Abstract
The magnetooptical Kerr effect (MOKE) is a powerful tool for studying changes in the magnetization of ferromagnetic materials. It works by measuring changes in the polarization of reflected light. However, because the conventional theoretical basis for interpreting a MOKE signal assumes measurement with continuouswave light^{1,2}, its use for understanding highspeed magnetization dynamics of a material probed with femtosecond optical pulses^{3,4} has been controversial^{5,6,7,8,9,10}. Here we establish a new paradigm for interpreting timeresolved MOKE measurements, through a firstprinciples investigation of ferromagnetic nickel. We show that the timeresolved optical and magnetic responses energetically follow their respective optical and magnetooptical susceptibilities. As a result, the onetoone correspondence between them sensitively depends on the incident photon energy. In nickel, for photon energies below 2 eV the magnetic response is faithfully reflected in the optical response, but above 2 eV they decouple. By constructing a phasesensitive polarization versus magnetization plot, we find that for short pulses the magnetic signals are delayed by 10 fs with respect to the optical signals. For longer pulses, the delay shortens and the behaviour approaches the continuouswave response. This finally resolves the longstanding dispute over the interpretation in the timeresolved MOKE measurements and lays a solid foundation for understanding femtomagnetism^{3,4}.
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Laserinduced femtosecond magnetism^{3} or femtomagnetism opens a new frontier for a faster magnetic storage device^{4,11}, but probing such a fast magnetization change is a big challenge experimentally and theoretically^{12,13,14}. Shortly after its discovery, secondharmonic generation was used to probe the ultrafast spin dynamics and revealed a faster magnetization dynamics of 300 fs (ref. 15), but a similar study showed that the magnetooptical secondharmonicgeneration signal change does not reflect magnetization change on a subpicosecond timescale^{16}. A very recent Xray magnetic circular dichroism study showed an even faster (120 fs) demagnetization^{10}. The majority of experimental investigations are based on the timeresolved magnetooptical Kerr effect (TRMOKE), inspired by the huge success of the static magnetooptical Kerr effect in probing magnetism; however, whether the ultrafast TRMOKE signal reflects the magnetization change is still under debate. Koopmans et al. ^{5} concluded from the difference between the Kerr ellipticity and rotation that the loss of magnetooptical contrast cannot directly be related to an instantaneous demagnetization, and other experiments showed no or only negligible difference between ellipticity and rotation^{6,7,8,9}. Up to now, although the experimental results start to converge, they leave behind a fundamental question: what does TRMOKE really probe, a simple optical excitation artefact (charge origin) or a genuine magnetic excitation (spin origin)? Failing to answer this question significantly affects our confidence in TRMOKE as a vital tool to probe ultrafast magnetization, particularly femtomagnetism^{17,18,19}. Thus, a deep theoretical understanding is imperative. Our first attempt was made using a model Hamiltonian^{12,20}, long before these new experimental results, and the magnetization occurs much faster than those observed experimentally, as our laser pulse is much shorter. The study by Oppeneer and Liebsch removed electrons from the valence band to conduction band to simulate the excitation^{21} and did not compare optical and magnetic responses, and neither did Vernes and Weinberger who developed a linear response theory^{22}. Up to now, no existing theories^{21,22}, including our latest published studies^{23,24,25}, have addressed this crucial question in the time domain.
A typical TRMOKE experiment measures ellipticity and rotation of the outgoing light beam with respect to the incident beam as a function of time, with a sample experimental geometry shown schematically in Fig. 1. The hope is that a change in ellipticity and rotation, which is an optical signal, links to the true magnetization in a sample. Its theoretical basis is that both the magnetic and optical responses share the common timedependent density matrix ρ_{k}(t) through and , respectively, where the summation is over the crystal momentum k, but, as the spin matrix S and dipole matrix D differ, their intrinsic correlation is masked by taking their traces. Here we aim to establish such a correlation by carrying out a massively parallel^{26} and firstprinciples simulation^{27} in ferromagnetic nickel^{24,25}, first solving nearly half a million (k points) Liouville equations of density matrices (see Supplementary Information), and then carefully comparing the polarization with magnetization^{25}. Our theory does not include the electron correlation effect^{12} beyond the density functional theory and is valid for a weak laser field in a temporal region before thermalization. We choose a Gaussianshaped laser pulse with a duration of 12 fs and a laser field amplitude of 0.05 V Å^{−1}. Figure 2a,b presents a first and comprehensive picture of the timedependent optical and magnetic responses as a function of the incident photon energy. The opencircled line denotes the results obtained with the experimental laser energy^{3} of 2 eV, and all the curves are vertically shifted for a better view. The firstorder offdiagonal polarization (Im[P_{x y}^{(1)}]), which is responsible for the magnetooptical Kerr effect, shows a systematic change (Fig. 2a). On the short timescale it consistently shows a valley at time t=0 fs, similar to the laser pulse (see the inset), but on the longer timescale more structures appear for smaller incident energies. With an increase in laser energy, the valley becomes shallower, and the fluctuation on the long timescale diminishes.
The firstorder magnetization Im[M_{x y}^{(1)}], which is inaccessible experimentally, shows a stronger photon energy dispersion (Fig. 2b): for the photon energy far below 2 eV, Im[M_{x y}^{(1)}] also has a similar valley around 0 fs; as the energy increases, the valley becomes deeper and wider, and shifts to a later time. However, once it is over 2 eV, the valley becomes shallower and shifts to an earlier time, where the original right shoulder now gradually develops into a small hump. As a result, the change in Im[M_{x y}^{(1)}] with ℏω leaves a crescentlike trace with respect to time.
A comparison between the magnetic and optical responses leads to our first main result: for laser energy below 2 eV Im[P_{x y}^{(1)}] and Im[M_{x y}^{(1)}] correlate with each other fairly well, but when the excitation energy is above 2 eV there is a large discrepancy between the magnetic and optical responses. Im[P_{x y}^{(1)}] still shows a valley at 0 fs, but Im[M_{x y}^{(1)}] now shows a peak. This finding is significant, as most laser experiments use only a single central wavelength to probe the dynamics. The threshold energy of 2 eV here reflects the magnetic contribution of the density of states across the Fermi level and may differ in different materials, that is, may be material specific. For a new material, it is necessary to scan different energy windows to avoid an accidental mismatch between polarization and magnetization or bleaching effect. The bleaching effect is defined as how far the magnetooptical response (Im[P_{x y}^{(1)}]) mirrors a true demagnetization (Im[M_{x y}^{(1)}]). This explains a previous experimental finding observed in Fe where the probe wavelength shows a significant effect on the TRMOKE signal^{28}, whereas CoPt_{3} shows a smaller energy dependence^{7}. However, is it possible to remove this uncertainty beforehand?
The answer is yes, which was noticed in ferromagnetic perovskite La_{0.6}Sr_{0.4}MnO_{3} (ref. 29), but the authors looked into only the optical response, not the magnetic. We find that the frequencydomain information of both optical and magnetic responses, induced by cw light, provides good guidance. Figure 2c shows the offdiagonal optical absorption (solid line) and conductivity (dashed line) as a reference. Figure 2d plots the magnetooptical susceptibility, which is defined as (see Supplementary Information)
where ℏ is Planck’s constant over 2π, the summation is over k points and band states (n,m), ρ is the density matrix, S^{x} is the xcomponent of the spin matrix, D^{y} is the ycomponent of the dipole operator, ℏω_{k;n m} is the band energy difference between states n and m and Γ is the damping (0.2 eV), which takes into account effects beyond density functional theory. Figure 2c,d explains the energy dependence observed in Fig. 2a,b, respectively. The decrease in the valley with energy in Fig. 2a is directly connected to the decrease in the absorption spectrum in this region, and the energy dependence of Im[M_{x y}^{(1)}] is consistent with the change of the magnetooptical susceptibility. The four arrows denote the energy region used to excite the system.
To quantify the correlation between polarization and magnetization, we develop a sensitive phase diagram, a P_{x y}^{(1)}versusM_{x y}^{(1)} plot, which can monitor phase, amplitude and period differences simultaneously. We align P along the x axis and M along the y axis. Their trace carries rich information about the relation between P_{x y}^{(1)} and M_{x y}^{(1)} (see the inset above Fig. 3c). For instance, if P and M match perfectly, their trace should be a straight line in the first and third quadrants; if they mismatch, the line becomes a loop. The width of the loop reflects the level of the match between P and M. If P and M were completely out of phase, the line or loop would fall in the second and fourth quadrants.
Our findings are truly insightful. Figure 3 shows both P and M to start at −40 fs with zero value and progress in a counterclockwise fashion, which demonstrates that P_{x y}^{(1)} precedes M_{x y}^{(1)}. The arrows represent the time directions, and numbers close to the curve denote the times. The laser pulse duration is 12 fs, and the energy is 2 eV. The two dashed lines identify the zero values. P_{x y}^{(1)} reaches its negative maximum around 0 fs, whereas M_{x y}^{(1)} delays by 10 fs, which is comparable to the charge dephasing time^{30}. After 30 fs, both the magnetic and optical signals come back to zero. This time delay does not lead to a change, even if we increase the laser intensity tenfold from 0.05 to 0.50 V Å^{−1} (Fig. 3b) while keeping the pulse duration at 12 fs (see Supplementary Information). Therefore, we arrive at our second conclusion: there is an intrinsic mismatch between the optical and magnetic responses. At the laser energy of 2 eV, this mismatch is maximized.
Next, the laser pulse duration has a substantial effect on the correlation between the magnetic and optical responses. When we increase the laser duration to 64 fs, we find that the original loop collapses into a narrow and symmetric loop (see Fig. 3c), where both P_{x y}^{(1)} and M_{x y}^{(1)} reach their respective negative maxima at the same time and follow each other faithfully over 300 fs. This reveals a new paradigm for the magnetooptical Kerr effect: the optical response reflects the magnetic signal if the laser pulse is longer than the charge dephasing time^{30}. As most of the experiments use much longer pulses than ours, the observed Kerr signal does reflect the magnetization change, provided that the time dependence is dominated by the offdiagonal polarization.
The reason why the pulse duration has a substantial influence on the correlation between the magnetic and optical responses can be understood from the nature of state excitation and dephasing. It is known that states with larger transition energies dominate the dynamics for the first few femtoseconds. However, states which make substantial magnetic contributions have lower transition energies. A long laser pulse can wait for those states of large transition energies to decay before it induces a substantial magnetic response. Once the initial dephasing is over, the optical and magnetic responses are driven by similar sets of states and tend to correlate with each other much better. This result finally resolves a decadelong debate as to whether TRMOKE really probes the magnetization, and also, importantly, it establishes a solid theoretical foundation for femtomagnetism.
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Acknowledgements
This work was supported by the US Department of Energy under contract No DEFG0206ER46304 and US Army Research Office under contract W911NF0410383, and was also supported by a Promising Scholars grant from Indiana State University. In addition, we acknowledge part of the work as done on Indiana State University’s high performance computers. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy under contract No DEAC0205CH11231. W.H. and G.L. acknowledge support from Priority Programmes 1133 and 1153 of the German Research Foundation. Initial studies used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the US Department of Energy under contract No DEAC0206CH11357.
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G.P.Z. drafted the paper, and W.H., G.L., Y.B. and T.F.G. modified it. G.P.Z. computed the results, and G.P.Z., W.H. and G.L. analysed the data. Y.B. implemented the parallelization of the source code.
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Zhang, G., Hübner, W., Lefkidis, G. et al. Paradigm of the timeresolved magnetooptical Kerr effect for femtosecond magnetism. Nature Phys 5, 499–502 (2009). https://doi.org/10.1038/nphys1315
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DOI: https://doi.org/10.1038/nphys1315
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