We study the spin-dependent transport and thermoelectric properties of ferromagnetic zigzag-edge phos- phorene 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 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 al- most 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.

中文翻译：

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电场下铁磁锯齿形磷烯纳米带的自旋相关输运和热电性能的奇偶效应


我们研究了由垂直电场（PEF）调制的具有N链（N-ZPNR）的铁磁锯齿形边缘磷烯纳米带的自旋相关输运和热电性质。采用哈伯德模型哈密顿量结合自洽计算，利用非平衡格林函数方法，得到了能带结构、自旋相关传输系数和塞贝克系数。结果表明，在铁磁态下，12-ZPNR 和 13-ZPNR 的能带都被自旋向上和向下的带分裂，并且两个 ZPNR 都表现出自旋半导体行为，自旋间隙值为 1 eV。有趣的是，当应用 PEF 时，自旋相关的输运和热电性质会出现奇偶效应，这源于 N-ZPNR 的结构对称性，并且可能导致偶数和奇数情况下边缘态的不同行为。电场提升了 12-ZPNR 中自旋向上和自旋向下通道的两个边缘带的简并性，然而，它仅使自旋向上边缘带弯曲并将自旋向下边缘带向上移动，而在 13-ZPNR 中没有分裂。此外，我们发现两个 ZPNR 的自旋塞贝克系数表现出宽阔的平台，并且其最大值可以达到前所未有的巨大 5 mV/K。两个 ZPNR 的自旋塞贝克系数随着温度升高而逐渐减小。随着电场的增加，12-ZPNR 自旋塞贝克系数的平台值逐渐减小，然而，在这种情况下，13-ZPNR 的平台几乎没有变化。 偶数和奇数情况下自旋塞贝克系数对 PEF 的不同响应源于对电场的不同边缘态响应，我们从相应的自旋向上和自旋向下塞贝克系数提供了解释。我们的结果表明，铁磁 ZPNR 具有完美的自旋极化，是高效自旋热电子学的有希望的候选者。