Spin-wave excitations in presence of nanoclusters of magnetic impurities

Nanoscale inhomogeneities and impurity clustering are often found to drastically affect the magnetic and transport properties in disordered/diluted systems, giving rise to rich and complex phenomena. However, the physics of these systems still remains to be explored in more details as can be seen from the scarce literature available. We present a detailed theoretical analysis of the effects of nanoscale inhomogeneities on the spin excitation spectrum in diluted magnetic systems. The calculations are performed on relatively large systems (up to $N$=$66^3$). It is found that even low concentrations of inhomogeneities have drastic effects on both the magnon density of states and magnon excitations. These effects become even more pronounced in the case of short ranged magnetic interactions between the impurities. In contrast to the increase of critical temperatures $T_C$, reported in previous studies, the spin-stiffness $D$ is systematically suppressed in the presence of nanoscale inhomogeneities. Moreover $D$ is found to strongly depend on the inhomogeneities' concentration, the cluster size, as well as the range of the magnetic interactions. The findings are discussed in the prospect of potential spintronics applications. We believe that this detailed numerical work could initiate future experimental studies to probe this rich physics with the most appropriate tool, Inelastic Neutron Scattering (INS).

Disordered magnetic systems, such as transition metal alloys 1-3 , diluted magnetic semiconductors (DMSs) [4][5][6] , diluted magnetic oxides (DMOs) 7,8 , and manganites [9][10][11] have continued to attract a great deal of attention ever since their respective discoveries. The prospects of these materials for novel spintronic devices led to a plethora of work from both the experimental as well as the theoretical point of view. For a long time the major focus of spintronics research was mostly on homogeneously diluted systems (with magnetic impurities distributed randomly on the host lattice), which were believed to give rise to the much coveted room-temperature ferromagnetism. For that purpose huge efforts to grow systems as "clean" as possible are made. However, in contrast to the common belief, this does not systematically appear to be the optimal route to room-temperature ferromagnetism. For instance, in homogeneously or well-annealed Mn doped III-V DMSs it has been impossible, so far, to go beyond the critical temperature of 140 K for 5% Mn [12][13][14] .
These findings led people to look into other possible avenues in the ultimate quest for the room-temperature phenomenon. Transmission electron microscopy (TEM) experiments revealed the presence of coherent Mn-rich spherical nanocrystals in (Ga,Mn)As 15,16 , which exhibited a Curie temperature of ∼360 K. Similar nanoscale clusters were also reportedly observed in other diluted materials like (Ge,Mn) [17][18][19] and (Zn,Co)O 20 , exhibiting critical temperatures in excess of 300 K. The nanoscale spinodal decomposition into regions with high and low concentration of magnetic ions was speculated to be the possible reason for the apparently high Curie temperatures in these compounds. Nanoscale inhomogeneities have also often been detected in other families of compounds. In manganites, inhomogeneities arise due to the interplay between the charge, spin, orbital, and lattice degrees of freedom, leading to the coexistence of metallic and insulating phases. For example, scanning tunneling spectroscopy (STS) images of La 1−x Ca x MnO 3 revealed a clear phase separation just below the critical temperature 21 . It is widely believed that the phase separation is at the origin of the well known colossal magneto-resistance (CMR) effect 22,23 . In high T C superconductors experimental studies often indicate the crucial role of inhomogeneities [24][25][26] . For example in Ref. 26, it was found that regions with weak superconductivity can persist to higher temperatures if bordered by regions of strong superconductivity. The possibility of increasing the maximum transition temperature by controlled distribution of the dopants has also been suggested.
Despite the existence of numerous experimental studies, the effects of nanoscale inhomo-geneities in disordered and diluted systems still remain largely unexplored on the theoretical front. Because of the large supercells required density functional theory (DFT) based calculations for inhomogeneous systems are considerably difficult. For the same reason the essentially exact Monte Carlo studies are unfortunately incapable to deal with these systems. Moreover the crucial importance of both thermal and transverse fluctuations calls for the inevitable need of an exact treatment of the disorder effects, which implies a real space treatment. Recently using a real-space RPA approach, able to handle considerably large system sizes, it has been demonstrated that nanoscale inhomogeneities can lead to a dramatic increase of the Curie temperatures 27 . The question which naturally arises is how impurity clustering affects the spin excitation spectrum? Indeed the spin-wave excitations provide valuable insight into the underlying spin dynamics of the system. Inelastic neutron scattering is a powerful experimental tool in this context, as the dispersion relation and the magnon damping can be measured directly and accurately. In contrast to non-dilute systems, such as manganites 28,29 , cobaltites 30,31 , multiferroics 32,33 , pnictides 34,35 , etc., one can find very few detailed studies 36 devoted to spin dynamics in dilute magnetic systems. Note that till now no experimental data has been reported for spin excitations in III-V DMSs, such as (Ga,Mn)As.
The aim of the current work is to provide such a detailed theoretical account of the spin excitation spectrum of diluted magnetic systems in the presence of nanoscale inhomogeneities. We show that nanoclusters of magnetic impurities can have drastic effects on the magnon density of states (DOS), the dynamical spectral function, as well as the spin-stiffness in these systems, when compared to the homogeneously diluted case. In our calculations, we have assumed for the lattice a simple cubic structure with periodic boundary conditions. In order to avoid additional parameters, the total concentration of impurities in the whole system is fixed to x=0.07. The inhomogeneities are assumed to be spherical of radii r 0 and the concentration inside each of these nanospheres is denoted by x in . For simplicity we also fix x in =0.8 for all cases considered in this study. We define the concentration of nanospheres in the system as x ns = N S /N, where N S is the total number of sites included in all the nanospheres and N = L 3 is the total number of sites. A variable P N = (x in /x)x ns , is used to represent the percentage of total impurities contained within the nanospheres. For a particular disorder configuration the nanospheres are distributed in a random manner on the lattice, the only restriction imposed is to avoid any overlapping between them. a lattice of N sites, which is given by the diluted random Heisenberg model where the sum ij runs over all sites and the random variable p i is 1 if the site is occupied by an impurity, otherwise it is 0. The above Hamiltonian (Eq.1) is treated within the self-consistent local random phase approximation (SC-LRPA), which is a semi-analytical approach based on finite temperature Green's functions. Within this approach, the retarded Green's func- where S z j are the local magnetizations which have to be calculated self-consistently. |Ψ R,c α and |Ψ L,c α are respectively the right and left eigenvectors, associated with the same eigenvalue ω c α of the effective Hamiltonian H c ef f , whose matrix elements are given by At finite T , H c ef f is non-Hermitian (real non-symmetric) but it is bi-orthogonal 37 . Hence one needs to define the left and right eigenvectors of the effective Hamiltonian. However, since in the present case all calculations are performed at T = 0 K, the matrix H c ef f is real symmetric and the left and right eigenvectors are identical. For more details on the SC-LRPA approach one can see for example Refs. 5,38,39. The SC-LRPA was previously successfully implemented to calculate the magnetic excitation spectra in the nearest-neighbor diluted Heisenberg model 40 as well as in the case of optimally annealed (Ga,Mn)As 41 . In the latter case, an especially good agreement with experiments 42,43 was also found. Now it is well known from first principles based studies as well as model calculations that the exchange couplings in III-V DMS compounds are relatively short-ranged and exponentially decaying in nature 41,44,45 . Thus, without any loss of generality, we have assumed the interactions between the magnetic impurities to be of the exponential form J ij =J 0 exp(- where λ is the damping parameter. For this damping parameter, we focus here on two particular values, λ = a and a/2 (a is the lattice spacing), corresponding to relatively long-ranged and short-ranged couplings respectively. In fact it was only recently shown that just varying λ within this scale can give rise to rather spectacular effects on the Curie temperatures in these inhomogeneous diluted systems 27 . Our current goal is to provide a detailed analysis of the effects of nanoscale inhomogeneities, in the presence of these interactions, on the dynamical magnetic properties of diluted systems.
We first start with the calculation of the magnon DOS for the homogeneously diluted case. The average magnon DOS is given by ρ avg (ω) = 1 In what follows ... c denotes the average over disorder configurations. Figure 1 shows ρ avg as a function of the energy ω, for the two different values of λ mentioned above. The average over disorder is performed for a few hundred configurations in all the following calculations. ρ avg is found to exhibit a regular Gaussian-like shape for the case of λ=a. The broad peak is located at 0.42W with a half-width of about 0.36W (W is the magnon excitation bandwidth). For longer ranged couplings, ρ avg remains essentially similar to that of λ = a. On the other hand, for shortranged couplings (λ = a/2), ρ avg has a more irregular and richer structure. The peak in ρ avg is now located at much lower energy, 0.06W , and a clear long tail extending toward the high energies with multiple shoulders appears. These additional features result from clusters of impurities weakly coupled to the rest of the system. These shoulders become even more pronounced for shorter ranged interactions. It is interesting to note that a similar kind of magnon DOS was obtained in the case of (Ga,Mn)As 41 . However, in Ref. 41 the exchange couplings used had been directly calculated from the "V -J" model. It has been found that λ = a/3 provides a very good fit for these couplings.
In order to analyze the effects of inhomogeneities, we calculate in addition to ρ avg , the local magnon DOS inside (ρ in ) and outside (ρ out ) the nanospheres. Unlike ρ avg , the local DOS contains information on the nature of the magnon states. We consider the particular case of nanospheres with fixed radii r 0 =2a, concentration inside x in =0.8, and four different concentrations of nanospheres x ns . The results for λ=a are depicted in Fig. 2. Let us first focus on ρ avg . From Fig. 2(a), we immediately notice that a relatively small concentration of inhomogeneities (x ns ∼0.02) causes a significant change in the magnon DOS. Indeed, in comparison to the homogeneous case, the excitations spectrum bandwidth is now doubled, and ρ avg has a bimodal structure, with a broader peak at higher energies. With increasing x ns , we observe a gradual transfer of weight from the low to high energy peak. The low energy peak shifts to smaller energies which is consistent with the decrease in the concentration of impurities outside the nanospheres. In order to have a better understanding of the features seen in ρ avg , we now analyze ρ in and ρ out . We observe that ρ in remains unchanged in all cases and exhibit a very small weight from 0 to 0.7W . Thus the high energy peak seen in ρ avg can clearly be attributed to the nanocluster modes. A careful analysis of a single isolated cluster reveals that the first non-zero eigenmodes are located at 0.7W , which explains the very small weight in ρ in below this value. Note that in Fig. 2(a), the shaded regions correspond to the discrete spectrum of an isolated single nanosphere, which is calculated over a few hundred configurations (random position of the impurities inside the nanosphere). The weak variation of ρ in with respect to x ns , indicates that the disappearance of the discreteness in ρ in (as seen in the isolated nanosphere spectrum) results mainly from the interactions between the cluster impurities and those outside. The above discussion of ρ avg and ρ in explains naturally the behavior of ρ out . In the case of more extended couplings, it is expected that (i) ρ avg loses progressively the bimodal nature, (ii) the pseudo-gap in ρ in at low energies is filled gradually, and (iii) the second peak in ρ in becomes narrower and shifts to higher energies with respect to the spectrum of a single isolated cluster.
We now discuss the effects of short ranged interactions (λ=a/2) on the average and local magnon DOS. The results are shown in Fig. 3. The magnon bandwidth increases by 250% with respect to that of the homogeneous system. As seen before, we observe a clear transfer of weight in ρ avg , from the low to higher energies with increasing x ns . In contrast to the bimodal nature observed for λ=a, ρ avg now exhibits a long wavy tail, extending toward higher energies. ρ in shows (i) a clear multiple peak structure now, (ii) is independent of x ns , and (iii) a well defined gap of approximately 0.5W is observed. The reasons for the appearance of these multiple peaks in ρ in are the enhanced discreteness (larger sub-gaps) of the eigenmodes of the single isolated nanosphere and the reduced interactions of the cluster impurities with those outside. Concerning ρ out , besides a shift to lower energies as seen for λ=a, we now observe that the peak becomes narrower with increase in x ns . (The latter feature was absent for the long ranged couplings). The reason for this is with increasing x ns , the concentration of impurities outside decreases and the effective interactions between them become weaker. This effect will be even more pronounced for shorter ranged couplings.
Even though the couplings are comparable, drastic changes between Fig. 2 and Fig. 3 We propose now to focus on the low energy excitation spectrum in these systems. For this purpose we evaluate the dynamical spectral function, which provides deeper insight into the underlying spin dynamics. This physical quantity can be directly and accurately probed by inelastic neutron scattering ( is the total average magnetization over all spin sites, and G c (q, ω) is the Fourier transform of G c ij (ω) given in Eq. 2. The averaged dynamical spectral function is evaluated from the followinḡ and where λ j = S z j S z is the temperature dependent local parameter. Note that at T = 0 K, all λ j 's = 1. We start with the average spectral function for the homogeneously diluted systems. It is interesting to note that these effects could hardly be anticipated from the magnon DOS results (Fig. 2). In fact the analyses of the DOS suggested that the low energy excitations should be weakly affected by the inhomogeneities. In the following, we discuss the spectral function in the presence of short-ranged interactions, shown in Fig. 6. As in the previous case, well-defined excitations exist only for small values of the momentum. However, here we find that the shift toward the lower energies is strongly enhanced. If we consider the particular case of q x a ≈ 0.12π, the magnon energies are shifted by 30% and 60% respectively, for x ns =0.02 and 0.04, with respect to that of the homogeneous case. The R(q)'s for this value of q x are 0.4, 0.8, and 1.3 for x ns =0, 0.02, and 0.04. This indicates that the excitations have dramatically lost their well defined character as compared to the previous case (Fig. 5). From theĀ(q, ω) calculations, we now extract both the magnon dispersion ω(q) and the spin-stiffness D inh . We remind that in the long wavelength limit, ω(q)=D inh q 2 . In order to gauge the effects of the inhomogeneities, we define the normalized spin-stiffness coefficient D n =D inh /D hom , where D inh denotes the spin-stiffness of the inhomogeneous systems, and D hom that of the homogeneously diluted system. The values of the spin-stiffness D hom are found to be 2.9J 0 and 0.07J 0 , for λ=a and a/2 respectively, for x=0.07. We would like to stress that both ω(q) and D inh were obtained from the first non-zero q excitations (q=( 2π La , 0, 0)) corresponding to system sizes N = L 3 , where L varies from 36 to 52. Figs. 7(a) and 7(b) show ω(q) as a function of q for various x ns , corresponding to λ=a, and a/2 respectively. First, in both cases, we observe that the inhomogeneities suppress the magnon dispersion. This suppression is even more drastic in the case of short-ranged interactions.
In addition, the quadratic behavior persists up to large q for λ=a/2, whilst a clear softening of the magnon modes can be seen in the case of the extended couplings, for q x /π ≥0.2. This is consistent with the discussions of Figs. 5 and 6. The well-defined magnon region shrinks significantly on going from λ=a to λ=a/2. This tendency is enhanced on increasing x ns .
In order to evaluate more quantitatively the effects of the inhomogeneities, the normalized spin-stiffness D n is plotted in Figs. 7(c) and 7(d), as a function of x ns , for various radii of the nanospheres. As anticipated from the previous figures, one observes for both λ=a and a/2, a monotonous decrease of the spin-stiffness with increasing x ns . D n decreases almost linearly up to x ns ≈0.04. The slope for λ=a/2 is twice that of λ=a. We find that the suppression of the spin-stiffness is greater for larger radii of the nanospheres. For instance, for the particular x ns =0.03, the spin-stiffness is reduced by 15% for r 0 = √ 2a, and by almost 30% for r 0 = √ 6a, for λ=a ( Fig. 7(c)), although the concentration outside the nanospheres is almost the same. In the case of the short-ranged couplings this effect is even stronger.
The effects of the nanoscale inhomogeneities on the spin-stiffness are in striking contrast to that observed on the critical temperatures, where an increase by more than one order of magnitude was found 27 .
To conclude, in this work we have shed light on the effects of nanoscale inhomogeneities on the magnetic excitation spectrum of dilute systems. To our knowledge, no such detailed numerical study has been performed so far. Compared to homogeneously diluted systems, even low concentrations of nanoscale inhomogeneities is sufficient to induce drastic changes in the magnon DOS. The magnon dispersion is found to vary significantly with the concentration of inhomogeneities, leading to a strong suppression of the spin-stiffness. This is in contrast to what has been reported about the critical temperatures. We also find a strong increase of the half-width of the magnon excitations in the long wavelength limit. These features are strongly enhanced in the case of short-ranged couplings. Now it would be interesting to study the effects of temperature on the spin dynamics of these inhomogeneous systems 46 . We expect the temperature dependence of the spin-stiffness, especially in systems with short-ranged interactions, to be unconventional compared to that of the homogeneous systems. Finally, we believe this detailed study would serve to motivate future experimental works on these inhomogeneous compounds.