Even–odd effect of spin-dependent transport and thermoelectric properties for ferromagnetic zigzag phosphorene nanoribbons under an electric field

Abstract

We study the spin-dependent transport and thermoelectric properties of ferromagnetic zigzag-edge phosphorene nanoribbon with N chains (N-ZPNR) modulated by a perpendicular electric field (PEF). By adopting the Hubbard model Hamiltonian combined with the self-consistent calculation, and using the nonequilibrium Green’s function method, we obtain the band structure, spin-dependent transmission coefficients and Seebeck coefficients. It is demonstrated that the bands are split with spin-up and -down ones for both 12- and 13-ZPNR in the ferromagnetic state, and both ZPNRs show spin-semiconducting behavior with the spin gap of value 1 eV. Interestingly, there is an even–odd effect on the spin-dependent transport and thermoelectric properties when a PEF is applied, which originates from the structure symmetry of N-ZPNR and can result in different behavior of edge states for even and odd cases. The electric field lifts the degeneracy of the two edge bands for both spin-up and -down channels in 12-ZPNR, however, it only bends the spin-up edge bands and shifts the spin-down ones upward without splitting in 13-ZPNR. In addition, we find that the spin Seebeck coefficients exhibit wide plateaus and their maximum values can reach unprecedented giant 5 mV K−1 for both ZPNRs. The spin Seebeck coefficients for both ZPNRs decrease gradually as temperature increasing. As the electric field increasing, the value of the plateau in the spin Seebeck coefficient for 12-ZPNR decreases gradually, however, the plateau almost has no variation for 13-ZPNR in this case. This different responses of spin Seebeck coefficients to the PEF between even and odd cases originate from the different edge state responses to the electric field, and we provide the explanation from the corresponding spin-up and spin-down Seebeck coefficients. Our results reveal that ferromagnetic ZPNRs are perfectly spin-polarized, and are promising candidates for spin caloritronics with high efficiency.