Model of ultrafast demagnetization driven by spin-orbit coupling in a photoexcited antiferromagnetic insulator Cr2O3
MModel of ultrafast demagnetization driven by spin-orbit coupling in a photoexcitedantiferromagnetic insulator Cr O Feng Guo, Na Zhang, Wei Jin, Jun Chang ∗ College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710119, China
We theoretically study the dynamic time evolution following laser pulse pumping in an antiferro-magnetic insulator Cr O . From the photoexcited high-spin quartet states to the long-lived low-spindoublet states, the ultrafast demagnetization processes are investigated by solving the dissipativeSchr¨odinger equation. We find that the demagnetization times are of the order of hundreds offemtosecond, in good agreement with recent experiments. The switching times could be strongly re-duced by properly tuning the energy gaps between the multiplet energy levels of Cr . Furthermore,the relaxation times also depend on the hybridization of atomic orbitals in the first photoexcitedstate. Our results suggest that the selective manipulation of electronic structure by engineeringstress-strain or chemical substitution allows effective control of the magnetic state switching inphotoexcited insulating transition-metal oxides. PACS numbers: 75.78.Jp, 82.50.-m, 82.53.-k, 63.20.kd
INTRODUCTION
In recent years, growing attention has been drawn tothe photodriven ultrafast control of the quantum statesand the physical properties in solid-state and molecularsystems. In addition to the great theoretical interest inunderstanding the nonequilibrium dynamics in materials,it could be applied technically to the magnetic or elec-tronic recording.
The photoinduced change of physicalproperties is often attributed to thermal effects becausethe photon energy eventually is redistributed among in-teracting charge, spin, and lattice degrees of freedom,and increases the system temperature instantly. On theother hand, photoirradiation may induce non-thermalmetastable states or transient phases with optical, mag-netic and electric properties distinct from that of theground states.
Among these light-responsive materials, the ferromag-netic materials have been brought into sharp focus bylaser-induced demagnetization since Bigot and cowork-ers found the ultrafast dropping of magnetization innickel film following optical pulses in 1996. Until re-cently, the ultrashort pulses of light are applied to manip-ulate the ultrafast processes in the antiferromagnets.
Indeed, antiferromagnetic (AFM) materials have moreadvantages than ferromagnets. For example, theyare insensitive to external magnetic fields, stable inminiaturization and much faster in controlling spindynamics. AFM insulator Chromium oxide (Cr O ) has been thesubject of study since the 1960s and its electronic andstatic optical properties are now well understood. However, the ultrafast dynamic demagnetization pro-cesses were not probed until recently.
The time-resolved second harmonic generation is applied to probethe time evolution of the magnetic and structural statefollowing laser illuminations in the AFM insulator. Vari-ations in the pump photon-energy lead to either local-ized transitions within the metal-centered states of the Cr ion or charge transfer between Cr and O. Despite itsrelevance to industrial technology, the ultrafast processesof demagnetization are not well understood from quan-tum nonequilibrilium dynamics. To selectively controlthe demagnetization rate is still at a tentative stage inexperiments.In this paper, we first construct a local quantum-mechanical demagnetization model of the photoinducedelectron states in Cr O . The effects of the energydissipations are taken into account by a dissipativeSchr¨odinger equation. We simulate the time evolutionof the excited states following 1.8 eV and 3.0 eV lightillumination and find that the decay times from the high-spin quartet states to the low-spin doublet states rangefrom 300 femtoseconds (fs) to 450 fs, in line with theexperiments. We show that the ratio of the energy gapto the electron-phonon self-energy has a marked impacton the demagnetization times. The decay times are alsoinfluenced by the hybridization of atomic orbitals in thefirst photoexcited state.
DEMAGNETIZATION MODEL
A typical static energy-level scheme of a Cr metalion is shown in Fig. 1. The metal ion is in close prox-imity to oxygen octahedral surrounding and the five-folddegenerate 3 d orbitals are split into a lower threefold-degenerate t g and an upper twofold-degenerate e g or-bitals by the crystal field with O h symmetry. Due to theHund coupling, the ground state, less than half filled, isa high-spin ( S = 3 / A ( t g ) configuration. Early in1963, McClure reported the polarized optical absorptionspectrum of Cr O with the wavelength ranging from 300to 800 nm in thin single-crystal plates. Two broad ab-sorption bands are observed in the range of 400–800 nmcorresponding to the transitions of the 3 d electron shellfrom the A ground-state level to the excited-state lev-els, T ( t g e g ) and T ( t g e g ), respectively. Between a r X i v : . [ c ond - m a t . s t r- e l ] J un FIG. 1. On the left: schematic absorbance of chromium oxidebased on the spectrum measurements found in Ref 15. Twobroad absorption bands located in the range of 1.8-3.0 eV cor-respond to the spin-allowed transition from the A groundstate to the T and T quartet states. The three sharp linesare associated with the spin-forbidden transitions to the T , E and T doublet states. On the right: energy-level schemeof Cr O . R represents the coordination along the metal-ligand coordinate. ∆ and ε are the energy gap between thelowest vibrational levels and the electron-phonon self-energydifference between two oscillation states, respectively. Thecentral energy value E i of the absorption spectrum is indi-cated by a line segment. In some cases, E i is different fromthe energy value of the lowest vibrational level owing to differ-ent electron-phonon couplings, e.g. in the T and T states.Here, we have set the ground state energy to zero. the T and T absorption bands, there is a sharp line as-sociated to the spin-forbidden transition to the T ( t g )doublet. Two other sharp lines link to the transitions tothe low-lying E ( t g ) and T ( t g ) doublets. During the ultrafast photodirven demagnetization pro-cess from the high-spin to low-spin states in Cr oxides,the first localized excited state triggered by laser irradi-ation does not directly return to the ground state butfollows a complex route of intermediate states accompa-nying with changes in spin and lattice parameters. Thespin-orbit coupling (SOC) could flip the spin of d -orbitelectrons in the intermediate states. The redistribution ofanisotropic d -orbital occupations often leads to geometricdeformation or structural phase transition. Meanwhile,the locally excited state dissipates energy to its surround-ings by emission of phonons and/or photons. Since therelaxation time of fluorescence is on a nanosecond (ns)time scale, then a phonon continuum dominates the en-ergy dissipation in the ultrafast demagnetization. To elu-cidate this dynamical process, we introduce a model withelectronic multiplet levels at energies E i , coupled to aphonon bath. Due to the strong electron-phonon cou-pling and the substantial bath memory effects in a pho-todriven system, a Born-Markov master equation failsto effectively describe the ultrafast electron dynamics. Therefore, we first map the spin-boson-like model to analternative model, where the electronic levels are cou-pled to a single harmonic mode damped by an Ohmicbath. The memory effects could be effectively takeninto account by the time evolution of the strength of theharmonic mode. Here, the correlations between electronsare taken into account by the renormalization of the elec-tronic state energies. The local system Hamiltonian iswritten as H S = (cid:88) i E i c † i c i + ¯ hωa † a + (cid:88) i λ i c † i c i ( a † + a )+ (cid:88) ij V ij ( c † i c j + c † j c i ) , (1)where c † i c i gives the occupation of the multiplet i , V ij is the coupling constant that causes a transition be-tween energy level j and i , a + is the creation oper-ator for the harmonic phonon with frequency ω . Wefurther define the electron-phonon self-energy difference ε ij = ( λ i − λ j ) / ¯ hω and the energy gap ∆ ij = ( E i − λ i / ¯ hω ) − ( E j − λ j / ¯ hω ) between two states, where varia-tions in the electron-phonon coupling strength λ i changethe equilibrium positions of different states. (see Fig. 1).We describe the time evolution of the local openquantum system with the dissipative Schr¨odingerequation i ¯ h d | ψ ( t ) >dt = ( H + iD ) | ψ ( t ) >, (2)where H is the the Fr¨ohlich transformation of the Hamil-tonian H S , D is a dissipative operator that describes thebath induced state transfer D = ¯ h (cid:88) k d ln P k ( t ) dt | ψ k >< ψ k | . (3)Here, P k ( t ) = | c k ( t ) | is the state probability and | ψ ( t ) > = (cid:80) k c k ( t ) | ψ k > . The time evolution of theprobability of multiplet i with n excited phonon modesis given by dP in ( t ) dt = − n Γ P in ( t ) + 2( n + 1)Γ P i, ( n +1) ( t ) , (4)with Γ = π ¯ ρ ¯ V / ¯ h , the environmental phonon relax-ation constant. According to the Jablonski energydiagram, (2Γ) − ranges from 0.01 picosecond (ps) to 10ps. In this paper, we set (2Γ) − = 0 . DEMAGNETIZATION PROCESS
Since the spin-flip is forbidden in photoexcitation, thefirst photoexcited states starting from the A groundstate are T , T , and the metal-ligand charge transfer M LCT ( t g L ) quartet states, depending on the ener-gies of photoexciation. Here, L denotes that an elec-tron transfers to ligands. After the light illumination,the system decays to a relatively long-lived metastablestate, e.g., the T and E doublet states in Cr O .Importantly, a long lifetime of the excited energy levelpromises to be a candidate to develop a potential laserdevice. In order to understand the dynamic process, wefirst need to determine the energies of the states involvedin the cascading process. According to the absorptionspectra, the central energies E i of the E , T , T , T , T , and M LCT states locate around 1.7, 1.76,2.1, 2.45, 2.75, 3.3 eV, respectively.
Next, we needto establish an appropriate range of interaction parame-ters. The strength of the interaction between the Cr ionand its surrounding oxygen anions can be obtained from ab initio calculations. The change in energy for differ-ent configurations is close to parabolic for an adiabaticchange in the Cr-ligand distance. From the change inequilibrium distance or the optical absorption and lumi-nescence spectra, we can obtain the Huang-Rhys factor g ≈ E , T and T ) andthe quartet states (e.g. T and T ). We assume theHuang-Rhys factor g ≈ M LCT according to the sharp absorption line. Corre-spondingly, the difference of electron-phonon self-energy ε ij is equal to g ¯ hω and the electron-phonon coupling con-stant | λ i − λ j | = (cid:112) ε ij ¯ hω . The spin changes during thetransfer from a quartet to a doublet state, and the cou-pling V between the two different spin states is generallyaccepted to be due to the SOC. We take the strengh ofSOC around 0.03 eV in Cr ions. Strongly coupled tothe optically excited electrons, the optical phonon modescould be observed by Raman spectroscopy. Owing to thesymmetry of Cr O , there are seven Raman modes, twowith A g symmetry and five with E g symmetry, and thelonger wavelengths corresponding to the E g modes. Wetake the E g mode value ¯ hω = 0 .
075 eV, which dominatesthe relaxation at the 1.8 eV pumping, and A g mode¯ hω = 0 .
065 eV, the main damping phonon at 3.0 eVphoton excitation. The 1.8 eV photoexcitation results in the transitionfrom the A ground state to the T excited state. Anelectron in the t g orbital is locally excited to the e g or-bital by the illumination. Such a transition yields anelongation of the Cr–O band length of several tenths ofan ˚Angstrom since the change from a t g to an e g chargedistribution leads to a stronger repulsion between the Crand the O ligands. The bond length change leads to dif-ferent electron-phonon couplings between the two states,thereby forming a Franck-Condon continuum. Underthe action of SOC and electron phonon interaction, thefirst excited state relaxes to the long lived states, namely,the T and E doublet states. The energy gap between T and E is small, e.g. around 0.06 eV, therefore the T and E populations are often combined for kineticpurposes. In Fig. 2, we show the time evolution of
FIG. 2. The probability of finding the quartet and doubletstates as a function of time at 1.8 eV photon excitation. Thedashed line (blue) P q ( t ) gives the T quartet state probabil-ity, the solid line (green) P d ( t ) shows the sum of the proba-bility of the T and E doublet states. There are two oscil-lations in the state probabilities with the periods around 100fs and 40 fs, corresponding to the 0.04 eV and 0.1 eV energylevel gaps between two doublet states and the quartet state,respectively. the three states involved in the ultrafast demagnetizationprocess by solving the dissipative Schr¨odinger equation.The starting state is T , excited from the ground state.The E g phonon mode with ¯ hω = 0 .
075 eV dominates therelaxation. Therefore, the self-energy difference ε be-tween the metal-centered quartet and doublet is 0.3 eV,with the Huang-Rhys factor g = 4. The quartet stateand the doublet state are mediated by SOC, i.e. V =0.03eV. The energy gap ∆ between T and T is 0.04 eV,and 0.1 eV between T and E . The 0.04 eV and 0.1 eVenergy gaps are indicated in the obvious oscillations withthe periods around 100 fs and 40 fs, respectively in thetime evolution of both the quartet and doublet states.We find that the probability of the T state falls quicklyand the sum of the probability of the T and E statesincreases in the first 0.5 ps. Fitting the curves using ki-netic rate equations, the rise time constant of the sum ofthe two doublet states are around 400 fs, which agreeswell with the experiments by the time-resolved secondharmonic generation. The decay time of a photoexcited state strongly de-pends on the ratio of the energy gap to the electron-phonon self-energy difference, which has been demon-strated in transition-metal complexes. When the ratio∆ /ε ranges from 0.5 to 1.5, the fastest decay occurs. Inengineering, the energy gap between the multiplets couldbe changed by distortion stress, strain or chemical sub-stitution of ligands, which provides a feasible approachto adjust the demagnetization time. For example, sincethe gap between T and T is very small, ∆ = 0 .
04 eV,it may result in a longer decay time. We found that pro-vided the gap increases 0.2 eV, close to ε = g ¯ hω = 0 . FIG. 3. The time evolution of the state probabilities withdifferent energy gaps between T and T at 1.8 eV pho-ton pumping. (a) The gap ∆ between T and T is 0.24eV, close to the corresponding ε =0.3 eV, the demagnetiza-tion time reduces to 100 fs; (b) ∆ = 0 .
34 eV, the relaxationtime is around 150 fs. The dashed line (blue) and the solidline (green) give the probabilities of finding T and the sumof T and E , respectively. On the other hand, if we only vary the photoexcitationenergies from 1.8 eV to 2.1 eV and keep all the other pa-rameters the same, the time evolutions of the three stateschange slightly from Fig. 2 (not shown).The 3.0 eV photon energy is supposed to excite the A ground state to the T metal-centered state and/or the M LCT metal-ligand charge transfer state. Due to theelectron-phonon interaction, SOC and orbital hybridiza-tion, the first excited state finally reaches the T and E doublet states via the transit T , T and T states. Inpure octahedral symmetry, there is no coupling between M LCT and T , , since the e g orbital states do notcouple to the ligand π ∗ states. Nevertheless, since thenonequilibrilium charge transfer often leads to a lowerstructural symmetry, a small hybridization between thetwo quartet states should be presented, depending onthe amount of distortion. The weak hopping energy be-tween the ligands π ∗ or π and the metal ion’s e g or-bitals is of the order of hundredths eV. Our numericalcalculations are not sensitive to the change in the hy-bridization from 0.03 eV to 0.09 eV. In Fig. 4, we setthe hybridization parameter 0.05 eV between π ∗ and e g .The metal-centered quartet and doublet states are sup-posed to be mediated by SOC, V =0.03 eV. There is nodirect coupling between M LCT and the doublet statessince no interaction allows spin-flip and charge transfersynchronously. The main damping phonon is the A g phonon mode with ¯ hω = 0 .
065 eV. Consequently, ε be-tween the metal-centered quartet and doublet states is0.26 eV with the Huang-Rhys factor g = 4. First, weassume that the electrons in the ground state is excitedto the M LCT state. From the time evolution of thestates, we find that the rise time constant of the dou-blets is around 360 fs by fitting the curves using kineticrate equations. Next, it has been pointed out that thefirst excited state at high energy excitation could be the
FIG. 4. The electron occupation probability in demagne-tization process at 3.0 eV photon pumping. The dashedlines (blue) denotes the time evolution of the sum of the MLCT , T and T quartet state probabilities, the solidline (green) refers to the sum of the probability of the T , T and E doublets. The first photoexcited state is a (cid:12)(cid:12) MLCT (cid:11) + √ − a (cid:12)(cid:12) T (cid:11) with a = 0 .
25. The fitting risetime value of the doublet probability is around 300 fs. mix of T and M LCT . Taking a mixed first excitedstate a (cid:12)(cid:12) M LCT (cid:11) + √ − a (cid:12)(cid:12) T (cid:11) , the fitting rise timevalues of the doublets are 450 fs with the mix constant a = 0 .
75, 380 fs for a = 0 .
5, and 300 fs for a = 0 . a = 0 .
25. Interestingly,the T state is often ignored in some literatures becauseits absorption is too narrow to be resolved in optical spec-tra. However, if the T state is omitted in our model,we find that the probability of T plateaus at value 0.1from 3 ps after the pumping. An extension of the timeevolution up to 0.1 ns confirms that the neglect of T re-sults in an incomplete decay of T , which is inconsistentwith the experiments. Therefore, the time scale of ultra-fast demagnetization depends not only on the pumpingenergy, but also on the electric configuration of energylevels, the spin-orbit and electron-phonon couplings. CONCLUSION
To conclude, we have presented a quantum-mechanicaldemagnetization model for the locally photoinduced elec-tron state in Cr O . Using the dissipative Schr¨odingerequation, the environmental enegy dissipations are con-sidered. We numerically simulated the time evolution ofthe excited states following 1.8 eV and 3.0 eV photonexcitation. The decay times are consistent with experi-ments on the order of hundreds of femtosecond from thehigh-spin quartet states to the low-spin doublet states.Both the spin-orbit coupling and electron-phonon cou-pling take important roles in the ultrafast demagneti-zation processes. We have shown that the ratio of theenergy gap to the electron-phonon self-energy has strongimpact on the demagnetization times. The hybridiza-tion of the atomic orbitals in the first photoexcited statealso affects the decay times. We further expect that thedemagnetization times could be selectively controlled bythe engineering stress-strain or chemical substitution ofligands in insulating transition-metal oxides. ACKNOWLEDGMENTS
We are thankful to Jize Zhao,Hantao Lu and Ning Lifor fruitful discussions. F. G. and J. C. are supported bythe Fundamental Research Funds for the Central Univer-sities, Grant No. GK201402011. W. J. is supported byNSFC 11504223. ∗ [email protected] O. Sato, Nature Chem. , 644 (2016). M. Cammarata, R. Bertoni, M. Lorenc, H. Cailleau,S. Di Matteo, C. Mauriac, S. F. Matar, H. Lemke, M. Chol-let, S. Ravy, et al., Phys. Rev. Lett. , 227402 (2014). W. Jin, C. Li, G. Lefkidis, and W. H¨ubner, Phys. Rev. B , 024419 (2014). A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. , 2731 (2010). E. Collet, M. H. Leme-Cailleau, C. M. Buron-Le, H. Cail-leau, M. Wulff, T. Luty, S. Y. Koshihara, M. Meyer,L. Toupet, and P. Rabiller, Science , 612 (2003). E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,Phys. Rev. Lett. , 4250 (1996). J. Zhang, X. Tan, M. Liu, S. W. Teitelbaum, K. W. Post,F. Jin, K. A. Nelson, D. N. Basov, W. Wu, and R. D.Averitt, Nat. Mater. , 965 (2016). H. Ehrke, R. I. Tobey, S. Wall, S. A. Cavill, M. F¨orst,V. Khanna, T. Garl, N. Stojanovic, D. Prabhakaran, A. T.Boothroyd, et al., Phys. Rev. Lett. , 217401 (2011). H. Ichikawa, S. Nozawa, T. Sato, A. Tomita, K. Ichiyanagi,M. Chollet, L. Guerin, N. Dean, A. Cavalleri, S.-i. Adachi,et al., Nat. Mater. , 101 (2011). X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He,R. J. Paull, J. D. Clarkson, J. Kudrnovsk, and I. Turek,Nat. Mater. , 367 (2014). S. Loth and A. J. Heinrich, Science , 196 (2012). M. F¨orst, R. I. Tobey, S. Wall, H. Bromberger, V. Khanna,A. L. Cavalieri, Y.-D. Chuang, W. S. Lee, R. Moore, W. F.Schlotter, et al., Phys. Rev. B , 241104 (2011). M. Fiebig, N. Phuc Duong, T. Satoh, B. B. Van Aken,K. Miyano, Y. Tomioka, and Y. Tokura, J. Phys. D: Appl.Phys. , 164005 (2008). A. V. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, andT. Rasing, Nature , 850 (2004). D. S. Mcclure, J. Chem. Phys. , 2289 (1963). R. M. Macfarlane, J. Chem. Phys. , 3118 (1963). C. M. Mo, W. L. Cai, G. Chen, X. M. Li, and L. D. Zhang,J. Phys.: Condens. Mat. , 6103 (1997). T. I. Y. Allos, R. R. Birss, M. R. Parker, E. Ellis, andD. W. Johnson, Solid. State. Commun. , 129 (1977). L. S. Forster and L. S. Forster, Coord. Chem. Rev. ,261 (2004). J. S. Dodge, A. B. Schumacher, J. Y. Bigot, D. S. Chemla,N. Ingle, and M. R. Beasley, Phys. Rev. Lett. , 4650(1999). V. N. Muthukumar, R. Valent, and C. Gros, Phys. Rev. B , 433 (1996). Y. Tanabe, M. Muto, M. Fiebig, and E. Hanamura, Phys.Rev. B , 8654 (1998). X. G. Wang, W. Weiss, S. K. Shaikhutdinov, M. Ritter,M. Petersen, F. Wagner, R. Schloegl, and M. Scheffler,Phys. Rev. Lett. , 1038 (1998). V. G. Sala, S. D. Conte, T. A. Miller, D. Viola, E. Luppi,V. Vniard, G. Cerullo, and S. Wall, Phys. Rev. B , 1113(2016). T. Satoh, B. B. V. Aken, N. P. Duong, T. Lottermoser,and M. Fiebig, Phys. Rev. B , 155406 (2007). G. Lefkidis, G. P. Zhang, and W. H¨ubner, Phys. Rev. Lett. , 217401 (2009). M. G. Brik, N. M. Avram, and C. N. Avram, Solid. State.Commun. , 831 (2004). B. B. Krichevtsov, V. V. Pavlov, R. V. Pisarev, and V. N.Gridnev, Phys. Rev. Lett. , 4628 (1996). K. Ogasawara, F. Alluqmani, and H. Nagoshi, ECS J. SolidState Sci. Technol. , R3191 (2016). G. A. Torchia, O. Martinez-Matos, N. M. Khaidukov, andJ. O. Tocho, Solid. State. Commun. , 159 (2004). J. Chang, I. Eremin, and J. Zhao, Phys. Rev. B , 104305(2014). A. Garg, J. N. Onuchic, and V. Ambegaokar, J. Chem.Phys. , 4491 (1985). M. van Veenendaal, J. Chang, and A. J. Fedro, Phys. Rev.Lett. , 067401 (2010). J. Chang, A. J. Fedro, and M. van Veenendaal, Phys. Rev.B , 075124 (2010). H. Dekker, Phys. Rep. , 1 (1981). J. Chang, A. J. Fedro, and M. van Veenendaal, Chem.Phys. , 65 (2012). M. G. Brik, N. M. Avram, and C. N. Avram, Solid. State.Commun. , 233 (2005). L. Zundu and H. Yidong, J. Phys.: Condens. Mat. , 9411(1993). M. Muto, Y. Tanabe, T. Iizuka-Sakano, and E. Hanamura,Phys. Rev. B , 9586 (1998). M. Stamenova, J. Simoni, and S. Sanvito, Phys. Rev. B , 760 (2016). S. H. Shim, T. S. Duffy, R. Jeanloz, C. S. Yoo, and V. Iota,Phys. Rev. B , 1124 (2004). L. D. Zhang, C. M. Mo, W. L. Cai, and G. Chen, Nanos-truct. Mater.9