev. B **102**, 054415 (2020)

The Einstein-de Haas Effect in Yttrium Iron Garnet at 3 MHz

The ability to image magnetodynamics has proved key to the advancement of spintronics technology [1]. As technological size scales reduce and speeds increase, there is a need to provide commensurate advancement in experimental tools to image magnetodynamics down to the atomic scale with ultrafast time resolution. In pursuit of this goal, we are developing a custom designed scanning tunneling microscope which will be paired with a THz light source to create a THz-STM that achieves ultrafast time resolution [2]. THz-STM experiments require significant acquisition time and present an extreme challenge. To make such an instrument practical, the STM must be designed with optical access, long-term stability, and rigidity in mind. Progress on the design and construction of a variable temperature scanning tunneling microscope purpose built for the observation of magnetodynamics will be presented. 

1 https://science.sciencemag.org/content/294/5546/1484
2 https://www.nature.com/articles/nphoton.2013.151

Imaging Magnetic Dynamics on the Atomic Scale

Since the magnetic skyrmion is a topologically protected particle-like domain in the ferromagnetic film, it can survive under thermal agitations and shows Brownian motion. The skyrmion system can be an ideal platform to design unconventional computers like stochastic/Brownian computers. Therefore, investigation of the Brownian motion of the magnetic skyrmion is now the subject of scientific and technical interest [1,2].
In this talk, experimental proof of a non-vanishing x-velocity to y-coordinate correlation will be shown. The result is explained by the Thiele equation including a stochastic field [3,4]. Description of the phenomena introduces the gyro-diffusion constant [5] that is special for the diffusion of the chiral particles like skyrmions. The existence of the x-velocity to y-coordinate correlation also evidences the perpetual rotation motion of the skyrmion. Theory shows that the result does not conflict with the 2nd law of thermodynamics and the van Leeuwen theorem.
Acknowledgments 
The authors acknowledge Mr. Y. Jibiki of Osaka University, Dr. S. Auffret, Dr. C. Baraduc, and Dr. Hélèn Béa of Spintech, France for their initial contributions. The authors also acknowledge Dr. M. Oogane and Dr. K. Saito. This research and development work was supported by the ULVAC, Inc., the Ministry of Internal Affairs and Communications, Basic research S (20H05666) of JSPS, and CREST (Non-classical Spin project) of JST.
References
[1] Y. Jibiki et al., Appl. Phys. Lett. 117, 082402 (2020).
[2] L. Zhao et al., Phys. Rev. Lett. 125, 027206 (2020).
[3] A. Thiele et al., Phys. Rev. Lett. 30, 230 (1973).
[4] C. Schuette et al., Phys. Rev. B 90, 174434 (2014)
[5] E. Tamura et al. https://arxiv.org/abs/2005.04860.

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Multi-frequency torque magnetometry: contribution of the Einstein-de Haas effect, and direct detection of overlapping magnetic and mechanical resonance modes

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Collective dynamics of antiferromagnetic domain walls in an optical cavity

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