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Grain size dependence of electrical and optical properties in Nb-doped anatase
4.B. O’Regan and M. Grätzel, Nature (London) 353, 737 (1991);
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12.The Scherrer equation is , where is the grain size, is the wavelength of the filament used in the XRD machine, is the width of a peak at half of its intensity, is the angle of the same peak.
16.In order to subtract the substrate effect, the transmission and reflection spectra of every substrate were deliberately measured twice before and after the film deposition, respectively. According to the Lambert law, , here and , and are the measured transmittance and reflectance of the samples before and after film deposition, respectively. is the interfacial reflectance between film and substrate. According to the Fresnel formula, , here , and are the refractive indexes of the air , substrate and film, respectively. For simplicity, we take the mean refractive index for and for anatase film. Then the absorption coefficient can be calculated easily from the transmission and reflection spectra by the equation of .
18.The BM shift (, where is the energy at the bottom of the conduction band) can be estimated from the equation . See also, K. W. Boer, Survey of Semiconductor Physics, p. 240 (Van Nostrand Reinhold, New York, 1990).
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