Ab initio investigation of Elliott-Yafet electron-phonon mechanism in laser-induced ultrafast demagnetization
AAb initio investigation of Elliott-Yafet electron-phonon mechanism in laser-inducedultrafast demagnetization
K. Carva,
1, 2, ∗ M. Battiato, and P. M. Oppeneer Department of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala, Sweden Charles University, Faculty of Mathematics and Physics,Department of Condensed Matter Physics, Ke Karlovu 5, CZ-12116 Prague 2, Czech Republic (Dated: November 1, 2018)The spin-flip (SF) Eliashberg function is calculated from first-principles for ferromagnetic Ni to accu-rately establish the contribution of Elliott-Yafet electron-phonon SF scattering to Ni’s femtosecondlaser-driven demagnetization. This is used to compute the SF probability and demagnetizationrate for laser-created thermalized as well as non-equilibrium electron distributions. Increased SFprobabilities are found for thermalized electrons, but the induced demagnetization rate is extremelysmall. A larger demagnetization rate is obtained for non-equilibrium electron distributions, but itscontribution is too small to account for femtosecond demagnetization.
PACS numbers: 78.47.J-, 78.20.Ls, 78.20.Bh, 71.70.Ej, 75.40.Gb
Ultrafast demagnetization of ferromagnetic metalsthrough excitation by a femtosecond laser pulse was dis-covered fifteen years ago by Beaurepaire et al. [1]. Inspite of intensive investigations the microscopic origin ofthe ultrafast demagnetization could not be disclosed andcontinues to be controversially debated (see [2] for a re-cent review). Several mechanisms have been proposed toexplain the observed ultrafast phenomenon [3–11]. Mostof these theories assume the existence of an ultrafast spin-flip (SF) channel, which would cause dissipation of spinangular momentum within a few hundred femtoseconds.Elliott-Yafet electron-phonon SF scattering has beenproposed as a mechanism for ultrafast spin-dissipation[4]. Strong support in favor of electron-phonon medi-ated spin-flips as the actual mediator of the femtosec-ond demagnetization was made in a very recent work,in which ab initio calculated SF probabilities for ther-malized electrons compared favorably to SF probabili-ties derived from pump-probe demagnetization measure-ments [8]. While these results definitely favor the Elliott-Yafet SF scattering mechanism, the calculation of theelectron-phonon scattering involved several serious ap-proximations. Applying the so-called Elliott approxima-tion [12] only spin-mixing due to spin-orbit coupling inthe ab initio wavefunctions was included, but no electron-phonon matrix elements and no real phonon dispersionspectrum was considered. The thus-obtained SF proba-bility is however not a direct measure of demagnetization.Recent model simulations for thermalized hot electrons[9] using the Landau-Lifshitz-Bloch equation [13] and as-suming a fitted SF parameter did reproduce the exper-imental magnetization response, but couldn’t assign theSF origin. Hence, it remains a crucial, open questionwhether laser-induced demagnetization can indeed be at-tributed to electron-phonon mediated SF scattering.Here we report an ab initio investigation to accu-rately establish the extent to which the Elliott-Yafetelectron-phonon SF scattering contributes to fs demag- netization. To this end we perform ab initio calculationsfor ferromagnetic Ni, which ultrafast magnetization de-cay is well documented [1, 8, 14]. We include the fullelectron-phonon matrix elements and phonon dispersionsin our calculations. Introducing an energy-dependentSF Eliashberg function we compute SF probabilities anddemagnetization rates for laser-heated thermalized elec-trons as well as laser-induced non-equilibrium electrondistributions, from which we draw qualified conclusionson the possibility of phonon-mediated demagnetization.To treat phonon-mediated SF scattering at variableelectron energies we define a generalized energy- andspin- dependent Eliashberg function, α σσ (cid:48) F ( E, Ω) = 12 M Ω (cid:88) ν,n,n (cid:48) (cid:90) (cid:90) d k d k (cid:48) g νσσ (cid:48) k n, k (cid:48) n (cid:48) ( q ) × δ ( ω q ν − | Ω | ) δ ( E σ k n − E ) δ ( E σ (cid:48) k (cid:48) n (cid:48) − E ) , (1)which comprises initial and final electron states withquantum numbers k n , k (cid:48) n (cid:48) that interact through aphonon with frequency Ω= ω q ν , ν and q denote its modeand wavevector. M is the ionic mass, σ = ↑ , ↓ denote thespin majority, miniority components. For E = E F (theFermi energy) the SF part α ↑↓ F ( E F , Ω) gives the SFEliashberg function [15] and the sum over all σσ (cid:48) corre-sponds to the standard Eliashberg function, α F ( E F , Ω)[16]. The (squared) electron-phonon matrix elements are g νσσ (cid:48) k n, k (cid:48) n (cid:48) ( q ) = | u q ν · (cid:104) Ψ σ k n |∇ R V | Ψ σ (cid:48) k (cid:48) n (cid:48) (cid:105)| , (2)where V is the potential, u q ν the phonon polarizationvector and | Ψ σ k n (cid:105) are the eigenstates in the ferromagnet.Momentum conservation requires q = k (cid:48) − k . SF scatter-ing becomes possible through the relativistic spin-orbitcoupling. The majority, minority Bloch states | Ψ ↑ k n (cid:105) and | Ψ ↓ k n (cid:105) can be decomposed in pure spinor components | Ψ ↑ k n (cid:105) = a ↑ k n ( )+ b ↑ k n ( ) , | Ψ ↓ k n (cid:105) = a ↓ k n ( )+ b ↓ k n ( ) , (3) a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t where the components b σ k n are nonzero only if spin-orbitcoupling is present and represent the degree of spin-mixing, which is a precondition for nonzero g ν ↑↓ k n, k (cid:48) n (cid:48) .To study demagnetization we consider two quantities,SF probabilities and spin-resolved transition rates. Thelatter are defined as [17] S σσ (cid:48) = (cid:90) (cid:90) α σσ (cid:48) F ( E, Ω) f σ ( E )(1 − f σ (cid:48) ( E + (cid:126) Ω)) × (Θ(Ω)+ N (Ω)) d Ω dE. (4)Here N (Ω) is the phononic Bose-Einstein distribution, f σ the Fermi distribution, and Θ(Ω) the Heaviside function.Important for the effective demagnetization is the spindecreasing rate S − , which corresponds to S ↑↓ , while theincreasing one S + corresponds to S ↓↑ .An approximation of Eq. (4) is helpful to achieve afaster evaluation and provide more insight in the pro-cess. Energy conservation during electron-phonon scat-tering requires E k (cid:48) n (cid:48) − E k n = (cid:126) Ω, but the phonon energy (cid:126)
Ω is usually very small ( < .
04 eV) compared to elec-tron related properties. Already in the standard Eliash-berg formulation Eq. (1) an energy difference betweeninitial and final states is neglected while the δ -functions δ ( E σ k n − E ) are broadened with a parameter (0.03 eV,here). Similarly, one can neglect the energy variationdue to (cid:126) Ω in the Fermi function f σ ( E + (cid:126) Ω), as long asthe temperature is high enough. We can then rewritespin-resolved transition rates in the form S σσ (cid:48) = (cid:90) w σσ (cid:48) ( E ) f σ ( E )(1 − f σ (cid:48) ( E )) dE, (5)where we introduced the energy- and spin- dependentspecific scattering rate for electrons w σσ (cid:48) given by w σσ (cid:48) ( E ) = (cid:90) ∞ d Ω α σσ (cid:48) F ( E, Ω)(1+ 2 N (Ω)) . (6)Note that w ↑↓ ( E ) = w ↓↑ ( E ). All calculations werechecked against a more accurate numeric implementationnot involving this approximation. The SF probability foran electron with energy E is defined as the ratio of theSF part to the corresponding total counterpart, p S ( E ) =2 w ↑↓ ( E ) / (cid:80) σσ (cid:48) w σσ (cid:48) ( E ). Analogously, the total SF prob-ability during a scattering event can be defined as P S = ( S − + S + ) / (cid:88) σσ (cid:48) S σσ (cid:48) . (7)Although the SF probability has been used in recent dis-cussions of laser induced-demagnetization [8, 18], it isactually not the crucial quantity (as a high but equal SFprobability for both spin channels would not cause a de-magnetization). We define therefore the normalized de-magnetization ratio, D S = ( S − − S + ) / (cid:80) σσ (cid:48) S σσ (cid:48) , whichtracks the difference of magnetic moment increasing anddecreasing SF contributions. To investigate phonon-induced demagnetization inlaser-excited Ni we proceed now in three steps. First, wecompute the ab initio SF probability P S for equilibriumNi, i.e., for E = E F . Second, we compute SF probabilities P S for laser-heated Ni, by treating a range of electron en-ergies that correspond to those in a hot, thermalized elec-tron gas after laser-excitation. Thermalization to elec-tron temperatures T e of a few thousand K occurs quicklywithin about 200 fs after the laser pulse, but the hot elec-trons are not in equilibrium with the lattice and the lat-tice temperature is not altered significantly. In the thirdstep we consider the SF probability for non-equlibrium(NEQ) electron distributions [19] that are expected to bepresent within ∼
100 fs after laser stimulation. Demag-netization ratios D S are subsequently evaluated for thesethree situations. The results obtained in these steps arefurthermore compared to values which we compute withthe so-called Elliott relation (see below).An ab initio evaluation of the SF probability of equi-librium Ni requires calculated phonon dispersions anda relativistic electronic structure. Such calculation haspreviously been done for paramagnetic Al [15], but hasnot yet been accomplished for ferromagnets. An approx-imation was introduced years ago by Elliott [12], whopointed out a possible source of SF scattering arisingfrom the spin-mixing of eigenstates. Employing severalassumptions, viz . a paramagnetic metal, nearly constantelectron-phonon matrix elements, b k n constant in theBrillouin zone, and b σ k n (cid:28) a σ k n , Elliot derived a rela-tion between the spin lifetime τ S for a general kind ofscattering event with lifetime τ . This so-called Elliottrelation uses the Fermi surface averaged spin-mixing ofeigenstates (cid:104) b (cid:105) = (cid:80) σ,n (cid:82) d k | b σ k n | δ ( E σ k n − E F ) and pre-dicts the SF probability P b S = ( τ S /τ ) − = 4 (cid:104) b (cid:105) .In a similar way as introduced above, the influence ofspin-mixing on the SF probability in laser-heated Ni canbe evaluated. We define a SF density of states (DOS) as n ↑↓ ( E ) = (cid:88) n,σ (cid:90) d k | b σ k n | δ ( E σ k n − E ) . (8)A generalized Elliott SF probability for an electron withenergy E is then given as P b S ( E ) = 4 n ↑↓ ( E ) / n ( E ) (with n ( E ) the total DOS) which yields the standard Elliott ex-pression (cid:104) b (cid:105) in the limit b σ k n (cid:28) a σ k n and E = E F . The to-tal SF probability P b S of a laser-heated system with elec-tron distribution f σ ( E ) is obtained from Eqs. (7) and (5),where w ↑↓ ( E ) is replaced by n ↑↓ ( E ) and w ( E ) by n ( E ).Note that although the treatment is intended for phononscattering the Elliott relation in fact does not take thecharacter of scattering involved into account. Also, theassumption of a paramagnetic material is essential in El-liott’s derivation as this permits SF scattering in each k point in the spin-degenerate majority, minority bands at E F . Experimentally the Elliott relation was found to bevalid up to a multiplication by a material specific con- Phonon energy (eV) E li a s hb e r g f un c ti on α F α F x 10 FIG. 1. (Color online)
Ab initio calculated Eliashberg α F ( E F , Ω) and SF Eliashberg α ↑↓ F ( E F , Ω) functions of Niin equilibrium. stant with variation smaller than one order of magnitudefor various paramagnetic metals [20]. Recently it has alsobeen applied to ferromagnetic metals [8, 18], even thoughfor exchange-split ferromagnetic bands there exist far less k points at which spin-degenerate bandcrossings occur.We have tested the implementation by computing firstAl and Ni in equilibrium at low temperature ( <
300 K).Our calculations are based on the density functional the-ory (DFT) within the local spin-density approximation(LSDA), see [21] for details. For Al our calculated α ↑↓ F is of the order of 10 smaller than α F and in good agree-ment with the existing previous result [15]. The ab initio calculated SF and non-SF Eliashberg functions of equi-librium Ni are shown in Fig. 1. For Ni the computed SF α ↑↓ F function is only about 50 times smaller than the or-dinary α F function; this is due to the larger spin-orbitcoupling. The resulting total SF probability, P S =0.04, isgiven in Table I. To estimate the accuracy of the Elliottapproximation we have calculated the Elliott SF proba-bility and obtain P b S =0 .
07. This value is in rough agree-ment with P b S =0 .
10 computed in Ref. [18]. Thus we findthat the Elliott relation overestimates the SF probabilityin equilibrium Ni by about a factor two.
TABLE I. Calculated spin-flip probabilities P S and demagne-tization ratios D S for laser-pumped Ni. Results are given forequilibrium (low T ), for thermalized electrons at a high Fermitemperature T e , and for the non-equilibrium (NEQ) electrondistribution created by fs laser-excitation. Results obtainedfor the approximate Elliott SF probability P b S (this work and[18]) are given for comparison. P b S P S D S Ni (low T ) 0.07 (0.10 [18]) 0 .
04 0Ni ( T e = 1500K) 0.08 0.05 0.002Ni ( T e = 3000K) 0.11 0.07 0.003Ni ( T e = 5000K) 0.12 0.10 0.004Ni (NEQ) 0.12 0.09 0.025 -3 -2 -1 0 1 Energy (eV) S F p r obab ili t y elec.-phonon SF prob.Elliott SF probability 0.10.2 S c a t. r a t e ( a r b . u . ) SF rate (x 10)Non-SF rate
FIG. 2. (Color online) Energy-resolved electron-phonon totaland SF scattering rates w ( E ) and w ↑↓ ( E ) of Ni, and normal-ized SF probability P S ( E ) and approximate SF probability P b S ( E ) obtained from the Elliott relation. Next we turn to the topic of current controversy,the actual amount of phonon-induced demagnetizationin laser-excited Ni. In Fig. 2(top) we show calculatedenergy-resolved SF and non-SF scattering rates ( w ↑↓ ( E )and w ( E )). Note the strong energy variations of w ( E ).In Fig. 2(bottom) we compare the computed electron-phonon SF probability P S ( E ) to that obtained from theElliott relation. At some energies, e.g., 0.5 - 1 eV, thesetwo quantities are nearly the same, but at other energiesthere is no direct relation other than that SF probabilityis large where band states are present. An interesting dif-ference in the context of ultrafast demagnetization is thesuppression of P S ( E ) around E F , which is not capturedby P b S ( E ). The features of P S ( E ) that are not capturedby P b S ( E ) can be understood by comparing Eqs. (1) and(8). One of the differences is the presence/absence ofsummation over destination eigenstates k (cid:48) n (cid:48) . The latterare restricted in Eq. (1) by the construction of g ν ↑↓ k n, k (cid:48) n (cid:48) to correspond to a different spin than the source state k n . The number of available end states is however nottaken into account in Elliott formula (which, derived fora paramagnetic metal, assumes that the same numberof states is available for both spins, and hence suppressesthis distinction). The mentioned discrepancy between P S and P b S above E F is thus easily explained by the lack ofstates with the same energy and opposite spin in the NiDOS (see Fig. 3). Hence, the Elliott relation fails forferromagnets in strongly exchange-split energy regions.After laser-excitation electrons equilibrate quickly dueto electron-electron scattering at a high electron tem-perature T e of the order of thousands K. To describethis situation we use appropriate f σ ( E ), but note thatthe chemical potential must be adjusted also. Spin con-servation leads to differences between f ↑ ( E ) and f ↓ ( E ),namely f ↓ ( E ) has a lower chemical potential than f ↑ ( E )in Ni due to the shape of its DOS. SF probabilities P S - - - E ne r g y ( e V ) N E Q T H E R M FIG. 3. (Color online) Spin-resolved DOS (filled areas) andphonon induced spin-flips (arrows) of NEQ and electron ther-malized Ni. The equilibrium DOS is shown by thin lines. SFtransitions are significantly different at energies above andbelow E F (=0 eV). The arrows thickness corresponds to thetransition rate, its direction and length give which direction isdominant and how much. The amount of laser redistributedelectrons has been enlarged to improve visibility. computed for several T e are given in Table I. With in-creasing T e P S increases, too. Also the Elliott SF proba-bility P b S increases with T e , but it deviates still from P S .A previous work [8] used a Gaussian smearing to stimu-late a thermalized system (without E F adjustment) andobtained P b S ≈ dM/dt =2 µ B ( S − − S + ) for ther-malized electron distributions we obtain quite small val-ues, of the order of 0.08 µ B /ps. The reason is that notjust a large SF probability, but also an imbalance between f ↑ ( E ) and f ↓ ( E ) is essential for a magnetization change.The distributions of spin populations specific to Ni implythat for thermalized electrons below E F most spin-flips increase the spin moment, spin-reducing transitions oc-cur only above E F . In that region the SF scattering rateis however very low (Fig. 2). The situation is illustratedin Fig. 3. As a consequence the spin-decreasing rate( S − − S + ) is thus much lower than the SF rate ( S − + S + ),and in addition it exhibits only a weak temperature de-pendence. Hence we find that phonon-mediated SF scat-tering in thermalized Ni cannot be the mechanism of theobserved ultrafast demagnetization.One remaining possibility for a fast demagnetizationis an enhanced SF rate in the NEQ distribution presentimmediately after the laser pulse. Previous ab initio cal-culations showed that minority-spin electrons are excitedmore than majority-spin ones, see [19]. Assuming a 1.5-eV pump-laser and a simplified step-like electron distri-bution reduced by about 5% in the 1.5-eV energy win-dow below E F , the calculated demagnetization ratio D S is higher than for thermalized distributions (Table I). A critical role is played here by holes deep below E F withhigh SF probability as well as a significant difference be-tween majority and minority occupations (see Fig. 3). Animportant yet unknown element in estimating the demag-netization is the laser fluence. Nonetheless, we find thatphonon-mediated demagnetization in Ni is much moreeffective in the NEQ state than in the thermalized state,as was proposed recently for Gd [22]. An important as-pect is the time scale on which the NEQ demagnetiza-tion is active. Electron thermalization proceeds fast inNi and transforms the initial NEQ distribution to a ther-malized one in ∼
200 fs. A rough estimate of the de-magnetization in this time-window is 0.1 µ B , i.e. smallerthan the observed experimental demagnetization. Theprecise amount of the demagnetization depends howeveron the time-evolution of the distributions, which requiresfurther investigations.Using relativistic ab initio calculations we have evalu-ated the phonon-induced SF probability and demagneti-zation in laser-pumped Ni. A strong dependence of thesequantities on the electron energy is observed, which isnot tracked by the Elliott approximation. In the electronthermalized state Elliott-Yafet phonon-mediated demag-netization is too small to explain the ultrafast demagneti-zation, despite reasonably large SF probabilities. We findthat Elliott-Yafet SF scattering contributes more to thedemagnetization for NEQ distributions immediately afterthe fs laser-excitation. We note lastly that the existenceof other fast SF channels [5–7, 11] cannot be excluded.We thank H.C. Schneider, J. K. Dewhurst and Th. Ras-ing for valuable discussions. This work has been sup-ported by the Swedish Research Council (VR), by FP7EU-ITN “FANTOMAS”, the G. Gustafsson Founda-tion, Czech Science Foundation (P204/11/P481), and theSwedish National Infrastructure for Computing (SNIC). ∗ [email protected][1] E. Beaurepaire et al. , Phys. Rev. Lett. , 4250 (1996).[2] A. Kirilyuk, A.V. Kimel, and Th. Rasing, Rev. Mod.Phys. , 2731 (2010).[3] G.P. Zhang and W. H¨ubner, Phys. Rev. Lett. , 3025(2000).[4] B. Koopmans et al. , Phys. Rev. Lett. , 267207 (2005).[5] E. Carpene et al. , Phys. Rev. B , 174422 (2008).[6] M. Krauss et al. , Phys. Rev. B , 180407(R) (2009).[7] J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Nature Phys. , 515 (2009).[8] B. Koopmans et al. , Nature Mater. , 259 (2010).[9] U. Atxitia et al. , Phys. Rev. B , 174401 (2010).[10] M. Battiato, K. Carva, and P.M. Oppeneer, Phys. Rev.Lett. , 027203 (2010).[11] A.B. Schmidt et al. , Phys. Rev. Lett. , 197401 (2010).[12] R.J. Elliott, Phys. Rev. , 266 (1954).[13] U. Atxitia et al. , Appl. Phys. Lett. , 232507 (2007).[14] C. Stamm et al. , Nature Mater. , 740 (2007).[15] J. Fabian and S. Das Sarma, Phys. Rev. Lett. , 1211(1999). [16] G. Grimvall, Electron-phonon interaction in metals (North-Holland, Amsterdam, 1981).[17] Y. Yafet, in
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