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How semiconducting are ferroelectrics: The fundamental, optical and transport gaps of Na$_{0.5}$Bi$_{0.5}$TiO$_3$-BaTiO$_3$ and NaNbO$_{3}$

The energy gap is a fundamental property of materials, directly related to their optical and electronic properties. The energy gap of ferroelectric compounds and its adjustment by compositional variation has particularly attracted attention in recent years due to potential application in energy conversion and/or catalytic devices. It is demonstrated that it is necessary to distinguish between the fundamental gap, $E_{\rm g}^{0}$, the optical gap, $E_{\rm g}^{\rm opt}$, and the transport gap, $E_{\rm g}^{\rm tr}$, of ferroelectrics, which can differ significantly. The situation is comparable to those in organic semiconductors and emerges from the presence of localized charges. The fundamental gap is a ground state property, i.e.\ the energy difference between the maximum of the fully occupied valence band and the minimum of the completely empty conduction band. In contrast, the optical and transport gaps are excited state properties involving localized (polaronic) electrons and/or holes at energies considerably different from the band edges. This work illustrates how the different energy gaps of ferroelectrics can be determined by combining optical measurements, X-ray photoelectron spectroscopy and temperature and oxygen partial pressure dependent electrical conductivity measurements. We determine fundamental gaps of $\approx 4.5\,$eV for both materials, optical gaps of $3.25-3.45\,$eV/$3.5\,$eV and electrical gaps of $\approx 1.4\,$eV/$3.3\,$eV for Na$_{0.5}$Bi$_{0.5}$TiO$_3$-BaTiO$_3$/NaNbO$_{3}$, respectively.