Lifetimes of Magnons in Two-Dimensional Diluted Ferromagnetic Systems
LLifetimes of Magnons in Two-Dimensional Diluted Ferromagnetic Systems
Akash Chakraborty, Paul Wenk, and John Schliemann
Institut f¨ur Theoretische Physik, Univerist¨at Regensburg, 93040 Regensburg, Germany (Dated: September 23, 2018)Spin dynamics in low dimensional magnetic systems has been of fundamental importance for along time and has currently received an impetus owing to the emerging field of nanoelectronics.Knowledge of the spin wave lifetimes, in particular, can be favorable for future potential appli-cations. We investigate the low-temperature spin wave excitations in two-dimensional disorderedferromagnetic systems, with a particular focus on the long wavelength magnon lifetimes. A semi-analytical Green’s functions based approach is used to determine the dynamical spectral functions,for different magnetic impurity concentrations, from which the intrinsic linewidth is extracted. Weobtain an unambiguous q scaling of the magnon linewidth which is ascribed to the disorder induceddamping of the spin waves, thereby settling a longstanding unresolved issue on the wave-vector de-pendence. Our findings are also in good agreement with previous theoretical studies on Heisenbergferromagnets. Additionally, we demonstrate the futility of using the low moments associated withthe spectral densities to evaluate the magnon dispersions and lifetimes. PACS numbers: 75.30.Ds, 73.21.-b, 75.50.Pp
I. INTRODUCTION
Two-dimensional magnetic systems have been a sub-ject of intensive investigation for almost half a centurynow. Both ferro- and anti-ferromagnetic systems havebeen studied extensively, experimentally as well as the-oretically, revealing a myriad of interesting propertiesincluding the discovery of high-temperature supercon-ductivity in doped two-dimensional cuprates . One ofthe important aspects of these systems which has con-tinued to attract interest is the magnetic excitationswhich are of fundamental relevance to understand thespin dynamics. Knowledge of the collective spin waveexcitations can provide valuable insight into their dy-namical response as well as their thermodynamic be-havior. The advancement of experimental techniques,such as ferromagnetic resonance spectroscopy and in-elastic neutron scattering, have been of immense helpin exploring this field meticulously . Inelastic neu-tron scattering is one of the most powerful and versa-tile tools as the long wavelength spin excitations, bet-ter known as magnons, can be probed directly andaccurately. One of the most thoroughly studied sys-tems, in this context, is the two-dimensional Heisenberganti-ferromagnet Rb Mn − x Mg x F , which was inves-tigated by means of neutron scattering techniques tostudy the low-temperature magnetic excitations, viz. themagnon dispersion, linewidths, and lineshapes, as well asthe critical exponents near the transition temperature.This led to similar studies on K Cu − x Zn x F , whichis a quasi bi-dimensional ferromagnet. In the aforemen-tioned studies, good agreement with numerical calcula-tions, available at that time, was also reported. However,despite the existence of innumerable studies, one impor-tant feature which has eluded understanding, over thedecades, is the wave-vector dependence of the magnonlifetime (inversely proportional to the linewidth), espe-cially in the long wavelength limit ( q → . The author calculated the spin waveenergies and the scattering cross section, within the Bornapproximation, and reported a q scaling of the magnonlifetime. The exchange interactions, in this case, were re-stricted to nearest neighbors only. Similar q dependencewas also found in amorphous Heisenberg ferromagnets,in the low-temperature and long wavelength limit, byusing an effective medium approximation . In this case,however, spatially dependent extended couplings were as-sumed between the magnetic sites. Based on Green’sfunctions calculations, Mano also predicted an identi-cal behavior of the lifetime in the long wavelength limit.The finite linewidth of the excitations, which increasedrapidly with decreasing wavelength, was attributed tothe randomness in the magnitude of the spins. Also thediscrepancy between the observed magnetization behav-ior and that predicted by elementary spin wave theorywas believed to originate from this finite linewidth of thespin waves. On the contrary, similar spin wave studiesin amorphous ferromagnets by Kaneyoshi led to a q dependence of the linewidth. This was an outcome of us-ing a quasi-crystalline approximation, which is essentiallya virtual-crystal-like approach. Within this approxima-tion, the magnon dispersion reduces to that of an idealcrystal, wherein the disorder effects are completely ne-glected. In yet another study, based on the two-magnoninteraction theory of Heisenberg ferromagnets, a leadingorder q scaling of the magnon lifetime was suggestedby Ishikawa et al. . However, in most of the afore-said studies, the systems under consideration were three-dimensional Heisenberg ferromagnets and there was noclear mention of the dimensional dependence. It was onlylater that Christou and Stinchcombe investigated thelow-temperature spin excitations in bond-diluted Heisen-berg ferromagnets from a more generalized perspective.Using a diagrammatic perturbation theory, the authors a r X i v : . [ c ond - m a t . d i s - nn ] M a r obtained a q d +2 ( d >
1, is the dimensionality) scalingof the magnon linewidth. Although the discussion wasextended to the more relevant site-diluted systems, theexchange interactions were again restricted to nearestneighbors only.Thus, the lack of a general accord on the issue oflinewidth scaling becomes apparent from the widely vary-ing predictions available in the literature. Moreover, con-siderable attention and interest have also been devotedto the case of anti-ferromagnets, including even lately .In a very recent study , on three-dimensional disorderedferromagnets, a q scaling of the magnon linewidth, in thelong wavelength limit, was reported using similar numer-ical approaches as implemented here. This served as afurther motivation behind the current study of the mag-netic excitations in two-dimensional ferromagnetic sys-tems, with a view to identify the dimensional dependenceof the scaling of the magnon lifetimes. Also a properknowledge of the lifetimes is not only of fundamentalinterest but can also be of practical importance, as weshall discuss later. In this article, we provide a com-prehensive and detailed analysis of the low-temperaturespin wave excitations in two-dimensional site-diluted fer-romagnets, in the presence of extended exchange inter-actions. The calculations have been performed on suffi-ciently large system sizes and a proper statistical sam-pling over disorder is also taken into account. We layspecial emphasis on the correct evaluation of the magnonlinewidths in the long wavelength limit. In the process,we demonstrate that determining the correct wave-vectordependence of the lifetimes constitutes a non-trivial task.In addition, we also discuss the nature of the magnondensity of states, the spectral functions, as well as themagnon dispersion over a relatively broad concentrationrange. II. HEISENBERG MODEL AND EXCHANGECOUPLINGS
We start with the Hamiltonian describing N imp inter-acting spins ( S i ) randomly distributed on a square latticeof N sites, given by the dilute Heisenberg model H = − (cid:88) i,j J ij p i p j S i · S j (1)where the sum i, j runs over all sites and the randomvariable p i =1 if the site is occupied by an impurity orotherwise zero. We consider classical spins ( | S i | = S )on a square lattice, with lattice spacing a , and withperiodic boundary conditions. The distribution of thespins, in this case, is completely random and uncor-related; in other words the probability of a spin tobe placed at site i is independent of the neighboringsites. This is in contrast to a previous study on themagnetic excitations in inhomogeneous diluted systems,where well-defined spherical clusters of spins were con-sidered. Spin-orbit coupling is neglected as this would lead to anisotropy in the system which is not the pri-mary focus here. The effects of spin-orbit on the magnonlifetimes in two-dimensional systems were studied in .All calculations, in the present work, are performed at T =0 K. The concentration of magnetic impurities in thesystem is denoted by x (= N imp /N ). The Hamiltonian,Eq. (1), is treated within the self-consistent local randomphase approximation (SC-LRPA), which is essentially asemi-analytical approach based on (finite temperature)Green’s functions. Within this approach, the retardedGreen’s functions are defined as G cij ( ω ) = (cid:90) ∞−∞ G cij ( t ) e iωt dt (2)where G cij ( t )= − iθ ( t ) (cid:104) [ S + i ( t ) , S − j (0)] (cid:105) , describe the trans-verse spin fluctuations, and (cid:104) . . . (cid:105) denotes the expectationvalue, and ‘ c ’ the disorder configuration index. After per-forming the Tyablikov decoupling (assuming mag-netization along the z -axis) of the higher-order Green’sfunctions which appear in the equation of motion of G cij ( ω ), we obtain( ω I − H ceff ) G c = D (3)where H ceff , G c , and D are N imp × N imp matrices. Theeffective Hamiltonian matrix elements are( H ceff ) ij = −(cid:104) S zi (cid:105) J ij + δ ij (cid:88) l (cid:104) S zl (cid:105) J lj (4)and the diagonal matrix D ij = 2 (cid:104) S zi (cid:105) δ ij . (5)For a given temperature and disorder configuration, thelocal magnetizations (cid:104) S zi (cid:105) ( i = 1 , , . . . , N imp ) have to becalculated self-consistently. However, since we are inter-ested at T =0 K, where the ground state is assumed tobe fully polarized, all (cid:104) S zi (cid:105) are equal to S in this case.We shall not go into further details of the method here,as the accuracy and reliability of the SC-LRPA to han-dle disorder (dilution) effects in different contexts havebeen discussed and established on numerous previous oc-casions (for details see Refs. 21–23). The virtual crystalapproximation, as a possible alternative approach, failsin these systems as will be discussed in Sec. V.The exchange interactions are assumed to be of theform J ij = J r − αij , where r ij = | r i − r j | . In most of theprevious studies the exchange couplings were restrictedto nearest neighbors only, but in realistic systems theseinteractions extend well beyond the nearest neighbors.Moreover, the choice of the couplings is motivated by thetheoretical proposition put forward in Ref. 24, whereinthe author extends the Mermin-Wagner theorem toHeisenberg and XY systems with long-range interac-tions. It is stated that a d -dimensional system ( d =1 or2) with monotonically decaying interactions as | J r |∝ r − α cannot have ferro- or anti-ferromagnetic long-range or-der at T >
0, if α ≥ d . For RKKY-like interactions(long-range oscillatory nature) magnetic order could bestrictly ruled out for the one-dimensional systems, butonly for certain cases in the two-dimensional ones. It waslater proved by Loss, Pedrocchi, and Leggett , againas an extension of the Mermin-Wagner theorem, thatno long-range magnetic order is possible in one or two-dimensional systems at a finite temperature, in the pres-ence of RKKY interactions. The choice of exponentiallydecaying couplings can also be ruled out in this caseas they satisfy the Mermin-Wagner theorem trivially.Hence, this led us to the choice of the exponent α =3 forthe couplings, which implies that in our two-dimensionalsystems long-range (ferro-)magnetic order at a finite tem-perature is not excluded by the Mermin-Wagner theorem.Also, since the couplings are isotropic and all ferromag-netic ( J ij >
0) there is no frustration expected and hencethe collinear state can be safely assumed to be the groundstate. This in turn leads to only positive eigenvalues inthe magnon spectrum which will become clear in the fol-lowing calculations of the magnon DOS.
III. MAGNON DENSITY OF STATES ANDSPECTRAL FUNCTION ω /J S ρ a v g x= 0.02x= 0.04x= 0.06x= 0.08x= 0.1 FIG. 1. (Color online) Average magnon DOS ρ avg as a func-tion of energy ω plotted for different concentrations x . From the retarded Green’s functions defined aboveone can calculate the average magnon density of states(DOS), which is given by ρ avg ( ω ) = (1 /N imp ) (cid:80) i ρ i ( ω ),where ρ i ( ω ) = − / (2 πS ) (cid:61) [ G ii ( ω )] is the local magnonDOS. Fig. 1 shows the average magnon DOS as a func-tion of the energy for different impurity concentrations.The DOS have been averaged over a hundred disorderconfigurations, although it was found that typically 25configurations were sufficient for each impurity concen-tration. We observe irregular features in the DOS whichbecome more pronounced with increase in dilution. Ondecreasing the concentration from x = 0 . x = 0 .
02, a significant increase in weight around the low energy endof the spectrum is observed. This increase in weight isattributed to the increase in the fraction of impuritieswhich are weakly connected to the rest. These isolatedimpurity regions have their own zero-energy modes whichin turn contribute to the DOS at the low energies. In or-der to gain a better insight into this behavior we look atthe distribution of the local DOS shown in Fig. 2(a) and2(b), at two different energies 2.2 J S and 3.2 J S ,respectively for x = 0 .
1. Here we can clearly identify cer-tain impurity regions, of typically two or three impurities,which are weakly coupled to the surrounding impurities.These can be seen to make a higher contribution to theDOS. (For more details see Fig. 7, App.). Note that thedistribution shown corresponds only to a part of the lat-tice from a 200 a × a system. With increasing dilutionthe average separation between the spins increases andhence the effective coupling decreases. This accounts forthe increase in the irregular features observed in the DOSfor x = 0 . (a) (b) (cid:45) (cid:45) (cid:64) Ρ loc (cid:68)(cid:144)(cid:64) arb . (cid:68) FIG. 2. (Color online) Distribution of the local magnon DOSat energies (a) ω/ ( J S ) = 2 .
2, and (b) ω/ ( J S ) = 3 .
2, foran impurity concentration of x = 0 .
1. Shown is a part of thelattice of size L =200 a in coordinate space. The dots indicatethe positions of the spins S i . The dynamical spectral function, also known as thestructure factor, provides valuable insight into the un-derlying spin dynamics of a system. Experimentally thiscan be probed by inelastic neutron scattering and ferro-magnetic resonance to a good accuracy. The averagedspectral function is defined by A ( q , ω ) := − (cid:28) πS (cid:61) [ G c ( q , ω )] (cid:29) c , (6)where G c ( q , ω ) is the Fourier transform of the retardedGreen’s function G cij ( t ), and (cid:104) . . . (cid:105) c denotes the configu-ration average. Fig. 3 shows the averaged spectral func-tions as a function of energy for four different concen-trations. The A ( q , ω )’s are averaged over a few hundred A ( q , ω ) ω /J S A ( q , ω ) ω /J S x=0.02 x=0.04x=0.06 x=0.08 q qq q FIG. 3. (Color online) Average spectral function A ( q , ω ) as a function of the energy with q = n (2 π/La ) { , } (cid:62) (where n ∈ N ),corresponding to four different concentrations x . The system size is L = 300 a × a . disorder configurations, and the results are plotted onlyin the [1 0] direction of the Brillouin zone, for progres-sively increasing momentum q , since the focus is on thelong wavelength regime here. It should be noted thatthe [0 1] direction is equivalent to the [1 0] direction inthis case, due to the lattice symmetry. Also note that for q (cid:29) π/ ( La ) the deviation from rotation invariance isnot negligible. Well-defined excitations are found to ex-ist only for small values of q , in each case. For increasing q , the excitation peaks become broader and develop atail extending toward the higher energies. On decreasingthe concentration from 0.08 to 0.02 the zone of stabilityof the well-defined magnon modes is found to decreaseby almost one order of magnitude. Also the excitationsbecome increasingly asymmetric with increase in the mo-mentum. This increase in asymmetry is associated withthe crossover from propagating low-energy spin wavesto localized or quasi-localized excitations (fractons) athigher energies. Here, the term localized implies thatthe excitations are quite broad in energy at fixed wave-vectors, or rather quantitatively the excitation energy ismuch larger than the linewidth (i.e. the full-width at half-maximum).The nature of the spectral functions is similar towhat was observed by neutron scattering experimentsin Mn x Zn − x F , which is a three-dimensional ran-domly diluted anti-ferromagnet. The authors measuredsharp spin waves near the zone center which broadened progressively with the wave vector approaching the zone-boundary. These findings were attributed to a crossoverfrom low-energy extended spin waves (magnons) to local-ized high-energy excitations (fractons), which was fur-ther consistent with the theoretical conjecture of frac-tons in disordered percolating networks . A recentnumerical study on site-diluted two-dimensional anti-ferromagnets also reveal the existence of localized excita-tions at high energies. The authors evaluated the inverseparticipation ratio (IPR), for different dilutions and dif-ferent system sizes in order to establish the nature (ex-tended or localized) of the states, although the largestsystem size studied was only 32 a × a . These studiesprovide relevance and also additional motivation to studythe two-dimensional ferromagnets from this aspect. Theproper and accurate evaluation of the spectral functions,as we shall see in what follows, constitutes a vital tasksince the magnon dispersion as well as the lifetime canbe directly extracted from them. IV. MOMENTS ANALYSIS
Before embarking into further details of the long wave-length magnon properties, we define the moments asso-ciated with the spectral density. The n -th moment is x ( D , D i n un it s o f J S ) q ω m DD x=0.03 FIG. 4. (Color online) Spin stiffness D and effective spinstiffness D as a function of x . ( m ( q ) ≈ D q , where m is the first moment associated to the spectral density). Theinset shows a comparison of the excitation energies extractedfrom A ( q , ω ) and the first moments m , respectively, for thecase of x = 0 . defined by m n ( q ) = (cid:90) ∞ ω n A ( q , ω ) dω (7)In the limit q →
0, it can be shown that m ( q ) ≈ D q ,where we call D as the effective spin wave stiffness.It is also well known that in the long wavelength limitthe dispersion in ferromagnetic systems is quadratic in q , ω ( q ) ≈ Dq , where D denotes the spin stiffnesscoefficient. The moments, as sometimes found in theliterature , are used in the spectral function analy-ses as a good approximation to estimate the excitationenergy and linewidth, especially in the presence of disor-der. Nonetheless, the accuracy and the viability of thisassumption is subject to further examination. In orderto address this, as a first step, we numerically calculatedthe dispersion from the first moment and then comparedit to the real excitation energy ω ( q ) extracted from the A ( q , ω ) peaks shown in Fig. 3. The results for the partic-ular case of x = 0 .
03 are plotted in the inset of Fig. 4. Ascan be seen, in the small q limit, both m ( q ) and ω ( q ) arelinear in q but the first moment fairly overestimates thereal magnon energies. This is better reflected when weextract the respective spin stiffness coefficients, D from m ( q ) and D from ω ( q ), and plot them against the con-centration as shown in Fig. 4. For all considered x , theeffective spin stiffness is larger than the actual spin stiff-ness, overestimating by 15-20% in each case. This clearlydemonstrates that the first moment is not a reliable quan-tity to evaluate the spin stiffness in these diluted systemsas it fails to reproduce the magnon energies precisely.The other relevant quantity of interest is the intrin-sic linewidth of the magnetic excitations. The linewidth qa γ ( q ) x=0.01x=0.03x=0.05x=0.07 γ (q)=C q FIG. 5. (Color online) Effective linewidth γ (in units of J S )as a function of q (in the [1 0] direction), for different con-centrations x , Eq. (8). The dashed lines indicate the linearfits. gives a measure of the excitations’ broadening due to dis-order, which maybe magnetic or structural disorder, ordue to the magnon-magnon interactions. One can ob-tain the linewidth from the second moment from therelation γ ( q ) = (cid:113) m ( q ) − m ( q ) (8)where γ ( q ) is the effective linewidth. In Fig. 5 we haveplotted this effective linewidth as a function of q for fourdifferent impurity concentrations. We find that in thesmall- q limit the linewidth is linear in q for all consid-ered x . The same holds true for all other intermediateconcentrations, which are not shown here. Consequently,we end up with ω ∝ q and γ ∝ q , in the limit q → γ > ω .However, this is somehow contrary to what we have ob-served in the spectral functions shown in Fig. 3, wherethe excitations are well defined for small q values. Hence,we can safely assume that the effective linewidth ob-tained from the moments does not correspond to the reallinewidth of the excitations. The same discrepancy wasalso demonstrated for the case of Ga − x Mn x As , a well-known III-V diluted ferromagnetic semiconductor, wherethe lattice has an fcc structure. Note that similar linear q -dependence, obtained from the moments analysis, wasreported by the authors in disordered double-exchangesystems . Determining the correct q -dependence of theintrinsic linewidth, in the long wavelength limit, requiresfurther detailed analysis which is elucidated in the fol-lowing. V. SCALING OF MAGNON LIFETIME
We extract the linewidth, which is the full-width athalf-maximum, directly from the magnon spectral func-tions (Fig. 3) corresponding to the first non-zero q valuesfrom different system sizes. The extracted linewidths areplotted as a function of the wave-vector in Figs. 6(a) and6(b), for x = 0 .
03 and 0.05, respectively. In order tohave sufficiently small q values, and also check for theprobable finite-size effects we have performed the calcu-lations on system sizes ranging from 200 a × a up to500 a × a . The linewidth data are averaged over onehundred disorder configurations and the error bars corre-sponding to the standard deviation are contained withinthe symbols. Now, since we are interested in the q → q values,(highlighted by the shaded regions in the plots), in or-der to give more weight to the smallest available q ’s. Weremark that the limit considered for the shaded regionsonly serve as an approximate value and not as a clear de-marcation of the q regime, defining the long wavelengthlimit. Note that the value of ln( qa ) ≈ − qa ≈ .
02. To determine the q -dependence weuse a linear fit of the form n ln( qa ) + C , (with n = 3,4, and 5) for the data within these shaded regions. Ascan be clearly seen for both cases, x = 0 .
03 and 0.05, itis the n = 4 fit (denoted by the solid line) which bestdescribes the linewidth behavior in this region. Beyondthis region, the linewidth begins to deviate from this be-havior although the deviations are less for x = 0 .
05 com-pared to x = 0 .
03. Also note that the same q -scalingwas observed for the other concentrations as well. Thisclearly shows that in the long wavelength limit and atlow temperatures the intrinsic linewidth actually scalesas q in these two-dimensional systems. Our findings areinterestingly in good agreement with the prediction of a q d +2 - dependence reported in Ref. 13. This agreement isnot obvious since in the latter work a different analyticalapproach, based on diagrammatic perturbation theory,was used and also the couplings were restricted to near-est neighbors only. Whereas our study is more general inthe sense that the couplings are extended, as well as thelinewidth is extracted directly from the magnon spec-tral functions. In this context, it is worth mentioningthat studies based on virtual-crystal-like approaches of-ten lead to an infinite lifetime, since the spin fluctuationsare unaccounted and the disorder effects are neglected,implying no mechanism for magnon decay. However, dis-order plays an essential role, as shown here, in leadingto a finite lifetime in these systems. From this q scalingwe infer that, in the long wavelength limit, the linewidthis actually smaller than the excitation energy which wasqualitatively clear from the well-defined peaks observedaround the Γ-point in the spectral functions. Neverthe-less, as we have seen, it is difficult to identify preciselythe values of q below which this behavior holds and thesevalues, in turn, should also depend on the concentration x . Similar difficulties were also demonstrated in the case of three-dimensional systems where a q behavior, in-stead of q , was observed if the considered q values werenot sufficiently small. Apparently, in the present casewe do not observe any clear crossover from the q scal-ing to any other form within the considered range of thewave-vectors. We also conclude that the scaling of thelinewidths does in fact depend on the dimensionality. -4.4 -4.3 -4.2 -4.1 -4 -3.9 -3.8 -3.7 -3.6 -3.5 ln (qa) -13-12-11-10 l n γ ( q ) n= 3n= 4n= 5 x=0.03 (a) -4.3 -4.2 -4.1 -4 -3.9 -3.8 -3.7 -3.6 ln (qa) -12-11-10 l n γ ( q ) n= 3n= 4n= 5 x=0.05 (b) FIG. 6. (Color online) Logarithm of the magnon linewidth γ (in units of J S ) as a function of ln ( qa ), for (a) x = 0 . x = 0 .
05. Dashed (red), solid (green), and dot-dashed(blue) lines indicate linear fits of the form n ln( qa )+ C , ( n = 3,4, and 5), for the linewidth data within the shaded region. As already mentioned, the energy and the linewidthcalculated from the moments do not coincide with thereal ones extracted from the spectral function. The rea-son behind is that moments can only reproduce the char-acteristic features of a distribution when they are per-fectly symmetric, such as a Gaussian or a Lorentzian. Inthe present case, the spectral functions are actually asym-metric and hence the moments prove to be inappropriateto estimate the real line shape and the peak positions. Inthe clean case (absence of any disorder) one is likely to getreliable results from the moments analysis of the spectralfunctions as the excitations can be completely symmet-ric. However, disorder leads to a strong asymmetry of theexcitations, as we have seen in the present case. Thereis a considerable broadening in the spectrum observedespecially close to the zone boundary. Thermal fluctua-tions also play an important role in these systems, butsince we focus only on the low-temperature excitationswe can neglect the thermal effects here. Further exper-imental studies to quantitatively examine the linewidthin these compounds could prove to be very useful.
VI. CONCLUSION
We have addressed the low temperature spin exci-tations in two-dimensional diluted Heisenberg systems,with a particular focus on the long wavelength limit. Aself-consistent Green’s functions based approach is usedto evaluate the magnon DOS and the dynamical spectralfunctions. Well-defined excitations are observed only ina restricted region of the Brillouin zone, around the Γ-point. It is demonstrated that determining the correctwave-vector dependence of the magnon linewidth in di-luted systems is not an ordinary task. Contrary to someprevious studies, we have shown that the moments as-sociated with the spectral function are inappropriate todetermine the linewidth or the excitation energies. Themoments overestimate the real spin stiffness as well asprovide a linear q -dependence of the linewidth, imply-ing incoherent excitations in the limit q →
0. How-ever, this is found to be inconsistent with the stiffnessand the linewidth extracted from the calculated spectralfunctions. In the long wavelength limit, the linewidth infact scales as q in two-dimensional systems, for a widerange of impurity concentrations. The discrepancy arisesdue to the inability of the moments to reproduce theasymmetry in the excitation peaks. The origin of thisasymmetry, and thus a finite lifetime, is ascribed to thedisorder induced broadening of the spin waves. Hence,this underlines the importance of the disorder effects inthese systems and we emphasize that the failure to prop-erly account for them will certainly result in an incorrectwave vector dependence of the linewidth.Most data storage devices, in nowadays spintronics,try to manipulate the dynamical motion of spins. Fromthis perspective, a precise knowledge of the excitations’lifetime (inversely proportional to the linewidth) could beof practical relevance. For example, a short lifetime is im-portant for memory devices to leave a bit in a steady stateafter a read-in or read-out operation. On the contrary, alonger lifetime is advantageous for the unhampered trans-mission of signals in inter-chip communications. It wouldbe equally interesting to look into the temperature effectson the spin dynamics, in particular the linewidth, wherein addition to disorder the thermal effects also play a vi-tal role. However, this is beyond the scope of the current work. The present findings provide qualitative insightsinto the low temperature excitations and the magnon life-times in two-dimensional ferromagnets, and could serveas a firm basis for future research on complex disorderedmagnets. More experimental studies oriented in this di-rection are also highly desirable to resolve the controversyarising from the numerous theoretical proposals. ACKNOWLEDGMENTS
We acknowledge financial support by DFG within thecollaborative research center SFB 689. AC would liketo thank Georges Bouzerar for insightful comments anddiscussions.
Appendix A: Impurity Configuration Energies (cid:72) a (cid:76) (cid:72) b (cid:76) (cid:72) c (cid:76)(cid:72) d (cid:76) (cid:72) e (cid:76) (cid:72) f (cid:76)(cid:72) g (cid:76) FIG. 7. Basic impurity configurations on a square latticewhich give a high contribution to the magnon DOS.
The peaks in the averaged magnon DOS in the dilutedcase, Fig. 1, can be related to different configurations ofsome few impurities in coordinate space on the squarelattice. This can be motivated by considering the distri-bution of the local magnon DOS in Fig. 2(a),2(b) whichreveals clusters of less than four impurities up to an en-ergy of ω/ ( J S ) ≈
3. Thus, the relevant small clusterswhich give rise to the magnon DOS peaks on a squarelattice are identified to be those plotted in Fig. 7. Theenergies corresponding to the configurations (a)-(g) aregiven by E (a);1 = 2 (A1) E (b);1 = 2 − α/ E (a);1 (A2) E (c);1 = 3 (A3) E (c);2 = 1 + 2 − ( α/ (A4) E (d);1 = 3 (A5) E (d);2 = 1 + 2 − α (A6) E (e);1 / = 1 + 2 − α/ + 5 − α/ ± − α/ [2 α − α/ (1 + 2 α/ ) + 5 α (1 − α/ + 2 α )] (A7) E (f);1 = 2 − α/ E (c);1 (A8) E (f);2 = 2 − α/ E (c);2 (A9) E (g);1 = 2 − α/ E (d);1 (A10) E (g);2 = 2 − α/ E (d);2 (A11)where the energy E (.); p is given in values of J S /a α , with a being the lattice constant and the index p the eigen-value number (the ground-state energy zero has beenexcluded). The energies E ( . ); p are indicated in Fig. 8 Ω (cid:144)(cid:72) J S (cid:76) Ρ a vg (a) Ω (cid:144)(cid:72) J S (cid:76) Ρ a vg (b) FIG. 8. (Color online) Averaged magnon DOS ρ avg of a smallsystem L = 10 a with only three impurities.(a) For specific maxima we show the corresponding impurityconfiguration in coordinate space on the square lattice. Theenergy corresponding to a configuration inset is on its LHSindicated by a solid line, according to Eq. A1- A11 with α = 3.(b) Zoom of plot (a) at small ω values. The solid lines indicatethe energies after rescaling the lattice constant by two, thedashed lines a rescaling by √ by vertical lines. Fig. 8 shows the magnon DOS of a10 a × a system with three impurities averaged over allpossible configurations. A comparison of this results withthe x = 0 .
02 case in Fig. 1 (system size of L = 1340 a ) reveals indeed that the relevant energies E (cid:38) . J S are given by the Eqs. (A1)-(A11), with α = 3. Manyof the other configurations can be generated by a simpleisotropic rescaling of the configurations shown in Fig. 7.The corresponding energies are indicated by dashed andsolid lines in Fig. 8(b). To answer the question how spe- Α Ω (cid:144)(cid:72) J S (cid:76) (cid:64)(cid:72) a (cid:76) ;1 (cid:68) (cid:64)(cid:72) b (cid:76) ;1 (cid:68) (cid:64)(cid:72) c (cid:76) ;1 (cid:68) , (cid:64)(cid:72) d (cid:76) ;1 (cid:68) (cid:64)(cid:72) d (cid:76) ;2 (cid:68)(cid:64)(cid:72) c (cid:76) ;2 (cid:68) (cid:64)(cid:72) e (cid:76) ;2 (cid:68) (cid:64)(cid:72) e (cid:76) ;1 (cid:68) (cid:64)(cid:72) f (cid:76) ;1 (cid:68)(cid:64)(cid:72) f (cid:76) ;2 (cid:68) (cid:64)(cid:72) g (cid:76) ;1 (cid:68) (cid:64)(cid:72) g (cid:76) ;2 (cid:68) FIG. 9. (Color online) Magnon energies given by Eq. A1-A11 for the corresponding impurity configurations plotted inFig. 8 as function of the exponent α with | J r |∝ r − α . cific the choice of the exponent α = 3 is, we plottedthe energies Eq. A1- A11 of the magnon DOS peaks fordifferent values of α in Fig. 9. We recall that a two-dimensional system with monotonically decaying interac-tions cannot have ferro- or anti-ferromagnetic long-rangeorder at T >
0, if α ≥
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