Determination of the spin-flip time in ferromagnetic SrRuO3 from time-resolved Kerr measurements
C.L.S. Kantner, M.C. Langner, W. Siemons, J.L. Blok, G. Koster, A.J.H.M. Rijnders, R. Ramesh, J. Orenstein
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Determination of the spin-flip time in ferromagnetic SrRuO from time-resolved Kerrmeasurements C.L.S. Kantner,
M.C. Langner,
W. Siemons, J.L. Blok, G. Koster, A.J.H.M. Rijnders, R. Ramesh,
1, 3 and J. Orenstein
1, 2 Department of Physics, University of California, Berkeley, CA 94720 Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720 MESA + Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands (Dated: December 6, 2018)We report time-resolved Kerr effect measurements of magnetization dynamics in ferromagnetic SrRuO . Weobserve that the demagnetization time slows substantially at temperatures within 15K of the Curie temperature,which is ∼ c . We also make a direct comparison of the spin flip rate and the Gilbert damping coefficientshowing that their ratio very close to k B T c , indicating a common origin for these phenomena. I: Introduction
There is increasing interest in controlling magnetism in fer-romagnets. Of particular interest are the related questions ofhow quickly and by what mechanism the magnetization can bechanged by external perturbations. In addition to advancingour basic understanding of magnetism, exploring the speedwith which the magnetic state can be changed is crucial to ap-plications such as ultrafast laser-writing techniques. Despiteits relevance, the time scale and mechanisms underlying de-magnetization are not well understood at a microscopic level.Before Beaurepaire et al.’s pioneering work on laser-excitedNi in 1996, it was thought that spins would take nanosecondsto rotate, with demagnetization resulting from the weak inter-action of spins with the lattice. The experiments on Ni showedthat this was not the case and that demagnetization could oc-cur on time scales significantly less than 1 ps . Since thendemagnetization is usually attributed to Elliott-Yafet mecha-nism, in which the rate of electron spin flips is proportionalto the momentum scattering rate. Recently Koopmans et al.have demonstrated that electron-phonon or electron-impurityscattering can be responsible for the wide range of demag-netization time scales observed in different materials . Alsorecently it has been proposed that electron-electron scatteringshould be included as well as a source of Elliott-Yafet spinflipping, and consequently, demagnetization . Although Ref. specifically refers to interband scattering at high energies, itis plausible that intraband electron scattering can lead to spinmemory loss as well.Time-resolved magneto-optical Kerr effect (TRMOKE)measurements have been demonstrated to be a useful probe ofultrafast laser-induced demagnetization . In this paper we re-port TRMOKE measurements on thin films of SRO/STO(111)between 5 and 165K. Below about 80 K we observe dampedferromagnetic resonance (FMR), from which we determine aGilbert damping parameter consistent with earlier measure-ments on SrTiO with (001) orientation . As the the Curietemperature ( ∼ K ) is approached the demagnetizationtime slows significantly, as has been observed in other mag-netic systems . The slowing dynamics have been attributedto critical slowing down, due to the similarities between thetemperature dependencies of the demagnetization time and the relaxation time . In this paper we develop an analyticalexpression relating the demagnetization time to the spin-fliptime near the Curie temperature. This provides a new methodof measuring the spin-flip time, which is essential to under-standing the dynamics of laser-induced demagnetization. II: Sample Growth and Characterization
SRO thin films were grown via pulsed laser deposition at700 ◦ C in 0.3 mbar of oxygen and argon (1:1) on TiO termi-nated STO(111) . A pressed pellet of SRO was used for thetarget material and the energy on the target was kept constantat 2.1 J/cm . High-pressure reflection high-energy electrondiffraction (RHEED) was used to monitor the growth speedand crystallinity of the SRO film in situ. RHEED patternsand atomic force microscopy imaging confirmed the presenceof smooth surfaces consisting of atomically flat terraces sep-arated by a single unit cell step (2.2 ˚Ain the [111] direction).X-ray diffraction indicated fully epitaxial films and x-ray re-flectometry was used to verify film thickness. Bulk magneti-zation measurements using a SQUID magnetometer indicateda Curie temperature, T c , of ∼ III: Experimental Methods
In the TRMOKE technique a magnetic sample is excited bythe absorption of a pump beam, resulting in a change of polar-ization angle, ∆Θ K (t), of a time delayed probe beam. The ul-trashort pulses from a Ti:Sapph laser are used to achieve sub-picosecond time resolution. Near normal incidence, as in thisexperiment, ∆Θ K is proportional to the ˆ z component of theperturbed magnetization, ∆ M z . ∆Θ K is measured via a bal-anced detection scheme. For additional sensitivity, the deriva-tive of ∆Θ K t) with respect to time is measured by locking intothe frequency of a small amplitude ( ∼
500 fs) fast scanning de-lay line in the probe beam path as time is stepped through onanother delay line.
IV.1: Experimental Results: Low Temperature
Fig. 1 shows the time derivative of ∆Θ K for an 18.5nmSRO/STO(111) sample for the 16ps following excitation by apump beam, for temperatures between 5 and 85K. Clear fer-romagnetic resonance (FMR) oscillations are present, gener- FIG. 1. Derivative of the change in Kerr rotation as a function of timedelay following pulsed photoexcitation, for 5 < T <
85 K ated by a sudden shift in easy axis direction upon thermal ex-citation by a pump beam . This motion is described by theLandau-Lifshitz-Gilbert equation with the frequency of oscil-lation proportional to the strength of the magnetocrystallineanisotropy field, and the damping described by dimension-less phenomenological parameter, α . The motion appears asa decaying oscillation to TRMOKE. The orientation of theanisotropy field, closer to in-plane with the sample surface inSRO/STO(111) than in SRO/STO(001), makes these oscilla-tions more prominent when observed with the polar Kerr ge-ometry compared to previous measurements.Attempting to model the time derivative of ∆Θ K with adamped cosine reveals that it cannot be fit by such a functionfor t < ∼ .5 ps) withthe amplitude of the subsequent oscillations (defined as thedifference between d ∆Θ K /dt at the peak at ∼ ∼ ∆Θ K /dt is comprised of a superposition of a tempera-ture independent, short-lived component with the longer liveddamped oscillations.Fitting the oscillatory portion of the signal to a dampedcosine, the temperature dependencies of the amplitude, fre-quency, and damping parameter are found, as shown in Fig. 3.Comparing these parameters for SRO/STO(111) to previouslypublished work on SRO/STO(001), the frequency is found tobe somewhat smaller and to change more with temperature.Of particular interest is α , which is also smaller in this ori-entation of SRO, consistent with the more pronounced FMRoscillations. Strikingly, in both orientations there is a dip in α around 45K, which is relatively stronger in SRO/STO(111).This further strengthens the link between α and the anoma-lous hall conductivity, speculated in that paper, through neardegeneracies in the band structure . IV.2: Experimental Results: High Temeperature
FIG. 2. Comparing amplitudes of the short time feature and the fer-romagnetic resonance oscillations
By taking the time derivative of ∆Θ K , the FMR oscilla-tions can be followed until they disappear at elevated temper-atures, at which point it becomes simpler to look at ∆Θ K thanits time derivative. Fig. 4 shows ∆Θ K as a function of timefor the first 38 ps after excitation by the pump laser, for tem-peratures between 120K and 165K. A property of a secondorder phase transitions is that the derivative of the order pa-rameter diverges near the transition temperature. The peak inmagnitude of ∆Θ K in figure 4, shown in figure 5, can be un-derstood as the result of the derivative of magnetization withrespect to temperature becoming steeper near the Curie tem-perature. A strong temperature dependence of the demagneti-zation time, τ M , is seen, with τ M significantly enhanced near150K, consistent with previous reports on SRO . ∆Θ K (t) in Fig. 4, normalized by the largest value of ∆Θ K (t) in the first 38 ps, can be fit with the following func-tion: for t < K ( t )∆Θ max ( t ) = 0 for t > K ( t )∆Θ max ( t ) = C − Ae − t/τ M (1)where the decay time is τ M . The resulting τ M is plottedas a function of temperature in Fig. 6. Notably, τ M increasesby a factor of 10 from 135K to 150K. Taking the fit value ofT c = 148.8K, as will be discussed later, τ M is plotted log-log as function of reduced temperature, t R = ( T c − T ) /T c .The result looks approximately linear, indicating a power lawdependence of τ M on the reduced temperature. V: Discussion of Results:
Efforts to explain demagnetization have been largely phe-nomenological thus far, understandably, given the dauntingchallenge of a full microscopic model. Beaurepaire et al. in-troduced the three temperature model (3TM) to describe de-magnetization resulting from the interactions of the electron,phonon, and spin baths . In 3TM the dynamics are determined FIG. 3. Temperature dependence of (a) Amplitude of oscillations,(b) FMR frequency, and, (c) damping parameterFIG. 4. Change in Kerr rotation as a function of time delay followingpulsed photoexcitation, for 120 < T <
165 K FIG. 5. Magnitude of change in Kerr rotation at 38ps as a functionof temperatureFIG. 6. Demagnetization time at high temperature by the specific heats of each bath as well as the couplingconstants between them. Demagnetization can generally bedescribed with the appropriate choice of coupling constants,providing a guide into the microscopic mechanism. Koop-mans et al. also offer a phenomenological description of de-magnetization considering three baths, but one that followsspin in addition to heat . Spin is treated as a two state systemwith energy levels separated by an exchange gap and Fermi’sgolden rule is used to relate demagnetization to electron scat-tering which flips a spin. Equations for coupling constants arederived based on parameters such as the density of states ofelectrons, phonons, and spins, the electron-phonon scatteringrate, and the probability of spin flip at a scattering event.In the following we attempt to understand the behavior ofthe demagnetization time near T c with an approach based onthe two spin state model. A general relationship between thelaser-induced τ M and the spin flip time, τ sf , can be derivednear the transition temperature based on the concept of de-tailed balance . In equilibrium, the ratio of the probability of FIG. 7. Log-log plot of demagnetization time as a function of re-duced temperature a spin flipping from majority to minority to the reverse of thisprocess is the Boltzmann factor, e − ∆ ex /kT , where ∆ ex is theexchange energy gap. The time derivative of the number ofmajority and minority electrons can then be written: ˙ N maj = − ˙ N min = N min τ sf − N maj τ sf e − ∆ ex /k B T (2)When the sample is thermally excited by a pump beam, theelectron temperature is increased by δT e . The rate of changeof spins is then altered in the following way: ˙ N maj = − ˙ N min = N min τ sf − N maj τ sf e − ∆ ex /k B ( T + δT e ) (3)The demagnetization time is related to the total change inspin, ∆ S , from initial to final temperature, where, setting ~ =1, S is defined by: S = 1 / N maj − N min ) /N total (4)Assuming that ∆ S , as a function of time, can be written: ∆ S ( t ) = [ S ( T f ) − S ( T i )](1 − e − t/τ M ) (5)the demagnetization time can be written as: τ M = ∆ S ˙ S (0) (6)where ˙ S (0) is the initial change in the time derivative of thespin.The total change in spin can be calculated by taking thederivative of S with respect to T , and multiplying by ∆ T eq ,the increase in temperature once electrons, phonons, and spins have come into thermal equilibrium with each other. S ( T ) and ∆ S can be written: S ( T ) = − tanh (cid:18) ∆2 kT (cid:19) (7)and: ∆ S = dSdT (cid:12)(cid:12)(cid:12) T = T ∆ T eq = − ∆ ex k B T (cid:20) T ∆ ′ ex ∆ ex − (cid:21) ∆ T eq (8)where we have relied on the fact that near the transition tem-perature, ∆ ex ≪ k B T and made the approximation that δT e ≪ T for low laser power. In the last equation T ′ ex ∆ ex ≫ near T c , so only the first term will be considered.The quantity ˙ S (0) , where ˙ S = 1 /
2( ˙ N maj − ˙ N min ) /N total ,can be found by taking the derivative of ˙ S (0) with respect to T e , since immediately after excitation the electron tempera-ture has increased, but the spin temperature, T , has not. ˙ S (0) = d ˙ SdT e (cid:12)(cid:12)(cid:12) T = T ∆ T eq = N maj N τ sf ∆ ex k B T ∆ T eq (9)Near the Curie temperature N maj ∼ N min ∼ N total .Using this approximations and equation (6), we find: τ M = (cid:18) ∆ ′ ex ∆ ex (cid:19) T c τ sf (10)where ∆ ′ ex is the derivative of ∆ ex with respect to temperatureand ∆ ex ∼ ( T c − T ) β , where β is the critical exponent ofthe order parameter. Taking the derivative, we find ∆ ′ ex ∼− β ( T c − T ) β − , and thus can write τ M = βτ sf (cid:18) T c T c − T (cid:19) (11)Therefore τ M is predicted to scale as / ( T c − T ) near thetransition temperature. A fit of T c ∼ /t R near the transition temperature re-gardless of the underlying mechanism of the demagnetization.Additionally, the critical exponent found is independent of β .It should also be noted that the current situation, where thesample has been excited by a laser, is distinct from criticalbehavior as typically considered. In general, divergent timescales are linked to divergent length scales, but here excita-tions of various length scales are not being excited. Insteadthe length scale is always effectively infinite, having been de-termined by the laser spot size. τ sf is plotted as a function oftemperature for the mean field value of β = 1 / , which hasbeen shown to be suitable for SRO , in Fig. 8. τ sf is revealedto be approximately 200 fs and nearly constant as a functionof temperature. FIG. 8. Spin flip time at high temperature
Previous reports of conductivity in SRO give a scatteringtime of ∼
20 fs near the transition temperature . A compar-ison of the spin flip time with the scattering time implies aprobability of 0.1 that a scattering events results in a spin flip.Though electron-phonon interactions are the most commonlyconsidered source of demagnetization, as mentioned previ-ously, Eliot Yafet-like electron-electron coulomb scatteringcan also result in demagnetization . This is especially true formaterials with strong spin orbit coupling, such as SRO. Ad-ditionally in SRO the interaction with the crystal field meansthat total spin is not conserved[Goodenough], so every elec-tron interaction can perturb the spin state.Having found a relationship between the demagnetizationtime and the spin flip time we would like to explore the rela-tionship between these parameters and the damping param-eter, α . Intuitively, the damping parameter should be pro-portional to the spin flip scattering rate, or inversely propor-tional to the spin flip scattering time: α ∼ /τ sf . Elliot-Yafettype scattering dissipates energy from motion described bythe LLG equation by disrupting the coherent, collective pre-cession of spins. Spins that have had their angular momen-tum changed through electron collisions must be pulled backinto the precession through the exchange interaction, repre- senting a transfer of energy away from the precessional mo-tion. These collision-mediated spin-orbit coupling effects arethought to be the primary source of Gilbert-type damping inferromagnets . Again, this should be particularly true in aferromagnet with strong spin orbit coupling.Combining the spin flip time and the damping parameterwith Planck’s constant reveals an energy scale, E , given bythe condition that: α ∼ E ~ τ sf (12)Noting that the values for α and τ sf found in figures 3 and7, respectively, are approximately constant as a function oftemperature, this energy scale for SRO is ∼ k B T c ∼
13 meV, is of the same order. This suggests an underlyingconnection between the critical temperature (and thus the ex-change energy), Gilbert damping, and spin flip scattering.A relationship similar to equation (12) has been found pre-viously between τ M (rather than τ sf ) and α by Koopmans etal. at low temperature: τ M = 14 ~ k B T c α (13)Applying this equation to SRO at 5K yields τ m ∼ ∼ . Whether the fundamental re-lationship is between transition temperature and the demagne-tization time or the spin-flip scattering time remains a questionfor a microscopic model to resolve. ACKNOWLEDGMENTS
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