We investigate ultrafast spin dynamics due to exchange, electron–phonon and Elliott–Yafet spin-flip scattering in a model with a simple band structure and ferromagnetically coupled electronic sublattices (or more generally, subsystems). We show that this incoherent model of electronic dynamics leads to sublattice magnetization changes in opposite directions after ultrashort-pulse excitation. This prominent feature on an ultrafast timescale is related to a transfer of energy and angular momentum between the subsystems due to exchange scattering. Our calculations illustrate a possible incoherent mechanism that works in addition to the coherent optically induced spin transfer mechanism.

The concept of exchange interactions has been used extensively in the quantum mechanical explanation of magnetic behavior in solids. It plays a prominent role in magnetic solids that can be described by models using localized spins and also in magnetic materials with extended electronic states. While it is strictly speaking not an independent mechanism, but a consequence of the interplay of spin, statistics, and the Coulomb interaction, it can be viewed as setting the intrinsic energy scale of magnetism. Only in the last decades, however, has it been possible to probe the magnetization dynamics on timescales corresponding to this exchange-energy scale via excitation by ultrashort optical excitation and detection by magneto-optical probes.^{1} Electronic dynamics influenced by exchange interactions and exchange scattering processes in a variety of ferromagnetic materials have also been studied extensively.^{2–6} Materials containing *d* and *f* electrons have been of particular interest for magneto-optical applications and basic science, as these allow one to tailor the exchange interaction and, thus, the magnetic behavior via their chemical composition.^{7} Starting with antiferromagnetically coupled two-sublattice systems, such as GdFe(Co), it has been possible to image the dynamics of the sublattices.^{8} The transient ferromagnetic-like state, which is observed after excitation with ultrashort pulses, has been explained by the interplay of inter-sublattice exchange and sublattice demagnetization.^{9–11} Even more recently, it was shown that magnetic materials that are characterized by a ferromagnetic coupling between their different constituents, such as NiFe and FeCo alloys, exhibit a different element-specific magnetization dynamics after ultrashort-pulse excitation, usually with a delay between the two element specific magnetization traces and slow remagnetization dynamics.^{12–15} Based on a theoretical proposal for the coherent optically induced spin transfer (OISTR) mechanism,^{16} it was realized that there is also an element specific signature at very early times after/during the pulse where the sublattice magnetizations change in different directions.^{17–19}

In this paper, we theoretically analyze a different mechanism of spin-dependent electron dynamics that occurs due to the ferromagnetic coupling of electronic subsystems, namely, incoherent electron scattering dynamics. Our model is closely related to the one analyzed for antiferromagnetic coupling in Ref. 11. Figure 1 depicts the electronic states and interaction processes included. We employ a simple band structure that allows us to account for the effect of exchange spin-flip scattering (Fig. 1, bottom) and spin-flips due to spin–orbit coupling (Fig. 1, right). Both effects are included in the Hamiltonian and are, therefore, treated in a consistent manner in the equilibrium properties of the model and its electronic scattering dynamics.

We believe that our model captures important generic properties of the incoherent dynamics of ferromagnetically coupled subsystems, even though the simple band curvatures are rather artificial. Our results suggest that there is an incoherent mechanism due to the exchange interaction that leads to similar fingerprints as, but works independently of the coherent field-induced dynamics, which is behind OISTR. The incoherent approach to be presented in this paper is more in line with studies of incoherent electronic dynamics in (bulk or thin film) elemental ferromagnets.^{20–22}

This paper is organized as follows. We first briefly introduce our model band structure and the dynamical equations for the reduced density matrix that lead to the different scattering terms as sketched in Fig. 1. From the spin-dependent distribution functions, we can obtain any ensemble-averaged single-particle quantity, such as ensemble spin expectation value/magnetization and energy. We then present results for a two-sublattice system with a ferromagnetic coupling and discuss their relation to experiments.

For the electronic structure, we employ a model that combines electrons in different bands, includes the effect of spin–orbit coupling, and contains an exchange interaction between the different electronic subsystems, thus allowing for spin-dependent electron dynamics. The spin–orbit coupling in combination with spin-independent electron–phonon scattering is sometimes called an Elliott–Yafet spin flip.^{21,23,24} The details are as follows. For numerical simplicity, the electronic states live in a two-dimensional *k*-space and the band structure comprises a spin-1/2 subsystem and a spin-1 subsystem, which we will call Sub1 and Sub2, respectively.^{25,26} To be specific, Sub1 has a parabolic dispersion described by $H\u0302kin=\u210f2k22m*$ with an effective mass $m*$ and a two-dimensional electronic wave number *k* and, hence, resembles itinerant bands. The spin-1 subsystem is idealized as localized with a flat dispersion. This band structure is chosen here only for its simplicity, i.e., in order to reflect a difference in the band curvature and the spin in the two subsystems while only relying on few material parameters. It can be replaced by a more complicated and realistic one without changing the mean-field approach and dynamical equations, which we are going to discuss next. Our model bears some resemblance to intra-atomic d–f exchange in Gd, but we do not study a localized system with a 4f splitting of 10 eV.^{27,28} This choice of model band structure necessarily makes phonon coupling much more efficient in Sub1, and we, therefore, include it only there.

Ferromagnetism is included in our approach on two levels, the first is a mean-field Stoner Hamiltonian only for Sub1, $H\u0302Stoner=43Ueff\u27e8S\u03021z\u27e9S\u03021z$, that splits the ↑ and ↓ bands of Sub1. Here, $Ueff$ is an effective Coulomb potential related to the Stoner parameter, which is a material constant and used as a parameter in our system, $S\u03021z$ is the dimensionless spin operator in Sub1 in *z*-direction, $S\u03021z\u2261\sigma \u0302z/2$, where $\sigma \u0302z$ is the third Pauli matrix, and $\u27e8\cdots \u27e9$ denotes the ensemble average. We call $S1\u2261\u27e8S\u03021z\u27e9$ the ensemble spin of Sub1, where we already used that $\u27e8S\u03021x\u27e9=\u27e8S\u03021y\u27e9=0$ since our system is isotropic.

The second magnetic contribution is an exchange Hamiltonian that results in a spin-dependent splitting of the bands in both subsystems, $H\u0302exchange=\u2212J[\u27e8S\u03022z\u27e9S\u03021z+\u27e8S\u03021z\u27e9S\u03022z]=H\u0302exchange(1)+H\u0302exchange(2)$. Here, the exchange constant *J* is used as a parameter and $S\u03022z$ denotes the dimensionless spin operator *S _{z}* of Sub2 and we call $S2\u2261\u27e8S\u03022z\u27e9$ the ensemble spin of Sub2. Since we chose Sub2 with a spin length of 1, the eigenstates of $S\u03022z$ yield three bands that will be labeled $+,0,\u2212$ in the following.

The complete single-particle Hamiltonian for Sub1 and Sub2 is $H\u03021=H\u0302kin+H\u0302Stoner+H\u0302exchange(1)$ and $H\u03022=H\u0302exchange(2)$, respectively. Spin–orbit coupling is not included in the Hamiltonian. It is straightforward to include as in Ref. 25, but to keep the numeric simpler, we will add an effective spin-flip term in Eq. (2) below.

The simple band structures chosen here are particularly useful for the numerical evaluation of the scattering integrals below. In detail, we have for the dispersions of Sub1

at a point **k** in the two-dimensional *k*-space. The bands of Sub1 are labeled by $\nu \u2208{\u2191,\u2193}$. Sub2 has the “band structure” $\epsilon \lambda $ with bands $\lambda \u2208{+,0,\u2212}$, i.e., $\epsilon +/0/\u2212=\u2212JS1/0/JS1$. The corresponding eigenstates for both subsystems are the unit vectors. These explicit expressions show how the band structures of the subsystems are obtained in a mean-field fashion and how they depend on the spin expectation values of Sub1 and Sub2, *S*_{1} and *S*_{2}. For an equilibrium at temperature $Teq$, we determine the splitting and ensemble spins self-consistently with Fermi–Dirac distributions of temperature $Teq$ occupying the states of the mean-field band structure. For the dynamics, we make an adiabatic approximation and assume that the band structure at each time *t* depends on the instantaneous spin expectation values at that time.

In Fig. 1, we show the band structures and Fermi energy $EF$ of Sub1 that result from the self-consistent calculation for $T1=T2=Teq$. The chosen parameters lead to a weak influence of the ferromagnetic contribution ($Ueff<J$) in Sub1 so that the magnetic properties are essentially determined by the exchange coupling. We see that this parameter choice leads to a small splitting of the Sub1 bands, which in turn leads to a partial occupation of the ↑-band, which is shifted up by the mean-field Stoner and exchange contributions. Note that the energy scales of Sub1 and Sub2 are independent, i.e., energy zero in Sub1 is in the middle of the gap at *k *=* *0 and in Sub2 the energy of the 0-band.

The dynamics of electrons in the two-sublattice system are described by occupation numbers $nk\nu $ and $n\lambda $ for bands of Sub1 and Sub2, respectively. Their equation of motion (EOM) has the form

and $\u2202\u2202tn\lambda =\u2202\u2202tn\lambda |exchange$. The first term in Eq. (2) describes spin-conserving electron–phonon (e–pn) scattering, it is given by Eq. (3) in Ref. 25 where also the model parameters for electron–phonon coupling are given.

The effects of spin–orbit coupling which result in an Elliott–Yafet-like electron-spin flip are included in an effective fashion with the last term in Eq. (2). *τ* is a relaxation time constant, and $fk\nu eq(T1)$ is an equilibrium (i.e., Fermi–Dirac) distribution with the temperature of Sub1 $T1(t)$ at time *t*. Hence, this term allows a redistribution of occupation in between the bands of Sub1 (resulting in a change of the ensemble spin *S*_{1}) without additional cooling effects. This approach to the electron–phonon scattering has been compared to calculations, which include spin–orbit coupling explicitly together with a spin-conserving e–pn scattering mechanism.^{26} From our earlier studies,^{25} we conclude that the timescale is mainly set by spin-independent scattering mechanisms, such as electron–phonon (and electron–electron) scattering and that an effective treatment is sufficient for the purpose of this paper. We, thus, fix *τ* = 200 fs.

Finally, the exchange scattering terms in Eq. (2) and below read

respectively, where the exchange matrix element is given by $Wk\u2192\nu ,k\u2192\u2032\nu \u2032\lambda \lambda \u2032=J\u27e8k\u2192,\nu |S\u2192\u03021|k\u2192\u2032,\nu \u2032\u27e9\xb7\u27e8\lambda |S\u2192\u03022|\lambda \u2032\u27e9$. Importantly, these scattering terms have been derived from the same single-particle Hamiltonian and, thus, the exchange constant *J* is the same as in the eigenenergies. The imaginary part terms of the EOMs (3) and (4) reduce to energy conserving delta-distributions in the limit $\gamma \u21920$. In this case, the kinetic energy of the combined system of Sub1 and Sub2 is conserved.

The different scattering terms in Eq. (2) and below are illustrated in Fig. 1. (1) Exchange scattering of Sub1 electrons $\u2191\u2192\u2193$ with Sub2-electrons $+\u21920$; (2) EY spin-flip scattering $\u2193\u2192\u2191$; (3) spin-conserving electron–phonon scattering $\u2193\u2192\u2193$. The gray arrows signify phonon creation, which leads to a transfer of energy of the electron system to the phonon bath at fixed temperature $TEq$. The other processes, such as e–pn scattering with phonon annihilation or exchange scattering with opposite spin-flips, are also included, but become mostly important for the longest timescale where the equilibrium is reached. The exchange scattering leads to opposite spin-flips in both subsystems such that the total spin is conserved. The e–pn scattering primarily cools Sub1, while Sub2 is not coupled directly to a phonon bath.

The main macroscopic quantities that we use to characterize the dynamics are the ensemble spins of the subsystems and the subsystem temperatures. We will sometimes refer to the ensemble spin expectation values as subsystem magnetization, as we will always be using normalized quantities and we do not introduce different magnetic moments for the sublattices. We use the concept of “temperature” not in the strict thermodynamic sense but we determine the time-dependent temperatures *T*_{1} and *T*_{2} simply by fitting Fermi–Dirac distributions to the dynamic distributions in order to have a characterization of the level of electronic excitation. For the determination of *T*_{1}, we assume only a single chemical potential for ↑ and ↓ bands in Sub1.^{29}

The rest of the paper is devoted to a presentation of the results of our calculations for the sublattice magnetization dynamics in response to an excitation that we idealize as an instantaneous increase in energy starting from the equilibrium configuration shown in Fig. 1. As excitation, we have in mind an ultrashort optical pulse that is predominantly absorbed via interband dipole transitions in Sub1. Such a process by itself does not lead to magnetization change.^{21} We approximately describe the effects of this excitation in our simple two-band model by instantaneously changing only the electronic distributions in Sub1 to “hot” Fermi–Dirac electron distributions that have the same *S*_{1} (i.e., leaving the magnetization in Sub1 unchanged). The excited distributions are determined by assuming an increase in electronic (kinetic) energy $T1\u2192Te\u226bTeq$, where $Te$ is a parameter that measures the excitation strength. The chemical potentials $\mu C\u2191/\u2193$ are adjusted such that *S*_{1} is unchanged, i.e., the density in each band must not change, and that the total density is conserved. The excited distributions with the shifted chemical potentials are shown as the $t=0\u2009ps$ distributions in Fig. 3(a).

Figure 2(a) shows the normalized sublattice ensemble spin, i.e., essentially the relative change of the magnetization, and Fig. 2(b) the respective temperatures for Sub1 and Sub2 on all relevant time scales after the instantaneous excitation process. On an ultrashort timescale of some femtoseconds, Figs. 2(a) and 2(b) show that the magnetization of Sub1 increases rapidly while its temperature decreases; the opposite happens for Sub2 where the average spin decreases and the subsystem heats up. Between *t *=* *10 and 20 fs, *S*_{1} reaches a maximum before the system exhibits the dynamics shown on the right side and in the inset of Fig. 2, which will be discussed below. The microscopic picture of the behavior on the ultrashort timescale, which is the intrinsic timescale of the exchange interaction, is contained in Fig. 3 as snapshots of the distribution functions with the same color/symbol code as Fig. 1: Fig. 3(a) shows the distributions in Sub1 and Sub2 after the instantaneous excitation process in Sub1, which can be compared to the equilibrium distributions reached again for long times in subfigure (d). The distributions of ↑ and ↓ bands in Sub1 in Fig. 3(a) are obviously “hotter” due to the excitation, which corresponds to the elevated temperature at *t *=* *0 in Fig. 2(b), while Sub2 remains unexcited with the temperature of the lattice. During the next few femtoseconds leading up to Fig. 3(b), electrons transition in Sub1 from initial states in the ↑(minority)-band to final states in the ↓(majority)-band via exchange scattering processes in which the electrons in Sub2 scatter from initial “−” band states to final 0-band states. In Sub1, these transitions are possible for all electrons in the ↑ band and can reach electronic final states in the ↓ band that have a small energy difference ($\epsilon 0\u2212\epsilon \u2212$) to the initial states. However, states close to the bottom of the ↓ band cannot be reached by these transitions so that the electronic distribution in that energy range is left unchanged and a kink develops in the ↓ distribution in Fig. 3(b). There also appears a kink in the ↑-band as a result of the band structure being adjusted to the instantaneous ensemble spins according to Eq. (1) and below. The details of the evolution of the distribution functions and development of these kinks are best illustrated in the video file found in the supplementary material: During the increase in *S*_{1}, the splitting in Sub2 increases, while it decreases in Sub1 accordingly. Due to this effect, states at the lowest part of the ↑ band can scatter into ↓ states as they become available due to the band dynamics, leading to the kink in the ↑ distribution functions.

A sublattice magnetization that increases compared to its equilibrium value on this ultrashort timescale may seem counterintuitive. However, this behavior is inherently included in the exchange interaction due to the energy and angular momentum conservation and, therefore, also present in the exchange scattering processes. The exchange scattering, which is in effect a Coulomb scattering, “wants to transfer” energy to Sub2 where an energy increase necessarily leads to an increase in angular momentum. This angular momentum is supplied in the exchange scattering by the transition in Sub1 and, thus, decreases the spin angular momentum there. Figures 2 and 3 provide a valuable picture of how the exchange scattering causes this transfer of energy and angular momentum at the microscopic level of electronic distributions. We want to point out again that during the exchange-scattering dominated phase, the spin splittings in Sub1 and Sub2, which are dynamically calculated according to Eq. (1) and below, decrease and increase, respectively. Last, but not least, Fig. 2(a) demonstrates that on these timescales, the total spin is approximately conserved as the SOC induced change in the total ensemble Spin $S1+S2$ plays no role yet.

The increase in one sublattice magnetization on the timescale of the exchange scattering after what is essentially an ultrafast heating process that we find in our simple model is a robust result directly related to a conservation law and should, therefore, occur in other systems with ferromagnetically coupled itinerant electron bands. Conducive to this effect are materials in which electrons can be optically excited in energy regions that are connected to a high density of empty majority electron states via exchange scattering. For such band structure conditions, exchange scattering serves as a possible incoherent mechanism that may show some similar characteristics as OISTR, but takes place independently of the coherent nonlinear OISTR mechanism (which is not included in our calculation).

We now move on to the dynamics on time scales that are longer than the one dominated by exchange scattering. We start with the right side of Fig. 2, which covers the dynamics up to 4 ps. Figure 2(a) shows that both normalized sublattice magnetizations and the normalized total ensemble spin $S1+S2$ decrease to around 60% of their equilibrium values. Figure 2(b) shows that when the sublattice magnetizations reach their minimum, the temperature *T*_{1} has dropped by about 1700 K and becomes essentially equal to *T*_{2} but still stays larger than the lattice temperature. On this timescale, all three processes sketched in Fig. 1 work together. The exchange scattering between the sublattices continues to create nonequilibrium majority electrons in Sub1. Now, the Elliott–Yafet like process drives the nonequilibrium ensemble spin in Sub1 toward a quasi-equilibrium magnetization that corresponds to the instantaneous temperature $T1(t)$ of Sub1. In addition, e–pn scattering between the bands in Sub1 transfers energy from Sub1 to the lattice as long as the electrons in Sub1 are “hotter” than the lattice. The resulting dynamics is essentially an Elliott–Yafet like demagnetization of Sub1 with the added effect of an exchange coupling to Sub2, as has been thoroughly investigated see, e.g., Refs. 20–22 and 25. Note that the exchange-dominated dynamics at very short times, during which *S*_{1} and *S*_{2} move in different directions, is followed by typical demagnetization-remagnetization dynamics with the same minimal values of the relative sublattice magnetizations, but these curves are offset by a delay of 0.8 ps.

In order to discuss the nature of the remaining nonequilibrium in the two subsystems for this timescale we refer to Fig. 3(c). The snapshot of the occupation numbers at *t *=* *1.8 ps is taken at the minimum of the ensemble spin *S*_{1}, hence with minimal spin splitting in Sub2. Sub1 has reached a quasi-equilibrium at $T1\u2243300$ K with chemical potentials for ↑ and ↓ bands that are almost equal. The temperature *T*_{2} of Sub2 is roughly the same, but Sub2 is only indirectly coupled to the phonon bath via exchange scattering with Sub1, so that the cooling of Sub2 is mediated by this indirect coupling to the phonon bath via the directly cooled Sub1. As mentioned before, we included only a coupling to the phonon bath via Sub1, because it is much more efficient as a coupling to Sub2 with its relatively large spin splittings, where a phonon-induced spin flip would cost too much energy.

The behavior of the system on the longest timescale of some 100 ps is shown in the insets of Fig. 2. On this timescale, spin and temperature return to their respective equilibrium values in both subsystems. The delay of 0.8 ps persists even on these very long times. The steady state of the distributions is shown in Fig. 3(d). The distributions are identical to those obtained for the equilibrium state at *T *=* *70 K. A more detailed picture of the distribution function dynamics and the band structure shifts can be obtained by watching the video linked in Fig. S3 in the supplementary material.

We now examine how the results on the magnetic dynamics discussed so far depend on some of the parameters that define our model. In Fig. 4, we show the equivalent of Fig. 2 for a stronger ferromagnetic ordering tendency within Sub1, i.e., a Stoner parameter $Ueff=420\u2009meV$ instead of $Ueff=150\u2009meV$. This leads to a different band structure, see the supplementary material for more details and, hence, to different electron dynamics, which will be discussed in the following.

On the ultrashort timescale shown on the left side of Fig. 4, the exchange scattering dominates also in the case of a more rigid ferromagnetism in Sub1, albeit with a smaller magnitude of the change in the normalized ensemble spins *S*_{1} and *S*_{2} on this timescale of roughly 2% and 1%, respectively. The qualitative results discussed above still hold: One subsystem shows a decrease in magnetization while the other one shows an increase while energy is transferred from the “hotter” to the “cooler” system. The demagnetization and remagnetization dynamics, cf. the right side of Fig. 4(a), qualitatively differ from the ones shown in Fig. 2(a) for both subsystems. Instead of simple demagnetization curves with a delay between *S*_{1} and *S*_{2}, the normalized ensemble spins of Sub1 and Sub2 reach different values at their respective minima and show a larger difference between the normalized *S*_{1} and *S*_{2} values during remagnetization. This occurs at intermediate and very long times, as the inset of Fig. 4 shows. The reason for the different dynamics is the larger Stoner parameter, which to some extent competes with the exchange interaction and leads to a stronger asymmetry between the magnetic behavior of the sublattices. It results in a different behavior of the sublattice bands in both the equilibrium and the instantaneous band structure, as explained in more detail in the supplementary material. For the parameters explored in the present paper, our model predicts a relatively robust signature of the exchange scattering on very short timescales that is essentially related to the effective exchange coupling between the different sublattices. On longer timescales, however, the details of the band structure and the different coupling mechanisms become more important.

In conclusion, we investigated the consequences of incoherent exchange scattering in the band picture of a ferromagnetic two-subsystem model for ultrafast spin dynamics. The model includes basic features of different sublattice band structures, i.e., different sublattice spin lengths and dispersions, while keeping the subsystems as simple as possible by making one of them itinerant and one localized. The electronic dynamics are described by distribution functions so that exchange and electron–phonon scattering can be included at the level of Boltzmann scattering integrals, as well as a simplified treatment of an Elliott–Yafet like spin relaxation process. The behavior of the dynamic sublattice magnetizations after impulsive excitation is dominated on ultrashort timescales by the exchange scattering between the sublattices, which increases one sublattice magnetization (compared to equilibrium) at the expense of the other. For ferromagnetic coupling, the transfer of angular momentum follows the flow of energy between the subsystems. On longer timescales, an imprint of the early-time dynamics remains in the offset between the two sublattice magnetization curves. These results show the similar trends as recent measurements on ferromagnetically coupled alloys, and we believe that they establish an incoherent mechanism that is complementary to the coherent OISTR mechanism.

See the supplementary material that consists of a pdf with details on the band structures and temperature dependent magnetization that results from the self-consistent mean-field calculation and a “movie” that shows the complete time dependence of all the quantities, which are shown as snapshots in Figs. 2 and 3.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—TRR 173/2—268565370 Spin+X (Project No. A08).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.