Single and Double Orbital Excitations Probed by Resonant Inelastic X-ray Scattering

Filomena Forte, Luuk J.P. Ament and Jeroen van den Brink Institute-Lorentz for Theoretical Physics, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: August 1, 2008) The dispersion of the elusive elementary excitations of orbital ordered systems, orbitions, has escaped detection so far. The recent advances in Resonant Inelastic X-ray Scattering (RIXS) tech- niques have made it in principle a powerful new probe of orbiton dynamics. We compute the detailed traces that orbitons leave in RIXS for an eg orbital ordered system, using the ultrashort core-hole life-time expansion for RIXS. We observe that both single and two orbiton excitations are allowed, with the former having sharper features, appearing lower in energy. The rich energy and momentum dependent intensity variations that we observe make clear that RIXS is an ideal method to identify and map out orbiton dispersions.

Introduction The exotic phases and phenomena exhib- ergy. Orbiton scattering causes a characteristic energy ited by many transition metal oxides originate from the and momentum dependent intensity variations in RIXS interplay of their electronic , charge and orbital de- with certain selection rules. Matrix element effects also grees of freedom, coupled to the lattice dynamics [1, 2]. make for instance the two orbiton scattering intensity The orbital degree of freedom, originating from the un- very different from the bare two-orbiton density of states. lifted or only partially lifted local orbital degeneracy of This bolsters the case that from a theoretical perspective the 3d , plays a particularly important role in RIXS is ideally suited to map out the orbiton disper- for instance the physics of Colossal-Magneto-Resistence sions [15, 16]. (CMR) manganites [3–5]. The orbitals also stand out be- In RIXS a material is resonantly excited by tuning the cause –in contrast to the other degrees of freedom– the energy of incoming X-rays to an atomic absorption edge, dynamics of these elementary excitations is still far from in manganites for instance the Mn K- or L-edge. At the being fully understood. The main reason is that it has Mn K-edge the incoming promotes a 1s proven very difficult to access orbital excitations experi- from the core into the 4p state far above the Fermi energy, mentally. see Fig. 1. In this way a 1s core-hole is created, which The first claim of the observation of these elusive or- interacts with the 3d electrons, that tend to screen the bital excitations, orbitons, in LaMnO3 by optical Raman core-hole potential. After a very short time, the photo- scattering [6] is very controversial [7]. Irrespective of the excited electron decays, leaving the 3d electrons in an interpretation of these data, however, a severe limitation excited state. From the kinetics of the outgoing X-rays, of the optical Raman technique is its selectiveness to ex- the energy and momentum of the collective excitations citations carrying zero momentum. This method is thus of the solid can be determined directly [17]. A great ad- intrinsically unsuitable to map out orbiton dispersions. vantage of RIXS is that the high energy X-rays carry a Other evidence for the existence of orbital excitations momentum that is comparable to the crystal momentum comes from very recent optical pump-probe experiments in the Brioullin zone so that, unlike optical Raman exper- on manganites [8]. Even if very ingenious, also these ex- iments, RIXS can probe the full dispersion of excitations periments cannot provide information on the momentum in solids. The present experimental resolution at the Mn dependence of orbitons. edges is ∼ 100 meV. In the very near future instrumen- A technique that is in principle capable of detecting tation with an improvement with an order of magnitude the full dynamics of orbitons is Resonant Inelastic X- in resolution at the K edge will become feasible [18], al- ray Scattering (RIXS). Recently, astounding progress was lowing the detection of low-lying orbital excitations. made in the energy and momentum resolution of RIXS, In this Letter we determine the orbiton RIXS spec- allowing for instance the observation of magnon excita- trum for an orbital ordered system with orbitals of eg tions and their dispersions in copper oxides [9]. The suc- symmetry, but the approach that we outline can equally cess of theory in describing the dispersive magnetic RIXS well be used for other orbital ordering symmetries. In data [10–12] and in particular the success of the so-called order to compute the RIXS spectrum it is key to deter- ultra-short core-hole lifetime (UCL) expansion [13, 14], mine how the intermediate state core-hole modifies the provides the motivation to uncover also the signatures orbital dependent superexchange processes between the of orbital excitations using this theoretical framework. 3d electrons. After doing so, our calculations based on We therefore set out to compute and predict the detailed the UCL expansion will show how such modifications fingerprints of the orbitons in RIXS, finding that both give rise to both single and double orbiton features in single and two orbiton excitations are allowed, with the the RIXS spectrum. The one-orbiton part turns out to former having sharper features, appearing lower in en- carry most spectral weight. This is in stark contrast with 2

a local Coulomb repulsion U between the electrons in the eg subspace. In the resulting Kugel-Khomskii model [4] the orbitals of classical antiparallel spins decouple if one neglects the Hund’s rule exchange compared to the on- site Coulomb repulsion. As the quantum fluctuations as- sociated with the large Mn core-spin on the A-type AFM structure are typically small, in leading order the orbital degrees of freedom effectively decouple along the c-axis, simplifying the orbital dispersion to a two-dimensional one. The orbitals in the ab-plane are described by pseudo- spins, where pseudo-spin√ up corresponds to the orbital 2 2 |zi ∼ (3√r − z )/ 6 occupied and down to |xi ∼ (x2 − y2)/ 2. The in-plane hopping integrals are |txx| = √ 3 zz 1 xz 3 zz 4 t, |t | = 4 t, |t | = 4 t with reference t = |t | along the z-direction. After a rotation in pseudo-spin space over an angle θ = π/4, the orbital model Hamiltonian is 0 J P 0 H = Hij with FIG. 1: The effect of the core-hole on the orbital exchange. 2 hiji 0 An X-ray with energy ωin and momentum qin excites the 1s electron to a 4p state. Via an intermediate state, the sys- √ 0 z z x x z x x z tem reaches a final state and the core-hole decays, emitting Hij = 3Ti Tj + Ti Tj ± 3 Ti Tj + Ti Tj , (1) 0 a photon of energy ωout and momentum qout. In the fig- ure the matrix element h↓ ↑ | H |↑ ↑ i is considered in pres- i j i j √ ence of a core-hole. There are two alternative intermediate where J = t2/U [15]. The prefactor of the 3-term is states to reach the final state. In the upper case, the ampli- xz xx positive in the x-direction and negative in the y-direction. tude is proportional to t t /(U + Uc) and in the lower case xz zz The classical ground state has electrons alternately oc- to t t /(U − Uc). Adding these gives a modified exchange 0 cupying the orbitals √1 (|xi + |zi) and √1 (|xi − |zi). We J = J (1 + η) where η depends on Uc. 2 2 introduce sublattice A with pseudo-spin up and B with + pseudo-spin down and Holstein-Primakoff bosons Ti∈A = magnetic RIXS, where only two-magnon scattering is al- z † + † z † ai,Ti∈A = 1/2−ai ai and Tj∈B = aj,Tj∈B = ajaj −1/2. lowed at zero temperature [10, 12]. The computed or- To obtain the orbital excitations we retain the terms up biton spectrum for the eg orbital ordering of LaMnO3 to quadratic order in the boson operators. After Fourier shows, besides the orbiton dispersions, also strong mo- transforming the Hamiltonian and a Bogoliubov trans- mentum dependence of scattering intensity, with in par- formation, the orbiton Hamiltonian is H0 = const. + q ticular a vanishing of it at q=(0,0) and (π,π) for all but P  α† α with  = 3J 1 + 1 (cos k + cos k ). The one orbiton branch. The orbiton is also expected to have k k k k k 6 x y orbiton spectrum is gapped: as our orbital Hamiltonian phonon sidebands [19, 20], observable in RIXS as well. does not have a continuous symmetry, Goldstone modes Model Hamiltonian We focus on orbital excitations are absent. in a system with staggered eg orbital order, such as LaMnO3, the mother compound of the CMR mangan- Modifications by core-hole In the RIXS intermediate ites. The methods used below can without restriction be state a core-hole is present and the Hamiltonian becomes 0 core applied to other orbital ordered eg or t2g systems as well, H = H + H , which includes the interaction be- including for instance the orbital ordered titanates that tween the core-hole and the orbital degrees of freedom. are currently being investigated [21]. In the undoped The main effect of the core-hole potential is to lower manganite three 3d electrons occupy the Mn t2g orbitals the Coulomb repulsion U between two eg electrons at and a fourth 3d electron can be in either one of the two the core-hole site by an amount Uc, disrupting the su- Mn eg orbitals. Below 780 K, the eg orbitals order in perexchange processes [12]. This effect is substantial as an antiferro-orbital fashion. At lower temperatures, the Uc ≈ 7 eV [25]. To calculate the matrix elements of spins order in an A-type magnetic structure, where the Hcore, we consider how the core-hole changes the su- ferromagnetic, orbital ordered planes are stacked antifer- perexchange processes for all different pseudo-spin ori- romagnetically along the c-axis [15, 22]. entations. In Fig. 1 the two exchange paths for the spe- The orbital physics of LaMnO3 can be cast in a simple cific case h↓i↑j| H |↑i↑ji are shown. The upper process pseudo-spin model, where the pseudo-spin represents the involves txztxx/U where U is increased in presence of a xz zz Mn 3d eg orbital that is occupied. It is derived starting core-hole with Uc. The lower process involves t t /U from a generic Kondo lattice Hamiltonian [23, 24], with and decreases the intermediate energy U by Uc. These 3

ωfi) with

X hf| Dˆ |ni hn| Dˆ |ii A = ω . (4) fi res ω − E − iΓ n in n

Here Dˆ is the dipole operator, describing the excitation from the initial state |ii with energy Ei to an interme- diate state |ni with energy En and the subsequent evo- lution into a final state |fi with energy Ef , after the decay of the core-hole. The energies En are measured with respect to the resonance energy ωres. The ultra- short core-hole life time is taken into account through FIG. 2: The indirect RIXS spectrum for a cut through the an energy broadening Γ for the intermediate state. The Brillouin zone. The lower two branches originate from sin- delta function in the cross section ensures energy conser- gle orbiton excitations, the upper, continuous spectrum from vation: the energy gained by the system ωfi = Ef − Ei double orbiton scattering. Selection rules are such that at 0 0 equals the energy lost by the photon ω = ωin −ωout. The q = (0, 0) all spectral weight vanishes and at q = (π, π) only 0 one single-orbiton branch is active. energy ωin = ωin − ωres represents the detuning of the photon energy from the resonant edge [28]. When the energy of the incoming photon is tuned two processes result in to the resonance (ωin = 0), the amplitude Afi can be  xz zz xz xx  expanded in a powerseries in En/Γ [13, 14]. The ze- core t t t t − z h↓↑| Hij |↑↑i = 2 − h↓↑| Ti Tj |↑↑i roth order term gives only elastic scattering and is thus U − Uc U + Uc √ omitted in the following. We can retain only the low- 3 est order terms in η J/Γ ≈ 0.2 in the expansion of = ± J 0 h↓↑| T −T z |↑↑i (2) 1,2 4 i j Hl = (H0 + Hcore)l. This approximation is controlled by

0 Uc(Uc−2U) 0 the large core-hole broadening (Γ ≈ 0.58 eV at the Mn with J /J = 1 + 2 2 . Note that J is in general U −Uc K-edge, 1.3 eV at the Mn L1-edge and 0.16 eV at the Mn different for each matrix element. Collecting the matrix L2,3-edges [29]) and the values of J ≈ 25 meV [25, 30] elements, one finds H = H0 + J P Hcores s† with 2 ij i i and Uc/U ≈ 1.1. With this the expression for the scat- ωres 1 ˆ h √ i tering simplifies to Afi = hf| Oq |ii, with the Hcore = η H0 + η T x − T x ∓ 3 T z − T z , (3) iΓ iΓ+ω ij 1 ij 2 j i j i ˆ J P iq·Ri core effective scattering operator Oq = 2 e Hij . We evaluate this expression in terms of the boson cre- where si creates a core-hole and the dimensionless cou- 2 Uc UUc ation and annihilation operators, in linear spinwave ap- pling constants are η1 = 2 2 and η2 = 2 2 . The ∓- U −Uc U −Uc proximation. After Fourier transforming and introducing sign is − for bonds along the x-direction and + along the the Bogoliubov transformed orbiton operators, we obtain y-direction. The first term in Hcore is similar to the one ij the single- and double-orbiton scattering operators, Oˆ(1) encountered in magnetic RIXS scattering. Physically it is q ˆ(2) P ˆ(2) due to the fact that the core-hole modifies the strength of and Oq = k Ok,q, respectively. At T = 0, the single- the superexchange bonds to all neighboring sites. Its an- orbiton scattering operator is alytic form implies the selection rule that RIXS intensity √ ˆ(1) η1 3N − † vanishes for q = (0, 0), as at zero momentum transfer Oq = − Jq (u¯q − v¯q) α−¯q the scattering operator is proportional to the Hamilto- √8 nian H0 and thus commutes with it [10, 12]. The second η2 N † + (Jq − J0)(uq − vq) α−q (5) term, with coefficient η2, contains single orbital opera- 4 tors and is specific for core-hole orbital coupling –in spin and the double orbiton scattering operator systems such a coupling is not allowed by conservation of Sz . The presence of this term will allow the observation ˆ(2) η1 tot Ok,q = − [(6(Jq + J0) + Jk + Jk+q) ukvk+q of single-orbiton scattering. 8 † † Scattering Cross section. Having derived the Hamilto- − Jk+q (ukuk+q + vkvk+q)] αkα−k−q √ nian, we can compute the RIXS spectrum using the ultra- η 3 short core-hole life-time (UCL) expansion [13, 14]. The + 2 J −u v α† α† , (6) 2 q k k+¯q k −k−¯q resonant X-ray cross section for an incoming/outgoing 0 0 photon with momentum qin/qout and energy ωin/ωout where uk and vk are the coefficients of the Bogoliubov x(y) can be expressed by the Kramers-Heisenberg relation [26, transformation, ¯q = q+(π, π), Jk = 2J cos kx(y),Jk = 27] as a function of energy loss ω = ω0 −ω0 and momen- x y − x y in out Jk + Jk and Jk = Jk − Jk . As expected, since the z- d2σ P 2 z P z tum transfer q = qout − qin as dΩdω ∝ f |Afi| δ(ω − component of the total pseudospin Ttot = i Ti is not 4

Acknowledgements We thank Sumio Ishihara, Giniyat Khaliullin and Jan Zaanen for stimulating discussions.

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