Spin transfer torque driven higher-order propagating spin waves in nano-contact magnetic tunnel junctions

Short wavelength exchange-dominated propagating spin waves will enable magnonic devices to operate at higher frequencies and higher data transmission rates. While giant magnetoresistance (GMR)-based magnetic nanocontacts are efficient injectors of propagating spin waves, the generated wavelengths are 2.6 times the nano-contact diameter, and the electrical signal strength remains too weak for applications. Here we demonstrate nano-contact-based spin wave generation in magnetic tunnel junctions and observe large-frequency steps consistent with the hitherto ignored possibility of second- and third-order propagating spin waves with wavelengths of 120 and 74 nm, i.e., much smaller than the 150-nm nanocontact. Mutual synchronization is also observed on all three propagating modes. These higher-order propagating spin waves will enable magnonic devices to operate at much higher frequencies and greatly increase their transmission rates and spin wave propagating lengths, both proportional to the much higher group velocity.

S teady-state large-angle magnetization dynamics can be generated via spin-transfer torque (STT) 1-3 in a class of devices commonly referred to as spin torque nanooscillators (STNOs) [4][5][6][7][8][9][10] . The typical building block of an STNO is a thin-film trilayer stack, where two magnetic layers are separated by a nonmagnetic spacer. The charge current becomes partially spin polarized by the magnetic layers and can act as positive or negative spin wave (SW) damping depending on its polarity. Above a certain critical current density, the negative damping can locally overcome the intrinsic damping, resulting in auto-oscillations on one or more SW modes of the system. To sustain such auto-oscillations, a large current density of the order of 10 6 −10 8 A/cm 2 is required, which can be achieved by spatial constriction of the current, e.g. using a nanocontact (NC) on top of a GMR trilayer stack. Such NC-based STNOs are also the most effective SW injectors for miniaturized magnonic devices [11][12][13][14] , in particular for short wavelength, exchange-dominated SWs, since the wave vector (k) is inversely proportional to the NC radius (r NC ) through the Slonczewski relation k = 1.2/r NC . As the SW group velocity, which governs the data transmission rate, scales with k, and the operating frequency with k 2 , future ultra-high data rate magnonic devices will have to push the SW wavelength down to a few 10s of nanometers 15 .
For efficient electrical SW readout, magnonic devices will also have to be based on magnetic tunnel junctions (MTJs), as tunneling magnetoresistance (TMR) is one or more orders of magnitude higher than GMR 16,17 . The relatively low conductivity of the MTJ tunneling barrier compared with the top metal layers leads to large lateral current shunting for an ordinary NC (Fig. 1a). To force more of the current through the MTJ, we instead fabricate so-called sombrero NCs (Fig. 1b), in which the MTJ cap layer is gradually thinned as it extends away from the NC 18,19 , and use a MgO layer with a low-resistance area (RA) product of 1.5 Ω m 2 to further promote tunneling through the barrier. The resulting devices exhibit the typical SW modes associated with NC STNOs, such as the spin-wave bullet 20-23 and the Slonczewski propagating SW mode 3 . In addition, we observe two additional, higher-frequency modes, which we identify as the second-and third-order propagating SW modes mentioned in Ref. 3 , but never previously observed. We estimate the wavelengths of these two modes to be 120 and 74 nm, i.e., much smaller than the 150-nm nanocontact. Using double nanocontact devices, we furthermore observe mutual synchronization on all three propagating modes, corroborating their propagating character.

Results
Magnetostatics. Figure 1c shows the magnetic hysteresis loop of the unpatterned MTJ stack in a magnetic field applied along the in-plane easy axis (EA) of the magnetic layers (for details see Methods). Figure 1d shows the corresponding resistance (R) of an MTJ-STNO with a nominal diameter of d NC = 150 nm, displaying a magnetoresistance (MR) of 36%, confirming that a significant fraction of the current indeed tunnels through the MgO barrier. The very good agreement between the field dependence of the unpatterned stack and the fully processed MTJ-STNO suggests minimal process-induced changes of the magnetic layers, a strong indication that the free layer (FL) remains intact.
Magnetodynamics. Figure 2 shows the generated power spectral density (PSD) vs. field strength during auto-oscillations at six different drive currents, with the field angle fixed to θ ex = 85°(the color plot is assembled from PSD measurements at constant current and field, a few of those shown in Supplementary At stronger negative currents, I dc = −7 mA, the PSD in the highfield region changes dramatically, as a much stronger mode appears with a frequency much higher than f FMR . This change is accompanied by additional low-frequency noise, indicative of mode hopping. Further increasing the negative current strength to I dc = −8 mA, first modifies this new mode, after which another sharp jump up to an even higher frequency is observed at about 1.6 T. As we increase the current magnitude further to −9 and −10 mA, this new mode dominates the entire high-field region. The lowfrequency noise is now concentrated to the field region just above the critical field, where the 2nd and 3rd modes appear to be competing. To analyze this behavior, we draw renewed attention to the higher-order propagating SW modes put forward by Slonczewski 3 but up to this point entirely overlooked in experiments. The excited propagating SWs have a discrete set of possible wave vectors r NC k ≃ 1.2, 4.7, 7.7..., where only the first-order mode (r NC k ≃ 1.2) is discussed in the literature because of its lower threshold current. Taking the literature value 25 for the free-layer exchange stiffness, A ex = 23 × 10 −12 J/m, and allowing for a reasonable lateral current spread 26 (an effective NC radius of r NC = 90 nm), we find that we can fit the field-dependent frequencies of both the second-and third-mode almost perfectly using the predicted k = 4.7/r NC and k = 7.7/r NC . The ordinary first-order mode can be equally well fitted (not shown). It is noteworthy that increasingly higher currents are required to excite the higher mode numbers, in agreement with Slonczewski's original expectations 3 . We also point out that in the original derivation only radial modes were considered, excluding any azimuthal modes. The additional smaller frequency step observed within the 2 nd radial mode in Fig. 2(d) could hence be related to a further increase in the exchange energy of a possible azimuthal mode 27 .
In all fits, we allowed M s to be a function of I dc and used the same M s to calculate f FMR , f SWR , and f i¼1;2;3 PSW . This allows us to estimate the amount of heating due to the drive current. Figure 3a shows the variation of M s as a function of temperature (blue circles) measured from 10 to 340 K using temperature-dependent FMR spectroscopy on unpatterned areas of the MTJ stack. We can fit M s (T) to a Bloch law function and extrapolate this dependence to higher temperatures (black solid line). The red triangles in Fig. 3a then show the extracted M s values at each I dc placed on the extrapolated fit, which allow us to extract the local temperature of the FL underneath the NC. As can be seen in the inset, the temperature shows a parabolic rise with increasing current, indicative of Joule heating. The current-induced temperature rise at e.g., I dc = −9 mA is about 220 K, which is consistent with literature values of nanoscale temperature gradients in similar structures sustaining similar current densities 28 .
We then show how we can control which propagating mode to excite by varying the current at constant applied field (Fig. 3b). We can again fit the three modes very accurately using the currentdependent M s (I dc ) extracted from Fig. 3a. The weak current tunability of our MTJ-based NC-STNOs is consistent with the weak non-linearity values found in the literature on MTJ pillars 29,30 and is advantageous as it reduces any non-linearity driven increase in phase noise from amplitude noise 31,32 . It is also consistent with theoretical predictions that the non-linear frequency shift, at a constant current, should decrease strongly with increasing NC size and only be prominent for sub 100-nm NCs 33 . Since our large MTJ-based STNOs can auto-oscillate at about the same currents as  Mutual synchronization. In Fig. 4, we show experimentally that it is also possible to achieve spin-wave-mediated mutual synchronization 34-37 on all three modes, further corroborating their propagating character. Figure 4a shows the PSD vs. field for a double-NC MTJ-STNO auto-oscillating on the first Slonczewski mode (the double NCs were fabricated on the same type of stack as the device above). At fields below 1.22 T, the PSD shows two distinct peaks at high frequency and substantial microwave noise below 2.5 GHz. Above 1.22 T, the two signals instead merge into a single signal with a frequency in between the first two, and the microwave noise disappears. These observations are consistent with highly interacting but individually auto-oscillating regions below 1.22 T and a robust mutually synchronized state above 1.22 T.
As the original double NC devices did not survive the currents required for auto-oscillations on the higher modes, we instead made a second batch of devices based on an improved MTJ stack with a thicker bottom electrode (150 nm CuN instead of 30 nm) to promote more of the current to pass through the tunneling barrier and further reduce the lateral current spread 26 . As a result, we could effectively reduce the threshold current density for all three modes. Figure 4b shows the PSD vs. current from one such optimized double nano-contact device operating at 1.18 T. The NCs first auto-oscillate on the first mode at relatively close frequencies (Fig. 4b). They then jump to the second mode at different currents: one NC at about −13 mA and the other at about −17 mA. As soon as both NCs auto-oscillate on the second mode, their frequencies approach each other, and at about −18 mA, both the first and the second harmonic indicate mutual synchronization. Figure 4c shows the same device at a higher-field of 1.45 T. At this field, the second mode does not show mutual synchronization at any current. However, when both NCs jump to the third mode, their overlapping signals indicate that mutual synchronization on the third mode is also possible.

Discussion
The possibility of generating higher-order Slonczewski modes has a number of important implications. Their much shorter wavelengths, in our case estimated to 120 nm (2nd mode) and 74 nm (3rd mode), already bring them into the important sub 100 nm range 15 , which only a few years ago was considered out-of-reach for magnonics 38 . As the SW group velocity increases linearly with the wave vector as v gr ≃ 4γA ex k/M s , much faster transmissions can be achieved in magnonic devices and the SWs can travel significantly farther before being damped out. The calculated group velocities for the three observed modes are v 1 = 258 m/s, v 2 = 1010 m/s, and v 3 = 1655 m/s. This is particularly beneficial for mutual synchronization of multiple MTJ-based NCs. For example, one can find the maximum distance of synchronization, a max , between the two coupled oscillators, using the method developed by Slavin and Tiberkevich 39 . Using typical parameters of coupling strength, Δ max = ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ν 2 p ¼ 50 MHz, and a Gilbert damping of α G = 0.015, we find a max = 240, 350, and 420 nm for the PSWs with the corresponding frequencies f PSW = 13.5, 17.7, and 24.7 GHz observed at μ 0 H ext = 1.6 T (Fig. 3). As the drive current of both single and mutually synchronized STNOs can be modulated very rapidly [40][41][42][43][44][45] , high data rate frequency shift keying 46 will likely be possible using only a small modulation amplitude of the drive current. In addition, novel modulation concepts such as wave vector keying could be readily realized, with possible use in magnonic devices.
We conclude by pointing out that both the nominal NC diameter (150 nm) and the estimated effective NC diameter (180 nm) are much larger than what could be realized using state-of-the-art MTJ lithography. We see no fundamental reason against fabricating NCs down to 30 nm, which would then translate to wavelenghts down to 15 nm and SW frequencies well beyond 300 GHz. The use of higher-order propagating SW modes might therefore be the preferred route toward ultra-high frequency STNOs.  (7), with thicknesses in nanometer. For the demonstration of mutual synchronization on the higher spin wave modes, the same stack but with a thicker CuN(150) layer was used.

Methods
Nanocontact fabrication. After stack deposition, 16 μm × 8 μm mesas are defined using photolithography. To make the hybrid NC structure, electron-beam lithography (EBL) with a negative tone resist is used to define nanocontacts with a nominal diameter of 150 nm. The negative tone resist is used as an etching mask in the ion beam etching (IBE) process. Etching of the cap layers in IBE is carefully monitored by in situ secondary ion mass spectroscopy to prevent any damage to the layers underneath the cap. After this step, a structure similar to that shown in Fig. 1b is realized. Following the etching process, 30 nm of SiO 2 is deposited to provide electrical insulation between the cap and top contact. The remaining negative tone resist acts as a lift-off layer this time. The devices are left in a hot bath of resist remover combined with a high-energy ultrasonic machine for a successful liftoff. In order to provide electrical access to the devices, top contacts are defined using photolithography.
Static characterization. The static magnetic states throughout key points of the reversal are highlighted as insets in Fig. 1c and d. Decreasing the field from a fully saturated state (1) allows the RL to gradually rotate to be anti-parallel (2) with the PL due to the strong antiferromagnetic coupling (AFC). In going from state 2 → 3, the FL switches rapidly in a relatively small field and once again becomes parallel to the RL, hence a minimum R is restored. Note that the FL minor loop is shifted toward positive fields in both Fig. 1c and d, indicating some weak ferromagnetic coupling to the RL. Upon further decreasing the field, the magnetic state moves from 3 → 4, as the PL, working against the strong AFC and weaker exchange bias, slowly switches to be parallel to the RL. In Fig. 1d, we find small increases in R when moving from states 3 → 4 and 4 → 5. These can be attributed to minor scissoring of the RL and PL layers due to their strong AFC. As one goes from state 5 → 6 → 1, the FL switches to align with the applied field, followed by the RL switching at high field.
Ferromagnetic resonance (FMR) measurements. The magnetodynamic properties of the free layer (CoFeB) are determined using an unpatterned thin-film stack. The inset of Fig. 1d shows the extracted FL resonance field from broadband FMR measurements (blue squares), fitted with the standard Kittel equation (red line). From the fit, we extract the values of the gyromagnetic ratio, γ/2π = 29.7 GHz/T, and effective saturation magnetization, μ 0 M eff = 1.41 T. Subsequent microwave measurement are performed such that the in-plane component of the field lies along the EA of the MTJ stack. We also study the temperature dependence of the magnetodynamcis at low temperature using a NanOsc Instrument CryoFMR system. The low-temperature measurements are performed between 10 K-340 K. At each temperature, the FMR response was measured at several frequencies over the range 4-16 GHz, where an external magnetic field is applied in the film plane. At each frequency, the resonance field (H res ) is extracted by fitting the FMR to a Lorentzian function. We extracted the effective magnetization (M eff ) of CoFeB thin films by fitting the dispersion relation (frequency vs. field) to the Kittel equation , where γ 2π is the gyromagnetic ratio. We fit the variation of M eff with the temperature to a Bloch's law to extract M eff at higher temperatures (T > 340 K).
Microwave measurements. All measurements were performed at room temperature. In-plane magnetization hysteresis loops of the blanket MTJ multilayer film stacks were measured using an alternating gradient magnetometer (AGM). The MR was measured using a custom-built four-point probe station. The magnetodynamic properties of the unpatterned FL were determined from using a NanOsc Instruments PhaseFMR spectrometer.
Microwave measurements were performed using a probe station with a permanent magnet Halbach array producing a uniform and rotatable out-of-plane field with a fixed magnitude μ 0 H = 0.965 T. A direct electric current, I dc , was applied to the devices through a bias tee, and the resulting magnetodynamic response was first amplified using a low-noise amplifier and then measured electrically using a 40 GHz spectrum analyzer (see Supplementary Information). Microwave measurements at higher fields were performed using another custom-built setup capable of providing a uniform magnetic field of up to μ 0 H = 1.8 T.
Bullet frequency and PSW spectrum calculation. The angular dependence of the nonlinear frequency coefficient, N, is calculated from the following expression 24 : where f FMR is the FMR frequency. f H ¼ γ 2π μ 0 H int , f M ¼ γ 2π μ 0 M s and finally H int and θ int are the internal magnetic field magnitudes and out-of-plane angles, respectively. H int and θ int are extracted using a magnetostatic approximation: The frequency of the Slavin-Tiberkevich bullet mode is calculated from 20 : where f SWB is the bullet angular frequency and B 0 is the characteristic spin-wave amplitude. The calculated f SWB quantitatively describes the measured field dependence by setting B 0 = 0.46, providing further evidence that this mode is, in fact, a solitonic bullet. The value of B 0 is calculated according to Tyberkevych et al. 20 and reaches B 0 = 0.46, which is the upper limit of the theory. The spectrum of the propagating spin waves in the linear limit is defined as 50 where D = 2A ex /(μ 0 M eff ) is the dispersion coefficient and A ex is the exchange stiffness constant.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.