Lattice structure dependence of laser-induced ultrafast magnetization switching in ferrimagnets

Fast, reliable, and inexpensive data manipulation and storage is the cornerstone for innovation and progress of our information-technology-based society. Ultrafast magnetism holds promise for fast and low energy data manipulation solutions.^1–5 The field was initiated by the discovery of femtosecond laser pulse induced subpicosecond demagnetization in Ni.⁶ To this breakthrough followed the demonstration of field-free magnetization switching in GdFeCo alloys using a train of circularly polarized pulses.⁷ Later on, a combined theoretical/experimental study showed the possibility of single pulse switching using linearly polarized light.^8,9 This finding uncovered the purely thermal origin of the switching process, which in turn was used to demonstrate that ultrafast heating by picosecond electric pulses is sufficient to achieve switching in GdFeCo.¹⁰ Single-pulse switching can also be accomplished in CoTb alloys,¹¹ Gd-based ferrimagnetic multilayers,^12,13 magnetic tunnel junctions of Tb/Co,¹⁴ and the rare-earth-free Heusler alloy Mn₂Ru_xGa.^15,16 For future information technologies, ultrafast magnetization manipulation promises high potential; as such, further understanding of the microscopic origin of single-pulse magnetization switching is key for ultrafast spintronics applications.¹⁷ 

Single pulse magnetization switching in ferrimagnets has been described using computer simulations based on atomistic spin dynamics (ASD)^18–21 and phenomenological models.^22–25 A recent work has merged these models into an unified macroscopic theory that describes magnetization dynamics and switching of two-sublattice ferrimagnets upon femtosecond laser excitation.^26,27 Within this theory, the switching process becomes possible due to an enhancement of the exchange relaxation—angular momentum exchange between sublattices—when the magnetization of the sublattices is reduced below a certain threshold that depends on material parameters. After femtosecond laser photo-excitation, the electron system enters a high temperature regime at which angular momentum dissipation into electron or phonon degrees of freedom dominates. This so-called relativistic relaxation is related to the spin–orbit coupling connecting spin and orbital degrees of freedom. Depending on the laser power, the sublattice magnetization can reduce the threshold that leads to magnetic switching. Once the laser pulse is gone, the electron system starts to cooldown, which leads to a local recovery of magnetic order due to the exchange coupling between spins. In two sublattice magnets, the so-called exchange relaxation, through local exchange of angular momentum between sublattices, drives magnetic switching. Previous computational works using ASD methods have investigated how switching depends on a variety of parameters, such as element-specific damping,^20,21 rare-earth concentration,¹¹ duration of the laser pulse,^20,28 or the role of the initial temperature.²⁹ 

Amorphous transition metal rare-earth alloys have been synthesized before by various techniques.^30–32 For the same GdFeCo concentration, it was found that melt-spun technique³¹ GdFe alloys show a 100 K higher Curie temperature than those using thermal evaporation.³⁰ The reason might be the different local lattice structure. One can also find strong modifications of the local structure in monolayers. The impact of the number of exchange-coupled neighbors on the switching behaviors in GdFeCo has remained, however, unexplored. In this work, we provide insights about how the single-pulse magnetic switching of the ferrimagnetic alloy GdFeCo depends on the lattice structure, in terms of number of exchange-coupled spins. We use both a semi-phenomenological theory and atomistic spin dynamics simulations to demonstrate that by reducing the number of exchange-coupled spins, the laser energy necessary to switch the magnetic state of GdFeCo reduces significantly.

Experimental observations combined with ASD simulations suggest that in rare earth transition metal alloys, the damping values are element specific.^20,21 Specifically, it was found that using $λ Fe = 0.06$ and $λ Gd = 0.01$ in the ASD simulations, one can qualitatively reproduce the ultrafast magnetization dynamics and switching of the GdFeCo alloys in a range of Gd concentrations.²⁰ Similarly, in Gd_22−xTb_xCo₇₈ alloys, it was found element-specific damping values (⁠ $λ Co = λ Tb = 0.05$ and $λ Gd = 0.005 − 0.05$⁠) could describe switching as a function of Tb content.²¹ For the sake of simplicity, we illustrate the impact of element-specific damping on the relation between relaxation parameters for a particular case, z = 6. Figure 1(b) shows $α ex = α$ in solid lines, which corresponds to Fig. 1(a), and $α a / α b = 5$⁠, which agree to experimental observations. For element-specific damping values, each element (sublattice) will enter the exchange dominated regime under different conditions, namely, sublattice a when $α ex = α a$ and sublattice b when $α ex = α b$⁠. Our model predicts that under those circumstances the magnetization relaxation of the Gd sublattice will be quickly dominated by the exchange relaxation, i.e., by transfer of angular momentum to the Fe sublattice, whereas Fe sublattice angular momentum relaxation remains mostly relativistic—transfer of angular momentum to other degrees of freedom.

It is worth noting that the magnetization relaxation dynamics described by Eq. (1) not only scales with the value of the relaxation parameters but also with the non-equilibrium effective field H_a [Eq. (4)]. In particular, the relaxation of the sublattice magnetization Eq. (1) can be split into two contributions, the relativistic relaxation rate: $Γ a r = α a μ a H a$⁠, and exchange relaxation rate $Γ ex = α ex ( μ a H a − μ b H b )$⁠. After the application of a laser pulse, the temperature quickly increases beyond the critical temperature and the dynamics is dominated by the thermal fields, which translates into $Γ a r ∼ λ a k B T m a$ whereas $Γ a ex ∼ λ a k B T / z$⁠. Thus, a prerequisite for switching is that the laser pulse produces a high temperature profile (see Fig. 2, top) to quickly reduce the magnetization m_a so that the exchange relaxation takes over, $Γ ex > Γ a r$⁠. One would expect a more efficient switching process for lower values of z and a highly non-efficient for high values of z. One can rationalize this by analyzing the equations of motion for some limiting situations. For the same value of the intrinsic damping parameter $λ Fe = λ Gd$⁠, by assuming that one of the sublattice demagnetizes faster (Fe in GdFe alloys) than the other, soon after the application of a fs laser pulse one finds that $m a ≪ m b$⁠, from the condition $Γ ex > Γ a r$⁠, one gets $m a < m b / 2 z$⁠, and from $Γ ex > Γ b r$ one gets $m a ≤ 1 / 2 z$⁠.²⁷ For example, for a lattice with fcc + bcc structure, z = 20, $m a < 1 / 40 = 0.15$⁠, while for a two-dimensional magnet with a square lattice structure, z = 4, $m a < 1 / 8 = 0.35$⁠, or a spin chain, z = 2, $m a < 1 / 4 = 0.5$⁠.

This finding is illustrated in Fig. 2, bottom, where the magnetization dynamics of each sublattice is calculated using Eq. (1) coupled to the so-called two-temperature model (TTM), which describes the dynamics of the electron and phonon temperature.^35,36 For all z values, we use the same parameters for the TTM, and thus, the electron and phonon temperatures dynamics are the same (Fig. 2, top). In the TTM, we use the following values for the electron specific heat $c e = γ e T e$ (⁠ $γ e = 700$ J/K² m³), phonon specific heat $c ph = 3 × 10 6$ J/K m³ and electron–phonon coupling $g e – ph = 6 × 10 17$ J/s K m³.²⁷ The heat-bath to which the spins are coupled is represented by the electron system. Figure 2, bottom, shows the magnetization dynamics of the Fe sublattice for different number of neighbors z. We set the exchange parameters in such a way that the Curie temperature of all systems is the same. For the same energy input from the laser pulse, the system with less exchange-coupled spins z = 4 switches faster than for example z = 6. For larger number of z, switching does not occur since the magnetization threshold for switching is not achieved. In the limit $z → ∞$⁠, the exchange relaxation parameter is null, and consequently switching becomes impossible.

Within the MFA used so far, the temperature-dependence of the equilibrium magnetization is independent of the number of neighbors. Differently, a model based on atomistic spins correctly accounts for the differences in the spin correlations for the different lattice structures. In the one hand, for the same values of the exchange parameters, due to the higher number of exchange-coupled spins, the critical temperature, T_c, will also be higher. For example, in a simple ferromagnet with z n.n. and exchange coupling J, one finds that in the MFA, $k B T c = z J / 3$⁠, while in ASD simulations $ε k B T c = z J / 3$⁠, where $ε < 1$ and accounts for the spin fluctuations neglected in the MFA.³³ For ferrimagnets, the same arguments apply, and more n.n. leads to a higher T_c and the shape of the temperature dependent equilibrium magnetization is sensitive to the lattice dimension and structure and to the form of spin interactions.³⁷ For this reason, we rely on atomistic spin dynamics simulations to conduct more accurate comparison between different lattice structures.

Figure 3(a) shows the temperature-dependent total equilibrium magnetization $M tot = M Gd − M Fe$⁠, where $M a = q μ a m a$ (q_a concentration of element a) for the three cases studied here. In the three cases, the so-called magnetization compensation temperature T_M, the temperature at which $M tot = 0$⁠, slightly depends on the lattice structure, in comparison with the MFA for which all three cases behave the same. We use ASD computer simulations to investigate the minimum laser energy $P sw$ necessary to switch the magnetic state in GdFeCo. Since the critical role of T_M in the switching behavior of GdFeCo has been discussed in the literature before, we determine $P sw$ as a function of the initial temperature. We find three scenarios depending on the final magnetization of the sublattices: (i) switching is achieved when the magnetization of the sublattices have changed sign and are larger than 0.1 in length 30 ps after the laser pulse is gone, (ii) no-switching, the sign of the sublattice magnetization remains the same and larger than 0.1, and (iii) thermal demagnetization, otherwise, namely, the magnetization remains lower than 0.1 for large time scales. The colored area in Fig. 3(b) represents the laser energy that induces switching. For a particular system with z exchange-coupled neighbors, the minimum energy necessary to switch decreases almost linearly as the initial temperature increases. For z = 8, $P sw ( z = 8 )$ is larger than $P sw ( z = 6 )$ but proportional to it. By looking at Eq. (5), one sees that the exchange relaxation rate scales as z, that is, $α ex ( z 1 ) / α ex ( z 2 )$⁠. We find that the ration $P sw ( z 1 ) / P sw ( z 2 ) ∼ z 1 / z 2$ describes well this proportionality. For initial temperature above around 400 K, the system is close to the critical temperature T_c, and the final temperature after the laser pulse goes beyond T_c. Therefore, the magnetic system ends up into a thermal demagnetization state. For even larger number of neighbors, $z ≫ 20$⁠, the laser energy would be too large so that the system will end up also into a thermal demagnetization state.

To summarize, we have demonstrated that for ferrimagnetic GdFeCo alloys, the single-pulse magnetic switching is sensitive to the lattice dimension and structure and to the form of spin interactions. We have used a semi-phenomenological theory for a detailed discussion of the dependence of the relaxation terms—exchange and relativistic—on the number of exchange-coupled spins. We have validated these insights by computational simulations of the switching process using an atomistic spin model for GdFeCo. For example, for the limiting case of an infinity number of exchange-coupled spin, we predict that switching is not possible in ferrimagnets. At the same time, by reducing the number of exchange-coupled spins, the laser energy necessary to switch the magnetic state of GdFeCo reduces significantly.

U.A. gratefully acknowledges support by Grant No. PID2021-122980OB-C55 and Grant No. RYC-2020-030605-I funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and “ESF Investing in your future.”