Giant Optical Anisotropy in High Temperature Superconducting Cuprate Bi₂Sr₂CaCu₂O_8+δ

Kun Zhang¹ , Masataka Matsumoto² , Kenta Nakagawa^3,4 , Azusa Matsuda¹, Genki Shino⁵ , Sota Sato⁶ , Takuji Ikeda⁷ , and Toru Asahi^1,8,9,

Understanding the mechanism of high-\(T_{\text{c}}\) superconductivity (HTSC) is one of the most important and challenging issues in condensed matter physics. To address the same, various studies on magnetic, electronic, and optical properties, among others, have been reported over a few decades. However, their nature is still not fully understood. In recent years, the pseudo-gap state,^1–3) which is observed in the underdoped region in the phase diagram, has been a central topic in the study of HTSC. It is believed that an understanding of the unique properties of the pseudo-gap state would reveal the nature of HTSC. In particular, spatial and time-reversal symmetry breakings in the pseudo-gap and superconducting phase have been actively discussed because these are deeply related to the electronic and magnetic orders in the materials. Bi₂Sr₂CaCu₂O\(_{8+\delta}\) (Bi2212) is a typical compound that exhibits HTSC. Bi2212 is a cuprate superconductor with unique layers of copper oxide (CuO₂). Although several studies on the symmetry breaking in Bi2212 have been published, whether the time reversal and/or spatial symmetry are broken is still controversial, as briefly shown in Table S·I of the Supplemental Materials.⁴⁾ Angle-resolved photoemission spectroscopy for Bi2212 thin films has revealed that the time-reversal symmetry is broken below the characteristic temperature \(T^{*}\) at which the pseudo-gap state is observed.⁵⁾ This result is supported by another experiment that was conducted using polarized neutron scattering.⁶⁾ Conversely, an optical study in visible light regions on Bi2212 single crystals, with an extended high-accuracy universal polarimeter, demonstrated that the spatial symmetry was broken below \(T_{\text{c}}\) while the time-reversal symmetry was not.⁷⁾ Moreover, Kubota et al. reported that X-ray natural circular dichroism was detected below \(T^{*}\), whereas non-reciprocal X-ray linear dichroism could not be observed, and therefore insisted that the time-reversal symmetry was preserved in the pseudo-gap phase, but the spatial symmetry was broken.⁸⁾

The optical study mentioned above is a powerful tool for investigating spatial symmetry breaking and/or time-reversal symmetry breaking. Optical activity (OA) and circular dichroism (CD) are related to spatial and time-reversal symmetry. Natural OA and CD, which are reciprocal signals, were observed when the spatial symmetry was broken. On the other hand, Faraday rotation and magnetic circular dichroism, which are non-reciprocal signals, were observed when the time-reversal symmetry was broken.⁹⁾ Here, we emphasize the consideration of linear birefringence (LB) and linear dichroism (LD) in the measurement of OA and CD in anisotropic crystals. LB and LD are optical anisotropies that denote the difference in refractive indices and absorptions, respectively, between two orthogonally polarized lights in an anisotropic crystal. Schellman et al.¹⁰⁾ reported that LB and LD signals are usually \(10^{3}\)–\(10^{5}\) times larger in magnitude than OA and CD signals. Therefore, the accurate separation of OA and CD from LB and LD is a challenge. Furthermore, Shindo et al.¹¹⁾ pointed out that optical elements, devices, and detectors installed in polarization modulation spectroscopy cause serious systematic errors and non-negligible artifacts. This critical issue has been widely recognized in the field of chiral science. Many research groups have developed optical measurement theories and polarimeters to overcome this issue. Eventually, OA and CD in the anisotropic condensed matter were successfully measured.^12–20) Regardless, some research groups neglected optical anisotropies, systematic errors, and artifacts in OA and CD measurements, leading to controversial conclusions regarding the symmetry breaking in Bi2212 crystals.

Bi2212 crystals usually exhibit anisotropy caused by the incommensurably modulated structure along the b axis.^21,22) Therefore, we carefully dealt with the optical anisotropy when we investigated the spatial and time-reversal symmetry breakings using optical measurements. Optical reflectivity measurements revealed that the Bi2212 crystal exhibits a strong anisotropy between the b and c axes.²³⁾ In addition, strong phonon anisotropy has been reported to have appeared in the ab plane.²⁴⁾ However, the measurements using the reflected light are limited to the infrared light region. In other words, the wavelength dependence of the optical properties in Bi2212 has not been measured using the transmitted light in the ultraviolet and visible light region.

Herein, we analyzed the optical anisotropy of the Bi2212 crystal using the generalized-high-accuracy universal polarimeter (G-HAUP),^{7,12,14,16,17)} which enabled us to measure the optical anisotropies and chiroptical properties of an anisotropic medium in visible and ultraviolet light regions. We found that the Bi2212 single crystal exhibited large optical anisotropies, approximately 350 nm, in the ultraviolet light region at 298 K. The results explicitly suggest careful consideration of their influence on the OA and CD measurements while discussing the spatial and time-reversal symmetry breakings.

The optical system of the G-HAUP is composed of only two optical elements, a polarizer and an analyzer, in which the polarization axes are set in the crossed-Nicols configuration. A schematic of the G-HAUP is shown in Fig. 1. In the G-HAUP measurement, the sample was set between the polarizer and the analyzer, and the transmitted light intensity from these elements was measured as a function of the azimuth angles of the polarizer \(\theta'\) and analyzer \(Y'\). The bulk properties of a single crystal were expected to be obtained as the optical system of the G-HAUP is a transmission type. In our measurements, very thin plate specimens of Bi2212 crystal were prepared such that visible and ultraviolet lights could be transmitted through the crystal. The cleavage property of the Bi2212 crystal in the (001) plane was utilized to obtain ultrathin plate specimens.

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Figure 1. (Color) The G-HAUP is composed of only two optical elements, a polarizer and an analyzer, which the axes are set in the crossed-Nicols configuration. The transmitted light intensity through these elements is measured as a function of the azimuth angle \(\theta'\) of the polarizer and the rotation angle \(Y'\) of the analyzer. The origin of \(\theta'\) and \(Y'\) coordinate system is the extinction position.

Single crystals were grown using the traveling solvent floating zone method with a feed rod of Bi2212 as shown in Fig. 2(a). Powder X-ray diffraction measurements using an X-ray with an energy of 12.4 keV (SPring-8 beamline BL02B2) confirmed the Bi2212 crystal growth by comparing experimental values with those in the literature as shown in Fig. S·1(a).²⁵⁾ To vary the hole concentration p, the post-annealing processes for the grown crystals were performed under an oxygen atmosphere at 500 °C for 10 h. The values of p in the crystals were estimated by measuring \(T_{\text{c}}\) using a superconducting quantum interference device (Quantum Design VSM SQUID). \(T_{\text{c}}\) was confirmed to be 90.6 K, which indicates that the Bi2212 crystals used in this study are optimally doped as shown in Fig. S·1(b).²⁵⁾

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Figure 2. (Color) Photographs of the Bi2212 crystal (a), ultrathin specimen of (001) prepared by exfoliating the Bi2212 crystal with a water-soluble tape (b), and polarizing microscopic image of the ultrathin Bi2212 specimen mounted on the pinhole plate used in the G-HAUP.

Thin (001) plate specimens with a diameter of ca. 0.2 mm were prepared by exfoliating the crystal as shown in Fig. 2(b). In the G-HAUP optical system, the specimens were mounted on a stainless-steel pinhole plate, with a diameter of 0.2 mm, to improve the signal-to-noise ratio of the transmitted light intensity. It was confirmed that the specimens consisted of a single domain using a polarizing microscope (Leica DMLP) with a λ plate. As shown in Fig. 2(c), we obtained single-domain samples with high homogeneity. Notably, we could not obtain a specimen with multiple domains during the preparation of ultrathin specimens with exfoliation. The high-homogeneity specimens enabled us to perform high-precision measurements on the LB and LD.

We applied the G-HAUP method to the Bi2212 specimen to measure its optical anisotropies, namely LB and LD, at 298 K. The measurement theory of the G-HAUP is described using the relative ratio Γ of the transmitted light intensity I to the incident light intensity \(I_{0}\), which is given as \begin{equation} \Gamma = I/I_{0} = | \mathbf{A}^{\mathbf{T}}\mathbf{M}_{\mathbf{H}}\mathbf{P}|^{2}, \end{equation} (1) where \(\mathbf{P}\) and \(\mathbf{A}^{\mathbf{T}}\) represent a Jones vector of the polarizer and a transposed vector of the analyzer, respectively, and \(\mathbf{M}_{\mathbf{H}}\) represents the Jones matrix of the sample. The derivation is fully explained in Refs. 7, 12, 14, 16, and 17, and is briefly shown in Sect. 1 of the Supplemental Materials.²⁶⁾ The G-HAUP theory reveals that LB and LD are related to the total phase difference Δ and the total linear dichroism E of the specimen, respectively, as follows: \begin{equation} \mathrm{LB}= \frac{\Delta \lambda}{2\pi d}, \end{equation} (2) and \begin{equation} \mathrm{LD}= \frac{E\lambda}{2\pi d}, \end{equation} (3) where d and λ are the thicknesses of the specimen and the wavelength of the incident light, respectively.²⁶⁾

In this study, we prepared five ultrathin (001) plate specimens with thicknesses of 272, 381.5, 140.6, 69.3, and 165.4 nm. These thicknesses were estimated using the LB value of 0.157 at 488 nm, which was determined in a previous study.⁷⁾ The G-HAUP revealed that the wavelength dependency of Δ and E along the c axis for these five specimens exhibit qualitatively similar behaviors at 298 K but are not identical quantitatively, as shown in Fig. 3, due to the difference in thickness. The values of Δ and E have been found to exhibit strong peaks at about \(\lambda = 345\) and 330 nm, respectively. The wavelength dependences of LB and LD along the c axis at 298 K were obtained by normalizing Δ and E using the specimen thickness, as shown in Figs. 4(a) and 4(b), respectively. LB and LD also attained strong peaks at \(\lambda = 345\) and 330 nm, respectively. The error bars were estimated from the standard deviation for LB and LD of five samples with different thicknesses, as further indicated in Figs. 4(a) and 4(b). The standard errors in LB and LD of the G-HAUP, observed in this study were found to be below \(5\times 10^{-3}\), as shown in Figs. S·2(a) and S·2(b).²⁷⁾ By contrast, LD broadly attained small values in the range above 380 nm. The absorbance spectrum, \(-{\log I}/I_{0}\), is also indicated in Fig. 4(c), which is obtained from the absorption spectrum over the entire wavelength range of the measurement region. The absorbance spectrum exhibits a large anomaly and a very small broad peak at approximately 310 and 450 nm, respectively.

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Figure 3. (Color) Wavelength dependences of Δ and E of Bi2212 ultrathin crystals along the c axis at 298 K. The thicknesses of the specimens are 381.5 nm (red), 272 nm (black), 165.4 nm (purple), 140.6 nm (blue), and 69.3 nm (green), respectively.

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Figure 4. (Color) Wavelength dependences of LB (a), LD (b), and absorbance (c) in Bi2212 crystal along the c axis at 298 K. The values and error bars of LB and LD in the figures were obtained from the averaged value and the standard deviation of G-HAUP measurements for five specimens with different thicknesses, respectively.

Here, we emphasize that the giant anisotropies in the Bi2212 crystal were scarcely reported in previous optical studies and most researchers focused on the charge-transfer absorption, which corresponds to the optical transition between the O 2p band and Cu 3d upper Hubbard band.²⁸⁾ A clear anisotropy of reflectivity in the ab plane of Bi2212 single crystals was reported by Liu et al.²⁹⁾ However, they did not demonstrate the LB and LD peaks, which were observed at about 345 and 330 nm, respectively in this study.

In summary, we studied the wavelength dependence of optical anisotropy in an optimally doped Bi2212 crystal at 298 K using G-HAUP. We found that the LB and LD spectra exhibited large peaks at 298 K at about \(\lambda = 345\) and 330 nm, respectively. As discussed in the introduction, the optical anisotropy signals are generally much larger than the chiroptical signals, such as OA and CD. Therefore, the information of the present results is extremely important, as we have confirmed the imminent removal of the influence of optical anisotropy while verifying the spatial and/or time-reversal symmetry breaking of Bi2212 in optical measurement. In a previous study related to OA and CD in Bi2212,⁵⁾ the optical anisotropy effects were completely ignored. The origin of optical anisotropy in Bi2212 crystals is possibly related to the incommensurate modulation along the b axis.^21,22,29) To confirm this anticipation in a further study, LB and LD spectra from this study would be compared with those in Pb-doped Bi2212 crystals that have no incommensurate modulation.³⁰⁾ Furthermore, our study should be extended to the measurement of the temperature dependence of LB, LD, OA, and CD to investigate the symmetry breaking in the pseudo-gap and superconducting phases.

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