Two-component anomalous Hall and Nernst effects in anisotropic Fe$_{4-x}$Ge$_x$N thin films

A series of thin films Fe$_{4-x}$Ge$_x$N (x=0-1) was fabricated onto MgO substrates by magnetron sputtering with the aim of studying the possible enhancement of the anomalous Nernst effect (ANE), envisaged based on Density Functional Theory (DFT) calculations. The Nernst and Hall effects of the series were systematically analyzed, complemented with resistivity, magnetic, electron microscopy and M\"ossbauer experiments, and DFT calculations including elastic properties. The Fe$_4$N phase crystallizes in the cubic symmetry with Pm3m space group, whereas a small tetragonal distortion is realized in for x>0.35. From the comparison of the experimental isomer shift with DFT calculations, we conclude that Ge occupies the 4b site in the tetragonal I4/mcm tructure. Ferromagnetic T$_C$ decreases rapidly from 750 K for x=0 to 100 K for x=1. The tetragonal samples with x=0.8 and 1 display two-component behavior in the Hall and Nernst effects hysteresis loops, which can be analyzed as a sum of positive and negative loops with different saturation fields. This unusual behavior is a product of a combination of several factors. (1) Co-existence of two different crystallographic orientations in the tetragonal thin film, namely with the majority of c-axis and minority of a-axis normal to the film surface. (2) Opposite sign of the anomalous Hall and Nernst effects for the direction of magnetization along the a and c-axis revealed by DFT calculation. (3) The magnetocrystalline anisotropy characterized by an easy ab-plane, which is responsible for the different saturation fields for a and c-axis. The maximum ANE was determined to be 0.9 $\mu$V/K for x=0 at room temperature, and -0.85 $\mu$V/K for x=1 at T=50 K. The rapid increase of ANE of Fe$_3$GeN from low temperatures indicates that, were it not for its low T$_C$, it could surpass ANE of Fe$_4$N.