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Unconventional crystal field splitting in non-centrosymmetric BaTiO$_3$ thin films

Understanding the crystal field splitting and orbital polarization in non-centrosymmetric systems such as ferroelectric materials is fundamentally important. In this study, taking BaTiO$_3$ (BTO) as a representative material we investigate titanium crystal field splitting and orbital polarization in non-centrosymmetric TiO$_6$ octahedra with resonant X-ray linear dichroism at Ti $L_{2,3}$-edge. The high-quality BaTiO$_3$ thin films were deposited on DyScO$_3$ (110) single crystal substrates in a layer-by-layer way by pulsed laser deposition. The reflection high-energy electron diffraction (RHEED) and element specific X-ray absorption spectroscopy (XAS) were performed to characterize the structural and electronic properties of the films. In sharp contrast to conventional crystal field splitting and orbital configuration ($d_{xz}$/$d_{yz}$ $<$ $d_{xy}$ $<$ $d_{3z^2-r^2}$ $<$ $d_{x^2-y^2}$ or $d_{xy}$ $<$ $d_{xz}$/$d_{yz}$ $<$ $d_{x^2-y^2}$ $<$ $d_{3z^2-r^2}$) according to Jahn-Teller effect, it is revealed that $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals are nearly degenerate, whereas $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ orbitals are split with an energy gap $\sim$ 100 meV in the epitaxial BTO films. The unexpected degenerate states $d_{xz}$/$d_{yz}$/$d_{xy}$ are coupled to Ti-O displacements resulting from competition between polar and Jahn-Teller distortions in non-centrosymmetric TiO$_6$ octhedra of BTO films. Our results provide a route to manipulate orbital degree of freedom by switching electric polarization in ferroelectric materials.