Speed limits of the laser-induced phase transition in FeRh Open Access

First-order phase transitions are characterized by an abrupt change of structural, electronic, or/and magnetic properties and a co-existence of multiple phases that introduces nucleation and domain growth to the kinetics of the phase transition.^1–10 

The abrupt change of properties accompanying the emerging phase as a consequence of a fine interplay of spin, charge, orbital, and lattice degrees of freedom^6,11 predestine materials featuring first-order phase transitions for ultrafast laser control of functionalities.¹² In this context, the first-order antiferromagnetic to ferromagnetic (AFM-FM) phase transition of FeRh at 370K attracted considerable attention in terms of ultrafast generation of ferromagnetic order^13,14 that extends the more extensively studied ultrafast demagnetization^15,16 and magnetization reversal.^17–19 

The metamagnetic phase transition in FeRh is parameterized by the expansion of the unit cell,²⁰ the change in the electronic band structure,^21,22 and the arising magnetization,^23,24 which each serve as order parameters for different aspects of the ultrafast laser-induced phase transition. Time-resolved photoelectron spectroscopy experiments reveal the formation of an electronic signature of the ferromagnetic state by a photo-induced change of the band structure on a sub-picosecond timescale.²⁵ X-ray magnetic circular dichroism,^26,27 magneto-optical Kerr effect (MOKE),²⁸ and double-pulse THz emission spectroscopy²⁹ report a subsequent formation of an in-plane magnetization on a 100 ps timescale. While the rise of a macroscopic magnetization is dominated by the slow coalescence and alignment of the nucleated domains in an external magnetic field,^14,29 the large FM lattice constant as a structural order parameter is independent of the orientation of the arising magnetization.²⁸ Hence, ultrafast x-ray diffraction (UXRD) directly accesses the nucleation and growth of FM domains, which also determines the rise of the laser-induced magnetization within the first picoseconds.^28,29 Previous UXRD studies report nucleation timescales ranging from 15 to 90 ps, depending on the probing depth and fluence.^28,30 Thus, the kinetics of the nucleation and growth of the laser-induced FM phase remains unclear and controversial.

Here, we use UXRD experiments on a homogeneously optically excited 12 nm thin FeRh film to identify an intrinsic fluence-, temperature-, and field-independent 8 ps rise time of the structural order parameter. This nucleation of the FM phase is not limited by the time that it takes for strain waves to traverse the film at the speed of sound, which requires only 2.5 ps. For the inhomogeneously excited 44 nm thick film, we observe the same intrinsic 8 ps nucleation timescale and an additional delayed slow rise for high fluences when the deposited energy is sufficient to heat the lower lying parts beyond the transition temperature T_T by heat diffusion. This unlocks a growth of the FM phase into the depth of the film. Modeling the UXRD results shows that this growth driven by near-equilibrium heat transport is considerably slower than the 8 ps nucleation timescale after heating above T_T, indicating the crucial role of optically induced non-thermal states for the kinetics of the phase transition. The heat transport timescale is cross-checked by a buried tungsten detection layer, which measures the energy transmitted through FeRh. To complement the insights from the structural order parameter, we probe the subsequent formation of a macroscopic out-of-plane magnetization within 180 ps by using polar tr-MOKE. When we excite the thick FeRh film from the backside, we find a considerably slower and fluence-dependent rise compared to frontside excitation, which experimentally verifies the slow out-of-plane growth of the FM phase.

The two samples are shown in Figs. 1(a) and 1(b) and consist of a 12.6 nm thick FeRh film on MgO(001) and a 43.8 nm thick FeRh film embedded in a metallic heterostructure on MgO(001) consisting of a 2 nm Pt capping and an 8 nm W buffer layer. The thickness of the layers has been characterized via x-ray reflectivity (XRR), and the FeRh films were grown using magnetron sputtering from an equiatomic FeRh target.²⁰ The W layer is purposely grown beneath the thick FeRh as an optically unexcited detection layer for the coherently excited strain pulses. Its strain response accesses the initial stress profile and calibrates the energy transport within the FeRh layer on longer timescales.³¹ Both the FeRh films exhibit the first-order phase transition characterized by the temperature-dependent magnetization (solid lines) and average out-of-plane lattice constant d (symbols) shown in Figs. 1(c) and 1(d). They were measured in thermal equilibrium via vibrating sample magnetometry (VSM) using a QuantumDesign VersaLab magnetometer and XRD performed at the KMC-3 XPP endstation at BESSY II ,³² respectively.

The thin film exhibits a reduced mean transition temperature of T_T = 365 K in comparison to the thick film (375 K) and a residual FM phase fraction of around 20% originating from interface effects.^33–35 This reduces the relative expansion associated with the AFM-FM phase transition from 0.6% in the thick film to 0.48% in the thin film. While the magnetization and lattice constant as order parameters of the temperature-induced phase transition nicely agree for the thick film, the inhomogeneity of the thin film results in a narrower hysteresis for the locally probed lattice constant in contrast to the global magnetization determined by VSM.

In the combined UXRD and tr-MOKE experiment, the FeRh layers are excited by p-polarized pump pulses with a central wavelength of 800 nm and 100 fs pulse duration that are incident under 50° with respect to the sample surface. Utilizing UXRD, we probe the transient out-of-plane strain response of the FeRh layers via reciprocal space mapping³⁶ of the FeRh(002) Bragg peak at a table-top laser-driven plasma x-ray source³⁷ providing 200 fs hard x-ray pulses with a photon energy of $≈8keV$⁠. The Bragg peak position in reciprocal space is given by the average out-of-plane lattice constant d of the FeRh films via q_z = 4π/d. Therefore, the laser-induced peak shift accesses its change Δd determining the lattice strain η_FeRh = Δd/d₀ as the relative change with respect to its value d₀ before excitation.

Figure 2 shows the laser-induced strain response of both the FeRh films and the buried W layer to a weak sub-threshold excitation, which is not able to drive the AFM-FM phase transition.^27,28 Thus, the strain response is the superposition of only two contributions: a quasi-static expansion due to heating and coherently driven propagating strain pulses (partially) reflected at the surface and interfaces. In the nearly homogeneously excited 12 nm FeRh film, the laser excitation launches a strain pulse that is reflected at the surface and partially transmitted onto the substrate. This results in a decaying oscillation with a period of 2L_FeRh/v_s determined by the layer thickness L_FeRh and the sound velocity v_s³¹ that is superimposed with a decreasing quasi-static expansion due to heat transport into the substrate [see Fig. 2(a)]. For the thick film sample, the bipolar strain pulse launched at the optically excited surface is partially reflected at the FeRh–W interface, which leads to a more complex shape of the oscillations in the FeRh strain response [see Fig. 2(b)]. The initial compression of the W layer is caused by the leading part of the bipolar strain pulse generated in FeRh and, hence essentially senses the energy profile.³¹ The W strain turns positive when the expansive part of the strain pulse enters and the compressive part again exits the layer. The slowly rising expansion of W accesses the heat transport from FeRh into W on tens of picoseconds. These thermoelastic strain contributions scale linearly with the deposited energy.³¹ 

In the following, we utilize this calibration of the thermoelastic strain response in the absence of a laser-induced AFM-FM phase transition to extract the transient FM volume fraction V_FM from the strain response to above-threshold excitations that additionally contains a strain signature from the forming FM phase. In order to subtract scaled noise-free sub-threshold data from the above-threshold measurements and in order to access the inhomogeneous spatiotemporal temperature profile within the 44 nm-thick FeRh film, we modeled the strain response by the modular Python library udkm1Dsim³⁸ utilizing literature values for the thermo-elastic properties presented in Table S1. We calculate the absorption profile and solve the one-dimensional heat diffusion equation determining the spatiotemporal temperature that determines the stress on the lattice.³¹ By solving the linear one-dimensional elastic wave equation for this stress, we finally calculate the strain response.³¹ As an approximation, we use a simple one temperature model, which implies that electron-propagation within the electron–phonon coupling time³⁹ is negligible for the driving stress. Since the compression of W in the first 10 ps shown in Fig. 2(b) clearly indicates that the driving electron–phonon stress is still confined to about 13 nm, we consider this approximation useful. The supplementary material S1 describes the details of the strain modeling. Figures 2(c) and 2(d) show the spatiotemporal strain η and temperature increase ΔT for the incident fluence of F_st = 1.6 mJ/cm². Averaging the strain η(z, t) over a respective layer yields the layer-specific strain response measured in our UXRD experiment. We find excellent agreement of the modeled strain response (solid lines) with the measurements in both samples. The excellent agreement with both the 44 nm FeRh and the buried W layer for a single set of parameters pinpoints the modeled spatiotemporal temperature shown in Fig. 2(c). The shape of the initial compression of W quantifies the absorption profile and its slowly rising expansion probes the heat propagated through the inhomogeneously excited FeRh layer.³¹ 

We complement the insights of the UXRD measurements by measuring the transient out-of-plane magnetization by polar MOKE in the very same experimental setup under identical excitation conditions. We applied a maximum out-of-plane magnetic field of 1 T provided by an electromagnet. The transient magnetization is extracted by the difference of the transient MOKE signal for opposite field-polarities. The 100 fs-long p-polarized probe pulse with a central wavelength of 400 nm is focused through the pole of the magnet and is incident under less than 2° with respect to the sample normal.

The transient out-of-plane magnetization shown in Fig. 3(c) verifies this fluence dependence of $VFM*$⁠. Up to the threshold fluence of F_th = 0.6 mJ/cm², we observe no macroscopic magnetization at all and a fluence of 2.1 mJ/cm² fully saturates the laser-induced magnetization. We observe a fluence-independent rise time with the maximum signal at 180 ps for 1 T, i.e., much slower than the nucleation of the FM domains. This slow formation of a macroscopic magnetization by the alignment of the local magnetization of the nucleated domains was reported previously.^27–29 The local magnetization initially lies in the sample plane along four equal directions determined by a cubic anisotropy of around 100 mT that represent the magnetic easy axes due to the shape anisotropy field of 1.38 T for thin FeRh films.^28,43 For an in-plane magnetic field, a macroscopic in-plane magnetization is formed by field-driven coalescence of the FM domains via domain wall motion in agreement with the observed linear increase in the growth rate of the macroscopic magnetization with the field.²⁹ In contrast, for the out-of-plane magnetization, we observe the maximum to be established faster for smaller out-of-plane fields (Fig. S4). This is consistent with precessionally tilting¹⁴ the magnetization of the nucleated domains out of the sample plane. Figures 3(d)–3(h) show the series of events.

These MOKE results upon direct optical excitation of the thin film serve as a reference for the inhomogeneously excited thick film where the redistribution of energy within the layer by heat transport extends the dynamics of the phase transition. We utilize the finite probing depth of MOKE to gain insights into the out-of-plane evolution of the FM phase by comparing the rising magnetization at the sample surface for front- and backside excitation shown in Figs. 4(c) and 4(d). For frontside excitation, we observe the same fluence-independent rise time as for the thin film shown in Fig. 3(c). In contrast, exciting through the substrate leads to a delayed and slower rise. In addition, the rise time strongly depends on the fluence (see Fig. S3 for a comparison normalized to the maximum). The delayed and much slower rise for backside excitation is fluence-dependent, indicating a slow out-of-plane growth of V_FM via near-equilibrium heat transport that brings the upper part of the film above the transition temperature. However, the tr-MOKE measurements cannot quantify whether it is the heat transport or different kinetics of the phase transition in near-equilibrium that yield the increased timescale.

In order to determine the speed at which the FM phase nucleates and grows in the thick film, we again subtract the fluence-scaled strain response to a sub-threshold excitation F_st shown in Fig. 4(a) measured by UXRD from the strain measured above the threshold. The resulting transient FM volume fraction in Fig. 4(b) shows the change from a single exponential rise for low fluences to a two-step rise for high fluences $(>6mJ/cm2)$⁠. For all fluences, V_FM rises according to Eq. (1) with the intrinsic 8 ps timescale already observed in the thin film during the first 30 ps [see the solid colored lines shown in Fig. 4(b)]. The amplitude of the additional contribution in the high fluence regime increases with the fluence. The maximum V_FM is reached at increasing delays up to 250 ps for the highest fluence, where the complete FeRh film is driven across the AFM-FM phase transition. Reaching the full transformation to the FM phase in Fig. 4(b) shows that for high fluences, the temperature at the backside finally exceeds the transition temperature T_T, thus enabling the phase transition in near-equilibrium. This results in the two-step rise of V_FM for high fluences in contrast to a phase transition on the 8 ps timescale limited to the optically excited near-surface region for low fluences, as shown in Figs. 4(e)–4(h). Note that the laterally homogeneous growth of the FM phase into the depth, which follows the inhomogeneous optical excitation, differs from the dynamics observed for homogeneous equilibrium heating,² where columns through the FeRh film are formed after the nucleation of FM domains at both interfaces before the in-plane domain growth.

To cross-check our interpretation, we repeated the UXRD varying the initial sample temperature (cf. Fig. S2). At temperatures only slightly below T_T, the laser excitation is sufficient to drive the phase transition in the not directly excited bottom part of FeRh, which results in a two-step rise of V_FM. At lower temperatures, the heating is insufficient and we observe only a rise on the 8 ps timescale. In this temperature-dependent experiment, we directly accessed V_FM by the transient relative amplitudes of the structural Bragg peaks of the AFM and FM phases that are separated in a reciprocal space due to the excellent collimation of the x-ray beam at the KMC-3 XPP endstation at BESSY II.³² This parameter-free analysis also applied by Mariager and co-workers²⁸ serves as a cross-check of our findings shown in Figs. 3 and 4. The slow growth of the FM phase into the depth unlocked by strong excitations explains the different timescales for different fluences and probing depth reported in previous UXRD experiments on inhomogeneously excited FeRh films.^28,30 The exponential probing profile in grazing incidence geometry²⁸ may have masked the clear two-step behavior observed in our experiments.

We obtain additional insights into the growth of the FM phase within the inhomogeneously excited FeRh layer from the modeled spatiotemporal temperature profile T(z, t) shown in Fig. 2(c). From this analysis, we obtain the gray solid line shown in Fig. 4(b), which denotes the fraction of the film transiently heated above T_T = 375 K for an excitation of 11.9 mJ/cm². We explicitly take into account the required additional energy that is consumed by the latent heat of the phase transition, which would otherwise raise the lattice temperature by about 8 K for the full transition. This analysis shows that already within the first 30 ps a considerably large fraction and, within 80 ps, the complete film is heated above the transition temperature. This is much faster than the observed rise of V_FM within 250 ps and indicates that the growth of the FM phase does not simply follow the heating of the backside above T_T but exhibits intrinsic kinetics. If we assume the FM phase to locally rise with an 8 ps timescale as soon as the local temperature exceeds T_T, the modeled rise of V_FM (gray dashed line) is still significantly faster than in the measurement. Therefore, establishing the FM phase by only near-equilibrium heating must be considerably slower than the nucleation in the optically excited near-surface part of the film. The gray dotted line shown in Fig. 4(b) represents the combination of an 8 ps nucleation timescale for the optically heated unit cells and a second 50 ps timescale for the formation of the FM phase for unit cells heated above T_T by heat transport. This approach matches the delay when $VFM*$ is reached and provides an estimation of the intrinsic timescale related to the phase transition driven by near-equilibrium heating via thermal electrons and phonons. The deviation of our simple model from the experiment between 40 and 80 ps may originate from a more complex heat transport within FeRh due to the large latent heat of the phase transition⁴⁴ and an interplay of the locally rising FM phase upon direct photo-excitation and near-equilibrium heating.

In summary, we discovered an intrinsic fluence-, temperature-, and field-independent 8 ps timescale for locally establishing macroscopic properties of the FM phase in FeRh via the nucleation of domains upon direct optical excitation. This timescale is not limited by the relaxation of the lattice with sound velocity as stated previously^28,41 but represents intrinsic kinetics of the first-order phase transition. Previous photoemission spectroscopy experiments reported a change in the electronic band structure within the first picosecond upon laser excitation.²⁵ In the vicinity of a thick Cu layer serving as a spin bath, Kang and co-workers⁴¹ recently observed a fast rise in the magnetization of FeRh within the first picoseconds in contrast to the previously reported latency in the formation of a macroscopic magnetization.^28,29 This hints that the fast dynamics observed by Kang and co-workers may be related to the spin transport into the thick Cu layer after the ultrafast formation of the intermediate non-equilibrium state of FeRh indicated by the change in the electronic band structure. However, in our experiments, we only probe the subsequent formation of the equilibrium FM phase via the nucleation of domains. In addition to the photo-induced nucleation of FM domains in the near-surface region, we observe a slow growth of the FM phase into the depth of an inhomogenously excited FeRh layer driven by near-equilibrium heat transport in case of high fluences and sample temperatures near the transition temperature. This additional slow contribution enabled by high fluences explains the different rise times of the structural order parameter in previous UXRD experiments^28,30 and clarifies the timescale of the formation of a local equilibrium FM phase. Our modeling reveals that the FM phase rises approximately on a 50 ps timescale after heating above the transition temperature by near-equilibrium heat transport, which is substantially slower than the nucleation upon direct optical excitation in the near-surface region. This hints to the crucial role of modifying the electronic band structure within the first picosecond via photoexcited electrons²⁵ for the kinetics of the subsequent formation of the equilibrium FM phase.

The supplementary material contains a detailed description for modeling the thermoelastic strain response, including all parameters. Additional temperature dependent UXRD data reconfirm conclusions drawn from the fluence dependence shown in Fig. 4. MOKE measurements for varied external magnetic field are presented.

We acknowledge the DFG for financial support via Grant No. BA 2281/11-1 and Project-No. 328545488 – TRR 227, Project No. A10. V. U. acknowledges Project No. CZ.02.01.01/00/22_008/0004594. Access to the CzechNanoLab Research Infrastructure was supported by the MEYS CR (Grant No. LM2023051). Beamtimes at the KMC-3 XPP endstation of the synchrotron radiation facility BESSY II at the Helmholtz Zentrum Berlin were required for thorough sample characterization and parameter-free crosscheck measurements. Open access funding is provided by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 491466077.

The authors have no conflicts to disclose.

M. Mattern: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (equal); Validation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). J. Jarecki: Investigation (supporting); Writing – original draft (supporting). J. A. Arregi: Investigation (equal); Methodology (equal); Resources (equal); Writing – original draft (supporting); Writing – review & editing (supporting). V. Uhlíř: Funding acquisition (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (supporting); Writing – review & editing (supporting). M. Rössle: Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (supporting); Writing – review & editing (supporting). M. Bargheer: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (lead); Project administration (lead); Resources (lead); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

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