Dynamic control of spin-wave propagation

In this work we present a method to dynamically control the propagation of spin-wave packets. By altering an external magnetic field the refraction of the spin wave at a temporal inhomogeneity is enabled. Since the inhomogeneity is spatially invariant, the spin-wave impulse remains conserved while the frequency is shifted. We demonstrate the stopping and rebound of a traveling Backward-Volume type spin-wave packet.

www.nature.com/scientificreports/ assuming translation invariance and thus wave-vector conservation. If spatial heterogeneities of the medium or magnetic field occur (e.g. at the borders of a magnetic film) the wave-vector composition of the wave packet will be altered.

Methods
A set of micromagnetic simulations, using OOMMF 19 , was conducted to verify this concept. Here we consider a 20 nm thick permalloy film, which is 320 µm long in the Backward-Volume direction (hereafter x direction).
In the out-of-plane direction (here after z direction) the simulation volume is constrained to a single cell. In y direction the simulation volume extents over 4 cells ( 20 nm ) and translation symmetry is approximated by periodic boundary conditions. The exchange constant used is A = 1.3 × 10 −11 J m −3 ; the saturation magnetization was set as M S = 803.7 kA m −1 . Our material of choice for purpose of demonstration is permalloy to stress that the demonstrated approach does not depend on the choice of the specific material. To enhance the quality of the simulations we chose α = 2 × 10 −3 as Gilbert damping parameter, a value which is only a quarter of the real Gilbert damping parameter in permalloy. It is still achievable using other materials, such as Cobalt alloys 20 . A modulated Gaussian pulse of the out of plane magnetic field ( H z ) with a modulation frequency of f c = 4 GHz and FWHM of 3.3 ns excites a Backward-Volume pulse with a central wave vector of k c = 15.8 µm −1 . The spatial distribution of the excitation field is of Gaussian shape with a FWHM of 133 nm in the center of the film ( x = 0 ), exciting two wave packets propagating in opposite directions. For the example of stopping a wave packet, the magnetic field is increased from µ 0 H 0 = 16 mT to µ 0 H stop = 139 mT over 4 ns , beginning 8 ns after the pulse peak. The time dependences of the exciting magnetic field pulse and the external magnetic field are depicted in Fig. 1c. For the demonstration of propagation reversal, the target magnetic field strength is substituted with µ 0 H rev = 350 mT.
Due to the finite size of the simulated film in the direction of the magnetic field (x) an in-plane demagnetization field remains, weakening the total internal field. The demagnetization field is calculated following Joseph and Schlömann 21 , retaining a value of µ 0 H demag = 1.0 mT at x = 0 . A corrected form of the dispersion relation for thin films by Kalinikos and Slavin 18 was obtained by subtracting the demagnetization field from the externally applied field, yielding where H ext − H demag is the static internal field, with the gyroscopic ratio γ = 176 GHz T −1 , the film thickness t and the vacuum permeability µ 0 = 1.625 µH m −1 . The in-plane wave-vectors parallel and orthogonal to the static magnetic field are denoted by k || and k ⊥ , respectively. Furthermore, F is given by www.nature.com/scientificreports/ Here we neglect dynamic in-plane demagnetization fields arising due to the finite lateral extent of the ferromagnetic film, since the static correction is sufficient for the purpose of this paper.

Results
An excerpt of the simulation is depicted in Fig. 2. In the left hand panel the increase of the external field to µ 0 H stop = 139 mT is plotted, while the right hand panel shows the simulation for an increase to µ 0 H rev = 350 mT . The spin-wave packets are formed at x = 0 , propagating in both directions. The packets separate from another and continue to propagate with constant velocity. After the field increase (beginning at t = 15.5 ns ) the envelope of the wave packet remains fixed for the left hand panel, while the direction of propagation is reversed for the right hand panel.  www.nature.com/scientificreports/ It is noteworthy that the spin-wave packets decay quicker after the field increase, especially for H rev . The cause for the accelerated attenuation is found in the frequency dependence of the magnon lifetime 22,23 . Since the lifetime decreases with increasing spin-wave frequency, and the frequency of a spin wave with fixed wave number increases with the magnetic field, the lifetime is reduced when the magnetic field grows.
The wave-number distribution of the spin waves is depicted in Fig. 4. The excited central wave number is determined from the spatial Fourier transform as k c = 15.8 µm −1 , which is also the intended central wave number. Most importantly, in both simulations, the central wave number remains unaltered during and after the field increase in agreement with the postulated conservation of the wave number. In Fig. 4c,d profiles of the wavevector spectra are depicted for times t = 14 ns just before and t = 21 ns after the field increase.
Additionally, in Fig. 4b a boost of the Fourier amplitude is discernible during the field increase, attributed to the raising of the ratio between the external bias field and the dynamic out-of-plane demagnetization field. Since the out-of-plane demagnetization field reduces the out-of-plane component M z of the spin-wave, and thus the relative strength of the dynamic out-of-plane demagnetization field, the out-of-plane spin-wave amplitude increases with the external magnetic field.

Extension to redirection of spin waves
Based on the demonstrated wave-number conservation the concept of the dynamic propagation control can be extended to the two-dimensional case. For demonstration purposes we will again discuss a BV wave packet with a central frequency of f c = 4 GHz and the same medium as before. Here it is helpful to use the concept of the isofrequency curve, which is the set of wave vectors with identical frequency, according to the dispersion relation. The group velocity, which is the gradient of the frequency in k space, is always perpendicular on the isofrequency curve. In Fig. 5a reversal of the wave-packet propagation is depicted. The black line marks the isofrequency curve for the wave packet's initial central frequency of f c = 4 GHz at an external field of µ 0 H 0 = 16 m T , while the red curve marks the isofrequency curve at an external magnetic field of µ 0 H rev = 350 m T . Both isofrequency curves encompass the central wave vector k c (gray dot) as required by the wave-vector conservation. The initial group velocity v 0 is marked by a black arrow and group velocity v rev after the field increase is marked by a red arrow. Figure 5b illustrates a rotation of the external magnetic field. Again, the black line depicts the isofrequency curve for the initial situation, i.e., the field is oriented in x direction ( θ = 0 ). All parameters are identical to the precedent simulations. If, for example, the external field is tilted by 10 • , the static orientation of the magnetization adapts accordingly and the long axis of the isofrequency curve is also tilted by 10 • . Considering a spatially homogeneous field-orientation change, the wave vector, again, remains conserved and the frequency of the spin-wave packet is altered. This means, the isofrequency curve for the new central frequency and field orientation will still include the unaltered central wave vector, as marked by the grey dot. The central frequency changes accordingly to f c = 4.36 GHz . The according isofrequency curve is depicted in Fig. 5 in blue color. The group velocity v 10 • belonging to k c is now tilted by 70 • . It is thus possible to remotely steer spin-wave packets by rotating an external magnetic field. In combination with altering the field strength any orientation of the group velocity is achievable. Furthermore a Backward-Volume spin wave can be converted to a Damon-Eshbach spin wave (strongly boosting the frequency) and vice versa by rotating the external field 90 • .

Conclusion
In summary we propose a method to steer spin-wave packets utilizing dynamic alterations of an external magnetic field. Micromagnetic simulations validate the introduced concept for the propagation control of Backward-Volume wave packets. By choosing the target magnetic field the resulting group velocity of the wave packet can be chosen. It is possible to stop or even reverse the packet propagation. The results of the simulations are in good agreement with the model, especially the wave number remains conserved. The method is expandable to the two-dimensional case, where spin-wave packets can be remotely steered via orientation and strength of the external field. Due to recent advances both in the generation of suitable Backward-Volume packets 24,25 , and the change of the magnetic field in the timescale of few nanoseconds 26 , the proposed concept appears feasible although still challenging. The initial and resulting group velocities are denoted as black and blue arrows, respectively.