Manifestation of intra-atomic 5d6s-4f exchange coupling in photoexcited gadolinium
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Manifestation of intra-atomic 5d6s-4f exchange coupling inphotoexcited gadolinium
G. P. Zhang ∗ , T. Jenkins, and M. Bennett Department of Physics, Indiana State University, Terre Haute, Indiana 47809, USA
Y. H. Bai
Office of Information Technology, Indiana StateUniversity, Terre Haute, Indiana 47809, USA (Dated: December 19, 2017)
Abstract
Intra-atomic exchange couplings (IEC) between 5 d s and 4 f electrons are ubiquitous in rare-earth metals and play a critical role in spin dynamics. However, detecting them in real time domainhas been difficult. Here we show the direct evidence of IEC between 5 d s and 4 f electrons ingadolinium. Upon femtosecond laser excitation, 5 d s electrons are directly excited; their majoritybands shift toward the Fermi level while their minority bands do the opposite. For the first time,our first-principles minority shift now agrees with the experiment quantitatively. Excited 5 d s electrons lower the exchange potential barrier for 4 f electrons, so the 4 f states are also shiftedin energy, a prediction that can be tested experimentally. Although a significant number of 5 d s electrons, some several eV below the Fermi level, are excited out of the Fermi sea, there is nochange in the 4 f states, a clear manifestation of intra-atomic exchange coupling. Based on ourresults, we propose that the demagnetization time of a material be inversely proportional to thedensity of states at the Fermi level and the excited, not the whole, spin moment. This can betested experimentally. PACS numbers: 75.40.Gb, 78.20.Ls, 75.70.-i, 78.47.J-Keywords: (December 19, 2017) . INTRODUCTION Gadolinium (Gd) is the only rare-earth metal that belongs to a group of elementaryferromagnets for magnetic storage devices , and is one of the well studied ferromagnetsboth experimentally and theoretically . Gd is also the key element for all-optical spinswitching . One key feature of rare-earth metals is that their 4 f states, deep below theFermi energy, are highly localized but contribute a major part of the spin moment. In Gd,7 µ B out of 7.55 µ B are from half-filled 4 f electrons . Thus Gd could be considered asan ideal system for the Heisenberg model, but this is oversimplified. 4 f wavefunctions inGd have little overlap with 4 f wavefunctions on neighboring sites, so the direct exchangeinteraction between 4 f states is almost zero. It is the 5 d s electrons that mediate the intra-atomic exchange coupling among 4 f electron spins. 5 d s electrons are itinerant and acrossthe Fermi level, rendering Gd metallic and optically active. In contrast to 3 d transition met-als, where the same 3 d s electrons are responsible for both magnetic and optical properties,in Gd 4 f and 5 d s electrons are respectively responsible for the magnetic properties andoptical and transport properties. The apparent disconnection between magnetic and opticalresponses presents an opportunity to investigate intra-atomic exchange coupling (IEC) intime domain.Vaterlaus et al. carried out a time-resolved spin-polarized photoemission and foundthat the spin-lattice relaxation time in Gd is 100 ±
80 ps. H¨ubner and Bennemann soonpointed out that the origin of this time scale is the spin-orbit induced magnetocrystallineanisotropy energy and showed a theoretical value of 48 ps, thus supporting the experimentalfinding. However, Vaterlaus’ pulse duration was too long to resolve IEC. Beaurepaire andhis coworkers undertook an unprecedented investigation of ultrafast spin dynamics in fer-romagnetic nickel films, thus opening a new frontier of femtomagnetism . Time-resolvedsecond harmonic generation (SHG) was first employed to detect the phonon-modulatedcoherent spin dynamics in Gd(0001) ferromagnetic metal surfaces . Lisowski et al. per-formed the time-resolved photoemission measurement and found that the spin polarizationof the surface state is reduced by half upon laser excitation, while the exchange splittingremains unchanged. The linewidth of the surface states is broadened . A complete reviewon these earlier results is given by Bovensiepen . These SHG studies are useful for surfacestates, but they do not have access to the 4 f core level (Fig. 1), thus IEC. Melnikov et al. (December 19, 2017) mployed the magnetic linear dichroism to investigate the 4 f core level and showed thatupon optical excitation of the 5 d s valence electrons, the magnetic order in the 4 f spin isreduced, from which the intra-atomic exchange effect can be inferred. Koopmans et al. compared ultrafast demagnetization between Ni and Gd through time-resolved magneto-optical Kerr effect. Despite the importance of IEC , theoretical investigations have beenscarce, in sharp contrast to other studies . Sandratskii did an interesting calculationon the exchange splitting in surface and bulk states by a static noncollinear configuration of4 f spins, but did not investigate the IEC dynamics, neither did Oroszlany . Thus, a studyof IEC is timely.In this paper, we employ a time-dependent Liouville density functional theory(TDLDFT) to study ultrafast inter-atomic exchange in Gd. We show that upon laserexcitation, the 5 d s majority band indeed shifts toward the Fermi level, while the minor-ity band shifts away from the Fermi level by 0.10 eV, in quantitative agreement with theexperiment . The excited 5 d s electrons generate a new potential for otherwise op-tically silent 4 f electrons, so 4 f states are also shifted. In the many-body physics, thiscorresponds to the intra-atomic exchange coupling, but is now manifested in the time do-main. We examine the occupancy at the Γ point before and after laser excitation andnotice that electrons even a few eV below the Fermi level are excited out of the Fermi sea.However, at the energy window where 4 f states appear, there is no population loss. Thisproves that the effect on 4 f states is indirect . We scan along the Γ-M direction or Σ linein the reciprocal lattice space and notice that the population loss is much stronger at the M point than that at the Γ point. We believe that our current study breaks new groundby putting the theory at a semi-quantitative or quantitative level, so a direct comparisonbetween experiment and theory is now possible.The rest of the paper is arranged as follows. In Sect. II, we present our theoreticalformalism. The results and discussions are presented in Section III. Finally, we concludethis paper in Section IV. II. THEORETICAL FORMALISM
Ground state properties of Gd have been thoroughly studied for a long time. Gd has astandard hcp structure with lattice constants of a = 3 . c = 5 . (December 19, 2017) nd with two atoms per unit cell . We directly use the experimental lattice constants. Thecomputed lattice parameters are very close to the experimental ones and within 1% (0.58%for PBE functional and 0.25% with + U ) . Traditionally, 4 f states can be treated as corestates or valence states, but Kurz et al. found that it is more accurate to treat it as valencestates. We employ the full-potential augmented plane wave method as implemented in theWien2k code . We adopt the generalized gradient approximation (GGA) for our densityfunctional (Perdew, Burke and Ernzerhof, 1996) , and include spin-orbit coupling (SO).Both GGA+SO and GGA+SO+U calculations are performed, and our results (density ofstates and magnetic moments) are fully consistent with the prior investigations . All theGGA+SO+U results, U and J values, and other details are presented in the supplementarymaterials.To investigate laser-excited dynamics, we solve the Liouville equation of motion for thedensity matrices ρ at each k point i ¯ h ˙ ρ = [ H, ρ ] , (1)where H is the Hamiltonian and consists of two terms, one is for the system and the otheris for the interaction between the laser and system. We choose the velocity gauge and the p · A ( t ) operator, where p is the momentum operator and A ( t ) is the vector potential of thelaser field in the unit of Vfs / ˚A. We consider a circularly polarized laser pulse propagatingalong the − z axis (see Fig. 1). Our laser has a Gaussian shape with duration τ = 48 fs andphoton energy of ¯ hω = 1 . , once the densityis obtained, one can directly compute the spin moment by tracing over the product of thedensity matrix and spin operator. However, doing so misses the important impact of theexcited states on the system itself and disregards the relaxation of the band structure .For each time step, we feed the excited density back into the Kohn-Sham equation andperform a self-consistent calculation under a constraint excited potential where the electronoccupation is held fixed. This allows the excited state to create a new potential for theentire system, so electrons, not directly excited optically, are affected as well. This provesto be the key step to our method (for details see Ref. ). This also partially overcomesthe weak demagnetization in time-dependent density functional theory . We should notethat none of the current theories is able to reproduce the same amount of the spin momentreduction under the same experimental condition. Our TDLDFT theory represents a small4 (December 19, 2017) tep forward by introducing a spin-scaling functional, so the excited state spin informationis fed back into the density . III. RESULTS AND DISCUSSIONS
Figure 2(a) shows an ultrafast reduction in the spin moment under left ( σ − ) and right( σ + ) polarized light. The laser helicity affects the amount of the reduction. σ + reducesmore due to the selection rule . Figure 2(b) shows the energy absorbed into the systemduring laser excitation. For the same laser parameter, Gd absorbs more energy than fccNi. Since our laser photon energy is 1.6 eV, the major excitation is within 5 d s electronsaround the Fermi level. The fast response is mainly due to the 5 d electrons. This can beseen clearly in the partial density of states for 5 d electrons in Fig. 3. Figure 3(a) shows themajority spin states with a low binding energy move toward the Fermi level by 0.03 eV (seethe arrow), a trend that is consistent with experimental findings . We set the Fermilevel at 0 eV. Not all the parts of the density of states behave similarly. Around -1 eV, thereis a clear modification in the structure of DOS. Quite surprisingly, this is very similar to alatest report on the band mirroring effect found in fct Co , but this result comes out of ourfirst-principles calculation naturally, without invoking other mechanisms. Figure 3(b) showsthat our minority band moves away from the Fermi level by -0.10 eV, again consistent withthe experiments . This shift is larger than that for the majority state since the minoritychannel above the Fermi level has a larger phase space and can easily receive electrons whilethe majority channel has a big gap between 0 eV and 0.7 eV (see Fig. 3(a)).Our majority and minority band shifts should be compared with the experimentalresults . Carley et al. found that the majority band shifts by 0.13 eV. This is muchlarger than our theoretical results. Their minority band shifts by 0.097 eV and matches ourtheoretical result almost quantitatively. The difference between our theory and their ex-periment is understandable. Experimentally, the photoemission probes the exchange alongthe Γ-M direction or Σ line, but theoretically, our results are from all directions; and thetheoretical results exactly along the Σ line are difficult to obtain since our k mesh is alwaysslightly shifted. It is likely that there is a dispersion along different crystal momentum direc-tions, a conjecture that can be tested in future experiments. To reduce the space chargingeffect, the experimental pulse duration is stretched to 300 fs, but our theoretical duration5 (December 19, 2017) s 48 fs, similar to Wietstruk et al. who used 50 fs. In addition, Frietsch et al. recentlyfound that the exchange splitting depends on the laser fluence. The stronger the fluence is,the larger the shift in bands becomes. Considering these differences, the agreement betweenthe theory and experiment is very satisfactory and gives us confidence in our method .A central goal of our investigation is to understand how the optically silent 4 f states arechanged through IEC during laser excitation. Figures 3(c) and (d) compare the density ofstates for 4 f states before and after laser excitation. A sharp narrow peak, consisting of 7electrons, is the hallmark of 4 f states in Gd . Before laser excitation, the majority 4 f statesare located around -4.5 eV below the Fermi level (see Fig. 3(c)), while the minority bandis empty and 0.5 eV above the Fermi level. Because of this special energy arrangement, 4 f electrons cannot be directly excited optically in Gd, different from Tb . Figure 3(d) showsthat at 193 fs after the laser pulse peaks, both the majority and minority peaks are shiftedto a high energy side, and the partial density of states changes its shape. The majority bandshifts by 1.1 eV while the minority shifts 0.63 eV. As a result the spin polarization is reduced.Wietstruk et al. detected this trend, but their data were noisy and not accurate enough tomake a comparison with our theory. We should point out that the large shift in 4 f statesis mainly due to the overestimated itinerancy of the 4 f states in the density functionaltheory (both LDA and GGA levels) . The enhanced 4 f itinerancy increases the intra-atomic exchange interaction, so the shift in the 4 f state becomes larger. The supplementarymaterials show a smaller shift in 4 f states under GGA+SO+U approximation since U termpushes the 4 f states away from the Fermi level and the intra-atomic interaction between 4 f and 5 d s states is reduced. Had we adopted the rigid band approximation, the 4 f energylevel would never have been changed.We can prove that there is no direct excitation of 4 f electrons. Figure 4 illustrates thedensity of states along the Σ line (a) before and (b) after laser excitation. For clarity, theoccupations for the middle point and M point are vertically shifted. We superimpose the4 f -partial density of states at the bottom of Figs. 4(a) and (b) so we can see where the 4 f electrons are located energetically. Note that this figure uses Rydberg as its energy unit andthe Fermi energy is not set at zero but instead is denoted by a long-dashed line. Before laserexcitation, all k points have a normal Fermi distribution (Fig. 4(a)) and have no occupationabove the Fermi level. At 193 fs after laser excitation, the distribution function is non-Fermi-like, so our Fermi energy is approximate. Figure 4(b) shows that electrons several eV6 (December 19, 2017) elow the “Fermi energy” are excited out of the Fermi sea, and excitation in each part of theoccupation is non-uniform. The electrons start to accumulate above the “Fermi energy” witha long tail. In our calculation, we include 91 states covering 4 Rydberg, but this may not beenough since we see that even original highest unoccupied states have nonzero occupations.Importantly, at 4 f states, there is no population loss (see Fig. 4(b)). The entire distributionis no longer like a Fermi distribution. The change becomes more pronounced as we moveaway from Γ point. At the M point, there is a big loss below the Fermi level, and electronspile up above the original Fermi level. We should add that there are different methodsto visualize the electron redistribution. An interesting one is the crystal orbital overlappopulation method where one plots the weighted charge distribution . Another one is tosee how the hybridization changes upon laser excitation.Finally we address a key question: whether or not time-resolved photoemission (TRPE)really detects a true magnetization change. From the above comparison of the bindingenergy change, we see that our theoretical value agrees with two experiments nearly semi-quantitatively. This agreement could be fortuitous, but the fact that both σ + and σ − lightcome to the same conclusion suggests that our TDLDFT calculation catches importantphysics. Additional evidence comes from the partial agreement between our GGA+SO+Ucalculation and experiment (see the supplementary material). Within our theory limit, weconclude that TRPE does probe demagnetization. This conclusion is conditional. First ofall, most TRPE probes emitted electrons only along one crystal momentum direction. Ifbands are narrow and flat and have little dispersion, such as d and f bands, the spin changeprobed along one direction in TRPE can be representative for the entire Brillouin zone.Second, there is a crucial and unsettling difference in demagnetization time between pho-toemission and magneto-optics and magnetic dichroism. In magneto-optics and magneticdichroism , one probes the bulk magnetization change. However, photoemission probes thespin polarization of the emitted electrons (the number of spin up electrons minus the num-ber of spin down electrons), not the spin moment in the sample. In Gd magneto-optics (MOKE) and magnetic circular dichroism (MCD) agree on the demagnetization time of750 fs within the error margin of laser pulse duration. In TRPE, the time constants ofthe binding energy shift are 200 fs and 900 fs for minority and majority spins . Neitherof these matches 750 fs. However, the exchange splitting decreases within 860 ±
100 fs ,which matches MOKE and MCD demagnetization time. This suggests that the exchange7 (December 19, 2017) plitting reduction in TRPE, not the binding energy shift, is related to demagnetization.This finding needs additional experimental investigations. Caution must be taken that oneshould not expect a similar exchange splitting collapsing in rare-earth metals as that intransition metals since 4 f electrons in rare-earth metals strongly polarize 5 d electrons. Thisis reminiscent of an early study in Co , where the spin-resolved inelastic lifetime is as shortas 20 fs, 10 times shorter than the current established demagnetization time of 220 fs for fccCo .However, what determines demagnetization times in a sample has no simple answer be-cause both intrinsic and extrinsic factors play a role. Current theories are unable to givea quantitative answer. The bulk demagnetization time of Gd is found around 0.7 ps andincreases by 10% within a fluence change up to 1 mJ/cm . Wietstruk et al. showed thatboth Gd and Tb have a similar ultrafast demagnetization time of 750 fs with an uncertaintyof 250 fs. Koopmans et al. suggested a simple expression that relates the demagnetizationtime to the Curie temperature T c and magnetic moment µ at , T c /µ at of a sample. However,when Wietstruk et al. applied it to Gd and Tb, they could not explain a similar demagne-tization time in Gd and Tb . Our above simulation singles out the importance of excitedcharge density in demagnetization. Both Gd and Tb involve the same type of 5 d s electrons.We propose an alternative relation that the demagnetization time should be proportional to τ m ∝ ρ ( E f ) µ x , (2)where µ x refers to the spin moment of the excited electrons , not all the electrons, and ρ ( E f )is the density of states at the Fermi level. Quantitatively, the 3 d density of states in Niat its Fermi level is about 1.5 states/eV and the spin moment is 0.6 µ B , while in Gd, the5 d density of states in Gd is about 0.3 states/eV and the spin moment of 5 d is about 0.58 µ B . According to Mathias et al. , the demagnetization time for Ni is 157 fs, so this givesthe demagnetization time for Gd, 785 fs. This differs from Wietstruk’s results by 35 fs,well within their pulse duration of 100 fs . Given that these two experiments use entirelydifferent techniques, this quantitative agreement is truly gratifying, and should be tested inother systems. Physically, the above expression appears to be more reasonable since it isconsistent with the basic theory of fundamental excitation in metals. If there is low densityof states at the Fermi level, the relaxation among the electrons themselves is going to beslow. In half-metals such as CrO , one channel is shut off, so the relaxation must be slow .8 (December 19, 2017) ery recently, Frietsch et al. reported two different dynamics for 5 d (800 fs) and 4 f electrons (14 ps). This is fully expected since 4 f electrons have nearly zero density of statesat the Fermi level and are indirectly excited through the intra-atomic exchange interaction.The short time dynamics is associated with the 5 d s electrons. While an extensive study onthis is beyond the scope of this paper, we propose to measure the density of states in thoseprior samples and reexamine those experimental results . IV. CONCLUSION
Our time-dependent first-principles calculation has shed new light on how intra-atomicexchange correlation develops in Gd after laser excitation. It starts with the 5 d s electronsbecause of their proximity to the Fermi level. Because of differences in their phase spaces,the majority and minority 5 d s electrons respond differently: The majority band shifts to-ward the Fermi level, while the minority moves away from the Fermi level. Our theory nowagrees with the experimental results qualitatively and even quantitatively for the minoritybands. To the best of our knowledge, this has never been attempted. The excited 5 d s elec-trons not only affect themselves, but also generate a new potential for optically inaccessible4 f electrons, so 4 f states feel the impact of laser excitation. We find that the 4 f -density ofstates changes its original shape, also seen in Co , and shifts its position as well, a predictionthat must be tested experimentally. Although 5 d s electrons are excited intensively, thereis no change at 4 f states, a hallmark of the intra-atomic exchange correlation. Our studyrepresents a beginning and is expected to have a broad impact on future research. In par-ticular, nearly a quantitative agreement between the density functional calculation and theexperiments allows an unbiased comparison. This will certainly encourage new theoreticaland experimental efforts in ultrafast demagnetization in rare-earth materials. Acknowledgments
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FIG. 1: (Left) Schematic of the electron density of states in hcp Gd. 5 d s states are around theFermi level and optically accessible, while 4 f states are deeply below the Fermi energy and silentoptically. (Right) Laser pulses excites 5 d s electrons first and, through the inter-atomic exchangecoupling, affects highly localized 4 f electrons.D. Ebke, V. Drewello, G. Reiss and A. Thomas, Insights into ultrafast demagnetization inpseudogap half-metals, Phys. Rev. X , 041008 (2012). (December 19, 2017)
100 0 100 200Time (fs)00.10.20.30.4 ∆ E ( R y d ) σ − σ + −100 0 100 200Time (fs)−0.5−0.4−0.3−0.2−0.10.0 ∆ M ( µ B ) σ − σ + (a) (b) FIG. 2: (a) Magnetic spin moment reduction ∆ M as a function of time for left ( σ − , solid line)and right ( σ + , dashed line) circularly polarized light. The laser pulse duration is 48 fs, its photonenergy is 1.6 eV and the field amplitude is 0 . / ˚A. The spin minimum appears after the laserpeaks. (b) Energy absorbed into the system. It follows the spin dynamics closely. (December 19, 2017) f ) (eV) − D en s i t y o f s t a t e s −2 −1 0 1(E−E f ) (eV) − D en s i t y o f s t a t e s Original@193 fs −6 −4 −2 0 2(E−E f ) (eV) f − D en s i t y o f s t a t e s −6 −4 −2 0 2(E−E f ) (eV) f − D en s i t o f s t a t e s (c) (d)(a)(b) −0.10 eV5d−Majority5d−Minority 4f−Majority4f−Minority FIG. 3: Densities of states (DOS) before (solid line) and after laser excitation (dashed line).All DOS are computed at the GGA level and under σ + excitation. The laser field amplitude is0 . / ˚A and pulse duration is 48 fs. The Fermi level is at 0 eV. (a) Upon laser excitation, 5 d -majority DOS is shifted toward the Fermi level by 0.03 eV. This is smaller than the experimentalvalue , but the trend is correct. (b) 5 d -Minority spin DOS is shifted away from the Fermi levelby -0.10 eV. This value quantitatively agrees with the experimental results . (c) 4 f -Majority spinDOS is shifted toward the Fermi level by 1.1 eV. (d) 4 f -Minority spin DOS is shifted to the higherenergy by 0.4 eV. These two shifts need experimental verification. (December 19, 2017) O cc upa t i on Original −2 −1 0 1 2E(Ryd) 0123 O cc upa t i on at 193 fsE f "E f " Γ Mmiddle Γ middleM (a) (b) Σ Σ f − m a j o r i t y f − m i no r i t y f − m a j o r i t y f − m i no r i t y FIG. 4: Electron occupancy along the Γ-M direction (Σ line). (a) Before laser excitation, thedistributions are a typical Fermi-Dirac distribution. Our Γ point is approximate since our k gridmesh is slightly shifted to improve convergence. Different from Fig. 3, the energy scale is inRydberg. The Fermi energy is not at 0; instead it is denoted by the vertical long-dashed line.The occupancies for the middle point and M point are vertically shifted for clarity. (Bottom inset)Both the majority and minority 4 f partial densities of states are superimposed on the occupation(dotted line). (b) After laser excitation, the electrons are excited out of the Fermi sea. The electronexcitation is much stronger at the M point than at the Γ point. States several eV below the Fermilevel are excited, but not at the 4 f location.18