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Transient Exchange Interaction in a Helical Antiferromagnet

We have performed time-resolved resonant x-ray scattering studies in the Lanthanide metal Dy to reveal the dynamic response of the helical order exchange coupling to injection of unpolarized spins. The observed spin dynamics are significantly slower than that exhibited by the ferromagnetic phase in Lanthanide metals and are strongly dependent on temperature and excitation fluence. This unique behavior results from transient changes in the shape of the conduction electron Fermi surface and subsequent scattering events that transfer the excitation to the core spin.

Lanthanide metals exhibit a variety of magnetic phases due to competition between spin-orbit coupling, magneto-elastic effects, and long-range exchange coupling mediated by the indirect RKKY (Ruderman-Kittel-Kasuya-Yoshida) interaction [1]. The nature of the exchange interaction creates a composite spin system comprised of the closely coupled conduction and the core electron spins that account for the majority of the magnetic moment. Helical or conically ordered phases, where the magnetic structure is characterized by a non-zero ordering wavevector, are created by competing symmetric and antisymmetric long-range exchange interactions, where the exchange interactions are determined by the spatial distribution of the conducting electron wave-functions.
Ultrafast demagnetization mechanisms have been previously explored in transition metal and rare earth magnets using all-optical and x-ray dichroism techniques [1][2][3][4][5]. These studies have focused on the angular momentum transfer between core spins, conducting spins, and lattice, and addressed the dynamics of the uniform ferromagnetic phase. Ultrafast optical pump/x-ray probe measurements reveal the coupling mechanisms between the constituent spin systems in the Lanthanide magnets by observing the dynamics of the core spins in response to excitation of the conducting electrons responsible for the exchange interaction. An as yet unexplored aspect of these magnetic systems are the dynamics in the helical phase, where transiently altering the conduction electron distribution can have a profound influence on the long-range magnetic structure.
In this study, we measure the dynamics of the inner shell f-electron spin helix in Dysprosium in response to transient injection of conduction electrons with unpolarized spins. We observe a reduction in the amplitude of the helical order parameter and a shift in the helical wavevector q on disparate time-scales that are strongly dependent on both the pump fluence and sample temperature. Notably, the dynamics in the helical antiferromagnetic phase (HAF) of Dy are significantly slower than those observed in the ferromagnetic (FM) phase of other Lanthanide metals [2,3,5].
We attribute these anomalous dynamics to the relationship between the wave vector of the electronic excitation k and the wave vector of the core magnetic ordering q. In the FM phase, the electronic excitation at k = 0 is closely coupled to the magnetic ordering at q = 0. In the HAF phase, the core spin helix is concomitant with a nesting of the Fermi surface (FS). Initial ultrafast scattering of spins from the k = 0 excitation does not directly change the energy-minimizing configuration of the system, and instead the helical dynamics are driven by changes to the shape of the FS and subsequent changes to the electron distribution through scattering events.
The spin helix was investigated in an epitaxially grown yttrium(Y)/dysprosium(Dy)/yttrium(Y) multilayer film that exhibits a second-order phase transition to a helical antiferromagnetic phase below T N = 180 K [6]. A magnetostriction driven first order phase transition to a ferromagnetic phase occurs at T C = 60 K, with a critical temperature reduced from the bulk T C by the strain induced by the underlying yttrium layer. Between T N and T C , the pitch of the spin helix, θ = qa (where a is the lattice constant), changes continuously with temperature, from 46 degrees (q=2.24 nm −1 ) at T N to 30 degrees (q=1.46 nm −1 ) at T C [8].
The helical order parameter was was probed using resonant x-ray scattering, which provides a direct probe of the core-spin helical ordering through direct optical transitions between atomic core levels and the valence f-orbital [8]. The sample is excited with an optical pump pulse with a photon energy of 1.5 eV and a duration of 100 fs. The 20 nm top Y layer absorbs ∼ 90 % of the pump energy, creating a hot electron distribution in the Y layer and resulting in ultrafast injection of unpolarized spins into the 500 nm Dy film via nonequilibrium diffusion [2,3,9].    with only a moderate increase in q, indicates that the dynamics observed on time-scales faster than ∼ 20 ns are non-thermal, and the helical system at short time scales cannot be described with an effective spin temperature.
The microscopic origin of the HAF dynamics is clarified by considering the relationship between the core spin ordering and the conduction electron FS. In equilibrium, the FS has extended regions in which electrons can scatter from φ (k) to φ (k + q), which combined with the exchange interaction between core and conduction spins, leads to helical ordering of the core spins with wave vector q [10]. The temperature dependence of q results primarily from the dependence of the helical ordering energy gap on thermal fluctuations of the basal plane magnetization, leading to a re-shaping of the FS as the thermal fluctuations are frozen out [11][12][13][14].
Magneto-optical Kerr measurements indicate a sub-ps injection of hot unpolarized electrons from the Y layer [15], consistent with dynamics observed for the same process in other magnetic materials [2,3,9]. The hot electrons create an excitation at k = 0 in the conduction spins, perturbing the magnetic ordering but preserving the FS nesting q. The induced disorder of the conduction spins reduces the number of quasi-elastic k → k + q scattering events where spin angular momentum is conserved, and increases k → k + q scattering involving changes in angular momentum. The core/conduction exchange couples this angular momentum scattering to the core helix, propagating spin disorder to the core spins, but not initially changing the favored helical wave-vector.  The response of the core-spins to the perturbed conduction electron distribution mimics the reduction in the magnetization due to thermal excitations, and the equilibrium wave vector q for the FS nesting and core helix changes due to the dependence of q on the basal plane magnetiza-tion. This process is diagrammed in figure 3. The shift in the nesting vector of the FS creates a mismatch between the FS and the electron distribution. The electron distribution relaxes via inelastic scattering events that couple electrons with excess energy (above E F ermi ) with hole states below E F ermi , with transfer of spin angular momentum occurring through exchange and spin-orbit coupling interactions. This situation is analogous to the breathing Fermi surface model used to describe damping of the precession in ferromagnets, in which the dynamical pointing of the ferromagnetism alters the FS through spin-orbit coupling [16][17][18]. The excitation time-scales observed in the helical system are similar to damping time-scales in other magnetic systems, limited by the scattering rate and efficiency of angular momentum transfer during scattering events. The three temperature model is often invoked to describe the dynamics of magnetic systems in terms of energy transferred between electronic, lattice, and spin degrees of freedom. In a ferromagnet, a single temperature defines the state of a global, uniform magnetization, whereas in the HAF phase of Dy, the magnetic state is defined by both the strength of the order parameter and the helical wavevector. Dynamically, the relation between I and q differs from the I/q relationship in equilibrium, thus simple models based on an effective spin temperature are inadequate for describing this system. Instead, we employ a Gross-Pitaevski (GP) model, wherein the dynamics of I and q are coupled, resulting from changes in the nested FS and subsequent interactions between core and conducting electrons [19,20]. We use a Hamiltonian given by where m is the strength of the order parameter of the HAF, proportional to the experimentally measured I. This Hamiltonian contains the exchange terms J 1 and J 2 , which are the effective nearest and next-nearest neighbor coupling between core spins that stabilize the helical phase. J 2 , appears through θ, which is the equilibrium turn angle of the helix given by cos(θ) = J 1 /4J 2 . The helix is the lowest energy state of this system when J 1 and J 2 have opposite sign.
For a one-dimensional helical magnetic structure with wave-vector q, the above Hamiltonian leads to an effective free-energy given by: . The first two terms, with factors α and β, are the lowest-order terms in the free energy expansion that stabilize the order parameter for T < T N , and a is the lattice constant. Changes to the free energy of the core helix arising from excitation of the conduction electrons are considered through the parameters J 1 and J 2 . Note that θ and q are retained as variables to distinguish the equilibrium helix turn angle from the measured dynamic variable q.
The dynamics can be calculated by using the Hamiltonian to describe an effective action of the helical system parameterized by m and q. From equation 2, we can write the equations of motion as Due to the absence of oscillations in the data, we neglect the second derivative terms in the above equation. Additionally, we have introduced phenomenological damping terms with parameters γ m and γ q to account for relaxation of the system back to equilibrium.
The behavior of m and q with different rheonomic constraints provides insight into the origins of the dynamics of the spin helix. Within the GP model, the dynamics of the parameters J 1,2 emulate the dynamics of the FS, and the damping parameters model the scattering mechanisms that drive the spin-ordering to match the FS. The observed dynamics I and q are described by introducing time-dependent parameters J 1 (t) and J 2 (t) into equations 3 and 4 such that the derived m and q match the data.
We choose J 1 (t) and J 2 (t) to be consistent with our microscopic picture in the following ways.
The uniform (k = 0) excitation of the conduction electrons is modeled as a proportional reduction in both exchange parameters, such that initially the equilibrium q-vector is unchanged. We introduce two recovery time-constants to describe the observed recovery times in the data. We introduce parameters σ and ρ to account for the symmetric and asymmetric recovery amplitudes of the exchange constants. Within this model, the exchange parameters vary according to are treated as partial asymmetric recoveries of either exchange parameter. The black lines in figure   2 show fits to the data using m(t) and q(t) calculated with the GP model.   figure 4 (b).
The initial shift in q occurs through a relatively fast recovery of J 2 through the term ρ 2 . This parameter is shown in figure 4 (c). The shift in q for time delays greater than 20 ns is treated through J T,1 , which is a small remnant reduction in J 1 representing an increased sample temperature. Note that {J T,1 , ρ 2 } ≪ {δJi, σ}; the terms representing anti-symmetric dynamics in J 1 and J 2 that shift the spiral q are much smaller than the symmetric changes in the exchange constants, resulting in overall dynamics of J 1 and J 2 that are nearly symmetric [15]. The modeled time-constants become slower with increasing fluence ( figure 4(b)) and also become slower as T is increased from T C to T N ( figure 5). Both of these trends suggest a link between the magnetic ordering and the recovery time-scales of the spin helix. This behavior is consistent with changes in the shape of the FS, with a corresponding reduction in the effective exchange coupling between core spins, resulting from a reduction of the basal plane magnetization m [11][12][13][14].
The equilibrium helical phase results from the interaction between the core spins and the nested scales similar to those observed in previous ultrafast measurements of rare-earth magnetism [1][2][3][4][5].
A sharp change in amplitude at T C is not observed in the reflectivity measurements.

LATTICE STRAIN
The c-axis lattice constant changes with temperature, corresponding to the changes in the spiral ordering [6]. The lattice constant is reduced by δc/c = 0.003 as the sample is heated through the helical phase, making the observed shifts in the magnetic wavevector q with laser excitation too large to be attributed to a thermal change in the lattice constant. In our analysis of the dynamical data, changes in the lattice constant are considered indirectly through the effective exchange constants.
On long time-scales (> 20 ns ), the changes in the diffraction peak are consistent with an increase in the sample temperature, with little change in the peak intensity at intermediate tem- peratures and an increase in wavevector. The increase in q-vector with fluence is shown in figure   S2 at 105 K. The linear dependence on fluence is expected for a regime in which the specific heat is roughly constant with photo-induced change in temperature with a temperature increase of ∼ 5 K at the highest fluence. At times shorter than ∼ 20 ns, the relatively large decrease in scattering intensity indicates that the dynamics are non-thermal.  Figure S3 shows the temperature dependence of the spin-ordering peak dynamics. The dynamics of the exchange constants become progressively slower at higher temperature (figure 5 of the main text), with little change in the reduction amplitude except near T N , as shown in figure S4. TIME-DEPENDENCE OF J 1 AND J 2 Figure S5 shows the time dependence of J 1 and J 2 for data at 105 K with a pump-fluence of 0.66 mJ/cm 2 . The J 2 exchange parameter partially recovers on a relatively fast time-scale, initiating the shift in q, while a partially unrecovered J 1 is used to describe the change in q due to an increase in temperature of the sample.