Magnonic Goos–Hänchen Effect Induced by 1D Solitons

A field-effect transistor, which is the key element of modern complementary metal oxide semiconductor (CMOS) technology, is used in microprocessors, static random access memory and other digital logic circuits. Nevertheless, the future of CMOS is not clear, since both the miniaturization of single-element sizes and the operational speed will reach their ultimate limits in the near future1. Therefore, one of the primary tasks facing modern science is the search for alternative concepts to CMOS. Distinct achievements in this direction are the development of transistors based on carbon nanotubes and graphene nanoribbons2 3, spin torque transistors4 5 as well as three-dimensional spintronic circuits6. However, these approaches do not resolve another drawback of CMOS technology: the generation of waste heat during switching that is responsible for an increase in the power consumption of CMOS devices. Moreover, the waste heat increases with increasing data-processing speed due to the high switching frequencies. This fundamental drawback is inherent not only in CMOS, but in electronics in general since it is associated with a translational motion of electrons. Thus, there is a strong need for the development of new particle-less technologies for data transport and processing. Magnons, which are the quanta of spin waves7 8, are excellent candidates for carriers in such technologies. Magnonics, the field of science dealing with magnon-based data operations9 10 11 12, encompasses a full spectrum of phenomena used in general wave-based signal processing13 14 15 16 17. The data can be coded into magnon phase or density and processed using wave effects such as interference. This approach has already been realized in spin-wave logic gates performing XNOR and NAND operations18 19 20 21. The main drawback of these gates is that the input data were coded in a form of direct current electric pulses manipulating magnon phases, while the output signal was carried by the magnons themselves. Obviously, that made it impossible to combine two logic gates without additional magnon-to-voltage converters. Moreover, the processing of large amounts of data has to be made on the same magnetic chip exclusively within the magnonic system. This fact stimulated a search for a way to control one magnon by another magnon and for the development of the all-magnon device presented in this paper. In addition, recent discoveries in the fields of spin transfer torque22 23, spin pumping and inverse spin Hall effects24 25 26 made it possible to perform interconversion of currents of magnons to electron-carried spin- and charge-currents and combine, in such a way, magnonic circuits with spintronic or CMOS devices. Here we report on the realization of an insulator-based magnon transistor. The information is carried and processed in this three-terminal device using magnons and is fully decoupled from free electrons. The device demonstrated here has the potential to be scaled down27 to the sub-ten nanometer scale using exchange magnons26 28. Regarding frequency, there is large potential for ultra-fast data processing since magnon frequencies can reach up into the THz range28 29 30. Results Transistor’s design The magnon transistor is shown schematically in Fig. 1a. Its main element is an artificial magnetic material—a magnonic crystal31 32 33 34 35 36 37 38 designed in the form of a yttrium iron garnet (YIG) strip with periodic modulation of its thickness33 38: in our demonstrator, an array of 20 parallel grooves was etched into the surface of the strip (see Methods). The flow of magnons propagating through the crystal is partially (around one per cent33) reflected from each groove and the influence of the grooves is negligible for most of the magnons. Nevertheless, the magnons which have wavelengths satisfying the Bragg conditions k a=m·π/a (where m is an integer and a=300 μm is the crystal lattice constant) will be resonantly scattered back resulting in the generation of rejection bands (band gaps) in a spin-wave spectrum over which magnon propagation is fully prohibited10 11 12 31 32 33 34 35 36 37 38. The measured magnonic-crystal transmission spectrum shows pronounced band gaps and is displayed in the inset of Fig. 2b. The YIG magnonic crystal is magnetized along its long axis, and thus along the magnon propagation direction, by an applied biasing magnetic field B=177 mT. This provides the conditions for the excitation and propagation of backward volume magnetostatic waves8, which are known for their high scattering efficiency in grooved magnonic crystal39. In order to inject the magnons into and to collect them from the transistor, as well as to control them by the gate magnons, three identical microstrip antennas10 are placed at equal distances 4 mm apart. The antenna that injects magnons into the transistor’s gate is placed at the middle of the magnonic crystal area, while the source and the drain antennas are placed outside in the non-structured YIG film areas. It has to be emphasized that all antennas are only used as an instrument for magnon injection and detection, and thus they are not part of the transistor. In forthcoming all-magnon circuits, the antennas will be absent and the magnons will be injected into or collected from another magnon data-processing unit via magnon conduits. A promising way for this might be, for example, the usage of T-shaped structures, which allow for the design of two-dimensional magnon circuits40. Operational principle of the magnon transistor The operational principle of the transistor is shown in Fig. 1c. The idea is as follows: magnons are injected into the transistor’s source (S-magnons shown in blue in the scheme in Fig. 1c) at a frequency corresponding to the magnon transmission band, that is, f S=7.025 GHz, see the inset in Fig. 2b. In the case of absence of magnons in the transistor’s gate, the S-magnons propagate towards the drain practically without distortion33. Therefore, the density of the magnons at the drain (D-magnons) n D is given by the density of the S-magnons n S with additional consideration being given to magnetic losses due to weak magnon–phonon scattering8. In order to manipulate the magnon current flowing from the transistor’s source to the drain, magnons are injected into the gate region (G-magnons, shown in red in Fig. 1c). The frequency of the G-magnons f G=7 GHz is chosen to be in the centre of the magnonic crystal band gap that prohibits the flow of the G-magnons out of the magnonic crystal area. This confinement allows for a large increase of the G-magnon concentration and, consequently, for an increase in the related efficiency of nonlinear phenomena. The spatial distribution of the G-magnons in the gate region in the absence of S-magnons was measured using the technique of Brillouin Light Scattering10 41 and is shown in Fig. 1b. The localization of the G-magnons in the central part of the gate is clearly visible. Further, the S-magnons injected into the source region, while propagating through the transistor’s gate populated with G-magnons, are scattered and, therefore, are able to reach the transistor’s drain only partially (see Fig. 1c). To demonstrate this principle experimentally, we applied a pulsed microwave signal of 50-ns duration at time t=0 to the source antenna. The injected S-magnon packets propagate a distance of 8 mm to the transistor’s drain in approximately 300 ns, inducing the output microwave signal at the drain antenna (this value corresponds well with the calculated magnon velocity of 28 km s−1, which results in a propagation time of 286 ns). The applied rectangular input pulse as well as the transmitted magnon signal, which is slightly distorted from the rectangular shape due to a dispersion spreading, are shown in Fig. 2a. The detected D-magnon density is maximal in the absence of gate magnons, that is, n G=0 mW, and its maximum density is denoted as n D0. (Please note that for simplicity, we measure the magnon densities in Watts corresponding directly to the power of the microwave signals applied to or measured at the antennas.) The application of a continuous-wave signal to the gate antenna and the injection of the G-magnons result in the suppression of the transmitted signal as shown in the figure. The profile of the transmitted signal subject to suppression remains undisturbed thus ensuring information preservation. For relatively large magnon densities of n G≥80 mW, almost no magnons are able to propagate through the gate region, and the peak density of the D-magnons (marked as n D) approaches zero. The dependence of the D-magnon density n D normalized to the original density n D0 is shown in Fig. 2b as a function of the G-magnon density. It can be seen that the suppression increases monotonically with an increase in the G-magnon density and spans almost three orders of magnitude. For the purpose of digital data processing, we use the approach used in CMOS assuming that logic ‘0’ corresponds to the amplitude of the signal smaller than 1/3 of the signal maximum (n D/n D0 2/3). This approach allows for the minimization of possible errors in definition of the bit state. Figure 2b shows that it is sufficient to inject G-magnons having a density of only 25 mW in order to switch the binary signal from ‘1’ to ‘0’. Underlying physical mechanism In order to determine the physical mechanism responsible for the magnon scattering in the transistor’s gate, we analyse the magnon dispersion characteristics as shown in Fig. 3a. Dispersions are plotted for 40 spin-wave thickness modes propagating along the source-to-drain axis in both directions thus having positive and negative wavenumbers k. The originally injected S-magnons are shown by a blue-filled circle located on the lowest thickness mode having the maximum excitation efficiency. The G-magnons, injected with a Bragg wavenumber of k G=±k a (red-filled circles) propagate in both directions simultaneously until they undergo Bragg scattering. The standing nature of the G-magnons is shown in the figure with a red-dashed horizontal line. The key issue of nonlinear interactions in magnetic media is the density of magnons42. In our case, we have artificially increased the density drastically due to the fact that the G-magnons are not able to leave the magnonic crystal and are all localized under the injecting antenna just in the middle of the gate region (see Fig. 1b). If the concentration of the G-magnons is high enough, the four-magnon scattering mechanism becomes pronounced8 42 43: an S-magnon collides with a G-magnon creating a pair of secondary magnons, see orange ovals in Fig. 3a. The frequencies f 3, f 4 and the corresponding wavenumbers k 3, k 4 of the secondary magnons are determined by the energy and momentum conservation laws: f S+f G=f 3+f 4; k S +k G =k 3 +k 4 . Considering that f S≈f G, one might assume that one of the most probable scattering mechanisms is scattering with frequency conservation, that is, f 3=f 4=(f S+f G)/2 as is shown in the figure. Nevertheless, we cannot exclude other mechanisms with one of the secondary magnons having a lower and the other one a higher frequency. Taking into account the Bose–Einstein distribution of energies for exchange magnons, it can be shown8 that the density of states (and thus the four-magnon scattering probability) is proportional to k 3 2=k 4 2. This density of states is shown in Fig. 3b demonstrating that the scattering probability increases quadratically with the wavenumber. Further, the group velocity, which is responsible for the flow of energy out of the over-populated magnon zone44, also influences the efficiency of the scattering since it determines the concentration of the secondary magnons in the gate region. The average of the group velocity of all modes (presented in Fig. 3a) is shown in Fig. 3b. Well-pronounced velocity minima are visible for wavenumbers of about ±105 rad cm−1. Thus, magnons with wavenumbers equal to or larger than 105 rad cm−1 have the highest probability of participation in four-magnon scattering as shown in Fig. 3a (orange ellipses). Since the magnons having such high wavenumbers are not detectable in our experimental conditions, this conclusion is supported by the experimental fact that we were unable to see any additional magnons at the drain antenna (see Fig. 2a). We also performed an additional experiment measuring the magnons reflected back from the transistor’s gate region. However, no magnons reflected by G-magnons have been detected at the source antenna corroborating our model. In addition, due to the high efficiency of the observed suppression of the source-to-drain magnon current, we might assume that avalanche-like processes also take place when the secondary magnons generated after scattering play the role of new scatterers for the source magnons. In order to determine the time required for the four-magnon scattering process to develop, we have performed additional experiments with a continuous microwave signal applied to the source antenna and a pulsed control signal at the gate antenna. The results show that the time required for the transistor to be closed decreases rapidly with an increase in the G-magnon density and varies between 170 ns and 70 ns for densities of n G=10 mW and n G=100 mW, respectively. Conversely, the time the system needs to recover increases with increase of n G. However, this dependence is not so pronounced: The fastest opening of the transistor took about 50 ns for n G=10 mW and the slowest 110 ns for n G=100 mW. The four-magnon scattering process is not the only mechanism that might take place in the magnon transistor. With further increase of the magnon density, the YIG saturation magnetization decreases resulting in a shift of the dispersion relation and the magnon spectrum towards lower frequencies37. To prove the absence of this mechanism in our experiments, we have measured the magnon transmission spectrum in the presence of G-magnons, see Fig. 3c, where the first band gap and the two neighbouring transmission bands are shown. One can see that a suppression of the magnon transmission, rather than a shift of the band gap as shown in ref. 37, takes place. This proves that the four-magnon scattering mechanism occurring in the transistor is more efficient and requires smaller magnon densities. Additionally, Fig. 3c shows that no heating effects influence our experiments since these would result in a shift of the band gap45. From an applications point of view, the equivalent suppression of the source-to-drain magnon current in the transmission bands independently from the magnon frequency allows for the transistor to operate simultaneously with several source signals separated in frequency. Amplification of magnon-carried signals Semiconductor transistors are widely used not only for logical operations but also for the amplification of electronic signals. Such amplification is based on the control of a large signal applied to the transistor’s source by a weak one at the gate. To demonstrate the possibility of this kind of amplification using our device, we have performed a set of n D(n G) measurements for different densities of S-magnons at the signal source. The results are shown in Fig. 4a, where the suppression rate of the source-to-drain magnon current ζ=1−n D/n D0 (where n D and n D0 are the densities of the drain magnons with and without signal at the gate, respectively) is presented as a function of the gate magnon density n G normalized to the source magnon density n S. One sees that the suppression mechanism does not depend on the density of the S-magnons: the same percentage of S-magnons is scattered in the transistor’s gate for a given G-magnon density. This can be considered a good operational characteristic of the device. The vertical-dashed line in Fig. 4a separates two regions with n S being smaller and larger with respect to the gate magnon density n G. It can indeed be observed that a high density of S-magnons can be manipulated by a small density of G-magnons, that is, n G n G is fulfilled (see Fig. 4a), amplification of the magnonic current can be realized. In the approximation of negligible magnetic loss, the output signal n O can be calculated as , where the D-magnon density can be found as n D=n S (1−ζ). The gain factor A is then given by the ratio A=n O/n I. We have estimated the possible gains of our concrete realization of the transistor for the S-magnon density n S of 200 mW. The gain A is shown in the inset of Fig. 4b as a function of the G-magnon density n G. The gain initially increases with an increase in n G but after reaching some maximum value, it slowly decreases (since if the transistor is practically closed, further increase in n G in all practical terms does not affect the output n O). The maximum gain factor A achievable in our case is A max=1.8 but it can be considerably improved by an increase in concentrations of G-magnons during the course of further miniaturization/optimization of the transistor. Logic operation on the basis of a magnon transistor The main motivation behind the development of the magnon transistor is the possibility to perform all-magnon processing of digital data. As was shown in the original papers on spin-wave logic gates18 19 20, the usage of Mach–Zehnder interferometer allows for the realization of different types of logic gates. Here we also propose the use of a similar concept, but with two magnon transistors embedded into the arms of the interferometer. The operational principle of such a logic gate is given with the example of a XOR gate and is shown in Fig. 4c. First, the feeding magnon current, which is sent to the interferometer (the left arrow in the figure), is divided into two identical currents. These currents n S1 and n S2 are directly applied to the transistors’ sources S1 and S2 and are independently controlled by the input magnon signals I 1=n G1 and I 2=n G2 injected into the gates G1 and G2. The input signal I=‘1’ corresponds to the critical gate magnon density n G crit large enough to decrease the magnon density at the drain to the value n D