Ferromagnetic Resonance of a 2D Array of Nanomagnets

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arXiv:1710.07452v2 [cond-mat.mes-hall] 8 Jun 2018 nano-spheres ewe h aoee n arsoi cls Ferro- scales. gap (FMR) macroscopic the resonance and magnetic bridging nanometer further the thus spatial between to resolution, regards temporal with progress and ex- measure- great known hand, and have other characterization ments the of the On techniques size, organization. perimental the spatial as and such shape parameters controllable struc- and well-defined with for fundamental tures demand as long-standing In welcome a are meet achievements they on. hy- these in so research, recording, interest theoretical and magnetic of catalysis as are perthermia, such that applications properties practical magnetic nano-elements tunable magnetic of of arrays characterizing and ing riso h aeil o ntne h ioa interac- dipolar the prop- instance, For absorption microwave material. the the control of be to erties material’s can as parameters the these so This characterize hand, varied other to the field. 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We also cluding the effects of both dipolar interactions and sur- discuss the effect of the array size on the difference in fre- face anisotropy and their competition [see Section III]. quency shift between the results of the two approaches. Therefore, the main objective of the present work is to Finally, we discuss the competition between DI and sur- i) derive the correction to the resonance frequency due face effects in two situations with a positive or negative to DI using perturbation theory, beyond the point-dipole contribution from the latter. The paper ends with our approximation, i.e. taking account of the shape and size conclusions and two short appendices. of the nano-elements (or dipoles) and ii) derive the shift of the resonance frequency due to surface effects using the effective model for each nano-element18,19. Then, we A. Energy apply this formalism to the prototypical case of an array of thin disks and derive the corresponding approximate Here we define the systems targeted by this study and expression for the frequency shift induced by DI. Next, discuss the various contributions to their energy, with a we analyze the contribution from surface anisotropy to special focus on the dipolar interactions. For simplicity, the FMR frequency and compare with the DI-induced the discussion of the contribution to the energy from the shift. nano-element surface anisotropy is postponed to Section Another general approach has been developed in Ref. III. 22 for studying the collective dipolar (or magnonic) spin- Consider a monodisperse array of magnetic nano- wave excitations in a two dimensional array of mag- elements each (of volume V ) carrying a magnetic mo- netic nano-dots, in the absence of magneto-crystalline ment mi = misi, i =1,..., N of magnitude mi = MsV anisotropy. Other similar works and approaches can be and direction si, with |si| = 1, Ms being the saturation found in the literature23–25 which deal with collective magnetization. The energy (in S.I. units) of the magnetic effects in assemblies of nano-elements. In the present moment mi is given by work we adopt a general but simple approach that al- (0) lows us to take into account surface effects as well as Ei = Ei + EDI,i, (1) dipolar interactions and to study their competition, in di- (0) lute and monodisperse assemblies with oriented magneto- where Ei is the energy of the non-interacting nano- crystalline (effective) uniaxial anisotropy. Moreover, for elements that comprises the Zeeman and (effective) practical reasons related with the possibility to com- anisotropy energies, and the second term EDI is the pare with experiments, we focus on FMR resonance DI contribution. The total energy of the system is and provide explicit analytical expressions for the fre- N E = i=1 Ei. quency shift induced by dipolar interactions and surface For two magnetic nano-elements carrying macroscopic P anisotropy. As was discussed above, today several ex- moments mi and mj, located at two arbitrary sites i and 5,6,8,10,23–30 perimental groups are able to fabricate well j, the dipolar interaction reads (in SI units) organized and almost monodisperse assemblies of cobalt or iron-oxide nano-particles and aim at measuring their µ0 ferromagnetic resonance frequency and resonance field. EDI,i,j ≡ mi · Dij · mj (2) It is then desirable to have at least approximate but sim- 4π ple analytical formulas to compare with the experimental where Dij is the corresponding tensor [see Eq. (12)]. results and to infer rough estimates of the most relevant Summing over all pair-wise interactions, avoiding double physical parameters, such as the elements size and sepa- counting, yields the energy of a magnetic moment at site ration. i due to its interaction with all other moments in the This paper has been organized as follows. In Section assembly with the corresponding energy I, we define the system and its energy, focusing on the µ contribution from the dipolar interactions. In Section II E ≡ 0 m · D · m . (3) DI,i 4π i ij j we present our general formalism in a matrix form and j

2 2 ix,iy ∈ N. Then, Rij = (xi − xj ) + (yi − yj ) while 2 q 2 rij = (ix − jx) + (iy − jy) = Rij /d. Therefore, when dealingq with DI on a super-lattice, it is quite natural to introduce the parameter

µ m2 λ ≡ 0 , (4) 4π d3 to characterize the strength of the DI in the system since the dipolar energy in Eq. (2) scales with λ. In the next section, we will apply our formalism to spe- Figure 1: A pair of magnetic nano-elements belonging to the cific situations where the expressions of all contributions 2D array (in the yz plane), of diameter D = 2R, height L and to the energy can be explicitly written. separation d. The standard system of spherical coordinates In the present work, we develop a general formalism (θ, ϕ) is also shown together with the setup of the magnetic H e that can be applied to an arbitrary system of interacting field and (magneto-crystalline) anisotropy easy axes . arrays of nano-magnets. However, here it will be applied (0) to the specific case of a two-dimensional mono-disperse In Eq. (1) the energy density Ei of an isolated nano- array of nano-magnets, with the main objective to derive element (ignoring its SE) is given by explicit expressions for the FMR frequency shift induced by the DI and SE. These calculations are based on pertur- E(0) = −µ M H · s − K (s · e )2 + E (6) bation theory for a dilute assembly but are valid for nano- i 0 s i 2 i i demag magnets of arbitrary size and shape, thus going beyond where K2 is the magneto-crystalline (uniaxial) anisotropy the simple point-dipole approximation (PDA). On other constant, ei the uniaxial anisotropy easy axis directed hand, in Section III we discuss in detail surface effects along the x axis (see Fig. 1). The term Edemag in Eq. that come into play when the size of the nano-elements (6) is the magnetostatic energy density becomes small enough (with the number of surface atoms µ µ E = − 0 M H · s = 0 M 2s N · s (7) exceeding 50%). In the framework of the effective model demag 2 s d i 2 s i i discussed earlier, we will discuss how the expressions in N H the present section are extended to include the contribu- where is the demagnetization tensor and d the de- tion from surface anisotropy. magnetizing field. A rigorous evaluation of the demagnetization tensor To summarize, our simplification only refer to: i) a for uniformly magnetized particles with cylindrical sym- collective condition which assumes that the assembly is metry was provided in Refs. [31] using elliptical inte- diluted. That is to say the center-to-center distance be- grals [see Eq. (74) therein]. However, as already empha- tween the nanoelements is much larger than the linear di- sized earlier, our goal here is to derive an approximate mension of the nanoelements. ii) An intrinsic condition analytical expression for the FMR frequency shift due requiring that the magnetic state of the nanoelements is to DI. Now, FMR measurements are performed under nearly saturated by the applied magnetic field. Now, the a DC magnetic field that is strong enough for saturat- geometry of the nanoelements themselves or that of the ing the magnetic system, and this leads to a smoothing sample (i.e. the assembly thereof) are brought in by the out of the spin non-colinearities that usually occur in a dipolar tensor D and the distribution of the distances rij . magnetic system, especially when its aspect ratio differs Therefore, we consider a 2D array of magnetic nano- from unity. In addition, in our system setup, the exter- elements which we assume to be lying in the yz plane, for nal magnetic field is applied in the direction of uniaxial mathematical convenience. The applied magnetic field anisotropy, thus leading to a strong effective field along and the (magneto-crystalline) uniaxial anisotropy easy the cylinder axis. In such a situation, the calculations e H e axes i are all directed along the x axis, i.e. = H x of the demagnetizing field greatly simplify, as is exempli- e e and i = x for all i =1,..., N . As shown in Fig. 1, we fied by Eqs. (26) and (27) of Ref. [32]. Consequently, adopt the usual spherical coordinates for the magnetiza- after averaging over the sample’s length, the following tion orientation approximate expression for the demagnetization factors for a nano-element with cylindrical symmetry about the ⊥ 33 six = sin θi cos ϕi = si cos ϕi, x axis (as is the case here) is obtained s ⊥ i = siy = sin θi sin ϕi = si sin ϕi, (5) 2 siz = cos θi = cos θi, Nx = (1+ δ) − 1+ δ ,

1 p 2  Ny =Nz = 1+ δ − δ , with |si| =1. 2 p 4 where δ ≡ R/L, with R being the radius of the cylinder is the usual DI dyadic and rîj the unit vector connecting and L its length (or thickness). In particular, for a very the sites i and j, i.e. rîj = rij /rij . Jij is the diagonal long cylinder with R ≪ L (δ ≪ 1), the longitudinal de- matrix magnetization factor Nx → 0 while the transverse factors Φij 0 0 Nz = Ny → 1/2. In the opposite limit, for a very thin Jij = 0 10 disk, R ≫ L (δ ≫ 1), Nx → 1 and Nz = Ny → 0.  0 01  Therefore, using |si| = 1 the demagnetizing energy density becomes (up to a constant)   and Φij is a function of the size and shape of the nano- µ0 2 2 elements as well as their separation [see Fig. 1 (right)]. E = M (N − N ) s . (8) 23,34 demag 2 s x z i,x It is defined by

The effective field H = − 1 δE(0)/δs , normalized eff,i Ms i i h with respect to the anisotropy field Jd (ηij , τ) Φij (ηij , τ) ≡ h (13) Id (ηij , τ) 2K2 µ0HK = , (9) with Ms H h H namely eff,i −→ eff,i ≡ eff,i/HK, and upon drop- ∞ −2qτ ping the index i (for simplicity), reads h 2 dq 2 1 − e Id (η, τ) = 16η τ 2 J1 (q) J1 (2ητq) 1 − , ˆ q " 2qτ # heff = [h + ksx + hdsx] ex (10) 0 ∞ where h ≡ H/H ,h ≡ −µ M (N − N ) /H and k dq K d 0 s x z K J h (η, τ) = 16η3τ J 2 (q) J (2ητq) 1 − e−2τq . (14) (=0 or 1) is a label merely introduced for keeping track d ˆ q2 1 0 of the contribution from magneto-crystalline anisotropy. 0 The angular frequency of an isolated nano-element is J and J are the well known Bessel functions. We (0) 11 −1 0 1 given by ω = γHeff , with γ ≃ 1.76×10 (T.s) being have also introduced the following geometrical or “aspect- the gyromagnetic ratio. Since, in the present setup, the ratio” parameters minimum-energy state of the nano-magnet corresponds to having its magnetic moment along the x axis, we have rij d L (0) ηij ≡ , τ ≡ . (15) ω = ωK (h + k + hd), where ωK ≡ γHK . Thus, for L 2R convenience we also introduce the dimensionless angular frequency Note that rij d is the center-to-center distance between the pair of nano-elements on sites i, j, with d being the ω(0) (super-lattice) step of the array (rij are dimensionless ̟(0) ≡ = h + k + h (11) ω d real numbers). Finally, the DI coefficient ξij , appearing K in Eq. (12), is given by [see Eq. (4)] In the sequel, we shall measure all frequencies in units λ of ωK, i.e, ̟ ≡ ω/ωK. h ξij ≡ Id (ηij , τ) . (16) 2K2V Now, we discuss the equilibrium state of the system. B. Dipolar interactions beyond the point-dipole approximation In general this is obtained by minimizing the total energy of the system with respect to all degrees of freedom. In the present case, we would have to minimize the energy For a pair of magnetic nano-elements belonging to the (1), summed over the whole lattice, with respect to the 2D array, as shown in Fig. 1, the DI interaction was 2N angles θi, ϕi,i =1,..., N . In general, it is well know obtained in this horizontal configuration in Ref. 34, see to the numerical-computing community that minimiza- also Ref. 23. It is given by tion of such a multi-variate function is a formidable task that requires a lot of efforts and high computing powers. N N Apart from the latter, one of the reasons is that there are EDI 1 EDI ≡ = ξij si · Dij sj (12) no “automatic” algorithms for finding the absolute min- 2K2V 2 i=1 imum of the function and for each situation, one has to X jX=1 j 6= i guide the solver through some prescribed path(s). Since the interactions are pair-wise, one could also proceed by where obtaining the energy minima for a dimer and then sum over the lattice. The equilibrium state of a dimer with DI in several configurations of anisotropy and applied field r r Jij − (2+Φij )îj îj were thoroughly studied in Ref. 34 and the various ex- Dij = 3 rij trema were found in a closed form. However, it is clear 5 that for interactions of arbitrary intensity the state of where heff,i now comprises both the free and interacting an (N +1)-body system does not necessarily include the parts of the system’s energy. α (. 1) the phenomenologi- state of the N-body system. For this reason the states cal damping parameter and tr is the (dimensionless) time −1 obtained in Ref. 34 cannot just be extended to the array defined by tr = t/ts, where ts = ωK is the nanoelement’s studied here by summing over the lattice index. As such, characteristic timescale. and as stated earlier, we resort to an analytical treatment In order to compute the spectrum of the system ex- based on a few simplifying assumptions. Accordingly, we citations we may proceed by linearizing the LLE (17) first assume that the DI are weak enough as to allow for about the equilibrium state. Indeed, we assume that the such an extension. In fact, in FMR measurements, the equilibrium state, which minimizes the system’s energy, applied magnetic field is usually strong enough as to sat- s(0) denoted by i , has been determined with the urate the magnetic state of the system, and this leads i=1,...,N to the nearly linear branch of the resonance frequency help of somen analyticalo or numerical technique. Then, we s s(0) s as a function of the amplitude of the applied field. This may write i ≃ i + δ i and transform the differential 37 is the situation that we adopt here. Usually the DI in equation (17) into the following equation such a 2D array (oblate assembly) favor a net magnetic N moment in the plane of the array. However, the strong d (δsi) = [H I (α)] δs , i =1,..., N (18) effective anisotropy and the (relatively) strong magnetic dτ ik k field are parallel to each other and perpendicular to the Xk=1 array’s plane, and should then lead to a reorientation of with the (pseudo-)Hessian all magnetic moments towards their direction. For this reason, the setup in Fig. 1 leads to the equilibrium state: ∂2 E 1 ∂2 E θkθi sin θ θkϕi π i θi = 2 , ϕi =0,i =1,..., N . Hik [E] ≡ (19) The effect of dipolar interactions in ordered and disor-  1 ∂2 E 1 ∂2 E  sin θk ϕkθi sin θk sin θi ϕkϕi dered low-dimensional nanoparticle assemblies have been   studied by many authors35. In dense assemblies, the DI whose matrix elements are second-order derivatives of the lead to various local magnetic orders that depend on energy with respect to the spherical angles of (θi, ϕi) that 35,36 the lattice structure . Under some conditions, they determine the direction si. All these matrix elements are may also induce long-range order leading to the so-called s(0) evaluated at the equilibrium state i . The super-ferromagnetic state28. Obviously, the situation we i=1,...,N symbol ∂2 stands for the secondn derivativeo with re- consider here is quite different in that the concentration αk βi we assume is not high enough as to lead to the onset of spect to the angles αk,βi. assembly-wide collective states. In addition, as argued The matrix earlier we assume that the DI do not modify the equilib- α −1 I (α) ≡ rium state as determined by a competition between the 1 α applied field and the anisotropy. However, even such a weak intensity of DI would be important to the dynam- stems from the double cross product in Eq. (17). ics of the assembly since then the energy barriers and Next, one may seek solutions of Eq. (18) in the form iΩτ thereby the relaxation rates would be affected. δsk = δsk(0) e leading to the following eigenvalue problem

N II. FMR SPECTRUM : GENERAL FORMALISM [HikI (α) − iΩ1] δsk = 0 (20) k=1 In this section, we present the general formalism we X have developed in order to derive approximate analytical whose set of roots {Ωn}1≤n≤N yields the system’s eigen- Ωn ωn expressions for the shift of the FMR frequency induced frequencies, fn = −i 2π = 2π , where ωn is the real angu- by dipolar interactions. lar frequency (in rad/s). Here, 1 is the identity matrix 1µν µν with matrix elements ik = δikδ . In the general case of an array of nano-elements with arbitrary DI, it is not possible to determine the system’s A. Landau-Lifshitz equation and FMR eigenvalue s(0) problem exact equilibrium state i that minimizes the i=1,...,N total energy of the system,n includingo the (core and sur- The time evolution of the magnetization orientation si face) anisotropy, the DI and the applied field. In ad- is governed by the damped (norm-conserving) Landau- dition, it is not an easy matter to solve the eigenvalue Lifshitz equation (LLE) problem (20) in its full generality. Of course, these tasks can be numerically accomplished to some extent. How- dsi ever, as stated earlier, our objective here is to obtain an = −si × heff,i + α si × (si × heff,i) , (17) dtr analytical expression for the FMR frequency. In order to 6 do so, we restrict ourselves to the case of dilute assem- further assumption that the equilibrium state is not al- blies of nanomagnets, and as such, we solve the eigenvalue tered by the dipolar interactions. More precisely, we as- problem (20) using perturbation theory that we present sume that the main equilibrium of the system is setup by now. the competition between the strong (effective) anisotropy and the external DC magnetic field. Of course, the DI of arbitrary strength would change both the energy min- B. DI correction to FMR frequency : perturbation ima and saddle points of the system, and thereby signif- theory icantly change its dynamics. Here we restrict ourselves to the situation where the DI only contribute through In the following we only consider the undamped case, the second term in Eq. (21), which is regarded as a cor- i.e. with α =0 [see Eq. (17)], we present our formalism rection to the first term. This assumption is experimen- in the general case. tally relevant for dilute assemblies. For instance, it has As we have seen, the excitation spectrum can be ob- been demonstrated14 that inter-particle interactions in tained by diagonalizing the matrix Hik (α) ≡ HikI (α) of assemblies of core-shell (Fe3O4/SiO2) nanoparticles can matrix elements be tuned by modifying the thickness of the shell. Therefore, in spin components the LLE (18) reads Hµν µρ ρν ik (α)= Hik I (α) . ρ=θ,ϕ µ N X d (δs ) µν i = F 1 + F −1Ξ δsν . (22) A word is in order regarding the various indices. The dt ik k ν=θ,ϕ k=1 problem being studied here is an array of magnetic mo- X X ments located at the nodes of a super-lattice. Hence, there are two kinds of indices. The first one, using the Regarding the various indices discussed above, the matrix F of the non-interacting array can be written as F = Greek letters µ,ν,ρ, refers to the components of a mag- 1 µν µν F2×2 ⊗ N ×N , or in components Fik = F2×2 i δik = netic moment in the system of spherical coordinates and µν δ H E(0) I (α) . Using the matrix F introduced ear- thus assumes the values θ, ϕ. The second index, using the ik ii Roman letters i, j, k, refers to the lattice site and assumes lier, the matrix F explicitly reads the values 1,..., N . Therefore, the full “phase space” is a direct product of the two sub-spaces corresponding to θϕ θθ Fii −Fii Fik = δik ⊗ 1. the two kinds of variables. Likewise, the matrices in- F ϕϕ −F ϕθ volved in these calculations are tensor products of the ii ii corresponding sub-matrices. Note that the 2 × 2 matrix above has two eigenval- If we focus on dilute assemblies with relatively weak (0) (0) DI we can write the total energy as the sum of the en- ues ±i̟i , where ̟i is the (normalized) resonance ergy of the non-interacting assembly and the interaction frequency of the magnetic moment at site i in the non- (0) interacting case. In fact, for the mono-disperse assem- contribution, i.e. E = E + EDI, with E ≡ E/ (2K2V ) and similarly for each contribution. Then, the pseudo- blies considered here all these frequencies are identi- (0) (0) Hessian Hik(Ei) can also be correspondingly split as fol- cal, i.e. ̟i ≡ ̟ , defined in Eq. (11). Hence, 2 lows N (0) (0) 2N det F = i ̟i = ̟ . Hµν µν µν On the other hand, the matrix Ξ introduced above and ik (α)= Fik + Ξik (21) Q which contains the DI contribution can also be written where Fik is the contribution in the absence of interac- explicitly to some limit. Again, using the matrix Θ in- tions given by the same matrix as in Eq. (19), upon troduced earlier, the 2 × 2 diagonal block of the matrix (0) substituting E for E, and multiplied by I (α). Thus, Ξ= H [EDI] I (α) reads (0) F = H E I (α). Similarly, Ξik is the DI contribution given by the matrix in Eq. (19), with sub- θϕ θθ Θii −Θii stitution of EDI for E, multiplied by I (α), i.e. Ξ = ϕϕ ϕθ , i =1, 2,..., N . (23) Θii −Θii H [EDI] I (α). For later use, we introduce the two ma- (0) trices F ≡ H E , Θ ≡ H [EDI]. It is understood that wherever they appear all matrix One should note that these DI matrix elements with elements have to be evaluated at the equilibrium state identical lattice sites are not equal to zero even if they s(0) s(0) correspond to pair-wise interactions. Indeed, in the most i with i = (1, 0, 0). In the situation of i=1,...,N general situation, the second derivatives of the energy are nrelativelyo weak coupling considered here, we make the given by 7

2 s e e h e e h ∂θiθk E = δik [ i ·− θi · ( θi · ∇i)] eff,i − (1 − δik) θi · [ θk · ∇k] eff,i, 2 s e e e h e e h ∂ϕkϕi E = δik sin θi [(sin θi i + cos θi θi ) − sin θi ϕi · ( ϕi · ∇i)] eff,i − (1 − δik) sin θi sin θk ϕi · [ ϕk · ∇k] eff,i, 2 e e e h e e h ∂θkϕi E = −δik [cos θi ϕi · + sin θi ϕi · ( θi · ∇i)] eff,i − (1 − δik) sin θi ϕi · [ θk · ∇k] eff,i, 2 e e e h e e h ∂ϕkθi E = −δik [cos θi ϕi · + sin θi θi · [ ϕi · ∇i]] eff,i − (1 − δik) sin θk θi · ( ϕk · ∇k) eff,i. (24)

Explicit expressions for the DI energy only are given and thereby we obtain in Appendix A. Note then that because of the first term µν in each line of Eq. (24), the second derivatives Θii do N not vanish even for the DI contribution, and using (19, −1 1 θθ ϕϕ Tr F Ξ = 2 F Θ A1) we do see that Θµν 6=0. However, we stress that the (̟(0)) ii ii ii i=1 DI contribution to heff,i contains a sum over the whole X lattice except (for the site i) and thus the DI coefficient ϕθ θϕ θϕ ϕθ ϕϕ θθ − Fii Θii + Fii Θii + Fii Θii . entering heff,i involves a sum over j with j 6= i. Obvi- ously, the matrix Ξ has also nonzero off-diagonal blocks i Thus, Eq. (26) becomes which are of the same form as in (23) but with distinct indices i, k =1, 2,..., N , i.e. i 6= k. Now, we introduce the new tensor Λ ≡F 1 + F −1Ξ N N 1 (0) 1 θθ ϕϕ log ̟i = log ̟ + 2 F Θ N 2N (̟(0)) ii ii det Λ = det F × det 1 + F −1Ξ . (25) i=1 i=1 X X (0) − F ϕθΘθϕ + F θϕΘϕθ + F ϕϕΘθθ . and set to compute its determinant. Similarly to ̟n ii ii ii ii ii ii (the eigenvalues of F) we introduce the eigenvalues ̟n i as the resonance frequencies with the index n running Next, it is quite reasonable to drop the sum on the left- through all the 2N (collective) modes of the interacting hand side of the equation above as long as one considers nano-element arrays which are large enough and spatially system. This leads to detΛ = N ̟2 . Then, let us n=1 n isotropic. Then, upon expanding with respect to the examine the last determinant in Eq. (25). The product (0) −1 Q small parameter ̟i/̟ . 1, we obtain the final expres- F Ξ scales with the ratio λ/H, i.e. the ratio of the (0) µ0 2 3 sion for the DI-induced frequency shift ∆̟DI ≡ ̟−̟ DI intensity λ = 4π m /d to the static magnetic field H. This ratio is obviously small for a dilute assembly, especially for standard FMR measurements where the N 1 1 θθ ϕϕ DC field is usually taken strong enough to saturate the ∆̟DI ≃ 2̟(0) N Fii Θii sample (usually between 0.3T and 1T). Hence, it is i=1 X (27) justified to make an expansion with respect to F −1Ξ. ϕθ θϕ θϕ ϕθ ϕϕ θθ For this, we apply the logarithm and use the expansion − Fii Θii + Fii Θii + Fii Θii . log (1 + x) ≃ x (for operators) together with the identity i log det A = Tr log A. Doing so, we obtain We recall here again that the various matrix elements appearing above are second derivatives of the energy with N N 2 respect to the system coordinates, evaluated at the equi- 2 (0) −1 (0) (0) log ̟i ≃ log ̟i + Tr F Ξ librium state s , with s = (1, 0, 0). In some partic- i=1 i=1 ular situations of anisotropy and field setup, the matrix Ξ X X =2N log ̟(0) + Tr F −1Ξ . (26) can be explicitly computed, thus directly rendering the correction to the FMR frequency. Accordingly, in the In order to compute the trace above, we only need to next section we give explicit results for the specific case collect the (block) diagonals of the matrix F −1Ξ whose of nano-elements with effective uniaxial anisotropy along first block is as follows (showing only the diagonal ele- the field direction. ments)

C. FMR frequency of a 2D array of nanomagnets ϕθ θθ θϕ θθ 1 −Fii Fii Θii −Θii 2 (̟(0)) −F ϕϕ F θϕ Θϕϕ −Θϕθ 1. DI-induced frequency shift ii ii ii ii 

θθ ϕϕ ϕθ θϕ 1 Fii Θii − Fii Θii ∗ Now, we come to the evaluation of the matrix elements = 2 (̟(0)) ∗ F ϕϕΘθθ − F θϕΘϕθ appearing in Eq. (27). ii ii ii ii 8

For the non-interacting case we have present work we perform these calculations in the case of thin disks. For thin disks (τ ≪ 1), the integrands in Eq. (14) decay (0) 1 (0) ∂2 E ∂2 E to zero for q & 3. Hence, we can expand the exponential θi i sin θi θiϕi i ̟(0) 0 Fii =   = (0) . in these integrals up to the first order and then expand 1 2 (0) 1 2 (0) 0 ̟ ∂ E 2 ∂ E the integrals in powers of 1/κ (for κ> 1). This yields  sin θi ϕiθi i sin θi ϕi i    2 For the DI contribution, both derivatives ∂θ EDI and i h 9 1 2 J (κij , τ) ≃ 1+ × , ∂ϕi EDI survive in Eq. (27) when evaluated at the equilib- d 2 2 16κ rij rium state (θi = π/2, ϕi = 0), whereas the cross derivatives vanish. Consequently, we obtain h 3 1 Id (κij , τ) ≃ 1+ 2 × 2 , (29) 16κ rij N h θθ ϕϕ λ Id (ηij , τ) Θii =Θii = − 3 Φij . and 2K2V rij jX=1 j 6= i h Jd (κij , τ) 3 1 Φ (κij , τ)= h ≃ 1+ 2 × 2 . (30) Next, using (13) and introducing the geometrical factor Id (κij , τ) 8κ rij κ as the ratio of the inter-element separation d to their diameter D = 2R, i.e. κ = d/D [see Eq. (4)], we may To be specific, we consider the FeV disks of Ref. 38 rewrite the result above as follows with D = 600 nm,L = 26.7 nm and a center-to-center separation d = 1600nm, we have τ = 1/ (2δ) = L/D = N h θθ ϕϕ A Jd (ηij , τ) 0.0445, η = d/L ≃ 60 and thereby κ = ητ = d/D = Θii =Θii = − 3 3 2.667. Therefore, the condition for the validity of the κ rij jX=1 results above, i.e. κ > 1 is satisfied even in the most j 6= i unfavorable case. Consequently, we obtain the frequency shift where we have introduced the material-dependent constant A 9 ∆̟DI ≃− 3 C3 + 2 C5 (31) 2 3 2 κ 16κ µ0 m /D µ0 Ms V A ≡ = 3 . 4π 2K2V 8π K2D where we have introduced the lattice sum Then, substituting this result in Eq. (27) we arrive 1 N N 1 at the explicit DI correction to the FMR (dimensionless) C ≡ . n N rn angular frequency ̟ of the array of nano-magnets i=1 ij X jX=1 N N j 6= i A 1 J h (η , τ) ∆̟ ≃− × d ij . (28) DI κ3 N r3 =1 ij We thus have to evaluate two lattice sums, i j =1 N N X X the well-known one39 C ≡ 1/ N r3 , j 6= i 3 i k,k6=i ik which is equal to C3 ≃ 9 and the other C5 ≡ N N 5 P P i k,k6=i 1/ N rik , equal to C5 ≃ 5.1, both in the thermodynamic limit. 2. 2D array of nano-disks PWeP may then rewrite Eq. (31) as follows (assuming C3 6=0, i.e. excluding spheres and cubes) The DI correction to the FMR frequency given in Eq. (28) is an implicit expression that depends on various parameters pertaining both to the nano-elements them- 9 C5 ∆̟DI ≃ ∆̟pda 1+ 2 (32) selves (size, shape, energy) and to the assembly (spa- 16κ C3 tial arrangement and shape). In particular, the nano- elements separation d enters this expression via the inte- where we have singled out the contribution ∆̟pda ≡ h 3 gral Jd (ηij , τ) and the parameter κ. In order to derive − A/κ C3 that obtains within the PDA. As such, we an explicit (analytical) expression for the frequency shift see more explicitly the correction to the FMR frequency in terms of the nano-elements separation d (or the param- due to the size and shape of the nanomagnets. Both eter κ), one has to (numerically) compute the integrals contributions in Eq. (32) are in the form of a dipolar- h h Id (η, τ) and Jd (η, τ). However, this can also be analyt- like term multiplied by a lattice sum. While the PDA ically done in some limiting cases of the parameters η and term ∆̟pda scales with the nano-elements separation d τ, namely τ ≪ 1 for thin disks (or platelets) or τ ≫ 1 for as 1/d3, the term that stems from size and shape effects long cylinders (or wires). By way of illustration, in the scales with d as 1/d5. The power 5 here arises from the 9

6 which cannot be eﬃciently dealt with even with the help of optimized numerical approaches. Nonetheless, in the 5 limiting case of not-too-strong surface eﬀects, inasmuch (%)

DI as the spin conﬁguration inside of the nanomagnet can ∆ϖ 4 be regarded as quasi-collinear, the static and dynamic )/

pda properties of the nanomagnet may be recovered with the

∆ϖ 3 help of an eﬀective macroscopic model for the net mag- - netic moment of the nanomagnet. More precisely, it has DI

∆ϖ 2 been shown that a many-spin nanomagnet of a given lat-

= ( tice structure and energy parameters (on-site core and pda 1 surface anisotropy, local exchange interactions) may be δϖ modeled by a macroscopic magnetic moment m evolving 20 0 in an effective potential . The latter is, in principle an 3 4 5 6 7 infinite polynomial in the components of m, but whose κ leading terms are of two types, one is a quadratic and the other a quartic contribution with coefficients K and K Figure 2: Relative variation of the frequency shift between 2 4 ̟ ̟ that strongly depends on the microscopic parameters, as |∆ DI−∆ pda| PDA and cylindrical nanomagnets δ̟pda = ̟ as ∆ DI well as on the shape and size of the nanomagnet. Here, a function of the distance κ. we would like to emphasize in passing the fact that the quartic term is a pure surface contribution, that appears even in the absence of core anisotropy [see Ref. 20,21] 3-dimensional space coordinates of the individual nano- and which may renormalize the cubic anisotropy of the particles, plus the 2 space dimensions arising from the (underlying) magnetic material the nanomagnet is made shape of the disks, for which the thickness is ignored (in of. However, there remains the question as to how one the current thin-disk approximation). Likewise, the ex- can distinguish this surface-induced fourth-order contri- pansions of the shape integrals (29) and (30) also exhibit bution from the (usually weak) cubic anisotropy found in a point-dipole contribution together with a 2-dimensional 2 magnetic materials. At least for thin disks where the ef- shape correction that scales as 1/κ . In Fig. 2 we plot fective anisotropy is mostly of (boundary) surface origin, the relative difference between ∆̟DI and ∆̟pda, namely this quartic contribution may become dominant. An ex- |∆̟DI−∆̟pda| δ̟pda = . The results confirm that, for ∆̟DI ample of this situation was provided by cobalt nano-dots not-too-dense assemblies, i.e. for 2.5 . κ . 3, there with enhanced edge magnetic anisotropy41. is a variation (δ̟pda ≃ 5%) of the frequency shift due In the present work, we assume that the uniaxial to the fact that the nanomagnets are not simple point anisotropy in Eq. (6), with coefficient K2, is an effective dipoles. This variation should be accessible to experi- anisotropy that already includes the (small) renormaliza- ments. Obviously, for very dilute assemblies (κ & 7) the tion effect from surface anisotropy. On the other hand, PDA provides a correct description of the physics up to the strongest contribution induced by surface effects is an error less than 1%. given by 1 E(SE) = K s4 . (33) i 2 4 i,α α=x,y,z III. SURFACE EFFECTS X where K4 is a constant that scales with the square of 20 In this section we discuss the impact of surface effects the surface anisotropy constant . K4 may be posi- on the results obtained above. In an assembly of nano- tive or negative, depending on the underlying magnetic magnets the intrinsic features of the latter, such as sur- material19. In the sequel, we will use the more relevant face anisotropy (SA), are generally smoothed out by the parameter ζ ≡ K4/K2. Consequently, adding this con- distributions of size and (easy axis) orientation. However, tribution to the free-particle energy (6) adds the term 3 e in some situations, e.g. of monodisperse assemblies with −ζ α=x,y,z mi,α α to the effective field (10) and thereby oriented anisotropy, as is considered here, SE may lead the angular frequency of an isolated nanomagnet be- P (0) to a non negligible contribution to the magnetic prop- comes ω = ωK (h + k + hd − ζ). Likewise, the cor- erties of the nano-elements, especially FMR frequency. responding dimensionless angular frequency is now given (0) (0) Many examples of such assemblies have been fabricated by ̟˜ = h + k + hd − ζ ≡ ̟ + ̟SE [see Eq. (11)]. by several experimental groups around the world, see We see that due to the surface anisotropy contribution, the already cited works in the introduction as well Refs. the FMR frequency of a single nanomagnet may either 26,29,30,40. Surface effects are local effects whose study increase or decrease according to the sign of ζ. In particu- requires recourse to an atomic approach that accounts for lar, it is interesting to investigate how surface effects may the local atomic environment. However, from the com- make up for the frequency red-shift induced by dipolar putational point of view, taking account of such effects in interactions, as discussed earlier. Accordingly, the total an interacting assembly leads to tremendous difficulties frequency shift, due to both DI and surface effects, is 10

model developed in Ref. 42 for multiple interacting mag- (0) 5.4 f = 5.39733 GHz netic moments. The dynamical ﬁelds arising from the (a) dipolar coupling, which are necessary for calculating the FMR spectra of the nanoparticle array using the semi- f (GHz) 5.35 th analytical model, are given in Appendix B. fsa f For the FeV disks, the thin-disk regime (L/ (2R) ≪ 1) 5.3 applies and thereby the frequency shift can be calculated 27,38 using Eq. (31). The materials parameters are : Ms = 2 6 4 3 (b) 1.353 × 10 A/m, H = 1.72 T,Kv = 4.1 × 10 J/m , (%) (0) 1.5 |∆ωth| /ω (0) and from Eq. (9) we can infer HK ≃ 0.0606 T and ωK =

(0) |∆ω | /ω sa 9 −1 th γHK ≃ 10.67 × 10 rad.s . Next, from Fig. 3(c) of δωsa /ω 1

| Ref. 38 we can read oﬀ the frequency of the isolated (0) ω 0.5 ∆ωlw/ω (0) (0) 9 −1

∆ elements, f ≃ 5.35 GHz or ωexp ≃ 33.62 × 10 rad.s . | 0 We can also compute the effective field using Eq. (10). δ = R/L ≃ 11.24 leading to N = 0.956,N = 0.022 3 4 5 6κ 7 x z and Hd ≃−1.59 T. Note that for an infinitely thin disk (Nx → 1 and Nz → 0) we would obtain |Hd| ≃ 1.7 T. Figure 3: (a) Resonance frequency of an interacting 20 × 20 Then, since H > |Hd| we may consider the magnetic square array of FeV nanodisks as a function of the rela- moment of the disks to be aligned along the direction of tive nano-disk separation κ. The solid black line represents the applied magnetic field, i.e. sx ≃ 1 and thereby the (0) fth = ω +∆ω / (2π), where the frequency shift is ob- effective field evaluates to Heff ≃ 0.193 T. This yields the tained from Eq. (28). In red circles we present the semi- (theoretical) frequency of non-interacting nano-elements 42 analytical uniform mode, obtained from the model of Ref. . (0) 9 −1 ωth = γHeff ≃ 33.91 × 10 rad.s , which is in good Horizontal dashed line represents the resonance frequency in (0) the non-interacting case. (b) Relative variation of the fre- agreement with the experimental value ωexp. (0) (0) quency shifts ∆ωth/ω and ∆ωsa/ω obtained from theory Now, regarding the comparison between our work and th and semi-analytical calculations, respectively. δωsa is the dif- the experiments of Ref. 38, beyond the agreement of the ference between the two approaches. orders of magnitude, an important warning is necessary. In Fig. 4 of this reference, the authors plot the difference in frequency between the anti-binding and binding modes given by [see Eq. (31)] as a function of the nano-elements separation. Apart from the fact that only 3 values of the latter were avail- A 9 able, and despite the (apparent) qualitative agreement ∆̟ = −̟SE + ∆̟DI = ζ − C3 + C5 . (34) κ3 16κ2 with our theory, it is not possible to compare these experiments with our theory. Indeed, as discussed earlier, For instance, for ζ > 0 we see that surface anisotropy our approach only renders the frequency of the collec- may compete with dipolar interactions. This will be dis- tive mode, which is here the binding mode, and it is not cussed in Section IVB. possible to derive the frequency of the anti-binding mode as this would require the full solution of the eigenvalue problem. On the other hand, the individual frequencies IV. RESULTS AND DISCUSSION of the two modes cannot be extracted from these experiments because the nano-disks are not fully identical and Let us now discuss some of the results that can be in- their distances to the sensor are not equal either. ferred from Eq. (31) for the effect of DI and Eq. (34) In Fig. 3(a) we plot the resonance frequency ob- (0) when SE are included, especially in what regards the de- tained from Eq. (31) (fth = ω + ∆ω / (2π) - black pendence of the shift in frequency on the parameter κ, solid line) as a function of the relative distance param- that is the ratio of the nano-elements separation d to eter κ, together with the frequency (fsa) of the uniform their diameter D. resonance mode of the system obtained from the semi- analytical model42 (red dashed line with circles). As expected, the dipolar coupling reduces the frequency rela- A. Effects of dipolar interactions (ignoring surface tive to the non-interacting case (horizontal dashed line). effects) The shift decreases as the distance between the disks increases, which is equivalent to a decrease in the dipolar For an order of magnitude and a comparison with other coupling. Both theoretical calculations render the same theoretical models, we consider for instance the ferromag- qualitative behavior, with some quantitative discrepan- netic resonance of a finite 20 × 20 square array. We com- cies, especially for stronger DI. This is due to the several pare the results from Eq. (28) with C3 = 7.50253 (for a approximations and expansions used in the derivation of square 20 × 20 array) and those from the semi-analytical Eq. (31). Nonetheless, we can clearly see that the differ- 11 ence is reduced as κ increases, reaching a good agreement 5.42 for κ ≥ 5. (0) 5.4 f Fig. 3(b), shows the variation of the relative fre- (0) quency shift |∆ω| /ω of each approach, and the dif- 5.38 sh (0) ference δωsa ≡ (∆ωth − ∆ωsa) /ω . ∆ωth and ∆ωsa are the absolute frequency shifts induced by the dipolar

(GHz) 5.36 coupling, obtained from Eq. (31) and the semi-analytical f model, respectively. We see that the DI induce small fre- ζ=K /K = 0.00 5.34 4 2 quency shifts on the order of 1.5 % or even lower for the ζ=K /K =-0.01 explored distances. Furthermore, we can see that when 4 2 ζ=K4/K2= 0.01 Eq. (31) becomes a good approximation (κ ≥ 5), the rel- 5.32 ative frequency shifts are on the order of 0.3 %. This relative variation expressed as a percentage is below typ- 3 4 5 6 7 ical relative experimental linewidths in similar systems κ (0) 38 (linewidth ∆ωlw/ (2π) ≃ 20 MHz, ω / (2π) ≃ 5 GHz , Figure 5: FMR frequency as a function of the relative nan- thus ∆ω /ω(0) =0.4 %). However, increasing ω(0) (e.g. lw odisk distance κ for various surface anisotropies ζ = K4/K2. by increasing the applied field) reduces the relative fre- (0) sh quency shift |∆ω| /ω and the error δωsa . As a conse- quence, the validity of our formalism [see Eq. (31)] ex- tends to stronger interactions (smaller κ), making it pos- B. Effects of dipolar interactions including surface sible to reach the regime where the predicted frequency effects shifts can be measured in experiments. Eq. (34) clearly reveals a competition between the effects of surface anisotropy and DI on the FMR frequency. In order to better assess the role of SE, we consider an

th interacting 20×20 array of nano-disks similar to the sam- δω (%) sa ple of Fig. 3. The results are shown in Fig. 5 where we 0.7 have restricted our investigation to the case of small sur- 0.6 face anisotropy i.e. |ζ| ≪ 1 in order to remain within the limits of the effective macrospin model [see discus- 0.5 18,19 20 × 20 sion in Section III] . First, analyzing the effect of sur- 15 × 15 face anisotropy alone, we see that changing the value and 0.4 10 × 10 sign of ζ has a large effect on the FMR frequency, taking 5 × 5 (0) 0.3 f as a reference. Now, for 2D square arrays the dipolar interactions tend to maintain the magnetic moments 0.2 within the plane. In contrast, depending on the sign of ζ, SE favor a magnetic alignment along the cube facets 0.1 (ζ < 0), or along the cube diagonals (ζ > 0). Therefore, for materials with one may expect a competition 0 ζ > 0 between SE and DI. This is what is observed in Fig. 5: 3 4 5 6 7 κ at high densities (small κ) DI dominate the correction to the FMR frequency and induce a red-shift, whereas for very dilute assemblies (large κ) each nanodisk behaves Figure 4: Error in the relative shift in frequency as a function 30 of the relative distance κ for different sizes of a square array like an isolated entity and SE dominate and induce a of disks. blue-shift. At leading orders in κ, the critical value κc marking the crossover from a red- to a blue-shift is given 1/3 by κc ≃ (AC3/ζ) . This is the point where the blue line crosses the dashed line in Fig. 5, implying that SE th compensate for the DI. For the FeV thin disks considered The dependence of δωsa on κ for different sizes of the array is shown in Fig. 4. It can be clearly seen that here, κc ≃ 3.9. This corresponds to an inter-element the error decreases for smaller arrays. Indeed, decreasing separation of 4 times the element diameter, i.e. a center- the size of the array decreases the overall dipolar contri- to-center distance of 2400 nm. butions, thus making the different approximations more The value of ζ taken here is rather small as compared precise. Furthermore, it can also be seen that the error to the estimates obtained by other authors in cobalt and tends to stabilize as the size of the array increases, and no iron-oxide elements15–17. For such higher values (an or- important variations are expected for arrays larger than der of magnitude larger) of surface anisotropy, compensa- 20 × 20. tion of the DI effects should occur for much closer nano- 12 elements, or equivalently denser assemblies. However, Appendix A: The pseudo-Hessian matrix elements this reasoning cannot be taken too far, at least in the for the DI contribution framework of our approach, since our treatment is limited to dilute assemblies and not-too-strong surface disorder. 37 Nonetheless, it does confirm the screening effect of DI by The following general expressions are used in the cal- surface disorder studied earlier by the authors43,44. culation of the second derivatives of the DI contribution with respect to the angular variables (θi, ϕi)

V. CONCLUSION N ∂2 E = −δ ξ (1 − δ ) s · D · s (A1) We have developed a general formalism for deriving θiθk DI ik ij ij i ij j j=1 practical analytical formulas for the shift in FMR fre- X e e quency induced by both dipolar interactions and surface +ξik (1 − δik) ( θi · Dik · θk ) disorder in an array of magnetic nano-elements. Even N though this has been done with the help of perturbation 2 ∂ϕiϕk EDI = −δik ξij (1 − δij ) × theory, which only applies to relatively dilute assemblies, j=1 X or equivalently for well separated nano-elements, the gen- sin2 θ s + sin θ cos θ e · D · s eral character of this formalism resides in the fact that i i i i θi ij j +ξ (1 − δ ) sin θ sin θ (e · D · e ) it applies to nano-elements of arbitrary shape and size, ik ik i k ϕi ik ϕk and as such, it deals with the dipolar interactions beyond N 2 e s the point-dipole approximation. An analytical expres- ∂θkϕi EDI = δik ξij (1 − δij )cos θi ( ϕi · Dij · j ) j=1 sion for the frequency shift induced by dipolar interac- X e e tions has been explicitly derived for an arbitrary array +ξik (1 − δik) sin θi ( ϕi · Dik · θk ) of monodisperse elements, and the contribution due to N their shape and size has been singled out. Next, this 2 e s ∂ϕkθi EDI = δik ξij (1 − δij )cos θi ( ϕi · Dij · j ) formalism has been applied to the limiting case of thin j=1 disks of FeV, recently investigated by the technique of X +ξ (1 − δ ) sin θ (e · D · e ) , Magnetic Resonance Force Microscopy. We have clearly ik ik k θi ik ϕk shown that the contribution of dipolar interactions to the FMR frequency of a 2D array of nano-elements is a linear with function of the parameter ξ which scales as the inverse of the third power of the elements separation. In addition to this contribution, that obtains within the point-dipole approximation, we also obtain a contribution from the nano-elements size and shape which scales with the in- cos θi cos ϕi verse fifth power of the nano-elements separation. We e = ∂ s = cos θ sin ϕ , θi θi i  i i  have also studied the effect of the array size on the fre- − sin θi quency shift and have found that the red-shift of the reso-  − sin ϕ nance is smaller for smaller arrays. The effects of surface 1 i e = ∂ s = cos ϕ . anisotropy on the frequency shift have been taken into ϕi ϕi i i sin θi  0  account with the help of an effective macroscopic model for the isolated nano-elements. Depending on the sign   of the corresponding contribution, which changes with the properties pertaining to the nano-element itself, we may obtain either a blue-shift or a red-shift of the FMR frequency. Correspondingly, this may lead to a competi- Appendix B: Dynamical dipolar fields tion or a concomitant effect with the dipolar interactions. This means that surface anisotropy and dipolar interactions provide us with a handle for adjusting the resonance The dynamical fields due to the dipolar coupling can frequency of nano-magnet assemblies. be calculated in the context of the model presented in42 for the propagation of in-plane spin waves in multilayer systems. This model can be applied to the array of interacting nano-particles presented here for the zero wave- Acknowledgments vector (uniform mode). Starting from Eq. (12) (in SI units), and following a similar procedure as the one pre- A. F. Franco acknowledges financial support from the sented in42, the following dipolar dynamical fields were FONDECYT postdoctoral project No 3150180. obtained 13

± with ϕi,j ≡ ϕi ± ϕj . j d Ms Hxixi = − 3 [Φij cos θi cos θj + sin θi sin θj 4πrij Xj6=i − + × cos ϕi,j − (2+Φij ) rij,xrij,y sin ϕi,j 2 2 +rij,x cos ϕi cos ϕj + rij,j sin ϕi sin ϕj d d Hyiyi = Hxixi j d Ms Hxixj = 3 [Φij sin θi sin θj + cos θi cos θj 4πrij − + × cos ϕi,j − (2+Φij ) rij,xrij,y sin ϕi,j 2 2 +rij,x cos ϕi cos ϕj + rij,y sin ϕi sin ϕj j d Ms − 2 Hyiyj = 3 cos ϕi,j − (2+Φij ) rij,x sin ϕi sin ϕj 4πrij 2 − +rij,y cos ϕi cos ϕj − rij,xrij,y sin ϕi,j j d Ms − Hxiyj = 3 cos θi sin ϕi,j +(2+Φij ) 4πrij 2 2 × rij,x cos ϕi sin ϕj − rij,y sin ϕi cos ϕj + −rij,xrij,y cos ϕi,j j d Ms − Hyixj = − 3 cos θj sin ϕi,j − (2+Φij ) 4πrij 2 2 × rij,x sin ϕi cos ϕj − rij,y cos ϕi sin ϕj + −rij,xrij,y cos ϕi,j ,

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