One-dimensional long-period superlattices in alloys

One-dimensional (1D) long-period superlattices (LPSLs) in alloys, which consist of a periodic array of antiphase boundaries in the L10 or L12 ordered structure, have been examined theoretically from the viewpoint of the appearance of charge-density waves (CDWs) in a three-dimensional system. The Fermi surface of the LPSL has flat portions along the 110 directions, and the Fermi-surface nesting results in two nesting vectors in the L10 structure and four vectors in the L12 structure. This means that the 1D LPSL is not a single first-order CDW state but is characterized by the superposition of multiple CDWs. The response of the lattice system to the CDW is further understood to take place in two ways: One is the introduction of a periodic array of antiphase boundaries as the atomic arrangement, and the other is the periodic lattice distortion as the atomic displacement. Unlike low-dimensional materials such as 2H-TaSe2, the former way plays a particularly important role in the response for the LPSLs. As a result of the appearance of the CDWs, the change in the period with respect to the composition of an alloy, the electron-atom ratio, is determined by the size of the Fermi surface along the 110 directions and can be basically explained on the basis of the same equation as that in the Sato-Toth theory [Phys. Rev. 124, 1833 (1961)]. In addition, a Ginzburg-Landau free-energy functional is proposed in order to understand the features of an incommensurate structure, such as the temperature dependence of the period and the discommensurate structures in the LPSLs. It is assumed in the theory that the normal structure is the normal ordered structure and the order parameter is the CDW. Note that the temperature dependence of the period is closely related to the phase modulation of the first-order CDW by means of the higher-order CDWs produced from the higher-order harmonics via the umklapp process. Hence the LPSL is classified into two groups in the L10 structure and three groups in the L12 structure on the basis of the positional relation between the first- and third-order spots in diffraction patterns. The temperature dependence of the period and the phase modulation of the first-order wave are then calculated by the derived free-energy expression for each group, and are found to be in good agreement with those obtained experimentally. That is, the present theory can well reproduce overall features of the LPSLs. This means that the 1D LPSLs are concluded to be the CDW state in the three-dimensional system.