Nonlinear Decay of Quantum Confined Magnons in Itinerant Ferromagnets
Kh. Zakeri, A. Hjelt, I. V. Maznichenko, P. Buczek, A. Ernst
NNonlinear Decay of Quantum Confined Magnons in Itinerant Ferromagnets
Kh. Zakeri, ∗ A. Hjelt, I. V. Maznichenko, P. Buczek, and A. Ernst
3, 4 Heisenberg Spin-dynamics Group, Physikalisches Institut,Karlsruhe Institute of Technology, Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe, Germany Department of Engineering and Computer Sciences,Hamburg University of Applied Sciences, Berliner Tor 7, D-20099 Hamburg, Germany Institute for Theoretical Physics, Johannes Kepler University, Altenberger Str. 69, A-4040 Linz, Austria Max-Planck-Institut f¨ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
Quantum confinement leads to the emergence of several magnon modes in ultrathin layered mag-netic structures. We probe the lifetime of these quantum confined modes in a model system com-posed of three atomic layers of Co grown on different surfaces. We demonstrate that the quantumconfined magnons exhibit nonlinear decay rates, which strongly depend on the mode number, insharp contrast to what is assumed in the classical dynamics. Combining the experimental resultswith those of linear-response density functional calculations we provide a quantitative explanationfor this nonlinear damping effect. The results provide new insights into the decay mechanism of spinexcitations in ultrathin films and multilayers and pave the way for tuning the dynamical propertiesof such structures.
Understanding the processes behind the excitation andrelaxation of spin excitations in low-dimensional mag-netic structures is one of the most intriguing researchdirections in solid-state physics. A detailed knowledgeon the fundamental mechanisms involved is such pro-cesses is the key to understand many different phenom-ena. Examples are ultrafast magnetization reversal byultrashort magnetic field pulses [1] or by torque transferfrom spin-polarized currents [2–4], vortex core gyrationdriven magnon emission [5], subpicosecond demagnetiza-tion by ultrafast photon pulses [6–9], and strong spin-dependence of image potential states at ferromagneticsurfaces [10]. In addition to its fundamental impact, acomplete understanding of magnetic relaxation mecha-nisms is of great importance for designing efficient spin-based devices as the power consumption of such devicesis determined by the magnetic damping [11–13].The excited state of a magnetic system is describedby magnons, the quanta of spin waves. The relaxation ofsuch an excited state can, in principle, involve the dissipa-tion of magnetic energy in several different ways. In themost common approach, based on the classical dynam-ics, damping is described by a phenomenological damp-ing parameter, commonly referred to as Gilbert damp-ing [14–16]. This description is only valid in the case ofuniform ferromagnetic resonance, i.e. the magnons withzero wavevector, under some circumstances [17–21]. Formagnons with a nonzero wavevector it has been assumedthat the relaxation of a magnon with a certain wavevec-tor can involve its dissipation to other magnons with dif-ferent wavevectors (multi-magnon scattering process) or,in the case of itinerant magnets, their dissipation intothe single-particle electron-hole pair excitations, knownas Stoner excitations. In both cases one would expectan increase of the magnon decay rate with the magnonenergy, since usually the density of both magnon as wellas Stoner states increases with energy. In structurally well-defined low-dimensional magneticstructures one would expect additional magnon modes asa result of quantum confinement. For example in ultra-thin magnetic films or multilayers composed of a finitenumber of atomic layers N one can show that there exist n = 0 , , ...., N − n . This observation is incontrast to what is commonly discussed in the frame-work of the classical dynamics. Combining the results oflinear-response time-dependent density functional theorycalculations with adiabatic spin dynamics calculations weprovide a quantitative explanation for the damping ofquantum confined magnons.Experiments are performed by means of spin-polarized high-resolution electron energy-loss spec-troscopy (SPEELS) on ultrathin Co films with a thick-ness of three atomic layers epitaxially grown of Ir(001),Ir(111) and Cu(001). Figure 1(a) shows typical SPEELSspectra recorded on a 3 monolayer (ML) Co film onIr(001). The spectra are recorded for the two possible in-coming spin states. I ↓ ( I ↑ ) represents the intensity spec-trum when the spin polarization of the incoming beam isparallel (antiparallel) to the sample magnetization. Thedifference spectrum, shown in the lower panel, containsall the possible spin-flip excitations of down to up, in-cluding magnons [26, 35]. The data are recorded at ( q x , q y )=(0.3 ˚A − , 0.3 ˚A − ), corresponding to q (cid:107) = 0 .
42 ˚A − a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b ) V e ( y g r en E no i t a t i cx E nonga M M Γ X Γ (c) ) V e ( y g r en E no i t a t i cx E nonga M Im. Susceptibility (arb. units)n=0n=1n=2 (d)
Theory
I I x ( y t i s ne t n I ) . c e s / s t nuo C Electron Energy Loss (eV)
I - I (a) ) V e ( ss oL y g r en E no r t c e l E Norm. Intensity(arb. units)Exp.n=0 n=1 n=2 (b) n=2 n=1 n=0 FIG. 1. (a) SPEELS spectra recorded on 3 ML Co/Ir(001) at ( q x , q y )=(0.3 ˚A − , 0.3˚A − ). I ↓ ( I ↑ ) represents the spectrum whenthe spin polarization of the incoming electron beam is parallel (antiparallel) to the magnetization. The difference spectrum I ↓ − I ↑ is shown in the lower panel. (b) The difference spectrum and the fits used to extract the dispersion relation and thelifetime of different quantum confined magnons indicated by n = 0, 1 and 2. (c) The dispersion relation of all confined magnonmodes. The experimental data are shown by symbols and the calculated magnon Bloch spectral function using our adiabaticapproach is presented as the color map. The dotted line shows the place in the surface Brillouin zone, where the spectra shownin (a) and (b) are recorded. (d) The imaginary part of the dynamical susceptibility calculated at (0.3, 0.3) by LRTDDFT. along the ¯Γ– ¯M direction. In order to unambiguously de-termine the magnon excitation energy and the lifetime,the difference spectra recorded at different wavevectorsare fitted with three lines, corresponding to the expectedthree magnon modes of the system. Each line includes aLorentzian lineshape, convoluted with the experimentalbroadening [36, 37]. An example is shown in Fig. 1(b),where the experimental difference spectrum ( I ↓ − I ↑ ) isshown together with the fits. The magnon dispersion re-lation was measured by probing the magnons at differentwavevectors and along different symmetry directions ofthe surface Brillouin zone and the results are summarizedin Fig. 1(c). For a quantitative description the magnonproperties were calculated based on first-principles. Ithas been shown that a quantitative description of theexperimental magnon bands can only be provided whenspin-dependent many-body correlation effects on the ma-jority Co spins are taken into consideration [32, 38]. InFig. 1(c) the magnonic band structure calculated us-ing this approach is presented as the color map. Theband structure is presented by plotting the magnon Blochspectral function. Since the adiabatic approach does notaccount for the decay of magnons, the spectral functionexhibits sharp peak at the places where different magnonmodes exist. The approach accounts, however, for all thedetails of the geometrical structure (e.g., the reconstruc-tion of the Ir surface) and provides an unambiguous wayfor the determination of magnon properties e.g., theirreal space localization and density of magnon states [34].In order to understand the decay mechanism ofmagnons we calculated, based on first principles, the fre-quency ω and momentum q dependent transverse dy-namical spin susceptibility χ ( ω, q ), using linear-responsetime dependent density functional theory (LRTDDFT) [39–42]. In Fig. 1(d) we provide an example of Im χ ( ω, q )at ( q x , q y )=(0.3, 0.3). Since in this approach bothmagnons and Stoner excitations are taken into account,this quantity can directly be compared to the differencespectrum shown in Fig. 1(b) [43–49]. In order to havea better comparison, we convolute the results with theexperimental resolution. Similar to the experiment, oneobserves all three confined magnon modes of the systemand different magnon modes exhibit different decay rates.In order to quantify the decay rates of different quan-tum confined magnon modes we carefully analyze theintrinsic broadening of the modes and the results aresummarized in Fig. 2(a), indicating that different quan-tum confined magnon modes show different decay rates.Moreover, to address the effects of the symmetry andhybridization effects between the electronic structures ofthe film with those of the underlying substrate similarexperiments were performed on Co films grown on theCu(001) and Ir(111) surfaces [26, 50–52]. The results arepresented along with those of the Co on Ir(001) system.We note that due to the geometrical consideration thesesystems possess different magnonic band structures. Themain aim here is to understand the decay rate of differ-ent magnon modes, with respect to the mode numberand mode energy. In a similar manner we also analyzethe line broadening of different magnon modes as cal-culated by LRTDDFT and the results are summarizedin Fig. 2, indicating a very good agreement with theexperimental results. A careful analysis of the experi-mental and theoretical results indicates that, (i) differ-ent magnon modes possess different decay rates, (ii) forall studied systems the magnon mode with n = 0 ex-hibits the lowest damping among all the others, and (iii)the magnon linewidth versus energy is nonlinear, which ) V e ( h t d i w en i L nonga M n=0n=1 n=2 n=0n=1 n=2 (a) (b) Magnon Excitation Energy (eV)
CoIr CoCu
FIG. 2. The magnon linewidth versus energy for all quantumconfined magnon modes in 3 ML Co on (a) Ir(001) and (b)Cu(001). The experimental results are shown by diamondsand the results of LRTDDFT calculations are shown by cir-cles. The filled (open) symbols represent the data along the¯Γ– ¯M (¯Γ– ¯X) direction. The black diamonds in (a) representthe results of the n = 0 magnon mode of 3 ML Co/Ir(111). becomes more pronounced for the higher order modes.Generally the main source of damping in itinerant fer-romagnets, such as systems studies here, is the resultof the decay of these collective modes into the single-particle Stoner excitations, a mechanism known as Lan-dau damping. It has been shown that in the case of ultra-thin magnetic films on nonmagnetic substrates, the hy-bridizations of the electronic states of the film with thoseof the substrate can open additional decay channels andlead to stronger damping of the magnons, meaning thatthe Landau damping in layered ferromagnets on nonmag-netic substrates can be very complex, compared to thatof the single element bulk ferromagnes [41, 42, 53, 54].Hence, in order to adequately describe the magnon damp-ing a proper description of the electronic structures isrequired. In our LRTDDFT calculations, we start withthe electronic structures calculated based on the experi-mental geometrical structure. Since the approach treatsboth magnons and Stoner excitations on the same basis,the Landau damping is fully taken into consideration.Looking at the results shown in Fig. 2, for eachmagnon mode the linewidth increases with energy in anonlinear fashion. The increase of the magnon linewidthwith energy is due to the fact that the probability that amagnon decays into a Stoner pair increases with energy.This probability depends on the details of the involvedelectronic bands and hence is not a simple linear functionwith energy. This scenario is even more complex whenconsidering the fact that the electronic states of the filmhybridize very strongly with those of the substrate. Theso-called Landau hot spots, where the decay of magnonsto Stoner excitations takes place become increasingly im-portant for higher energy magnons [41, 42, 53]. As it isapparent from Fig. 2 both the experimental and theoret-ical results indicate that different magnon modes exhibitdifferent (nonlinear) decay rates. This implies that the ) V e ( y g r en E no i t a t i cx E nonga M X Γ X M Γ M ) V e ( y g r en E no i t a t i cx E nonga M Magnon DOS 3 ML 4 ML (a) (b) n=3n=2n=1n=0 FIG. 3. (a) The magnon Bloch spectral function of 3 MLCo on Ir(001) based on adiabatic calculations. The magnon-ics bands of a 4 ML film are also shown by the light-graycurves. The multimagnon scattering process of the n = 1magnon mode within the 3 ML terraces are shown by thesolid arrows. The decay process within the terraces of differ-ent thicknesses (3 and 4 ML) are shown by the dashed arrows.(b) The magnonic density of states for a 3 ML and 4 ML filmcalculated based on the adiabatic approach. The direct decayof the n = 1 magnon mode of the 3 ML film to the n = 0and n = 1 of a 4 ML film is illustrated with the dashed ar-rows. The indirect decay of the n = 2 magnon mode of the3 ML film to the n = 2 and n = 1 modes of 4 ML film isschematically shown by the dotted black arrows. origin of the mode-dependent decay rate lies in the Lan-dau damping. The effect can be understood based onthe fact that for the n (cid:54) = 0 magnon modes the perpen-dicular magnon momentum is nonzero ( q ⊥ (cid:54) = 0). Hencethe Stoner pairs with nonzero perpendicular momentumwill enter the picture and, therefore, the possibility of thedecay into such modes shortens the lifetime of magnonswith n (cid:54) = 0. In the calculations based on LRTDDFT boththe magnons and Stoner excitations are put on an equalfooting and should therefore adequately describe the de-cay rates of different magnon modes caused by their dis-sipation into the Stoner excitations. Hence this effect isclearly observed in the calculated decay rates.The second possible decay channel is the decay of acertain magnon mode to the other possible magnon (orphonon) modes which share the same energy. As an ex-ample in Fig. 3(a) the processes of magnon decay of the n = 1 magnon mode at the ¯Γ-point is shown by the solidarrows. In addition, small variations in the film thicknesscan lead additional magnon modes, which may also sharethe same energy with this mode [55]. In order to mimicthis effect in Fig. 3(a) we also show the magnonic bandsof a film with the same geometrical structure but witha thickness of 4 ML. The results are shown by the light-gray color in Fig. 3(a). Since the n = 1 mode can alsodegenerate with the modes of such a system one needs toconsider such decay rates. The decay rates of this kindare shown by the dashed arrows in Fig. 3(a). In theLRTDDFT calculations such effects are not taken intoaccount. In order to generalize the decay process of acertain magnon mode to all the other possible quasipar-ticles the damping may be written as [53, 56]Γ( q , ω ) = (cid:88) n,m (cid:90) Ω BZ P n k P m k − q δ (cid:0) E − E n k − E m k − q (cid:1) d k , (1)where P n k and P m k − q denote the probability of finding amagnon, phonon or electron with the wavevector k in the n th and k − q in the m th band, E n k ( E m k − q ) describes theenergy dispersion of the n th ( m th) quasiparticle band.The term shall account for all the possible decay chan-nels, which a magnon with the energy (cid:126) ω and momen-tum q can decay into single-particle Stoner pairs, othermagnons and phonons, satisfying the energy conserva-tion rule E q = (cid:126) ω = E k − E k − q . Note that in the caseof electronic bands the transition should also account forthe conservation of magnon’s total angular momentum,meaning that only the transitions between the bands withopposite spins should be considered. In order to describethe decay of magnons into bosonic quasiparticles, e.g.,phonons and other possible magnon states, according toEq. (1), one should be able to analyze the different possi-bilities of such decays in the energy–momentum space. Inmetallic ferromagnets the phonon energies are rather low( <
20 meV) and the magnon-phonon coupling is ratherweak. Hence, the magnon decay by phonons becomes ofminor importance for high-energy magnons. The maindecay channel of magnon-boson kind is the one associ-ated with their decay into other magnon modes. In orderto estimate the strength of such decay rates we calculatedthe magnonic density of states (DOS). The results of suchcalculations are presented in Fig. 3(b). We also presentthe magnon DOS of a film composed of four atomic lay-ers. The probability of finding magnons in a given state(indicated by circles) can therefore be simply estimatedby analyzing the magnon DOS. The magnon–magnon de-cay is directly proportional to the number of initial andfinal magnon states, which may contribute to such a pro-cess ( P n k and P m k − q ). If such states are largely availablethe magnon decay can occur with a large probability.Looking at the data presented in Fig. 3 one realizesthat such decay process can occur with a large proba-bility, since there are enough initial and final magnonstates, which can contribute to this kind of magnon de-cays (solids arrows in Fig. 3). In addition to the intrinsicmagnon-magnon decay, the variation in the film thicknesscan also lead to a magnon decay. For example if the filmis composed of terraces with the thickness of 3 and 4 ML,the n = 1 magnon mode of the 3 ML terraces can decayinto the n = 1 and n = 0 magnon modes of the terraceswith the thickness of 4 ML. Such a process can happenwith a large probability as shown by the dashed arrows in Fig. 3. Interestingly the n = 2 mode of the 3 ML re-gion can, in principle, decay to the other modes of the 4ML region via an inelastic process in which the magnonsare decayed in a two step process [dotted arrows in Fig.3(b)]. The process can occur with a high probability be-cause near the states of the n = 2 mode of the 3 MLregion there exist a large number of states caused by the n = 2 and n = 1 modes of the 4 ML region. Due to thereconstruction of the Ir(001) surface the roughness of Cofilms on this surface is larger than that of the Co filmson Cu(001). This leads to a larger magnon decay of thiskind and explains the larger experimental linewidth ofthis mode as compared to the Co/Cu(001) system andalso to the results of LRTDDFT.In summary, aiming on a fundamental understandingof the decay processes of quantum confined magnonsin layered ferromagnets, we investigated the lifetime ofthese excitations in a model system composed of 3 ML Cogrown on different surfaces over a wide rage of energy andmomentum. It was observed that the quantum confinedmagnons exhibit nonlinear decay rates. The decay ratesstrongly depend on the mode number. In phenomeno-logical approach of classical dynamics the decay rate isassumed to be linear. Such an assumption is not validfor the quantum confined magnons. Combining the ex-perimental results with those of LRTDDFT calculationswe provide a quantitative explanation for this nonlineardamping. In addition, since the quantum confinementleads to the emergence of several magnon branches, thedecay processes as a result of magnon-magnon scatter-ing become also important. These multimagnon decayprocesses become stronger due to variations in the filmthickness. Our results indicate that the main source ofdamping in layered structures made of itinerant ferro-magnets is due to the Landau damping as a result oftheir decay into Stoner excitations. Hence in order todesign layered ferromagnets with low damping, first theelectronic structure should be tuned such that the Lan-dau damping is suppressed. Moreover, atomically flatfilms are required to achieve a low damping. In addi-tion to the fact that our results provide new insights intothe decay mechanism of spin excitations in ultrathin filmsand multilayers, they provide guideline regarding how thedynamical properties of layered structures can be tuned. ACKNOWLEDGEMENTS
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