The exchange stiffness coefficient, Aex, represents the strength of direct exchange interactions among neighboring spins. Aex is linked to most of the magnetic properties such as skyrmion formation, magnetic vortex, magnetic domain wall width, and exchange length. Hence, the quantification of Aex is essential to understanding fundamental magnetic properties, but little is known for the dynamics of Aex on a sub-picosecond timescale. We report the ultrafast dynamcis of Aex in an ordered magnetic state in Co/Pt ferromagnetic multilayer. Time-resolved magneto-optical Kerr effect and reflectivity measurements were analyzed for various pump fluences. We reveal that the significant dynamical reduction of Aex is responsible for the dramatic increase of remagnetization time for high fluences. The analysis shows that Aex dynamically varies, strongly affecting overall ultrafast demagnetization/remagnetization process. The investigation demonstrates the possibility of Aex engineering in femtosecond timescale and thereby provides a way to design ultrafast spintronic devices.
The exchange interaction is a fundamental aspect of ferromagnetism1,2,3; this interaction, underpinning the existence of ordered magnetic states, allows the alignment of neighboring spins in a system. The origin of the direct exchange interaction is ascribed to the overlapping of electron wave functions among neighboring atoms with no classical analogy, and its strength depends sensitively on the atomic and lattice structure4,5, while there also exist other exchange mechanisms such as Dzyaloshinskii-Moriya interaction6,7, and superexchange interaction8. The strength of the direct exchange interaction is represented by the exchange-stiffness coefficient Aex, which is a material parameter mostly depending on the atomic and lattice structure and detailed characteristics of which are determined by the electronic band structure9,10,11. Fundamental mechanisms of novel spin phenomena, such as magnetic vortices12,13 and skyrmions14,15, might be understood based on a quantitative analysis of Aex. It has been known that Aex is temperature-dependent16,17. As the temperature increases, thermal agitation reduces the degree of the ordering of neighboring spins, effectively lowering the value of Aex.
Although the temperature-dependence of Aex is relatively well recognized in static cases17, very little is known regarding how Aex varies on an in the ultrafast timescale. In case of ultrafast photo-induced demagnetization, as the fluence FP of a pump laser increases, the disorder in a spin system should increase so that the effective spin temperature should also increase. It is then expected that the remagnetization time of the system from an excited disordered state to a stable equilibrium state should increase as FP increases. Recent reports of the dynamics of the exchange interaction on ultrafast timescale have shown that a fundamental exchange interaction varies on scales of several tens of femtoseconds in ferromagnetic NiFe alloy18 and antiferromagnetic KNiF319. A possibility of ultrafast control of exchange interaction by using a femtosecond pump laser has been proposed theoretically20,21,22. All of these results indicate that the exchange interaction or exchange splitting dramatically changes by a femtosecond pump laser. However, to our best knowledge, no systematic study has been conducted to quantify the dynamics of Aex on ultrafast timescale.
Here, we report, the dynamics of Aex on a femtosecond timescale in Co/Pt multilayers for a range of FP, demonstrating that Aex varies rapidly, affecting spin dynamics and its variation can be controlled by the pump fluence. While the electron-spin interaction strength is kept constant all the time in the conventional three-temperature model (3TM) in the study of ultrafast magnetism, the dynamic change of Aex is considered, adopting the generalized three-temperature model (G-3TM) developed by A. Manchon et al23.
Fluence dependent remagnetization time
We performed time-resolved magneto-optical Kerr effect (TR-MOKE) measurements for [Co (6.2 Å)/Pt (7.7 Å)]5 multilayer film, of which the magnetic properties such as perpendicular magnetic anisotropy and saturation magnetization are well known7,24,25,26,27. A detailed experimental configuration was reported elsewhere28. TR-MOKE signals were measured for 1.7 ≤ FP ≤ 28.5 mJ cm−2 for time delays of up to 30 ps. To exclude the dichroic bleaching effect, the experiments are carried out by pump beam at both 400 and 800 nm wavelength (λpump). TR-MOKE vs. time at different fluences is plotted in Fig. 1a for the case of λpump = 800 nm, while no significant difference is observed in the overall trend for the case of λpump = 400 nm. The signals were normalized by their maximum changes to compare the dynamical behaviors in remagnetization for different conditions. An external magnetic field of 1.7 kOe was applied normally to the film surface. All the measured curves exhibit clearly the dynamics of photo-induced demagnetization and subsequent remagnetization. The maximal change of demagnetization is observed around t = 0.3 ps for all the fluence cases. In case of λpump = 800 nm, as FP was increased from 1.7 to 28.5 mJ cm−2, the remagnetization was slowed down for FP > 9.9 mJ cm−2. The remagnetization behavior at 1.7 ≤ FP ≤ 16.5 mJ cm−2 well-fitted with a single exponential curve (Fig. 1a, dotted lines), yielding the characteristic time τR of remagnetization (open square in Fig. 1b). We note that τR increased drastically as FP increased by only a factor of a few. Fitting with a single exponential curve was not valid for FP > 16.5 mJ cm−2. A similar trend is observed for the case of λpump = 400 nm, where τR is fitted with a single exponential curve as well.
where Te, Tl, and Ts are the electron, lattice, and spin temperatures, respectively. Ce, Cl, and Cs are the specific heats of the electron, lattice, and spin, respectively. Gel, Ges, and Gls are the electron-lattice, electron-spin, and lattice-spin interaction channels, respectively. P(t) is a laser source term with a Gaussian temporal profile. The term that contains Kl represents lattice thermal diffusion, which is modeled to be proportional to the third power of the temperature increase of lattice system30. A typical relation between the magnetization and the spin temperature: ∝ (1 – (Ts / TC))0.5, where TC is Curie temperature of 1131 K17, to match the normalized TR-MOKE signal (|∆θKerr|/|∆θpeak|).
In conventional 3TM, Gel, Ges, and Gls have been set to be constant over time. However, in our study, for correct analysis, the electron-spin interaction channel(Ges) was allowed to change over time. Adopting the G-3TM, Ges can be written as
where a is a lattice constant, Aex0 an exchange-stiffness coefficient at 0 K, V unit cell volume, TF Fermi temperature, S = 3/2 spin quantum number, M magnetization and G2 a function based on the second-order Debye function23 (Eq. (3)). TF is Fermi temperature, chosen to be that of fcc Co (TF = 16.87 Ry/kB)31. D = S M[Ts] qm2 a2, where qm is magnon wave number qm = kF = (6π2)1/3a−1, and kF is Fermi wave number. Features of Ge[Te, Ts] is described in detail in Supplementary Note 1. Ges0 (Eq. (4)) is a temperature-independent electron-spin interaction channel. In static case, it is well known that Aex \( \propto \) (Jex a−1) <S2 > , where Jex is an exchange interaction constant, a is a lattice constant. The proportionality depends on material parameters such as a periodic lattice configuration. Since Aex can be easily measured rather than Jex, we focus on qauntifying Aex on an ultrafast timescale. When the relation between Aex and Jex is extended, we have put Aex(T) \( \propto \) Aex0 < M(T)2 > \( \propto \) (Jex a−1) <S(T)2 > , where the temperature T dependence is included in <M(T)2 > without affecting Jex. For simplicity, we have used the approximation, Jex ~ 2aAex016. In the G-3TM, Gel is still assumed to be constant because the relaxation rate between the electron and the lattice is expected to be simply proportional to the temperature difference. Gls was also set to be constant throughout the simulations.
The G-3TM is composed of several free parameters, so the fitting should be processed with care. First, time-resolved reflectivity R(t) data were utilized to estimate values for Ce, Cl, and Gel, considering only the electron and lattice, based on the 2-temperature model32,33. In the full analysis using the G-3TM, the reflectivity and MOKE data were fitted. As a constraint in the analysis, the measured values for the degree of demagnetization (Ddemag) were used (Fig. 2a, d). Hysteresis loop measurement is the best way to estimate the Ddemag. The hysteresis loops were measured at t = −2 ps and 0.3 ps (the maximal demagnetization) using the same TR-MOKE setup with probe-beam modulation for all the FP (Methods section). An example of measurements for FP = 13.2 mJ cm−2 (Fig. 2a, inset) shows that a Ddemag is 70%. The excellent match has been established in all the cases (See Supplementary Note 2, where the utilization of R(t) measurement and the Ddemag for fitting is described).
The G-3TM fitting determines temporal evolutions of spin, electron, and lattice temperature at wavelength of pump pulse λpump = 800 nm (Fig. 2b, c) and λpump = 400 nm (Fig. 2e, f). The cases for very high FP corresponding to Ts being very close to Curie temperature (1131 K) are not considered, where the G-3TM may not be valid. The fitted value of Ce was 1.8 ~ 2.1 × 103 J (m3 K2)−1 and, Cl was 1.8 ~ 5.0 × 106 J (m3 K)−1 34,35. (all the fitting parameters are summarized in Supplementary Note 2). Cs should depend on the spin temperature Ts. In the original Manchon’s paper23 which the G-3TM on, Cs is determined from the numerical derivative of the spin energy. In fitting our data, we have found that the fitting becomes quite good if Cs is smaller than ~104 J (m3K)−1 in all cases. Thus, we used a small value of Cs = 100 J (m3K)−1 for all cases. The upper limit of fitted Cs value (~104 J (m3K)−1) in the present work seems to be a little bit smaller than the reported values determined from 3TM. For instance, in Ref. 35, Cs of Ni and FeCuPt are 0.2 × 106 J (m3 K)−1 and 0.17 × 106 J (m3 K)−1.
The maximum values of electron temperature Temax and spin temperature Tsmax at t = 0.3 ps increased as FP increased; e.g., Tsmax at t = 0.3 ps changes from 564 to 1040 K as FP increases from 1.7 to 9.9 mJ cm−2 (λpump = 800 nm). High FP increases the amount of energy transferred to the subsystems, so the increase of Temax and Tsmax is expected. The equilibrium temperature at which Te = Tl = Ts also increased consistently as FP increased, but it is very interesting to note that the difference between Temax and Tsmax got larger substantially as FP increased (Fig. 2b, c, Fig. 2e, f). In the context of the G-3TM, this observation indicates that the interaction channel Ges between the electron and spin subsystem is reduced, resulting in the increase of the thermal separation of the spin system from the electron subsystem as well as thereby the increase of the time required to reach thermal equilibrium. This phenomenon may be a reason for the increase of τR as FP increases as observed in Fig. 1b.
Figure 3a is the plot of Ges (Eq. (4)) as a function of Te and Ts. As Ts increases, Ges increases then decreases for a given electron temperature. The values Te, Ts, and Ges determined from fitting to our experimental data at λpump = 800 nm are shown in a gray curved line in Fig. 3a, and again in Fig. 3b for various FP. The case of Ts = Te is also presented as a dotted curve for guidance in Fig. 3b. The non-monotonic nature of Ges with respect to Ts is a direct consequence of Eq. (3). Ges increased monotonically with an increase in Ts at low FP = 1.7 and 3.3 mJ cm−2, but the increasing-then-decreasing behavior is observed at high FP > 6.6 mJ cm−2. We suspect that diverse experimental results of ultrafast demagnetization dynamics might be originated from this different trend of Ges at high FP36,37,38.
The dynamical variation of Ges on a femtosecond timescale is plotted for various FP in Fig. 3c. At low FP, the simple increase-and-decrease behavior of Ges is observed, with a maximum at t = 0.3 ps. The time of the maximum Ges coincides with the time at which the Ddemag is the greatest. At high FP, Ges quickly reaches the first peak right after the arrival of a pump pulse, decreased to a minimum at around the time of maximal Ddemag (t = 0.3 ps), and then increased again. A comparison between the behaviors of Ges, Gls, and Gel at t = 0.3 ps under various FP (Fig. 3d) reveals that Gel is the strongest channel, Ges increased at low FP, but decreases at high FP; this trend may be the result from the feature of Ges (Fig. 3a, b). On the other hand, Gls is the weakest channel (as often neglected) but becomes comparable to Ges as FP increases. Gls is involved with spin-orbit coupling23, which might get stronger as Te or Ddemag 39.
The above discussion indicates that the dynamics of the photo-induced demagnetization and remagnetization in Co/Pt spin system is mostly governed by Ges and Gls. Figure 3e shows a ratio of Ges to Gls at t = 0.3 ps for various fluences. Ges/Gls is larger than 10 for FP < 6.6 mJ cm−2 for λpump = 800 and 400 nm. This imbalance implies that the spin-electron interaction is dominant in this FP regime. For FP ≥ 6.6 mJ cm−2, Ges/Gls approaches unity asymptotically, indicating that spin-lattice interaction becomes increasingly important. The G-3TM fitting yields the values for Ges. Equations (2) and (4) allow us to calculate Aex0, temperature-independent exchange-stiffness coefficient. The estimated value of Aex0 turns out to be 10.01 pJ m−1 at all the FP; this value agrees well with a reported value for a Co/Pt multilayer40,41. Other analysis methods42,43 could also reproduce the slow rate of magnetization at high fluences. It should be commented that the G-3TM based on the Hamiltonian for laser-induced demagnetization23 allows us to separately monitor time-dependent Ges and Gls as well as their ratio Ges/Gls. In this work, we note that the ratio particularly seems to play an important role in determining the energy-excessive spin dynamics on a sub-ps timescale.
The previous studies17,44,45,46 in static cases have shown that the temperature dependence of Aex is expressed as power of M with a scaling exponent ranging from 1.79 to 1.82 in case of Co. We set the exponent to be 1.8 and write the temporal variance of Aex as
Based on Eq. (5), time-dependent Aex is plotted in Fig. 4. The increase in FP results in the reduction of Aex, as generally expected in static cases. However, the recovery of reduced Aex depends sensitively on FP. Aex decreases asymptotically as FP increases, and saturates at FP ≈ 9.9 mJ cm−2 (λpump = 800 nm) or 12.1 mJ cm−2 (λpump = 400 nm) without further decrease with respect to the fluence higher than this value, which is expected from the saturated behavior of the Ddemag at high FP. At FP > 9.9 mJ cm−2, Aex was ~1 pJ m−1. The maximal decrease of Aex occurred at t = 0.3 ps when the maximum Ddemag occurs in the TR-MOKE measurement. TR-MOKE data (Fig. 1a) show a similar trend to the trend in Aex (Fig. 4). The magnetization M and Aex recovered quickly at low FP whereas the recovery becomes significantly slow for high FP.
It should be noted that Eq. (5) is valid for the steady-state case and we use the very rough assumption that considering that even in the out-of-equilibrium case, there could be a rough relation between Aex and temperature-dependent M9. Indeed, although we use 3TM23,28,29, 3 temperatures are not fundamentally well defined in the out-of-equilibrium state and only phenomenologically defined once 3TM is used. On the other hand, we consider that the M[T] might not be totally different compared to the steady-state case, since the estimated spin temperature is still below TC. The pump pulse excites the electrons around the Fermi energy so that the excited electrons occupy the allowed energy levels above the Fermi energy, while remaining electrons still follow the Fermi-Dirac distribution. Moreover, the thermal equilibrium among 3 temperatures is achieved around ~10 ps and thus, the M[T] will be soon replaced back to the equilibrium case after this timescale. Therefore, we think that there might be a deviation of M[T] from the steady-state case, but the M[T] can be roughly approximated based on the Eq. (5). We have varied the exponent value from 1.6 to 2.0 in our analysis, where no significant difference in the analysis result is observed.
Possible mechanisms of Aex reduction might be involved with Stoner exchange splitting reduction9, where it has been reported that dynamic exchange splitting is determined by time-dependent magnetization M(t). On the other hand, magnon generation should be also an important factor47, where it has been reported that the magnon contribution to demagnetization is dominant only on a very short (700 fs) timescale. Thus, in our case, we consider that the magnon contribution could exist on a sub-ps timescale, while the Stoner exchange splitting reduction is lasting longer up to few tens of ps since there is still a substantial amount of demagnetized M(t) in the present work. It should be also noted that G-3TM, which our whole analysis is based on, includes the magnon generation by hot electron as a key mechanism in the model. In G-3TM, electron-spin interaction Hamiltonian intrinsically deals with the effect of magnon generation, which might be reflected in the Aex dynamics, particularly on the sub-ps timescale. The effective Stoner exchange splitting reduction is understood based on the reduced M(t) over the whole process of demagnetization and remagnetization. It should be mentioned that our film is prepared on a Si substrate with no doping, which can be approximated to be an insulator so that the spin diffusion effect could be negligible in the process of demagnetization.
The above analysis reveals that the significant increase of the remagnetization time (τR) (Fig. 1b) for high fluence is directly related to the reduction of Aex. In order to confirm how much Aex or demagnetization state affects the remagnetization process, we have carried out another series of independent micromagnetic simulations. The micromagnetic simulation was performed using the Object-Oriented Micromagnetic Framework48 based on the Landau-Lifshitz-Gilbert (LLG) equation:
where the gyromagnetic ratio γ = 2.210 × 105 m (A s)−1 and Heff is the effective magnetic field. Since the micromagnetic simulation does not consider the temperature variation, it is not suitable to dynamics study but still provides valuable information about the material properties for remagnetization at a fixed temperature. We set the initial degree of magnetization according to the measurement and simulated how the remagnetization proceeds for different magnetic parameters such as Aex and magnetic anisotropy. In the simulations, an external magnetic field of 1.7 kOe was applied with an angle of 0° to the surface normal of the film as in the experiments. The saturation magnetization Ms of the film was set as 103 kA m−1. Magnetic anisotropy constant K was set as 6 × 105 J m−3. Gilbert damping constant α was set as 0.05. The cell size was 0.5 × 0.5 × 0.5 nm3 and the sample size was 50 × 50 × 7.5 nm3. The initial demagnetization state is set by the experimentally-measured Ddemag for various FP (Fig. 1b is re-plotted with respect to Ddemag corresponding to various FP (black open squares) as shown in Fig. 5a). As the Ddemag increases, τR increases, drastically at larger Ddemag than 60 %. To determine parameters that are most responsible for the abrupt increase of τR with the increase of FP (or high Ddemag), we performed micromagnetic simulations for Aex = 1, 8, and 15 pJ m−1. In each simulation, Aex was fixed throughout the simulation. Simulations with Aex = 8 (Fig. 5a, green triangle) and 15 pJ m−1 (Fig. 5a, red circle) agree well with experiments at low FP (or low Ddemag). The literature value40,41 of Aex of Co/Pt multilayer for a static case is ~ 10 pJ m−1, which is consistent with the range of our simulation parameter. In the simulation with Aex = 1 pJ m−1, τR was substantially higher than the experimental observations (low Ddemag).
Magnetic anisotropy is another important parameter that might affect τR. The magnetic anisotropy is determined mostly by the crystal structure, sample shape, and multilayer interfaces. This anisotropy produces a perpendicular magnetic anisotropy in the Co/Pt multilayer used in the present study. The micromagnetic simulation of τR for various magnetic anisotropy constants K for various Ddemag from 47 to 80 % showed no significant change of τR at 0.01 ≤ K ≤ 1.2 MJ m−3 (Fig. 5b). The measured value of K for the Co/Pt multilayer in the present study is K = 0.6 MJ m−3, which is within our simulation range. Thus, we infer that the variation in K is not responsible for the increase of τR. This independence is expected because the variation of K will mostly affect the total effective field without directly modifying spin-spin or spin-electron interactions.
We also systematically changed Aex in micromagnetic simulations with the Ddemag being fixed at 80% (Fig. 5c). As Aex was varied from 4.0 to 0.1 pJ m−1, τR increased from 10.8 to 30.1 ps. In particular, at Aex < 1 pJ m−1, τR increases rapidly in a similar manner to the experimental observation (near high FP in Fig. 1b, or near high Ddemag in Fig. 5a). The increase in FP can be expected to reduce Aex significantly, resulting in a large increase of τR. The decrease in Aex is generally expected to cause the increase in τR, because the reduced spin-spin interaction weakens the ordering among neighboring spins. This micromagnetic simulation also confirms the analysis by G-3TM.
We have carried out a simulation with the variation of damping parameter (α), as seen in Fig. 5d. Aex is set to be 11 pJ m−1, anisotropy constant K is 0.6 MJ m−3, and Ddemag is set to be 80%. τR is found to be insensitive if α is larger than 0.005, as seen in the figure. For α smaller than 0.005, τR drastically increases due to the significant contribution of precessional oscillation. α of Co/Pt multilayer is reported in the range of 0.02–0.149,50, which is much larger than 0.005.
In summary, we have investigated the dynamical variation of Aex on an ultrafast timescale, by TR-MOKE and reflectivity measurements in a Co/Pt multilayer for various FP. Our phenomenological analysis suggests that the ultrafast remagnetization mechanisms may be governed by the dynamically changing Aex, which is also closely related to Ges, and Gls also becomes non-negligible in case of high FP. Our comprehensive micromagnetic simulations implies that significantly reduced Aex is responsible for the large remagnetization time. These results demonstrate the possibility of engineering magnetic properties on an ultrafast timescale by modifying Ges, Gls, and Gel.
TR-MOKE measurements with a pump-probe stroboscope were performed on a Co/Pt multilayer. We used the femtosecond laser pulses generated by a Ti:sapphire multipass amplifier operating at a repetition rate of 3 kHz with a center wavelength of 800 nm and a pulse duration of 25 fs. We employed two pump wavelengths of 800 nm and 400 nm obtained from BBO (BaB2O4) crystal. As probe pulses, one wavelength of 800 nm was used. FP was varied from 1.7 to 28.5 mJ cm−2 and probe fluence was 0.3 mJ cm−2. For TR-MOKE measurements, the pump beam was modulated using a mechanical chopper at 500 Hz. An external magnetic field of 1.7 kOe was applied throughout the measurements, with an angle of 0° to the surface normal of the film to keep the initial sample condition saturated before a subsequent pump pulse.
To estimate the Ddemag, we conducted a series of hysteresis measurements at times of t = −2 ps and 0.3 ps under the same TR-MOKE setup with only probe-beam modulation at 500 Hz, while sweeping a magnetic field from −1.7 kOe to 1.7 kOe. The hysteresis measurement at t = −2 ps gives the magnetization of an intact sample.
[Co(6.2 Å)/Pt(7.7 Å)]5 multilayer films were deposited on Si substrates by dc magnetron sputtering, then capped by a 22-Å Pt layer to prevent the oxidation of the surface. The structure of the Co/Pt multilayers with well-defined interfaces was confirmed by a low angle X-ray diffraction and extended X-ray absorption fine structure analysis. The film had a perpendicular magnetic anisotropy (K = 0.63 MJ m−3) and saturation magnetization (Ms = 1.04 × 103 kA m−1), which are similar to literature values24,25,26,27.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The source codes used for G-3TM fitting/micromagnetic simulation are available from the corresponding author upon reasonable request.
Kittel, C. Physical theory of ferromagnetic domains. Rev. Mod. Phys. 21, 541–583 (1949).
Anderson, P. W. Generalizations of the Weiss molecular field theory of antiferromagnetism. Phys. Rev. 79, 705 (1950).
Zakeri, K. et al. Direct probing of the exchange interaction at buried interfaces. Nat. Nanotechnol. 8, 853–858 (2013).
Zener, C. Interaction between the d-Shells in the Transition. Met. Phy. Rev. 81, 440 (1951).
Yan, B. et al. Observation of dipolar spin-exchange interactions with lattice confined polar molecules. Nature 501, 521–525 (2013).
Dzyaloshinskii, I. Ie dzyaloshinskii. Sov. Phys. JETP 5, 1259 (1957).
Yang, H., Thiaville, A., Rohart, S., Fert, A. & Chshiev, M. Anatomy of Dzyaloshinskii–Moriya interaction at Co/Pt interfaces. Phys. Rev. Lett. 115, 267210 (2015).
Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91–98 (1960).
Mueller, B. Y. et al. Feedback effect during ultrafast demagnetization dynamics in ferromagnets. Phys. Rev. Lett. 111, 167204 (2013).
Rhie, H.-S., Dürr, H. A. & Eberhardt, W. Femtosecond electron and spin dynamics in Ni/W(110) films. Phys. Rev. Lett. 90, 247201 (2003).
Zhang, G. P., Bai, Y. H. & George, T. F. Ultrafast reduction of exchange splitting in ferromagnetic nickel. J. Phys. Condens. Matter 28, 236004 (2016).
Jain, S. et al. From chaos to selective ordering of vortex cores in interacting mesomagnets. Nat. Commun. 3, 1330 (2012).
Wintz, S. et al. Magnetic vortex cores as tunable spin-wave emitters. Nat. Nanotech. 11, 948–953 (2016).
Zhang, X., Zhou, Y. & Ezawa, M. Magnetic bilayer-skyrmions without skyrmion Hall effect. Nat. Commun. 7, 10293 (2016).
Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. Nanotech. 8, 899–911 (2013).
Atxitia, U. et al. Multiscale modeling of magnetic materials: temperature dependence of the exchange stiffness. Phy. Rev. B 82, 134440 (2010).
Moreno, R. et al. Temperature-dependent exchange stiffness and domain wall width in Co. Phy. Rev. B 94, 104433 (2016).
Mathias, S. et al. Probing the timescale of the exchange interaction in a ferromagnetic alloy. Proc. Natl Acad. Sci. USA 109, 4792–4797 (2012).
Batignani, G. et al. Probing ultrafast photo-induced dynamics of the exchange energy in a Heisenberg antiferromagnet. Nat. Photon. 9, 506–510 (2015).
Zhang, G. P., Gu, M. & Wu, X. S. Ultrafast reduction in exchange interaction by a laser pulse: alternative path to femtomagnetism. J. Phys. Cond. Matter 26, 376001 (2014).
Mentink, J. H., Balzer, K. & Eckstein, M. Ultrafast and reversible control of the exchange interaction in Mott insulators. Nat. Commun. 6, 6708 (2015).
Mikhaylovskiy, R. V. et al. Ultrafast optical modification of exchange interactions in iron oxides. Nat. Commun. 6, 8190 (2015).
Manchon, A., Li, Q., Xu, L. & Zhang, S. Theory of laser-induced demagnetization at high temperatures. Phy. Rev. B 85, 064408 (2012).
Zeper, W. B., Greidanus, F. J. A. M., Carcia, P. F. & Fincher, C. R. Perpendicular magnetic anisotropy and magneto optical Kerr effect of vapor deposited Co/Pt-layered structures. J. Appl. Phys. 65, 4971 (1989).
Davies, J. E. et al. Magnetization reversal of Co/Pt multilayers: microscopic origin of high-field magnetic irreversibility. Phys. Rev. B 70, 224434 (2004).
San Emeterio Alvarez, L. et al. Spin-transfer-torque-assisted domain-wall creep in a Co/Pt multilayer wire. Phys. Rev. Lett. 104, 137205 (2010).
Quach, D.-T. et al. Minor hysteresis patterns with a rounded/sharpened reversing behavior in ferromagnetic multilayer. Sci. Rep. 8, 4461 (2018).
Shim, J.-H. et al. Ultrafast giant magnetic cooling effect in ferromagnetic Co/Pt multilayers. Nat. Comm. 8, 796 (2017).
Beaurepaire, E., Merle, J.-C., Daunois, A. & Bigot, J.-Y. Ultrafast spin dynamics in ferromagnetic nickel. Phys. Rev. Lett. 76, 4250 (1996).
Lu, G., Wang, X. D., Duan, Y. Y. & Li, X. W. Effects of non-ideal structures and high temperatures on the insulation properties of aerogel-based composite materials. J. Non-Cryst. Sol. 357, 3822 (2011).
Bagayoko, D. et al. Band structure of bcc cobalt. Phys. Rev. B 27, 7046 (1983).
Sun, C.-K., Vallée, F., Acioli, L. H., Ippen, E. P. & Fujimoto, J. G. Femtosecondtunable measurement of electron thermalization in gold. Phys. Rev. B 50, 15337–15348 (1994).
Kruglyak, V., Hicken, R., Matousek, P. & Towrie, M. Spectroscopic study of optically induced ultrafast electron dynamics in gold. Phys. Rev. B 75, 035410 (2007).
Kimling, J. et al. Ultrafast demagnetization of FePt:Cu thin films and the role of heat capacity. Phys. Rev. B 90, 224408 (2014).
Iihama, S. et al. Ultrafast demagnetization of L10 FePt and FePd ordered alloys. J. Phys. D. Appl. Phys. 49, 035002 (2016).
Atxitia, U., Chubykalo-Fesenko, O., Walowski, J., Mann, A. & Münzenberg, M. Evidence for thermal mechanisms in laser-induced femtosecond spin dynamics. Phys. Rev. B 81, 174401 (2010).
Roth, T. et al. Temperature dependence of laser-induced demagnetization in Ni: a key for identifying the underlying mechanism. Phys. Rev. X 2, 021006 (2012).
Kuiper, K. C. et al. Spin-orbit enhanced demagnetization rate in Co/Pt-multilayers. Appl. Phys. Lett. 105, 202402 (2014).
Stamm, C., Pontius, N., Kachel, T., Wietstruk, M. & Dürr, H. A. Femtosecond x-ray absorption spectroscopy of spin and orbital angular momentum in photoexcited Ni films during ultrafast demagnetization. Phy. Rev. B 81, 104425 (2010).
Eimüller, T. et al. Spin-reorientation transition in Co/Pt multilayers on nanospheres. Phy. Rev. B 77, 134415 (2008).
Okamoto, S., Kikuchi, N., Furuta, M., Kitakami, O. & Shimatsu, T. Switching behaviors and its dynamics of a Co/Pt nanodot under the assistance of rf fields. Phy. Rev. Lett. 109, 237209 (2012).
Koopmans, B. et al. Explaining the paradoxical diversity of ultrafast laser-induced demagnetization. Nat. Mater. 9, 259–265 (2010).
Atxitia, U. & Chubykalo-Fesenko, O. Ultrafast magnetization dynamics rates within the Landau-Lifshitz-Bloch model. Phys. Rev. B 84, 144414 (2011).
Turek, I., Kudrnovsky, J., Drchal, V., Bruno, P. & Blügel, S. Ab initio theory of exchange interactions in itinerant magnets. Phys. Status Solidi (b) 236, 318 (2003).
Pajda, M., Kudrnovský, J., Turek, I., Drchal, V. & Bruno, P. Ab initio calculations of exchange interactions, spin-wave stiffness constants, and Curie temperatures of Fe, Co, and Ni. Phys. Rev. B 64, 174402 (2001).
Kvashnin, Y. O., Sun, W., Marco, I. D. & Eriksson, O. Electronic topological transition and noncollinear magnetism in compressed hcp Co. Phys. Rev. B 92, 134422 (2015).
Turgut, E. et al. Stoner versus Heisenberg: ultrafast exchange reduction and magnon generation during laser-induced demagnetization. Phys. Rev. B 94, 220408 (2016).
Donahue, M. J. & Porter, D. G. OOMMF User’s Guide. http://math.nist.gov/oommf (2002).
Fujita, N. et al. Damping constant of Co/Pt multilayer thin-film media. J. Magn. Magn. Mater. 320, 3019–3022 (2008).
Barman, A. et al. Ultrafast magnetization dynamics in high perpendicular anisotropy [Co/Pt]n multilayers. J. Appl. Phys. 101, 09D102 (2007).
This research has been supported in part by Max Planck POSTECH/KOREA Research Initiative Program [Grant No 2016K1A4A4A01922028] through the National Research Foundation of Korea (NRF) funded by Ministry of Science, ICT & Future Planning, partly by Korea Institute for Advancement of Technology (KIAT) grant funded by the Korea Government (MOTIE) (P0008763, The Competency Development Program for Industry Specialist), partly by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education [Grant No 2017R1A6A3A04011173 and 2017R1A6A3A11032995] and partly by Korea Research Foundation (NRF) grant No. 2018R1A2B3009569 and a KBSI Grant D39614.
The authors declare no competing interests.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Shim, J., Syed, A.A., Shin, Y. et al. Ultrafast dynamics of exchange stiffness in Co/Pt multilayer. Commun Phys 3, 74 (2020). https://doi.org/10.1038/s42005-020-0346-y