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We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel–Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition and prove that its global degree is
. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel–Prym map is
Four epitaxial ScN(001) thin films were successfully deposited on MgO(001) substrates by dc reactive magnetron sputtering at 2, 5, 10, and 20 mTorr in an Ar/N2 ambient atmosphere at 650 °C. The microstructure of the resultant films was analyzed by x-ray diffraction, scanning electron microscopy, and transmission electron microscopy. Electrical resistivity, electron mobility and concentration were measured using the room temperature Hall technique, and temperature dependent in-plain measurements of the thermoelectric properties of the ScN thin films were performed. The surface morphology and film crystallinity significantly degrade with increasing deposition pressure. The ScN thin film deposited at 20 mTorr exhibits the presence of <221> oriented secondary grains resulting in decreased electric properties and a low thermoelectric power factor of 0.5 W/mK2 at 800 K. The ScN thin films grown at 5 and 10 mTorr are single crystalline, yielding the power factor of approximately 2.5 W/mK2 at 800 K. The deposition performed at 2 mTorr produces the highest quality ScN thin film with the electron mobility of 98 cm2 V−1 s−1 and the power factor of 3.3 W/mK2 at 800 K.