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Unconventional superconductivity often emerges in proximity to various competing or coexisting states. In cuprate superconductors, these states include spin order, charge order, the pseudogap state, and the strange metal phase. A comprehensive understanding of their relationship is fundamental to establishing the mechanism of high-temperature superconductivity. While spin dynamics in cuprates has been extensively investigated using inelastic neutron scattering, charge correlations remain much less understood. The recent advancement of resonant X-ray scattering (RXS) has enabled the detection of charge correlations with unprecedented sensitivity. A series of RXS studies have revealed the universal existence of charge correlations in cuprate materials, which extend across a wide range of the phase diagram. Resonant inelastic X-ray scattering (RIXS) experiments further unveiled the dynamical behaviors of charge order. These findings highlight the important influence of charge correlations on the properties of cuprates. In this article, we review the latest progress in the study of charge order in cuprates using RXS, with a particular emphasis on RIXS experiments. Our focus includes recent works on dynamical charge correlations at high temperature, and uniaxial strain tuning of charge order. We discuss topics including the underlying interactions, microscopic structure and symmetry, and possible influence of charge order on both the superconducting and normal states.
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图 1 空穴型掺杂铜氧化物超导体的典型相图. 灰色长虚线标记赝能隙出现的温度$ T^* $. 灰色短虚线指示奇异金属和费米液体相的过渡区域. 灰绿色阴影示意电荷密度涨落出现的区域. AF表示反铁磁有序
Figure 1. Typical phase diagram for hole-doped cuprates. The gray dashed line marks the pseudogap onset temperature $T^* $. The gray dotted line indicates the crossover from the strange metal phase to the Fermi liquid state.
图 2 X射线散射原理示意图. (a) 散射几何. (b) 共振X射线散射过程[19]. 共振X射线将芯电子激发至价带未占据态, 使体系短暂处在具有芯空穴和激发价电子的中间态, 随后占据态的电子返回填充芯电子并发射出X射线
Figure 2. Schematic illustration of the principle of X-ray scattering. (a) Scattering geometry. (b) The process of resonant X-ray scattering[19]. Upon the absorption of an X-ray photon, a core electron is promoted to the valence state, creating a core hole and an excited valence electron in the intermediate state. In the final state, a valence electron fills the core hole, accompanied by the emission of an X-ray photon.
图 3 关联电子体系中元激发及其特征能量尺度示意图. 利用RIXS的能量分辨可以区分准弹性与非弹性散射贡献, 从而实现对电荷序的灵敏探测. 弹性散射的能量宽度取决于仪器的分辨率
Figure 3. Schematic illustration of the energy scales of elementary excitations in correlated electron systems. The energy resolution of RIXS allows for the separation of elastic from inelastic scattering processes, enhancing the sensitivity in probing charge order. The energy width of the elastic line is determined by the instrumental energy resolution.
图 4 中子散射在Nd-LSCO中观测到的条纹序衍射峰[24]. (a) CuO2平面内自旋-电荷条纹模型示意图. 圆圈表示铜的位置. 箭头代表铜上的自旋. 灰色圆圈指示空穴掺杂位置. (b) 倒空间$ (H, K, 0) $ 面内自旋和电荷序的衍射峰 (实心圆), 以及晶格的布拉格峰 (空心圆). (c, d) 中子散射测量到的(c)自旋序和(d)电荷序衍射峰. 动量空间中的扫描方向如图(b)中箭头所示
Figure 4. Neutron scattering observation of stripe-order reflections[24]. (a) Schematic illustration of the stripe pattern in the CuO2 plane. Circles denote the Cu sites. Arrows indicate the spins and gray circles indicate the doped holes. (b) Illustration of the $ (H, K, 0) $ plane in the reciprocal space, where spin and charge diffraction peaks were scanned and displayed in (c) and (d), respectively.
图 5 铜氧化物体系中电荷序(a)及低能自旋涨落或自旋序(b)波矢随掺杂演化关系. 数据来自X射线或中子散射的倒空间测量. 电荷序测量结果来自文献[26-52]. 自旋序或低能自旋涨落测量结果来自文献[24,26,29,53-62]
Figure 5. Doping evolution of wave vectors associated with (a) charge and (b) low-energy spin fluctuations for hole-doped cuprates, determined from X-ray or neutron scattering experiments. Data in (a) are taken from Refs. [26-52]. Data in (b) are taken from Refs. [24,26,29,53-62].
图 6 Bi2201体系中电荷序和赝能隙态下“费米弧”的嵌套[41]. (a) Bi2201中共振X射线测量得到的电荷序衍射峰. (b) 铜$ L_3 $边电荷序的共振行为. (c) ARPES数据显示电荷序波矢量连接费米弧尖端
Figure 6. Charge order and nesting of Fermi arc tips in the pseudogap state of Bi2201[41]. (a) Charge order peak in Bi2201 measured by REXS (b) Resonant behavior of charge order at the Cu $L_3 $-edge. (c) ARPES data showing charge ordering wave vector connects the Fermi arc-tips.
图 7 YBCO和Bi2201中电荷序的单向结构[73,74]. (a) YBCO中电荷序的动量结构示意图. 左右插图分别展示了H和K方向上电荷序峰部分方位角下的RXS动量扫描. (b) K方向电荷序峰强随方位角α的变化. 黑色横线表示峰宽$ \Delta Q $. (c) YBa2Cu3$ \text{O}_{6.51} $ (Y651), YBa2Cu3$ \text{O}_{6.67} $ (Y667) 和 YBa2Cu3$ \text{O}_{6.75} $ (Y675) 中$ \Delta Q $随方位角α变化的极坐标图[73]. (d) Bi2201的转角RIXS实验示意图. 通过改变样品面内转角ϕ和面外转角θ实现对电荷序峰从不同方位角α进行扫描. (e) 电荷序峰沿不同角度α的动量宽度表现出各向异性[74]
Figure 7. Unidirectional charge order in YBCO and Bi2201[73,74]. (a) Schematic of the momentum structure of charge order in YBCO. Left and right insets display RXS momentum scans along H and K directions at selected azimuthal angles α, respectively. (b) Intensity of the charge order peak along K as a function of azimuthal angle. Black bars indicate the peak width $ \Delta Q $. (c) Polar plots of $ \Delta Q $ versus α for YBa2Cu3$ \text{O}_{6.51} $ (Y651), YBa2Cu3$ \text{O}_{6.67} $ (Y667), and YBa2Cu3$ \text{O}_{6.75} $ (Y675)[73]. (d) Schematic of the angular-dependent RIXS experiment on Bi2201. The charge order is scanned at different azimuthal angles α by varying the in-plane and out-of-plane sample rotation angle ϕ and θ. (e) Scans at different α reveal the anisotropic structure of the charge order peaks[74].
图 8 电荷序的轨道对称性[75,76]. (a) 共振X射线散射实验及电荷序不同轨道对称性示意图. (b) LBCO[75] 与 (c) YBCO[76]中不同极化入射光下电荷序强度的方位角依赖及模型比较[74]
Figure 8. Orbital symmetry of charge order[75,76]. (a) Schematics of the RXS geometry and different orbital symmetries of charge order. (b, c) Intensity of the charge order peak as a function of azimuthal angle ϕ using different incident light polarizations for (b) LBCO[75] and (c) YBCO[76].
图 9 条纹序铜氧材料的高温电荷关联[28]. (a), (b) RIXS测量得到的$ 1/8 $空穴掺杂Eu-LSCO中不同温度下的电荷序峰. (c) Eu-LSCO中电荷序峰强的温度依赖. 在20 K以上峰强正比于$ T^{-2} $ (灰色虚线). 插图中$ T_s $和$ T^{*} $ 分别代表低温四方结构相和赝能隙态的起始温度. (d) 镧214体系中条纹电荷序峰强与关联长度的关系
Figure 9. High-temperature charge correlations in stripe-ordered cuprates[28]. (a), (b) Charge order peak in Eu-LSCO at various temperatures measured by RIXS. (c) Temperature evolution of the charge order peak amplitude, which decays roughly as $T^{-2} $ (gray dashed line). (d) Relationship between charge order peak amplitude and correlation length in La-214 compounds.
图 10 镧系铜氧材料中的短程电荷关联[34]. (a) 不同能量分辨率RIXS对LSCO $ x = 0.145 $测量得到的能谱. 下图中更高分辨率使得声子和电荷序的贡献得以区分. (b), (c) LSCO $ x = 0.145 $ 和 $ x = 0.16 $中电荷序的温度依赖
Figure 10. Short-range charge correlations in La-based cuprates[34]. (a), (b) RIXS spectra on LSCO $x = 0.145$ obtained with different energy resolutions. The improved resolution in (b) allows for resolving the phonon branch. (c), (d) Temperature dependence of charge order in LSCO $x = 0.145$ and $x = 0.16$.
图 11 电荷序有关的RIXS声子异常[19,111]. (a), (b) Eu-LSCO中15 K和200 K时电荷序波矢附近的RIXS能谱. 低温下零能量附近的信号峰来自电荷序. 黑色圆圈标记了键伸缩(bond-stretching)声子位置. (c)—(f) Eu-LSCO中电荷序波矢附近的RIXS声子软化及强度增强[19]. (g), (h) Bi2201 中电荷序波矢附近的铜L边和氧K边RIXS能谱. 氧K边具有更高能量分辨率, 因此可以分辨键伸缩和更低能量的键屈曲(bond-buckling)声子, 但是能覆盖的动量范围更小. (i)—(l) Bi2201中的RIXS声子软化及强度异常[111]
Figure 11. RIXS phonon anomaly associated with charge order[19,111]. (a), (b) RIXS spectra around the charge ordering wave vector in Eu-LSCO at 15 K and 200 K. The intense elastic peak at low temperature originates from charge order. Black dots mark the bond-stretching phonon position. (c)–(f) RIXS phonon energy softening and intensity enhancement near the charge ordering wave vector in Eu-LSCO[19]. (g), (h) Cu L-edge and O K-edge RIXS spectra around the charge ordering wave vector in Bi2201. (i)–(l) RIXS phonon softening and intensity anomaly in Bi2201[111].
图 12 电荷密度涨落[112,113]. (a) YBCO和Bi2212中$ (H, 0) $和$ (H, H) $方向RIXS信号作差. (b) 总结YBCO, NBCO及Bi2212体系中RIXS结果得到的电荷密度涨落的特征温度与空穴掺杂量的关系[112]. (c) LSCO $ x = 0.15 $中电荷序波矢附近不同温度下的RIXS 能谱. $ 100 $ meV以下可以分辨三支声子及更低能量的电荷激发模式. (d) LSCO $ x = 0.15 $中电荷激发的强度、特征能量及寿命倒数的温度依赖[113]
Figure 12. Charge density fluctuations[112,113]. (a) Intensity difference between RIXS spectra taken along $ (H, 0) $ and $ (H, H) $ directions in YBCO and Bi2212. (b) Doping evolution of the characteristic temperature of charge density fluctuations obtained from RIXS data on YBCO, NBCO, and Bi2212[112]. (c) RIXS spectra around the charge ordering wave vector in LSCO $ x = 0.15 $ at different temperatures. (d) The intensity, characteristic energy, and inverse lifetime of the charge excitation as a function of temperature in in LSCO $ x = 0.15 $[113].
图 14 YBCO中二维电荷序的应力响应[124]. (a) H 和 (b) K 方向电荷序对沿a方向施加应力的响应. (c) H 和 (d) K 方向电荷序对沿b方向施加应力的响应. (e), (f) 单轴应力下电荷序畴的实空间示意图
Figure 14. Strain response of the 2D charge order in YBCO[124]. (a), (b) Evolution of the charge order peaks along (a) H and (b) K directions under the a-axis compression. (c), (d) The charge order peaks along (c) H and (d) K directions under the b-axis compression. (e), (f) Schematics of the real-space charge order domains under uniaxial strain.
图 15 LSCO中应力引发的条纹序旋转[126]. (a) 镧214铜氧材料结构示意图[28]. 左图为高温四方 (HTT) 相结构. 右图示意LTO与LTT相内铜氧八面体的旋转畸变. (b) 常压及单轴压下LSCO中电荷序的动量结构示意图. 橙色和红色实线分别示意(c)和(d)中动量空间扫描方向. (c), (d) 电荷序峰的(c)横向与(d)纵向动量依赖随外加应力的变化. (e) 电荷序关联长度和 (f) 非公度波矢随外加应力的变化关系
Figure 15. Strain-induced stripe order rotation in LSCO[126]. (a) Schematic illustration of the crystal structure of La-based cuprates[28]. Left: structure of the HTT phase; right: distortion of the CuO6 octahedra in the LTO and LTT phases. (b) Momentum structure of the charge order peak in LSCO under ambient and compressive strain. The orange and red solid lines indicate the momentum scan directions in (c) and (d), respectively. (c), (d) Strain evolution of the (c) transverse and (d) longitudinal momentum scans of charge order peak. (e) Correlation length and (f) incommensurability of charge order as a function of applied strain.
图 16 LTT结构中条纹电荷序对单轴应力的响应[130,131]. (a), (b) 对Nd-LSCO沿面内Cu-O键方向施加拉伸应力时LTT结构相变及电荷序的变化[130]. (c)—(f) 沿面内Cu-O键方向施加压缩应力时Nd-LSCO中电荷序的变化[131]
Figure 16. Response of charge-stripe order to uniaxial strain applied in the LTT structure[130,131]. (a), (b) Evolution of charge order and LTT phase in Nd-LSCO upon the application of tensile strain along the in-plane Cu-O bond direction[130]. (c)–(f) Response of charge order in Nd-LSCO to compressive strain applied along the Cu-O bond direction[131].
图 17 LSCO中电荷序及低能激发的应力响应[127]. (a) 代表性氧K边RIXS谱及拟合. (b) 未施加应力时的动量依赖RIXS激发谱. (c), (d) 外加面内Cu-O键方向压缩应力时沿(c) H (垂直于应力) 及(d) K (平行于应力) 方向的动量依赖RIXS激发谱. (b)—(d)中已减去拟合得到的弹性散射信号. (e) 键伸缩声子和低能电荷激发模式的色散关系的应力依赖. (f)—(h) 电荷序峰及低能电荷激发强度的应力依赖
Figure 17. Strain response of charge order and low-energy excitations in LSCO[127]. (a) Representative RIXS spectrum fitting. (b) Momentum-dependent RIXS excitations without strain application. (c), (d) RIXS excitation spectra along (c) H (perpendicular to strain) and K (parallel to strain) directions under compressive strain applied along the in-plane Cu-O bond direction. The fitted elastic signal has been removed in (b)–(d). (e) Strain dependence of the bond-stretching phonon and low-energy charge excitation dispersion. (f)–(h) Strain dependence of the charge order and charge excitation intensities.
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