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Investigation of the multiplet features of SrTiO3 in X-ray absorption spectra based on configuration interaction calculations

aFujian Provincial Key Laboratory of Semiconductors and Applications, Collaborative Innovation Center for Optoelectronic Semiconductors and Efficient Devices, Department of Physics, Xiamen University, Xiamen 361005, People's Republic of China, bCenter for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, USA, cBeijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China, and dXiamen University Malaysia, 439000 Sepang, Selangor, Malaysia
*Correspondence e-mail: jczheng@xmu.edu.cn, hqwang@xmu.edu.cn

Edited by R. W. Strange, University of Essex, UK (Received 11 January 2018; accepted 12 March 2018; online 24 April 2018)

Synchrotron-based L2,3-edge absorption spectra show strong sensitivities to the local electronic structure and chemical environment. However, detailed physical information cannot be extracted easily without computational aids. Here, using the experimental Ti L2,3-edges absorption spectrum of SrTiO3 as a fingerprint and considering full multiplet effects, calculations yield different energy parameters characterizing local ground state properties. The peak splitting and intensity ratios of the L3 and L2 set of peaks are carefully analyzed quantitatively, giving rise to a small hybridization energy around 1.2 eV, and the different hybridization energy values reported in the literature are further addressed. Finally, absorption spectra with different linearly polarized photons under various tetragonal crystal fields are investigated, revealing a non-linear orbital–lattice interaction, and a theoretical guidance for material engineering of SrTiO3-based thin films and heterostructures is offered. Detailed analysis of spectrum shifts with different tetragonal crystal fields suggests that the eg crystal field splitting is a necessary parameter for a thorough analysis of the spectra, even though it is not relevant for the ground state properties.

1. Introduction

SrTiO3 (STO) compound has attracted longstanding attention due to its fascinating properties both for fundamental research and applications in functional oxide electronics (Cen et al., 2009[Cen, C., Thiel, S., Mannhart, J. & Levy, J. (2009). Science, 323, 1026-1030.]; Choi et al., 2009[Choi, T., Lee, S., Choi, Y. J., Kiryukhin, V. & Cheong, S.-W. (2009). Science, 324, 63-66.]), as well as in the field of renewable energy (Marschall, 2014[Marschall, R. (2014). Adv. Funct. Mater. 24, 2421-2440.]; Comes et al., 2015[Comes, R. B., Xu, P., Jalan, B. & Chambers, S. A. (2015). Appl. Phys. Lett. 107, 131601.]). Bulk STO, Ti4+ with 3d0 electronic configuration, has a perovskite crystal structure with a cubic lattice constant of 3.905 Å. It is usually used as a substrate in thin-film engineering due to its structural and chemical compatibility with other perovskite transition metal oxides (Schlom et al., 2007[Schlom, D. G., Chen, L.-Q., Eom, C.-B., Rabe, K. M., Streiffer, S. K. & Triscone, J.-M. (2007). Annu. Rev. Mater. Res. 37, 589-626.]). STO bulk and related thin films show novel physical properties: for instance, the quantum paraelectric and the incipient ferroelectrics at the surfaces and heterostructures of STO (Haeni et al., 2004[Haeni, J. H., Irvin, P., Chang, W., Uecker, R., Reiche, P., Li, Y. L., Choudhury, S., Tian, W., Hawley, M. E., Craigo, B., Tagantsev, A. K., Pan, X. Q., Streiffer, S. K., Chen, L. Q., Kirchoefer, S. W., Levy, J. & Schlom, D. G. (2004). Nature (London), 430, 758-761.]; Dawber et al., 2005[Dawber, M., Rabe, K. M. & Scott, J. F. (2005). Rev. Mod. Phys. 77, 1083-1130.]); the high mobility of two-dimensional electron gas exhibiting quantum oscillations in magneto­transport measurements at the interfaces of LaAlO3–STO heterostructures (Ohtomo & Hwang, 2004[Ohtomo, A. & Hwang, H. Y. (2004). Nature (London), 427, 423-426.]; Hwang et al., 2012[Hwang, H. Y., Iwasa, Y., Kawasaki, M., Keimer, B., Nagaosa, N. & Tokura, Y. (2012). Nat. Mater. 11, 103-113.]); the unexpected superconductivity at Tc = 0.2 K at the interface between LaAlO3 and STO (Reyren et al., 2007[Reyren, N., Thiel, S., Caviglia, A. D., Kourkoutis, L. F., Hammerl, G., Richter, C., Schneider, C. W., Kopp, T., Rüetschi, A.-S., Jaccard, D., Gabay, M., Muller, D. A., Triscone, J.-M. & Mannhart, J. (2007). Science, 317, 1196-1199.]), etc.

Synchrotron radiation soft X-ray absorption measurements for 3d transition metal compounds (L2,3-edges, typically with photon energies in the 400–1000 eV range) are sensitive to the local valence state, orbital occupation and spin arrangement, which is important for exploring the correlated interactions among different degrees of freedom, e.g. charge, spin, orbital, lattice, etc. However, detailed structural and electronic information cannot be extracted from the L2,3-edge absorption spectrum without computational aids. Density functional theory (DFT) is highly desirable but has been a longstanding theoretical challenge due to strong multiplet effects such that the spectral function calculated from first principles fails.

The soft X-ray absorption spectrum of single-crystal STO shows no exception, where the intensity ratios at the Ti L2,3-edges are still under debate. For instance, the theoretical spectra of STO bulk and thin films based on DFT can reproduce well the peak energy positions and the polarization-dependent peak shifts for STO thin films, but the spectra show opposite tendencies for the peak intensities of the L3 set of peaks (Woicik et al., 2007[Woicik, J. C., Shirley, E. L., Hellberg, C. S., Andersen, K. E., Sambasivan, S., Fischer, D. A., Chapman, B. D., Stern, E. A., Ryan, P., Ederer, D. L. & Li, H. (2007). Phys. Rev. B, 75, 140103.]). Further, spectra calculated by DFT in the independent particle approximation taking three different types of one-electron potentials into account all give incorrect peak intensities (Krüger, 2010[Krüger, P. (2010). Phys. Rev. B, 81, 125121.]). Many calculation approaches have been developed over the last decades to try to understand the L2,3-edge absorption spectrum: for example, time-dependent DFT, which to our knowledge has not been applied to STO but to other 3d0 oxides such as V2O5 (Fronzoni et al., 2012[Fronzoni, G., De Francesco, R. & Stener, M. (2012). J. Chem. Phys. 137, 224308.]) and FeTiO3 (Bunău & Joly, 2012[Bunău, O. & Joly, Y. (2012). Phys. Rev. B, 85, 155121.]) with fairly good agreements with experimental absorption spectra; the Bethe–Salpeter equation, where the theoretical spectra either lack capturing of the peak intensities at the STO Ti L2,3-edges (Laskowski & Blaha, 2010[Laskowski, R. & Blaha, P. (2010). Phys. Rev. B, 82, 205104.]; Gilmore et al., 2015[Gilmore, K., Vinson, J., Shirley, E., Prendergast, D., Pemmaraju, C., Kas, J., Vila, F. & Rehr, J. (2015). Comput. Phys. Commun. 197, 109-117.]) or exhibit an incorrect peak splitting between the leading peaks and the excitation peaks (Gilmore et al., 2015[Gilmore, K., Vinson, J., Shirley, E., Prendergast, D., Pemmaraju, C., Kas, J., Vila, F. & Rehr, J. (2015). Comput. Phys. Commun. 197, 109-117.]; Vinson et al., 2011[Vinson, J., Rehr, J. J., Kas, J. J. & Shirley, E. L. (2011). Phys. Rev. B, 83, 115106.]); multichannel multiple-scattering calculations, where the calculated spectra show variations in relative peak splitting compared with the experimental STO spectrum (Krüger, 2010[Krüger, P. (2010). Phys. Rev. B, 81, 125121.]; Krüger & Natoli, 2016[Krüger, P. & Natoli, C. R. (2016). J. Phys. Conf. Ser. 712, 012007.]); and ab initio full multiplet calculations (Ikeno et al., 2009[Ikeno, H., de Groot, F. M. F., Stavitski, E. & Tanaka, I. (2009). J. Phys. Condens. Matter, 21, 104208.]; Haverkort et al., 2012[Haverkort, M. W., Zwierzycki, M. & Andersen, O. K. (2012). Phys. Rev. B, 85, 165113.]; Ramanantoanina & Daul, 2017[Ramanantoanina, H. & Daul, C. (2017). Phys. Chem. Chem. Phys. 19, 20919-20929.]), where the Ti L2,3-edges show a better description of the experimental spectrum in terms of relative peak positions and intensities. We note that all of these approaches can reproduce the STO Ti L2,3 absorption spectrum reasonably well but demand large computational efforts which shows the advantages of understanding the spectrum of large electron systems. Furthermore, none has heretofore quantitatively accounted for the variation of the intensity ratios, which will be mainly addressed in the present work.

For the STO of interest here, the L2,3 spectrum can also be reproduced well using configuration interaction (CI) cluster calculations and a regular PC. For instance, de Groot et al. reported a detailed calculation of the L2,3-absorption spectrum for d0 compounds including K+, Ca2+, Sc3+ and Ti4+ in octahedral symmetry (de Groot et al., 1990[Groot, F. M. F. de, Fuggle, J. C., Thole, B. T. & Sawatzky, G. A. (1990). Phys. Rev. B, 41, 928-937.]). Tanaka & Jo reported the calculated resonant photoemission spectra for different 3d compounds, including the TiO2 L2,3-absorption spectrum as a 3d0 compound example (Tanaka & Jo, 1994[Tanaka, A. & Jo, T. (1994). J. Phys. Soc. Jpn, 63, 2788-2807.]). Bocquet et al. performed CI calculations for a wide range of early transition metal compounds by analyzing the core-level X-ray photoemission spectra (Bocquet et al., 1996[Bocquet, A. E., Mizokawa, T., Morikawa, K., Fujimori, A., Barman, S. R., Maiti, K., Sarma, D. D., Tokura, Y. & Onoda, M. (1996). Phys. Rev. B, 53, 1161-1170.]). Kroll et al. reported the final-state projection method based on CI which allows separation of the charge transfer effect on each possible final-state configuration taking the explicit STO L2,3-absorption spectrum as the model system (Kroll et al., 2015[Kroll, T., Solomon, E. I. & de Groot, F. M. F. (2015). J. Phys. Chem. B, 119, 13852-13858.]). We note that the Ti L2,3-absorption spectrum shows sensitivities to small structural and electronic variations so that different fine structures appear in anatase and rutile TiO2 (Maganas et al., 2014[Maganas, D., DeBeer, S. & Neese, F. (2014). Inorg. Chem. 53, 6374-6385.]), BaTiO3 (Ootsuki et al., 2014[Ootsuki, S., Ikeno, H., Umeda, Y., Yonezawa, Y., Moriwake, H., Kuwabara, A., Kido, O., Ueda, S., Tanaka, I., Fujikawa, Y. & Mizoguchi, T. (2014). Microscopy, 63, 249-254.]), PbTiO3 (Torres-Pardo et al., 2011[Torres-Pardo, A., Gloter, A., Zubko, P., Jecklin, N., Lichtensteiger, C., Colliex, C., Triscone, J.-M. & Stéphan, O. (2011). Phys. Rev. B, 84, 220102.]), PbZr0.2Ti0.8O3 (Arenholz et al., 2010[Arenholz, E., van der Laan, G., Fraile-Rodríguez, A., Yu, P., He, Q. & Ramesh, R. (2010). Phys. Rev. B, 82, 140103.]) and STO (Uehara et al., 1997[Uehara, Y., Lindle, D., Callcott, T., Terminello, L., Himpsel, F., Ederer, D., Underwood, J. H., Gullikson, E. M. & Perera, R. (1997). Appl. Phys. A, 65, 179-182.]) as well as their thin films with lower symmetries. Moreover, none of the above CI calculations are specialized for STO compound or solving the debate regarding the intensity ratios at the Ti L2,3-edges.

We therefore aim to provide a comprehensive understanding of the multiplet structures and the local ground state properties of STO compound. In particular, we analyze the intensity ratios quantitatively based on CI cluster calculations. In combination with our previous studies of LaTiO3 compound with a TiO6 cluster of 3d1 electronic configuration (Wu et al., 2017[Wu, M., Zheng, J.-C. & Wang, H.-Q. (2017). J. Appl. Cryst. 50, 576-584.]), our present calculations provide a comparison between LaTiO3 and STO compounds. These detailed studies of the absorption spectra with different valence states also offer the potential to understand the interfacial physics of the heterostructure composed of LaTi3+O3 and ST4+O and the ground state of the (La,Sr)TiO3 bulk solid solution. Furthermore, we present the photon polarization-dependent absorption spectra with different tetragonal crystal fields, relating the linear dichroic spectrum to the character of the d states and the local electronic structures. This study reveals the orbital–lattice interactions of the system and provides guidance for the experimental design in strain engineering of STO-based thin films and heterostructures. Nevertheless, cluster calculations are proposed to reproduce and to help understand the experimental spectra, which will provide a variety of energy parameters representing the ground state properties. CI cluster calculations are thus complementary and verify other calculations.

In the following we introduce simple principles of CI cluster calculations in §2[link]. In §3[link], we start from a direct comparison between the simulated spectrum and the experimentally measured STO Ti L2,3-edges absorption spectrum. In §3.2[link], we investigate the effect of different energy parameters on multiplet structures. In particular, we analyze the peak splitting and peak intensity ratios at the Ti L2,3-edges which give rise to the crystal field splitting energy 10Dq and hybridization strength pdσ values. In §3.3[link], we focus on the photon polarization-dependent absorption spectra with different tetragonal crystal fields. A conclusion is given §4[link].

2. CI cluster calculation

The theory for CI cluster calculations of transition metal compounds was developed from ligand field calculations taking the full multiplet effects into account (Haverkort et al., 2012[Haverkort, M. W., Zwierzycki, M. & Andersen, O. K. (2012). Phys. Rev. B, 85, 165113.]; van der Laan et al., 1981[Laan, G. van der, Westra, C., Haas, C. & Sawatzky, G. A. (1981). Phys. Rev. B, 23, 4369-4380.], 1986[Laan, G. van der, Zaanen, J., Sawatzky, G. A., Karnatak, R. & Esteva, J.-M. (1986). Phys. Rev. B, 33, 4253-4263.]; Haverkort, 2016[Haverkort, M. W. (2016). Quanty - a quantum many-body script language, http://www.quanty.org/.]), showing the advantages of interpreting the L2,3-edges excitation spectrum over the last decades. Each calculation is carried out using the following procedure: a many-electron configuration basis is firstly built for the initial and the final state separately. We concern only the three lowest energy configurations in our present calculations. The Hamiltonian matrix comprises different energy parameters, including the crystal field of 3d orbitals, the crystal field of ligand p orbitals, the hopping term and the Coulomb interaction plus the charge transfer energy term for the initial-state Hamiltonian. The final-state Hamiltonian takes into account the attractive interaction Upd between the core hole and the 3d electron as well as spin–orbit coupling effects. Finally, the transition probability is calculated for each final state from a given initial state with an authorized incident photon polarization based on Fermi's Golden rule with the electron dipole approximation. Regarding the general background and details on performing the CI cluster calculations, refer to the literature mentioned above.

3. Results and discussions

3.1. Comparison with experiment

We measured the L2,3-edges X-ray absorption spectrum for a STO single-crystal at beamline 4B9B of the Beijing Synchrotron Radiation Facility. The spectrum was measured in total electron yield mode, as shown in Fig. 1[link]. The experimental spectrum comprises two leading peaks which are typical for 3d0 compounds, and the L3, L2 set of peaks owing to the spin–orbit coupling of the Ti 2p levels. We denote the L3 and L2 absorption peaks as a1(E = 457.6 eV), a2(E = 459.9 eV), b1(E = 462.9 eV) and b2(E = 465.3 eV), corresponding to excitations of the form 2pj=3/2t2g, 2pj=3/2eg, 2pj=1/2t2g and 2pj=1/2eg, respectively. One electron picture without taking the electron–electron correlations into account would provide the intensity ratio I(a1):I(a2):I(b1):I(b2) = 6:4:3:2 by considering the intensity ratio of 4:2 from excitations of, respectively, 2pj=3/2 and 2pj=1/2 levels and the intensity ratio of 6:4 between excitations to, respectively, t2g and eg states, which are obviously unresolved from experimental measurement. This indicates that the multipet features in the absorption spectrum are complicated and related to combined covalent effects and Coulomb interactions. We denote the intensity ratios of the L3 and L2 set of peaks as IL3 = I(a2)/I(a1) = 1.1 ± 0.05 and IL2 = I(b2)/I(b1) = 1.5 ± 0.1. The intensity of each peak is obtained by fitting the multiple peaks with Lorentzian profiles. Furthermore, from the energy position of the absorption peaks in the experimental spectrum, we obtain the energy splitting between the L3 and L2 set of peaks, i.e. [\Delta_{{L3}}] = E(a2)-E(a1) = 2.3 eV and [\Delta_{{L2}}] = E(b2)-E(b1) = 2.4 eV. The X-ray absorption spectra contain unique features and thus are usually used as a fingerprint for the ground state properties. Here, both the [\Delta_{{L3,2}}] and IL3,2 values at the L3- and L2-edges serve as good quantities for testing the multiplet effects and to confirm the energy parameters of our calculations, as will be discussed in detail below.

[Figure 1]
Figure 1
Comparison between the experimental X-ray absorption spectrum for a STO single-crystal (miscut angle <0.05°, purchased from Crystec) and the calculated spectrum of STO (bottom).

We carried out CI cluster calculations for a TiO6 cluster with 3d0 electronic configuration. The simulated spectrum shown in Fig. 1[link] is calculated with optimized parameters which can reproduce the experimental spectrum well. The parameters for the present calculations are (in units of eV): Udd = 6.0, Upd = 8.0, [\Delta] = 6.0, 10Dq = 1.8 and [pd\sigma] = −1.2. We will discuss the magnitudes of these energy parameters and their effects on the multiplet structure in the next section.

3.2. Effects on the multiplet structures

In this section, we vary each energy parameter separately, while keeping the others at the optimized values as mentioned above, which could help in understanding the effect of each energy parameter on the multiplet structures. The multipole interactions of the Coulomb interaction are treated with the Hartree–Fock approximation where the radial wavefunctions are related to different Slater parameters, i.e. Fdd 2, Fdd 4 Slater parameters for dd interactions and Fpd 2, Gpd 1, Gpd 3 Slater parameters for pd interactions. Fig. 2(a)[link] shows the simulated absorption spectra calculated with different Slater parameters. The Slater parameters are rescaled to 0%, 25%, 50%, 60%, 70%, 80%, 90% and 100% for the top to the bottom spectra which show dramatic differences. The absorption spectrum with 0% scaling factor corresponds to a single-particle scenario while that with 100% scaling factor corresponds to the spectrum for a free ion. Spectra with a scaling factor of less than 100% indicate the existence of the covalent screening and therefore the intra-atomic interactions of the solid state are considered. As shown in Fig. 2(a)[link], the two leading peaks only exist if one considers a strong solid effect, i.e. values of the Slater integrals larger than 50% of the ionic values. Furthermore, the reductions of the Slater parameters not only reduce the relative intensities for excitations to eg levels and shift to t2g levels but also shift the leading peaks closer to the rest of the spectra, which is consistent with the decreasing intra-atomic interactions. We use the energy splitting between the leading peaks and the rest of the excitation peaks to determine the Slater parameters, which suggests a correction of 70–80% of the Slater parameters for the best agreement with the experimental results. This correction factor is consistent with the value pointed out by de Groot et al. to account for the intra-atomic configuration interaction inside a solid (de Groot et al., 1990[Groot, F. M. F. de, Fuggle, J. C., Thole, B. T. & Sawatzky, G. A. (1990). Phys. Rev. B, 41, 928-937.]). Finally, we note that the small leading peaks keep their intensities and energy splitting almost constant when varying other energy parameters, e.g. the charge transfer energy Δ and Coulomb interaction Udd as shown in Figs. 2(b)–2(c)[link], as well as the crystal field splitting energy 10Dq in Fig. 3[link], which are shown as general small leading peaks in X-ray absorption spectra for d0 compounds.

[Figure 2]
Figure 2
(a) Simulated absorption spectra for a TiO6 cluster with different Slater parameters. Each subsequent spectrum has the value of the Slater integrals further rescaled by 0, 25, 40, 50, 60, 70, 80, 90 and 100% from top to bottom. The inset shows an enlarged plot of the two leading peaks regions. (b) Calculated isotropic spectra for different Δ from −1 to 10 eV. (c) The effect of dd Coulomb interaction energy Udd from 4 to 10 eV on the simulated isotropic spectra. All spectra are shifted to the same first excitation peak (peak a1 shown in Fig. 1[link]).
[Figure 3]
Figure 3
(a) Isotropic spectra with different crystal field splitting energies. (b) Energy splitting [\Delta_{L3}] as a function of 10Dq. The grey dashed line corresponds to the condition that the peak energy splitting [\Delta_{L3}] is equal to 10Dq. The black dot-dash line is reproduced from de Groot et al. (1990[Groot, F. M. F. de, Fuggle, J. C., Thole, B. T. & Sawatzky, G. A. (1990). Phys. Rev. B, 41, 928-937.]). (c) Relative intensities of different excitation peaks a1, a2, b1 and b2 versus 10Dq values. The intensities are normalized by a2 peak intensity calculated at 10Dq = 0 eV.

Figs. 2(b) and 2(c)[link] show the simulated absorption spectra with different charge transfer energy Δ ranging from −1 eV to 10 eV and on-site electron–electron Coulomb interaction energy Udd from 4 eV to 10 eV, both of which characterize the energy differences among different electronic configuration bases. We note a contraction of the multiplet structures as Δ or Udd increases. However, the differences among the simulated spectra are small for Δ above 4 eV as well as for Udd values above 5 eV. Our simulations thus are insensitive to the Δ and Udd values, similar to the large uncertainties of Δ and Udd values reported by Haverkort et al. (2012[Haverkort, M. W., Zwierzycki, M. & Andersen, O. K. (2012). Phys. Rev. B, 85, 165113.]), Tanaka & Jo (1994[Tanaka, A. & Jo, T. (1994). J. Phys. Soc. Jpn, 63, 2788-2807.]) or Bocquet et al. (1996[Bocquet, A. E., Mizokawa, T., Morikawa, K., Fujimori, A., Barman, S. R., Maiti, K., Sarma, D. D., Tokura, Y. & Onoda, M. (1996). Phys. Rev. B, 53, 1161-1170.]). In the present calculations, we use the typical values Udd = 6.0, [\Delta] = 6.0 for STO compound.

The crystal field effect is treated in a mean-field approximation and directly reflects the symmetry of the ground states. Fig. 3[link] shows the absorption spectra calculated with different crystal field splitting 10Dq values for the TiO6 octahedron with cubic Oh symmetry. The L3- and L2-absorption edges split into two peaks as 10Dq increases, which is often imaged as a splitting of the L3- and L2-edges into t2g and eg levels. One might expect that the energy difference between the t2g and eg levels is equal to the crystal field splitting energy 10Dq intuitively. Fig. 3(b)[link] shows the averaged energy splitting between the L3 set of peaks, i.e. [\Delta_{{L3}}] as a function of 10Dq ([\Delta_{{L2}}] is within the error bar), which obviously deviates from the grey dashed line representing the same magnitudes between the peak energy splitting [\Delta_{{L3}}] and 10Dq. The black dash-dot line represents the peak splitting [\Delta_{{L3}}] and [\Delta_{{L2}}] versus 10Dq which is reproduced from de Groot et al. with CI calculations for 3d0 compounds in general (de Groot et al., 1990[Groot, F. M. F. de, Fuggle, J. C., Thole, B. T. & Sawatzky, G. A. (1990). Phys. Rev. B, 41, 928-937.]). Both curves show very similar tendencies but small derivations which might be related to the optimization of other energy parameters in the present calculations. A comparison of [\Delta_{{L3}}] = 2.3 eV and [\Delta_{{L2}}] = 2.4 eV from the experimental spectrum suggests a crystal field splitting energy 10Dq = 1.8–2.0 eV from our calculations, which is consistent with other CI calculations performed for STO compound. Fig. 3(c)[link] shows the relative intensities of each peak separately. The peak intensities show no clear correlation, indicating that the emerged peaks as 10Dq switched on cannot be visualized as the splitting of atomic lines into two but as a multiplet effect, i.e. a redistribution of peak intensities over all lines.

In CI-based cluster calculations, the hopping term has been taken into account which enters the off-diagonal terms of the Hamiltonian and describes the hybridization strength between the transition metal ion and oxygen. The hopping term is expressed using the Slater–Koster parameters pdσ and pdπ, with [pd\sigma/pd\pi] = −2.17 reflecting the anisotropic hybridization strengths (Bocquet et al., 1996[Bocquet, A. E., Mizokawa, T., Morikawa, K., Fujimori, A., Barman, S. R., Maiti, K., Sarma, D. D., Tokura, Y. & Onoda, M. (1996). Phys. Rev. B, 53, 1161-1170.]; Slater & Koster, 1954[Slater, J. C. & Koster, G. F. (1954). Phys. Rev. 94, 1498-1524.]). Fig. 4(a)[link] shows the effect of hybridization strength pdσ ranging from 0 to 4 eV on the simulated isotropic spectra, which causes dramatic changes in the absorption spectra including the energy splitting between the L3 and L2 set of peaks, as well as the redistributions of peak intensities. The energy splitting [\Delta_{{L3}}] and [\Delta_{{L2}}] exhibit a monotonically increase as pdσ increases (Fig. 4b[link]), which agrees well with the theoretical prediction of the increasing energy splitting as hybridization increases by solving the determinant of the Hamiltonian (Bocquet et al., 1996[Bocquet, A. E., Mizokawa, T., Morikawa, K., Fujimori, A., Barman, S. R., Maiti, K., Sarma, D. D., Tokura, Y. & Onoda, M. (1996). Phys. Rev. B, 53, 1161-1170.]; Okada et al., 1994[Okada, K., Uozumi, T. & Kotani, A. (1994). J. Phys. Soc. Jpn, 63, 3176-3184.]). A further comparison between the experimental [\Delta_{{L3}}] and [\Delta_{{L2}}] values and the theoretical values suggests that pdσ is in the energy range between 1.0 eV and 1.4 eV. Fig. 4(c)[link] shows the intensity ratios IL3 = I(a2)/I(a1) and IL2 = I(b2)/I(b1) versus the hybridization energy parameter pdσ, both of which decrease as pdσ increases. A pdσ value between 1.0 eV and 1.4 eV corresponds to IL3 = I(a2)/I(a1) [\in] (1.3, 1.7) and IL2 = I(b2)/I(b1) [\in] (1.2, 1.5), which is consistent with the IL2 value obtained from the experimental spectrum. The experimental IL3 value, however, is smaller and falls into the pdσ energy range between 1.5 eV and 2 eV. A comprehensive comparison among the energy splitting [\Delta_{{L3}}] and [\Delta_{{L2}}] values, the intensity ratios IL3 and IL2, as well as the lineshapes between the L3 and L2 excitation peaks prefers a pdσ value in the energy range between 1.0 eV and 1.4 eV; we therefore use [pd\sigma] = 1.2 eV for our calculations.

[Figure 4]
Figure 4
Isotropic spectra with different hybridization strengths. The effect of hybridization strength [pd\sigma] (ranging from 0 to 4 eV with an interval of 0.5 eV) on the simulated isotropic spectra [(a)]. (b) Energy splitting [\Delta_{L3}] and [\Delta_{L2}] versus [pd\sigma] values. The straight lines denote the experimental [\Delta_{L3,{\rm{exp}}}] and [\Delta_{L2,{\rm{exp}}}] values. (c) Calculated intensity ratio IL3 and IL2 as a function of different [pd\sigma] values.

We note that different pdσ values have been reported for STO compounds in theoretical calculations. Bocquet et al. (1996[Bocquet, A. E., Mizokawa, T., Morikawa, K., Fujimori, A., Barman, S. R., Maiti, K., Sarma, D. D., Tokura, Y. & Onoda, M. (1996). Phys. Rev. B, 53, 1161-1170.]) reported [pd\sigma] = 2.6 ± 0.1 eV based on the CI cluster model analysis of the core-level metal 2p X-ray photoemission spectrum. Haverkort et al. (2012[Haverkort, M. W., Zwierzycki, M. & Andersen, O. K. (2012). Phys. Rev. B, 85, 165113.]) reported a pdσ value around 2.3 eV based on ab initio multiplet ligand-field theory. This discrepancy might be related to the size of the basis set used and/or the different calculation methods. Our smaller pdσ value, however, is consistent with the 1.15 eV reported by Ikeno et al. (2009[Ikeno, H., de Groot, F. M. F., Stavitski, E. & Tanaka, I. (2009). J. Phys. Condens. Matter, 21, 104208.]) and by Kroll et al. (2015[Kroll, T., Solomon, E. I. & de Groot, F. M. F. (2015). J. Phys. Chem. B, 119, 13852-13858.]). The early 3d transition metal oxide typically shows a large charge transfer energy. The strong covalent effect thus is related to the large hybridization energy. For the STO compound of interest here, the large charge transfer energy and the small hybridization energy resolved by CI cluster calculations suggest a relatively smaller covalent effect in the system than other Ti and V oxides. This is different as suggested by the CI calculation analysis of the STO X-ray photoemission spectrum (Bocquet et al., 1996[Bocquet, A. E., Mizokawa, T., Morikawa, K., Fujimori, A., Barman, S. R., Maiti, K., Sarma, D. D., Tokura, Y. & Onoda, M. (1996). Phys. Rev. B, 53, 1161-1170.]). Finally, we note that our spectra have been broadened with a Gaussian broadening of 0.15 eV full width at half-maximum (FWHM) and a Lorentzian lifetime broadening with energy-dependent FWHM. The leading peaks, peaks a1, a2, b1 and b2 have FWHM intervals from 0.1 eV to 0.3 eV, 0.3 eV to 0.5 eV, 0.5 eV to 0.7 eV, 0.7 eV to 1.0 eV and 1.0 eV to 1.2 eV, respectively.

3.3. Crystal field effects

Accompanying the fast development of thin-film deposition techniques, transition metal thin films and heterostructures have expanded into a booming field. Strain induced by the lattice mismatch between thin film and substrate generally lowers the crystal symmetry from cubic Oh to tetragonal D4h. Strain engineering has been proposed to tailor the electronic properties in perovskite thin films and heterostructures (Rondinelli & Fennie, 2012[Rondinelli, J. M. & Fennie, C. J. (2012). Adv. Mater. 24, 1918.]); for example, the enhancement of ferroelectric properties (Haeni et al., 2004[Haeni, J. H., Irvin, P., Chang, W., Uecker, R., Reiche, P., Li, Y. L., Choudhury, S., Tian, W., Hawley, M. E., Craigo, B., Tagantsev, A. K., Pan, X. Q., Streiffer, S. K., Chen, L. Q., Kirchoefer, S. W., Levy, J. & Schlom, D. G. (2004). Nature (London), 430, 758-761.]), the manipulation of orbital occupancies (Chakhalian et al., 2011[Chakhalian, J., Rondinelli, J. M., Liu, J., Gray, B. A., Kareev, M., Moon, E. J., Prasai, N., Cohn, J. L., Varela, M., Tung, I. C., Bedzyk, M. J., Altendorf, S. G., Strigari, F., Dabrowski, B., Tjeng, L. H., Ryan, P. J. & Freeland, J. W. (2011). Phys. Rev. Lett. 107, 116805.]; Wu et al., 2013[Wu, M., Benckiser, E., Haverkort, M. W., Frano, A., Lu, Y., Nwankwo, U., Brück, S., Audehm, P., Goering, E., Macke, S., Hinkov, V., Wochner, P., Christiani, G., Heinze, S., Logvenov, G., Habermeier, H.-U. & Keimer, B. (2013). Phys. Rev. B, 88, 125124.]), the modification of octahedral rotations which affect the electric and magnetic properties (Zayak et al., 2006[Zayak, A. T., Huang, X., Neaton, J. B. & Rabe, K. M. (2006). Phys. Rev. B, 74, 094104.]; Rondinelli & Spaldin, 2009[Rondinelli, J. M. & Spaldin, N. A. (2009). Phys. Rev. B, 79, 054409.]), etc. We thus consider explicitly a TiO6 octahedron with D4h tetragonal symmetry, to offer theoretical guidance for understanding the experimental L2,3-edge X-ray absorption and electron energy-loss spectra based on CI cluster calculations.

A tetragonal distortion generally can be modelled by two additional energy parameters, i.e. [\Delta_{{t2g}}] and [\Delta_{{eg}}], representing the energy splitting of t2g levels [[\Delta_{{t2g}}] = E(dyz,dxz)-E(dxy)] and eg levels [[\Delta_{{eg}}] = E(d3z2-r2)-E(dx2-y2)], respectively. Fig. 5(a)[link] shows the polarization-dependent simulated spectra at different [\Delta_{{t2g}}] values ranging from −300 meV to 300 meV. The spectra show strong sensitivities to [\Delta_{{t2g}}] and obvious polarization dependence. One might expect that the spectra do not change with different [\Delta_{{eg}}] values intuitively since the eg levels are empty and far away from the antibonding bands near the Fermi level. This, however, is not the case. Different from the [\Delta_{{t2g}}]-dependent spectra where the excitations to t2g levels are activated (peak a1 and b1), the peaks a2 and b2 with corresponding excitations to eg levels show considerable linear dichroic effects with different [\Delta_{{eg}}] values, as shown in Fig. 5(b)[link], which we attribute to a change of the final states as [\Delta_{{eg}}] varies.

[Figure 5]
Figure 5
(a, b) Calculated soft X-ray absorption spectra for a TiO6 cluster with photon polarization parallel ([E\parallel c]) and perpendicular ([E\perp c]) to the crystal c-axis at different [\Delta_{t2g}] values and different [\Delta_{eg}] values, i.e. −300, −240, −120, −60, 0, 60, 120, 240 and 300 meV from top to bottom spectra. The crystal field splitting value is shown beside the corresponding plot in units of meV. (c, d) Normalized linear dichroic spectra, i.e. ([{\rm{XAS}}_{E\perp,c}][{\rm{XAS}}_{E\parallel c}])/[(2Ix+Iz)/3] for different [\Delta_{t2g}] and [\Delta_{eg}] values.

Fig. 5(c)[link] shows the normalized dichroic spectra for negative and positive [\Delta_{{t2g}}] values, where the signs and magnitudes of the natural linear dichroism change substantially. The leading peaks exhibit strong natural dichroism due to the intrinsic predominantly triplet characters which are mixed into the main L3 set of peaks through spin–orbit interactions and Coulomb repulsion interactions (de Groot et al., 1990[Groot, F. M. F. de, Fuggle, J. C., Thole, B. T. & Sawatzky, G. A. (1990). Phys. Rev. B, 41, 928-937.]). The two linear dichroic spectra, although showing opposite signs to each other, are not interchangeable simply by reversing their respective signs which is related to the complicated multiplet effects. Fig. 5(d)[link] shows the normalized dichroic spectra calculated at different [\Delta_{{eg}}] values. The linear dichroic spectra are almost symmetric for positive and negative [\Delta_{{eg}}] values, consistent with the isotropic nature of the eg orbitals when [\Delta_{{eg}}] varies.

We further calculate the spectra with both [\Delta_{{t2g}}] and [\Delta_{{eg}}] energy parameters. In principle, any [\Delta_{{t2g}}] and [\Delta_{{eg}}] values can be used to parameterize the different spectra, depending on the details of the TiO6 octahedral distortion. For instance, an expansion of the out-of-plane Ti—O bond length and a compression of the Ti—O in-plane bond length results in decreased/increased Ti—O overlapping, which could happen when STO thin films are under compressive strain. The different hybridization strengths between eg and t2g orbitals are essentially attributed to the eg orbitals pointing towards the nearest O p orbitals. The hopping magnitude of the eg–O p orbitals is twice as large as that of the t2g–O p orbitals as mentioned above. We thus took the value of [\Delta_{{eg}}] as [2\Delta_{{t2g}}] in our calculations in the first approximation, considering only the anisotropic hybridization strength.

Fig. 6(a)[link] shows the polarization-dependent simulated spectra at different [\Delta_{{t2g}}] and [\Delta_{{eg}}] values. The opposite dichroic effects appear at very small positive and negative crystal field values, which can be related to the insulating nature of the STO compound. This is different from metallic systems, where the considerable bandwidth should be taken into account. A clear energy shift, i.e. a band splitting of the t2g and eg sub-levels, has been observed for the spectra with both positive and negative [\Delta_{{t2g}}] and [\Delta_{{eg}}] values. We denote the energy shift as the spectra splitting between photons with parallel ([E\parallel c]) and perpendicular ([E\perp c]) polarizations. The positive (negative) energy shift indicates that the dxy (dxz, dyz) and the dx2-y2 (d3z2-r2) orbitals are lower in energy. The energy shifts for both t2g-excited (a1 and b1) peaks and eg-excited (a2 and b2) peaks are opposite in response to the positive and negative [\Delta_{{t2g}}] and [\Delta_{{eg}}] values but do not show a linear dependence. The simulated spectra show larger energy shifts for negative crystal field splitting, suggesting that compressive strain is a more efficient parameter for manipulating the orbital engineering of STO-based thin films.

[Figure 6]
Figure 6
(a) Calculated soft X-ray absorption spectra for [E\parallel c] and [E\perp c] photon polarizations at different crystal field splitting values, i.e. [\Delta_{t2g}] of −150, −120, −90, −60, −30, 30, 60, 90, 120 and 150 meV from top to bottom spectra and [\Delta_{eg}] = [2\Delta_{t2g}]. (b) Comparison of averaged energy shift of peak a1 and b1 for ([E\parallel c]) and ([E\perp c]) polarizations between the spectra calculated considering both [\Delta_{t2g}] and [\Delta_{eg}] values and the spectra calculated considering only the splitting of t2g levels, i.e. [\Delta_{t2g}]. (c) Comparison of averaged energy shift of peak a2 and b2 between the spectra calculated for both [\Delta_{t2g}] and [\Delta_{eg}] values and the spectra calculated considering only [\Delta_{eg}].

We further compare the energy shifts calculated by taking both [\Delta_{{t2g}}] and [\Delta_{{eg}}] values into account and the energy shifts calculated considering only [\Delta_{{t2g}}] or [\Delta_{{eg}}] separately. Fig. 6(b)[link] shows the averaged energy shift of peak a1 and b1 between the spectra calculated considering both [\Delta_{{t2g}}] and [\Delta_{{eg}}] values and the spectra calculated with [\Delta_{{t2g}}] alone. The energy splits of t2g-excited peaks are not equal in both cases, whereas the energy shifts of eg-excited peaks for the [\Delta_{{eg}}] case are comparable [as shown in Fig. 6(c)[link]]. This observation indicates that a proper understanding of the experimentally resolved orbital selective energy shifts should include the eg crystal field splitting, even though the eg orbitals are not relevant for the local ground state properties, at least in the STO-related thin films investigated here.

4. Conclusion

In summary, we carried out CI cluster calculations for a TiO6 octahedron of STO compound, which can reproduce well the experimentally measured L2,3-edges absorption spectrum. Detailed investigations between the experimental spectrum and the simulated spectra yield the values of different energy parameters, e.g. a correction of 70–80% of the Slater parameters needs to be considered for inclusion of intra-atomic interactions. The energy splitting of the L2,3 set of peaks in the X-ray absorption spectrum is not equal to the crystal field splitting energy 10Dq with Oh symmetry directly, which, however, can be obtained through CI calculations, i.e. 10Dq = 1.8–2 eV for STO oxide. Different pdσ values had been reported for STO compounds in other theoretical calculations where the discrepancy might be related to the size of the basis set used and/or the different calculation methods. Our careful analyses of the energy splitting [\Delta_{{L3,L2}}] values and the peak intensity ratios quantitatively, taking the lineshapes of the L3 and L2 excitation peaks into account, yield pdσ = 1.2 eV, indicative of a smaller covalent effect of STO compound compared with other Ti oxides, e.g. TiO2 and LaTiO3 compounds. We finally emphasize the photon polarization-dependent absorption spectra with different tetragonal crystal field splitting. For symmetry reasons, the spectra show dramatic natural linear dichroism. The energy shifts do not show linear dependences in response to the positive and negative [\Delta_{{t2g}}] and [\Delta_{{eg}}] values. The larger energy shifts for negative crystal field splitting suggests that compressive strain is a more efficient parameter for manipulating the orbital engineering of STO-based thin films. A detailed investigation of the energy shifts in different [\Delta_{{t2g}}] and [\Delta_{{eg}}] crystal fields suggests that a thorough understanding of the experimentally resolved energy shifts should include the eg crystal field splitting, even though the eg orbitals are not relevant for the local ground state properties. Our simulations show the powerful features of CI cluster calculations and its potential in understanding the full absorption spectra.

Acknowledgements

We acknowledge the valuable discussions with M. W. Haverkort, X. P. Yang, and the provision of synchrotron radiation beam time at the 4B9B beamline of the Beijing Synchrotron Radiation Facility.

Funding information

Funding for this research was provided by: National Natural Science Foundation of China (grant No. 11704317; grant No. U1332105); China Postdoctoral Science Foundation (grant No. 2016M602064); Fundamental Research Funds for Central Universities (grant No. 20720160020); Center for Functional Nanomaterials, which is a US DOE Office of Science Facility, at Brookhaven National Laboratory (contract No. DE-SC0012704l to HLX).

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