Transferring Abrikosov vortex lattices generate sub-40-nm magnons

Fabrication of the microwave nano-antenna

The fabrication of the experimental system started with the deposition of a 40/5 nm Au/Cr movie onto a Si (100 nm, p-doped)/SiO₂ (200 nm) substrate and its patterning for electrical d.c. present and microwave measurements. Within the sputtering course of, the substrate temperature was 22 °C, the expansion charges had been 0.055 nm s⁻¹ and 0.25 nm s⁻¹, and the Ar pressures had been 2 × 10⁻³ mbar and seven × 10⁻³ mbar for the Cr and Au layers, respectively. The microwave ladder antenna was fabricated from the Au/Cr movie by targeted ion beam milling at 30 kV/30 pA in a dual-beam scanning electron microscope (FEI Nova Nanolab 600). The multi-element antenna consisted of ten nanowires linked in parallel between the sign and floor leads of a 50-Ω-matched microwave transmission line. The antenna had a interval p = 108 nm with the nanowire width equal to the nanowire spacing in order that its Fourier rework contained solely odd spatial harmonics with(_=p) and ( _=p/3=36,), which made it delicate to spin-wave wavelengths of 36 ± 2 nm in our experiments (Supplementary Fig. 1).

Fabrication and properties of the Nb–C microstrip

The ladder antenna’s fabrication was adopted by direct writing of the superconducting strip at 2 μm (edge to edge) from the microwave antenna. The 45-nm-thick Nb–C microstrip was fabricated by targeted ion beam-induced deposition. Centered ion beam-induced deposition was carried out at 30 kV/10 pA, 30 nm pitch and 200 ns dwell time using Nb(NMe₂)₃(N-t-Bu) as precursor gasoline²⁶. The superconducting strip and the ladder antenna had been coated with a 3-nm-thick insulating Nb–C layer ready by targeted electron beam-induced deposition (FEBID). Earlier than the Co–Fe magnonic waveguide deposition, a 48-nm-thick insulating Nb–C–FEBID layer was deposited to compensate for the construction peak variations between the antenna and the Nb–C strip. The basic composition within the Nb–C strip was 45 ± 2 at.% C, 29 ± 2 at.% Nb, 15 ± 2 at.% Ga and 13 ± 2 at.% N, as inferred from energy-dispersive X-ray spectroscopy on thicker microstrips written with the identical deposition parameters. The Nb–C strip had well-defined easy edges and a root imply squared floor roughness of <0.3 nm, as deduced from atomic power microscopy scans over its 1 μm × 1 μm energetic half earlier than the deposition of the Co–Fe layer. The 2 ends of the strip had rounded sections to stop present crowding results on the sharp strip edges, which can result in an undesirable discount of the experimentally measured crucial present and the instability present⁴⁵.

The resistivity of the Nb–C microstrip at 7 Ok was (rho_=551,upmu ,). The microstrip transitioned to a superconducting state under the transition temperature T_c = 5.60 Ok, deduced utilizing a 50% resistance drop criterion. Utility of a magnetic subject H_ext ≈ 2 T led to a lower of T_c(0) to T_c(2 T) ≈ 5.1 Ok. Close to T_c, the crucial subject slope ((mathrm_{}/mathrm T) _{_{{rm }}}=-2.19) T Ok⁻¹ corresponds, within the soiled superconductor, to the electron diffusivity (D=-4_mathrm varDelta /(pi e(mathrm {H}_{{rm{c2}}}/mathrm{d}T) _{{T}_{{rm{c}}}})approx 0.5) cm² s⁻¹ with the extrapolated zero-temperature higher crucial subject worth H_c2(0) ≈ 12.3 T. Right here, okay_B is the Boltzmann fixed and e the elementary cost. The coherence size and the penetration depth at zero temperature had been estimated⁴⁶ as ({xi }_{mathrm{c}}={(hslash D/{okay}_{mathrm{B}}{T}_{mathrm{c}})}^{1/2}approx 9,{mathrm{nm}}) (comparable to ({xi}(0)={xi}_{rm{c}}{(1.76)}^{-1/2}approx7,{rm{nm}})) and (lambda (0)=1.05times 1{0}^{-3}{({rho }_{{rm{7K}}}{okay}_mathrm{B}/varDelta (0))}^{1/2}approx textual content{1,040},{mathrm{nm}}). Right here, ({varDelta}(0)) is the zero-temperature superconducting power hole and (hslash) is the Planck fixed.

Fabrication and properties of the Co–Fe conduit

The Co–Fe magnonic conduit was 1μm huge, 5 μm lengthy and 30 nm thick. We fabricated it by FEBID using HCo₃Fe(CO)₁₂ because the precursor gasoline⁴⁷. FEBID was carried out with 5 kV/1.6 nA, 20 nm pitch and 1 μs dwell time. The fabric composition within the magnonic waveguide is 61 ± 3 at.% Co, 20 ± 3 at.% Fe, 11 ± 3 at.% C and eight ± 3 at.% C. The oxygen and carbon are residues from the precursor within the FEBID course of⁴⁷. The Co–Fe conduit consisted of a dominating bcc Co₃Fe part blended with a minor quantity of FeCo₂O₄ spinel oxide part with nanograins of about 5 nm in diameter. The random orientation of Co–Fe grains within the carbonaceous matrix ensured negligible magnetocrystalline anisotropy. Additional particulars on the microstructural and magneto-transport properties of Co–Fe–FEBID had been reported beforehand⁴⁷.

Electrical resistance measurements

The I–V curves had been recorded in current-driven mode inside a ⁴He cryostat fitted with a superconducting solenoid. The exterior magnetic subject H_ext was tilted at a small angle (beta={5}^{circ}) with respect to the conventional to the pattern aircraft (z axis) within the aircraft perpendicular to the path of the transport present. The small subject tilt angle (beta) ensures that the sphere part H_ext,z appearing alongside the z axis is simply negligibly smaller than H_ext with (H_ext − H_ext,z)/H_ext ≤ 0.5%. The transport present utilized alongside the y axis in a magnetic subject H ≈ H_ext = H_z exerts on vortices a Lorentz power appearing alongside the x axis²⁸. The voltage induced by the vortex movement throughout the superconducting microstrip was measured with a nanovoltmeter. A collection of reference measurements was taken earlier than the deposition of the Co–Fe magnonic conduit on prime of the Nb–C strip. No voltage steps had been revealed within the I–V curves of the naked Nb–C strip. Against this, constant-voltage steps within the I–V curves had been revealed after the deposition of the Co–Fe magnonic conduit on prime of the Nb–C strip. The hardly ever achieved mixture of weak quantity pinning, together with close-to-perfect edge limitations and a quick leisure of nonequilibrium electrons, permits for ultrafast movement of Abrikosov vortices within the Nb–C superconductor³¹.

Microwave detection of spin waves

The microwave detection of spin waves was carried out utilizing a microwave ladder nano-antenna linked to a spectrometer system. This allowed detecting alerts at energy ranges right down to 10⁻¹⁶ W in a 25 MHz bandwidth³². The detector system consisted of a spectrum analyser (Keysight Applied sciences N9020B, 10–50 GHz), a semirigid coaxial cable (SS304/BeCu, d.c.–61 GHz) and an ultrawide-band low-noise amplifier (RF-Lambda RLNA00M54GA, 0.05–54 GHz, +20 dB achieve).

Micromagnetic simulations

The micromagnetic simulations had been carried out utilizing the graphics processing unit-accelerated simulation package deal MuMax3 to calculate the investigated buildings’ space- and time-dependent magnetization dynamics⁴¹. The simulations had been carried out for the next Co–Fe parameters: saturation magnetization M_s = 1.4–1.5 MA m⁻¹, trade fixed A = 15–18 pJ m⁻¹ and Gilbert damping ({alpha}=0.01). The very best match of the simulation outcomes with the experimental information has been revealed for M_s = 1.45 MA m⁻¹ and A = 15 pJ m⁻¹. The mesh was set to 2 × 2 nm², which is smaller than the trade size of Co–Fe (~5 nm) and fulfils the necessities for micromagnetic simulations. The simulations had been validated by evaluating with the outcomes obtained throughout the Kalinikos–Slavin principle⁴⁸ (Supplementary Fig. 3). The estimated attenuation size of the generated magnons (at okay_SW ≈ 175 rad μm⁻¹) is round 600 nm (ref. ⁴⁸).

An exterior subject H_ext within the vary 1.75–1.95 T, enough to magnetize the construction to saturation, was utilized at a small angle (beta) relative to the z axis within the x–z aircraft (Supplementary Fig. 4). A quick-moving periodic subject modulation was used to imitate the impact of the transferring vortex lattice. The oscillations m_x(x, y, t) had been calculated for all cells and all instances through m_x(x, y, t) = M_x(x, y, t) − M_x(x, y, 0), the place M_x(x, y, 0) corresponds to the bottom state (totally relaxed state with none transferring magnetic subject supply). The dispersion curves had been obtained by performing two-dimensional quick Fourier transformations of the time- and space-dependent information. The spin-wave spectra had been calculated by performing a quick Fourier transformation of the info in a area at a distance of 1 μm from the spin-wave excitation area (Supplementary Fig. 5). Full particulars on the micromagnetic simulations are given in Supplementary Word 1. The evolution of the magnon era situation upon variation of the magnetization, trade stiffness and thickness of the Co–Fe conduit is illustrated in Supplementary Figs. 6–9. The noticed magnon era could also be interpreted by two eventualities, specifically the coherent fluxon-magnon coupling and the magnonic Cherenkov impact. The relation between the Cherenkov impact for single and a number of periodically organized transferring particles is illustrated in Supplementary Fig. 10.

Ginzburg–Landau simulations

The evolution of the superconducting order parameter ({varDelta}=|{varDelta}|{e}^{i{varPhi}}) was analysed relying upon a numerical answer of the modified TDGL equation⁴⁹, solved together with the heat-balance equation, to account for attainable heating results

the place ({xi }_{{rm{mod}}}^{2}=pi sqrt{2}hslash D/(8{okay}_{{rm{B}}}{T}_{{rm{c}}}sqrt{1+{T}_{{rm{e}}}/{T}_{{rm{c}}}})), ({varDelta }_{{rm{mod}}}^{2}={({varDelta }_{0}tanh (1.74sqrt{{T}_{{rm{c}}}/{T}_{{rm{e}}}-1}))}^{2})(/(1-{T}_{{rm{e}}}/{T}_{{rm{c}}})), A is the vector potential, φ is the electrostatic potential, D is the diffusion coefficient, ({sigma}_{rm{n}}=2{e}^{2}DN(0)) is the normal-state conductivity with N(0) being the single-spin density of states on the Fermi degree, T_e and T_p are the electron and phonon temperatures, and ({{bf{j}}}_{{rm{s}}}^{{rm{Us}}}) and ({{bf{j}}}_{s}^{{rm{GL}}}) are the superconducting present densities within the Usadel and Ginzburg–Landau fashions

the place ({bf{q}}_{rm{s}}={nabla}{varphi}+2{pi}{bf{A}}/{varPhi}_{0}) and ({{bf{j}}}_{{rm{s}}}^{{rm{GL}}}=frac{pi {sigma }_{{rm{n}}}| varDelta ^{2}}{4e{okay}_{{rm{B}}}{T}_{{rm{c}}}hslash }{{bf{q}}}_{{rm{s}}}).

The vector potential A = (0, A_y, 0) within the TDGL equation consists of two components: A_y = H_extx + A_m, the place H_ext is the exterior magnetic subject and A_m is the vector potential of the magnetic subject induced within the superconducting strip by spin waves.

The modelled size of the microstrip is L = 4w, the width (w=50{xi}_{c}), and the parameter ({B}_{0}={varPhi}_{0}/(2{{pi}}{xi}_{rm{c}}^{2})approx 4.15,{rm{T}}), the place ({xi}_{c}=8.9,{rm{nm}}). The calculations had been carried out with parameters (gamma=9) and ({tau}_{0}=925,{rm{ns}}) for NbN, as their values for NbC are unknown however are presupposed to be of the identical order of magnitude. A variation of (gamma) and (tau) solely results in quantitative modifications within the I–V curves and doesn’t qualitatively change the vortex dynamics. In simulations, dA was various between 0 (no ferromagnet layer) and (0.1{varPhi}_{0}/(2{pi}{xi}_{rm{c}})), which corresponds to about 1/4 of the depairing velocity for superconducting cost carriers (Cooper pairs) or crucial (q_{rm{sc}}0.35,{varPhi}_{0}/2{pi}{xi}_{rm{c}}) of the superconducting strip at B = 0 and T = 0.8T_c. The parameters a_x and a_y had been chosen to mannequin a triangular transferring vortex lattice with out the ferromagnetic layer and much from the instability level. We current the outcomes for ({v}=110{xi}_{rm{c}}/{tau}_{0},,{a}_{x}=5.5{xi}_{rm{c}}) and ({a}_{y}=9.2{xi}_{rm{c}}(a_{rm{VL}}=4.9{xi}_{rm{c}},{rm{at}}, B=0.3,B_{0})). We discover that the width and the slope of the plateau within the I–V curve weakly fluctuate with small variations of a_x and a_y, whereas the worth of v_m controls the voltage plateau place.

Additional particulars on the TDGL simulations are given in Supplementary Word 2. The TDGL modelling outcomes are introduced in Prolonged Information Fig. 1. The animated spatiotemporal evolutions of the superconducting order parameter are proven in Supplementary Movies 4–12.