# Magnetic phase dependency of the thermal conductivity of FeRh from thermoreflectance experiments and numerical simulations

Aloïs Castellano, Kévin Alhada-Lahbabi, Jon Ander Arregi, Vojtěch Uhlíř,

Bernard Perrin, Catherine Gourdon, Daniele Fournier, Matthieu Jean

Verstraete, L. Thevenard

► To cite this version:

Aloïs Castellano, Kévin Alhada-Lahbabi, Jon Ander Arregi, Vojtěch Uhlíř, Bernard Perrin, et al. Magnetic phase dependency of the thermal conductivity of FeRh from thermoreflectance experiments and numerical simulations. 2024. hal-04531324

HAL Id: hal-04531324

https://hal.science/hal-04531324

Preprint submitted on 3 Apr 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Magnetic phase dependency of the thermal conductivity of FeRh from thermoreflectance experiments and numerical simulations

A. Castellano,¹ K. Alhada-Lahbabi,² J. A. Arregi,³ V. Uhlíř,³,⁴ B. Perrin,²

C. Gourdon,² D. Fournier,² M. J. Verstraete,¹,⁵ and L. Thevenard²

¹NanoMat/Q-Mat Université de Liège, and European Theoretical Spectroscopy Facility, B-4000 Liège, Belgium

²Sorbonne Université, CNRS, Institut des Nanosciences de Paris, 4 place Jussieu, 75252 Paris, France

³CEITEC BUT, Brno University of Technology, Purkyňova 123, 612 00 Brno, Czech Republic

⁴Institute of Physical Engineering, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic

⁵ITP, Physics Department, Utrecht University 3508 TA Utrecht, The Netherlands

(*thevenard@insp.jussieu.fr)

(Dated: April 3, 2024)

FeRh is well known in its bulk form for a temperature-driven antiferromagnetic (AF) to ferromagnetic (FM) transition near room temperature. It has aroused renewed interest in thin film form, with particular focus on its biaxial AF magnetic anisotropy which could serve for data encoding, and the possibility to investigate laser-assisted phase transitions, with varying contributions from electrons, phonons and magnons. In order to estimate the typical temperature increase occurring in these experiments, we performed modulated thermoreflectance microscopy to determine the thermal conductivity $\kappa$ of FeRh. As often occurs upon alloying, and despite the good crystallinity of the layer, $\kappa$ was found to be lower than the thermal conductivities of its constituting elements. More unexpectedly given the electrically more conducting nature of the FM phase, it turned out to be three times lower in the FM phase compared to the AF phase. This trend was confirmed by examining the temporal decay of incoherent phonons generated by a pulsed laser in both phases. To elucidate these results, first and second principles simulations were performed to estimate the phonon, magnon and electron contributions to the thermal conductivity. They were found to be of the same order of magnitude, and to give a quantitative rendering of the experimentally observed $\kappa_{AF}$. In the FM phase however, simulations overestimate the low experimental values, implying very different (shorter) electron and magnon lifetimes.

INTRODUCTION

The FeRh alloy near equiatomic composition is a unique magnetic material, studied long ago in bulk form¹,², and which has recently sparked a renewed interest in thin film form for technological applications. The first-order metamagnetic phase transition of FeRh from the antiferromagnetic (AF) to the ferromagnetic (FM) state happens close to room temperature, and the associated changes in magnetic, electrical and thermodynamical properties have been utilized to propose novel approaches for memory cells³, heat-assisted magnetic recording⁴, and magnetic refrigeration⁵. FeRh thin films are also good candidates to investigate laser-assisted transient⁶⁻¹² or irreversible¹³ phase transitions.

Excitation of the AF phase by a femtosecond laser pulse generates an out-of-equilibrium population of electrons and then phonons, whose temperatures eventually equilibrate, typically on picosecond time scales. The respective roles of electrons and phonons at the onset of a transient FM state are still much debated⁷,⁸,¹⁰,¹¹,¹⁴. After this initial step on the ps timescale the local temperature decreases through heat diffusion. This evolution determines the nucleation, expansion and coalescence of FM domains as well as the decay of the transient magnetic moment upon cooling back to the AF phase¹². However, the heat diffusion coefficient $D$ of FeRh is poorly known. It is related to the thermal conductivity $\kappa$ by $D = \kappa/(\rho C)$ where $\rho$ is the mass density and $C$ the specific heat. Both $\rho$ and $C$ are known to vary with temperature and magnetic phase¹⁵,¹⁶. The heat conductivity also determines the stationary temperature rise of the sample when the pulsed laser repetition rate is too high for the system to relax

back to its base temperature between pulses⁶,⁸,¹⁷⁻¹⁹, or when a CW laser or current is used to induce a local nucleation of FM domains¹³,²⁰. Whereas specific heat measurements are reported for FeRh bulk¹,²,²¹ and thin film samples¹⁶, the thermal conductivity is usually estimated from the electrical conductivity $\sigma$ using the Wiedemann-Franz law $\kappa_e = L_0 T \sigma$, with $L_0$ being the Lorenz number⁷. A noteworthy exception is a recent paper by Ahn et al.¹⁹ in which they evaluate indirectly $\kappa_{AF}=11.8$ and $\kappa_{FM}=22.4 \text{ W m}^{-1} \text{ K}^{-1}$.

There are numerous reports of the material's electronic and phononic band structures, some expliciting calculations of its thermal properties²²⁻²⁷. Since FeRh is metallic, one could expect the electronic contribution to the thermal conductivity $\kappa_e$ to be dominant over the phononic and magnonic ($\kappa_{ph}$, $\kappa_m$) ones, and to govern how $\kappa$ should scale between its two magnetic phases. Relating in first approximation $\kappa_e$ to the electrical conductivity using the Wiedemann-Franz law mentioned above gives a value almost twice larger in the FM phase as in the AF phase²⁸. As for the phononic contribution, recent DFT calculations by Cazorla et al.²⁶ predict it to be very slightly smaller in the AF than the FM phase. Finally, there is no record yet – either experimental or numerical – for the magnonic contribution $\kappa_m$. In pure BCC Fe it has been estimated to represent about 10% of the total thermal conductivity²⁹. Overall, calculations so far suggest the FeRh FM phase to be thermally more conductive than the AF one.

In this paper we determine the total thermal conductivity of a 195 nm FeRh thin film, using modulated thermoreflectance (TR) microscopy. It is an ideal technique to measure the thermal properties of thin films deposited on a substrate, when
Magnetic phase dependency of the thermal conductivity of FeRh from thermoreflectance experiments and numerical simulations

A. Castellano,¹ K. Alhada-Lahbabi,² J. A. Arregi,³ V. Uhlíř,³,⁴ B. Perrin,²

C. Gourdon,² D. Fournier,² M. J. Verstraete,¹,⁵ and L. Thevenard²

¹NanoMat/Q-Mat Université de Liège, and European Theoretical Spectroscopy Facility, B-4000 Liège, Belgium

²Sorbonne Université, CNRS, Institut des Nanosciences de Paris, 4 place Jussieu, 75252 Paris, France

³CEITEC BUT, Brno University of Technology, Purkyňova 123, 612 00 Brno, Czech Republic

⁴Institute of Physical Engineering, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic

⁵ITP, Physics Department, Utrecht University 3508 TA Utrecht, The Netherlands

(*thevenard@insp.jussieu.fr)

(Dated: April 3, 2024)

FeRh is well known in its bulk form for a temperature-driven antiferromagnetic (AF) to ferromagnetic (FM) transition near room temperature. It has aroused renewed interest in thin film form, with particular focus on its biaxial AF magnetic anisotropy which could serve for data encoding, and the possibility to investigate laser-assisted phase transitions, with varying contributions from electrons, phonons and magnons. In order to estimate the typical temperature increase occurring in these experiments, we performed modulated thermoreflectance microscopy to determine the thermal conductivity $κ$ of FeRh. As often occurs upon alloying, and despite the good crystallinity of the layer, $κ$ was found to be lower than the thermal conductivities of its constituting elements. More unexpectedly given the electrically more conducting nature of the FM phase, it turned out to be three times lower in the FM phase compared to the AF phase. This trend was confirmed by examining the temporal decay of incoherent phonons generated by a pulsed laser in both phases. To elucidate these results, first and second principles simulations were performed to estimate the phonon, magnon and electron contributions to the thermal conductivity. They were found to be of the same order of magnitude, and to give a quantitative rendering of the experimentally observed $κ_{AF}$. In the FM phase however, simulations overestimate the low experimental values, implying very different (shorter) electron and magnon lifetimes.

INTRODUCTION

The FeRh alloy near equiatomic composition is a unique magnetic material, studied long ago in bulk form¹,², and which has recently sparked a renewed interest in thin film form for technological applications. The first-order metamagnetic phase transition of FeRh from the antiferromagnetic (AF) to the ferromagnetic (FM) state happens close to room temperature, and the associated changes in magnetic, electrical and thermodynamical properties have been utilized to propose novel approaches for memory cells³, heat-assisted magnetic recording⁴, and magnetic refrigeration⁵. FeRh thin films are also good candidates to investigate laser-assisted transient⁶⁻¹² or irreversible¹³ phase transitions.

Excitation of the AF phase by a femtosecond laser pulse generates an out-of-equilibrium population of electrons and then phonons, whose temperatures eventually equilibrate, typically on picosecond time scales. The respective roles of electrons and phonons at the onset of a transient FM state are still much debated⁷,⁸,¹⁰,¹¹,¹⁴. After this initial step on the ps timescale the local temperature decreases through heat diffusion. This evolution determines the nucleation, expansion and coalescence of FM domains as well as the decay of the transient magnetic moment upon cooling back to the AF phase¹². However, the heat diffusion coefficient *D* of FeRh is poorly known. It is related to the thermal conductivity $κ$ by $D = κ/(ρC)$ where ρ is the mass density and C the specific heat. Both ρ and C are known to vary with temperature and magnetic phase¹⁵,¹⁶. The heat conductivity also determines the stationary temperature rise of the sample when the pulsed laser repetition rate is too high for the system to relax

back to its base temperature between pulses⁶,⁸,¹⁷⁻¹⁹, or when a CW laser or current is used to induce a local nucleation of FM domains¹³,²⁰. Whereas specific heat measurements are reported for FeRh bulk¹,²,²¹ and thin film samples¹⁶, the thermal conductivity is usually estimated from the electrical conductivity $\sigma$ using the Wiedemann-Franz law $\kappa_e = L_0 T \sigma$, with $L_0$ being the Lorenz number⁷. A noteworthy exception is a recent paper by Ahn et al.¹⁹ in which they evaluate indirectly $\kappa_{AF}=11.8$ and $\kappa_{FM}=22.4$ W m⁻¹ K⁻¹.

There are numerous reports of the material's electronic and phononic band structures, some expliciting calculations of its thermal properties²²⁻²⁷. Since FeRh is metallic, one could expect the electronic contribution to the thermal conductivity $\kappa_e$ to be dominant over the phononic and magnonic ($\kappa_{ph}$, $\kappa_m$) ones, and to govern how $\kappa$ should scale between its two magnetic phases. Relating in first approximation $\kappa_e$ to the electrical conductivity using the Wiedemann-Franz law mentioned above gives a value almost twice larger in the FM phase as in the AF phase²⁸. As for the phononic contribution, recent DFT calculations by Cazorla et al.²⁶ predict it to be very slightly smaller in the AF than the FM phase. Finally, there is no record yet – either experimental or numerical – for the magnonic contribution $\kappa_m$. In pure BCC Fe it has been estimated to represent about 10% of the total thermal conductivity²⁹. Overall, calculations so far suggest the FeRh FM phase to be thermally more conductive than the AF one.

In this paper we determine the total thermal conductivity of a 195 nm FeRh thin film, using modulated thermoreflectance (TR) microscopy. It is an ideal technique to measure the thermal properties of thin films deposited on a substrate, when
<figure><img src="image_1.png" /><figcaption>FIG. 1. Determination of the transition temperature of the film. Left: Global measurement of magnetization using vibrating sample magnetometry (dashed line). Right: Local measurement of the DC reflectance (continuous line) under probe beam only ($\lambda$ = 488 nm, 0.8 µm spatial resolution) with a temperature ramp of 2 K/min.</figcaption></figure>

the volume is too small to be probed by standard calorimetry techniques$^{30-33}$. This enables us to estimate quantitatively transient and stationary laser-driven temperature rises. We confirm the results on thermal conductivity by monitoring the time decay of interferometric measurements in a pump-probe setup in both AF and FM phases. In parallel, we compute the electronic, magnetic and vibrational properties of the material, and the different contributions to the thermal conductivity in both phases. We then discuss how the total value compares to our data.

97

## SAMPLE

An FeRh film with a thickness of $h$ = 195 nm was deposited on a 500-µm-thick MgO(001) substrate via magnetron sputtering of an equiatomic FeRh target. The film was grown at 430°C after preheating the substrate for 60 min at the same temperature. We used an Ar pressure of $2.7 \times 10^{-3}$ mbar and a sputtering power of 50 W, leading to a deposition rate of $2 \text{ nm min}^{-1}$. Post-growth annealing in high vacuum at 780°C for 80 min yielded a good quality and homogeneous CsCl-type structure of the film$^{15}$. It is nearly fully relaxed, with a +0.03% strain in the out-of-plane direction at room temperature. Finally, a 2-nm-thick Pt cap was grown after cooling the sample below 120°C, to protect the film from oxidation.

The temperature-dependent magnetization data of the film measured via vibrating sample magnetometry (VSM) (Fig. 1) show a heating transition from AF to FM state between 87 and 108°C (360 and 381 K), and a cooling transition between 100 and 80°C (373 and 353 K). The very weak residual ferromagnetic contribution in the AF phase (4% of the magnetization at 400 K) is a signature of the excellent quality of the sample. The AF-FM transition can also be monitored locally by reflectance microscopy at variable temperature. For this the sample is glued with thermal paste on the holder of a Linkam cryostat adapted to a microscopy set-up. As ex-

pected for FeRh$^{34-36}$, a 3-4% hysteretic change of the phase-dependent reflectance $R(T)$ is observed at the AF-FM transition (Fig. 1). The slight difference in shape and position of the transition compared to the curve measured by VSM reflects the much smaller probed volume (a few µm$^2$ wide, and about 10 nm deep) and the spatial variations of the magnetic properties.

## MODULATED THERMOREFLECTANCE EXPERIMENTS: METHODOLOGY AND RESULTS

We now briefly describe the home-built thermoreflectance microscope that allows to access the thermal properties of the sample.

<figure><img src="image_2.png" /><figcaption>FIG. 2. (a) Schematics of the modulated thermoreflectance set-up. (b) $10 \times 10$ µm$^2$ map of the amplitude of the modulated thermoreflectance signal.</figcaption></figure>

Light from a $\lambda$ = 488 nm (blue) continuous wave (CW) laser (probe beam) is focused onto its surface to a diffraction-limited spot by a $\times$50 long-working-distance objective (numerical aperture NA = 0.5, working distance 10.6 mm). The typical power impinging on the sample is 120 µW. A heat source is provided by a 532 nm (green) CW laser passing through an acousto-optic modulator driven by a square modulation at frequency $f$ and focused onto the sample by the objective ($\approx$5.3 mW on the sample). This green laser spot is raster-scanned on the sample surface while the reflected blue probe beam is collected by the photodiode which monitors the reflectance of the layer. The green light reflected back towards the detector is blocked by an interference filter. The signal is fed into a lock-in amplifier that returns the amplitude and phase of the AC component of the reflectance at the frequency of the modulated heat source, $\Delta R_f$, as well as the DC component $R_0$ thanks to an Analog to Digital Converter (ADC). A typical map of the modulated reflectance is shown in Fig. 2(b) (amplitude component).

As described in detail in Ref. 30, the reflectance is assumed to depend on the temperature increase to the first order $R(\mathbf{r},t)=R_0 + \frac{\partial R_{\text{probe}}}{\partial T} I_{\text{probe}}(\mathbf{r}) * \Delta T(\mathbf{r},t)$, where the temperature profile is convolved with the probe spot profile $I_{\text{probe}}(\mathbf{r})$. The modulated reflectance $\Delta R_f$ at frequency $f$ reflects the $f$-harmonic of the temperature increase $\Delta T_f$ induced by the pump laser. Modulation frequencies are moreover assumed to be low enough to consider the system at thermodynamic equilibrium, with the thermal flux being proportional to the tem-
<figure><img src="image_2.png" /><figcaption>FIG. 1. Determination of the transition temperature of the film. Left: Global measurement of magnetization using vibrating sample magnetometry (dashed line). Right: Local measurement of the DC reflectance (continuous line) under probe beam only ($\lambda$ = 488 nm, 0.8 µm spatial resolution) with a temperature ramp of 2 K/min.</figcaption></figure>

the volume is too small to be probed by standard calorimetry techniques$^{30-33}$. This enables us to estimate quantitatively transient and stationary laser-driven temperature rises. We confirm the results on thermal conductivity by monitoring the time decay of interferometric measurements in a pump-probe setup in both AF and FM phases. In parallel, we compute the electronic, magnetic and vibrational properties of the material, and the different contributions to the thermal conductivity in both phases. We then discuss how the total value compares to our data.

97

## SAMPLE

An FeRh film with a thickness of $h$ = 195 nm was deposited on a 500-µm-thick MgO(001) substrate via magnetron sputtering of an equiatomic FeRh target. The film was grown at 430°C after preheating the substrate for 60 min at the same temperature. We used an Ar pressure of $2.7 \times 10^{-3}$ mbar and a sputtering power of 50 W, leading to a deposition rate of $2 \text{ nm min}^{-1}$. Post-growth annealing in high vacuum at 780°C for 80 min yielded a good quality and homogeneous CsCl-type structure of the film$^{15}$. It is nearly fully relaxed, with a +0.03% strain in the out-of-plane direction at room temperature. Finally, a 2-nm-thick Pt cap was grown after cooling the sample below 120°C, to protect the film from oxidation.

The temperature-dependent magnetization data of the film measured via vibrating sample magnetometry (VSM) (Fig. 1) show a heating transition from AF to FM state between 87 and 108°C (360 and 381 K), and a cooling transition between 100 and 80°C (373 and 353 K). The very weak residual ferromagnetic contribution in the AF phase (4% of the magnetization at 400 K) is a signature of the excellent quality of the sample. The AF-FM transition can also be monitored locally by reflectance microscopy at variable temperature. For this the sample is glued with thermal paste on the holder of a Linkam cryostat adapted to a microscopy set-up. As ex-

pected for FeRh$^{34-36}$, a 3-4% hysteretic change of the phase-dependent reflectance $R(T)$ is observed at the AF-FM transition (Fig. 1). The slight difference in shape and position of the transition compared to the curve measured by VSM reflects the much smaller probed volume (a few µm$^2$ wide, and about 10 nm deep) and the spatial variations of the magnetic properties.

## MODULATED THERMOREFLECTANCE EXPERIMENTS: METHODOLOGY AND RESULTS

We now briefly describe the home-built thermoreflectance microscope that allows to access the thermal properties of the sample.

<figure><img src="image_3.png" /><figcaption>FIG. 2. (a) Schematics of the modulated thermoreflectance set-up. (b) $10 \times 10$ µm$^2$ map of the amplitude of the modulated thermoreflectance signal.</figcaption></figure>

Light from a $\lambda$ = 488 nm (blue) continuous wave (CW) laser (probe beam) is focused onto its surface to a diffraction-limited spot by a $\times$50 long-working-distance objective (numerical aperture NA = 0.5, working distance 10.6 mm). The typical power impinging on the sample is 120 µW. A heat source is provided by a 532 nm (green) CW laser passing through an acousto-optic modulator driven by a square modulation at frequency $f$ and focused onto the sample by the objective ($\approx$5.3 mW on the sample). This green laser spot is raster-scanned on the sample surface while the reflected blue probe beam is collected by the photodiode which monitors the reflectance of the layer. The green light reflected back towards the detector is blocked by an interference filter. The signal is fed into a lock-in amplifier that returns the amplitude and phase of the AC component of the reflectance at the frequency of the modulated heat source, $\Delta R_f$, as well as the DC component $R_0$ thanks to an Analog to Digital Converter (ADC). A typical map of the modulated reflectance is shown in Fig. 2(b) (amplitude component).

As described in detail in Ref. 30, the reflectance is assumed to depend on the temperature increase to the first order $R(\mathbf{r},t)=R_0+\frac{\partial R_{\text{probe}}}{\partial t}I_{\text{probe}}(\mathbf{r}) * \Delta T(\mathbf{r},t)$, where the temperature profile is convolved with the probe spot profile $I_{\text{probe}}(\mathbf{r})$. The modulated reflectance $\Delta R_f$ at frequency $f$ reflects the $f$-harmonic of the temperature increase $\Delta T_f$ induced by the pump laser. Modulation frequencies are moreover assumed to be low enough to consider the system at thermodynamic equilibrium, with the thermal flux being proportional to the tem-
perature gradient³⁰. Adapting Ref. 37 and using cylindrical coordinates, the thermal field at the surface of the sample and

the modulated reflectance are given by inverse Hankel trans- forms as detailed in Eqs. (1-3).

$$ \Delta T_f(r) = \frac{Q_f}{2\pi} \int_0^\infty g(u) J_0(\mu r) u \exp\left(-\frac{d_{\text{pump}}^2 u^2}{32}\right) du, \quad (1) $$

$$ \Delta R_f(r) \propto \frac{\partial R_{\text{probe}}}{\partial T} \int_0^\infty g(u) J_0(\mu r) u A(u) \exp\left(-\frac{d_{\text{pump}}^2 u^2}{32}\right) du, \quad (2) $$

$$ g(u) = \frac{\alpha^2}{\alpha^2 - \Sigma_I^2} \frac{\left(1 - \frac{\kappa_0^* \Sigma_0}{\kappa_1 \alpha}\right) \left(\cosh(\Sigma_1 h) - \exp(-\alpha h)\right) + \left(\frac{\kappa_0^* \Sigma_0}{\kappa_1 \Sigma_I} - \frac{\Sigma_I}{\alpha}\right) \sinh(\Sigma_1 h)}{\kappa_0^* \Sigma_0 \cosh(\Sigma_1 h) + \kappa_1 \Sigma_I \sinh(\Sigma_1 h)}, \quad (3) $$

where $\Sigma_i(u, f) = \sqrt{u^2 + j \frac{2\pi f}{D_i}}$ is akin to a complex thermal wave-vector ($j = \sqrt{-1}$). Indices 0 and 1 refer to the MgO substrate and the FeRh thin film, respectively. $D_i = \kappa_i/(\rho_i C_i)$ is the heat diffusivity with $\rho_i$ and $C_i$ the mass density and specific heat, respectively. $Q_f = \frac{4}{\pi} P_{inc}(1-R)$ is the f-component of the incoming modulated heat power with R the reflection coefficient at the pump wavelength. $J_0$ is the Bessel function of the first kind, and $\alpha$ is the absorption coefficient of the pump beam in FeRh³⁸. The last two terms of Eq. (1) describe the finite diameter of the laser spots: $A(u)$ is the Hankel transform of the Airy pattern of the diffraction-limited probe spot, and $d_{pump}$ is the diameter at which the intensity of the Gaussian pump spot has fallen by $e^2$. Numerical values for known parameters are given in Appendix A.

The temperature increase includes all contributions to the thermal conductivity: phononic, electronic, magnonic, as well as the effect of any thermal interface resistance (TIR) $R_{th}$ between the substrate and the thin film due to a localized scattering of phonons. Taking into account the presence of a TIR amounts to using in Eq. (3) a modified thermal conductivity for the substrate $\kappa_0^* = \kappa_0/(1+R_{th}\kappa_0\Sigma_0)^{39}$. We now discuss possible values to give to $R_{th}$. The TIR is notoriously complicated to measure reliably⁴⁰, particularly in the case of a thermally insulating film/conducting substrate configuration. It may have intrinsic origins such as e.g. Umklapp processes⁴¹, or, more likely at play in our case, extrinsic origins such as interface roughness - expected to be minute in our epitaxial films. Applying the diffuse mismatch model to the FeRh/MgO interface yields $R_{th} \sim 4 \times 10^{-9} \text{ m}^2 \text{ K W}^{-1}$ for both magnetic phases (see below for numerical details on this simulation). This method is known to give a lower bound for thermal interface resistance, but matches reassuringly well with experimental values found in the literature for thin metallic films (e.g. Al or Cu) deposited on an electrically insulating substrate (e.g. SiO₂ or Al₂O₃)⁴²,⁴³. It is also quite close to the value $(2 \times 10^{-9} \text{ m}^2 \text{ K W}^{-1})$ estimated indirectly for anti-ferromagnetic FeRh/MgO by Ahn et al.¹⁹.

Modulated thermoreflectance spatial scans measured on FeRh/MgO are typical of a thermal insulator over conducting substrate configuration. In this case, heat diffuses rapidly to the substrate, and then back to the top layer. To illustrate this,

<table><caption>TABLE I. Thermal conductivity extracted from group fits of the log of the amplitude of the modulated reflectivity, imposing $R_{th} = 4 \times 10^{-9} \text{ m}^2 \text{ K W}^{-1}$ as explained in the text.</caption><thead><tr><th>Temperature (°C)</th><th>Phase</th><th>Thermal conductivity (W m<sup>-1</sup> K<sup>-1</sup>)</th></tr></thead><tbody><tr><td>5</td><td>AF</td><td>26 ± 3</td></tr><tr><td>40</td><td>AF</td><td>30 ± 5</td></tr><tr><td>60</td><td>AF</td><td>30 ± 5</td></tr><tr><td>100</td><td>FM</td><td>10 ± 1</td></tr><tr><td>130</td><td>FM</td><td>9 ± 1</td></tr></tbody></table>

we plot in Fig. 3(a) the amplitude of the temperature rise given by Eq. (2) at an example frequency of $f = 250$ kHz, with and without an FeRh layer ($h = 0$ or 195 nm). Far from the center, the behavior of the MgO substrate is recovered, with the slope essentially governed by the frequency-dependent thermal diffusion length $\mu_0 = \sqrt{\frac{D_0}{\pi f}}$. The central temperature rise, on the other hand, roughly scales as $h/\kappa_1 + R_{th}$ above the signal from the substrate (Fig. 3(a)). As a result, it is (i) quite challenging to measure reliably the thermal conductivity of very thin/highly conductive films, and justifies the use of a fairly thick (195 nm) film for this study, and (ii) almost impossible to determine independently $R_{th}$ and $\kappa_1$, as the two terms always appear together⁴⁴. To analyze our data, we thus chose to impose a value for $R_{th}$ (see discussion above). Finally, we verified that the $r = 0$ value of the amplitude scales linearly with the incident power.

In order to evaluate the thermal conductivity of FeRh at a given temperature, we record spatial scans of the amplitude and phase of the modulated thermoreflectance signal at three modulation frequencies (typically 100, 250 and 500 kHz), as shown for instance for $T = 100°C$ (373 K) in Fig. 3(b)-(c). We then perform a global fit of the logarithm of the normalized amplitude of the three curves, with the thermal conductivity of FeRh being the only free (shared) parameter and the pump beam diameter and TIR imposed. Starting from an identical value of $R_{th} = 4 \times 10^{-9} \text{ m}^2 \text{ K W}^{-1}$ in both AF and FM phases, we also explore the possibility of a much larger TIR in the FM phase, and use the quality of the fit to validate the chosen value. Note that no knowledge of $\partial R_{probe}/\partial T$ is necessary
161 perature gradient³⁰. Adapting Ref. 37 and using cylindrical
162 coordinates, the thermal field at the surface of the sample and

163 the modulated reflectance are given by inverse Hankel trans-
164 forms as detailed in Eqs. (1-3).

$$
\Delta T_f (r) = \frac{Q_f}{2\pi} \int_0^\infty g(u) J_0(\mu r) u \exp\left(-\frac{d_{\text{pump}}^2 u^2}{32}\right) du, \quad (1)
$$

$$
\Delta R_f (r) \propto \frac{\partial R_{\text{probe}}}{\partial T} \int_0^\infty g(u) J_0(\mu r) u A(u) \exp\left(-\frac{d_{\text{pump}}^2 u^2}{32}\right) du, \quad (2)
$$

$$
g(u) = \frac{\alpha^2 \left(1 - \frac{\kappa_0^* \Sigma_0}{\kappa_1 \alpha}\right) \left(\cosh(\Sigma_1 h) - \exp(-\alpha h)\right) + \left(\frac{\kappa_0^* \Sigma_0}{\kappa_1 \Sigma_1} - \frac{\Sigma_1}{\alpha}\right) \sinh(\Sigma_1 h)}{\alpha^2 - \Sigma_1^2 \left(\kappa_0^* \Sigma_0 \cosh(\Sigma_1 h) + \kappa_1 \Sigma_1 \sinh(\Sigma_1 h)\right)}, \quad (3)
$$

165 where Σᵢ(u, f) = √(u² + j(2πf/Dᵢ)) is akin to a complex thermal
166 wave-vector (j = √(-1)). Indices 0 and 1 refer to the MgO
167 substrate and the FeRh thin film, respectively. Dᵢ = κᵢ/(ρᵢCᵢ)
168 is the heat diffusivity with ρᵢ and Cᵢ the mass density and spe-
169 cific heat, respectively. Qᵣ = (4/π)Pᵢₘc(1-R) is the f-component
170 of the incoming modulated heat power with R the reflection
171 coefficient at the pump wavelength. J₀ is the Bessel func-
172 tion of the first kind, and α is the absorption coefficient of the
173 pump beam in FeRh³⁸. The last two terms of Eq. (1) describe
174 the finite diameter of the laser spots: A(u) is the Hankel trans-
175 form of the Airy pattern of the diffraction-limited probe spot,
176 and dₚump is the diameter at which the intensity of the Gaus-
177 sian pump spot has fallen by e². Numerical values for known
178 parameters are given in Appendix A.

179 The temperature increase includes *all* contributions to the
180 thermal conductivity: phononic, electronic, magnonic, as well
181 as the effect of any thermal interface resistance (TIR) R<sub>th</sub> be-
182 tween the substrate and the thin film due to a localized scat-
183 tering of phonons. Taking into account the presence of a TIR
184 amounts to using in Eq. (3) a modified thermal conductivity
185 for the substrate κ<sub>0</sub>* = κ<sub>0</sub>/(1 + R<sub>th</sub>κ<sub>0</sub>Σ<sub>0</sub>)<sup>39</sup>. We now discuss
186 possible values to give to R<sub>th</sub>. The TIR is notoriously compli-
187 cated to measure reliably<sup>40</sup>, particularly in the case of a ther-
188 mally insulating film/conducting substrate configuration. It
189 may have *intrinsic* origins such as *e.g.* Umklapp processes<sup>41</sup>,
190 or, more likely at play in our case, *extrinsic* origins such as
191 interface roughness - expected to be minute in our epitaxial
192 films. Applying the diffuse mismatch model to the FeRh/MgO
193 interface yields R<sub>th</sub> ~ 4 × 10<sup>-9</sup> m<sup>2</sup> K W<sup>-1</sup> for both magnetic
194 phases (see below for numerical details on this simulation).
195 This method is known to give a lower bound for thermal in-
196 terface resistance, but matches reassuringly well with exper-
197 imental values found in the literature for thin metallic films
198 (e.g. Al or Cu) deposited on an electrically insulating sub-
199 strate (e.g. SiO<sub>2</sub> or Al<sub>2</sub>O<sub>3</sub>)<sup>42,43</sup>. It is also quite close to
200 the value (2 × 10<sup>-9</sup> m<sup>2</sup> K W<sup>-1</sup>) estimated indirectly for anti-
201 ferromagnetic FeRh/MgO by Ahn *et al.*<sup>19</sup>.

202 Modulated thermoreflectance spatial scans measured on
203 FeRh/MgO are typical of a thermal insulator over conducting
204 substrate configuration. In this case, heat diffuses rapidly to
205 the substrate, and then back to the top layer. To illustrate this,

<table><caption>TABLE I. Thermal conductivity extracted from group fits of the
log of the amplitude of the modulated reflectivity, imposing R<sub>th</sub> =
4×10<sup>-9</sup> m<sup>2</sup>K W<sup>-1</sup> as explained in the text.</caption>
  <thead>
    <tr>
      <th>Temperature (°C)</th>
      <th>Phase</th>
      <th>Thermal conductivity (W m<sup>-1</sup> K<sup>-1</sup>)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>5</td>
      <td>AF</td>
      <td>26 ±3</td>
    </tr>
    <tr>
      <td>40</td>
      <td>AF</td>
      <td>30 ±5</td>
    </tr>
    <tr>
      <td>60</td>
      <td>AF</td>
      <td>30 ±5</td>
    </tr>
    <tr>
      <td>100</td>
      <td>FM</td>
      <td>10 ±1</td>
    </tr>
    <tr>
      <td>130</td>
      <td>FM</td>
      <td>9 ±1</td>
    </tr>
  </tbody>
</table>

206 we plot in Fig. 3(a) the amplitude of the temperature rise given
207 by Eq. (2) at an example frequency of f = 250 kHz, with and
208 without an FeRh layer (h = 0 or 195 nm). Far from the cen-
209 ter, the behavior of the MgO substrate is recovered, with the
210 slope essentially governed by the frequency-dependent ther-
211 mal diffusion length μ₀ = √(D₀/πf). The central temperature rise,
212 on the other hand, roughly scales as h/κ₁ + Rₜh above the signal
213 from the substrate (Fig. 3(a)). As a result, it is (i) quite chal-
214 lenging to measure reliably the thermal conductivity of very
215 thin/highly conductive films, and justifies the use of a fairly
216 thick (195 nm) film for this study, and (ii) almost impossible
217 to determine independently Rₜh and κ₁, as the two terms al-
218 ways appear together⁴⁴. To analyze our data, we thus chose
219 to impose a value for Rₜh (see discussion above). Finally, we
220 verified that the r = 0 value of the amplitude scales linearly
221 with the incident power.

222 In order to evaluate the thermal conductivity of FeRh at a given temperature, we record spatial scans of the amplitude and phase of the modulated thermoreflectance signal at three modulation frequencies (typically 100, 250 and 500 kHz), as shown for instance for T = 100°C (373 K) in Fig. 3(b)-(c). We then perform a global fit of the logarithm of the normalized amplitude of the three curves, with the thermal conductivity of FeRh being the only free (shared) parameter and the pump beam diameter and TIR imposed. Starting from an identical value of R<sub>th</sub> = 4 × 10<sup>-9</sup> m<sup>2</sup> K W<sup>-1</sup> in both AF and FM phases, we also explore the possibility of a much larger TIR in the FM phase, and use the quality of the fit to validate the chosen value. Note that no knowledge of ∂R<sub>probe</sub>/∂T is necessary
<figure><img src="image_3.png" /><figcaption>FIG. 3. (a) Illustration of the effect of the presence of a thin film over a better thermally conducting substrate: logarithm of the temperature rise induced by a 250 kHz modulated laser, with and without the 195-nm-thick FeRh film (Eq. 1). The "nose" contains information on the layer's thermal conductivity and on the thermal interface resistance. (b-c) Experimental spatial scans of amplitude and phase taken at three modulation frequencies in the uniform FM phase ($T = 100°C$, 373 K). The fit of $log|\Delta R|$ by Eq. (1) yields $\kappa_1 = 10 \pm 1$ W m$^{-1}$ K$^{-1}$. The response of the FeRh/MgO system (full line) and MgO substrate only (dashed line) are plotted using $d=2.1$ µm and $R_{th} = 4 \times 10^{-9}$ m$^2$K W$^{-1}$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 4. (a) Amplitude and (b) phase at 100 kHz modulation frequency in uniform FM phase ($T = 100°C$, 373 K). Fitting the data with different thermal interface resistances $R_{th}=0$, 4 or $40 \times 10^{-9}$ m$^2$K W$^{-1}$ gives $\kappa_1 = 9$, 10 or $20$ W m$^{-1}$ K$^{-1}$ (resp. green, blue and red curves). The amplitude and phase are then calculated with Eq. (2). Note that the difference between $R_{th}=0$ and $40 \times 10^{-9}$ m$^2$K W$^{-1}$ curves is minute, and that the agreement with the phase data is poor for the larger TIR.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 5. Amplitude of the modulated reflectivity measured at 500 kHz in the uniform AF (resp. FM) phases at $T=60°C$ in blue symbols (resp. $T=100°C$ in red symbols). Full lines are fit to the data with $\kappa_1=30$ Wm$^{-1}$K$^{-1}$ (AF) and $\kappa_1=10$ Wm$^{-1}$K$^{-1}$ (FM), while the dashed line is the contribution of the MgO substrate.</figcaption></figure>

to extract $\kappa_1$ from this analysis, since it is the relative spatial variations of the *f*-component of the reflectance that are of importance. Finally, complementary MatLab simulations including a third layer in the model confirm that we can neglect the presence of the thin Pt cap in the analysis of the data.

Table I shows the final temperature dependence of the thermal conductivity of the 195-nm-thick FeRh film, established using the group fit procedure described above. The overall trend is distinctly that of a smaller value of $\kappa_1(T)$ in the FM phase compared to the AF phase. More specifically, $\kappa_1$ is 25-30 W m$^{-1}$ K$^{-1}$ in the AF phase, and around 10 W m$^{-1}$ K$^{-1}$ in the FM phase, i.e., almost three times less. This shows up clearly when plotting together spatial scans measured at the highest AF temperature (60°C) and lowest FM temperature (100°C), in Fig. 5. The "nose" of the AF spatial scan is strikingly smaller than in the FM scan. This could either be due to a smaller thermal conductivity in the FM phase, or to a much larger interface resistance. To test the latter hypothesis, we fit the FM data imposing a very large value of $R_{th} = 4 \times 10^{-8}$

$m^2$ K W$^{-1}$ (red curve in Fig. 4). This does yield a higher $\kappa_{1,FM}$ of around 20 W m$^{-1}$ K$^{-1}$, but at the cost of a poorer fit quality, particularly visible on the phase (Fig. 4(b)). Note that this method sets an *upper* boundary to a likely value for $R_{th}$, while DFT simulations give a lower limit.

We can now estimate the heat diffusivity $D_1 = \frac{\kappa_1}{\rho_1 c_1}$ of FeRh in both phases. Since the volume density of FeRh $\rho_1$ varies appreciably with the magnetic phase (AF or FM, see Appendix A for numerical values), $D_1$ is divided by over two upon crossing the transition: from $\sim 8 \times 10^{-6}$ m$^2$ s$^{-1}$ in the AF phase at 5°C, to $\sim 3 \times 10^{-6}$ m$^2$ s$^{-1}$ in the FM phase at 130°C (403 K). Note that the former value is in the range of the diffusivity of $2 \times 10^{-5}$ m$^2$ s$^{-1}$ found by Bergman et al.⁸ by fitting the laser-induced transient reflectivity, late after the
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Illustration of the effect of the presence of a thin film over a better thermally conducting substrate: logarithm of the temperature rise induced by a 250 kHz modulated laser, with and without the 195-nm-thick FeRh film (Eq. 1). The "nose" contains information on the layer's thermal conductivity and on the thermal interface resistance. (b-c) Experimental spatial scans of amplitude and phase taken at three modulation frequencies in the uniform FM phase ($T = 100°C$, 373 K). The fit of $log|\Delta R|$ by Eq. (1) yields $\kappa_1 = 10 \pm 1$ W m$^{-1}$ K$^{-1}$. The response of the FeRh/MgO system (full line) and MgO substrate only (dashed line) are plotted using $d=2.1$ µm and $R_{th} = 4 \times 10^{-9}$ m$^2$K W$^{-1}$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 4. (a) Amplitude and (b) phase at 100 kHz modulation frequency in uniform FM phase ($T = 100°C$, 373 K). Fitting the data with different thermal interface resistances $R_{th}=0$, 4 or $40 \times 10^{-9}$ m$^2$K W$^{-1}$ gives $\kappa_1 = 9$, 10 or $20$ W m$^{-1}$ K$^{-1}$ (resp. green, blue and red curves). The amplitude and phase are then calculated with Eq. (2). Note that the difference between $R_{th}=0$ and $40 \times 10^{-9}$ m$^2$K W$^{-1}$ curves is minute, and that the agreement with the phase data is poor for the larger TIR.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 5. Amplitude of the modulated reflectivity measured at 500 kHz in the uniform AF (resp. FM) phases at $T=60°C$ in blue symbols (resp. $T=100°C$ in red symbols). Full lines are fit to the data with $\kappa_1=30$ Wm$^{-1}$K$^{-1}$ (AF) and $\kappa_1=10$ Wm$^{-1}$K$^{-1}$ (FM), while the dashed line is the contribution of the MgO substrate.</figcaption></figure>

to extract $\kappa_1$ from this analysis, since it is the relative spatial variations of the *f*-component of the reflectance that are of importance. Finally, complementary MatLab simulations including a third layer in the model confirm that we can neglect the presence of the thin Pt cap in the analysis of the data.

Table I shows the final temperature dependence of the thermal conductivity of the 195-nm-thick FeRh film, established using the group fit procedure described above. The overall trend is distinctly that of a smaller value of $\kappa_1(T)$ in the FM phase compared to the AF phase. More specifically, $\kappa_1$ is 25-30 W m$^{-1}$ K$^{-1}$ in the AF phase, and around 10 W m$^{-1}$ K$^{-1}$ in the FM phase, i.e., almost three times less. This shows up clearly when plotting together spatial scans measured at the highest AF temperature (60°C) and lowest FM temperature (100°C), in Fig. 5. The "nose" of the AF spatial scan is strikingly smaller than in the FM scan. This could either be due to a smaller thermal conductivity in the FM phase, or to a much larger interface resistance. To test the latter hypothesis, we fit the FM data imposing a very large value of $R_{th} = 4 \times 10^{-8}$

$m^2$ K W$^{-1}$ (red curve in Fig. 4). This does yield a higher $\kappa_{1,FM}$ of around 20 W m$^{-1}$ K$^{-1}$, but at the cost of a poorer fit quality, particularly visible on the phase (Fig. 4(b)). Note that this method sets an *upper* boundary to a likely value for $R_{th}$, while DFT simulations give a lower limit.

We can now estimate the heat diffusivity $D_1 = \frac{\kappa_1}{\rho_1 c_1}$ of FeRh in both phases. Since the volume density of FeRh $\rho_1$ varies appreciably with the magnetic phase (AF or FM, see Appendix A for numerical values), $D_1$ is divided by over two upon crossing the transition: from $\sim 8 \times 10^{-6}$ m$^2$ s$^{-1}$ in the AF phase at 5°C, to $\sim 3 \times 10^{-6}$ m$^2$ s$^{-1}$ in the FM phase at 130°C (403 K). Note that the former value is in the range of the diffusivity of $2 \times 10^{-5}$ m$^2$ s$^{-1}$ found by Bergman et al.⁸ by fitting the laser-induced transient reflectivity, late after the
<figure><img src="image_6.png" /><figcaption>FIG. 6. (a) Interferometric signal at different temperatures on either side of the transition obtained using a time-resolved pump-probe set-up (b)$^{45}$, with large diameter spots. Incoherent phonon generation occurs together with the launching of a thermo-elastically generated coherent strain wave that travels rapidly to the FeRh/MgO interface and back. The more rapid decay of the thermal background in the AF phase indicates a higher thermal conductivity as evidenced from (c) the calculated interferometric signal, given by Eq. 4 and normalized to its t = 0 value.</figcaption></figure>

274 for the heat to escape laterally a d ≈ 2 µm diameter heat spot
275 as τ ≈ d²/D₁, i.e., τAF ≈ 500 ns while τFM ≈ 1.2 µs.

CONFIRMATION OF $\kappa_{AF} > \kappa_{FM}$ USING PULSED GENERATION OF INCOHERENT PHONONS

The analysis of the thermoreflectance data points to a smaller thermal conductivity in the FM phase compared to the AF phase, even considering a larger thermal interface resistance at high temperature. This comes as a surprising result given that both the Wiedeman-Franz law and a previous experimental estimate¹⁹ seemed to indicate instead a larger thermal conductivity in the FM phase. To verify this, we performed picosecond acoustics pump-probe experiments (see Fig. 6, and Appendix B for experimental details). A 80 MHz femtosecond laser beam is split in a train of pump pulses that generate coherent strain waves by the thermoelastic effect, and delayed probe pulses that measure the dephasing of the reflected electric field of light by interferometry⁴⁵. After an electronic signal decaying within 10-20 ps, one observes a slower decay related to the cooling of the layer as the heat is evacuated from the 12-13 nm thick top layer in which the light has been absorbed (Fig. 6(a)). It is clearly slower in the FM compared to the AF phase, as it is also observed in the time-resolved strain measurements by Ahn et al. in Ref. 19 (see in particular their Supplemental Information⁴⁶). This indicates once more $\kappa_{1,AF} > \kappa_{1,FM}$. We can this time safely discard the influence of the interface resistance on the thermal background during the $t_{ps} = 300$ ps window of observation: whereas the longitudinal acoustic wave traveling at 4741 m s⁻¹ easily sees the FeRh interface twice, the thermal wave will only have reached around $l_{th} = \sqrt{Dt_{ps}} \approx 30-50$ nm deep into the FeRh layer. The interferometric signal is calculated as Im($\Delta r(t)/r$) where r is the amplitude reflection coefficient of the light electric field (Eq. 4). The first term contains the surface displacement $u(t, z = 0)$. $k_0$ equal to $2\pi/\lambda$ is the light wavevector. The next term describes the reabsorption and dephasing of the reflected light taking into account the change of the refractive index $n = n' + jn''$ induced by the propagating strain pulse $S_{prop}$ and the temperature variation $\Delta T$ arising from heat diffusion.

268 arrival of the pulse, when the layer is back into its AF phase.

269 This makes FeRh roughly as heat-diffusive as MgO in its AF

270 phase, but almost four times less in the FM phase. Typical

271 thermal diffusion lengths at 500 kHz are then of a few mi-

272 crons, much more than the film thickness. A rough estimate

273 in a 1D approximation moreover gives the typical time it takes

$$
\frac{\Delta r(t)}{r} = 2jk_0u(t,z=0) + jk_0 \frac{4n}{(1-n^2)} \int_0^\infty \left( \frac{dn}{dS}S_{prop}(t,z) + \frac{dn}{dT}\Delta T(t,z) \right) \exp(j2k_0nz) dz \quad (4)
$$

312 The detailed expressions of u(t, z = 0), S<sub>prop</sub>(t, z) and ΔT(t, z)

313 and the parameters used in the modelling are given in Ap-

314 pendix B. The surface displacement and propagating strain are

315 found to give smaller contributions to the interferometric sig-

316 nal as compared to heat diffusion. This signal is calculated

317 using the thermal diffusivity given by κ values in Table I and

318 shown in Fig. 6(c). It clearly evidences a faster decay in the

319 AF phase with a larger diffusivity, in good agreement with the

320 experimental results of Fig. 6(a).

DISCUSSION

322 Having confirmed the trend $\kappa_{1,AF} > \kappa_{1,FM}$ seen on the
323 modulated thermoreflectance data, we compare the val-
324 ues of the thermal conductivity in both phases (Table I),
325 with that of other materials at room temperature. The
326 absolute values are overall smaller than for⁴⁷ good met-
327 als (a few hundreds W m⁻¹ K⁻¹), semiconductors (≈ 10-
328 100 W m⁻¹ K⁻¹), or even an electrical insulator such as MgO
<figure><img src="image_7.png" /><figcaption>FIG. 6. (a) Interferometric signal at different temperatures on either side of the transition obtained using a time-resolved pump-probe set-up (b)$^{45}$, with large diameter spots. Incoherent phonon generation occurs together with the launching of a thermo-elastically generated coherent strain wave that travels rapidly to the FeRh/MgO interface and back. The more rapid decay of the thermal background in the AF phase indicates a higher thermal conductivity as evidenced from (c) the calculated interferometric signal, given by Eq. 4 and normalized to its t = 0 value.</figcaption></figure>

274 for the heat to escape laterally a d ≈ 2 µm diameter heat spot
275 as τ ≈ d²/D₁, i.e., τAF ≈ 500 ns while τFM ≈ 1.2 µs.

CONFIRMATION OF $\kappa_{AF} > \kappa_{FM}$ USING PULSED GENERATION OF INCOHERENT PHONONS

The analysis of the thermoreflectance data points to a smaller thermal conductivity in the FM phase compared to the AF phase, even considering a larger thermal interface resistance at high temperature. This comes as a surprising result given that both the Wiedeman-Franz law and a previous experimental estimate¹⁹ seemed to indicate instead a larger thermal conductivity in the FM phase. To verify this, we performed picosecond acoustics pump-probe experiments (see Fig. 6, and Appendix B for experimental details). A 80 MHz femtosecond laser beam is split in a train of pump pulses that generate coherent strain waves by the thermoelastic effect, and delayed probe pulses that measure the dephasing of the reflected electric field of light by interferometry⁴⁵. After an electronic signal decaying within 10-20 ps, one observes a slower decay related to the cooling of the layer as the heat is evacuated from the 12-13 nm thick top layer in which the light has been absorbed (Fig. 6(a)). It is clearly slower in the FM compared to the AF phase, as it is also observed in the time-resolved strain measurements by Ahn et al. in Ref. 19 (see in particular their Supplemental Information⁴⁶). This indicates once more $\kappa_{1,AF} > \kappa_{1,FM}$. We can this time safely discard the influence of the interface resistance on the thermal background during the $t_{ps} = 300$ ps window of observation: whereas the longitudinal acoustic wave traveling at 4741 m s⁻¹ easily sees the FeRh interface twice, the thermal wave will only have reached around $l_{th}=\sqrt{Dt_{ps}} \approx 30-50$ nm deep into the FeRh layer. The interferometric signal is calculated as Im($\Delta r(t)/r$) where r is the amplitude reflection coefficient of the light electric field (Eq. 4). The first term contains the surface displacement $u(t, z=0)$. $k_0$ equal to $2\pi/\lambda$ is the light wavevector. The next term describes the reabsorption and dephasing of the reflected light taking into account the change of the refractive index $n = n' + jn''$ induced by the propagating strain pulse $S_{prop}$ and the temperature variation $\Delta T$ arising from heat diffusion.

268 arrival of the pulse, when the layer is back into its AF phase.

269 This makes FeRh roughly as heat-diffusive as MgO in its AF

270 phase, but almost four times less in the FM phase. Typical

271 thermal diffusion lengths at 500 kHz are then of a few mi-

272 crons, much more than the film thickness. A rough estimate

273 in a 1D approximation moreover gives the typical time it takes

$$
\frac{\Delta r(t)}{r} = 2jk_0u(t,z=0) + jk_0 \frac{4n}{(1-n^2)} \int_0^\infty \left( \frac{dn}{dS} S_{prop}(t,z) + \frac{dn}{dT} \Delta T(t,z) \right) \exp(j2k_0nz) dz \quad (4)
$$

312 The detailed expressions of $u(t, z = 0)$, $S_{prop}(t, z)$ and $\Delta T(t, z)$

313 and the parameters used in the modelling are given in Ap-

314 pendix B. The surface displacement and propagating strain are

315 found to give smaller contributions to the interferometric sig-

316 nal as compared to heat diffusion. This signal is calculated

317 using the thermal diffusivity given by $\kappa$ values in Table I and

318 shown in Fig. 6(c). It clearly evidences a faster decay in the

319 AF phase with a larger diffusivity, in good agreement with the

320 experimental results of Fig. 6(a).

DISCUSSION

322 Having confirmed the trend $\kappa_{1,AF} > \kappa_{1,FM}$ seen on the

323 modulated thermoreflectance data, we compare the val-

324 ues of the thermal conductivity in both phases (Table I),

325 with that of other materials at room temperature. The

326 absolute values are overall smaller than for⁴⁷ good met-

327 als (a few hundreds W m⁻¹ K⁻¹), semiconductors (≈ 10–

328 100 W m⁻¹ K⁻¹), or even an electrical insulator such as MgO
(≈ 50 W m⁻¹ K⁻¹). As often occurs upon alloying⁴⁸, the ther-
mal conductivity of FeRh is also smaller than those of its
constituting single elements⁴⁷,⁴⁹: κ<sub>Fe</sub> ≈ 80 W m⁻¹ K⁻¹ and
κ<sub>Rh</sub> ≈ 135 W m⁻¹ K⁻¹.

We can moreover estimate the stationary temperature in-
crease due to the modulated green laser only, ΔT<sub>stat</sub>, using Eq.
(2) in which we set the frequency f = 0, r=0, and the incom-
ing heat flow as Q<sub>0</sub> = P<sub>inc</sub>(1 − R). For an incident average
laser power of P<sub>inc</sub> ≈ 5.3 mW, the stationary temperature rise
in the very center of a 2-µm-diameter pump spot will thus be
of ΔT<sub>stat</sub> ≈ 13 °C for a base temperature of 40 or 60 °C (313 or
333 K) in the AF phase (κ<sub>1</sub> ≈ 30 W m<sup>-1</sup> K<sup>-1</sup>). Note that this
estimate is strongly dependent on the probe spot diameter.

CALCULATIONS OF $\kappa_e$, $\kappa_{ph}$, $\kappa_m$ AND $R_{th}$

To dissect the experimentally observed and counter-
intuitive thermal conductivity of FeRh, we perform simula-
tions and combine with literature data to understand the indi-
vidual phonon, electron and magnon contributions for the FM
and AF phases.

Phonon thermal conductivity: anharmonic phonon calculations

From an *ab initio* setting, the lattice contribution to the
thermal conductivity can be computed by solving the phonon
Boltzmann equation⁵⁰⁻⁵². In order to obtain quantitative re-
sults, it is essential to include effects beyond the harmonic
approximation, such as thermal expansion and intrinsic an-
harmonicity, in the description of the system. This seems to
be particularly true for the AF phase of FeRh, where previous
theoretical studies predicted the appearance of an imaginary
mode²²,⁵³⁻⁵⁵, which prevents the use of the harmonic approx-
imation as a starting point for the Boltzmann equation. To
go beyond the harmonic approximation, we employ the tem-
perature dependent effective potential (TDEP) method⁵²,⁵⁶,⁵⁷
to include anharmonicity and renormalize phonon-phonon in-
teractions. To reduce the important simulation cost while
keeping the accuracy of DFT, we constructed two machine-
learning interatomic potential (MLIP⁵⁸⁻⁶⁰) models of FeRh
for the FM and AF phases. The details of the simulations
are described in Appendix C.

The resulting renormalized phonon band structures at room
temperature are represented in Fig. 7. In the FM phase, all
Fe atoms are equivalent by symmetry, so that the magnetic
unit cell coincides with the B2 structure which can be used to
compute phonon-related properties. This is not the case for
the AF phase, where the spin up or down occupation of Fe
atoms makes them inequivalent. Such a breaking of symme-
try can have an important impact on the phonon properties²⁶.
To account for this effect, the phonons in the AF phase were
computed using the magnetic unit cell.

For both phases, we find that the spectra are fully real,
demonstrating their dynamical stability. Some previous
works²²,²⁴ had suggested the possibility of an additional low

T phase for the AF phase, which our results do not support (at
least down to 50 K).

<figure><img src="image_7.png" /></figure>

FIG. 7. Room temperature phonon band structure for (up) the AF and (down) the FM phase of FeRh. For comparison, the magnon band structures by Gu$^{61}$ are also shown in red (assuming no magnetic anisotropy). The phonons in the FM phase are computed using the B2 structure as the unit cell, while the magnetic unit cell was used for the AF phase.

With these phonons and the third order phonon scattering
as input, we obtained the lattice thermal conductivity of the
two phases using the iterative solution of the Boltzmann equa-
tion. Our results give κ<sub>ph,AF</sub> = 11.84 W m<sup>-1</sup> K<sup>-1</sup> > κ<sub>ph,FM</sub> =
10.36 W m<sup>-1</sup> K<sup>-1</sup> at room temperature. Our calculation dif-
fers from previous work in the literature<sup>26</sup>, where the oppo-
site trend was found. The thermal conductivities computed by
Cazorla and Rurali<sup>26</sup> were based on the harmonic approxima-
tion at the ground-state volume and using a different exchange
and correlation functional. We show in Appendix C that be-
yond the differences in the DFT calculations that are minor,
the main difference in our results comes from the explicit vari-
ation of the interatomic force constants and volume with tem-
perature. The difference in lattice κ<sub>ph</sub> favors the AF phase, but
only slightly. However, the experimentally observed asymme-
try between FM and AF thermal conductivities also contains
electronic and magnonic contributions to κ, which we proceed
to estimate.

Electron thermal conductivity

The electronic thermal conductivity κe can be estimated
from the charge conductivity σ, through the Wiedemann-
Franz relationship. The latter is known not to hold univer-
sally, but breakdowns mainly appear for strongly correlated
metals, strange band structures, or sub-micron structures⁶².
In the present case if the Lorenz factor takes its ideal value of
L₀ = 2.44 × 10⁻⁸ W·Ω·K⁻², we can extract the electron ther-
mal conductivity as κe = L₀Tσ.
(≈ 50 W m⁻¹ K⁻¹). As often occurs upon alloying⁴⁸, the ther-
mal conductivity of FeRh is also smaller than those of its
constituting single elements⁴⁷,⁴⁹: κ<sub>Fe</sub> ≈ 80 W m⁻¹ K⁻¹ and
κ<sub>Rh</sub> ≈ 135 W m⁻¹ K⁻¹.

We can moreover estimate the stationary temperature in-
crease due to the modulated green laser only, ΔT<sub>stat</sub>, using Eq.
(2) in which we set the frequency f = 0, r=0, and the incom-
ing heat flow as Q<sub>0</sub> = P<sub>inc</sub>(1 − R). For an incident average
laser power of P<sub>inc</sub> ≈ 5.3 mW, the stationary temperature rise
in the very center of a 2-µm-diameter pump spot will thus be
of ΔT<sub>stat</sub> ≈ 13 °C for a base temperature of 40 or 60 °C (313 or
333 K) in the AF phase (κ<sub>1</sub> ≈ 30 W m<sup>-1</sup> K<sup>-1</sup>). Note that this
estimate is strongly dependent on the probe spot diameter.

CALCULATIONS OF $\kappa_e$, $\kappa_{ph}$, $\kappa_m$ AND $R_{th}$

To dissect the experimentally observed and counter-
intuitive thermal conductivity of FeRh, we perform simula-
tions and combine with literature data to understand the indi-
vidual phonon, electron and magnon contributions for the FM
and AF phases.

Phonon thermal conductivity: anharmonic phonon calculations

From an *ab initio* setting, the lattice contribution to the
thermal conductivity can be computed by solving the phonon
Boltzmann equation⁵⁰⁻⁵². In order to obtain quantitative re-
sults, it is essential to include effects beyond the harmonic
approximation, such as thermal expansion and intrinsic an-
harmonicity, in the description of the system. This seems to
be particularly true for the AF phase of FeRh, where previous
theoretical studies predicted the appearance of an imaginary
mode²²,⁵³⁻⁵⁵, which prevents the use of the harmonic approx-
imation as a starting point for the Boltzmann equation. To
go beyond the harmonic approximation, we employ the tem-
perature dependent effective potential (TDEP) method⁵²,⁵⁶,⁵⁷
to include anharmonicity and renormalize phonon-phonon in-
teractions. To reduce the important simulation cost while
keeping the accuracy of DFT, we constructed two machine-
learning interatomic potential (MLIP⁵⁸⁻⁶⁰) models of FeRh
for the FM and AF phases. The details of the simulations
are described in Appendix C.

The resulting renormalized phonon band structures at room
temperature are represented in Fig. 7. In the FM phase, all
Fe atoms are equivalent by symmetry, so that the magnetic
unit cell coincides with the B2 structure which can be used to
compute phonon-related properties. This is not the case for
the AF phase, where the spin up or down occupation of Fe
atoms makes them inequivalent. Such a breaking of symme-
try can have an important impact on the phonon properties²⁶.
To account for this effect, the phonons in the AF phase were
computed using the magnetic unit cell.

For both phases, we find that the spectra are fully real,
demonstrating their dynamical stability. Some previous
works²²,²⁴ had suggested the possibility of an additional low

T phase for the AF phase, which our results do not support (at
least down to 50 K).

<figure><img src="image_8.png" /></figure>

FIG. 7. Room temperature phonon band structure for (up) the AF and (down) the FM phase of FeRh. For comparison, the magnon band structures by Gu$^{61}$ are also shown in red (assuming no magnetic anisotropy). The phonons in the FM phase are computed using the B2 structure as the unit cell, while the magnetic unit cell was used for the AF phase.

With these phonons and the third order phonon scattering
as input, we obtained the lattice thermal conductivity of the
two phases using the iterative solution of the Boltzmann equa-
tion. Our results give κ<sub>ph,AF</sub> = 11.84 W m<sup>-1</sup> K<sup>-1</sup> > κ<sub>ph,FM</sub> =
10.36 W m<sup>-1</sup> K<sup>-1</sup> at room temperature. Our calculation dif-
fers from previous work in the literature<sup>26</sup>, where the oppo-
site trend was found. The thermal conductivities computed by
Cazorla and Rurali<sup>26</sup> were based on the harmonic approxima-
tion at the ground-state volume and using a different exchange
and correlation functional. We show in Appendix C that be-
yond the differences in the DFT calculations that are minor,
the main difference in our results comes from the explicit vari-
ation of the interatomic force constants and volume with tem-
perature. The difference in lattice κ<sub>ph</sub> favors the AF phase, but
only slightly. However, the experimentally observed asymme-
try between FM and AF thermal conductivities also contains
electronic and magnonic contributions to κ, which we proceed
to estimate.

Electron thermal conductivity

The electronic thermal conductivity κ_e can be estimated
from the charge conductivity σ, through the Wiedemann-
Franz relationship. The latter is known not to hold univer-
sally, but breakdowns mainly appear for strongly correlated
metals, strange band structures, or sub-micron structures⁶².
In the present case if the Lorenz factor takes its ideal value of
L₀ = 2.44 × 10⁻⁸ W·Ω·K⁻², we can extract the electron ther-
mal conductivity as κ_e = L₀Tσ.
We measure the temperature-dependent resistance of a 195-nm-thick FeRh film which was grown together with the sample utilized for the thermoreflectance measurements above. The resistance measurement was carried out using the 4-point probe method by contacting the film with four pins arranged in a square 4 mm on each side. The temperature was controlled using a custom-made Peltier holder and no magnetic field was applied during the measurement. In this way we find at 360 K $1/\sigma_{AF} = 1 \mu\Omega$ m and $1/\sigma_{FM} = 0.6 \mu\Omega$ m, very close to values from the literature$^{63}$. This implies $\kappa_{e,AF} \approx 9 \text{ W m}^{-1} \text{ K}^{-1}$ which is sensibly lower than $\kappa_{e,FM} \approx 15 \text{ W m}^{-1} \text{ K}^{-1}$. Theoretical calculations of $\kappa_e$ in FeRh based on its electronic band structure also show a slightly larger value for the FM phase: Jimenez et al.$^{25}$ find $\kappa_{e,AF} \approx 21 \text{ W m}^{-1} \text{ K}^{-1}$ and $\kappa_{e,FM} \approx 27 \text{ W m}^{-1} \text{ K}^{-1}$, using a somewhat arbitrary temperature and energy independent relaxation time of $10^{-14}$ s. In their calculations the Lorenz factor is not presumed constant (and depends on the electronic band structure) but the $\kappa_e$ is known only up to a rescaling by the relaxation time, which will be different for FM and AF phases, and should be energy/state dependent. Overall, we expect $\kappa_{e,FM}$ to be a bit larger than $\kappa_{e,AF}$, and of the order of $10-15 \text{ W m}^{-1} \text{ K}^{-1}$.

### Magnon thermal conductivity: model dispersion and lifetime

Finally, we wish to estimate the magnon contribution to the thermal conductivity. Wu et al.$^{29}$ calculate simultaneous lattice and spin dynamics for BCC Fe, and find that the magnon contribution to $\kappa$ ($\kappa_m = 15 \text{ W m}^{-1} \text{ K}^{-1}$) is larger than the lattice one ($\kappa_{ph} = 8 \text{ W m}^{-1} \text{ K}^{-1}$), though they are skeptical of their $\kappa_m$. We have re-implemented the magnon band model from Gu and Antropov$^{61}$ as a python script, and added the calculation of the group velocities $v_{q\lambda}$ and a lifetime given by $1/\tau_{q\lambda} = \alpha_G \omega_{q\lambda}$ which depends only on the frequency of mode $\lambda$ at wave vector $q$. Here $\alpha_G$ is the Gilbert damping parameter, which should vary with temperature and magnetic ordering (and in principle also with the magnon mode $\lambda, q$). Using these ingredients, one can calculate the thermal conductivity with a formula analogous to that for phonons (Eq. (11) in Appendix C):

$$ \tilde{\kappa}_m^{ab}(\omega) = \frac{1}{V} \sum_{q\lambda} v_{q\lambda}^a v_{q\lambda}^b C_v(\omega_{q\lambda}) \tau_{q\lambda} \delta(\omega - \omega_{q\lambda}) \quad (5) $$

$$ \kappa_m^{ab} = \int \tilde{\kappa}_m^{ab}(\omega) d\omega \quad (6) $$

where a, b are Cartesian components, $C_v$ is the mode specific heat ($[1/2x/(\sinh x/2)]^2$ with $x = \hbar\omega/k_B T$), and V is the unit cell volume. We call $\tilde{\kappa}$ the spectral thermal conductivity. The q integrations are carried out numerically on $100^3$ point grids for the primitive cubic Brillouin Zone. In this framework, the band structure strongly favors $\kappa_{m,FM}$, through the much larger DOS at low frequency (quadratic vs linear dispersion), which boosts the $C_v$ at low $\omega$, where $\tau$ is also large. This effect is stronger than that of the magnon group velocities, which favor the AF phase (constant acoustic velocity), over the FM phase (zero velocity at $\Gamma$ which grows linearly). The product

$C_v \tau \nu^2$ yields $\tilde{\kappa}_{m,FM} \propto \omega^{1/2}$ for the FM phase, while $\tilde{\kappa}_{m,AF} \propto \omega$. The Bose distribution weights most of $C_v$ and $\kappa_m$ at low frequencies, generically favoring $\kappa_{m,FM}$ over $\kappa_{m,AF}$.

An important ingredient of the calculation is the parameter $\alpha_G$, in both phases. The only measurements of FeRh magnon damping have been performed in the uniform FM phase, for which quite a large spread has been reported. Intrinsic damping values of around 0.001-0.004 have been measured$^{64}$ or computed using a multiple scattering method including position and spin fluctuations$^{63}$. Larger values between 0.03 and 0.1 have also been seen, and attributed to the spin-sink effect of either an adjacent Pt layer$^{65}$, or residual AF domains$^{66}$.

For the AF phase, a first principles calculation was made by Mahfouzi and Kioussis$^{67}$, and using their Eq. 5 we obtain values ranging from 0.03 to 0.3 between what they call low and high temperature. Simensen et al.$^{68}$ arrive at 0.3 with a magnon scattering theory.

Using a single conservative value $\alpha_G = 0.25$ for both phases, we obtain values of $\kappa_{m,AF} \sim 15 \text{ W m}^{-1} \text{ K}^{-1}$ and $\kappa_{m,FM} \sim 90 \text{ W m}^{-1} \text{ K}^{-1}$. The former is reasonable, but the latter is too large compared to our measurements. Examining the different ingredients for our model, the magnon dispersion compares favorably with experiments$^{69}$ and the calculations of Sandratskii and Buczek$^{70}$, except for a small but crucial underestimation of the FM phase stiffness at low frequency and q, noted by Castets$^{69}$. If the acoustic mode has a slight linear component (e.g. by mixing with phonon excitations) or much higher stiffness, the DOS, $C_v$ and $\kappa_{m,FM}$ will be strongly reduced. The main source of uncertainty is the simplistic lifetime model as $1/\omega\alpha_G$: the large DOS and lifetimes at low frequency give too much weight to the FM phase $\kappa_m$.

### Estimate of the thermal resistance at the FeRh/MgO interface

The thermal interface resistance was estimated theoretically based on the bulk phonon dispersions of FeRh (FM and AFM) combined with those of MgO. The diffuse mismatch model (DMM) was used$^{71,72}$, which has two main hypotheses: (i) the phonons are transmitted based on their frequency matching and their group velocity component normal to the interface; (ii) detailed balance and a steady state is achieved, which allows to calculate the transmission probability. The implementation was carried out in the Abinit$^{73}$ package following the full phonon description in Ref.$^{74}$ and was benchmarked against their values for interfaces between Si, Cu and Al. For (100) oriented FeRh on (100) MgO$^{75}$, the AFM phase has a room temperature interface resistance of $4.3 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$, and the FM phase of $4.2 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$. Other crystalline orientations tend to decrease the resistance, down to 2.3 (resp. $0.24 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$ for FM (resp. AF) (111) FeRh on (111) MgO. The main factor boosting the interface resistance is the small overlap in frequency between light MgO and heavier FeRh. To fit the thermoreflectance spatial scans, we used the value $R_{th} = 4 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$, but also considered the possibility of much larger values (see Fig. 4).
We measure the temperature-dependent resistance of a 195-nm-thick FeRh film which was grown together with the sample utilized for the thermoreflectance measurements above. The resistance measurement was carried out using the 4-point probe method by contacting the film with four pins arranged in a square 4 mm on each side. The temperature was controlled using a custom-made Peltier holder and no magnetic field was applied during the measurement. In this way we find at 360 K $1/\sigma_{AF} = 1 \mu\Omega$ m and $1/\sigma_{FM} = 0.6 \mu\Omega$ m, very close to values from the literature$^{63}$. This implies $\kappa_{e,AF} \approx 9 \text{ W m}^{-1} \text{ K}^{-1}$ which is sensibly lower than $\kappa_{e,FM} \approx 15 \text{ W m}^{-1} \text{ K}^{-1}$. Theoretical calculations of $\kappa_e$ in FeRh based on its electronic band structure also show a slightly larger value for the FM phase: Jimenez et al.$^{25}$ find $\kappa_{e,AF} \approx 21 \text{ W m}^{-1} \text{ K}^{-1}$ and $\kappa_{e,FM} \approx 27 \text{ W m}^{-1} \text{ K}^{-1}$, using a somewhat arbitrary temperature and energy independent relaxation time of $10^{-14}$ s. In their calculations the Lorenz factor is not presumed constant (and depends on the electronic band structure) but the $\kappa_e$ is known only up to a rescaling by the relaxation time, which will be different for FM and AF phases, and should be energy/state dependent. Overall, we expect $\kappa_{e,FM}$ to be a bit larger than $\kappa_{e,AF}$, and of the order of $10-15 \text{ W m}^{-1} \text{ K}^{-1}$.

### Magnon thermal conductivity: model dispersion and lifetime

Finally, we wish to estimate the magnon contribution to the thermal conductivity. Wu et al.$^{29}$ calculate simultaneous lattice and spin dynamics for BCC Fe, and find that the magnon contribution to $\kappa$ ($\kappa_m = 15 \text{ W m}^{-1} \text{ K}^{-1}$) is larger than the lattice one ($\kappa_{ph} = 8 \text{ W m}^{-1} \text{ K}^{-1}$), though they are skeptical of their $\kappa_m$. We have re-implemented the magnon band model from Gu and Antropov$^{61}$ as a python script, and added the calculation of the group velocities $v_{q\lambda}$ and a lifetime given by $1/\tau_{q\lambda} = \alpha_G \omega_{q\lambda}$ which depends only on the frequency of mode $\lambda$ at wave vector $q$. Here $\alpha_G$ is the Gilbert damping parameter, which should vary with temperature and magnetic ordering (and in principle also with the magnon mode $\lambda, q$). Using these ingredients, one can calculate the thermal conductivity with a formula analogous to that for phonons (Eq. (11) in Appendix C):

$$ \tilde{\kappa}_m^{ab}(\omega) = \frac{1}{V} \sum_{q\lambda} v_{q\lambda}^a v_{q\lambda}^b C_v(\omega_{q\lambda}) \tau_{q\lambda} \delta(\omega - \omega_{q\lambda}) \quad (5) $$

$$ \kappa_m^{ab} = \int \tilde{\kappa}_m^{ab}(\omega) d\omega \quad (6) $$

where a, b are Cartesian components, $C_v$ is the mode specific heat ($[1/2x/(\sinh x/2)]^2$ with $x = \hbar\omega/k_B T$), and V is the unit cell volume. We call $\tilde{\kappa}$ the spectral thermal conductivity. The q integrations are carried out numerically on $100^3$ point grids for the primitive cubic Brillouin Zone. In this framework, the band structure strongly favors $\kappa_{m,FM}$, through the much larger DOS at low frequency (quadratic vs linear dispersion), which boosts the $C_v$ at low $\omega$, where $\tau$ is also large. This effect is stronger than that of the magnon group velocities, which favor the AF phase (constant acoustic velocity), over the FM phase (zero velocity at $\Gamma$ which grows linearly). The product

$C_v \tau \nu^2$ yields $\tilde{\kappa}_{m,FM} \propto \omega^{1/2}$ for the FM phase, while $\tilde{\kappa}_{m,AF} \propto \omega$. The Bose distribution weights most of $C_v$ and $\kappa_m$ at low frequencies, generically favoring $\kappa_{m,FM}$ over $\kappa_{m,AF}$.

An important ingredient of the calculation is the parameter $\alpha_G$, in both phases. The only measurements of FeRh magnon damping have been performed in the uniform FM phase, for which quite a large spread has been reported. Intrinsic damping values of around 0.001–0.004 have been measured$^{64}$ or computed using a multiple scattering method including position and spin fluctuations$^{63}$. Larger values between 0.03 and 0.1 have also been seen, and attributed to the spin-sink effect of either an adjacent Pt layer$^{65}$, or residual AF domains$^{66}$.

For the AF phase, a first principles calculation was made by Mahfouzi and Kioussis$^{67}$, and using their Eq. 5 we obtain values ranging from 0.03 to 0.3 between what they call low and high temperature. Simensen et al.$^{68}$ arrive at 0.3 with a magnon scattering theory.

Using a single conservative value $\alpha_G = 0.25$ for both phases, we obtain values of $\kappa_{m,AF} \sim 15 \text{ W m}^{-1} \text{ K}^{-1}$ and $\kappa_{m,FM} \sim 90 \text{ W m}^{-1} \text{ K}^{-1}$. The former is reasonable, but the latter is too large compared to our measurements. Examining the different ingredients for our model, the magnon dispersion compares favorably with experiments$^{69}$ and the calculations of Sandratskii and Buczek$^{70}$, except for a small but crucial underestimation of the FM phase stiffness at low frequency and q, noted by Castets$^{69}$. If the acoustic mode has a slight linear component (e.g. by mixing with phonon excitations) or much higher stiffness, the DOS, $C_v$ and $\kappa_{m,FM}$ will be strongly reduced. The main source of uncertainty is the simplistic lifetime model as $1/\omega\alpha_G$: the large DOS and lifetimes at low frequency give too much weight to the FM phase $\kappa_m$.

### Estimate of the thermal resistance at the FeRh/MgO interface

The thermal interface resistance was estimated theoretically based on the bulk phonon dispersions of FeRh (FM and AFM) combined with those of MgO. The diffuse mismatch model (DMM) was used$^{71,72}$, which has two main hypotheses: (i) the phonons are transmitted based on their frequency matching and their group velocity component normal to the interface; (ii) detailed balance and a steady state is achieved, which allows to calculate the transmission probability. The implementation was carried out in the Abinit$^{73}$ package following the full phonon description in Ref.$^{74}$ and was benchmarked against their values for interfaces between Si, Cu and Al. For (100) oriented FeRh on (100) MgO$^{75}$, the AFM phase has a room temperature interface resistance of $4.3 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$, and the FM phase of $4.2 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$. Other crystalline orientations tend to decrease the resistance, down to 2.3 (resp. $0.24 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$ for FM (resp. AF) (111) FeRh on (111) MgO. The main factor boosting the interface resistance is the small overlap in frequency between light MgO and heavier FeRh. To fit the thermoreflectance spatial scans, we used the value $R_{th} = 4 \times 10^{-9} \text{ m}^2\text{K W}^{-1}$, but also considered the possibility of much larger values (see Fig. 4).
<table><caption>TABLE II. Numerical estimates of the different contributions to the thermal conductivity. The last column refers to the total value, to be compared to the experimentally measured one (Table I).</caption><thead><tr><th>Phase</th><th>κ<sub>ph</sub></th><th>κ<sub>e</sub></th><th>κ<sub>m</sub></th><th>κ (W m<sup>-1</sup> K<sup>-1</sup>)</th></tr></thead><tbody><tr><td>AF</td><td>12</td><td>9</td><td>15</td><td>36</td></tr><tr><td>FM</td><td>10</td><td>15</td><td>90</td><td>115</td></tr></tbody></table>

## DISCUSSION AND COMPARISON TO EXPERIMENTS

We find three comparable contributions for the thermal conductivity, between the lattice, electronic, and magnon terms (Table II). Each term has been estimated using available simulations or experimental input, and they are all on the order of 10 W m⁻¹ K⁻¹, with κ<sub>m,FM</sub> the only outlier. The sum renders quite nicely the measurements of κ<sub>AF</sub> ~ 30 W m⁻¹ K⁻¹, but the lower measured κ<sub>FM</sub> requires additional attention: the observed difference in the lattice κ<sub>ph</sub> provides 1–2 W m⁻¹ K⁻¹, but the other two terms favor FM over AF.

Several factors may contribute to lower κ<sub>FM</sub>:

* the multi-domain state of the layer in its FM phase could give an extra contribution from ferromagnetic domain-wall resistance<sup>76</sup>, but a large number of ferromagnetic domains is quite unlikely under the two-micron spot, due to the competition between exchange and magnetostatic energy cost.

* disorder will limit transport in both phases, but magnetic disorder (e.g., domains, impurities) will affect the FM phase transport more strongly as the dipolar interaction is longer ranged. The strong κ<sub>m</sub> advantage of FM over AF for long wavelengths will be reduced by defects and grain boundaries.

* the low-q magnon phonon coupling is bound to play a critical role (see Fig. 7). Due to their quadratic dispersion and very similar frequencies, the FM magnons will interfere with acoustic phonons, limiting both lattice κ<sub>ph</sub> and magnon κ<sub>m</sub>. The impact of this coupling on the latter could have a significant impact on the total thermal conductivity on the FM phase. Compounding this effect, the AF magnon group velocity is much higher and there will be no coupling of magnons to acoustic phonons.

* Our model for magnon lifetimes is quite crude, and may break down. Beyond the temperature and magnetic state, the “true” α<sub>G,qλ</sub> = 1/ω<sub>qλ</sub> τ<sub>qλ</sub> will also depend on the specific magnon mode. Any upper bound on the lifetimes at low q and low frequency will again reduce κ<sub>FM</sub> more strongly.

## CONCLUSIONS

Space and frequency-dependent thermoreflectance measurements were performed in order to determine the thermal

conductivity of a 195-nm-thick FeRh film in a wide temperature range of 5 to 130°C, across the AF to FM transition. Unexpectedly (based on rough estimates made with the Wiedemann-Franz law), the thermal conductivity is found overall three times larger in the AF phase compared to the FM phase, a trend confirmed by the temporal decay of laser-pulsed generated incoherent phonons. These values were then used to estimate the expected transient and stationary temperature rises induced by a laser. To explain why the total FM thermal conductivity is lower than the AF one, we estimated the phononic contribution to κ from anharmonic first-principles dynamics, the electronic contribution from the experimental resistivity using Wiedemann-Franz, and the magnonic contribution using a linear spin wave model from the literature, with a Gilbert-type relaxation time. In the AF phase, the three components were found to be of the same order of magnitude, with a total of around 36 W m⁻¹ K⁻¹, very close to the observed ≈ 30 W m⁻¹ K⁻¹. In the FM phase, however, the calculated thermal conductivity rockets to ≈ 115 W m⁻¹ K⁻¹, which is unrealistically large with regards to the observed ≈ 10 W m⁻¹ K⁻¹. More accurate modelling of electronic and magnonic contributions will be required: in particular the large uncertainties in the magnetic damping, the magnon-phonon coupling, and the effects of disorder lead to a strong overestimate of all three components of κ.

These results underscore the complexity of FeRh and the importance of the three-way couplings between electrons, phonons, and magnons, which are on equal footing in the total thermal conductivity.

## ACKNOWLEDGMENTS

This work has been partly supported by the French Agence Nationale de la Recherche (ANR ACAF 20-CE30-0027). We acknowledge M. Vabre (Institut des Nanosciences de Paris) for technical assistance. Access to the CEITEC Nano Research Infrastructure was supported by the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic under the project CzechNanoLab (LM2023051). AC and MJV acknowledge the Fonds de la Recherche Scientifique (FRS-FNRS Belgium) for PdR Grant No. T.0103.19 - ALPS, and ARC project DREAMS (G.A. 21/25-11) funded by Federation Wallonie Bruxelles and ULiege, and the Excellence of Science (EOS) program (grant 40007563-CONNECT) funded by the FWO and F.R.S.-FNRS. Simulation time was awarded by the Belgian share of EuroHPC in LUMI hosted by CSC in Finland, by the CECI (FRS-FNRS Belgium Grant No. 2.5020.11), as well as the Zenobe Tier-1 of the Fédération Wallonie-Bruxelles (Walloon Region grant agreement No. 1117545).

## APPENDIX A: PARAMETERS

The values for the thermal conductivity κ<sub>0</sub>(T), diffusivity D<sub>0</sub>(T) and specific heat C<sub>0</sub>(T) of MgO were taken from
<table><caption>TABLE II. Numerical estimates of the different contributions to the thermal conductivity. The last column refers to the total value, to be compared to the experimentally measured one (Table I).</caption><thead><tr><th>Phase</th><th>κ<sub>ph</sub></th><th>κ<sub>e</sub></th><th>κ<sub>m</sub></th><th>κ (W m<sup>-1</sup> K<sup>-1</sup>)</th></tr></thead><tbody><tr><td>AF</td><td>12</td><td>9</td><td>15</td><td>36</td></tr><tr><td>FM</td><td>10</td><td>15</td><td>90</td><td>115</td></tr></tbody></table>

## DISCUSSION AND COMPARISON TO EXPERIMENTS

We find three comparable contributions for the thermal conductivity, between the lattice, electronic, and magnon terms (Table II). Each term has been estimated using available simulations or experimental input, and they are all on the order of 10 W m⁻¹ K⁻¹, with κ<sub>m,FM</sub> the only outlier. The sum renders quite nicely the measurements of κ<sub>AF</sub> ~ 30 W m⁻¹ K⁻¹, but the lower measured κ<sub>FM</sub> requires additional attention: the observed difference in the lattice κ<sub>ph</sub> provides 1–2 W m⁻¹ K⁻¹, but the other two terms favor FM over AF.

Several factors may contribute to lower κ<sub>FM</sub>:

* the multi-domain state of the layer in its FM phase could give an extra contribution from ferromagnetic domain-wall resistance<sup>76</sup>, but a large number of ferromagnetic domains is quite unlikely under the two-micron spot, due to the competition between exchange and magnetostatic energy cost.

* disorder will limit transport in both phases, but magnetic disorder (e.g., domains, impurities) will affect the FM phase transport more strongly as the dipolar interaction is longer ranged. The strong κ<sub>m</sub> advantage of FM over AF for long wavelengths will be reduced by defects and grain boundaries.

* the low-q magnon phonon coupling is bound to play a critical role (see Fig. 7). Due to their quadratic dispersion and very similar frequencies, the FM magnons will interfere with acoustic phonons, limiting both lattice κ<sub>ph</sub> and magnon κ<sub>m</sub>. The impact of this coupling on the latter could have a significant impact on the total thermal conductivity on the FM phase. Compounding this effect, the AF magnon group velocity is much higher and there will be no coupling of magnons to acoustic phonons.

* Our model for magnon lifetimes is quite crude, and may break down. Beyond the temperature and magnetic state, the “true” α<sub>G,qλ</sub> = 1/ω<sub>qλ</sub> τ<sub>qλ</sub> will also depend on the specific magnon mode. Any upper bound on the lifetimes at low q and low frequency will again reduce κ<sub>FM</sub> more strongly.

## CONCLUSIONS

Space and frequency-dependent thermoreflectance measurements were performed in order to determine the thermal

conductivity of a 195-nm-thick FeRh film in a wide temperature range of 5 to 130°C, across the AF to FM transition. Unexpectedly (based on rough estimates made with the Wiedemann-Franz law), the thermal conductivity is found overall three times larger in the AF phase compared to the FM phase, a trend confirmed by the temporal decay of laser-pulsed generated incoherent phonons. These values were then used to estimate the expected transient and stationary temperature rises induced by a laser. To explain why the total FM thermal conductivity is lower than the AF one, we estimated the phononic contribution to κ from anharmonic first-principles dynamics, the electronic contribution from the experimental resistivity using Wiedemann-Franz, and the magnonic contribution using a linear spin wave model from the literature, with a Gilbert-type relaxation time. In the AF phase, the three components were found to be of the same order of magnitude, with a total of around 36 W m⁻¹ K⁻¹, very close to the observed ≈ 30 W m⁻¹ K⁻¹. In the FM phase, however, the calculated thermal conductivity rockets to ≈ 115 W m⁻¹ K⁻¹, which is unrealistically large with regards to the observed ≈ 10 W m⁻¹ K⁻¹. More accurate modelling of electronic and magnonic contributions will be required: in particular the large uncertainties in the magnetic damping, the magnon-phonon coupling, and the effects of disorder lead to a strong overestimate of all three components of κ.

These results underscore the complexity of FeRh and the importance of the three-way couplings between electrons, phonons, and magnons, which are on equal footing in the total thermal conductivity.

## ACKNOWLEDGMENTS

This work has been partly supported by the French Agence Nationale de la Recherche (ANR ACAF 20-CE30-0027). We acknowledge M. Vabre (Institut des Nanosciences de Paris) for technical assistance. Access to the CEITEC Nano Research Infrastructure was supported by the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic under the project CzechNanoLab (LM2023051). AC and MJV acknowledge the Fonds de la Recherche Scientifique (FRS-FNRS Belgium) for PdR Grant No. T.0103.19 - ALPS, and ARC project DREAMS (G.A. 21/25-11) funded by Federation Wallonie Bruxelles and ULiege, and the Excellence of Science (EOS) program (grant 40007563-CONNECT) funded by the FWO and F.R.S.-FNRS. Simulation time was awarded by the Belgian share of EuroHPC in LUMI hosted by CSC in Finland, by the CECI (FRS-FNRS Belgium Grant No. 2.5020.11), as well as the Zenobe Tier-1 of the Fédération Wallonie-Bruxelles (Walloon Region grant agreement No. 1117545).

## APPENDIX A: PARAMETERS

The values for the thermal conductivity κ<sub>0</sub>(T), diffusivity D<sub>0</sub>(T) and specific heat C<sub>0</sub>(T) of MgO were taken from
606 Ref. 77. The specific heat C₁(T) of FeRh was found in 626 Ref. 1, where they measured much thicker, polycrystalline 627 films. We believe any error on this parameter should im- 628 pact only mildly the determination of κ₁. Indeed this pa- 629 rameter intervenes through the calculation of the diffusivity 630 of FeRh, D₁(T), which has a minor effect on the analysis of 631 modulated thermoreflectance experiments in the "insulating- 632 over-conductive layer" configuration³⁰. The volume den- 633 sity of FeRh was taken phase-dependent and temperature- 634 independent⁷⁸ from Ref. 22: ρ₁,AF = 9744.51 kg m⁻³ and 635 ρ₁,FM = 9590.11 kg m⁻³. A standard knife-edge measure- 636 ment in our experimental geometry gives dₚump = 2.±0.1 µm. 637 The reflectance of FeRh was taken phase-dependent and esti- 638 mated as: RAF = 0.75 and RFM = 0.72 by comparing to the 639 reflectance of Aluminium. Finally, the light absorption coef- 640 ficient of FeRh was estimated in both phases by measuring 641 the light reflected from/transmitted through a 30 nm sample 642 grown in similar conditions. It corresponds to a penetration 643 depth of around 10 nm (resp. 12-13 nm) at 532 nm (resp. 644 773 nm), much less than the thickness of the film h = 195 nm. 645

641 where α, D₁, ν, ρ₁, C₁, R, are the absorption coef-
642 ficient, diffusivity, longitudinal acoustic velocity, specific
643 heat, and reflectivity coefficient of FeRh, respectively.
644 We take κAF (κFM) =30(10) W m⁻¹ K⁻¹, hence DAF (DFM) =
645 9.3(2.7) × 10⁻⁶ m² s⁻¹, and vAF (vFM) =4741 (4865) m s⁻¹⁸⁰.
646 F =0.2 mJ cm⁻² is the laser fluence. We note ωα = αν,
647 ωD = D₁α², x = ωα/ωD. The real and imaginary parts
648 of the refractive index n = n' + jn'', were obtained at
649 λ =773 nm as a function of temperature by ellipsometry
650 from 20°C to 120°C. For instance at 25°C for the AF
651 phase and 120°C for the FM phase: n'AF (n'FM) =4.1(4.2),
652 n''AF (n''FM) =5(4.4) and hence αAF (αFM) =8.1(7.2) × 10⁷ m⁻¹.
653 The derivatives are dn'AF /dT (dn'FM /dT) = 7.6(6.8) × 10⁻³
654 and dn''AF /dT (dn''FM /dT) = 4(2.8) × 10⁻³. S₀ is defined as
655 3βBα(1 − R)F / (ρ²₁C₁ν²) where β is the thermal expansion
656 coefficient, and B the bulk modulus. Using data from¹⁹,⁸⁰ we
657 have S₀AF (S₀FM) = 3.8(2) × 10⁻⁴. The derivative dn/dS is
658 not known but taking a large value of 30 consistent with data
659 from the literature⁸¹, we get a smaller contribution of the prop-
660 agating strain in the interferometric signal.

APPENDIX B: DETAILS ON THE PICOSECOND
ACOUSTICS EXPERIMENTS AND MODELLING

The picosecond acoustics pump-probe set-up is described
in Peronne et al.⁴⁵. More specifically to these measurements,
the laser repetition rate was 80 MHz, with a modulation of
1 MHz. Its wavelength was 773 nm, and the beam diameter
was of the order of ≈ 15 µm. The power of the pump beam
was around P=32 mW, that of the probe around 4 mW. The
delay line was scanned mechanically at 40 nm/ps. The model
is based on the 1D heat diffusion equation and the propagation
equation for elastic waves. Given the short time scale (300 ps),
the FeRh layer can be considered as a semi-infinite medium
since the thermal wave has not reached the substrate. The
surface displacement u(t,z = 0), propagating strain Sprop(t,z)
and temperature increase ΔT(t,z) are obtained as⁷⁹:

$$
u(0,t) = -\frac{S_0}{\alpha} \left( 1 - \frac{x \exp(-\omega_\alpha t)}{x+1} - \frac{x \exp(\omega_D t)}{x^2-1} \operatorname{Erfc}(\sqrt{\omega_D t}) + \frac{\exp(x \omega_\alpha t)}{x^2-1} \operatorname{Erfc}(\sqrt{x \omega_\alpha t}) \right) \quad (7)
$$

$$
S_{\text{prop}} = \begin{cases} z - vt > 0 & \frac{S_0}{1-x^2} (\exp(-\omega_D (z/v - t)) - x \exp(-\omega_\alpha (z/v - t))) \\ z - vt < 0 & \frac{S_0}{1-x^2} (\exp(\omega_D (t-z/v)) \operatorname{Erfc}(\sqrt{\omega_D (t-z/v)}) - x \exp(x\omega_\alpha (t-z/v)) \operatorname{Erfc}(\sqrt{x\omega_\alpha (t-z/v)})) \end{cases} \quad (8)
$$

$$
\Delta T(t,z) = \frac{F(1-R)\alpha}{2\rho_1 C_1} \exp(\omega_D t) \left( \exp(-\alpha z) \text{Erfc}\left(\frac{-z+2D_1\alpha t}{2\sqrt{D_1 t}}\right) + \exp(\alpha z) \text{Erfc}\left(\frac{z+2D_1\alpha t}{2\sqrt{D_1 t}}\right) \right), \quad (9)
$$

APPENDIX C: METHOD AND COMPUTATIONAL
DETAILS FOR LATTICE THERMAL CONDUCTIVITY

Temperature dependent effective potential

The temperature dependent effective potential⁵²,⁵⁶,⁵⁷
method is based on the construction of an effective anhar-
monic Hamiltonian of the form

$$
H = \sum_i \frac{p_i^2}{2M_i} + \frac{1}{2} \sum_{ij} \sum_{ab} \Theta_{ij}^{ab} u_i^a u_j^b + \frac{1}{3!} \sum_{ijk} \sum_{abc} \Psi_{ijk}^{abc} u_i^a u_j^b u_k^c \quad (10)
$$

where u<sup>a</sup><sub>i</sub> and p<sub>i</sub> are respectively the displacement along the
cartesian direction a and momentum of atom i with mass M<sub>i</sub>,
and Θ<sup>ab</sup><sub>ij</sub> and Ψ<sup>abc</sup><sub>ijk</sub> are the second and third order interatomic
force constants, which we fit to molecular dynamics data us-
ing ordinary least-squares methods. The effectiveness of this
Hamiltonian comes from the iterative nature of the fit, where
each order is fit on the residual of the previous order. While
this procedure ensures that the lowest orders intrinsically in-
clude most of the anharmonicity, it also produces, as the name
suggests, a temperature-dependent effective potential, so that
individual MD simulations have to be run for each tempera-
ture of interest.

Once the interatomic force constants are extracted from
606 Ref. 77. The specific heat $C_1(T)$ of FeRh was found in 626 films. We believe any error on this parameter should impact only mildly the determination of $\kappa_1$. Indeed this parameter intervenes through the calculation of the diffusivity 627 of FeRh, $D_1(T)$, which has a minor effect on the analysis of 628 modulated thermoreflectance experiments in the "insulating- 629 over-conductive layer" configuration³⁰. The volume density of FeRh was taken phase-dependent and temperature- 630 independent⁷⁸ from Ref. 22: $\rho_{1,AF} = 9744.51 \text{ kg m}^{-3}$ and 631 $\rho_{1,FM} = 9590.11 \text{ kg m}^{-3}$. A standard knife-edge measure- 632 ment in our experimental geometry gives $d_{pump} = 2 \pm 0.1 \mu\text{m}$. 633 The reflectance of FeRh was taken phase-dependent and esti- 634 mated as: $R_{AF} = 0.75$ and $R_{FM} = 0.72$ by comparing to the 635 reflectance of Aluminium. Finally, the light absorption coeffi- 636 cient of FeRh was estimated in both phases by measuring 637 the light reflected from/transmitted through a 30 nm sample 638 grown in similar conditions. It corresponds to a penetration 639 depth of around 10 nm (resp. 12-13 nm) at 532 nm (resp. 640 773 nm), much less than the thickness of the film $h = 195$ nm.

641 where $\alpha$, $D_1$, $v$, $\rho_1$, $C_1$, $R$, are the absorption coefficient, diffusivity, longitudinal acoustic velocity, specific 642 heat, and reflectivity coefficient of FeRh, respectively. 643 We take $\kappa_{AF}(\kappa_{FM}) = 30(10) \text{ W m}^{-1} \text{ K}^{-1}$, hence $D_{AF}(D_{FM}) = 644 9.3(2.7) \times 10^{-6} \text{ m}^2 \text{ s}^{-1}$, and $v_{AF}(v_{FM}) = 4741 (4865) \text{ m s}^{-180}$. 645 $F=0.2 \text{ mJ cm}^{-2}$ is the laser fluence. We note $\omega_\alpha = \alpha v$, 646 $\omega_D = D_1\alpha^2$, $x = \omega_\alpha/\omega_D$. The real and imaginary parts 647 of the refractive index $n = n' + jn''$ were obtained at 648 $\lambda = 773$ nm as a function of temperature by ellipsometry 649 from 20°C to 120°C. For instance at 25°C for the AF 650 phase and 120°C for the FM phase: $n'_{AF}(n'_{FM}) = 4.1(4.2)$, 651 $n''_{AF}(n''_{FM}) = 5(4.4)$ and hence $\alpha_{AF}(\alpha_{FM}) = 8.1(7.2) \times 10^7 \text{ m}^{-1}$. 652 The derivatives are $dn'_{AF}/dT(dn'_{FM}/dT) = 7.6(6.8) \times 10^{-3}$ 653 and $dn''_{AF}/dT(dn''_{FM}/dT) = 4(2.8) \times 10^{-3}$. $S_0$ is defined as 654 $3\beta B\alpha(1-R)F/(\rho_1^2 C_1 v^2)$ where $\beta$ is the thermal expansion 655 coefficient, and $B$ the bulk modulus. Using data from¹⁹,⁸⁰ we 656 have $S_{0AF}(S_{0FM}) = 3.8(2) \times 10^{-4}$. The derivative $dn/dS$ is 657 not known but taking a large value of 30 consistent with data 658 from the literature⁸¹, we get a smaller contribution of the prop- 659 agating strain in the interferometric signal.

## APPENDIX B: DETAILS ON THE PICOSECOND ACOUSTICS EXPERIMENTS AND MODELLING

The picosecond acoustics pump-probe set-up is described in Peronne et al.⁴⁵. More specifically to these measurements, the laser repetition rate was 80 MHz, with a modulation of 1 MHz. Its wavelength was 773 nm, and the beam diameter was of the order of ≈ 15 µm. The power of the pump beam was around $P=32$ mW, that of the probe around 4 mW. The delay line was scanned mechanically at 40 nm/ps. The model is based on the 1D heat diffusion equation and the propagation equation for elastic waves. Given the short time scale (300 ps), the FeRh layer can be considered as a semi-infinite medium since the thermal wave has not reached the substrate. The surface displacement $u(t, z=0)$, propagating strain $S_{prop}(t, z)$ and temperature increase $\Delta T(t, z)$ are obtained as⁷⁹:

$$u(0,t) = -\frac{S_0}{\alpha} \left( 1 - \frac{x\exp(-\omega_\alpha t)}{x+1} - \frac{x\exp(\omega_D t)}{x^2-1} \text{Erfc}(\sqrt{\omega_D t}) + \frac{\exp(x\omega_\alpha t)}{x^2-1} \text{Erfc}(\sqrt{x\omega_\alpha t}) \right) \quad (7)$$

$$S_{prop} = \begin{cases} z - vt > 0 & \frac{S_0}{1-x^2} (\exp(-\omega_D (z/v - t)) - x\exp(-\omega_\alpha (z/v - t))) \\ z - vt < 0 & \frac{S_0}{1-x^2} (\exp(\omega_D (t-z/v)) \text{Erfc}(\sqrt{\omega_D (t-z/v)}) - x\exp(x\omega_\alpha (t-z/v)) \text{Erfc}(\sqrt{x\omega_\alpha (t-z/v)}) \end{cases} \quad (8)$$

$$\Delta T(t,z) = \frac{F(1-R)\alpha}{2\rho_1 C_1} \exp(\omega_D t) \left( \exp(-\alpha z) \text{Erfc}\left(\frac{-z+2D_1\alpha t}{2\sqrt{D_1 t}}\right) + \exp(\alpha z) \text{Erfc}\left(\frac{z+2D_1\alpha t}{2\sqrt{D_1 t}}\right) \right), \quad (9)$$

## APPENDIX C: METHOD AND COMPUTATIONAL DETAILS FOR LATTICE THERMAL CONDUCTIVITY

### Temperature dependent effective potential

The temperature dependent effective potential⁵²,⁵⁶,⁵⁷ method is based on the construction of an effective anharmonic Hamiltonian of the form

$$H = \sum_i \frac{p_i^2}{2M_i} + \frac{1}{2} \sum_{ij} \sum_{ab} \Theta_{ij}^{ab} u_i^a u_j^b + \frac{1}{3!} \sum_{ijk} \sum_{abc} \Psi_{ijk}^{abc} u_i^a u_j^b u_k^c \quad (10)$$

where $u_i^a$ and $p_i$ are respectively the displacement along the cartesian direction $a$ and momentum of atom $i$ with mass $M_i$, and $\Theta_{ij}^{ab}$ and $\Psi_{ijk}^{abc}$ are the second and third order interatomic force constants, which we fit to molecular dynamics data using ordinary least-squares methods. The effectiveness of this Hamiltonian comes from the iterative nature of the fit, where each order is fit on the residual of the previous order. While this procedure ensures that the lowest orders intrinsically include most of the anharmonicity, it also produces, as the name suggests, a temperature-dependent effective potential, so that individual MD simulations have to be run for each temperature of interest.

Once the interatomic force constants are extracted from
MD simulations, the thermal conductivity can be computed using the Boltzmann equation. In this work, the solution of this equation was obtained with an iterative algorithm⁵⁰, after which the thermal conductivity tensor is written as

$$ \kappa_{ph}^{ab} = \frac{1}{V} \sum_{\lambda} c_{\lambda} v_{\lambda}^{a} F_{\lambda}^{b} \quad (11) $$

with V the volume of the unit cell, $c_λ$ the heat capacity associated with mode $λ$, $v_λ^a$ the group velocity of mode $λ$ along cartesian direction a, and $F_λ^a$ is the non-equilibrium phonon distribution accounting for the phonon relaxation, for which the BTE must be solved self-consistently. The Boltzmann equation was solved on a 25 × 25 × 25 q-point grid and using an adaptive gaussian scheme for the Dirac delta⁸².

For both the extraction of the interatomic force constants and the computation of the thermal conductivity, we used the implementation provided by the TDEP package⁸³.

## Machine-learning interatomic potential and Molecular dynamics

To decrease the important computational cost associated with the MD simulations, we used two machine-learning interatomic potential (MLIP) within the Moment Tensor Potential⁵⁹ formalism, one for the AF phase and the other one for the FM phase. The MLIP were fit on DFT calculations performed with the Abinit suite⁷³ using the PBE parameterization of the exchange and correlation functional⁸⁴ in the PAW formalism⁸⁵,⁸⁶. These MLIP were successfully used to study the elastic properties of FeRh and we refer to our previous work⁸⁰ for more details about the parameters and dataset used in their construction.

With the MLIP, we compute the effective anharmonic Hamiltonian from 100 to 500 K in steps of 100 K. For each temperature, we run two 100 ps MD simulations on 8 × 8 × 8 supercells, with a time step of 1 fs using the LAMMPS package⁸⁷. The first MD run is performed in the NPT ensemble, and is used to compute the average equilibrium volume, while the second one employs this equilibrium volume in the NVT ensemble, in order to compute the renormalized interatomic force constants. These force constants are computed using 900 uncorrelated configurations, extracted from the NVT MD trajectory after 10 ps of equilibration.

## Temperature dependent phonons

The temperature dependent phonon dispersions of both phases are plotted on Fig. 8. For the FM phase, we observe only a slight softening of the phonon frequencies with an increase in temperature, indicating a relatively small anharmonicity in the system even at a temperature of 500 K. It should be noted that most of this temperature dependence can be attributed to the thermal expansion. While most of the modes in the AF phase show similar behavior, this is not the case for the lowest energy mode located at the high symmetry

X point. This mode, which we find to be imaginary in the harmonic approximation (as in previous literature, e.g., 22,53–55) has instead the opposite trend: a strong hardening with temperature. We note that even at 100 K, which is the lowest temperature studied here, this mode and the whole spectrum are fully real, demonstrating the stability of the AF phase.

<figure><img src="image_8.png" /><figcaption>FIG. 8. Temperature dependent phonons for a) the FM and b) the AF phase of FeRh. In b) the harmonic phonons computed with finite difference to highlight the stabilization of the system by anharmonicity. The phonons in the FM phase are computed using the B2 structure as the unit cell, while the magnetic unit cell was used for the AF phase.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. 9. a) Theoretical lattice thermal conductivity of FeRh computed for the AF phase, in blue, and the FM phase, in red. The harmonic results, represented with dashed lines, are obtained with IFC computed at T = 25 K. b) Difference of thermal conductivity between the FM and the AF phases.</figcaption></figure>

Using these force constants as input, we computed the thermal conductivity by solving the iterative Boltzmann equation for each temperature. Our results, Fig. 9, show a slightly
MD simulations, the thermal conductivity can be computed using the Boltzmann equation. In this work, the solution of this equation was obtained with an iterative algorithm⁵⁰, after which the thermal conductivity tensor is written as

$$ \kappa_{ph}^{ab} = \frac{1}{V} \sum_{\lambda} c_{\lambda} v_{\lambda}^{a} F_{\lambda}^{b} \quad (11) $$

with V the volume of the unit cell, $c_λ$ the heat capacity associated with mode $λ$, $v_λ^a$ the group velocity of mode $λ$ along cartesian direction a, and $F_λ^a$ is the non-equilibrium phonon distribution accounting for the phonon relaxation, for which the BTE must be solved self-consistently. The Boltzmann equation was solved on a 25 × 25 × 25 q-point grid and using an adaptive gaussian scheme for the Dirac delta⁸².

For both the extraction of the interatomic force constants and the computation of the thermal conductivity, we used the implementation provided by the TDEP package⁸³.

## Machine-learning interatomic potential and Molecular dynamics

To decrease the important computational cost associated with the MD simulations, we used two machine-learning interatomic potential (MLIP) within the Moment Tensor Potential⁵⁹ formalism, one for the AF phase and the other one for the FM phase. The MLIP were fit on DFT calculations performed with the Abinit suite⁷³ using the PBE parameterization of the exchange and correlation functional⁸⁴ in the PAW formalism⁸⁵,⁸⁶. These MLIP were successfully used to study the elastic properties of FeRh and we refer to our previous work⁸⁰ for more details about the parameters and dataset used in their construction.

With the MLIP, we compute the effective anharmonic Hamiltonian from 100 to 500 K in steps of 100 K. For each temperature, we run two 100 ps MD simulations on 8 × 8 × 8 supercells, with a time step of 1 fs using the LAMMPS package⁸⁷. The first MD run is performed in the NPT ensemble, and is used to compute the average equilibrium volume, while the second one employs this equilibrium volume in the NVT ensemble, in order to compute the renormalized interatomic force constants. These force constants are computed using 900 uncorrelated configurations, extracted from the NVT MD trajectory after 10 ps of equilibration.

## Temperature dependent phonons

The temperature dependent phonon dispersions of both phases are plotted on Fig. 8. For the FM phase, we observe only a slight softening of the phonon frequencies with an increase in temperature, indicating a relatively small anharmonicity in the system even at a temperature of 500 K. It should be noted that most of this temperature dependence can be attributed to the thermal expansion. While most of the modes in the AF phase show similar behavior, this is not the case for the lowest energy mode located at the high symmetry

X point. This mode, which we find to be imaginary in the harmonic approximation (as in previous literature, e.g., 22,53–55) has instead the opposite trend: a strong hardening with temperature. We note that even at 100 K, which is the lowest temperature studied here, this mode and the whole spectrum are fully real, demonstrating the stability of the AF phase.

<figure><img src="image_9.png" /><figcaption>FIG. 8. Temperature dependent phonons for a) the FM and b) the AF phase of FeRh. In b) the harmonic phonons computed with finite difference to highlight the stabilization of the system by anharmonicity. The phonons in the FM phase are computed using the B2 structure as the unit cell, while the magnetic unit cell was used for the AF phase.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIG. 9. a) Theoretical lattice thermal conductivity of FeRh computed for the AF phase, in blue, and the FM phase, in red. The harmonic results, represented with dashed lines, are obtained with IFC computed at T = 25 K. b) Difference of thermal conductivity between the FM and the AF phases.</figcaption></figure>

Using these force constants as input, we computed the thermal conductivity by solving the iterative Boltzmann equation for each temperature. Our results, Fig. 9, show a slightly
<figure><img src="image_10.png" /><figcaption>FIG. 10. Phonon lifetimes at 300 K in a) the AF phase and b) the FM phase. In both figures, the harmonic results are taken without accounting for the temperature dependence of the phonons.</figcaption></figure>

739 larger $\kappa_{ph}$ for the AF phase than for the FM phase above 150 K.

740 K. As already stated in the main text, these finding differs

741 from the results of Cazorla and Rurali²⁶, where the opposite

742 trend was found based on the harmonic approximation at the

743 OK relaxed volume. To better understand the origin of this discrepancy, we computed the thermal conductivities with our MLIP using analogous approximations. Due to the imaginary mode in the AF phase, and in order to provide a meaningful comparison, we used IFC computed at a temperature of 25 K and at the reference relaxed volume for both phases, instead of the harmonic ones. As shown in Fig. 9, these approximations yield the same trend as in Ref. 26, with $\kappa_{ph,FM} > \kappa_{ph,AF}$ for all temperatures. To explain the opposite trend obtained with fully anharmonic IFC, we compare the T = 300 K phonon lifetimes obtained with the two approaches in Fig. 10.

745 The FM phase presents only a slight evolution of the lifetimes, confirming the low anharmonicity of this phase. On the contrary, the renormalization of the IFC brings an important increase of the phonon lifetimes in the AF phase. This counter-intuitive effect can be understood as the decrease of the scattering phase space due to the renormalization of the phonons⁸⁸. In the AF phase of FeRh, this effect is sufficiently strong for $\kappa_{ph,AF}$ to exceed $\kappa_{ph,FM}$.

762 ¹ M. Richardson, D. Melville, and J. A. R codeau, Physics Letters A 46, 153 (1973).

763 ² M. P. Annaorazov, K. A. Asatryan, G. Myalikgulyev, S. A. Nikitin, A. M. Tishin, and A. L. Tyurin, Cryogenics 32, 867 (1992).

767 ³ X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He, R. J. Paull, J. D. Clarkson, J. Kudrnovský, I. Turek, J. Kuneš, D. Yi, J.-H. Chu, C. T. Nelson, L. You, E. Arenholz, S. Salahuddin, J. Fontcuberta, T. Jungwirth, and R. Ramesh, Nature Materials 13, 367 (2014), arXiv:0402594v3 [arXiv:cond-mat].

772 ⁴ J.-U. Thiele, S. Maat, and E. E. Fullerton, Applied Physics Letters 82, 2859 (2003).

774 ⁵ S. Cervera, M. Trassinelli, M. Marangolo, C. Carrétéro, V. Garcia, S. Hidki, E. Jacquet, E. Lamour, A. Lévy, S. Macé, C. Prigent, J. P. Rozet, S. Steydli, and D. Vernhet, Physical Review Materials 1, 065402 (2017).

778 ⁶ J.-U. Thiele, M. Buess, and C. H. Back, Applied Physics Letters 85, 2857 (2004).

780 ⁷ G. Ju, J. Hohlfeld, B. Bergman, R. J. M. van de Veerdonk, O. N. Mryasov, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, Physical Review Letters 93, 197403 (2004).

783 ⁸ B. Bergman, G. Ju, J. Hohlfeld, R. J. M. van de Veerdonk, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, Physical Review B 73, 060407 (2006).

786 ⁹ I. Radu, C. Stamm, N. Pontius, T. Kachel, P. Ramm, J.-U. Thiele, H. A. Dürr, and C. H. Back, Physical Review B 81, 104415 (2010).

789 ¹⁰ F. Quirin, M. Vattilana, U. Shymanovich, A.-E. El-Kamhawy, A. Tarasevitch, J. Hohlfeld, D. von der Linde, and K. Sokolowski-Tinten, Physical Review B 85, 020103(R) (2012).

792 ¹¹ S. O. Mariager, F. Pressacco, G. Ingold, A. Caviezel, E. Möhr Vorobeva, P. Beaud, S. L. Johnson, C. J. Milne, E. Mancini, S. Moyerman, E. E. Fullerton, R. Feidenhans’l, C. H. Back, and C. Quitmann, Physical Review Letters 108, 087201 (2012).

796 ¹² F. Pressacco, D. Sangalli, V. Uhlíř, D. Kutnyakhov, J. A. Arregi, S. Y. Agustsson, G. Brenner, H. Redlin, M. Heber, D. Vasilyev, J. Demsar, G. Schönhense, M. Gatti, A. Marini, W. Wurth, and F. Sirotti, Nature Communications 12, 5088 (2021), arXiv:2102.09265.

800 ¹³ A. B. Mei, I. Gray, Y. Tang, J. Schubert, D. Werder, J. Bartell, D. C. Ralph, G. D. Fuchs, and D. G. Schlom, Advanced Materials 32, 2001080 (2020).

804 ¹⁴ G. Li, R. Medapalli, J. H. Mentink, R. V. Mikhaylovskiy, T. G. H. Blank, S. K. K. Patel, A. K. Zvezdin, T. Rasing, E. E. Fullerton, and A. V. Kimel, Nature Communications 13, 2998 (2022).

807 ¹⁵ J. A. Arregi, O. Caha, and V. Uhlíř, Physical Review B 101, 174413 (2020).

809 ¹⁶ D. W. Cooke, F. Hellman, C. Baldasseroni, C. Bordel, S. Moyerman, and E. E. Fullerton, Physical Review Letters 109, 255901 (2012).

812 ¹⁷ S. Shihab, L. Thevenard, A. Lemaître, J. Y. Duquesne, and C. Gourdon, Journal of Applied Physics 119, 153904 (2016).

814 ¹⁸ F. Pressacco, V. Uhlíř, M. Gatti, A. Nicolaou, A. Bendounan, J. A. Arregi, S. K. K. Patel, E. E. Fullerton, D. Krizmancic, and F. Sirotti, Structural Dynamics 5, 034501 (2018).

817 ¹⁹ Y. Ahn, J. Zhang, Z. Chu, D. A. Walko, S. O. Hruszkewycz, E. E. Fullerton, P. G. Evans, and H. Wen, ACS Nano 17, 18843 (2023).

819 ²⁰ T. Moriyama, N. Matsuzaki, K.-J. Kim, I. Suzuki, T. Taniyama, and T. Ono, Applied Physics Letters 107, 122403 (2015).

821 ²¹ B. Fogarassy, T. Kemény, L. Pál, and J. Tóth, Physical Review Letters 29, 288 (1972).

823 ²² U. Aschauer, R. Braddell, S. A. Brechbühl, P. M. Derlet, and N. A. Spaldin, Physical Review B 94, 014109 (2016), arXiv:1603.01827.

826 ²³ W. He, H. Huang, and X. Ma, Materials Letters 195, 156 (2017).

827 ²⁴ M. Wolloch, M. E. Gruner, W. Keune, P. Mohn, J. Redinger, F. Hofer, D. Suess, R. Podloucky, J. Landers, S. Salamon, F. Scheibel, D. Spoddig, R. Witte, B. Roldan Cuenya, O. Gutfleisch, M. Y. Hu, J. Zhao, T. Toellner, E. E. Alp, M. Siewert, P. Entel, R. Pentcheva, and H. Wende, Physical Review B 94, 174435 (2016).

833 ²⁵ M. J. Jiménez, A. B. Schvval, and G. F. Cabeza, Computational Materials Science 172, 109385 (2020).

834 ²⁶ C. Cazorla and R. Rurali, Physical Review B 105, 104401 (2022).

836 ²⁷ Y. Hao, L. Zhang, and J. Zhu, Zeitschrift für Naturforschung A 75, 789 (2020).
<figure><img src="image_11.png" /><figcaption>FIG. 10. Phonon lifetimes at 300 K in a) the AF phase and b) the FM phase. In both figures, the harmonic results are taken without accounting for the temperature dependence of the phonons.</figcaption></figure>

739 larger $\kappa_{ph}$ for the AF phase than for the FM phase above 150

740 K. As already stated in the main text, these finding differs

741 from the results of Cazorla and Rurali²⁶, where the opposite

742 trend was found based on the harmonic approximation at the

743 OK relaxed volume. To better understand the origin of this discrepancy, we computed the thermal conductivities with our MLIP using analogous approximations. Due to the imaginary mode in the AF phase, and in order to provide a meaningful comparison, we used IFC computed at a temperature of 25 K and at the reference relaxed volume for both phases, instead of the harmonic ones. As shown in Fig. 9, these approximations yield the same trend as in Ref. 26, with $\kappa_{ph,FM} > \kappa_{ph,AF}$ for all temperatures. To explain the opposite trend obtained with fully anharmonic IFC, we compare the $T = 300$ K phonon lifetimes obtained with the two approaches in Fig. 10.

745 The FM phase presents only a slight evolution of the lifetimes, confirming the low anharmonicity of this phase. On the contrary, the renormalization of the IFC brings an important increase of the phonon lifetimes in the AF phase. This counter-intuitive effect can be understood as the decrease of the scattering phase space due to the renormalization of the phonons⁸⁸. In the AF phase of FeRh, this effect is sufficiently strong for $\kappa_{ph,AF}$ to exceed $\kappa_{ph,FM}$.

762 ¹ M. Richardson, D. Melville, and J. A. R codeau, Physics Letters A 46, 153 (1973).

763 ² M. P. Annaorazov, K. A. Asatryan, G. Myalikgulyev, S. A. Nikitin, A. M. Tishin, and A. L. Tyurin, Cryogenics 32, 867 (1992).

767 ³ X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He, R. J. Paull, J. D. Clarkson, J. Kudrnovský, I. Turek, J. Kuneš, D. Yi, J.-H. Chu, C. T. Nelson, L. You, E. Arenholz, S. Salahuddin, J. Fontcuberta, T. Jungwirth, and R. Ramesh, Nature Materials 13, 367 (2014), arXiv:0402594v3 [arXiv:cond-mat].

772 ⁴ J.-U. Thiele, S. Maat, and E. E. Fullerton, Applied Physics Letters 82, 2859 (2003).

774 ⁵ S. Cervera, M. Trassinelli, M. Marangolo, C. Carrétéro, V. Garcia, S. Hidki, E. Jacquet, E. Lamour, A. Lévy, S. Macé, C. Prigent, J. P. Rozet, S. Steydli, and D. Vernhet, Physical Review Materials 1, 065402 (2017).

778 ⁶ J.-U. Thiele, M. Buess, and C. H. Back, Applied Physics Letters 85, 2857 (2004).

780 ⁷ G. Ju, J. Hohlfeld, B. Bergman, R. J. M. van de Veerdonk, O. N. Mryasov, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, Physical Review Letters 93, 197403 (2004).

783 ⁸ B. Bergman, G. Ju, J. Hohlfeld, R. J. M. van de Veerdonk, J.-Y. Kim, X. Wu, D. Weller, and B. Koopmans, Physical Review B 73, 060407 (2006).

786 ⁹ I. Radu, C. Stamm, N. Pontius, T. Kachel, P. Ramm, J.-U. Thiele, H. A. Dürr, and C. H. Back, Physical Review B 81, 104415 (2010).

789 ¹⁰ F. Quirin, M. Vattilana, U. Shymanovich, A.-E. El-Kamhawy, A. Tarasevitch, J. Hohlfeld, D. von der Linde, and K. Sokolowski-Tinten, Physical Review B 85, 020103(R) (2012).

792 ¹¹ S. O. Mariager, F. Pressacco, G. Ingold, A. Caviezel, E. Möhr Vorobeva, P. Beaud, S. L. Johnson, C. J. Milne, E. Mancini, S. Moyerman, E. E. Fullerton, R. Feidenhans’l, C. H. Back, and C. Quitmann, Physical Review Letters 108, 087201 (2012).

796 ¹² F. Pressacco, D. Sangalli, V. Uhlíř, D. Kutnyakhov, J. A. Arregi, S. Y. Agustsson, G. Brenner, H. Redlin, M. Heber, D. Vasilyev, J. Demsar, G. Schönhense, M. Gatti, A. Marini, W. Wurth, and F. Sirotti, Nature Communications 12, 5088 (2021), arXiv:2102.09265.

800 ¹³ A. B. Mei, I. Gray, Y. Tang, J. Schubert, D. Werder, J. Bartell, D. C. Ralph, G. D. Fuchs, and D. G. Schlom, Advanced Materials 32, 2001080 (2020).

804 ¹⁴ G. Li, R. Medapalli, J. H. Mentink, R. V. Mikhaylovskiy, T. G. H. Blank, S. K. K. Patel, A. K. Zvezdin, T. Rasing, E. E. Fullerton, and A. V. Kimel, Nature Communications 13, 2998 (2022).

807 ¹⁵ J. A. Arregi, O. Caha, and V. Uhlíř, Physical Review B 101, 174413 (2020).

809 ¹⁶ D. W. Cooke, F. Hellman, C. Baldasseroni, C. Bordel, S. Moyerman, and E. E. Fullerton, Physical Review Letters 109, 255901 (2012).

812 ¹⁷ S. Shihab, L. Thevenard, A. Lemaître, J. Y. Duquesne, and C. Gourdon, Journal of Applied Physics 119, 153904 (2016).

814 ¹⁸ F. Pressacco, V. Uhlíř, M. Gatti, A. Nicolaou, A. Bendounan, J. A. Arregi, S. K. K. Patel, E. E. Fullerton, D. Krizmancic, and F. Sirotti, Structural Dynamics 5, 034501 (2018).

817 ¹⁹ Y. Ahn, J. Zhang, Z. Chu, D. A. Walko, S. O. Hruszkewycz, E. E. Fullerton, P. G. Evans, and H. Wen, ACS Nano 17, 18843 (2023).

819 ²⁰ T. Moriyama, N. Matsuzaki, K.-J. Kim, I. Suzuki, T. Taniyama, and T. Ono, Applied Physics Letters 107, 122403 (2015).

821 ²¹ B. Fogarassy, T. Kemény, L. Pál, and J. Tóth, Physical Review Letters 29, 288 (1972).

823 ²² U. Aschauer, R. Braddell, S. A. Brechbühl, P. M. Derlet, and N. A. Spaldin, Physical Review B 94, 014109 (2016), arXiv:1603.01827.

826 ²³ W. He, H. Huang, and X. Ma, Materials Letters 195, 156 (2017).

827 ²⁴ M. Wolloch, M. E. Gruner, W. Keune, P. Mohn, J. Redinger, F. Hofer, D. Suess, R. Podloucky, J. Landers, S. Salamon, F. Scheibel, D. Spoddig, R. Witte, B. Roldan Cuenya, O. Gutfleisch, M. Y. Hu, J. Zhao, T. Toellner, E. E. Alp, M. Siewert, P. Entel, R. Pentcheva, and H. Wende, Physical Review B 94, 174435 (2016).

833 ²⁵ M. J. Jiménez, A. B. Schvval, and G. F. Cabeza, Computational Materials Science 172, 109385 (2020).

834 ²⁶ C. Cazorla and R. Rurali, Physical Review B 105, 104401 (2022).

836 ²⁷ Y. Hao, L. Zhang, and J. Zhu, Zeitschrift für Naturforschung A 75, 789 (2020).
838 28 V. Uhlíř, J. A. Arregi, and E. E. Fullerton, Nature Communications 7, 13113 (2016), arXiv:1605.06823.

839 29 X. Wu, Z. Liu, and T. Luo, Journal of Applied Physics 123, 085109 (2018), arXiv:1712.09002.

840 30 D. Fournier, M. Marangolo, and C. Frétigny, Journal of Applied Physics 128, 241101 (2020).

841 31 M. Rahimi, K. Sobnath, F. Mallet, P. Lafarge, C. Barraud, W. Daney De Marcillac, D. Fournier, and M. L. Della Rocca, Physical Review Applied 19, 1 (2023).

842 32 C. Frétigny, J. P. Roger, V. Reita, and D. Fournier, Journal of Applied Physics 102, 116104 (2007).

843 33 X. Qian, Z. Ding, J. Shin, A. J. Schmidt, and G. Chen, Review of Scientific Instruments 91 (2020), 10.1063/5.0003770.

844 34 V. Saidl, M. Brajer, L. Horák, H. Reichlová, K. Výborný, M. Veis, T. Janda, F. Trojánek, M. Maryško, I. Fina, X. Marti, T. Jungwirth, and P. Němec, New Journal of Physics 18, 083017 (2016).

845 35 S. P. Bennett, M. Currie, O. M. J. van 't Erve, and I. I. Mazin, Optical Materials Express 9, 2870 (2019).

846 36 J. A. Arregi, F. Ringe, J. Hajduček, O. Gomonay, T. Molnár, J. Jaskowiec, and V. Uhlíř, Journal of Physics: Materials 6, 034003 (2023).

847 37 A. Rosencwaig and A. Gersho, Journal of Applied Physics 47, 64 (1976).

848 38 With α tending to infinity, one recovers the usual expression used in thermoreflectance microscopy (Eq. (10) of Ref. 30). Ignoring the absorption of the pump beam leads to an overestimate of the thermal conductivity of about 5% when analyzing our data.

849 39 C. Frétigny, J.-Y. Duquesne, D. Fournier, and F. Xu, Journal of Applied Physics 111, 084313 (2012).

850 40 Y. Xian, P. Zhang, S. Zhai, P. Yuan, and D. Yang, Applied Thermal Engineering 130, 1530 (2018).

851 41 J.-Y. Duquesne, Physical Review B 79, 153304 (2009).

852 42 H.-K. Lyeo and D. G. Cahill, Physical Review B 73, 144301 (2006).

853 43 J. Zhu, D. Tang, W. Wang, J. Liu, K. W. Holub, and R. Yang, Journal of Applied Physics 108 (2010), 10.1063/1.3504213.

854 44 C. Frétigny, J. Y. Duquesne, and D. Fournier, International Journal of Thermophysics 36, 1281 (2015).

855 45 E. Péronne, N. Chuecos, L. Thevenard, and B. Perrin, Physical Review B 95, 064306 (2017).

856 46 In that case, the transient out-of-plane strain induced by a pulsed laser is measured using time-resolved X-ray diffraction. $\kappa_{FeRh}$ and $R_{th}$ are obtained simultaneously by fitting the time-dependent thickness-averaged strain variations using a layer-resolved 1D thermal transport model, and a DC measurement of the linear thermal expansion.

857 47 Y. S. Touloukian, Thermophysical Properties of Matter: Thermal Diffusivity (Plenum Press, 1973).

858 48 K.-W. Kim, H. W. Lee, K. J. Lee, and M. D. Stiles, Physical Review Letters 111, 1 (2013), arXiv:1308.1198.

859 49 G. K. White and S. B. Woods, Canadian Journal of Physics 35, 248 (1957).

860 50 M. Omini and A. Sparavigna, Physica B: Condensed Matter 212, 101 (1995).

861 51 A. J. H. McGaughey, A. Jain, H.-Y. Kim, and B. Fu, Journal of Applied Physics 125, 011101 (2019).

862 52 O. Hellman and D. A. Broido, Physical Review B 90, 134309 (2014).

863 53 J. Kim, R. Ramesh, and N. Kioussis, Physical Review B 94, 180407(R) (2016).

864 54 N. A. Zarkevich and D. D. Johnson, Physical Review B 97, 014202 (2018), arXiv:1710.04199.

865 55 M. P. Belov, A. B. Syzdykova, and I. A. Abrikosov, Physical Review B 101, 134303 (2020).

902 56 O. Hellman, I. A. Abrikosov, and S. I. Simak, Physical Review B 84, 180301 (2011).

903 57 O. Hellman and I. A. Abrikosov, Physical Review B 88, 144301 (2013).

904 58 I. S. Novikov, K. Gubaev, E. V. Podryabinkin, and A. V. Shapeev, Machine Learning: Science and Technology 2, 025002 (2021).

905 59 A. V. Shapeev, Multiscale Modeling & Simulation 14, 1153–1173 (2016).

906 60 Y. Zuo, C. Chen, X. Li, Z. Deng, Y. Chen, J. Behler, G. Csányi, A. V. Shapeev, A. P. Thompson, M. A. Wood, and S. P. Ong, J. Phys. Chem. A 124, 731 (2020).

907 61 R. Y. Gu and V. P. Antropov, Physical Review B 72, 012403 (2005).

908 62 B. Zink, A. Avery, R. Sultan, D. Bassett, and M. Pufall, Solid State Communications 150, 514 (2010).

909 63 S. Mankovsky, S. Polesya, K. Chadova, H. Ebert, J. B. Staunton, T. Gruenbaum, M. A. W. Schoen, C. H. Back, X. Z. Chen, and C. Song, Physical Review B 95, 155139 (2017).

910 64 E. Mancini, F. Pressacco, M. Haertinger, E. E. Fullerton, T. Suzuki, G. Woltersdorf, and C. H. Back, Journal of Physics D: Applied Physics 46, 245302 (2013).

911 65 K. Tanaka, T. Moriyama, T. Usami, T. Taniyama, and T. Ono, Applied Physics Express 11, 013008 (2017).

912 66 T. Usami, M. Itoh, and T. Taniyama, AIP Advances 11, 045302 (2021).

913 67 F. Mahfouzi and N. Kioussis, Physical Review B 98, 220410(R) (2018).

914 68 H. T. Simensen, A. Kamra, R. E. Troncoso, and A. Brataas, Physical Review B 101, 020403 (2020).

915 69 A. Castets, D. Tochetti, and B. Hennion, Physica B+C 86-88, 353 (1977).

916 70 L. M. Sandratskii and P. Buczek, Physical Review B - Condensed Matter and Materials Physics 85, 1 (2012).

917 71 E. T. Swartz and R. O. Pohl, Appl. Phys. Lett. 51, 200 (1987).

918 72 E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989).

919 73 X. Gonze, B. Amadon, G. Antonius, F. Arnardi, L. Baguet, J.-M. Beuken, J. Bieder, F. Bottin, J. Bouchet, E. Bousquet, N. Brouwer, F. Bruneval, G. Brunin, T. Cavignac, J.-B. Charraud, W. Chen, M. Côté, S. Cottenier, J. Denier, G. Geneste, P. Ghosez, M. Gi-antomassi, Y. Gillet, O. Gingras, D. R. Hamann, G. Hautier, X. He, N. Helbig, N. Holzwarth, Y. Jia, F. Jollet, W. Lafargue-Dit-Hauret, K. Lejaeghere, M. A. Marques, A. Martin, C. Martins, H. P. Miranda, F. Naccarato, K. Persson, G. Petretto, V. Planes, Y. Pouillon, S. Prokhorenko, F. Ricci, G.-M. Rignanese, A. H. Romero, M. M. Schmitt, M. Torrent, M. J. van Setten, B. V. Troeye, M. J. Verstraete, G. Zérah, and J. W. Zwanziger, Computer Physics Communications 248, 107042 (2020).

920 74 P. Reddy, K. Castelino, and A. Majumdar, Applied Physics Letters 87, 211908 (2005).

921 75 Note that in the DMM, the orientation around the axis normal (rotation of FeRh by 45° in our case) is not taken into account.

922 76 H. T. Huang, M. F. Lai, Y. F. Hou, and Z. H. Wei, Nano Letters 15, 2773 (2015).

923 77 A. M. Hofmeister, Physics and Chemistry of Minerals 41, 361 (2014).

924 78 This is a decent approximation given the weak temperature-dependence of a measured experimentally by Arregi et al.¹⁵.

925 79 B. Perrin, Microscale and Nanoscale Heat Transfer, edited by S. Volz, Topics in Applied Physics, Vol. 107 (Springer Berlin Heidelberg, 2007).

926 80 D. Ourdani, A. Castellano, A. K. Vythelingum, J. A. Arregi, V. Uhlíř, B. Perrin, M. Belmeguenai, Y. Roussigné, C. Gourdon, M. J. Verstraete, and L. Thevenard, “hal-04526444,” (2024), preprint.
838 28 V. Uhlíř, J. A. Arregi, and E. E. Fullerton, Nature Communications 7, 13113 (2016), arXiv:1605.06823.

839 29 X. Wu, Z. Liu, and T. Luo, Journal of Applied Physics 123, 085109 (2018), arXiv:1712.09002.

840 30 D. Fournier, M. Marangolo, and C. Frétigny, Journal of Applied Physics 128, 241101 (2020).

841 31 M. Rahimi, K. Sobnath, F. Mallet, P. Lafarge, C. Barraud, W. Daney De Marcillac, D. Fournier, and M. L. Della Rocca, Physical Review Applied 19, 1 (2023).

842 32 C. Frétigny, J. P. Roger, V. Reita, and D. Fournier, Journal of Applied Physics 102, 116104 (2007).

843 33 X. Qian, Z. Ding, J. Shin, A. J. Schmidt, and G. Chen, Review of Scientific Instruments 91 (2020), 10.1063/5.0003770.

844 34 V. Saidl, M. Brajer, L. Horák, H. Reichlová, K. Výborný, M. Veis, T. Janda, F. Trojánek, M. Maryško, I. Fina, X. Marti, T. Jungwirth, and P. Němec, New Journal of Physics 18, 083017 (2016).

845 35 S. P. Bennett, M. Currie, O. M. J. van 't Erve, and I. I. Mazin, Optical Materials Express 9, 2870 (2019).

846 36 J. A. Arregi, F. Ringe, J. Hajduček, O. Gomonay, T. Molnár, J. Jaskowiec, and V. Uhlíř, Journal of Physics: Materials 6, 034003 (2023).

847 37 A. Rosencwaig and A. Gersho, Journal of Applied Physics 47, 64 (1976).

848 38 With α tending to infinity, one recovers the usual expression used in thermoreflectance microscopy (Eq. (10) of Ref. 30). Ignoring the absorption of the pump beam leads to an overestimate of the thermal conductivity of about 5% when analyzing our data.

849 39 C. Frétigny, J.-Y. Duquesne, D. Fournier, and F. Xu, Journal of Applied Physics 111, 084313 (2012).

850 40 Y. Xian, P. Zhang, S. Zhai, P. Yuan, and D. Yang, Applied Thermal Engineering 130, 1530 (2018).

851 41 J.-Y. Duquesne, Physical Review B 79, 153304 (2009).

852 42 H.-K. Lyeo and D. G. Cahill, Physical Review B 73, 144301 (2006).

853 43 J. Zhu, D. Tang, W. Wang, J. Liu, K. W. Holub, and R. Yang, Journal of Applied Physics 108 (2010), 10.1063/1.3504213.

854 44 C. Frétigny, J. Y. Duquesne, and D. Fournier, International Journal of Thermophysics 36, 1281 (2015).

855 45 E. Péronne, N. Chuecos, L. Thevenard, and B. Perrin, Physical Review B 95, 064306 (2017).

856 46 In that case, the transient out-of-plane strain induced by a pulsed laser is measured using time-resolved X-ray diffraction. $\kappa_{FeRh}$ and $R_{th}$ are obtained simultaneously by fitting the time-dependent thickness-averaged strain variations using a layer-resolved 1D thermal transport model, and a DC measurement of the linear thermal expansion.

857 47 Y. S. Touloukian, Thermophysical Properties of Matter: Thermal Diffusivity (Plenum Press, 1973).

858 48 K.-W. Kim, H. W. Lee, K. J. Lee, and M. D. Stiles, Physical Review Letters 111, 1 (2013), arXiv:1308.1198.

859 49 G. K. White and S. B. Woods, Canadian Journal of Physics 35, 248 (1957).

860 50 M. Omini and A. Sparavigna, Physica B: Condensed Matter 212, 101 (1995).

861 51 A. J. H. McGaughey, A. Jain, H.-Y. Kim, and B. Fu, Journal of Applied Physics 125, 011101 (2019).

862 52 O. Hellman and D. A. Broido, Physical Review B 90, 134309 (2014).

863 53 J. Kim, R. Ramesh, and N. Kioussis, Physical Review B 94, 180407(R) (2016).

864 54 N. A. Zarkevich and D. D. Johnson, Physical Review B 97, 014202 (2018), arXiv:1710.04199.

865 55 M. P. Belov, A. B. Syzdykova, and I. A. Abrikosov, Physical Review B 101, 134303 (2020).

902 56 O. Hellman, I. A. Abrikosov, and S. I. Simak, Physical Review B 84, 180301 (2011).

903 57 O. Hellman and I. A. Abrikosov, Physical Review B 88, 144301 (2013).

904 58 I. S. Novikov, K. Gubaev, E. V. Podryabinkin, and A. V. Shapeev, Machine Learning: Science and Technology 2, 025002 (2021).

905 59 A. V. Shapeev, Multiscale Modeling & Simulation 14, 1153–1173 (2016).

906 60 Y. Zuo, C. Chen, X. Li, Z. Deng, Y. Chen, J. Behler, G. Csányi, A. V. Shapeev, A. P. Thompson, M. A. Wood, and S. P. Ong, J. Phys. Chem. A 124, 731 (2020).

907 61 R. Y. Gu and V. P. Antropov, Physical Review B 72, 012403 (2005).

908 62 B. Zink, A. Avery, R. Sultan, D. Bassett, and M. Pufall, Solid State Communications 150, 514 (2010).

909 63 S. Mankovsky, S. Polesya, K. Chadova, H. Ebert, J. B. Staunton, T. Gruenbaum, M. A. W. Schoen, C. H. Back, X. Z. Chen, and C. Song, Physical Review B 95, 155139 (2017).

910 64 E. Mancini, F. Pressacco, M. Haertinger, E. E. Fullerton, T. Suzuki, G. Woltersdorf, and C. H. Back, Journal of Physics D: Applied Physics 46, 245302 (2013).

911 65 K. Tanaka, T. Moriyama, T. Usami, T. Taniyama, and T. Ono, Applied Physics Express 11, 013008 (2017).

912 66 T. Usami, M. Itoh, and T. Taniyama, AIP Advances 11, 045302 (2021).

913 67 F. Mahfouzi and N. Kioussis, Physical Review B 98, 220410(R) (2018).

914 68 H. T. Simensen, A. Kamra, R. E. Troncoso, and A. Brataas, Physical Review B 101, 020403 (2020).

915 69 A. Castets, D. Tochetti, and B. Hennion, Physica B+C 86-88, 353 (1977).

916 70 L. M. Sandratskii and P. Buczek, Physical Review B - Condensed Matter and Materials Physics 85, 1 (2012).

917 71 E. T. Swartz and R. O. Pohl, Appl. Phys. Lett. 51, 200 (1987).

918 72 E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989).

919 73 X. Gonze, B. Amadon, G. Antonius, F. Arnardi, L. Baguet, J.-M. Beuken, J. Bieder, F. Bottin, J. Bouchet, E. Bousquet, N. Brouwer, F. Bruneval, G. Brunin, T. Cavignac, J.-B. Charraud, W. Chen, M. Côté, S. Cottenier, J. Denier, G. Geneste, P. Ghosez, M. Gionantomassi, Y. Gillet, O. Gingras, D. R. Hamann, G. Hautier, X. He, N. Helbig, N. Holzwarth, Y. Jia, F. Jollet, W. Lafargue-Dit-Hauret, K. Lejaeghere, M. A. Marques, A. Martin, C. Martins, H. P. Miranda, F. Naccarato, K. Persson, G. Petretto, V. Planes, Y. Pouillon, S. Prokhorenko, F. Ricci, G.-M. Rignanese, A. H. Romero, M. M. Schmitt, M. Torrent, M. J. van Setten, B. V. Troeye, M. J. Verstraete, G. Zérah, and J. W. Zwanziger, Computer Physics Communications 248, 107042 (2020).

920 74 P. Reddy, K. Castelino, and A. Majumdar, Applied Physics Letters 87, 211908 (2005).

921 75 Note that in the DMM, the orientation around the axis normal (rotation of FeRh by 45° in our case) is not taken into account.

922 76 H. T. Huang, M. F. Lai, Y. F. Hou, and Z. H. Wei, Nano Letters 15, 2773 (2015).

923 77 A. M. Hofmeister, Physics and Chemistry of Minerals 41, 361 (2014).

924 78 This is a decent approximation given the weak temperature-dependence of a measured experimentally by Arregi et al.¹⁵.

925 79 B. Perrin, Microscale and Nanoscale Heat Transfer, edited by S. Volz, Topics in Applied Physics, Vol. 107 (Springer Berlin Heidelberg, 2007).

926 80 D. Ourdani, A. Castellano, A. K. Vythelingum, J. A. Arregi, V. Uhlíř, B. Perrin, M. Belmeguenai, Y. Roussigné, C. Gourdon, M. J. Verstraete, and L. Thevenard, “hal-04526444,” (2024), preprint.
966 81 D. Royer and E. Dieulesaint, *Elastic Waves in Solids I: Free and Guided Propagation*, Advanced Texts in Physics (Springer, 2000).

969 82 J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, *Physical Review B* **75**, 195121 (2007).

971 83 F. Knoop, N. Shulumba, A. Castellano, J. P. A. Batista, R. Farris, M. J. Verstraete, M. Heine, D. Broido, D. S. Kim, J. Klarbring, I. A. Abrikosov, S. I. Simak, and O. Hellman, *Journal of Open Source Software* **9**, 6150 (2024).

975 84 J. P. Perdew, K. Burke, and M. Ernzerhof, *Physical Review Letters* **77**, 3865 (1996).

977 85 P. E. Blöchl, *Physical Review B* **50**, 17953 (1994).

978 86 F. Jollet, M. Torrent, and N. Holzwarth, *Computer Physics Communications* **185**, 1246 (2014).

979 87 A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in 't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, *Computer Physics Communications* **271**, 108171 (2022).

981 88 Y. Yang, L. Zhao, D. Yi, T. Xu, Y. Chai, C. Zhang, D. Jiang, Y. Ji, W. Jiang, J. Tang, P. Yu, H. Wu, and T. Nan, (2022), arXiv:2212.04049.
966 81 D. Royer and E. Dieulesaint, *Elastic Waves in Solids I: Free and Guided Propagation*, Advanced Texts in Physics (Springer, 2000).

969 82 J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, *Physical Review B* **75**, 195121 (2007).

971 83 F. Knoop, N. Shulumba, A. Castellano, J. P. A. Batista, R. Farris, M. J. Verstraete, M. Heine, D. Broido, D. S. Kim, J. Klarbring, I. A. Abrikosov, S. I. Simak, and O. Hellman, *Journal of Open Source Software* **9**, 6150 (2024).

975 84 J. P. Perdew, K. Burke, and M. Ernzerhof, *Physical Review Letters* **77**, 3865 (1996).

977 85 P. E. Blöchl, *Physical Review B* **50**, 17953 (1994).

978 86 F. Jollet, M. Torrent, and N. Holzwarth, *Computer Physics Communications* **185**, 1246 (2014).

979 87 A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in 't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, *Computer Physics Communications* **271**, 108171 (2022).

981 88 Y. Yang, L. Zhao, D. Yi, T. Xu, Y. Chai, C. Zhang, D. Jiang, Y. Ji, W. Jiang, J. Tang, P. Yu, H. Wu, and T. Nan, (2022), arXiv:2212.04049.