56 Physical Science Research Frontiers 2022 An X-ray can be scattered by electrons and lose energy. This process is called X-ray Compton scattering. The scattered X-ray carries information about the momentum distribution n ( k ) of electrons. In metals, n ( k ) exhibits a rapid change at the Fermi momentum k F , above (below) which electron occupation becomes low (high). The surface corresponding to k F is called the Fermi surface (FS), the shape of which reflects electron motions inside the metal. The FS is recognized as a fundamental concept in understanding material properties. High-temperature cuprate superconductors continue to be a central subject in condensed matter physics [1]. They are characterized by a stack of CuO 2 planes, and a vast number of studies using angle-resolved photoemission spectroscopy (ARPES) have confirmed the existence of the two-dimensional (2D) FS at least in the overdoped region — the critical temperature T c in cuprate superconductors is controlled by the carrier doping rate. In the underdoped region including the optimal doping at which T c becomes maximal, the FS was observed clearly only around k = (0.45 p , 0.45 p )— the so-called Fermi arcs, which were believed to be a portion of the underlying 2D FS [2]. To reveal the underlying FS, we perform Compton scattering for the La-based cuprate La 2– x Sr x CuO 4 with x = 0.08 (underdoped sample) at SPring-8 BL08W [3]. The obtained momentum distribution n ( k ) is shown in Fig. 1 in the first Brillouin zone – p < k x , k y < p . Is Fig. 1 consistent with the conventional 2D FS illustrated in Figs. 2(a) and 2(b)? There are two key criteria: n ( k ) should be nearly constant along the FS and the derivative of n ( k ), i.e., ∇ n ( k ), increases in magnitude there. In Fig. 2(c), we superpose the FS identified by ARPES (dots) [2] by focusing on the first quadrant of the Brillouin zone. Around k = (0.45 p , 0.45 p ), n ( k ) is nearly constant along the FS as expected. However, around k = ( p , 0) and (0, p ), n ( k ) varies along the FS (from yellow to green), implying that the underlying X-ray Compton scattering reveals electronic motions with a preferred direction in high-temperature cuprate superconductors Fig. 1. Electron momentum distribution measured by Compton scattering at 300 K for La 2– x Sr x CuO 4 with x = 0.08 [3]. Red (blue) indicates a higher (lower) occupation of electrons. Data around k = (0, 0) are removed because of low accuracy. [5] Fig. 2. Conventional idea. (a) 2D-like FS is realized in each CuO 2 plane. (b) Bulk FS. (c) Observed n ( k ) in the first quadrant of the Brillouin zone. The FS identified by ARPES (dots) is superposed. [5] – π – π π k x high low k y 300 K π (a) (b) (c) x y z A B A CuO 2 plane CuO 2 plane CuO 2 plane A B 0 k k x x k k x x k k x x π π π π – π – π – π 0 0 k k x x π – π 0 0 k k y y k ky y π – π 0 k k y y π – π 0 k k y y π 300 K 57 Research Frontiers 2022 FS cannot be the conventional one accepted thus far (Figs. 2(a) and 2(b)). We also examined the data of ∇ n ( k ), which indicated a curvature qualitatively different from that of the conventional FS [3]. The agreement between the results of the present work and ARPES around k = (0.45 p , 0.45 p ) is reasonable because ARPES signals are very clear there. However, the signals become weak and broad upon approaching k = ( p , 0) and (0, p ), which makes it difficult to identify the FS. Hence, the underlying FS was inferred by extrapolating the FS observed clearly around k = (0.45 p , 0.45 p ) to a region around k = ( p , 0) and (0, p ), as shown by dots in Fig. 2(c) [2]. However, such a FS cannot be reconciled with our data, especially around k = ( p , 0) and (0, p ). What does Compton scattering data Fig. 1 then imply? One possibility is an electronic nematic tendency — the electron motion has a preferred direction along the x - or y -direction in each CuO 2 plane; this tendency is enhanced in the underdoped region in cuprates [4]. For La-based cuprates, this theory [4] predicted that the electronic nematicity yields a quasi-one-dimensional FS in each CuO 2 plane and the direction of nematicity alternates along the z -direction through a coupling to the crystal structure; see Fig. 3(a). These FSs are hybridized via weak interlayer coupling, leading to 2D-like FSs, consisting of the inner and outer FS as shown in Fig. 3(b). We superpose those FSs on the observed n ( k ) in Fig. 3(c). The inner FS explains n ( k ), which is essentially constant (yellow) along the FS. Similarly, the outer FS may also explain n ( k ), although n ( k ) changes along the FS around k = ( p , 0) and (0, p ) from green to blue. We observed that the value of ∇ n ( k ) is very small there and thus the actual change in n ( k ) can be considered to be not sizable. We also performed the same analysis of data at 150 K and confirmed this conclusion. Upon reaching the overdoped region, it is expected that the FS in Fig. 3(a) will lose nematicity and become conventional 2D FS [4]. This expectation was also confirmed by performing Compton scattering at x = 0.15 and 0.30 [3]. In summary, X-ray Compton scattering reveals that, in contrast to the common belief (Fig. 2), electrons in La-based cuprates have a preferred direction along the x - or y -direction in each CuO 2 plane and the preferred direction alternates between the layers (Fig. 3). Compton scattering can be a powerful tool, especially when other probes such as ARPES cannot reveal the whole shape of the FS. Fig. 3. Electronic nematic scenario. (a) FS stacking where the preferred direction of the electron motion is rotated 90 degrees between adjacent layers. (b) Bulk FSs consist of the inner (black) and outer (purple) FSs. (c) Observed n ( k ) in the first quadrant of the Brillouin zone. The FS in (b) is superposed. [5] Hiroyuki Yamase Research Center of Materials Nanoarchitechtonics (MANA), National Institute for Materials Science (NIMS) Email: yamase.hiroyuki@nims.go.jp References [1] B. Keimer et al .: Nature 518 (2015) 179. [2] T. Yoshida et al .: J. Phys. Soc. Jpn. 81 (2012) 011006. [3] H. Yamase. Y. Sakurai, M. Fujita, S. Wakimoto and K. Yamada: Nat. Commun. 12 (2021) 2223. [4] H. Yamase and H. Kohno: J. Phys. Soc. Jpn. 69 (2000) 332; ibid. 2151. [5] H. Yamaes and Y. Sakurai: Isotope News 779 (2022) 38. (a) x y z A B A CuO 2 plane CuO 2 plane CuO 2 plane A k k x x π – π 0 π – π 0 k ky y B k k x x π – π 0 π – π 0 k k y y (c) (b) 0 k k x x π k k y y π 300 K k k x x π π – π – π 0 0 k k y y
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