Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals
Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals
复制标题
DOI：
10.1103/physrevb.74.024408
复制
发表时间：
2006-07-01
影响因子：
3.7
通讯作者：
Resta, R.
中科院分区：
文献类型：
--
作者：
Ceresoli, Davide;Thonhauser, T.;Resta, R.
我们推导出了正常周期绝缘体(即陈不变量或二维(2D)陈数为零的绝缘体)中轨道磁化强度的多带公式。遵循最近用来发展单波段形式主义的方法[Thonhauser，Ceresoli，Vanderbilt和Resta，Phys.莱特牧师。95,137205(2005年)]，我们在Wannier表示中工作，发现磁化由两个贡献组成，一个明显的与内部块状Wannier函数的内部循环有关，另一个意外的贡献来自表面附近的Wannier函数携带的净电流。与单带情况不同，在单带情况下，这些贡献中的每一个都是单独的规范不变的，而在多带形式中，只有两项的和是规范不变的。我们的轨道磁化强度的最终表达式可以改写为Bloch函数的块状属性，这使得它在现代代码包中实现起来很简单。对二维模型系统的倒易空间表达式进行了评估，并与有限样品磁化强度的计算结果进行了比较，结果得到了验证。最后，虽然我们的形式证明仅限于正常绝缘子，但我们也给出了一个启发式的推广到陈绝缘子(具有非零陈不变量)和金属。通过与二维模型系统中有限个样品的磁化强度的比较，再次验证了该扩展的有效性。我们发现了很好的一致性，从而为启发式公式的有效性提供了强有力的经验证据。
We derive a multi-band formulation of the orbital magnetization in a normal periodic insulator (i.e., one in which the Chern invariant, or in two dimensions (2D) the Chern number, vanishes). Following the approach used recently to develop the single-band formalism [Thonhauser, Ceresoli, Vanderbilt, and Resta, Phys. Rev. Lett. 95, 137205 (2005)], we work in the Wannier representation and find that the magnetization is comprised of two contributions, an obvious one associated with the internal circulation of bulklike Wannier functions in the interior and an unexpected one arising from net currents carried by Wannier functions near the surface. Unlike the single-band case, where each of these contributions is separately gauge invariant, in the multi-band formulation only the sum of both terms is gauge invariant. Our final expression for the orbital magnetization can be rewritten as a bulk property in terms of Bloch functions, making it simple to implement in modern code packages. The reciprocal-space expression is evaluated for 2D model systems and the results are verified by comparing to the magnetization computed for finite samples cut from the bulk. Finally, while our formal proof is limited to normal insulators, we also present a heuristic extension to Chern insulators (having nonzero Chern invariant) and to metals. The validity of this extension is again tested by comparing to the magnetization of finite samples cut from the bulk for 2D model systems. We find excellent agreement, thus providing strong empirical evidence in favor of the validity of the heuristic formula.