  Complex band structures (CBSs) are useful to characterize the static and
dynamical electronic properties of materials. Despite the intensive
developments, the first-principles calculation of CBS for over several hundred
atoms is still computationally demanding. We here propose an efficient and
scalable computational method to calculate CBSs. The basic idea is to express
the Kohn-Sham equation of the real-space grid scheme as a quadratic eigenvalue
problem and compute only the solutions which are necessary to construct the CBS
by Sakurai-Sugiura method. The serial performance of the proposed method shows
a significant advantage in both runtime and memory usage compared to the
conventional method. Furthermore, owing to the hierarchical parallelism in
Sakurai-Sugiura method and the domain-decomposition technique for real-space
grids, we can achieve an excellent scalability in the CBS calculation of a
boron and nitrogen doped carbon nanotube consisting of more than 10,000 atoms
using 2,048 nodes (139,264 cores) of Oakforest-PACS.
