Probing spin waves in Co 3 O 4 nanoparticles for magnonics applications †

The magnetic properties of spinel nanoparticles can be controlled by synthesizing particles of a speci ﬁ c shape and size. The synthesized nanorods, nanodots and cubic nanoparticles have di ﬀ erent crystal planes selectively exposed on the surface. The surface e ﬀ ects on the static magnetic properties are well documented, while their in ﬂ uence on spin waves dispersion is still being debated. Our ability to manipulate spin waves using surface and defect engineering in magnetic nanoparticles is the key to designing mag-nonic devices. We synthesized cubic and spherical nanoparticles of a classical antiferromagnetic material Co 3 O 4 to study the shape and size e ﬀ ects on their static and dynamic magnetic proprieties. Using a combination of experimental methods, we probed the magnetic and crystal structures of our samples and directly measured spin wave dispersions using inelastic neutron scattering. We found a weak, but unques-tionable, increase in exchange interactions for the cubic nanoparticles as compared to spherical nano-particle and bulk powder reference samples. Interestingly, the exchange interactions in spherical nano-particles have bulk-like properties, despite a ferromagnetic contribution from canted surface spins.


Introduction
The magnetic nanomaterials offer a unique opportunity to broaden our understanding of fundamental exchange interactions inside a solid-state matter on an atomic scale that underlines its macroscopic properties.The interactions are tailored by virtue of the finite-size effects resulting from a large fraction of the surface atoms.It is possible to further affect exchange interactions in nanoparticles by deliberately changing their shape to expose selected crystal planes.The nanoparticles of different shapes retain their chemical composition, in contrast to the bulk materials, in which the exchange interactions can only be altered by an isotope substitution, application of high pressure and extreme magnetic fields.
Here, we present a comprehensive study of the dynamic magnetic properties of the spherical and cubic Co 3 O 4 nanoparticles using a combination of advanced neutron scattering methods.2][3] The strength of the interactions is defined by the exchange constants J 1 and J 2 , corresponding to the AFM interactions between the nearest Co 2+ ions and weak ferro-magnetic (FM) interactions between the next-nearest Co 2+ ions.The strength of AFM interactions depends on Co-O bond lengths, the titling angle between them, and occupancies of the atoms within the exchange path. 4][19] The AFM ordering of the Co 3 O 4 makes it also possible to use the Co 3 O 4 nanoparticles in magnonics applications.5][26] The spin wave detection in magnetic nanoparticles is the next step in the same direction, aiming to miniaturize magnonic devices by bringing them to the nanoscale.8][29] However, the finite-size and shape effects on the spin wave properties are needed to be understood first.In the particular case of the Co 3 O 4 nanoparticles, a direct impact of surface engineering on exchange interactions has been demonstrated. 30,31The Co 3 O 4 nanoparticles with exposed (100) and (111) crystal planes showed an induced magnetic moment on nominally non-magnetic Co 3+ ions, while for (110) planes, Co 3+ ions remain non-magnetic.
We used inelastic neutron scattering to experimentally study spin waves dispersions and to directly measure J 1 and J 2 exchange constants in spherical and cubic Co 3 O 4 nanoparticles, as well as in their bulk powder counterpart.3][34] We also revised the experimental J 1 value for the bulk Co 3 O 4 powder, in order to provide an additional input for density functional theory calculations, that currently estimate values of J 1 varied by an order of magnitude. 35,36

Experimental
We synthesized cubic Co 3 O 4 nanoparticles using Co nanoparticles prepared by the thermal decomposition of Co 2 (CO) 8 in hot oleic acid as described in. 37,38Once the particles have fully formed, we annealed them at 400 °C for 4 hours inside a vacuum furnace.The use of the annealing temperature above the boiling point of the oleic acid (360 °C) ensured the absence of the surfactant coating of the cubic nanoparticles.
The spherical Co 3 O 4 nanoparticles and reference bulk powder of Co 3 O 4 were purchased from Sigma-Aldrich.
TEM measurements were carried out in the Ernst Ruska-Centre at Forschungszentrum Jülich with a Philips CM20 TEM and FEITecnai G2 F20 using an accelerating voltage of 200 kV.The SEM studies were performed using a Hitachi SU8000 instrument at an electron acceleration voltage of 20 kV.Additional HRTEM measurements were performed at Uppsala University.The samples were dispersed in toluene and drop-casted on 400 mesh carbon-coated copper grids.The particles were investigated using a FEI Titan Themis microscope with a Schottky field-emission gun operating at an accelerating voltage of 200 kV.The images were acquired using a BM-Ceta Camera.
The DC magnetization measurements were carried out using a Quantum Design Magnetic Property Measurement System (MPMS).For the measurements, 10 μL of the original aliquots were dispersed in 50 μL of liquid paraffin and then injected into a standard gelatin capsule to reduce inter-particle interactions.The gelatin capsules were fastened in plastic straws for immersion into the magnetometer.In order to measure the exchange bias field, a sample is cooled from 300 K to 5 K in an applied magnetic field of 70 kOe.The magnetization is then measured as a function of an applied magnetic field.The exchange bias field is calculated as , where H c1 and H c2 are the negative and positive coercive fields, respectively.The standard field-cooled (FC) and zero-field-cooled (ZFC) measurements of temperature-dependent magnetization were used to determine T N .FC and ZFC measurements were carried out in the field of 500 Oe.
The time-of-flight (TOF) neutron diffraction experiments were performed on the Nanoscale-Ordered Materials Diffractometer (NOMAD) at Spallation Neutron Source (SNS) at Oak Ridge National Laboratory. 39The low-temperature measurements were done using an Orange-type cryostat.NOMAD detectors were calibrated using scattering from diamond powder.The magnetic structure refinements of the data were done using Jana2006 software, 40 which was adopted for handling neutron TOF diffraction data from six detector banks.In order to obtain the structural factor S(Q) the scattering intensity was normalized to the scattering from a solid vanadium rod and the background was subtracted.The pairdistribution function (PDF) was obtained by the Fourier transform of S(Q) at Q min = 0.08 Å −1 and Q max = 31.5Å −1 : We used the Cold Neutron Chopper Spectrometer (CNCS) at the SNS at Oak Ridge National Laboratory 41 to perform zero field TOF inelastic neutron scattering experiments on nanoparticle powders with the incident wavelength of λ i = 2.34 Å (E i = 15 meV).Additional temperature-dependent inelastic neutron scattering measurements were carried out at TOFTOF instrument at Heinz Maier-Leibnitz Zentrum.Two incident wavelengths λ i,1 = 2 Å (E i,1 = 9 meV) and λ i,2 = 5 Å (E i,2 = 3.3 meV) were used for accessing the medium and high energy The vanadium measurements were used to define the resolution function of each instrument.The CNCS data have provided access to the energy gain and the energy loss parts of excitations (Fig. S5 †), while only the energy gain part of the TOFTOF data was analyzed.The TOFTOF data were reduced using Mantid software. 42oom temperature Raman measurements were conducted at Uppsala University using a Renishaw (inVia) spectrometer with a built-in optical microscope (Leica) and by employing two different excitation wavelengths of 532 and 785 nm, respectively.The latter were provided by different solid-state laser diodes.Prior to the analyses, the spectrometer was calibrated using a Si wafer to obtain a characteristic reference peak at ∼520 cm −1 .A low laser power was utilized for the respective measurements and the laser beam was focused onto the surface of the various samples via a 50× lens.

Results and discussion
Three samples were investigated in our work: spherical Co 3 O 4 nanoparticles, cubic Co 3 O 4 nanoparticles and reference bulk powder.The properties of all samples are summarized in Table 1.

Structural characterization
A combination of transmission electron microscopy (TEM) and scanning electron microscopy (SEM) measurements was used to characterize the overall size, shape, and crystallinity of our samples.Representative TEM and SEM images are shown in Fig. 1.The cubic sample shows polydisperse nanoparticles.The majority of the nanoparticles are best described by a rectangular shape with clear-cut edges (Fig. 1a).The spherical nanoparticles appear to be globular with a high degree of agglomeration (Fig. 1b).The spherical nanoparticles appeared to be amorphous, but high-resolution TEM (Fig. 1b, inset) and X-ray powder diffraction studies (Fig. S1 †) confirmed their crystallinity.The bulk sample shows polyhedron crystallites as large as several micrometers in size (Fig. 1c).

Magnetization measurements
The ZFC and FC magnetizations of the cubic and spherical nanoparticles peak at T N ∼45 K (Fig. 1d and e).At temperatures above T N , the long-range AFM order is lost and nanoparticles are transitioned into the paramagnetic phase.In this phase, the temperature dependence of the ZFC and FC magne-Table 1 Properties of Co 3 O 4 nanoparticles obtained from refinements of neutron scattering data.Here, d is the crystallite size, T N is the Néel temperature, m is the magnetic moment per Co 2+ at 10 K and a is the lattice constant at the same temperature.J 1 and J 2 are refined exchange constants for the nearest-neighbors AFM and next nearest-neighbors FM interactions, respectively.D S is the uniaxial single-ion anisotropy constant Sample d (nm)  In the case of cubic nanoparticles, we derived C = 1.16 × 10 −2 emu K g −1 Oe, θ = 109 K and χ 0 = 1.06 × 10 −4 emu g −1 Oe, in good agreement with the previous reports for the bulk Co 3 O 4 . 43,44For spherical nanoparticles we obtained C = 1.63 × 10 −2 emu K g −1 Oe, θ = 103 K and χ 0 = 1.57× 10 −5 emu g −1 Oe.The negative sign of θ suggests dominating AFM interactions for all samples.The strength of the interactions is rather similar because values of θ for nanoparticles are close 110 K, previously reported for the bulk Co 3 O 4 . 16,45he magnetic properties of cubic and spherical nanoparticles are distinctively different at low temperatures.ZFC magnetization of the spherical nanoparticles peaks at ∼25 K, while FC magnetization gradually increases and saturates below 25 K.Other researchers have observed similar DC magnetization dependence in Co 3 O 4 nanoparticles, attributed to the coexistence of AFM-ordered core and FM-like surface shell of uncompensated spins with a freezing temperature of 25 K. [46][47][48] The alternative explanation of this transition is a spin glass-like phase of the surface magnetic moments. 49,50owever, we found no signatures of the spin-glass phase below 25 K in our neutron diffraction and inelastic neutron scattering data, 51 at least within the experimental uncertainty.The existence of an FM-like surface shell is further confirmed by the observation of an exchange bias field (H EB ) below 25 K.The maximum field H EB = 357 (8) Oe was obtained at 5 K after cooling in 70 kOe field from the room temperature (Fig. 1f ).3][54][55][56][57] In the case of cubic nanoparticles, no exchange bias was observed under similar experimental conditions.

Raman scattering
We examine the structural properties of our samples with Raman scattering measurements at two wavelengths of 532 and 785 nm.A comparison of the normalized spectra for both nanoparticles and the bulk reference powder is presented in Fig. 2a and b, while their respective as-recorded spectral signals are shown in Fig. S2.9][60] The peaks for all samples are centered at around similar positions: 197 cm −1 (F 2g ), 483 cm −1 (E g ), 523 cm −1 (F 2g ), 621 cm −1 (F 2g ), and 692 cm −1 (A 1g ) as shown in Fig. 2a and b.The position of each peak is ascribed to a characteristic vibration mode (see ESI †).All normalized Raman spectra look very similar and spherical nanoparticles are the only sample that displays a slight redshift of ∼2 cm −1 for the A 1g mode when measured with a 532 nm wavelength (Fig. 2c).Moreover, the spherical nanoparticles show a moderate broadening of all the peaks compared to those displayed by the other two samples (Table S1 †).The peak broadening is less visible in the spectra recorded with 785 nm excitation wavelength (Table S2 †) and no shift is detected for any peak position in this case, while the intensities of the remaining modes appear more pronounced.The most likely explanation for both phenomena is a higher degree of a structural disorder in spherical nanoparticles, previously suggested for the similar Co 3 O 4 nanostructures. 59,61,62nterestingly, the slight red shift in Fig. 2c, d and Fig. S2a † affects only the A 1g mode at ∼690 cm −1 and the F 2g vibration at ∼196 cm −1 , whereas the positions of the other peaks are not influenced.If it were an optical phonon confinement effect, 63,64 the peak positions of the five active modes would sensibly be shifted toward lower wavenumbers, while this is not the case.Pronounced agglomeration for the spherical nanoparticles (see also Fig. 1b), due to their larger surface-tovolume ratio compared to the other samples, could influence their thermal response, 18 especially under conditions of elevated light absorption, e.g. during resonant Raman scattering (RRS).RRS causes excitation of electronic levels and substantial enhancement of some vibrational modes.Under such conditions and high absorption, the light penetration is typically limited to a few tens of nanometers, 64 thereby making this analysis more surface sensitive.Using 532 nm instead of 785 nm should in principle allow a shallower penetration depth (δ p = λ exc /4πnk) and thus probe more effectively the particle surface via RRS. 64Considering the reduced number of counts compared to the as-recorded spectra obtained with the 785 nm laser (see Fig. S2a and b †) and the negligible influence of thermal effects due to the purposeful use of low laser power, the results in Fig. 2a and Fig. S2a † suggest that surface defects, local lattice distortions and disorder represent the main cause for the slight changes in the spectral features of the spherical nanoparticles under RRS at 532 nm.In fact, the peaks remained sharp and hardly shifted to lower wavenumbers, while the peak broadening for the spherical nanoparticles appeared moderate, especially when compared to those reported for agglomerated Co 3 O 4 nanoparticles measured with similar laser power, whose spectral features sensibly red-shifted and exhibited much broader lines. 18,65herefore, this confirms that the surface properties of the spherical nanoparticles differ from those of the cubic nanoparticles and bulk powder.

Neutron diffraction studies
We combined neutron powder diffraction and pair-distribution function (PDF) measurements, to unambiguously determine our samples' crystal and magnetic structures.The quality of Rietveld refinements of the neutron diffraction data is demonstrated in Fig. 3a and b.The refined average crystal structure is cubic for all samples with the space group Fd3 ˉm. 66It is a normal spinel with the Co 2+ and Co 3+ ions occupying tetrahedral and octahedral sites, respectively.While positions of structural peaks are the same for all samples, the peak width of nanoscaled samples is broader as compared to the bulk counterpart due to the finite particle size.The crystalline sizes equal to 33(3) and 61(2) nm for spherical and cubic nanoparticles, respectively were derived from the Rietveld refinements (Tables S3 and S4 †).The most intriguing conclusion of the average crystal structure study is a difference in lattice constant values between the three samples shown in Fig. 3c.The lattice constant of cubic nanoparticles is higher than the bulk sample, while the lattice constant of spherical nanoparticles is smaller.
The magnetic structure of our samples was refined from neutron diffraction data collected below T N .In agreement with previous neutron diffraction studies of Co 3 O 4 powders, 1,46,67 we found that the magnetic structure is best described by a propagation vector k = (0, 0, 0), i.e. the magnetic unit cell is identical to the nuclear one.The two primary peaks below T N can be indexed to the (111) and (200) peaks of the bulk Co 3 O 4 structure.The first peak at Q 111 = 1.33 Å −1 constitutes structural and magnetic contributions, while the second peak Q 200 = 1.56 Å −1 is purely magnetic.Additional diffuse scattering had to be added in the modeling of the intensity of Q 111 peak for all three samples (see Fig. S3 †).This magnetic diffuse scattering was previously observed with neutron diffraction studies of the bulk Co 3 O 4 .It corresponds to the disordered shortrange magnetic state resembling spiral spin liquid.The correlation length of the disordered state was estimated using the Scherrer formula 68 ζ = d 2 111 /Δd 111 , where d 111 is the position and Δd 111 is the full width at half-maximum of the Q 111 peak.We found ζ equal to 9.8(2) nm, 6.7(3) nm and 12.4(1) nm, for the spherical, cubic and bulk samples.The correlation length in spherical nanoparticles is about 30% of their crystalline size, implying a non-negligible role of this magnetically disordered phase in their magnetic properties.
In our model, only Co 2+ ions bear magnetic moments.The magnetic moment of spherical nanoparticles is reduced down to 2.76(2)μ B from the value of 3.01(2)μ B observed for the bulk and cubic nanoparticles at 10 K. We found no deviation from a cubic symmetry in the crystal structural or additional magnetic phase transitions below T N in both of our nanoparticle samples.We obtained T N for each sample from mean-field fits of the magnetic moment temperature dependence (Fig. 3d).T N is decreasing from its bulk value of 42.4(8) K with the particle size down to 32.2(1) K and 30.1(2)K, for cubic and spherical particles, respectively.The result agrees well with the finitesize effects reduction of T N previously reported for Co 3 O 4 nanoparticles. 45,47,69,70e further investigate the local crystal structure of our samples using neutron PDF measurements.The purpose of PDF studies was to verify any deviations from the average crystal structure derived from Rietveld refinements.Nanoparticles are prone to dislocations, stress, vacancies and anti-phase boundaries, which all can influence local crystal structure and consequently, the strength of the AFM

Nanoscale Paper
This interactions. 71,72Most recent examples include modification the local structure of iron oxide nanoparticles due to vacancies and non-stoichiometry. 73,74Specifically for Co 3 O 4 samples, we used PDF measurements to investigate local structural transitions, which were suggested as one of the reasons for the incommensurate AFM order observed with muon spin rotation and relaxation method in bulk Co 3 O 4 . 43eutron PDF data for all three samples are qualitatively similar.We fitted neutron PDF data using the average crystal structure obtained with the Rietveld refinements as a starting model (Fig. 4a).The PDF analysis indicated no deviations of the local structure from the cubic symmetry or any signs of inverse spinel structure for both nanoparticle samples.The lattice constants obtained from the refinement of the PDF data are consistent with Rietveld refinement results (Fig. S4a †).The spherical nanoparticles have the smallest lattice constant.The most significant difference between the samples is observed for the spherical nanoparticles between 3.0 and 3.7 Å at 10 K.The intensities of the three peaks at 3.15, 3.35 and 3.57 AA are reduced (Fig. 4b).The position of the first peak at 3.15 Å   corresponds to an O-O atomic pair.The second peak is a superposition Co-Co Co-O atomic pairs, while the last peak corresponds to Co-O pair (Fig. S4b †).The reduction of intensity in PDF peaks compared to the bulk indicates that the atoms in corresponding pairs have a lower coordination number.We refined the occupancies of O and Co atoms while modeling the neutron PDF data (Table S4 †).The results of the PDF data refinements for cubic nanoparticles are shown in Fig. 4c and for the bulk in Fig. S4c.† We found that occupancies of O and Co atoms are reduced for the spherical nanoparticles, as compared to the bulk and cubic nanoparticles (Table S3 †).Neutron diffraction is not sensitive to the valence state of Co atoms, however previous density function theory (DFT) calculations confirmed that the formation of the Co 2+ defects is more likely than the formation of Co 3+ defects, because of the lower formation energy of Co 2+ vacancies. 59

Inelastic neutron scattering measurements
The spin wave properties of our samples were studied with TOF inelastic neutron scattering measurements (INS).We found no previous reports on TOF INS of polycrystalline a powder or single crystal of Co 3 O 4 .Growing a single crystal of Co 3 O 4 is extremely challenging.The main issue is the decomposition of Co 3 O 4 at 900 °C which limits applications of conventional crystal growth methods.The Co 3 O 4 is carcinogenic and extraordinary safety measures have to be taken in order to handle it at elevated temperatures. 75he overview of INS data is shown in Fig. 5.The energy and wave-vector dependence of the scattered intensity I(Q,E) is plotted for T = 5 K for the cubic nanoparticles and bulk powder (Fig. 5a and b).Similar for all three samples, the inelastic excitation peaked near 6 meV, and shows dispersion over the entire Q-range with its intensity decreasing strongly with increased wave vector Q.The dispersion is periodic and matches perfectly the periodicity of the Bravais magnetic lattice derived from the magnetic structure (Fig. S6 and S7 †).The excitations are only observed below T N for each individual sample.Since the excitation signals in the spectrum possess a characteristic dispersion in the low-energy regions and the scattering intensity decreases with increasing Q following the magnetic form factor, it is fair to conclude that the signals are contributions from magnetic spin waves.Similar spin wave dispersions for the nanoparticle and bulk samples confirm that they are inherently specific to the magnetic properties of Co 3 O 4 .We found no evidence of finite-size induced phonon dispersions nor excitations associated with the collective nanoparticle modes, at least within our experimental energy window of 15 meV.Fig. 5c and d show the energy E dependence of the scattered intensity I(E) for the cubic nanoparticles and bulk powder, integrated over wave vectors Q, for temperatures above and below T N .Above T N , the scattering is quasielastic (QE) for both samples.As the temperature is reduced, the QE linewidth is reduced and the overall scattering intensity decreases.QE scattering collapses below T N , for both samples and the inelastic peak begins to emerge.It changes its position and becomes more intense as the temperature further decreases.At the lowest temperature of 5 K, the peak position for the cubic nanoparticles is slightly but visibly increased to E = 6.2 meV, compared to the value of 6.0 meV for both bulk powder and spherical nanoparticles.The neutron diffraction studies of our samples showed a large magnetic moment of Co 2+ ions, therefore the spin wave dispersion can be analyzed by the Heisenberg model with a spin Hamiltonian: where, S i is the spin operator for the ith site, J ij denotes the exchange constants, and D S is the uniaxial single-ion anisotropy constant.According to eqn (2), a positive J ij corresponds to a FM interaction and a negative J ij corresponds to an AFM interaction, while a positive D S of a certain magnitude corresponds to the formation of an anisotropy-induced energy gap in the spin wave spectrum.In our analysis, only two exchange constants were used: J 1 to describe the nearest-neighbor interactions and J 2 to describe the next nearest-neighbor interactions between Co 2+ ions.The bosonic Hamiltonian in the momentum space was obtained by applying the Holstein-Primakoff transformation for spin operators.The Hamiltonian was diagonalized numerically to calculate the eigenvalues related to the spin wave dispersion and eigenvectors associated with a spin wave structure factor (see ESI †).The following neutron cross-section was used to model the experimental INS data: where, k f and k i are the final and the incident wave vectors, respectively.γr 0 = 5.39 fm is the magnetic scattering amplitude for an electron.g is the Lande factor of Co, F(Q) is the isotropic form factor for Co 2+ , and e −2W is the Debye-Waller factor.S α,β (Q,w) is the response function describing spin correlations. 76The calculated spectra are convoluted by a Gaussian function with FWHM of 0.21 meV to account for the instru-mental resolution, measured at room temperature with a solid vanadium rod.Error bars represent standard deviations.
In order to directly compare the calculations with the experimental INS data on the powder samples, the calculated INS cross-section (eqn (3)) was averaged over different Q vectors lying on a constant-Q sphere.The exchange constants for all samples were obtained by fitting the energy spectra at different momentum transfers (Fig. 6a).We found that for the bulk and spherical nanoparticle samples, J 1 = −7.60 meV, J 2 = 1.10 meV, and D S = 0.015 meV.Using these values we were able to calculate neutron scattering spectra that compare well and reproduce all the features of the experimental data (Fig. 6b  and c).However, the best fit of the energy spectra for the cubic nanoparticles was obtained with a slightly increased J 1 = −7.72 meV.The calculated neutron scattering spectra with these exchange parameters can reproduce the experimental data quite well (Fig. S8 †).Increased value of J 1 can also explain the shift of the inelastic peak position in I(E) data, experimentally observed for the cubic nanoparticles (Fig. 5c).The increase of J 1 value in the cubic nanoparticles is rather weak compared to the bulk value, presumably due to a large polydispersity of the cubic nanoparticles and their shape irregularities observed in TEM studies.Additional neutron scattering experiments on monodisperse cubic nanoparticles of various sizes might show an even larger increase of J 1 value.
The values obtained in this experiment are considerably higher than previously reported in the literature.][79][80] The only reported experimental value of J 1 obtained by INS measurements with a monochromatic beam on the polycrystalline Co 3 O 4 powder is equal to 0.63 meV. 813][84][85] Both effects can induce a net magnetic moment at the surface due to incomplete AFM alignment of the magnetic moments of the surface atoms.The exchange interactions between this FM-like magnetic moment and AFM inner core lead to non-zero H EB , only observed in the spherical nanoparticles.The lower coordination number of the surface atoms might lead to a defective crystal structure at the surface.Indeed, according to our neutron diffraction studies O and Co atoms deficiencies are the most prominent in the spherical nanoparticles.According to first-principle studies, oxygen vacancies lead to FM-like behavior in Co 3 O 4 thin films because they affect exchange interactions between metallic ions. 86The detailed model of the vacancies distribution in our samples is missing, but independent magnetization and Raman scattering measurements, both suggest that the vacancies are distributed at the surface of the spherical nanoparticles.Another consequence of the surface disorder in the spherical nanoparticles is the reduction of their lattice constant.The defective surface structure induces a tensile surface stress that in turn compresses the lattice constant to minimize the surface energy. 16,87,88Our explanation is supported by the previous X-ray absorption fine structure measurements and DFT calculations on similarly sized Co 3 O 4 nanostructures. 59The cubic nanoparticles appear to be more ordered than the spherical ones, showing only small deviations in the crystal and magnetic structures from their bulk counterpart.The most prominent structural feature of the cubic nanoparticles is their surface.They are exposing well-defined crystal planes formed by high-temperature annealing in a vacuum.Previous studies have concluded that for the cubic Co 3 O 4 nanoparticles, the most likely exposed planes at their surface are (111) and (100). 16These two planes have a larger fraction of Co 2+ ions, compared to a number of non-magnetic Co 3+ ions.The size of Co 2+ ions (0.745 Å) is larger than the size of Co 3+ ions (0.545 Å), hence negative stress is induced on the surface of those planes. 66,87That results in an increase of the lattice constant observed in our cubic sample and in previous studies of cubic Co 3 O 4 nanoparticles prepared by a similar hightemperature annealing method. 61,89ne would expect increased values of J 1 in nanoparticles with the smallest lattice constants due to a higher degree of 3d electron cloud overlap. 90Fig. 7a shows the nearest (L 1 ) and the next-nearest (L 2 ) neighbor distances between Co 2+ ions for all three samples obtained from neutron data refinements.The best Rietveld refinements were obtained with the position of all atoms fixed.Thus both distances follow the same dependencies as the lattice constants, i.e. the shortest bond lengths were found for the spherical nanoparticles.We further examine any possible local distortions of L 1 and L 2 by relaxing the positions of Co atoms in tetrahedral and octahedral sites during neutron PDF data boxcar refinements in the range r = 1.5-40Å, with a step of r max = 5 Å (Fig. 7b and c).The obtained values of L 1 and L 2 show no r max -dependence and remain constants within experimental error, as well as Co-O and O-O bond lengths (Fig. S9 †).We tried to relax and fit the positions of all atoms simultaneously, but it led to the over-parametrization of the model and the fits were unstable.In conclusion, we found no local distortions of the bond lengths in the superexchange path in our samples that could have explained the increased values of J 1 in the cubic nanoparticles.
Another parameter to consider when explaining the exchange interactions in a normal spinel is inversion. 86,91Our cubic nanoparticles might be susceptible to inversion because they were exposed to high temperatures of 400 °C.We accurately calculated the inversibility parameter using the spinel anion parameter u, according to 92 for all our samples, using lattice constants and bond lengths at 10 K derived from neutron diffraction studies.The value of 0.26(1) for all three samples is consistent with the bulk values of 0.26-0.27previously reported for a normal spinel structure. 92,93s we discussed in the preceding sections, the enhancement of J 1 observed in cubic nanoparticles cannot be explained by either decreased lattice constants, ligand interactions, or inversion.Thus, the shape effects must be considered.The previous reports have already indicated that the shape effects modify the magnetic structure of Co 3 O 4 nanoparticles in a significant way.For example, 7 nm Co 3 O 4 singlecrystal nanowires showed an induced FM-moment due to the symmetry breaking of Co 3+ ions on (111) surfaces. 52Similarly, cubic and platelet-like Co 3 O 4 nanoparticles displayed a variation in the net magnetic moment depending on the coordination number of Co ions at a dominant crystal plane. 16In the case of our cubic nanoparticles, the high-resolution TEM study verified that the dominant (100) (111) planes are exposed on the surface (Fig. 8).Presumably, exposure of these planes seen only in cubic nanoparticles is responsible for enhancing J 1 .Previous studies of larger Co 3 O 4 particles in the shape of cubes with the (001) plane, truncated octahedrons with (001) and (111) planes and an octahedron with the (111) show remarkably different physicochemical behaviors, as for example indicated by their respective electrochemical performances. 8Accordingly, the key role of specifically exposed surface planes in Co 3 O 4 nanoparticles with well-defined shapes must be considered.In the same vein, DFT calculations of the magnetic interactions on (100) and (111) surfaces of our cubic nanoparticles, including an experimental crystal structure can help to elucidate the origin of J 1 increase.However, given the relatively large size of the cubic nanoparticles (61 nm), those calculations are extremely time-consuming and difficult to handle.The large polydispersity of our cubic nanoparticles will further complicate the understanding of the DFT calculation results.Finally, we considered an alternative explanation of AFMorder in our samples and a possible reason for the FM-like phase observed in the spherical Co 3 O 4 nanoparticles.The idea was first put forward by W. Roth in his original work in 1964. 1 He suggested another exchange parameter J 3 between the nextnext nearest neighbor of Co 2+ ions separated by 6.68 Å and including 180°bond angle between O-Co 3+ -O (see Fig. 7a).This is a weak interaction, but it becomes dominant if the multiplicity of exchange paths is taken into account.Each of the FM and AFM exchange paths via 90°O-Co 3+ -O bonds have the same multiplicity of 24 and cancel each other out.Only the AFM exchange path via 180°O-Co 3+ -O bond with a multiplicity of 12 remains responsible for AFM order in Co 3 O 4 .However, in Co 3 O 4 nanoparticles with well-defined shapes, the multiplicity on the surface is broken and additional FM or AFM interactions are emerging. 16We found no experimental evidence of this scenario in our samples.Using additional exchange parameter J 3 in the spin Hamiltonian had no effect on the calculated spin wave spectrum for all three samples, however, the determination of J 3 might be hindered by the powder averaged data.Additional INS measurements of a single crystal are needed for a proper assessment of the J 3 contribution to the spin wave spectrum of Co 3 O 4 .

Conclusions
We carried out extensive experimental studies of the structural and magnetic properties of spherical and cubic Co 3 O 4 nanoparticles and compared them with the bulk powder.The spherical nanoparticles showed a pronounced effect of the canted surface spins, resulting in the appearance of an FM-like phase below 25 K. Despite, this surface disorder the exchange constants J 1 and J 2 of the spherical nanoparticles remained the same as in bulk, within the experimental error of our experiment.For the cubic nanoparticles, the lattice constant is increased as compared to the lattice constant of their bulk counterpart.It is a direct consequence of the negative surface stress caused by the high-temperature annealing.The importance of the particle's shape was further reinforced by the inelastic neutron scattering experiments, which showed increased exchange constant J 1 only for the cubic nanoparticles.Our results imply that magnetic nanoparticles with selectively exposed surface planes can strengthen the exchange interactions and hence be more attractive for magnonics applications.Those applications can also benefit from the use of the spherical Co 3 O 4 nanoparticles with magnetically disordered surfaces because they still retain their bulk-like spin waves dispersion.

Fig. 1
Fig. 1 Representative TEM and SEM images of (a) cubic, (b) spherical and (c) bulk Co 3 O 4 samples.The temperature-dependent ZFC (open symbols) and FC (filled symbols) magnetizations for (d) cubic and (e) spherical nanoparticles.Insets show inverse susceptibility χ as a function of temperature (open symbols) with the fit to Curie-Weiss law (solid line).(f ) Field-dependent DC magnetizations of spherical and cubic Co 3 O 4 nanoparticles, measured at 5 K and 30 K, respectively.

Fig. 2
Fig. 2 Comparison of normalized Raman spectra for all three samples obtained using different excitation wavelengths of (a) 532 nm and (b) 785 nm.Expanded views of the spectra in (a) show a slight red-shift and broadening of the most intense A 1g peak at high wavenumbers (c), and similarly of the weakest F 2g mode at low wavenumbers (d).Note also a decrease of the relative intensities of the peaks in (b) only for the spherical nanoparticles.

Fig. 3
Fig. 3 NOMAD TOF diffraction data for cubic Co 3 O 4 nanoparticles from the forward scattering bank at 10 K (a) and the backscattering bank at 100 K (b), with experimental data as blue open circles, the calculated pattern in red, and the difference curve in green.Inset shows an expanded view of the same data, with no indications of magnetic Bragg peaks.Temperature dependencies of the lattice constant (c) and the magnetic moment per Co 2+ (d) for the spherical, cubic and bulk samples.Solid lines are mean-field fits with T N listed in Table1.
Fig. 3 NOMAD TOF diffraction data for cubic Co 3 O 4 nanoparticles from the forward scattering bank at 10 K (a) and the backscattering bank at 100 K (b), with experimental data as blue open circles, the calculated pattern in red, and the difference curve in green.Inset shows an expanded view of the same data, with no indications of magnetic Bragg peaks.Temperature dependencies of the lattice constant (c) and the magnetic moment per Co 2+ (d) for the spherical, cubic and bulk samples.Solid lines are mean-field fits with T N listed in Table1.

Fig. 4
Fig. 4 (a) Magnetic and crystal structure of the bulk Co 3 O 4 obtained with Rietveld refinements.(b) Comparison of the experimental neutron PDF for all three samples at 10 K. (c) Analysis of the PDF data of cubic nanoparticles at 10 K with experimental data in open circles, calculated pattern in red, and difference curve in green.

Fig. 5
Fig. 5 The scattered intensity I(Q,E) at 5 K for (a) cubic nanoparticles and (b) bulk powder.The Q-integrated scattered intensity I(E) at different temperatures for (c) cubic nanoparticles and (d) bulk powder.

Fig. 6
Fig. 6 (a) Energy dependence of the inelastic neutron scattering intensity at different momentum transfers Q for the bulk powder (red open circles) and the cubic nanoparticles (green open squares).The solid lines are the calculated results, as described in the text.The calculated (b) and experimental (c) I(Q,E) at 5 K for the bulk Co 3 O 4 powder.

Fig. 7
Fig. 7 (a) The nearest L 1 , next-nearest neighbors L 2 and the next next-nearest neighbors L 3 distances for magnetic Co 2+ ions in Co 3 O 4 .(b) L 1 and (c) L 2 obtained from the neutron PDF data refinements at 10 K, as a function of r max .