Antiferromagnetic Spin Wave Field-Effect Transistor

In a collinear antiferromagnet with easy-axis anisotropy, symmetry dictates that the spin wave modes must be doubly degenerate. Theses two modes, distinguished by their opposite polarization and available only in antiferromagnets, give rise to a novel degree of freedom to encode and process information. We show that the spin wave polarization can be manipulated by an electric field induced Dzyaloshinskii-Moriya interaction and magnetic anisotropy. We propose a prototype spin wave field-effect transistor which realizes a gate-tunable magnonic analog of the Faraday effect, and demonstrate its application in THz signal modulation. Our findings open up the exciting possibility of digital data processing utilizing antiferromagnetic spin waves and enable the direct projection of optical computing concepts onto the mesoscopic scale.

Based on the Faraday rotation of AFM spin waves, we propose a gate-tunable field-effect transistor serving as the magnonic analog of the Datta-Das device of electrons 18 . We demonstrate its application in the amplitude-shift keying as a THz signal modulation. Finally, by including the field-induced anisotropy 19 , we can realize direct transitions between the two circularly-polarized modes, which, together with Faraday's rotation controlled by DMI, enable a complete control of spin wave states over the entire Bloch sphere. Possible electrical detections of the spin wave state on the Bloch sphere are also discussed. Our findings open up the possibility of digital data processing harnessing antiferromagnetic spin waves, and enable the direct projection of optical computing concepts onto the mesoscopic scale. . Consider a quasi-one dimensional nanostrip with an easy-axis along the ẑ-direction, as schematically shown in Fig. 2(a). A perpendicular gate voltage is applied to break the inversion symmetry and induce a DMI of the following form (see Supplementary S1) where  is the DMI strength and ∇ = × ∇ ∼ŷ with ŷ being the mirror plane normal. The DMI may be nonzero even without gating, as the geometry of the interface already breaks the mirror symmetry. In the exchange limit, Figure 1. Degenerate spin wave modes in a collinear antiferromagnet with easy-axis anisotropy. Red and blue arrows represent the two sublattice spins S A and S B in a unit cell, with ẑ the easy-axis. (a) and (b) The two circularly polarized modes are characterized by left-handed and right-handed precessions around the easy-axis; they also have slightly different ratios between the cone angles of S A and S B . (c) and (d) The linearly polarized modes consist of different combinations of the circular modes. While S A (S B ) is individually traveling counterclockwise (clockwise) on an elliptical orbit, the staggered field n = (S A − S B )/2S exhibits purely linear oscillation.
Scientific RepoRts | 6:24223 | DOI: 10.1038/srep24223  m n, we will drop the last term; the second term is a total derivative that does not affect the local dynamics. This leaves us with only the first term of Eq. (1). The total action in terms of n(x, t) is 20   (4) is plotted in Fig. 2(b). For an arbitrary given frequency ω above the antiferromagnetic resonance point at ω = c Z R , we find that the splitting of wave vectors is independent of ω. Let L be the length of the gated region, then the linear polarization of an AFM spin wave will rotate by π/2 when ΔkL = π, which can be regarded as the magnonic analogy of the Faraday rotation of an electromagnetic wave. This mechanism also has a direct analogy in the electron spin field-effect transistor (FET) 18 , where Δk is frequency independent and is proportional to the Rashba spin-orbit coupling.
AFM spin wave field-effect transistor. The key to realize an AFM spin wave field-effect transistor is the gate-tunable DMI 17,21 . It was shown that in transition metal compounds, an electric field E generates a DMI of the strength D E = JaeE/ so , where  so originates from the spin-orbit interaction and is typically on the order of 3 eV 21 . The electric field E can be produced by either a gate voltage or the interfacial mirror symmetry breaking. Recently, a gate-tunable DMI has been observed in a ferromagnetic insulator 22 ; but the same mechanism is applicable to AFMs as well. The condition ΔkL = π mentioned above is tantamount to an electric field The required E field can be scaled down by increasing L, but care should be taken that L not exceed the magnon phase coherence length φ  . Though the actual value of φ  depends on multiple factors, it suffices to consider the magnon dephasing due to the Gilbert damping. This allows for a simple estimate αω where v p is the phase velocity, α is the Gilbert damping constant, and ω is the spin wave frequency. Since v p = ω/k, we have α φ ~k 1/( ). For α ~ 0.005 and k < 2 μm −1 , we find that φ  exceeds 100 μm. This indicates that spin waves in AFMs can propagate over large distance without losing phase information and provides a loose upper limit for L. On the other hand, the need to suppress spiral formation sets a lower limit for L -maintaining the stability of the Néel ground state yields a maximum allowed electric field so max which depends on the ratio K/J that can differ by several orders of magnitude in different materials (0.1 ~ 10 −4 ) 23-26 . The value of E max then sets the corresponding lower bound on L via Eq. (6), which can be as short as a few nanometers.
THz data modulation. Since the resonance frequency of AFMs is typically in the THz range, the AFM spin wave FET has important applications in high frequency data modulations. As an example, we perform a numerical simulation of the amplitude shift keying (ASK) based on the device schematic of Fig. 2(a). Assume that the spin wave is generated at one end of the chain by an oscillating magnetic field along the y-direction at 1.4 THz. The spin wave is subsequently modulated by a 20 GHz square wave through the gate voltage V G , which covers 330 atomic sites (see Method). While the bit rate of such an ASK is as high as THz, the actual signal transfer rate (Baud rate) is limited by the relaxation time τ of the spin wave FET. As indicated by the dotted red line in Fig. 2(b), τ is the transient period that a transmitted spin wave adapts to an abrupt change of the gate voltage, so the maximum Baud rate cannot exceed 1/τ. Physically, the relaxation time can be regarded as the time required for the wavefront of an incident wave to traverse the gated region: τ must therefore scale as L/c. Since c = 2aJ/ħ, we have τ ω − L a ( / ) J 1 with ω J = J/ħ. In typical AFMs, ω J is around hundreds of THz, so τ is only a few picoseconds when L/a ~ 100. This allows reliable signal transfer at a Baud rate below 0.1 THz. The capacity of signal transfer can in principle be enlarged by reducing L as long as the ground state remains collinear. Figure 2(d) shows the spatial pattern of the staggered field. Long after the relaxation time, the trajectories of the Néel order on every 60 atomic sites (30 unit cells) are plotted for several periods. While the transmitted wave is essentially linearly-polarized, the reflected wave and the wave inside the gated region slightly open up into elliptical precessions. This fact implies that the relative ratio between the left-handed wave and the right-handed wave is close to 1 in the transmitted region, whereas it slightly deviates from 1 in the reflected and gated regions. By a straightforward wave matching calculation at the gate boundaries, we find that the deviation of this relative ratio from 1 is linear in J /  (= 0.5% in our simulation) in the reflected and gated regions, while it is proportional to  J ( / ) 3 in the transmitted region, which explains the observed pattern (see Supplementary S2).
Spin wave state on the Bloch sphere. Manipulations of AFM spin waves are not limited to the Faraday rotation, which is realized by the coupling of σ 3 via the DMI. Coupling to other Pauli matrices extends the attainability of spin wave states to the entire Bloch sphere (see Fig. 3). It has been shown that besides the DMI, an electric field could also induce magnetic anisotropy if it is tilted towards the easy axis 19 . This amounts to applying an artificial magnetic field along ê 1 in the internal degenerate space, which rotates the spin wave state in the −ê e 2 3 plane, toggling the spin wave chirality. In contrast, the DMI studied above behaves as a magnetic field along ê 3 that rotates the spin wave state along the equator. Under the basis of the circular modes, the equation of motion for spin waves is given by  with the effective Hamiltonian where ϑ G is the polar angle of the gate voltage direction with respect to the ẑ-axis, and ′  is the strength of the E-field induced anisotropy. This Hamiltonian enables us to explore the entire Bloch sphere spanned by the wave , where θ and φ are the spherical angles specifying the position on the Bloch sphere as shown in Fig. 3.
The mapping of the AFM spin wave onto a Bloch sphere places it on an equal footing as the electron spin, in the sense that it can store information through coherent superpositions of two states. Therefore, the AFM spin wave can be regarded as a classical analog of the quantum bit. However, unlike the challenging task of manipulating quantum information, manipulating AFM spin waves is much simpler. For example, by tunning the direction of the gate voltage (i.e., the angle ϑ G in Eq. (8)), the artificial fields that couple to σ 1 and σ 3 could be made equal, which mimics the behavior of a Hadamard gate in quantum computing.
Detection of spin wave state. Besides conventional optical approach 10,11 , a spin wave state (θ, φ) can also be read off by virtue of spin pumping 14 . Suppose that a heavy metal is deposited in direct contact to the AFM at the far end of a spin wave FET [see Fig. 2(a)]. Spin waves transmitted from the gate can pump spin current into the heavy metal, and this spin current is converted into the inverse spin Hall voltage 1,27 that is monitored by two voltmeters V 1 and V 2 28 . By measuring the DC component V 2 and the effective AC component ∼ V 1 (the root-mean-square value), one can determine θ and φ by (see Supplementary S3) where V m 2 is the maximum DC voltage along x (for purely right-handed spin wave), and ∼ V m 1 is the value of ∼ V 1 at zero gate voltage (also its maximum). In principle, the effective AC component can be determined by measuring the output power of the circuit.

Discussion
In contrast to FMs, AFMs are devoid of the long range dipolar interaction. In thin film FMs, the dipolar interaction substantially modifies the spectrum in the long wavelength limit 29 . In particular, the wave vector shift Δk due to the DMI tends to zero near the FM resonance frequency 30,31 . In AFMs, however, Δk is a constant proportional to the DMI strength as long as ω is greater than the resonance frequency, as shown above by Eq. (5). Therefore, the DMI induces the change of wave vectors more efficiently in AFMs than FMs.
Our proposal for the AFM spin wave FET hinges on the ability to generate sufficiently strong DMI via a gate voltage, which can be realized in compounds with heavy elements. Another crucial requirement is the availability of suitable AFMs with only easy-axis anisotropy. Possible candidates include MnF 2 23 , FeF 2 24 , and RbMnF 3 25 ; all are antiferromagnetic insulators. However, the Néel temperatures of these materials are too low (below 100 K) for room temperature applications. A promising replacement is NaOsO 3 with the Néel temperature around 410 K 26 . It is a G-type collinear AFM with a dominating easy-axis. To fully unlock the potential of AFM spin wave FETs thus calls for further development of room-temperature easy-axis AFMs.

Methods
The simulations were performed by solving the Landau-Lifshitz-Gilbert equation on a 1D chain using the Dormand-Prince method. We set the saturated staggered magnetization to be unity and scaled everything with frequency. The parameters were then taken as ω J = 100 THz, ω D = 500 GHz, and ω K = 10 GHz. In order to avoid effects on the gate system due to reflection from the chain boundaries, we made the length of the chain much larger than the gate system.