Spatially-resolved dynamic sampling of different phasic magnetic resonances of nanoparticle ensembles in a magnetotactic bacterium Magnetospirillum magnetotacticum

Nanoscaled magnetic particle ensembles are promising building blocks for realizing magnon based binary logic. Element-specific real-space monitoring of magnetic resonance modes with sampling rates in the GHz regime is imperative for the experimental verification of future complex magnonic devices. Here we present the observation of different phasic magnetic resonance modes using the element-specific technique of time-resolved scanning transmission x-ray microscopy within a chain of dipolarly coupled Fe3O4 nanoparticles (40–50 nm particle size) inside a single cell of a magnetotactic bacterium Magnetospirillum magnetotacticum. The particles are probed with 25 nm resolution at the Fe L3 x-ray absorption edge in response to a microwave excitation of 4.07 GHz. A plethora of resonance modes is observed within multiple particle segments oscillating in- and out-of-phase, well resembled by micromagnetic simulations.


Introduction
Global power consumption in its various ways results in the widely known increase of global warming utilizing more fossil resources on a yearly basis [1]. Information and communications technology (ICT) made many advances in terms of power-saving technologies in the last decades, still, it remains a major factor in contemporary and future power consumption [2,3], consuming up to 7% of the global electricity [2]. Without fundamental new approaches in ICT, its consumption will continue to increase at least in a projected timeframe till 2030 [2]. Connected to the ICT's power consumption, heat dissipation and its successive demand for power consuming cooling [4] needs to be overcome to reduce the overall energy footprint of ICT. Logic concepts relying on spin wave quanta (magnons) [5][6][7][8][9][10] are a low energy approach towards energy-saving computing with magnonic devices largely reducing power consumption and heat dissipation, as well as increasing the computational performance by up to three orders of magnitude higher operating frequencies in the THz regime [11]. Magnons as collective excitations in solid state bodies are governed by magnetic dipole interaction or exchange interaction after having been thermally excited, or by coupling in a high frequency magnetic field in the form of microwaves, with the magnon wavelength controlled by the material properties, dimensions, and shape of the employed solid-state materials. The employed energies are in the order of 10 µeV, which makes magnonic computing an ideal approach to overcome the immense power consumption of the present technologies, allowing long distance propagation of magnonic information without the issue of Joule heat influence. A plethora of magnonic concepts have been suggested ranging from lithographically created structures, e.g. the magnon transistor [12] and spin-wave majority gates [13], to Fe 3 O 4 nanoparticle ensembles within magnetotactic bacteria [14][15][16][17][18][19] grown by biomineralization [6,20]. By tailoring the bacteria DNA, the arrangement of the magnetite nanoparticles can be controlled, which largely impacts the magnonic dispersion, as shown earlier by ferromagnetic resonance (FMR) experiments and micromagnetic simulations [6,20].

Details of experiment and simulation
The sample consists of a chain of 29 Fe 3 O 4 [14,[17][18][19] truncated cube-shaped nanoparticles with a single particle edge length of 40-50 nm [14] in a question-mark shape like arrangement grown by biomineralization inside a bacterium Magnetospirillum magnetotacticum [14][15][16]. The nanoparticle chain is depicted in figure 1 by scanning electron microscopy (SEM) in (a), and by STXM recorded at the Advanced Light Source beamline 11.0.2.2 in (b). The origin of the bacterium is an actively growing culture from the Leibniz Institute DSMZ-German Collection of Microorganisms and Cell Cultures [34]. The bacteria culture was drop-casted on a transmission electron microscopy grid controlled by optical microscopy and SEM (figure 1(a)) and the region of interest containing the nanoparticle chain was consecutively cut out by focused ion beam milling. For the resonant excitation, the cutout was positioned on a Si 3 N 4 membrane inside the resonant loop of a micro-resonator [35].
The TR-STXM measurements were carried out at the MAXYMUS endstation at the BESSY II electron storage ring operated by the Helmholtz-Zentrum Berlin für Materialien und Energie. The sample was probed at room temperature at a pressure of 10 −8 mbar at a microwave excitation frequency selected in the scope of the employed pump-probe detection scheme as f mw = (f s /N)M = 4.07 GHz, where f s = 500 MHz is the synchrotron frequency, N = 7 corresponds to the number of sampled time points/recorded images, and M = 57 resembles the number of excitation periods in a measurement interval N/f s = 14 ns. The static magnetic bias field has been applied in-plane to the sample and was set to selected values between −0.008 T and 0.19 T, while the microwaves were linearly polarized parallel to the propagation direction of the x-rays and out-of-plane to the nanoparticle chain. The x-ray energy was set to the Fe L 3 edge with a nominal energy of 710 eV obtained from an energy scan. Taking the division of the recorded x-ray photon counts by the time-average reveals the dynamic magnetic contrast [24,26,36]. Consecutively the data was normalized to its average intensity followed by a minimum-maximum normalization. Fitting a sinusoidal to each row of equally positioned pixels was performed to extract the phase and amplitude response, coding the result in the hue-saturation-brightness format with the phase relative to time t = 0 ps as the color hue [37], the amplitude as the brightness and the fit accuracy in form of the p-value as the saturation [38].
A three-dimensional micromagnetic model was designed from SEM and STXM micrographs using a particle edge-length of 50 nm (figures 1(a) and (b)) in MuMax3 [39,40]. MuMax3 is a finite differences in time domain approach to solve a coupled system of Landau-Lifshitz equations of motion of the magnetization using the damping term formulation of Gilbert [39,41,42]. The magnetization is discretized into orthorhombic cells with each cell incorporating a magnetization vector as well as material parameters, e.g. cubic magnetocrystalline anisotropy [39]. For the simulation considering the imperfections in the stoichiometry of the Fe 3 O 4 [43] and in accordance with earlier studies [25], the saturation magnetization and the first order cubic magnetocrystalline anisotropy constant were set to M Sat = 465 kA m −1 , and K 1 = −1.0 × 10 4 J m −3 [44], respectively. A simulation grid of 200 × 200 × 12 cells with a cell size of (5 nm) 3 was defined with an exchange stiffness A = 1.32 × 10 −11 J m −1 [45]. For the dynamic simulation at 4.07 GHz, the   Figure 2 shows the simulated demagnetization and stray field intensity, and the stray field distribution of the 29 particle chain. The strong inter-particle magnetic dipolar coupling up to 0.55 T in the central fork of the chain can be seen. Due to the kinked chain structure and the large amount of displaced particles with respect to a linear chain arrangement, the majority of inter-particle coupling ranges from 0.25 T to 0.35 T. The inter-segment coupling is in general below 0.1 T except for upper left ensemble of three particles, which show a coupling of ∼0.1 T with respect to the right fork of the chain; the displaced outer left particle is coupled similarly to the right fork. The stray field distribution shows the presence of inter-particle and inter-segment dipolar coupling, with the latter being ⩽0.1 T. In the simulated angular-dependent FMR absorption spectrum (figure 3) a multitude of angular dependent resonance lines can be observed with visible periodicities of 90 • and 180 • , originating from the inter-particle and inter-segment dipolar coupling of the chain, resulting in a complex magnon dispersion relation incorporating different periodicities, as well as magnonic band gaps [6,20].    figure S1(d). Deviations between the calculated B Ext and the measured resonance fields can be attributed to the approximation of equally sized particles taken for the simulation, in contrary to the slightly differently sized and spaced real particles, which impacts on dipolar coupling strength and thus, the resonance fields. With extended measurement time the nanoparticle chain dissolved due to radiation damage to the lipid bilayer membrane of typically 5 nm thickness [46], encapsuling the nanoparticles. The following measurements were performed on the intact upper segment of the chain, as it is depicted in figure 4(d). At 0.106 T a resonant response uniform in phase (90 • ) of the chain segment in region R4 is visible, with a less intense resonance response showing the same relative phase at region R5 (figures 4(e) and (f)). Figure S2(b) depicts a high-intensity contrast oscillation in region R4 with a much weaker resonant response visible in region R5. Increasing B Ext to 0.112 T results in a higher intense and larger resonance area at region R5 (figures 5(b) and (c)), indicating that at 0.106 T this resonance was measured at its flank, while observing at 0.112 T a resonant response closer to its peak intensity, resembled by the animation shown in figure S3(b). At 0.116 T the resonant response at region R4 changes its relative phase to 60 • with some particles above and below and in the adjacent segments are remaining at 90 • (figures 5(e) and (f)). The animated TR-STXM images shown in figure S4(b) show corresponding oscillations in region R4, with the centrally located particles showing a different phasic resonant response. The different phased responses can be explained by the differing dipolar coupling strength between the upper segment and the two adjacent segments below (regions R5, R6), as it can be seen from the micromagnetic simulation in figure 2(a), ranging from 0.2 T to 0.45 T. Thereby, the obtained TR-STXM data experimentally proves spatially-resolved theoretically predictions of different phasic resonant responses in such nanoparticle chains [6,20].

Conclusion
In conclusion, FMRs from different segments of the studied dipolarly coupled Fe 3 O 4 nanoparticle chain inside a single cell of a magnetotactic bacterium Magnetospirillum magnetotacticum were detected using the element-specific and spatially resolved TR-STXM technique, showing resonant responses uniform in their relative phase distribution in different segments of the chain. By employing TR-STXM the resonant responses for resonance fields between 0.03 and 0.116 T could be localized. Even though the substantial radiation damage lead to dissolution of large segments of the chain, it was possible to observe in the remaining chain segment an FMR mode of a nanoparticle chain showing a different phasic response between the chain segments in resonance, and thus, confirming this theoretically predicted resonant behavior for comparable Fe 3 O 4 nanoparticle chains in [6,20].

Data availability statement
The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.