Cross-sublattice Spin Pumping and Magnon Level Attraction in van der Waals Antiferromagnets

Introduction.– The dawn of magnetic van der Waals (vdW) materials has renewed and invigorated interest in low-dimensional phenomena hosted by solid state systems Burch et al. (2018); Gibertini et al. (2019). These layered vdWs materials have been found to host various forms of magnetic order Huang et al. (2017); Deng et al. (2018); Huang et al. (2018); Song and Gabor (2018); Klein et al. (2018); Lado and Fernández-Rossier (2017) with antiferromagnets (AFs) taking a special place due to their various unique advantages Jungwirth et al. (2016); Baltz et al. (2018); Gomonay and Loktev (2014); Gomonay et al. (2018). Among these, a control over interfacial exchange coupling to an adjacent metal and canted or noncollinear ground state offers unprecedented pathways to achieve intriguing physics and applications Bender et al. (2017); Kamra and Belzig (2017); Troncoso et al. (2020); Flebus et al. (2019); Hellman et al. (2017); Johnsen et al. (2020); Rabinovich et al. (2019). The vdW magnets offer an effective control over both - interface and ground state - due to their layered structure and relatively weak interlayer antiferromagnetic exchange Huang et al. (2017); Deng et al. (2018); Wang et al. (2019).

Capitalizing on these features, magnon-magnon coupling resulting in level repulsion and hybridization has recently been observed in the vdW AF CrCl₃ MacNeill et al. (2019); Kapoor et al. (2020); Sklenar and Zhang (2020). Approaching the challenge from a different direction, similar magnonic hybridization via their mutual coupling Kamra et al. (2017); Rezende et al. (2019); Yu et al. (2020) has been discovered in carefully chosen platforms including compensated ferrimagnets Liensberger et al. (2019) and synthetic AFs Sud et al. (2020); Chen et al. (2018). These investigations have, in part, been driven by the desire to control magnonic systems for quantum information applications Lachance-Quirion et al. (2019) and were preceded by the realization of strong magnon-photon coupling Huebl et al. (2013); Harder and Hu (2018). The latter, being a relatively mature field, has started to explore dissipative magnon-photon coupling Harder et al. (2018); Wang and Hu (2020) resulting in intriguing phenomena that foray into the realm of non-Hermitian physics Ashida et al. (2020) providing a powerful model platform. While not explored thus far, such phenomena can also result from dissipative magnon-magnon coupling offering various advantages over the magnon-photon platform Kamra et al. (2020). Demonstrating these for a vdW AF interfaced with a heavy normal metal (NM) forms a key contribution of this work.

Heterostructures comprising a magnetic insulator interfaced with a thin NM layer have become basic building blocks in an emerging spin-based paradigm for information transport and processing Goennenwein et al. (2015); Cornelissen et al. (2015); Hou et al. (2019); Troncoso et al. (2020); Lebrun et al. (2018); Wimmer et al. (2019); Chumak et al. (2015); Althammer (2018); Nakata et al. (2017); Brataas et al. (2020). In such structures, magnonic spin in the magnetic insulator can be interfaced with the electronic spin in NM thereby allowing their integration with conventional electronics. Furthermore, spin generated in NM allows to control, and even negate Wimmer et al. (2019), dissipation in the magnetic system via spin transfer torques Ralph and Stiles (2008). Invigorated by recent breakthroughs, especially employing the magnonic spin in AFs Lebrun et al. (2018); Vaidya et al. (2020); Li et al. (2020); Troncoso et al. (2020), an interface-engineering and control of spin transfer from AF to NM via magnonic spin pumping Tserkovnyak et al. (2002, 2005); Cheng et al. (2014); Kamra and Belzig (2017); Bender et al. (2017) assumes a central role. While coherently driven spin pumping from an AF into NM has recently been observed in its collinear ground state Vaidya et al. (2020); Li et al. (2020), a noncollinear or canted AF should enable unique and novel phenomena emerging from cross-sublattice spin pumping Kamra and Belzig (2017); Kamra et al. (2018); Liu et al. (2017). Besides providing the much needed understanding of spin transfer across the AF-NM interface, cross-sublattice pumping may also offer a direct probe into the aforementioned dissipative and non-Hermitian magnon-magnon coupling phenomena, as shown in this work.

In this Letter, we demonstrate heterostructures comprising vdW AFs and a heavy NM to be a unique platform for observing intriguing phenomena emerging from cross-sublattice spin pumping and dissipative magnon-magnon coupling. This niche is enabled by the layered structure of vdW AFs resulting in the possibility of AF-NM interface engineering and canted AF ground states with application of relatively small magnetic fields. We show that in such a non-collinear ground state, the AF pumps spin into the NM along all three directions on excitation via an rf magnetic field. A detection of the two in-plane spin components via inverse spin Hall effect allows to determine the complete spin mixing conductance matrix of the interface. We further find an unconventional out-of-plane spin pumping component that results from a concerted effect of cross-sublattice spin pumping and a dissipative coupling resulting from the sublattice-symmetry breaking at AF-NM interface. Furthermore, the ensuing dissipative coupling is found to result in magnon-magnon level attraction and coalescence observable via typical magnetic resonance experiments. The system thus constitutes a novel and unique platform for investigating this interplay between magnon level repulsion, attraction, and non-Hermitian physics via in-situ damping matrix engineering by, for example, spin transfer torques.

Model.– We treat the vdW material as a two-sublattice magnet described by the magnetization fields ${\boldsymbol{M}}_{A}$ and ${\boldsymbol{M}}_{B}$ that correspond to the sublattices $A$ and $B$. We consider magnetic free-energy density MacNeill et al. (2019) $F=\mu_{0}H_{E}{\boldsymbol{M}}_{A}\cdot{\boldsymbol{M}}_{B}/M_{s}+{\mu_{0}}\left(M^{2}_{Az}+M^{2}_{Bz}\right)/2-\mu_{0}{\boldsymbol{H}}\cdot\left({\boldsymbol{M}}_{A}+{\boldsymbol{M}}_{B}\right)$, with $M_{s}$ the saturation magnetization of each sublattice. The inter-sublattice exchange coupling parametrized by $H_{E}>0$ favors antiferromagnetic order. In addition, an external dc magnetic field ${\boldsymbol{H}}$ is applied in-plane. The second term represents the easy-plane anisotropy. Gilbert damping is accounted for by the viscous Rayleigh dissipation functional Gilbert (2004); Kamra et al. (2018); Yuan et al. (2019) via the symmetric matrix $\eta_{\zeta\zeta^{\prime}}$: $R[\dot{\boldsymbol{M}}_{A},\dot{\boldsymbol{M}}_{B}]=\sum_{\zeta\zeta^{\prime}}\int_{V}d{\boldsymbol{r}}{\eta_{\zeta\zeta^{\prime}}}\dot{\boldsymbol{M}}_{\zeta}\cdot\dot{\boldsymbol{M}}_{\zeta^{\prime}}/2$, where $\{\zeta,\zeta^{\prime}\}=\{A,B\}$. The ensuing magnetization dynamics is described by the coupled Landau-Lifshitz-Gilbert (LLG) equations,

in terms of the unit vectors ${\boldsymbol{m}}_{\zeta}\equiv{\boldsymbol{M}}_{\zeta}/M_{s}$. The effective fields are given by ${\boldsymbol{h}}^{\text{eff}}_{\zeta}=(1/\mu_{0})\times\partial F/\partial{\boldsymbol{M}}_{\zeta}={\boldsymbol{H}}-H_{E}\sigma^{x}_{\zeta\zeta^{\prime}}{\boldsymbol{m}}_{\zeta^{\prime}}-M_{s}\left({\boldsymbol{m}}_{\zeta}\cdot{\hat{\boldsymbol{z}}}\right){\hat{\boldsymbol{z}}}$, with $\sigma^{x}$ the Pauli matrix and $\gamma>0$ is the gyromagnetic ratio magnitude. The Gilbert damping parameters are defined through $\alpha_{\zeta\zeta^{\prime}}\equiv\gamma M_{s}\eta_{\zeta\zeta^{\prime}}$ where, in particular, $\alpha_{AB}=\alpha_{BA}\equiv\alpha_{od}$. Note that sublattice asymmetry in our model is broken only by the Gilbert damping Kamra et al. (2018). It results from the AF-NM interface and spin pumping-mediated losses Tserkovnyak et al. (2002); Kamra et al. (2018). The magnetization dynamics may be excited by a time-dependent magnetic field ${\boldsymbol{h}}\equiv\mu_{0}\gamma{\boldsymbol{h}}_{\text{RF}}(t)$ that produces a torque ${\boldsymbol{\tau}}_{\zeta}={\boldsymbol{m}}_{\zeta}\times{\boldsymbol{h}}$.

Magnetization dynamics and magnon modes.– We now investigate the magnon modes in the material when the two sublattice magnetizations are non-collinear in their equilibrium configuration [Fig. 1(a)]. In the presence of an in-plane external magnetic field ${\boldsymbol{H}}=H\hat{\boldsymbol{y}}$, the magnetic ground state becomes ${\boldsymbol{m}}^{\text{eq}}_{\zeta}=\pm\cos\phi\hat{\boldsymbol{x}}+\sin\phi\hat{\boldsymbol{y}}$, where the non-collinearity is captured by the finite angle $\phi$ that satisfies $\sin\phi=H/2H_{E}$ [see Fig. 1(a)]. The magnetic ground state is invariant under a twofold rotational operation around the $y$ axis, ${\cal C}_{2y}$, in combination with sublattice exchange $A\leftrightarrow B$, i.e. ${\cal C}_{2y}{\boldsymbol{m}}^{\text{eq}}_{A}={\boldsymbol{m}}^{\text{eq}}_{B}$. Linearizing the LLG equations (1), considering ${\boldsymbol{m}}_{\zeta}={\boldsymbol{m}}^{\text{eq}}_{\zeta}+\delta{\boldsymbol{m}}_{\zeta}e^{i\omega t}$, the coupled dynamical equations become,

with the two magnetization dynamics or magnon modes described by the fields $\delta{\boldsymbol{m}}_{\pm}=\delta{\boldsymbol{m}}_{A}\pm{\cal C}_{2y}\delta{\boldsymbol{m}}_{B}$, the torques ${\boldsymbol{\tau}}_{\pm}={\boldsymbol{\tau}}_{A}\pm{\cal C}_{2y}{\boldsymbol{\tau}}_{B}$ and the operator ${\cal A}_{\pm}=\left(\mu_{0}\gamma H_{E}+i\omega\bar{\alpha}\right)\pm\left(\mu_{0}\gamma H_{E}+i\omega{\alpha}_{od}\right){\cal C}_{2y}$. Furthermore, we have reformulated the Gilbert damping parameters as $\alpha_{AA}=\bar{\alpha}+\Delta\bar{\alpha}$ and $\alpha_{BB}=\bar{\alpha}-\Delta\bar{\alpha}$. Note that when sublattice symmetry is assumed, i.e., $\alpha_{AA}=\alpha_{BB}$, the Eqs. (2a) and (2b) become decoupled since $\Delta\bar{\alpha}=0$. In the absence of dissipation, the magnon eigenmodes are captured well by the fields $\delta{\boldsymbol{m}}_{\pm}$, with the eigenfrequencies for the so-called optical and acoustic magnon modes being $\omega_{+}=\mu_{0}\gamma\sqrt{2M_{s}H_{E}\cos^{2}\phi}$ and $\omega_{-}=\mu_{0}\gamma\sqrt{2H_{E}\left(M_{s}+2H_{E}\right)}\sin\phi$, respectively. The two modes can be excited selectively by a careful choice of the rf-field ${\boldsymbol{h}}$ MacNeill et al. (2019). In general, the excitation of $(\pm)$-modes one at a time, which demands ${\boldsymbol{\tau}}_{\mp}=0$ for the torque, imposes the conditions ${\boldsymbol{h}}=\pm C_{2y}{\boldsymbol{h}}$, respectively.

Spin pumping.– We now investigate spin transport across the AF-NM interface resulting from the excitation of magnetization dynamics by rf magnetic field. The dc spin pumping current injected into the adjacent NM is given by Kamra and Belzig (2017)

where $\langle\cdots\rangle$ stands for the time-average over the period of oscillation. The diagonal elements of the matrix $g_{\zeta\zeta^{\prime}}$ describe the intra-sublattice spin mixing conductance. An asymmetric interfacial coupling Bender et al. (2017); Kamra and Belzig (2017); Troncoso et al. (2020); Flebus et al. (2019), resulting in $g_{AA}\neq g_{BB}$, occurs when the two magnetic sublattices are incommensurately exposed to the NM (see Fig. 1(c) for an example in which $g_{AA}=0$). The off-diagonal cross-sublattice conductance satisfy $g_{AB}=g_{BA}$ and are nonzero when both the sublattices are (partly) exposed to NM [Fig. 1(b) and (d)]. Such interfaces can be achieved with layered magnets Yin et al. (2018); Huang et al. (2018) (Fig. 1), but are not possible with synthetic AFs Sud et al. (2020). Our goal here is to establish the experimentally detectable spin pumping current as a direct probe of the $2\times 2$ spin mixing conductance matrix [Eq. (3)]. In particular, we are interested in establishing unique signatures of the cross-sublattice conductances $g_{AB}=g_{BA}$ that elude a direct experimental observation thus far.

In typical experimental setups (Fig. 1), the spin pumping current is detected via inverse spin-Hall effect (ISHE) Sinova et al. (2015); Ando et al. (2011); Valenzuela and Tinkham (2006); Saitoh et al. (2006). In metals with strong spin-orbit coupling, a nonequilibrium spin current ${\boldsymbol{j}}_{s}$ induces a transverse charge current ${\boldsymbol{j}}_{c}=\theta_{\text{SH}}(2e/\hbar){\boldsymbol{j}}_{s}\times\vec{\sigma}$, where $\vec{\sigma}$ denotes the spin polarization direction and $\theta_{\text{SH}}$ is the spin Hall angle Sinova et al. (2015). Under open circuit conditions, the generated charge current is countered by an induced inverse spin Hall voltage $V_{\text{ISHE}}$ proportional to the charge, and thus spin, current. The voltage thus generated is proportional to the spin current injected into the NM [Eq. (3)] and is directly detected in experiments Mosendz et al. (2010).

In order to relate spin mixing conductance matrix elements to experimental observables, we evaluate the spin pumping current Eq. (3) in the limit $\alpha_{AA}\approx\alpha_{BB}$, i.e., $\Delta\bar{\alpha}$ small. In this perturbative regime, Eqs. (2a) and (2b) decouple and the system eigenmodes are the optical and acoustic magnon modes. The corresponding spin pumping currents are evaluated to be

The $\pm$ index labels optical and acoustic modes, which are driven by the rf magnetic field with frequency $\omega$ and amplitude $h_{+,\phi}=2h_{y}\cos\phi$ and $h_{-,\phi}=2h_{x}\sin\phi$, respectively. The expression in Eq. (Cross-sublattice Spin Pumping and Magnon Level Attraction in van der Waals Antiferromagnets) represents a pure spin current that flows across the AF-NM interface (along $z$ axis). Its various components pertaining to directions in the spin space are proportional to the functions $\mathscr{F}^{j}_{\pm}(\omega)$, as detailed in the Supplemental Material in SM (see Eqs. (41)-(43)), which have been obtained to the first order in $\Delta\bar{\alpha}$. The in-plane components $\mathscr{F}^{x,y}_{\pm}$, satisfying $\mathscr{F}^{y}_{\pm}(\omega)=\tan\phi\mathscr{F}^{x}_{\pm}(\omega)$, are independent of $\Delta\bar{\alpha}$. However, the out-of-plane component of the spin current $\propto\mathscr{F}^{z}_{\pm}(\omega)$ scales linearly with $\Delta\bar{\alpha}$. The resulting field- and frequency-dependence of the spin current ${\bf I}^{\pm}_{z}$ components are plotted in Fig. 2 employing system parameters relevant for the vdW AF CrCl₃ MacNeill et al. (2019). The various spin current components are displayed in panels (a), (b) and (c) of Fig. 2 for the optical mode, and in panels (d), (e), and (f) for the acoustic mode.

In contrast with the case of spin pumping via collinear magnets, in which a dc spin current polarized along the equilibrium order is generated Tserkovnyak et al. (2002); Cheng et al. (2014); Mosendz et al. (2010); Saitoh et al. (2006); Vaidya et al. (2020); Li et al. (2020), the canted AF under consideration pumps spin with components along all three directions [Eq. (Cross-sublattice Spin Pumping and Magnon Level Attraction in van der Waals Antiferromagnets)]. As detailed in the Supplemental Material SM , $\mathscr{F}^{y,z}_{\pm}\propto\sin\phi$ implying $y$ and $z$ components of the spin pumping current vanish for $\phi=0$. Our result thus reduces to the existing understanding of collinear AFs Kamra and Belzig (2017); Troncoso et al. (2020). These additional components of the spin pumping current for the canted AF together with the existence of two independent magnon modes constitute some of the unique features and opportunities offered by this system. For example, detection of the ISHE voltage in two orthogonal directions [as depicted in Fig. 1(a)] enables determination of both in-plane spin current components. By detecting these while exciting the two magnon modes one at a time, we may determine the full spin mixing conductance matrix with the cross-sublattice term given by

which assumes the condition $g_{AA}\neq g_{BB}$. This accomplishes a key goal and constitutes a main result of this paper.

Furthermore, existence of the spin current $z$ component [Eq. (Cross-sublattice Spin Pumping and Magnon Level Attraction in van der Waals Antiferromagnets)] that we find is unconventional and counterintuitive as the magnon spin is expected to lie in the same plane as the equilibrium sublattice magnetizations Kamra et al. (2017); Rezende et al. (2019). However, this component is nonzero only when $\Delta\bar{\alpha}\neq 0$, $g_{AA}\neq g_{BB}$, $g_{AB}\neq 0$, and $\phi\neq 0$ implying that it results from a complex interplay of the sublattice-symmetry breaking dissipative coupling and a cross-sublattice interference effect. Such physics, especially dissipative coupling Harder et al. (2018); Wang and Hu (2020), appears to go beyond the magnon picture considered thus far and constitutes another key result of our work. While an out-of-plane spin component has not been measured in typical spin pumping experiments Saitoh et al. (2006); Sinova et al. (2015); Mosendz et al. (2010), the recent thermal drag-mediated detection of such an out-of-plane spin component Avci et al. (2020) provides one possible method for its direct observation.

Magnon level attraction.– We now discuss the magnon eigenmodes which become dissipatively coupled [see Eqs. (2a) and (2b)] due to the sublattice-symmetry breaking Gilbert damping Kamra et al. (2018), i.e. nonzero $\Delta\bar{\alpha}$. While a “reactive” coupling between magnon modes has been observed in various systems Chen et al. (2018); Liensberger et al. (2019); Sud et al. (2020); MacNeill et al. (2019); Kapoor et al. (2020), dissipative coupling remains less explored and invokes non-Hermitian physics Ashida et al. (2020); Flebus et al. (2020). Solving Eqs. (2a) and (2b) without an external rf drive, we obtain the complex eigenmode frequencies $\omega_{\pm}=\omega_{r\pm}+i\omega_{i\pm}$. $\omega_{r}$ and $\omega_{i}$, respectively, capture the energy and inverse lifetime of the magnon modes and have been plotted against the external dc magnetic field in Fig. 3 (a)-(c). Due to the dissipative nature of the coupling, a magnon-magnon level attraction is observed. The grey curve in Fig. 3 (b), corresponding to $\bar{\alpha}=\alpha_{od}=0$ and $\Delta\bar{\alpha}\neq 0$, depicts a perfect level coalescence or mode synchronization Harder et al. (2018); Wang and Hu (2020). Such values for Gilbert damping matrix require a dc spin transfer torque drive in the NM. An undriven system however imposes constraints $\alpha_{AA},\alpha_{BB}>0$, i.e. $\bar{\alpha}>\Delta\bar{\alpha}$ and $\alpha_{od}\leq\sqrt{\alpha_{AA}\alpha_{BB}}$ Kamra et al. (2018). In this scenario, we find a complex interplay of repulsion and attraction between the two modes. The resulting eigenfrequencies split slightly (see Fig. 3(a) when $\bar{\alpha}=0.1$, $\alpha_{od}=0.07$ and $\Delta\bar{\alpha}=0.07$) while coalescing at a specific point, the so-called exceptional point Tserkovnyak (2020). Furthermore, Fig. 3(d) depicts the imaginary part of dynamic susceptibility, which is directly accessible in experiments Liensberger et al. (2019); MacNeill et al. (2019); Berger et al. (2018). A peak in this susceptibility provides an additional experimental signature of the level attraction when $\Delta\bar{\alpha}$ is sufficiently large. Thus, vdW AFs under consideration constitute a rich platform for realizing non-Hermitian physics and magnon-magnon level attraction via AF-NM interface engineering and spin transfer torques exerted on AF by the NM.

Summary.– We have theoretically uncovered unique and intriguing cross-sublattice spin pumping and magnon-magnon level attraction effects in a model canted antiferromagnet. By providing guidance to experiments in extracting the interfacial spin mixing conductance matrix and key signatures of level attraction, we hope to establish van der Waals antiferromagnets interfaced with a heavy metal layer as a fertile and convenient platform for realizing and investigating unconventional non-Hermitian physics.

Note added: During the manuscript preparation, we noticed a recent related preprint Lu et al. (2020) that studies level-repulsion and hybridization of magnonic modes in bulk symmetry-breaking synthetic antiferromagnets. It however does not discuss spin pumping or the dissipative magnon level attraction - the two key novelties of our work.