Resonant inelastic X-ray scattering applications in quantum materials

Zhou Ke-Jin

doi:10.7498/aps.73.20241009

Abstract

The essence of quantum materials lies in the intricate coupling among charge, spin, orbital and lattice degrees of freedom. Although X-ray photoemission spectroscopy and inelastic neutron scattering have advantages in detecting fermionic single-particle spectral function and bosonic spin excitations in quantum materials, respectively, probing other bosonic collective excitations especially their coupling is not possible until the establishment of the advanced resonant inelastic X-ray scattering (RIXS). In the past decades, RIXS has flourished with continuously improved energy resolution which made a paradigm shift from measuring crystal-field splitting and the charge-transfer excitation, to probing collective excitations and the order parameters of all degrees of freedom. This review paper summarises the latest research progress of quantum materials studied by the soft X-ray RIXS. For instance, three-dimensional collective charge excitations, plasmons, were discovered experimentally by RIXS in both electron and hole doped cuprate superconductors. The collective orbital excitations and excitons were found in copper and nickel based quantum materials. For the newly discovered nickelate superconductors, RIXS has made substantial contributions to characterising their electronic and magnetic excitations and the related ordering phenomena critical for an in-depth understanding of the underlying superconducting mechanicsm. The RIXS is a unique tool in probing the higher-order spin excitations in quantum materials due to the strong spin-orbit coupling and the core-valence exchange interaction. The RIXS is also found to be superior in probing the Stoner magnetic excitations in magnetic metals and topological magnetic materials. Finally, the development of RIXS technology in Chinese large-scale research facilities is briefly prospected.

Keywords:

Authors and contacts

Zhou Ke-Jin

Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom

Corresponding author: Zhou Ke-Jin, kejin.zhou@diamond.ac.uk

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图 1 空穴掺杂铜氧超导体La_1.84Sr_0.16CuO₄和Bi₂Sr_1.6La_0.4CuO_6+d中三维类声学支等离子激发^[32]　(a)单层铜氧化物的结构示意图及RIXS散射几何结构; (b)单层铜氧化物的声学支等离子激发三维色散关系示意图; (c)空穴掺杂铜氧超导体La_1.84Sr_0.16CuO₄中类声学支等离子激发在l = 1.0时沿着h方向的色散关系, 其中绿色圆点和红色方块分别代表着等离子激发谱和双磁子激发谱的峰值; (d), (e)空穴掺杂铜氧超导体La_1.84Sr_0.16CuO₄和Bi₂Sr_1.6La_0.4CuO_6+d中类声学支等离子激发在h = 0.05时沿着l方向的色散关系, 其中绿色圆点和红色虚线分别代表等离子激发谱和双磁子激发谱的峰值

Figure 1. Three-dimensional acoustic-like plasmons in hole-doped cuprate superconductors La_1.84Sr_0.16CuO₄ and Bi₂Sr_1.6La_0.4CuO_6+d^[32]: (a) Schematic crystal structure of a single-layered cuprate and the experimental geometry of RIXS; (b) the schematic view of three-dimensional acoustic-like plasmons in (E, q)-space in a single-layered cuprate superconductor; (c) the acoustic-like plasmon dispersion in the hole-doped La_1.84Sr_0.16CuO₄ along h direction at a fixed l = 1.0, where green dots and red squares represent the peak positions of plasmons and bi-magnons respectively; (d), (e) the acoustic-like plasmon dispersion in the hole-doped La_1.84Sr_0.16CuO₄ and Bi₂Sr_1.6La_0.4CuO_6+d along l direction at a fixed h = 0.05, green dots and red dashed lines represent the peak positions of plasmons and bi-magnons respectively.

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图 2 低维铜氧化物中的集体轨道激发　(a)一维铜氧化物Sr₂CuO₃的RIXS实验谱, 0—0.8 eV能量范围为双自旋子连续激发, 2—3 eV为轨道子激发^[13]; (b) 二维铜氧化物CaCuO₂的RIXS实验谱, d_xy和d_xz/d_yz轨道激发呈现色散行为^[40]; (c) 二维铜氧化物CaCuO₂的RIXS实验谱(图(b))二阶求导结果, d_xy表现出50 meV左右的色散^[40]; (d) 二维铜晶格中纯粹轨道超交换机制^[40]

Figure 2. Collective orbital excitations in the low-dimensional copper-oxides: (a) Representative RIXS spectrum of a one-dimensional cuprate Sr₂CuO₃, the mode within 0–0.8 eV is the two-spinon continuum, the mode within 2–3 eV is the orbiton^[13]; (b) RIXS spectra of two-dimensional cuprate CaCuO₂ in which d_xyand d_xz/d_yz orbitals disperse in (E, q) space^[40]; (c) the second derivative of CaCuO₂ RIXS spectra shown in panel (b), the bandwidth of the d_xy orbital dispersion is about 50 meV^[40]; (d) pure orbital superexchange mechanism in two-dimensional copper lattice^[40].

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图 3 NiPS₃和NiBr₂中集体激子激发　(a) NiPS₃中自旋单重态激子激发¹A_1g沿着H方向的色散行为^[42]; (b) NiPS₃中自旋单重态激子激发¹A_1g沿着K方向的色散行为^[42]; (c) NiBr₂中自旋单重态激子激发¹A_1g沿着H方向的色散行为^[43]

Figure 3. Collective exciton excitations in NiPS₃ and NiBr₂: (a) Dispersion behaviour of the spin-singlet exciton ¹A_1g in NiPS₃ along H direction^[42]; (b) the dispersion behaviour of the spin-singlet exciton ¹A_1g in NiPS₃ along K direction^[42]; (c) the dispersion behaviour of the spin-singlet exciton ¹A_1g in NiBr₂ along H direction^[43].

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图 4 NdNiO₂的电子结构　(a) NdNiO₂的费米能级附近态密度示意图^[45]; (b) NdNiO₂的Ni L₃-RIXS实验谱, 黑线为Ni L₃-XAS实验谱, 黑色虚线长方形标出Ni 3d- Nd 5d杂化dd峰^[45]; (c) Nd_1–xSr_xNiO₂ (x = 0, 0.225) Ni L₃-XAS实验结果^[48]; (d) Nd_1–xSr_xNiO₂自旋单重态和自旋三重态示意图; (e) Nd_1–xSr_xNiO₂自旋单重态和自旋三重态理论计算结果^[48]

Figure 4. Electronic structure of NdNiO₂: (a) Schematic view of the density of states near Fermi level in NdNiO₂^[45]; (b) Ni L₃- energy-dependent RIXS spectra. The black solid line represents Ni L₃-XAS spectrum, and the black dashed rectangle highlights the hybridized dd peak between Ni 3d and Nd 5d orbitals^[45]; (c) the Ni L₃-XAS experimental spectra of Nd_1–xSr_xNiO₂ (x = 0, 0.225)^[48]; (d) schematic view of the spin-singlet and spin-triplet state in Nd_1–xSr_xNiO₂; (e) the theoretical calculation of the spin-singlet and spin-triplet XAS in Nd_1–xSr_xNiO₂^[48].

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图 5 (a) NdNiO₂和(b) La₂CuO₄中集体磁激发沿着动量空间高对称线的色散关系, 其中红色和蓝色实心圆点为磁激发拟合峰值, 虚线为线性自旋波拟合结果, 磁激发拟合峰值数据来源于文献[52,54]

Figure 5. (a) NdNiO₂ and (b) La₂CuO₄ magnon dispersion along the high-symmetry direction in the momentum space. The red dots and blue dots represent the fitted peak positions of magnons. Dotted lines stand for the fitting result of the linear spin-wave theory^[52,54] .

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图 6 La₃Ni₂O₇的XAS和RIXS结果^[68]　(a), (b) La₃Ni₂O₇和参考样品(NdNiO₃, NiO)O K边和Ni L₃边的XAS谱对比, 所有的吸收谱均采用了σ线偏振入射光; (c), (d) Ni L₃边入射能量依赖的RIXS谱, 其中白色实线为对应线偏振模式下的XAS吸收谱; (e) Ni L₃边RIXS集体磁激发沿着动量空间高对称线的色散关系, 插图中的红色箭头标出了动量空间的高对称线, 图中红色实心圆点为磁激发拟合峰值

Figure 6. Experimental XAS and RIXS spectra of La₃Ni₂O₇^[68]: (a), (b) The O K-XAS and Ni L₃-RIXS of La₃Ni₂O₇ as well as those from the references, all XAS spectra were taken using polarized X-rays; (c), (d) the Ni L₃-edge energy-dependent RIXS spectra using σ and πpolarization, the white solid lines represent the XAS spectra taken by the corresponding linear polarization in each configuration; (e) Ni L₃-RIXS magnon dispersion along the high-symmetry direction in the momentum space, the red arrows in the inset represent the scanning path, the red dots show the fitted peak positions of the magnons^[68].

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图 7 La₃Ni₂O₇的磁结构模型及自旋密度波^[68]　(a)自旋-电荷条纹相模型; (b)双自旋条纹相模型, 为简化模型, 示意图只显示了镍离子, 实线方格子为面内四方相晶格, 灰色立方体代表NiO₆八面体, 蓝色、红色、黑色圆点分别代表自旋向上Ni²⁺、自旋向下Ni²⁺以及Ni³⁺. 图中显示出的J₁S, J₂S, J_zS(垂直于双NiO₂面的超交换作用)为基于各个模型并结合RIXS磁激发的最佳拟合结果; (c)自旋密度波倒空间沿着(H, H)方向在不同线偏振入射光下的散射结果; (d)自旋密度波强度的温度依赖关系; (e)自旋密度波面内相干长度的温度依赖关系

Figure 7. Magnetic structure of La₃Ni₂O₇ and the spin-density-wave (SDW) results^[68]: (a) Spin-charge stripe (Stripe-1); (b) double spin stripe (Stripe-2), to simplify the models, only Ni ions are shown in the schematic view, solid squares represent the in-plane tetragonal lattice, grey shaded cubes stand for NiO₆ octahedra, blue, red, and black dots represent spin-up Ni²⁺, spin-down Ni²⁺, and spinless Ni³⁺, respectively, J₁S, J₂S, J_zS which is the superexchange interaction perpendicular to the layer are also shown in the figure following the optimized fitting to the experimental data; (c) SDW along (H, H) direction probed by σ and πlinear polarized X-rays; (d) temperature dependence of SDW order parameter; (e) the correlation length of CDW as function of temperature.

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图 8 SrCuO₂一维自旋链RIXS磁激发^[72]　(a) SrCuO₂一维自旋链的结构及RIXS实验示意图, 入射光采用π线偏振光, RIXS散射角固定在146°; (b1), (b2) 氧K边及铜 L₃边RIXS实验谱, (b3), (b4)氧K边及铜 L₃边基于t-J哈密顿量和RIXS Kramer-Heisenberg散射截面理论计算结果; (c1), (c2)铜 L₃边基于t-J哈密顿量和RIXS Kramer-Heisenberg散射截面理论得出的SC及NSC两个通道的磁激发结果, (c3)对应的为多重磁激发积分谱权重(图(c1), (c2)红色虚线部分为积分区域), (c4) NSC和SC理论多重激发谱加权平均结果与实验对比; (d)磁激发谱在UCL近似下展开后其一阶项(d1), (d2)和二阶项(d3), (d4)结果

Figure 8. RIXS magnon results in the one-dimensional SrCuO₂^[72]: (a) Schematic view of the lattice structure of SrCuO₂ and RIXS experimental geometry. Incident X-rays are πlinearly polarized. The scattering angle is fixed to 146°; (b1), (b2) O K- and Cu L₃- RIXS experimental spectra, (b3), (b4) theoretical calculations based on the Kramer-Heisenberg formula and t-J Hamiltonian; (c1), (c2) spin-conserving and Non-spin-conserving channels of calculated Cu L₃- RIXS spectra based on the Kramer-Heisenberg formula and t-J Hamiltonian, (c3) the spectral weight of multi-spinons in both SC and NSC channels integrated over the dotted red regions, (c4) the optimized sum of spectra weight with respect to the experimental data; (d) the first-order (d1), (d2) and the second-order (d3), (d4) component of the calculated spectra under the UCL approximation.

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图 9 Y₂BaNiO₅一维自旋链RIXS磁激发^[73]　(a) Y₂BaNiO₅一维自旋链沿着c轴链方向的Ni L₃边RIXS磁激发实验谱, 入射光采用π线偏振光, RIXS散射角固定在154°, 虚线为偶极磁激发色散关系$\omega^{2}(q_{//})=\varDelta_{\rm{H}}^{2}+v^{2} \sin ^{2} q_{//}+\alpha^{2} \cos ^{2}(q_{//}/2) $, 其中J = 24 meV, Δ_H = 0.41 J, v = 2.55 J, α = 1.1 J; (b)—(d)分别为基于密度矩阵重整化群理论计算得出的偶极磁激发ΔS₁, 四极磁激发 ΔS₂, 及自旋翻转为零多重磁激发ΔS₀, 其中(c)中的蓝色虚线标出了两个不相干的三重态激发的连续谱边界; (e) RIXS偶极磁激发和四极磁激发的示意图; (f) 一维Ni²⁺反铁磁自旋链的基态和偶极、四极、自旋翻转为零多重磁激发的示意图

Figure 9. Ni L₃-RIXS magnon in one-dimensional spin-chain compound Y₂BaNiO₅^[73]: (a) Momentum-dependent RIXS experimental spectra along the c-axis chain direction, incident X-rays are πpolarized, RIXS scattering angle is fixed at 154°, dotted line represents the dipolar magnon with a dispersion relationship $\omega^{2}(q_{/ /})=\varDelta_{\rm{H}}^{2}+v^{2} \sin ^{2} q_{/ /}+\alpha^{2} \cos ^{2}(q_{/ /}/2) $ in which J = 24 meV, Δ_H = 0.41 J, v = 2.55 J, α = 1.1 J; (b)–(d) the calculated dipolar-magnon spectra ΔS₁, quadrupolar-magnon spectra ΔS₂, and the multi-magnon spectra ΔS₀, respectively, the blue dashed lines highlight the boundary of the two independent spin-triplet excitations; (e) the schematic view of the creation of dipolar and quadrupolar magnons in the RIXS process; (f) the schematic view of the ground state, the dipolar-magnon, the quadrupolar magnon, and the multiple magnon with net zero spin-flips in the one-dimensional Ni²⁺ antiferromagnetic spin chain.

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图 10 铁磁金属LaCo₂P₂和CeCo₂P₂中Co L₃边RIXS磁激发^[78]　(a) 铁磁金属LaCo₂P₂和CeCo₂P₂的晶体结构示意图, 紫色箭头为自旋方向; (b) 铁磁金属中自旋波与自旋翻转的斯托纳激发在能量-动量空间的色散关系, 当自旋波与斯托纳激发相互作用时发生展宽; (c) LaCo₂P₂和CeCo₂P₂有代表性的RIXS磁激发实验结果

Figure 10. Co L₃-RIXS magnon experimental spectra in ferromagnetic LaCo₂P₂ and CeCo₂P₂^[78]: (a) Schematic view of the crystal structure of ferromagnetic LaCo₂P₂ and CeCo₂P₂, the purple arrows represent the spin direction; (b) spin-wave and the spin-flipped Stoner excitations in the energy-momentum space. Spin-wave becomes broadened when interacting with the Stoner excitations; (c) representative RIXS experimental spectra in LaCo₂P₂ and CeCo₂P₂.

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图 11 笼目晶格铁磁外尔半金属Co₃Sn₂S₂中Co L₃边RIXS自旋翻转平带斯托纳激发^[86]　(a) Co₃Sn₂S₂中六角CoSn平面结构、倒空间及RIXS实验示意图, 箭头标注Co自旋方向, 加粗实线标注了RIXS在动量空间的扫描路径, 入射光采用π线偏振光; (b) Co₃Sn₂S₂基于密度泛函理论计算得出的自旋向下的能带结构; (c) 自旋极化能带及自旋翻转斯托纳激发过程; (d) 对应于图(c)中自旋极化能带的自旋波和自旋翻转斯托纳激发谱示意图, 当能带变得扁平时, 对应的斯托纳激发也相应变窄; (e), (f) Co₃Sn₂S₂中Co L₃边RIXS自旋翻转平带斯托纳激发沿着h方向和l方向的色散关系; (g), (h)对应于图(e), (f)的基于密度泛函和动力学平均场理论得出的动态自旋关联函数

Figure 11. Near flat-band Stoner excitations probed by Co L₃-RIXS in Kagome ferromagnetic Weyl metal Co₃Sn₂S₂: (a) The in-plane hexagonal CoSn structure, the corresponding reciprocal space, and the RIXS experimental geometry, the arrow on Co atoms represents the spin direction, the thick solid lines highlight the RIXS scanning path in the momentum space, the incident X-rays are π linearly polarized; (b) spin-down polarized band structure of Co₃Sn₂S₂ calculated by density functional theory; (c) schematic view of spin-polarized bands and the allowed spin-flip excitations in the momentum space; (d) the corresponding spin-flipped Stoner excitations, when the band becomes flat the corresponding Stoner excitations gets narrow; (e), (f) Co₃Sn₂S₂ Co L₃-RIXS experimental near flat-band Stoner excitations along h and l directions, respectively; (g), (h) the theoretical spin-spin dynamical susceptibility (spin-spin correlation function) along h and l directions calculated based on density functional theory and dynamical mean-field theory.

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