Emergent magnonic singularities in anti parity-time symmetric synthetic antiferromagnets

Antiferromagnetic (AFM) materials with zero net magnetization offer an exciting platform for information handling that is robust against magnetic field perturbations and magnetic crosstalk between neighboring devices [1]. Here, we focus on synthetic antiferromagnetic (SyAFM) structures which are fabricated out of two ferromagnetic (FM) layers and are coupled via Ruderman–Kittel–Kasuya–Yosid (RKKY) interaction. SyAFMs allow for electrical manipulation of AFM ordering and are convenient to investigate using well-developed techniques for ferromagnets [2]. Recent research on their collective excitations, the spin waves (with magnons as quanta of excitations) [3–7], demonstrated their potential for logic gates, information processing, and sensory devices with low power operation [8–10]. Unlike FM systems with only right-handed spin wave modes, AFMs have two degenerate spin wave eigenmodes with opposite circular polarization, referred to as right- and left-handed magnons depending on the precessional handiness of the Néel vector [3, 11, 12] (cf figure 1). Any polarization state can be produced by a combination of these two eigenmodes, providing a way to encode information based on the spin wave polarization as well as on the amplitude and phase. Adverse however is that, the two-fold degeneracy is protected by the combined symmetry of time-reversal and sub-lattice exchange, and we have thus to seek controlled interactions that break either or both of the two symmetries. A way to achieve that is presented in this study.

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The magnetic dynamic is generically non-unitary due to the ubiquitous magnetic damping. Hence, the Hamiltonian governing the low energy (spin wave) modes can be intrinsically non-Hermitian, and may become defective at special conditions (exceptional points, EPs) depending on the external parameters. At these (degeneracy) points the number of eigenvalues (algebraic multiplicity) exceeds the number of eigenvectors (geometric multiplicity). For FM structure, various facets of how EPs affect the magnetic dynamics were discussed. We are concerned with SyAFMs. Such (interacting) systems are suitable for studying anti-parity-time (anti-$\mathcal{PT}$) symmetry behavior (in contrast, FM structures with balanced gain and loss magnetic damping are related to $\mathcal{PT}$ symmetry [13–30]). Different interlayer coupling (RKKY) strengths J correspond to different free-energy densities of the SyAFMs. At a certain value of the interaction strength J = J_c a degeneracy (or EP) in the relevant low-energy modes occur and the modes undergo a symmetry change: for J < J_c the anti-$\mathcal{PT}$ symmetry is broken (APTB phase), while for J > J_c the anti-$\mathcal{PT}$ symmetry (APT) is preserved. The degree of degeneracy is related to the order of the EP, which in turn can be tuned by structural design [31]. The value of J_c depends on other determining parameters of the free energy density such as the magnetic anisotropy and exchange energy as well as on magnetic damping and is given explicitly in the next section. In the APT phase the system has an imaginary spectrum. In the APTB phase the spectrum is complex [32–40].

The anti-$\mathcal{PT}$ phase transition point J_c can be reached by scanning J or the other parameters in the free energy density. Experimentally, electric tuning of J were realized in SyAFM structures, e.g. FeCoB/Ru/FeCoB and (Pt/Co)₂/Ru/(Pt/Co)₂ [41, 42], in which not only the amplitude but also the sign of RKKY interaction can be tuned by changing gate voltages. Thus, when we discuss below properties with varying J we implicitly mean that there is some external fields (such as gate voltage) which are varied affecting the respective changes in J (and hence the free energy density). Furthermore, the magneto-electric interactions upon interfacing an FM layer with a ferroelectrics allows for an electrical control of the magnetization damping [43–45] and the magnetic anisotropy [46, 47].

Considering SyAFMs with an interfacial Dzyaloshinskii–Moriya interaction (DMI), we find that a magnonic bound state in the continuum (BIC) is formed. This hybrid BIC cannot radiate away and consists of a maximally coherent superposition of modes of the two FM layer. The above statement are inferred analytically from a linearized model and confirmed with full numerical micromagnetic simulations.

Anti-$\mathcal{PT}$ symmetric SyAFM dynamics. The magnetic dynamics in a SyAFM is describable by coupled Landau–Lifshitz–Gilbert (LLG) equations [48]. For two coupled layers we write

$$\begin{equation}{\partial }_{t}{m}_{n}=-\frac{{\gamma }_{n}}{1+{\alpha }_{n}^{2}}{\mathbf{m}}_{n}\times \left[{\mathbf{H}}_{n}^{\text{eff}}+{\alpha }_{n}({\mathbf{m}}_{n}\times {\mathbf{H}}_{n}^{\text{eff}})\right].\end{equation} \tag{ 1 }$$

Here m_n (with $n=1,\bar{1}$) denote the magnetization direction for two FM sublayers, γ_n is the gyromagnetic ratio, and α_n is the Gilbert damping constant. The effective magnetic field acting locally on m_n reads

$$\begin{equation*}{\mathbf{H}}_{n}^{\text{eff}}=2({A}_{n}{\nabla }^{2}{\mathbf{m}}_{n}+{K}_{z}{m}_{n}^{z}{\hat{e}}_{z}-J{\mathbf{m}}_{\bar{n}}),\end{equation*}$$

where A_n is the Heisenberg exchange coupling constant within each m_n -layers, K_z is a magnetic anisotropy along the easy z-axis, and J is the RKKY interaction between the two sublayers.

When J = 0, the two layers are decoupled; each magnetization m_n prefers locally right-handed precessions around its own easy-direction, as sketched in figure 1(a).

A positive RKKY interaction (J > 0) tends to align the layers magnetization antiparallel, enforcing m₁ and ${\mathbf{m}}_{\bar{1}}$ to process in the same global manner, i.e. right- or left-handed precessions around the Néel vector $\mathbf{n}=({\mathbf{m}}_{1}-{\mathbf{m}}_{\bar{1}})/2$ (cf figure 1(b)). To show that the low-energy magnetic dynamics is anti-$\mathcal{PT}$ symmetric, we linearize the LLG equations by writing ${\mathbf{m}}_{n}={m}_{n}^{0}{\hat{\mathbf{e}}}_{z}+\delta {\mathbf{m}}_{n}$ with ${m}_{n}^{0}\approx \pm 1$ and |δ m_n | ≪ 1 and obtain the Schrödinger-type equation for the dynamical excitations of δ m_n

$$\begin{equation}\text{i}{\partial }_{t}{\Psi}(\mathbf{r},t)=\mathcal{H}{\Psi}(\mathbf{r},t)\end{equation} \tag{ 2 }$$

where ${\Psi}(\mathbf{r},t)={(\delta {m}_{1}^{x}-\text{i}\delta {m}_{1}^{y},\delta {m}_{\bar{1}}^{x}-\text{i}\delta {m}_{\bar{1}}^{y})}^{\mathrm{T}}$ and

$$\begin{equation}\mathcal{H}=\left[\begin{matrix}\hfill {H}_{1}({\tilde{\gamma }}_{1}-\text{i}{\tilde{\alpha }}_{1})\hfill & \hfill J({\tilde{\gamma }}_{1}-\text{i}{\tilde{\alpha }}_{1})\hfill \\ \hfill -J({\tilde{\gamma }}_{\bar{1}}+\text{i}{\tilde{\alpha }}_{\bar{1}})\hfill & \hfill -{H}_{\bar{1}}({\tilde{\gamma }}_{\bar{1}}+\text{i}{\tilde{\alpha }}_{\bar{1}})\hfill \end{matrix}\right]\end{equation} \tag{ 3 }$$

with renormalized ${\tilde{\gamma }}_{n}=2{\gamma }_{n}/(1+{\alpha }_{n}^{2})$ and ${\tilde{\alpha }}_{n}={\alpha }_{n}{\tilde{\gamma }}_{n}$. For J = 0, the magnetization dynamics of uncoupled layers are governed by their own Hamiltonian H_n = −A_n ∇² + K_z + J.

For SyAFMs with ${A}_{1}={A}_{\bar{1}}=A$, ${\gamma }_{1}={\gamma }_{\bar{1}}=\gamma$, and the same damping rates ${\alpha }_{1}={\alpha }_{\bar{1}}=\alpha$, $\mathcal{H}$ reduces to

$$\begin{equation}{\mathcal{H}}^{\prime }=\left[\begin{matrix}\hfill {H}_{k}(1-\text{i}\alpha )\hfill & \hfill J(1-\text{i}\alpha )\hfill \\ \hfill -J(1+\text{i}\alpha )\hfill & \hfill -{H}_{k}(1+\text{i}\alpha )\hfill \end{matrix}\right]\end{equation} \tag{ 4 }$$

where H_k = A k² + K_z + J in the long-wavelength limit of a plane-wave ansatz Ψ ∼e^{i(k⋅r−ωt)}, where k stands for the wave number of propagating spin waves [49].

Symmetry considerations. The eigenvalues of ${\mathcal{H}}^{\prime }$ are

$$\begin{equation}{\omega }_{\pm }={H}_{k}\left(-\text{i}\alpha \pm \sqrt{1-{\xi }_{k}^{2}}\right),\end{equation} \tag{ 5 }$$

where ${\xi }_{k}=J\sqrt{1+{\alpha }^{2}}/{H}_{k}$.

• (a)  
  For purely imaginary ω_± (when |ξ_k | > 1) the eigenvectors read

  $$\begin{equation}{{\Psi}}_{\mathrm{A}\mathrm{P}\mathrm{T}}^{\pm }=\left[\begin{matrix}\hfill \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\enspace \varphi \pm \text{i}\enspace \mathrm{tanh}\enspace \varphi \hfill \\ \hfill -{\text{e}}^{\text{i}\eta }\hfill \end{matrix}\right],\end{equation} \tag{ 6 }$$

  where $\mathrm{tanh}\enspace \varphi =\sqrt{1-1/{\xi }_{k}^{2}}$ and tan η = α. A parity operation $\hat{\mathcal{P}}$ corresponds to applying the Pauli operator ${\sigma }_{x}=\left[\begin{matrix}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{matrix}\right]$. The time-reversal $\hat{\mathcal{T}}$ corresponds to i → −i, t → −t and r → r. The states, given by equation (6), ${{\Psi}}_{\mathrm{A}\mathrm{P}\mathrm{T}}^{\pm }$ are also eigenvectors of the $\hat{\mathcal{P}}\hat{\mathcal{T}}$ operator [50], satisfying $\hat{\mathcal{P}}\hat{\mathcal{T}}{{\Psi}}_{\mathrm{A}\mathrm{P}\mathrm{T}}^{\pm }={\lambda }_{\pm }{{\Psi}}_{\mathrm{A}\mathrm{P}\mathrm{T}}^{\pm }$ with |λ_±|² = 1. The anticommutator $\left\{{\mathcal{H}}^{\prime },\hat{\mathcal{P}}\hat{\mathcal{T}}\right\}$ vanishes. Thus, ${\mathcal{H}}^{\prime }$ of a coupled SyAFM possesses an anti-$\mathcal{PT}$ symmetry. This symmetry is preserved till J reaches

  $$\begin{equation}{J}_{\text{c}}=\vert {H}_{k}/\sqrt{1+{\alpha }^{2}}\vert ,\end{equation} \tag{ 7 }$$

  where the EP occurs at $J={H}_{k}/\sqrt{1+{\alpha }^{2}}$ (cf figure 1(c)). Physically this means, the dynamic motions of m₁ and ${\mathbf{m}}_{\bar{1}}$ in the two magnetic layers are equator modes, sharing the same amplitude. No net dynamic magnetization is generated.
• (b)  
  For smaller J < J_c the system enters anti-$\mathcal{PT}$ symmetry-broken (APTB) phase with complex eigenvalues (|ξ_k | < 1). The corresponding eigenvectors [51] are given by

  $$\begin{equation}{{\Psi}}_{\mathrm{A}\mathrm{P}\mathrm{T}\mathrm{B}}^{+}=\left[\begin{matrix}\hfill \mathrm{cosh}\enspace \frac{\phi }{2}\hfill \\ \hfill -\mathrm{sinh}\enspace \frac{\phi }{2}\enspace {\text{e}}^{\text{i}\eta }\hfill \end{matrix}\right],\qquad {{\Psi}}_{\mathrm{A}\mathrm{P}\mathrm{T}\mathrm{B}}^{-}=\left[\begin{matrix}\hfill -\mathrm{sinh}\enspace \frac{\phi }{2}\hfill \\ \hfill \mathrm{cosh}\enspace \frac{\phi }{2}\enspace {\text{e}}^{\text{i}\eta }\hfill \end{matrix}\right]\end{equation} \tag{ 8 }$$

  where tanh ϕ = ξ_k . Considering ${\mathrm{cosh}}^{2}\enspace \frac{\phi }{2} > {\mathrm{sinh}}^{2}\enspace \frac{\phi }{2}$, one infers that m₁ and ${\mathbf{m}}_{\bar{1}}$ precesses with different cone angles in two FM sublayers. The left- and right-circularity polarized modes ${{\Psi}}_{\mathrm{A}\mathrm{P}\mathrm{T}\mathrm{B}}^{\pm }$ around the Néel vector n are dominated by the precession in the upper and lower sublayers. The precessional dynamics is accompanied by the emergence of a small intrinsic magnetization $\mathbf{m}=({\mathbf{m}}_{1}+{\mathbf{m}}_{\bar{1}})/2$ whose dynamics is governed by the temporal evolution of the Néel order, m ∝ (n × ∂_t n) [52]. Thus, the emergent magnetization in collinear SyAFM systems signals a breaking of anti-$\mathcal{PT}$ symmetry.

Generically, |ξ_k | < 1 and the anti-$\mathcal{PT}$ symmetry is broken. For isotropic SyAFMs (no magnetic anisotropy K_z = 0) and for a frequency at/below the ferromagnetic resonance (FMR) point H_k=0 = J, no spin waves are excited, leading to ${\xi }_{k}=\sqrt{1+{\alpha }^{2}}$ and the eigenvalues ω₊ = 0 and ω₋ = −2iJα. This APT phase persists in the presence of a small anisotropy, as long as ${K}_{z}/J< {\alpha }^{2}/(1+\sqrt{1+{\alpha }^{2}})$. Both eigenvalues become imaginary and their absolute values depend monotonically on K_z : |ω₋| (|ω₊|) approaches increasingly (decreasingly) the Gilbert damping constant α, as K_z increases. No precessional motions are found even at FMR point in the micromagnetic simulations and m_n decays with time exponentially to the easy-axis.

Generally, one may argue that due to the uniaxial magnetic anisotropy, the SU(2) spin rotational symmetry is reduced to SO(2)$\times {\mathbb{Z}}_{2}$. The low-lying excitations (Goldstone modes) are associated with the continuous SO(2) symmetry. Above the energy gap which is determined by ${\mathbb{Z}}_{2}$ symmetry, the excitation energy of the Goldstone mode should vanish when k → Q, where Q is the wavevector of magnetic ordering, for example, Q = 0 for the SyAFMs. This is indeed confirmed by figures 2(a) and (b)) (for α = 0 or J = 0). However, as the Gilbert damping constant α and/or RKKY interaction J increase in a way that $\sqrt{1+{\alpha }^{2}}-1 > {K}_{z}/J$ is satisfied, an EP emerges and the APT phase is spread out over the k-space. As a result, the real eigenfrequency of spin waves reaches its minimum value (zero) far away from the Q-point, as shown by the results in figure 2 which follow from micromagnetic simulations using the OOMMF package. A one-dimensional SyAFM system consisting of two 1000 × 1 × 1 nm³ FM layers is considered with a mesh size of 1 × 1 × 1 nm³. The ground state equilibrium antiparallel magnetization is oriented along the z-axis. To launch the magnetization excitations, we apply the rf magnetic fields, ${\tilde{h}}_{x}(t)={h}_{0}\enspace \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}({\omega }_{\text{c}}t){\hat{\mathbf{e}}}_{x}$ with the amplitude h₀ = 10 mT and the cut-off frequency ω_c = 60 GHz. The dispersion relation given by the fast Fourier transform (FFT) clearly indicates that not all momenta k can be excited into spin waves as long as ξ_k > 1 is fulfilled within a certain k-range, where the system is in the APT phase and the eigenfrequencies are pure imaginary. It is clear that such low-energy magnetization excitations are beyond the traditional Goldstone modes and can be realized by adjusting not only the Gilbert damping α but also the RKKY interactions J (cf the right column of figure 2). These full numerical simulations endorse the theoretical analysis as well as the experimental feasibility and relevance of the anti-$\mathcal{PT}$ symmetry breaking in SyAFM systems.

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BIC controlled by DMI. Besides EP, BIC is a (standing) wave that remains perfectly confined without any radiation even though it falls within the continuum spectrum. BICs were originally proposed by von Neumann and Wigner for electronic systems [53] and have been observed experimentally in electromagnetic, acoustic, water and elastic waves [54–63]. The unique properties of BICs have led to numerous applications, including slow light, sensors, filters, and quantum memory. Here, we show that magnonic BICs can be generated by introducing the asymmetric DMI. This new kind of singularity maybe useful for SyAFM magnonics.

For systems hosting interfacial DMI, we incorporate in the theory the interaction

$$\begin{equation}{\mathcal{H}}_{\mathrm{D}\mathrm{M}\mathrm{I}}=-D{\mathbf{m}}_{n}\cdot (\tilde{\nabla }\times {\mathbf{m}}_{n}),\end{equation} \tag{ 9 }$$

where $\tilde{\nabla }={\hat{\mathbf{e}}}_{n}\times \nabla$ with ${\hat{\mathbf{e}}}_{n}$ being the normal direction of DMI surfaces. For the one-dimensional SyAFM along the x-axis, the DMI contribute to the energy density with H_n = −A ∇² + K_z + J − iD_n ∇_x . Note that ${\hat{\mathbf{e}}}_{n}$ is the normal direction of the upper or the lower surface of each FM sublayers, depending on where the DMI is dominantly generated. For the sake of discussion, we have assigned an effective value D_n to each magnetic layers. Two distinct magnonic BICs are then found in SyAFMs depending on whether they are anti-$\mathcal{PT}$ symmetric or not:

• (a)  
  Asymmetric SyAFMs with ${D}_{1}=-{D}_{\bar{1}}=D$. The Hamiltonian ${\mathcal{H}}^{\prime }$ has no $\mathcal{PT}$ or anti-$\mathcal{PT}$ symmetry. The dispersion relation is asymmetric giving rise to nonreciprocal spin waves. The intrinsic Gilbert damping is balanced by the introduced DMIs, resulting in ω_± = 0, Friedrich–Wintgen (FW) [64] BICs (which occur in the vicinity of avoided crossing of two dispersion curves) appear as a pair, as shown in figure 3.
• (b)  
  Anti-$\mathcal{PT}$ symmetric SyAFMs with ${D}_{1}={D}_{\bar{1}}=D$. ${\mathcal{H}}^{\prime }$ now still satisfies $\left\{{\mathcal{H}}^{\prime },\hat{\mathcal{P}}\hat{\mathcal{T}}\right\}=0$. In the APT phase, near the EPs, FW BICs emerge at ${k}_{\mathrm{B}\mathrm{I}\mathrm{C}}=(-D\pm \sqrt{{D}^{2}-4A{K}_{z}})/2A$ provided that the uniform Néel-type AFM ordering is preserved in a narrow window beyond D² ⩾ 4AK_z for the geometrically constrained system discussed here. Unlike the above BICs in asymmetric SyAFMs, these anti-$\mathcal{PT}$ symmetry-protected BICs are equator modes that stem from the maximal coherent superposition of magnetization excitations with 50–50 contributions from the upper and lower FM layers, as demonstrated by equation (6). They are further confirmed by the dynamic magnetic susceptibility (χ(k, ω)) or dynamics spin correlations $({\mathcal{S}}_{n}(\mathbf{k},\omega )=\langle {S}_{n}^{-}(\mathbf{k}){S}_{n}^{+}{(-\mathbf{k})\rangle }_{\omega })$. Based on the linear response theory with ${S}_{1}^{-}=\delta {m}_{1}^{x}-\text{i}\delta {m}_{1}^{y}$ and ${S}_{1}^{+}=\delta {m}_{1}^{x}+\text{i}\delta {m}_{1}^{y}$, one has

  $$\begin{equation}\chi (\mathbf{k},\omega )\propto {\mathcal{S}}_{1}(\mathbf{k},\omega )=\frac{\omega +{H}_{n}(1+\text{i}\alpha )}{(\omega -{\omega }_{+})(\omega -{\omega }_{-})}.\end{equation} \tag{ 10 }$$

  As shown in figure 4, the imaginary part of the susceptibility $\mathfrak{I}\chi (\mathbf{k},\omega )$ diverges at the wavevector k_BIC, implying an infinite lifetime and a zero leakage rate of BICs. These unique hybridized modes are robust against the detuning parameters in the APT phase.

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As a typical dissipative system, the low-energy magnetization excitations in the balanced SyAFMs are shown to be eigenstates of the anti-$\mathcal{PT}$ symmetry operator for certain intrinsic parameters in the free energy density. Such parameters can be tuned by external probes or by material engineering, and the system can be driven to a phase with the low-energy modes are no longer eigenstates of the anti-$\mathcal{PT}$ operator. The phase transition between the symmetry-preserved and the symmetry-broken APT phase is accompanied with the emergence of a finite magnetization and bound states in the continuum with maximally coherent superposition for spin waves. These tunable singularities can be relevant to transport of magnons and imply numerous cross applications, such as magnon entanglement, slow spin waves, and magnonic quantum-information devices.

This work is supported by the National Natural Science Foundation of China (Nos. 12174164, 91963201, and 11834005), the German Research Foundation (SFB TRR 227), and the 111 Project under Grant No. B2006.