How closely does transient magnetic linear dichroism follow the spin moment?

TL;DR

This study examines the reliability of magnetic linear dichroism (MLD) as a probe of spin moments in out-of-equilibrium magnetic states, revealing its limitations in antiferromagnetic materials at high laser fluences.

Contribution

It demonstrates that MLD accurately tracks spin moments in ferromagnetic materials but fails in antiferromagnetic FePd under high laser excitation, highlighting the influence of electronic structure.

Findings

MLD's accuracy degrades in AFM FePd at high fluences.

MLD reliably tracks moments in ferromagnetic FePt.

Laser-induced excitations out of the $d$-band affect MLD response.

Abstract

In highly out-of-equilibrium states of matter, such as those induced by a pump laser, the applicability of well established spectroscopic probes of magnetic order are called into question. Here we address the validity of x-ray absorption techniques in pump laser conditions, focusing on magnetic linear dichroism (MLD), a crucial probe of antiferromagnetic (AFM) order. We directly compute the dynamics of the square of the spin moment and compare to those obtained via the MLD response. For AFM FePd the agreement between these distinct routes to the magnetic moment severely degrades at pulse fluences greater than 1 mJ/cm$^2$, indicating a breakdown of the MLD response as an accurate probe of the transient moment. This contrasts with the MLD for ferromagnetic FePt which reliably tracks the moment for fluences (and absorbed energies) up to an order of magnitude greater than the breakdown threshold for AFM FePd. The underlying microscopic reason for this we find to be increased laser induced excitations out of the $d$-band in AFM FePd, where this increase is made possible by the AFM pseudogap.

Equations

$$\mathbf{m}(\mathbf{r},t)=\sum _j{\psi }_j^{\ast }(\mathbf{r},t)\hat{\mathbf{s}}{\psi }_j(\mathbf{r},t),$$

$${\mathbf{M}}^2\propto \Im ({ϵ}_{\parallel }-{ϵ}_{\perp }),$$

Full text

Introduction: Antiferromagnets exhibit a variety of desirable characteristicsJungwirth et al. (2018); Olejník et al. (2018); Pal et al. (2022); Jungwirth et al. (2016); Vergès et al. (2020); Fabiani et al. (2021); Dannegger et al. (2021); Bossini et al. (2019); Němec et al. (2018); Wadley et al. (2016); Baltz et al. (2018), for instance reversible current-driven switching at THz speedOlejník et al. (2018), identifying them as key materials in future spintronics technologies. The magnetisation dynamics of antiferromagnetic (AFM) materials has, furthermore, proved to be exceptionally interesting at ultrafast timesFabiani et al. (2021); Dannegger et al. (2021); Bossini et al. (2019); Němec et al. (2018), with pump laser induced switching of AFM to ferromagnetic (FM) order on femtosecond time scales being recently proposed Dewhurst et al. (2018a). The early time spin dynamics of AFM matter thus presents rich possibilities for the control of magnetism by light. Investigation of this ultrafast regime, however, relies on accurate measurement techniques to probe the transient magnetic order. Unfortunately, the standard spectroscopic tool for probing ultrafast element-specific spin dynamics, magnetic circular dichroism (MCD), is inapplicable in situations of zero net magnetic moment. Magnetic linear dichroism (MLD), on the other hand, has quadratic dependence on the magnetisation, and so can be observed in both antiferromagnets and ferromagnets van der Laan et al. (1986); Thole et al. (1985); Stöhr and Siegmann (2006). MLD therefore represents a key spectroscopic technique for investigating laser pumped AFM order, and indeed a number of recent experiments probing the time-evolution of AFM order via MLD have been performedThielemann-Kühn et al. (2017); Zhao et al. (2021). It is thus timely to establish the reliability of this probe as a measure of transient moments in highly non-equilibrium AFM matter, and to identify what, if any, problems may arise in this context.

Both MCD and MLD are types of x-ray absorption spectroscopy, based on the principle that the magnetisation of a material can be estimated from its resonant x-ray absorption edges Stöhr and Siegmann (2006). For example, at the spin split $M$-edge or $L$-edge electrons are excited by a probe laser pulse from 3$p$ or 2$p$ states below the Fermi energy to empty $d$-states above the Fermi energy. The intensity of excitation allows one to count the number of empty $d$-states and, given that the $d$-band can always accommodate 5 spin up and 5 spin down electrons, this allows one to determine the moment on an atom. For a circularly polarized pulse the dichroic response (the difference in response to right and left circularly polarized light, i.e the MCD) is proportional to the magnetic moment, while for a linearly polarized probe pulse the difference between the parallel and perpendicular response functions, relative to the quantization axis, is proportional to the square of the magnetic moment. One should stress that this estimation of the moment from empty state counting, while exact for an isolated atom, becomes approximate in solids for which the lower symmetry crystallographic environment and concomitant hybridization of angular momentum channels complicates the “counting principle” underlying MCDStöhr and Siegmann (2006); Ebert et al. (1996). Nevertheless, numerous experiments attest to the success of MCD and MLD as probes of magnetic order in the ground state.

An investigation of the measurement of transient MCD Sharma et al. (2022) found that pulses with higher incident power densities trigger significant failures of the MCD to track the transient magnetization in the elemental ferromagnets Co and Ni. This failure could be attributed to excitation of charge from bands of $d$-character to bands of delocalized character, made possible by the hybridization of $d$ with relatively delocalized states in solidsSharma et al. (2022), resulting in violations of transient $d$-band state counting and overestimation of demagnetization by MCD. It is thus crucial to ask how this physics manifests in the MLD response of antiferromagnets, in particular given the quite different electronic structure in transition metal antiferromagnets.

To study the ability of MLD to accurately track the transient spin moment, we compare a prototypical FM, FePt, to its closest AFM analogue, FePd. Strikingly, we find that the failure of MLD to track the moment in AFM FePd occurs at an order of magnitude less absorbed energy than required to induced failure of the MCD response in FM FePt. The underlying microscopic reason we identify to be dramatically increased loss of $d$-band moment to delocalized states at the same absorbed energy, driven by increased $d$-$sp$ hybridization at the AFM pseudogap. As a corollary, we confirm that both MCD and MLD remain reliable tools to track the $d$-band moment on these timescales, even when failing to track the total moment.

Methods: In the present work we use the fully ab-initio method of time-dependent density functional theory (TD-DFT). This method has been shown to be very successful at describing the dynamics of spins in laser pumped solids Willems et al. (2020); Dewhurst et al. (2020); Hofherr et al. (2020). Within TD-DFT and its linear response extension Runge and Gross (1984); Sharma et al. (2014, 2011, 2022) two distinct schemes can be used to obtain the magnetic moment. In Scheme (i) the magnetization density, $\mathbf{m}(\mathbf{r},t)$, can be calculated directly from the expectation of the spin operator, $\hat{\mathbf{s}}$,

[TABLE]

where $\psi_{j}$ is the spinor-valued wavefunction with index $j$ obtained from solution of the TD-DFT Hamiltonian Sharma et al. (2014). The magnetic moment, M(t), can then be calculated by integrating $\mathbf{m}(\mathbf{r},t)$ over space, $\mathbf{r}$. This method of determining the magnetic moment represents the gold-standard within TD-DFT, in the sense that it is completely free from approximation for a given exchange-correlation functional. In Scheme (ii) the magnitudes of the local magnetic moments can be calculated from the responses of the material to probe pulses (x-ray beams). Such a response is given by a dielectric response tensor, $\epsilon$, which is the change in density of the material upon perturbation by the probe pulse. Scheme (ii) follows a pump-probe experimental technique for obtaining the moments, but we compute rather than measure the transient response for various pump-probe delays. Our method of calculating the transient response function, $\epsilon$, of a pumped system at the $M$ and $L$ resonant absorption edges within TD-DFT has already been studied extensively and demonstrated to be in very good agreement with experiment Willems et al. (2020); Dewhurst et al. (2020); Sharma et al. (2022). For magnetic linear dichroism (MLD) the relation between the response and the magnetisation is Kuneš et al. (2004),

[TABLE]

where $\parallel$ and $\perp$ are defined relative to the direction of the magnetic moment $\mathbf{M}$ (for example $\epsilon_{\parallel}$ is the dielectric response parallel to the direction of $\mathbf{M}$). For MCD a different relationship exists, see Ref. Sharma et al., 2022; Stöhr and Siegmann, 2006.

Computational details: All calculations were performed using the highly accurate full potential linearized augmented-plane-wave methodSingh (1994), as implemented in the ELKDewhurst et al. (2018b, 2016) code. A face centred cubic unit cell with lattice parameter of $3.21$\mathrm{\SIUnitSymbolAngstrom} was used for AFM FePd and AFM FePt, while for FM FePt a lattice parameter of $3.53$\mathrm{\SIUnitSymbolAngstrom} was used. The magnitudes of the groundstate Fe moments are identical in AFM FePd and FM FePt, with a value of 2.77 $\mu_{\rm B}$, where in the case of FePt the Pt atom acquires a small induced moment of 0.35 $\mu_{\rm B}$.

The Brillouin zone was sampled with a $10\times 10\times 8$ $k$-point mesh for both materials. To calculate the spin dynamics we have used the ab-initio state-of-the-art fully non-collinear spin-dependent version Krieger et al. (2015); Dewhurst et al. (2016) of time dependent density functional theory (TDDFT)Runge and Gross (1984) with the adiabatic local spin density approximation for the exchange-correlation potential. Spin-orbit coupling was included in all calculations. For time propagation the algorithm detailed in Ref. Dewhurst et al., 2016 was used with a time-step of $2.42$ attoseconds. We evaluate the response functions at the L3 peakKuneš et al. (2004), for which we employ the bootstrap kernelSharma et al. (2011) to treat excitonic effects, with a smearing of 0.9 eV, a value derived from the $GW$-calculated average width of the semi-core $p_{3/2}$ and $p_{1/2}$ $L$ peaks. By use of a scissor operator the $2p$ Kohn-Sham states of the DFT calculations were rigidly shifted in energy according to the outcome of the $GW$ calculations, and these energy-corrected states were used in calculations of the response functions. The $GW$ calculations were performed at a temperature of 500 K using the all-electron, spin-polarized $GW$ methodHedin (1965) as implemented in the Elk codeDewhurst et al. (2018b). A Matsubara cutoff of 12 Ha was used. The spectral function on the real axis was constructed using a Pade approximation Vidberg and Serene (1977). We find that, in contrast to spectra obtained at the $M$-edgeDewhurst et al. (2020), local field effects are not important for the $L$-edge spectra that we study here.

Results: Firstly, to give important context to the assessment of the error of MLD for FePt and FePd, we compare the accuracy of MLD to the that of MCD for a range of pump pulse fluences. In each case the central frequency (always 1.55 eV) and the duration (24 fs) are held fixed (see Fig. 1(a)). This comparison is possible only for materials with FM ordering, such as FePt, since MCD, proportional to $\bm{\mathrm{M}}$, is identically zero for AFM materials. Our calculations of the time-evolution of $M$ and $M^{2}$ for FM FePt during and after the pump laser pulse are shown in Fig. 1(b) and (c) respectively. For each pulse fluence we present data obtained both from Scheme (i), evaluation via the spin operator, as well as from Scheme (ii), calculation via the $L_{3}$ edge response functions. Whilst our two distinct schemes do not produce identical results, the similarity is sufficient to conclude a meaningful evaluation of the transient magnetisation could be made from the dichroism of the response (MLD or MCD). For both MLD and MCD we also find an increasing deviation of the two schemes as pump fluence is increased further beyond the displayed values, consistent with the finding observed in the case of MCD applied to elemental ferromagnets Sharma et al. (2022). However, in stark contrast to this previous finding Sharma et al. (2022) for elemental ferromagnets we find here the moment is overestimated by the dichroism, not underestimated.

Having established the accuracy of MLD as a probe in FM FePt for a range of pump fluences, we now turn to the important case of AFM FePd. Our calculations of $M^{2}$ for the Fe atom in AFM FePd are shown in Fig. 1 (d). We note that for 4.7 mJ/cm2 incident fluence AFM FePd absorbs approximately 15% more energy while 40% more charge is excited. In striking contrast to the MLD results for FM FePt (Fig. 1 (c)), the failure of the MLD to accurately track the magnetisation in AFM FePd is dramatic: only at the lowest incident fluence (1.1 mJ/cm2) is the difference between the two methods for obtaining $M^{2}$ tolerable. For the pulses with 2.3 mJ/cm2 and 4.7 mJ/cm2 incident fluence direct evaluation of $M^{2}$ reveals far greater demagnetisation than indicated by the MLD, with for the 4.7 mJ/cm2 case the MLD normalized $M^{2}$ amounting to only 50% of the directly calculated value, equivalent to an overestimation of the magnitude of the Fe moments by 70%. The message of these results is thus clear: MLD is, somewhat unexpectedly, profoundly less accurate for AFM FePd than for FM FePt.

We find that, as is the case for MCDSharma et al. (2022), the failure of MLD occurs as transient moment is not only reduced by the pump laser, but also delocalized. This is shown in Fig. 2(a) in which can be seen that the $M^{2}$ of $3d$ character significantly differs from the total transient $M^{2}$, indicating a strong delocalization. This invalidates the central assumption underpinning the response based moment calculation, that of the $d$-band state countingStöhr and Siegmann (2006), where this finding is reinforced by the fact that the transient $d$-band $M^{2}$ agrees very well with the transient $M^{2}$ obtained from the MLD calculation.

The remarkable scale of the difference in error between FM FePt and AFM FePd can be made clear by considering the incident power density required to reproduce, in FePt, the error found in FePd at a fluence of 2.3 mJ/cm2. As can be seen in Fig. 2(b) this requires increasing the fluence by more than an order of magnitude to 34 mJ/cm2 (with the absorbed energy $\sim 8$ times larger).

We now wish to understand the origin of this remarkable difference on an even more fundamental level. First, we seek to thoroughly define which of the distinctions between FM FePt and AFM FePd might be important. Pt and Pd are isoelectronic elements, and FePt shares its crystal structure with FePd, so the clearest point of distinction is the magnetic ordering. Nevertheless, we wish to rigorously exclude the possibility that the difference between Pt and Pd could be an important factor, so we now show in Fig. 3 a calculation of the magnetization dynamics of FePt when it is artificially constrained to an AFM coupled state. This conclusively demonstrates that the higher error in AFM FePd is a direct result of the different magnetic coupling.

Next, we must ask why the error is higher, or, equivalently, why a greater amount of charge (and magnetisation) is excited to outside the $d$-states if there is AFM coupling. AFM materials have previously been shown to demagnetise faster and to a greater extent, because AFM ordering permits a demagnetisation mechanism whereby charge can be driven rapidly from occupied majority states on one atom to the unoccupied minority states on the other (an OISTR Thielemann-Kühn et al. (2017); Dewhurst et al. (2018a)). It would thus be simplest to assume that the greater error is simply a result of this greater, faster, demagnetisation. In actual fact, we find that this would be a false assumption: the size of the error is not simply proportional to the size of the magnetisation quenching, as can be seen by comparison of Fig. 2(a) and (b). Rather, we can clarify the reason for the greater amount of charge excitation to non-$d$-character states, and therefore understand the error level, through a consideration of the energy-dependent density of states, which is shown in Fig. 4 for FM FePt, AFM FePd, and AFM FePt, where contributions to the total density of states from states with each atom and angular momentum character are shown separately.

The clearest difference between the FM and AFM materials is the presence of the AFM pseudogap, with associated magnetic bonding and anti-bonding exchange split $d$-bands, at -2.2 eV and 0.5 eV respectively. This results in a very high peak in minority $d$ states on the Fe atom in the energy domain at the Fermi energy and up to 1 eV above it. Crucially, driven by the marked reduction of states in the pseudogap, a significant increase in $sp$-states is also found in the same energy window (more than double that found for the case of FM order), and it is the concomitant increase in allowed $d\to sp$ transitions that underpins the increased delocalization seen in AFM FePd when a pump pulse is applied. Having its origin in the pseudogap associated with sub-lattice magnetic ordering (see e.g. Ref. Shallcross et al., 2007 or Ref. Kübler, 2000 for a discussion of the importance of the pseudogap in magnetic order) the dramatically reduced threshold for failure of MLD in FePd will, likely, be a general feature of transition metal antiferromagnets.

Conclusion: Our investigation of FePt (a ferromagnet) and FePd (an antiferromagnet) has revealed profoundly different failure thresholds exist for response function based methods of determining the transient moment, with massive ($>$50%) MLD errors being generated in FePd at fluences (4.7 mJ/cm2) for which MLD and MCD capture very well the transient moment in FePt. This failure results from the fact that pump laser pulses not only reduce but also delocalize the angular momentum of magnetic order, and in FePd the AFM pseudogap, that results in significantly increased $d$-$sp$ hybridization, aids this delocalization. As response function based methods rely on a $d$-band state counting principle, the excitation of significant moment to states of non-$d$-character guarantees their failure; in FePd the greater tendency to excite out of the $d$-band thus results in failure to record the total transient moment at much lower pulse intensity. A corollary of this, however, is that such methods will always capture the $d$-band moment, and our calculations confirm this: the MLD and MCD derived moments reproduce excellently the underlying transient $d$-band moments at all pulse intensities. Importantly, therefore, a valid protocol exists for collaboration between theory and experiment: comparison via the transient $d$-band moment. In summary, our work highlights the critical role that electronic state hybridization has in affecting the accuracy of MLD and MCD in tracking transient moments in ultrafast pump laser experiments, suggesting that a cautious approach is needed in the interpretation of experimental spectra as unveiling total transient moments.

Acknowledgements: E. I. Harris-Lee, J. K. Dewhurst and S. Sharma would like to thank the DFG for funding through project-ID 328545488 TRR227 (project A04). S. Shallcross and S. Sharma would like to thank Leibniz Professorin Program (SAW P118/2021) for funding. The authors acknowledge the North-German Supercomputing Alliance (HLRN) for providing HPC resources that have contributed to the research results reported in this paper.