Magneto-electronic specific heat of germanene nanoribbons

Electronic properties of zigzag and armchair germanene nanoribbons (ZGeNR and AGeNR) in magnetic field are calculated by the tight-binding model including the spin-orbit coupling (SOC). The SOC induces spin-split states changing energy dispersion and band-gap. As field strength increases, band-gap increases for ZGeNRs, but decreases for AGeNRs. At zero field, temperature-dependent electronic specific heat strongly depends on nanoribbon's boundary and width. For ZGeNRs, specific heat declines simply proportional to the inverse of width; however, it depends on the dimer-line number for AGeNRs. At low-temperature, width-dependent specific heat of ZGeNRs rapidly drops with increasing field strength, whereas it exhibits an oscillatory behavior for AGeNRs. As temperature increases, such an oscillatory behavior is smeared out or not that is profoundly related to the dimer-line number of AGeNRs. Low-temperature specific heat with varying magnetic field exhibits peak, peak-dip-peak, or smoothly increasing behaviors that strongly reveals geometry dependence of field-modulated electronic properties.