Abstract
Strongly correlated insulators are broadly divided into two classes: Mott–Hubbard insulators, where the insulating gap is driven by the Coulomb repulsion U on the transitionmetal cation, and chargetransfer insulators, where the gap is driven by the chargetransfer energy Δ between the cation and the ligand anions. The relative magnitudes of U and Δ determine which class a material belongs to, and subsequently the nature of its lowenergy excitations. These energy scales are typically understood through the local chemistry of the active ions. Here we show that the situation is more complex in the lowdimensional chargetransfer insulator Li_{2}CuO_{2}, where Δ has a large nonelectronic component. Combining resonant inelastic Xray scattering with detailed modelling, we determine how the elementary lattice, charge, spin and orbital excitations are entangled in this material. This results in a large latticedriven renormalization of Δ, which significantly reshapes the fundamental electronic properties of Li_{2}CuO_{2}.
Introduction
The celebrated Zaanen–Sawatzky–Allen classification scheme^{1} divides strongly correlated insulators, such as transitionmetal oxides (TMOs), into two broad categories: chargetransfer (CT) or Mott–Hubbard insulators. Two fundamental energy scales determine the boundary between these categories. The first is the Coulomb repulsion U associated with the transitionmetal cation site, which parameterizes the energy cost for (d^{n−1}d^{n+1})type charge excitations. The second is the CT energy Δ associated with (d^{n−1}L)type charge excitations, where a hole moves from the cation site to the ligand anions L. When these atomic energy scales dominate over electron itinerancy, the emerging insulator is of the CT type when Δ<U and of the Mott–Hubbard type when Δ>U (ref. 1).
Determining which factors set the magnitude of these scales is important for the most basic understanding of the behaviour of TMOs. In an ionic picture, the onsite Coulomb interaction U sets the splitting of the lower and upper Hubbard bands^{1,2}, while the CT energy is typically set by the relative electronegativity of the oxygen (O) anions and the ionization energy of the transitionmetal cation^{2}. As such, copper oxides are typically classified as CT insulators, where their conduction band is derived from the copper (Cu) states forming the upper Hubbard band, while the valence band is derived from the O 2p states. This dichotomy creates a fundamental asymmetry between electron and hole doping processes, as reflected for example in the phase diagram of the hightemperature superconducting cuprates^{3,4}.
Properly classifying a real material is a challenging task experimentally. One needs to be able to determine the size of Δ and U in the presence of complications such as hybridization effects and additional interactions. Resonant inelastic Xray scattering (RIXS) is a powerful spectroscopic tool in this context^{5,6}. It is capable of directly probing charge^{7,8,9,10}, orbital^{11}, spin^{12,13,14,15} and, as most recently discovered, lattice excitations^{16,17,18,19}. The observation of the latter is particularly exciting, as RIXS can access the electron–phonon (eph) coupling strength directly^{17}, and with element specificity^{18}. This opens a direct means to study the influence of lattice dynamics on the fundamental electronic energy scales.
In this work we perform such a study for the edgeshared CT insulator Li_{2}CuO_{2} (LCO) to determine how the eph interaction helps to shape the CT energy in this quasionedimensional spinchain cuprate. The active electronic degrees of freedom in LCO are formed from edgeshared CuO_{4} plaquettes with a central Cu 3d^{9} cation^{20,21,22}. As a result, LCO harbours ZhangRice singlet (ZRS) charge excitons similar to those found in the highT_{c} cuprates^{8,9,23}. The eph interaction is also expected to play a role in this system. This was recently demonstrated for the related edgeshared cuprate Ca_{2+x}Y_{2−x}Cu_{5}O_{10} (CYCO), where charge carriers couple strongly to Cu–O bondstretching phonon modes polarized perpendicular to the chain direction^{18,19}. We demonstrate here that a similar eph interaction occurs in LCO. More importantly, however, we show that this interaction provides a substantial contribution to Δ, accounting for ≈54% of its total value. This result is obtained from a comprehensive analysis of highresolution oxygen Kedge RIXS^{5,6} data that resolves individual phonon, dd, and CT excitations (including the ZRS exciton). This in turn allows us to disentangle the elementary spin, charge, orbital and lattice excitations over an energy range of ∼10 eV. If the eph interaction is omitted in our analysis, the spectra imply a value Δ≈4.6 eV; however, when the eph interaction is properly accounted for, this value separates into a purely electronic contribution of Δ_{el}≈2.1 eV, and a very substantial phononic contribution of about the same size Δ_{ph}≈2.5 eV. As such, the elementary excitations across the CT gap in LCO couple strongly to specific phonon modes, enhancing their total energy cost. This result places the basic classification of LCO in a new light, where the relevant energy scales are shaped not only by the local chemistry of the atoms but also dynamically by interactions with phonons that are relevant for many TMOs^{24,25,26,27}.
Results
RIXS at the oxygen Kedge in LCO
The oxygen Kedge RIXS process is sketched in Fig. 1. During the experiment, photons with energy ℏω_{in} and momentum ℏ k_{in} are absorbed by the system in its initial state via an O 1s→2p dipole transition. This creates an intermediate state with an O 1s core hole and an additional electron in the conduction band. The resulting intermediate state then evolves in time under the influence of the corehole potential and the excited electronic configuration. A number of elementary excitations are created in the system during this time until the corehole decays, emitting an outgoing photon (momentum ℏ k_{out} and energy ℏω_{out}) and leaving the system in an excited final state f〉.
To understand how the eph interaction enters this process it is important to examine further the states involved. The electronic ground state in LCO, and other spinchain cuprates, is largely of i_{el}〉∼αd^{9}〉+βd^{10}L〉 character, where L denotes a hole on the ligand O. This state, however, couples strongly to Cu–O bondstretching phonons like the transverse mode sketched in Fig. 1b. This coupling can occur in two ways. For instance, the bondstretching modes directly modulate the Cu–O hopping integral. Alternatively, these modes can modulate the Madelung energy of the central Cu atom, effectively lowering/raising the energy of the Cu site as the O atoms move closer to/further from it. This latter mechanism cannot be effectively screened in lower dimensions, and turns out to be the relevant coupling mechanism for our analysis^{24,28}. Since the electronic contribution to the CT energy (in hole language) in this system is , we can view the phonon modes as modulating the CT energy^{18}. This is confirmed in Fig. 1c, where we plot the linear variation in Δ_{el} obtained from a static point charge model under uniform expansions/compressions of the CuO plaquettes in the direction perpendicular to the chain (Methods section).
The physical interpretation of this result is as follows. The lighter O atoms, in an effort to eliminate the firstorder eph coupling and minimize the energy of the system, shift to new equilibrium positions located closer towards the Cu atoms. Subsequently, the new ground state of the system involves a coherent state of phonon quanta {n_{q}} that describes the distorted structure. The new equilibrium positions also produce changes in the Madelung energy of the Cu site, increasing the CT energy in comparison to the value obtained in the absence of the interaction. This renormalization of the CT energy is a bulk property of the crystal arising from the eph interaction with the Cu 3d^{9} hole present in the ground state. As such, it will manifest in many bulk spectroscopies including RIXS (this work), optical conductivity (Supplementary Fig. 1), and inelastic neutron scattering (Supplementary Note 1). It is important to note, however, that this renormalization is inherently dynamic, as the oxygen atoms are free to respond to changes in Cu hole density. This has observable consequences in the RIXS spectra, as we now demonstrate.
The RIXS process for LCO’s initial state dressed by the phonon excitations is sketched in Fig. 1b. At low temperatures it is now predominantly i〉∼αd^{9},{n_{q}}〉+βd^{10}L,{n_{q}}〉 in character. The intermediate state is formed after the creation of a core hole on the O site, through an O 1s→2p transition. This creates an intermediate state of m〉∼βd^{10}p^{6},{n_{q}}〉 character, which corresponds to an upper Hubbard band excitation, where the number of holes on the Cu site has changed. In response, the ligand O atoms begin to relax towards new positions until the corehole decays. Ultimately, this leaves the system in a final state with both excited electronic and lattice configurations .
It is important to stress that here the corehole provides us with a lens through which we can view the eph interaction using RIXS. The corehole does not create the interaction. While the lattice excitations we probe are being generated in the intermediate state, they carry information about the strength of the eph interaction that is present in the initial and final states. The change in carrier density introduced by the creation of the core hole excites the lattice, but the way in which the lattice responds depends on strength and details of the interaction.
Electron–phonon coupling in the RIXS data
The presence of the eph interaction in LCO is confirmed in our measured RIXS spectra, shown in Fig. 2a. The Xray absorption spectroscopy (XAS) spectrum (inset) has a prominent peak centred at 529.7 eV, which corresponds to the discussed excitation into the upper Hubbard band. The RIXS spectra, taken with incident photons detuned slightly from this energy (ℏω_{in}=530.08 eV, indicated by the arrow), are rich. (Here we have shown data detuned from the UHB resonance since the intensity of the ZRS excitation is largest for this incident photon energy^{9}.) We observe a number of features, including a long tail of intensity extending from the elastic line comprised of several phonon excitations; two nearly Tindependent peaks at ∼1.7 and ∼2.1 eV, which correspond to now wellknown dd excitations^{23,29}; a Tdependent peak at ∼3.2 eV, which corresponds to a ZRS excitation^{8,9}; and, finally, a band of CT excitations for ℏΩ=ℏω_{out}−ℏω_{in}>4. Here, we are using the term CT excitation as an umbrella term for any excitation where a Cu 3d hole has been transferred to the O 2p orbitals, with the exception of the ZRS excitation. As such, CT excitations include the fluorescence excitations. We have explicitly confirmed each of these identifications by examining the character of the final state wave functions obtained from our model calculations.
The phonon excitations are more apparent in the highresolution measurements of the quasielastic and ddexcitation energy range, shown in Fig. 2d,f, respectively. We observe clear harmonic phonon excitations separated in energy by ℏΩ_{ph}∼74 meV, consistent with those reported for CYCO^{18,19}. This demonstrates that the eph coupling is a common phenomenon in the spinchain cuprates. Another important aspect of the data is the positions of the ZRS and CT excitations, which are determined by the CT energy. From these data we infer Δ∼4.6 eV, which is significantly >3.2 eV obtained from Madelung energy estimates based solely on local chemistry considerations^{22}. This discrepancy can be accounted for by including the bondstretching phonons implied by the observed harmonic excitations in Fig. 2d,f.
Electron–phonon contribution to the CT energy
We assessed the phonon contribution to Δ by modelling the RIXS spectra within the Kramers–Heisenberg formalism^{5,6}. The initial, intermediate and final states were obtained from small cluster exact diagonalization calculations that included the lattice degrees of freedom^{9,18}. The electronic model and its parameters are the same as those used in a previous LCO study^{9}, however, we have extended this model to include additional Cu 3d orbitals and kept the bare CT energy as a fitting parameter. This number represents the size of the CT energy in the absence of the eph interaction. The model for the lattice degrees of freedom is similar to ref. 18 but with an eph coupling strength parameterized by g and the phonon energy ℏΩ_{ph}=74 meV, as determined from our data (Methods section). The calculated spectra are shown in Fig. 2b,e,g, where we have set Δ_{el}=2.14 eV and g=0.2 eV. This choice produces the best global agreement between the theory and experiment both in terms of the positions of the CT and ZRS excitations, as well as the intensities of the harmonic excitations in the dd and quasielastic regions. We also stress that the remaining parameters of the model were held fixed during our fitting procedure, as their values are heavily constrained by optical conductivity^{30,31}, electron energyloss spectroscopy (EELS)^{31}, inelastic neutron scattering^{32} and RIXS (this work and ref. 9) measurements.
A closer inspection of Fig. 2b reveals our main finding: the observed positions of the ZRS and CT excitations are not set by the purely electronic value of Δ_{el}=2.14 eV but rather the total Δ=Δ_{el}+Δ_{ph}∼4.6 eV with (Methods section). In other words, the electronlattice interaction is responsible for half of the effective CT energy in LCO. To stress this point, Fig. 2c shows results obtained from a similar model where the eph interaction is taken out of the analysis. To even qualitatively reproduce the positions of the ZRS and CT excitations in this electroniconly model, Δ_{el} must now be increased to 4.6 eV. Furthermore, the phonon satellite peaks are absent in the quasielastic and dd excitation regions. This is a clear deficiency in the electroniconly model that is corrected only when the eph interaction is included. It is clear that the electronlattice coupling plays a very significant role in establishing the effective value of Δ in LCO.
We note that there is a small discrepancy between the theory and experiment; namely, the relative intensity of the observed second phonon line with respect to the first one is slightly stronger than the one captured by the cluster calculation. While increasing the value of g does increase the intensity of the second phonon excitation relative to the first^{18}, the singlemode model we have adopted always produces a diminishing intensity in successive phonon excitations. (We have also examined nonlinear eph interactions but these are unable to account for this discrepancy.) We therefore speculate that increased intensity in the second phonon excitation is due to multiphonon processes that cannot be included in our calculations due to the necessary truncation of the phonon Hilbert space (Methods section). For these reasons, we selected g=0.2 eV, which is consistent with the Madelung energy considerations. This value also provides a conservative estimate for the lattice contribution to Δ.
In Fig. 3 we compare the measured incident photon energy dependence to the predictions of the eph coupled model as an additional verification. Here a resonance behaviour in the experimental data is observed, where the phonon excitations emanating from the elastic line and dd excitations persist to higher energy losses as ℏΩ_{in} resonates with the upper Hubbard band excitation in the XAS. Our experimental observations are in agreement with prior O Kedge measurements on the related CYCO system^{18}. (In both materials, the observed resonance behaviour is damped with respect to similar behaviour observed in gasphase oxygen molecules^{33}. This is due to the increased number of corehole decay channels present in the solid^{18}.) Our model with the eph interaction reproduces these features well. In contrast, the electronic model without the eph coupling fails to capture these features. This underscores once more the importance of the eph interaction for understanding the RIXS spectra on even the qualitative level.
Discussion
We have performed oxygen Kedge RIXS measurements on the edgeshared onedimensional cuprate LCO, revealing clear phonon excitations in the RIXS spectra. These excitations are well captured by a model that includes coupling to a Cu–O bondstretching optical phonon mode, which modulates the onsite energy of the Cu orbitals and leads to a substantial renormalization of the effective CT energy. This renormalization is not a simple effect related to the formation of the core hole. The nonzero eph interaction that we infer here is present in the system regardless of the existence of the core hole. Thus the corresponding renormalization of the CT energy will also be present in other spectroscopies such as optical conductivity (Supplementary Fig. 1)^{22,30}, EELS^{31} and inelastic neutron scattering^{32} (Supplementary Note 1).
Our results show that the eph interaction is of relevance to the Zaanen–Sawatzky–Allen classification of this material, where the lattice contribution to the CT energy accounts for nearly half of the total value. Since the ensuing renormalizations can be very large in materials possessing substantial eph couplings, we expect that such considerations will prove to be important in other families of quasionedimensional correlated systems, where the lattice motion cannot be effectively screened. For example, the related spinchain system CYCO likely has a large lattice contribution to the CT energy.
Methods
Sample preparation
LCO samples were grown under elevated gas pressure (in a gas mixture of Ar:O_{2} with a ratio of 4:1 at the total pressure of 50 bar) in a vertical travelling solventfloating zone facility with optical heating^{34}. The powder for the feed rods of LCO was prepared by grinding and sintering LiOH (Isotec, 99.9% of ^{7}LiOH powder was used) and CuO (Chempur 99.99%) at 750 °C. Because the powder was single phase after the first sintering, no further annealing was done to avoid vaporization of lithium. The singlephase powder was pressed to polycrystalline rods (EPSI Engineered Pressure Systems; 3,500 bar) in latex tubes and sintered again at 800 °C for 34 h.
RIXS measurements
The RIXS experiments were performed at the ADRESS beamline of the Swiss Light Source, Paul Scherrer Institut, using the SAXES spectrometer^{35,36}. All spectra were recorded with σpolarized light in the scattering geometry shown in Fig. 1a (the scattering angle was 130°, with an incidence angle of 65°). No momentum was transferred into the system along the direction of the chain using this geometry. The combined energy resolution was between 50 and 60 meV at the oxygen Kedge (ℏω_{in}∼530 eV). About 150 photons were collected on the dd excitations (maximum intensity) during 2 h of data acquisition at an energy resolution of 60 meV (RIXS spectra of Fig. 2a). About 300 photons were collected on the dd excitations (maximum intensity) during 8 h of data acquisition at an energy resolution of 50 meV (RIXS spectra of Fig. 2d,f). The samples were cleaved in situ at a pressure of ∼5 × 10^{−10} mbar and a temperature T=20 K. The surface of the crystal was perpendicular to the [101] axis such that the CuO_{4} plaquettes were tilted 21° from the surface.
XAS and RIXS intensities
The RIXS spectra at the oxygen (O) Kedge (1s→2p) were calculated using the Kramers–Heisenberg formula^{5,6,37}. If the incoming and outgoing photons have energies (polarizations) ℏω_{in} () and ℏω_{out} (), respectively, then the RIXS intensity is given by
Here, ℏΩ=ℏω_{out}−ℏω_{in} is the energy loss; i〉, m〉 and f〉 denote the initial, intermediate and final states of the RIXS process, respectively, with eigenenergies E_{i}, E_{m} and E_{f}, respectively; and Γ is the lifetime of the corehole, which we assume is independent of the intermediate state.
The 1s→2p transition is induced by the dipole operator D_{μ}. If no momentum is transferred to the sample (q=0) by the incoming photon, then the dipole operator is given by
where creates (annihilates) a 1s corehole of spin σ on O site i, creates (annihilates) a spin σ hole in the O 2p_{β} orbital on the same site and is the projection of the photon polarization onto the orientation of the O 2p_{β} orbital. For the scattering geometry shown in Fig. 1a, the transition operators are
for σ and πpolarized light, respectively. (Note that the p_{z} orbitals do not appear in these operators since we do not include them in our Hilbert space, see below.) Since the polarization of the outgoing photon was not measured in the experiment, the total intensity is given by an incoherent sum over outgoing polarizations I_{σ}=∑_{μ} I_{μ,σ} and I_{π}=∑_{μ} I_{μ,π}. Here, the reader should not confuse the polarization index σ with the spin index. In the main text we show results calculations I_{σ} polarization.
Model Hamiltonian
The eigenstates i〉, m〉 and f〉 were obtained from exact diagonalization of a small Cu_{3}O_{8} cluster with an edgeshared geometry and open boundary conditions, as shown in Fig. 1b. The orbital basis contains the 3d_{xy}, and orbitals on each Cu site, and the O 2p_{x,y} orbitals on each O site. Throughout, α and α^{′} are used to index Cu orbitals, β and β′ are used to index O orbitals and the roman indices i, j index the lattice sites.
The full Hamiltonian is H=H_{0}+H_{e−e}+H_{ph}+H_{e−ph}, where H_{0} and H_{ph} contain the noninteracting terms for the electronic and lattice degrees of freedom, respectively, H_{e−e} contains the electron–electron interactions, and H_{e−ph} contains the eph interactions.
The noninteracting terms for electronic degrees of freedom are
where the Cu operators and O operators create (annihilate) a hole of spin σ in orbital α (or β) on atomic site i. In Equation (4) and are the onsite energies of the Cu and O orbitals, respectively, while and are the Cu–O and O–O hopping integrals, respectively.
The electron–electron interactions include the onsite inter and intraorbital interactions on each Cu and O site, the nearestneighbor Cu–O repulsion and exchange interactions , and the nearestneighbor Cu–Cu repulsion . The Cu onsite interactions take the form
The form of onsite O interactions, , is the same. The nearestneighbour Cu–O interactions take a similar form , where the sum is over nearestneighbor Cu and O sites and
Finally, the Cu–Cu nearestneighbor repulsion is given by
For the lattice model H_{ph} and H_{e−ph}, we considered a singleoxygen Cu–O bondstretching mode that compresses the Cu–O bond in the direction perpendicular to the chain direction, as indicated by the arrows in Fig. 1b. The reduction to a singlephonon mode is required to maintain a manageable Hilbert space for the problem; however, this approximation is sufficient to describe the phonons in the related system CYCO (ref. 18). In principle, these bondstretching phonons couple to the carriers in the chain via two microscopic mechanisms: the first is via the direct modulation of the interchain hopping integrals. The second is via a modification of the Cu site energies. The magnitude of the former can be estimated from the distance dependence of the atomic hopping parameters. The magnitude of the latter can be estimated using an electrostatic point charge model for the Madelung energies^{2,22}. We carried out such calculations using known structural data^{20} and obtained the distance dependence of (neglecting crystal field effects) for static compressions of the CuO_{2} chain, as shown in Fig. 1b. The results are shown in Fig. 1c, where we obtain an eph coupling strength g∼0.24 eV. Calculations were then carried out for both coupling mechanisms and the Cu site energy modulation was found to have the the largest impact on the calculated RIXS spectra. We therefore neglected the modulation of the hopping integrals here for simplicity and introduced a Holsteinlike coupling to the Cu site energies. Within this model the Hamiltonian for the lattice degrees of freedom is
where b^{†} (b) creates (annihilates) a phonon quanta of the compression mode. The hilbert space for the lattice degrees of freedom is truncated at a large number of allowed phonon quanta (∼200). We have checked to ensure that our results are not significantly changed for further increases in this cutoff.
Finally, when calculating the intermediate states in Equation (1), the Hamiltonian is augmented with the appropriate terms describing the Coulomb interaction with the corehole^{8}. Specifically, we add
where is the number operator for the 1s core level on oxygen site i, is the energy of the O 1s corehole and U_{q} is the corehole potential.
Model parameters
The multiband Hamiltonian has a number of parameters that can be adjusted; however, we are constrained by multiple experimental probes. To this end we have a wellestablished set given in ref. 9, which simultaneously reproduces highenergy features in the RIXS data^{9}, Cu–Cu exchange interactions inferred from inelastic neutron scattering measurements^{32}, and optical conductivity and EELS measurements^{31} in LCO. Given this level of descriptive power, we adopt the same parameter set here.
When the eph interaction is included in the calculation we take (in units of eV) , , , and . The Cu–O hopping integrals are (in eV) , , , , , and . The O–O hopping integrals are (in eV) (0.240) and , for hopping parallel (perpendicular) to the chain direction. The Hubbard and Hunds interactions for the Cu sites are given by the Racah parameterization^{38} with A=6.45, B=0.25 and C=0.35. The oxygen interactions are U_{p}=4.1, and . The extended interactions are U_{dd}=0.4, U_{pd}=0.8, and J_{pd}=0.096. The phonon energy is taken to be ℏΩ_{ph}=74 meV, and the eph coupling strength g is taken as a variable. The corehole parameters are U_{q}=4.3 eV and Γ=150 meV for the oxygen Kedge.
All of the parameters remain the same for the calculations performed without eph coupling with the exception of Γ=300 meV, , and . It should be noted that this parameter set assumes a larger value for the CT energy in comparison to ref. 9, and fails to capture the phonon features in the RIXS data (Fig. 2c). To correct this, we take the bare CT energy and the bare eph interaction strength g as fitting parameters and keep all other model parameters to be the same as those listed above when the eph interaction is included. We therefore regard the CT energy Δ used in ref. 9 as an effective CT energy, which includes the effects of the eph interaction.
Madelung energies
The coupling to the phonon mode enters into our calculations to first order in displacement via the modulation of the Cu and O Madelung energies V_{Cu} and V_{O}, respectively. The Madelung energy for a given site i can be estimated using an ionic model, and is given by , where Z_{j}e is the formal charge associated with the atom at site j. Neglecting crystal field effects, the difference between the Cu and O site energies is related to the difference in Madelung energies ΔV_{M}=V_{O}−V_{Cu} by^{2}
where A_{O}(2) is the second electron affinity of oxygen, I_{Cu}(3) is the third ionization energy of Cu, d is the Cu–O distance and is the highfrequency dielectric constant. The distance dependence of Δ can be estimated by calculating ΔV_{M} using the Ewald summation technique and the known structural data^{20}. Assuming and , we arrive at Δ=3.2 eV for the experimental lattice parameters, in agreement with ref. 22. This value, however, is substantially lower than the value inferred from our RIXS study if the eph interaction is excluded.
To estimate the strength of the eph interaction, we performed calculations where the Cu–O plaquettes were compressed by a distance u in the directions indicated by the arrows in Fig. 1b. The resulting distance dependence of Δ(u) is plotted in Fig. 1c, where a linear dependence of Δ occurs over a wide range of displacements. To capture this, we parameterize the Cu site energy as , where M_{O} is the mass of oxygen. This results in an eph coupling of the form given in equation (8) with . A linear fit to Δ(u) (shown in Fig. 1c) gives , which yields g∼0.24 eV. It should be stressed that this value of g is an estimate based on a point charge model, however, it gives us an idea of the expected coupling strength.
Renormalization of the chargetransfer energy
As discussed in the main text, in the ground state of the LCO chain the oxygen atoms will shift to new equilibrium positions in response to the linear eph coupling terms of the Hamiltonian. This situation can be qualitatively understood by introducing shifted phonon operators and , where is the average number of holes on the Cu site in the ground state. These new operators yield a shifted atomic position given by . This shift of position is responsible for the renormalization of the CT energy. After this transformation is made the phonon and eph coupled terms of the Hamiltonian (equation (8)) reduce to
where we have dropped an overall constant. The second term describes the coupling to the lattice in the new equilibrium position, which is proportional to the fluctuation in Cu charge density from its ground state value. The third term can be folded into the definition of the Cu site energy with . This gives an effective CT energy Δ_{eff}=Δ_{el}+Δ_{ph} where and . From these considerations one can also see that no isotope effect is predicted for Δ_{ph}, since both g^{2} and ℏΩ_{ph} are proportional to the inverse of the mass of oxygen.
Additional information
How to cite this article: Johnston, S. et al. Electronlattice interactions strongly renormalize the chargetransfer energy in the spinchain cuprate Li_{2}CuO_{2}. Nat. Commun. 7:10563 doi: 10.1038/ncomms10563 (2016).
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Acknowledgements
We thank M. Berciu, T.P. Devereaux, W.S. Lee, B. Moritz and G. Sawatzky for useful discussions. This research has been funded by the Swiss National Science Foundation and the German Science Foundation within the DACH programme (SNSF Research Grant 200021L 141325 and Grant GE 1647/31). This work is supported by SFB 1143 of the Deutsche Forschungsgemeinschaft. C.M. also acknowledges support by the Swiss National Science Foundation under grant no. PZ00P2 154867. Further support has been provided by the Swiss National Science Foundation through the Sinergia network Mott Physics Beyond the Heisenberg Model (MPBH). J.G. gratefully acknowledge the financial support through the EmmyNoether programme of the German Research Foundation (grant no. GE1647/21). The experiments were performed at the ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institut.
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C.M., V.B., K.J. Z., R. K., V. N. S., J. G. and T. S. performed the RIXS experiments, G. B. grew the singlecrystalline samples. S.J., C.M., T.S., J.M., S.L.D. and J. v.d.B analysed the data and developed the model. S.J. performed the cluster calculations. J.v.d.B., J. G., T.S., S.J. and C.M. conceived and managed the project. S.J. and J.v.d.B. formulated the manuscript with the assistance of all other coauthors.
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Johnston, S., Monney, C., Bisogni, V. et al. Electronlattice interactions strongly renormalize the chargetransfer energy in the spinchain cuprate Li_{2}CuO_{2}. Nat Commun 7, 10563 (2016). https://doi.org/10.1038/ncomms10563
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DOI: https://doi.org/10.1038/ncomms10563
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