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In the 38 years since the discovery of cuprate superconductors, the theoretical mechanism of high-temperature superconductivity remains unresolved. Recent experimental progress has focused on exploring microscopic mechanisms by using novel characterization techniques. The development of synchrotron radiation has driven significant progress in spectroscopic methods. Resonant inelastic X-ray scattering (RIXS), based on synchrotron radiation, has been widely used to study cuprate superconductors due to its ability to perform bulk measurements, provide energy-momentum resolution, and directly probe various elemental excitations. The RIXS can measure phonons, which bind Cooper pairs in the BCS theory, and magnetic fluctuations and competing orders predicted by the Hubbard model in strongly correlated systems, allowing for the study of their interrelationships. This paper reviews the progress in using RIXS to measure charge density waves and related low-energy excitations, including phonon anomalies, in cuprate superconductors. It also examines the relationship between magnetic excitation and the highest superconducting transition temperature, and provides prospects for future research directions and challenges.
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Keywords:
- resonant inelastic X-ray scattering /
- cuprate superconductor /
- charge density wave /
- phonon /
- magnetic excitation
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图 1 共振X射线散射过程与材料内的元激发 (a) 共振的X射线使内层电子跃迁至导带, 另一个占据态的外层电子返回内层能级, 材料末态为激发态[2]; (b) RIXS谱示例, 包含弹性峰、声子、磁激发和轨道激发等特征
Figure 1. Resonant X-ray scattering and the excitations in the material: (a) The incoming X-rays excite an electron from a core level into the empty valence band, while another electron from the occupied states emits a photon and returns back to the core layer energy level[2]; (b) example of a RIXS spectrum shows the features of an elastic peak, phonon, paramagnon, and dd excitations.
图 2 过掺杂LSCO (x = 0.45)薄膜中观测到电荷序关联[17] (a) 沿着(H, 0)正负两个方向和(H, H)方向的弹性峰积分强度的动量依赖, 入射光为σ偏振. 红色和蓝色曲线为使用洛伦兹函数和一个多项式背底进行拟合的结果. (b) 在33 K和250 K两个温度下测量的入射光为σ和π偏振依赖. (c) 在ZRS峰附近的X射线吸收谱, 入射光垂直样品表面且为σ偏振. 蓝色虚线和红色虚线分别为弹性峰和轨道激发积分强度的入射光能量依赖
Figure 2. Charge order correlation in overdoped LSCO (x = 0.45) film[17]: (a) Integrated intensity of elastic peaks for positive and negative (H, 0) and (H, H) directions, using σ-polarization. Red and blue curves are Lorentzian peak fits to the data with a polynomial background. (b) Polarization measurements with σ-and π-polarized X-ray, collected at 33 K and 250 K. (c) XAS spectra near the ZRS absorption peak with σ polarization at normal incidence. The blue dashed line and the red dashed line correspond to the elastic peak and the orbital excitation, respectively.
图 3 LSCO (x = 0.45)薄膜中电荷序关联的REIXS结果[17] (a)入射光为Cu L边和O K边时, 在(0, –K)方向均观测到电荷序; (b)电荷序峰的偏振依赖, 入射能量为930 eV, L = 1.1 r.l.u.; (c)在Cu L边附近改变入射能量的测量; (d)电荷序的L依赖, 入射光为σ偏振, 930 eV入射能量可实现的L范围为[1.1, 1.8] r.l.u.
Figure 3. REIXS studies of charge order correlation in LSCO (x = 0.45) film[17]: (a) Observation of charge order correlation along (0, –K) direction at both Cu L edge and O K edge; (b) polarization dependence of the charge order peak, collected at 930 eV and L = 1.1 r.l.u.; (c) detuning measurements near the Cu L edge; (d) L dependence of charge order correlation within the accessible range of [1.1, 1.8] r.l.u. at 930 eV, collected with σ polarization.
图 4 过掺杂LSCO薄膜中电荷序的掺杂依赖与拓展相图[17] (a)—(c)在LSCO (x = 0.35, 0.45, 0.6)中由Cu L边REIXS测量的电荷序, L被固定在1.1 r.l.u., 电荷序在300 K未发生变化. (d)铜氧化合物的电荷序拓展相图, 显示了由临界温度$ {T}_{\mathrm{c }}$定义的超导拱形, 由${T}_{\mathrm{N}} $定义的反铁磁相[30], 由Nernst系数定义的赝能隙[11], 欠掺杂区域的电荷序和电荷涨落[5,19,29]. (e) LSCO中电荷序波矢的掺杂依赖和由线性响应理论计算得到的嵌套波矢
Figure 4. Doping dependence of charge order correlation in overdoped LSCO film and the extended phase diagram[17]: (a)–(c) Charge order peak profiles measured by Cu L edge REIXS in LSCO with (x = 0.35, 0.45, 0.6), respectively. L is fixed at 1.1 r.l.u. and the peak is nearly temperature independent up to 300 K. (d) The extended CO phase diagram of cuprates, showing superconducting dome defined by $ {T}_{\mathrm{c}} $, antiferromagnetism (AFM) defined by $ {T}_{\mathrm{N}} $[30], pseudogap determined from the Nernst coefficient[11], underdoped charge order and charge fluctuation[5,19,29], and overdoped charge order. (e) Doping dependence of the CO wave vector in LSCO and nesting vector obtained from Lindhard function.
图 5 过掺杂Bi2201单晶中的电荷序[31] (a)在过掺杂Tc为11 K的Bi2201单晶中, 观测到了沿着$ \left(H, 0\right) $方向对称, 波矢约为0.14 r.l.u.的电荷序; (b)电荷序在Cu L边具有共振行为; (c)电荷序的积分强度随温度的变化; (d)费米面随掺杂的变化, 在P ≈ 0.22时出现范霍夫奇点; (e)电荷序的波矢、关联长度和强度随掺杂的变化
Figure 5. Charge order in overdoped Bi2201 single crystal[31]: (a) Charge order with a wave vector of 0.14 r.l.u. along $ \left(H, 0\right) $ direction in Tc = 11 K Bi2201 single crystal; (b) charge order resonant at Cu L edge; (c) intensity of charge order is temperature independent; (d) doping dependence of Fermi surface, with the Lifshitz transition at P ≈ 0.22; (e) doping dependence of the wave vector, coherence length and intensity of charge order.
图 6 La-Bi2201 x = 0.6 (UD23 K)单晶中电荷密度波相干长度各向异性[34] (a)实验构型图, 从不同方向切过电荷密度波卫星峰而获得特定方向的相干长度; (b), (c) Cu L边与O K边在(0.25, 0)的RIXS谱图, 可通过拟合提取出弹性峰强度; (d), (e)沿$ \left(H, 0\right) $方向经过(0.25, 0)所得弹性峰强度随动量的变化, 可以通过二次背景和高斯峰很好的拟合, 可见两个共振边给出一致的电荷密度波峰; (f), (g)沿着不同方向经过电荷密度峰所得结果, 可见峰宽具有各向异性, 且在传播方向上更长程; (h)电荷密度波峰沿不同方向的半高全宽, H-CDW和K-CDW分别沿着传播方向更锐, 呈现90°对称, 且不同共振能量给出结果一致
Figure 6. Anisotropic charge density wave (CDW) correlation in La-Bi2201 (x = 0.6 UD23 K) single crystal[34]: (a) Schematic illustration of the experiment. Charge density wave coherence length in a specific direction can be obtained by cutting through the charge density wave peaks along that direction. (b), (c) RIXS spectra of Cu L and O K edge at (0.25, 0), with elastic peak obtained by fitting. (d), (e) Momentum dependence of the intensity of elastic peak along $ \left(H, 0\right) $ direction through (0.25, 0), which can be well fitted by the Gaussian peaks and quadratic background. (f), (g) Results for different directions show anisotropic peak width and longer-range ordering in the direction of the wave vector. (h) FWHM of the charge density wave along different directions, H-CDW and K-CDW, respectively, are sharper along the propagation direction. And different resonant energies give consistent results.
图 7 电荷密度波的基本单元对称性与其峰型形状对应关系[34] (a)—(d)对应具有二重旋转对称的条纹序或四重旋转对称性棋盘序, 实空间相干长度为各向同性或各向异性下的实空间模拟图像, 其中插图为基本单元的模式示意图; (e)—(h)对实空间图进行傅里叶变换得到的动量空间形状
Figure 7. Correspondence between the symmetry of charge density wave and the shape of peaks in reciprocal space[34]: (a)–(d) Correspond to the real-space map under isotropy or anisotropy with a stripe modulation with twofold or fourfold rotational symmetry, and the illustration show the model diagram of basic unit; (e)–(h) correspond to charge density wave peak patterns in reciprocal space, obtained by direct Fourier transformation of panels (a)–(d), respectively.
图 8 NBCO中的电荷密度涨落与相图[19] (a) 250 K下弹性峰沿着$ \left(H, 0\right) $和$ \left(H, H\right) $方向随动量的变化; (b) 60 K下弹性峰沿着$ \left(H, 0\right) $和$ \left(H, H\right) $方向随动量的变化; (c) NBCO的相图; (d)电荷密度涨落具有有限能量
Figure 8. Charge density fluctuation (CDF) and phase diagram of NBCO[19]: (a) Momentum dependence of elastic peak intensity along $ \left(H, 0\right) $ and $ \left(H, H\right) $ directions at 250 K; (b) momentum dependence of elastic peak intensity along $ \left(H, 0\right) $ and $ \left(H, H\right) $ direction at 60 K; (c) phase diagram of NBCO; (d) finite excitation energy of CDF.
图 9 不同铜氧化合物中电荷密度涨落的掺杂依赖[42] (a) Bi2212的RIXS谱与低能区间拟合, 按照能量由低到高依次为弹性峰、电荷密度涨落、Bond-Stretching声子、多声子与磁激发; (b)电荷密度涨落的强度随掺杂的变化; (c)电荷密度涨落的能量随掺杂的变化; (d)电荷密度涨落的能量对应的温度与线性电阻的温度区间一致
Figure 9. Doping dependence of charge density fluctuation (CDF) in different cuprate families[42]: (a) RIXS spectrum of Bi2212 fitted with elastic peak, CDF peak, Bond-Stretching phonon, multi-phonon, and magnon; (b) doping dependence of the intensity of CDF; (c) doping dependence of the energy of CDF; (d) the energy of the CDF corresponds to a temperature that coincides with the temperature range of the linear resistance.
图 10 RIXS在铜氧超导体中观测到的声子与电声耦合强度的测量方法 (a) Bond-Buckling (BB) 声子和Bond-Stretching (BS) 声子的示意图, 均为$ \text{CuO}_2 $面的振动模, 在RIXS实验中被广泛观测到[45–47]; (b)在Cu L附近测量的入射X光能量依赖的RIXS谱, 与拟合出的BS声子(红色阴影部分), 其余虚线部分包括弹性峰、磁激发和轨道激发[49]; (c)利用声子的共振依赖估算电声耦合强度[49]
Figure 10. Electron-phonon coupling strength measured by RIXS in cuprate: (a) Schematic representation of Bond-Buckling (BB) phonons and Bond-Stretching (BS) phonons, both of which are vibrational modes of the $ \text{CuO}_2 $-plane and have been widely observed in the RIXS experiment[45–47]; (b) RIXS spectra near Cu L edge. The red shaded peak is the BS phonon, and the remaining dashed lines are the elastic peaks, magnon and orbital excitation[49]; (c) estimation of electron-phonon coupling strength using resonant behavior of phonon[49].
图 11 电声耦合的掺杂依赖[51] (a)不同掺杂的Bi2212的O K边X射线吸收谱; (b) OP Bi2212中入射X光能量依赖的RIXS测量; (c)低能区域利用双声子模型进行拟合, 获得电声耦合强度(M); (d)电声耦合的掺杂依赖
Figure 11. Doping dependence of electron-phonon coupling strength[51]: (a) O K edge XAS for Bi2212 with different dopings; (b) RIXS spectra of OP Bi2212 with different incidence photon energies; (c) electron-phonon coupling strength (M) was obtained by fitting the RIXS spectrum with a model of two phonons; (d) doping dependence of electron-phonon coupling strength.
图 12 电荷密度波与声子异常[47] (a)低温下LSCO单晶的动量依赖RIXS谱; (b)扣除弹性峰以后的低能谱, 声子与电荷激发相干涉; (c)通过拟合得出的声子能量色散关系, 四个声子能量由低到高分别是La/Sr声学声子, B1g, A1g和Breathing声子; (d)电荷密度波随温度的变化
Figure 12. Charge density wave and phonon anomaly[47]: (a) Momentum dependence of RIXS spectra of LSCO at 24 K; (b) RIXS map obtained by subtracting the elastic peak from raw data, where the phonon interferes with the charge order; (c) phonon dispersion obtained by fitting. The four phonon energies from lowest to highest are the La/Sr acoustic phonon, B1g, A1g, and the breathing phonon; (d) temperature dependence of charge density wave peak.
图 13 LSCO (x = 0.45)薄膜中各向异性的声子强度和有色散的电荷激发[17] (a), (c)弹性强度二维图和(b), (d)非弹性强度二维图来比较沿着$ \left(H, 0\right) $和$ \left(H, H\right) $方向的电荷序和声子; (e)—(g)沿着$ \left(H, 0\right) $和$ \left(H, H\right) $两个方向BB声子、La/Sr的声学支声子和电荷序的强度积分
Figure 13. Anisotropic momentum dependence of phonon intensity and dispersive charge order excitations in LSCO (x = 0.45) film[17]: (a), (c) The elastic intensity map and (b), (d) the inelastic RIXS intensity map for visualizing the charge order correlation and phonon branches along $ \left(H, 0\right) $ direction and $ \left(H, H\right) $ direction, respectively; (e)–(g) integrated intensity for buckling phonon, acoustic phonon, and charge order correlation, respectively, along both directions.
图 14 铜氧化合物中具有色散的磁激发与理论模型[59] (a) 不同铜氧化合物中沿着$ \left(H, 0\right) $和$ \left(H, H\right) $方向的磁激发RIXS谱(CCO, CaCuO2); (b)最近邻Heisenberg模型(点线)、单带Hubbard模型(虚线)和含有四个最近邻交换参数的线性自旋波Heisenberg模型(实线)对于磁激发色散曲线的拟合情况. 右边为不同模型的示意图
Figure 14. Dispersive magnon in cuprate and the theoretical models[59]: (a) RIXS spectra along $ \left(H, 0\right) $ direction and $ \left(H, H\right) $ direction for different cuprate families (CCO, CaCuO2); (b) fits of the nearest-neighbor Heisenberg model (dotted line), the single-band Hubbard model (dashed line), and the linear spin-wave Heisenberg model containing four nearest-neighbor exchange parameters (solid line) to the magnon dispersion curve. Schematic plot of the different models are shown on the right.
图 15 磁激发色散与轨道激发和交换作用常数的关联[59] (a)不同铜氧化合物中dd激发谱; (b) $ {r}_J $和$ \text{d}_{z^2} $能量与$ \Delta E_{{\mathrm{MBZB}}} $的关联
Figure 15. Correlations of the spin-wave dispersions, dd excitations and exchange range parameters[59]: (a) The dd excitations of different cuprate families; (b) relationships between $ {r}_J $, the energy of $ \text{d}_{z^2} $ and $ \Delta E_{{\mathrm{MBZB}}} $.
图 16 磁激发随层数的变化[62] (a)在Hg1201和Hg1212中沿着$ \left(H, H\right) $和$ \left(H, 0\right) $方向的磁激发; (b)两个样品的低温减高温的Raman谱测量结果; (c)不同铜氧化合物的$ {T}_{{\mathrm{c, max}}} $与交换作用J的关系
Figure 16. Magnetic excitations in different layers cuprates[62]: (a) Magnetic exciations of Hg1201 and Hg1212 along $ \left(H, H\right) $ direction and $ \left(H, 0\right) $ direction; (b) high and low temperature differences in Raman spectra of the two samples; (c) relationship of $ {T}_{{\mathrm{c, max}}} $ and exchange interaction J for different cuprate families.
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