Magnon spin transport in the van der Waals antiferromagnet CrPS4 for non-collinear and collinear magnetization
Dennis K. de Wal Muhammad Zohaib Bart J. van Wees


APS/123-QED


d.k.de.wal@rug.nl

Zernike Institute for Advanced Materials, University of Groningen,
Groningen, the Netherlands











Zernike Institute for Advanced Materials, University of Groningen,
Groningen, the Netherlands



§ ABSTRACT 
We investigate the injection, transport and detection of magnon spins in the van der Waals antiferromagnet chromium thiophosphate (CrPS4). We electrically and thermally inject magnon spins by platinum contacts and examine the non-local resistance as a function of in-plane magnetic field up to 12 Tesla. We observe a large non-local resistance from both the electrically and thermally excited magnon modes above the spin-flip field where CrPS4 is in the collinear state. At 25 K for an in-plane field of ranging 5 - 12 T, we extract the magnon relaxation length λ_m ranging 200-800 nm and a typical magnon conductivity of σ_m≈1×10^4 Sm-1, which is one order of magnitude smaller than in yttrium iron garnet (YIG) films at room temperature. Moreover, we find that σ_m is almost zero for CrPS4 in the non-collinear state. Our results open up the way to understanding the role of the antiferromagnetic magnon modes on spin injection into antiferromagnets and implementation of two-dimensional magnets for scalable magnonic circuits.










                              
Magnon spin transport in the van der Waals antiferromagnet CrPS4 for non-collinear and collinear magnetization
    Bart J. van Wees
    September 9, 2024
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§ INTRODUCTION

Magnon spin transport has been extensively studied in insulating ferro- and ferrimagnets, by spin pumping<cit.>, the spin Seebeck effect (SSE)<cit.> and electrical injection and detection<cit.>. Antiferromagnets possess several advantages over ferromagnets for spintronic applications, such as stability to external fields<cit.> and operation frequencies up to terahertz scale<cit.>. Long distance magnon transport has been demonstrated in YFeO3<cit.> and antiferromagnetic hematite<cit.>, as well as coherent control of magnon spin dynamics<cit.>. The discovery of insulating ferro- and antiferromagnetic van der Waals materials such as the chromium trihalides (CrX3, X = Cl, Br, I) and transition metal phosphates (MPS3, M = Fe, Mn, Ni, Co) allows for the study of magnon spin transport in the quasi two-dimensional (2D) limit. These materials can be isolated into monolayer or few-layer thicknesses and show diverse inter- and intralayer exchange couplings and magnetic anisotropies. The resulting rich spin textures make 2D antiferromagnetic van der Waals materials a promising platform for the study of spin wave properties and transport.



In these materials the investigation of antiferromagnetic resonance (AFMR) reveals the existence of acoustic and optical magnon modes <cit.>. However, AFMR studies are typically limited to sub-40 GHz excitation frequencies at external fields below 1.5 T, whereas the critical fields and precession frequencies of antiferromagnets often exceed these values. Moreover, AFMR does not resolve the role of the magnon modes in spin transport. The study of the propagation of magnons by employing a non-local geometry unveils information about the transport properties such as the magnon relaxation and magnon conductivity as well as the role of the antiferromagnetic magnon modes on the transport. Magnon transport driven by a thermal gradient (SSE) has been reported in both ferro- and antiferromagnetic van der Waals materials<cit.>. Yet these thermally driven magnons provide only convoluted information about the magnon transport properties. 




On the other hand, “all electrical” magnon transport does not suffer these problems, as the exact locations of magnon exitation and detection are clear, allowing for direct electrical control over the magnon currents. This enables the study of the spins carried by the different AFM magnon modes and their effect on transport. Yet, so far in antiferromagnetic van der Waals materials, field tunable all electrical long distance magnon transport has only been shown in CrPS4<cit.>. 

Here, we expand this work and perform a detailed study of the injection, transport and detection of spins by exciting the antiferromagnetic magnon modes for magnetic field perpendicular to the anisotropy axis (c-axis) of CrPS4 through investigating the non-local resistance as a function of field and temperature. 



§ EXPERIMENTAL DETAILS

In this work, we employ a non-local geometry of 7 nm thick platinum strips on top of a ∼160 nm thick exfoliated CrPS4 flake (Fig. <ref>). A low frequency (<20Hz) AC charge current I through an injector Pt strip generates an in-plane spin accumulation perpendicular to I via the spin Hall effect (SHE)<cit.>. Moreover, in the same Pt strip the current I via by Joule heating, creates both vertical and lateral thermal gradients, in the CrPS4 flake, which drives a spin Seebeck generated magnon current. Lastly, in an adjacent detector Pt strip, the two effects both generate non-local voltages V_el and V_th via the inverse SHE. The first comes from the injected spin current and the second from the spin Seebeck effect. 

The time-dependent voltage response for I(t) = I_0 sin(ω t) can be expanded as V(t)=R_1I(t) + R_2I^2(t) + ... where R_1 and R_2 are the first and second order resistances. The non-local resistance R_nl=Vdetector/Iinjector can therefore be separated as R_nl^1ω=V_el/I and R_nl^2ω=V_th/I^2. Lebrun et al. <cit.> demonstrate that for the easy axis antiferromagnet hematite (α-Fe2O3), at fields below the spin-flop transition, the magnonic spin currents carry spin polarized along the Néel vector 𝐍 = (𝐦_1 -𝐦_2)/2, where 𝐦_1 and 𝐦_2 are the sublattice magnetizations. Yet, they show that thermal spin currents scale with the net (field-induced) magnetic moment 𝐌 = (𝐦_1 + 𝐦_2)/2. However, the effect of the magnon modes themselves on the transport is not discussed.

In figure <ref>, the (k=0) magnon modes for CrPS4 are depicted for non-zero out-of-plane (oop) and in-plane (ip) external field. Figure <ref>a and <ref>b show the modes before and after the spin-flop transition (oop), respectively. 
For the ω_∥1 mode (acoustic) the Néel vector is in the y-direction (Fig. <ref>b). Under the assumption that the ip anisotropies H_a=H_b (where the subscript indicated the crystal axis), the frequency of ω_∥2 (optical) vanishes at the spin-flop. We expect that ω_∥2 becomes a soft magnon mode, with zero frequency. In this case only the acoustic mode ω_∥1 has a non-zero frequency<cit.>. Figure <ref>d and <ref>e show the non-collinear (canted AFM) and the collinear (FM) state (for ip), before and after the spin-flip field above which the two sublattices align with the field, respectively. For ip fields, the ip projection (y-direction) of the net magnetization 𝐌 increases linearly with increasing field till the spin-flip field (H_E⊥)<cit.>. The Néel vector 𝐍 is along the z-axis and decreases with increasing field. 
The canting angle sinθ_⊥ = H/H_E⊥ is the angle between the static sublattice magnetization and the anisotropy axis (z-axis). 

At H>H_E⊥ the system is in the collinear state. The acoustic and optical modes persist; the first, ω_⊥1, is a FM-mode, which is the same as for an uniaxial ferromagnet (also known as Kittel mode).
The latter, ω_⊥2, is the AFM-mode,
where 𝐌 is static and 𝐍 still oscillates in the xz-plane. The dispersion relation at the band edge (k=0) as a function of the external field (at T=2 K) is shown in figure <ref>c for both the in-plane and out-of-plane magnon modes. 

A recent study on spin pumping in CrCl3 reveals that in the spin-flopped state the acoustic mode (ω_∥1) the spin current is driven by 𝐌, yet for the optical mode (ω_∥2) it is driven by 𝐍<cit.>. Further, a recent theoretical study by P. Tang et al.<cit.> on spin pumping in the non-collinear AFMs shows that when the two modes contribute equally, the spin current follows 𝐌, whereas the spin current component along 𝐍 only plays a role when the modes do not contribute equally (uncompensated Pt/AFM interface). 
Despite this understanding, the effect of the antiferromagnetic magnon modes on the transport of magnons, especially in the non-collinear regime with field perpendicular to the anisotropy axis, remains unclear.

Therefore, we explore here which antiferromagnetic magnon modes can be excited and detected by a spin accumulation μ. In the Pt strips, at the Pt/CrPS4 interface, produced by the SHE, the spin accumulation μ can only be generated in y-direction. 
Hence, when a magnon mode can carry spin in the y-direction it can absorb μ. For CrPS4 where we apply field H, we can distinguish 5 phases: For oop fields: 1. H<H_sf; spin injection is only possible for H∥μ and μ∥ M,N, 2. H_sf<H<H_E∥ & 3. H>H_E∥ and for ip fields 4. 0<H<H_E⊥ & 5. H>E⊥; spin injection is only possible for μ∥ H and H∥ M, H⊥ N. 
Secondly, only the magnon modes in which 𝐌 or 𝐍 precess can pump spins into the Pt. Since μ in the Pt is only collinear with H in phase 4. and 5. and only  for the acoustic magnon mode ω_⊥1 𝐌 precesses, only ω_⊥1 contributes to spin pumping. By reciprocity, electrical injection of spin can therefore only excite ω_⊥1.



Nonetheless, the above depiction only holds under the assumption that both magnetic sublattices contribute equally to the spin pumping (and injection). With CrPS4 being an A-type antiferromagnet, the second sublattice has a greater distance to the Pt/CrPS4 interface than the first. This could lead to a magnetically uncompensated interface
, for which precession of 𝐌 or 𝐍 orthogonal to the y-axis will contribute to the spin pumping as well<cit.>. The two spin currents from CrPS4 to the Pt as a result of spin pumping originating from the two individual sublattices carry spin collinear to their static sublattice magnetization (dashed line in figure <ref>d). The spin polarization of these spin currents over the interface have equal projection on the y-axis, but opposite on the z-axis. Hence, when both sublattices do not couple equally, the optical magnon mode ω_⊥2 can contribute to the spin pumping (and injection) proportional to oscillating 𝐍.





§  RESULTS AND DISCUSSION

In figure <ref>a, R_nl^1ω is plotted at 25 K as a function of magnetic field along the y-axis (ip). We find that at H< 6 T, the system is in the non-collinear state where R_nl^1ω is zero. At 6 T <H< 8 T, R_nl^1ω increases till it saturates at H> 8 T. This behavior agrees with the magnetization of the CrPS4 at 25 K, given in the Supporting Information (SI)<cit.>, where 𝐌 for an in-plane field gradually saturates for 6 T <H< 8 T<cit.>. R_nl^1ω in figure <ref> is given for three injector detector spacings d, which is the edge-edge distance between the Pt strips. These results agree well with the R_nl^1ω obtained from ADMR measurements in our earlier work<cit.>. 

At 25 K, k_BT ≫ħω, with ω being the frequency of the magnon modes, hence thermal equilibrium magnons populate both magnon modes (see figure <ref>c) at all field strengths shown in figure <ref>a. Therefore, both modes (ω_⊥1 and ω_⊥2) could contribute to transport. Regardless of which magnon mode contributes, the absence of R_nl^1ω below H_E⊥ is surprising. Possibly, the spin is not conserved when CrPS4 is in the canted AFM state, due to the axial symmetry breaking<cit.>. We discuss other possible reasons later. 

The thermally generated magnon transport signal shows a very different trend as a function of field. In figure <ref>b the local (right axis) and non-local (left axis) R^2ω are shown for different d. The advantage of the local over the non-local SSE signal is that the first contains direct information on effect of the magnon modes on the thermal spin pumping. Whereas the latter contains convoluted information on the magnon transport as well. For the local SSE (R^2ω_l), in the non-collinear state, the number of thermally excited magnons increases with increasing sinθ_⊥, i.e. follows the net magnetization (ferromagnetic SSE)<cit.>. However, at fields close to H_E⊥ R^2ω_l increases non-linearly w.r.t sinθ_⊥. Upon saturation of the sublattice magnetization, the signal starts to saturate, yet for H>H_E⊥ the saturation continues up to higher fields than for R_nl^1ω (Fig. <ref>a).

For the non-local SSE signal R_nl^2ω, at H<H_E⊥, (Fig. <ref>b), the dependence on the field is similar to that of the local SSE signal. However, the sign of R_nl^2ω is opposite to R_l^2ω and the amplitude is two orders of magnitude smaller. The former indicates that, driven by the SSE, the magnon chemical potential is opposite. The behavior of R_nl^2ω, can be understood as follows: A thermal magnon current (proportional to ∇ T) is driven away from the injector by the SSE, effectively creating a depletion of magnons at the injector and a magnon accumulation away from the injector. This accumulation drives diffusive magnon currents, proportional to ∇μ_m,  towards the injector and detector<cit.>. For H<H_E⊥, the absence of a R_nl^1ω is this state suggests that diffusive transport lengths are very small and the similarity of R_nl^2ω to R_l^2ω and in this state points out that the SSE drives the magnons till just under the detector, where the magnon accumulation is detected. 



The increase at H_E⊥, which is very similar to that in R_nl^1ω (figure <ref>a), shows the onset of diffusive magnon transport. Yet, where R_nl^1ω clearly saturates at larger fields, R_nl^2ω decreases and even changes sign at fields far above H_E⊥ for the shortest d. This can be understood by the competing thermal (SSE-driven) and diffusive magnon current in the system, i.e. both the SSE generated magnon accumulation and diffusive magnon currents contribute to R_nl^2ω. The decrease in R_nl^2ω at largest field strengths could indicate that the diffusion lengths increase with increasing field. To our surprise, for larger d, R_nl^2ω actually becomes larger. In YIG a sign change of R_nl^2ω is observed as a function of d. At larger d (where d≪λ_m still holds), R_nl^2ω increases with increasing d<cit.>. Similar behavior could be possible in CrPS4.

In figure <ref>c, R_nl^1ω as a function of field at different oop angles (β) is shown. Note that β=90^∘-θ_⊥. Within the non-collinear regime R_nl^1ω remains zero, whereas in the collinear regime the R_nl^1ω scales with the projection of 𝐌 on the y-axis (cos^2β, as being dependent on the electron spin accumulation at both the injector and detector). For R_nl^2ω, (figure <ref>d), the signal also scales with the projection of 𝐌 (cosβ). For β=90^∘, where the spin-flop is predicted around 0.8 T, neither for R_nl^1ω, nor for R_nl^2ω this transition is observed. As we cannot see any contribution of oop spins and since μ∥𝐍, following the spin pumping in CrCl3<cit.>, only the optical mode (ω_∥2) could contribute to spin pumping. However, this mode is a soft mode in our system. 

In figure <ref>, R_nl^1ω and R_nl^2ω are given as a function of temperature. In the non-collinear regime R_nl^1ω is zero for all temperatures above and below the Néel temperature. In the collinear regime a non-monotonous dependence is observed. For temperatures <10 K (at H>7 T) the acoustic magnon mode (ω_⊥1) is possibly not occupied, but R_nl^1ω is non-zero. This suggests that the optical magnon mode (ω_⊥2) does contribute spin pumping, indicating an unequal coupling of the sublattices. Around 25 K, R_nl^1ω is maximum and at higher temperature R_nl^1ω decreases, diminishing just above the Néel temperature, TN = 38 K) (at larger field strenghts, the magnetic ordering is maintained up to slightly higher temperatures, see magnetization behavior in the SI). 
However, the number of magnons and the magnon transport properties are highly temperature dependent, therefore, information on the contributions by the different magnon modes cannot be directly extracted.

In addition, R_nl^2ω shows a very different behavior as a function of temperature. The measured R_nl^2ω changes sign for low temperature (<15 K), where the effect is strongest in the collinear regime. The local SSE R_l^2ω, given in the SI, does not show this behavior. The occupation of the acoustic magnon mode (ω_⊥1) is affected by the temperature, whereas the optical mode (ω_⊥2) will remain populated for all temperatures in figure <ref>b. For fields strength of ≤7 T, the eigen energy ħω_⊥1<k_BT, the acoustic mode could contribute spin pumping. However, the effect of the temperature on the non-local SSE in CrPS4 and the effect of the magnon modes on transport are not fully understood. 






Altogether, the injection (excitation) of magnon spins and detection (incoherent spin pumping), and the transport of magnon spins by the magnon modes are entirely different processes. Spin is not a conserved quantity. This holds for injection and detection, and also for magnon spin transport. Thus, magnons excited by a spin current at the injector can diffuse towards the detector where they again create a spin current via spin pumping, but the spin is not necessarily conserved during transport (or injection and detection). Therefore, in our transport measurements we cannot determine if the magnon modes `carry' the spins. Furthermore, the magnon relaxation might be affected by the non-collinearity of the AFM. When the two sublattices are non-collinear, the strong exchange interaction possibly suppresses the magnon conductivity σ_m and relaxation length λ_m. At the spin flip field, the exchange energy is overcome and the sublattices become collinear, possibly allowing a strong increase in both σ_m and λ_m<cit.>.  


In figure <ref>a, the R_nl^1ω in the collinear state (at 7 T) is given as a function of edge-edge distance, d, between the injector and detector Pt strip, measured for multiple devices. The model for diffusive magnon transport leads to a decay in R_nl^1ω with increasing d as a function of the λ_m. Under the assumption of a large enough effective interface spin mixing conductance, R_nl^1ω is given by:

    R_nl^1ω = C/λ_mexpd/λ_m/1-exp2d/λ_m,

where C is a constant capturing all distance-independent pre-factors, such as the magnon conductivity σ_m. For d<λ_m the transport is purely diffusive and R_nl^1ω∼ 1/d, with the transport being Ohmic<cit.>, whereas for d>λ_m, R_nl^1ω decays exponentially as the magnon relaxation sets in. From equation <ref> we extract λ^1ω_m= 490±30 nm with σ_m≈1×10^4 Sm^-1 , at 7 T, under the assumptions elaborated in SI.

In figure <ref>b, λ_m and σ_m are extracted using equation <ref>, in similar fashion as in Fig. <ref>a, as functions of the ip field around the spin-flip transition. At 5 T, λ_m is smaller than the smallest d on our devices and σ_m is (close to) zero. Here, the magnon transport is heavily suppressed. Increasing in field, between 6-8 T, both transport parameters increase. σ_m increases sharply and saturates at field >8 T, whereas the increase in λ_m is less abrupt and, in our measurements, does not seem to saturate. In fact, >8 T, the decay of R_nl^1ω as a function of d seems to be fully Ohmic, yet due to insufficient data we cannot extract λ_m at these fields (see SI). For these fields we extract σ_m by assuming λ_m ≫ d, see SI.

The effect of the different magnon modes on magnon spin transport is still not entirely disclosed. For systems using the non-local geometries the measured non-local resistance depends on several factors, the SHE and ISHE in the Pt contacts, the transparency of the Pt/CrPS4 interface and the magnon conductivity and relaxation. 
The large spin Hall magnetoresistance measured on these samples (see SI) and in previous work<cit.> indicates a transparent interface.   



§ CONCLUSION

The exact effect of the antiferromagnetic magnon modes on magnon spin transport in the uniaxial antiferromagnet CrPS4 with field orthogonal to the anisotropy axis, remains so far unclear. In the non-collinear regime the magnon relaxation length λ_m and the magnon condutivity σ_m are almost zero. We find that both λ_m and σ_m strongly increase at the spin-flip transition. At 7 T, we find λ_m= 490±30 nm and σ_m≈1×10^4 Sm^-1, the latter is one order of magnitude smaller than the typical values found in 210 nm thick YIG at room temperature<cit.>. The thermally generated magnons via the SSE indicates that both magnon modes contribute to the non-local resistance, yet their individual contributions to transport continues to be unresolved. Obviously, currently we do not have a full comprehension of the role of the various modes for magnon spin transport. Nevertheless, these result prepare the way to understanding and using the antiferromagnetic magnon modes for long-distance magnon spin transport in 3D and 2D van der Waals antiferromagnets.  


We want to express our special gratitude towards G.E.W. Bauer, P. Tang and J. Barker for insightful discussions and suggestions. We acknowledge the technical support from J. G. Holstein, H. Adema, H. H. de Vries, and F. H. van der Velde. We acknowledge the financial support of the Zernike Institute for Advanced Materials and the European Union’s Horizon 2020 research and innovation program under Grant Agreements No. 785219 and No. 881603 (Graphene Flagship Core 2 and Core 3). This project is also financed by the NWO Spinoza prize awarded to B.J.W. by the NWO and has received funding from the European Research Council (ERC)
under the European Union’s 2DMAGSPIN (Grant Agreement No. 101053054). 































































































































 














































