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High-Tc cuprates, iron-based superconductors, heavy-fermion superconductors and κ-type layered organic superconductors share some common features − the proximity of the superconducting state to the magnetic ordered state and the non-s-wave superconducting pairing function. It is generally believed that the Cooper pairings in these unconventional superconductors are mediated by spin fluctuations. In this paper, we present a brief overview on the spin dynamics and unconventional pairing, focusing on high-Tc cuprates and iron-based superconductors. In particular, we will overview the properties of the neutron spin resonance and its possible origin, the pairing mechanism in Hubbard model within the weak-coupling framework and its application to the aforesaid unconventional superconductors. We point out that the interplay between magnetism and superconductivity is still an area of active research.
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Keywords:
- unconventional superconductivity /
- spin excitation /
- superconducting pairing symmetry /
- spin resonance
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图 1 铜氧化物高温超导La2–xSrxCuO4/Na2–xCexCuO4、铁基超导Ba1–xKxFe2As2/Ba(Fe1–xCox)2As2、重费米子超导CeCo(In1–xCdx)5以及层状有机超导κ-(BEDT-TTF)2X (标记为Mott insulator的相区在这里表示反铁磁绝缘体)的典型相图. 图分别来自文献[14,16,23,24]
Figure 1. Typical phase diagrams for high-Tc cuprates, iron-based superconductors, heavy fermion superconductors and κ-typed layered organic superconductors. The figures are reproduced from Refs. [14,16,23,24].
图 2 铜氧化物高温超导母体La2CuO4的自旋波色散和自旋波强度与二维波矢的依赖关系, 波矢方向见插图. (a)和(c)表示色散, (b)和(d)表示自旋波强度. 其中的实线是线性自旋波理论的计算结果(见文中介绍). 图(a)和图(b)来自文献[35], 图(c)和图(d)来自文献[36]
Figure 2. (a), (c) Spin-wave dispersion in La2CuO4 along high symmetry directions in the two dimensional Brillouin zone as indicated in the inset. (b), (d) Spin-wave intensity as a function of the wave vector. Line is the prediction of the linear spin-wave theory. Fig. (a) and Fig. (b) are reproduced from Ref. [35], Fig. (c) and Fig. (d) from Ref. [36].
图 3 中子散射实验揭示的掺杂铜氧化物高温超导自旋激发普适色散—沙漏状色散. 图(b)用于比较掺杂和未掺杂体系自旋激发谱的变化, 其中的实线表示未掺杂体系的自旋激发色散(纵轴的单位为meV). 图(a)和图(b)分别来自文献[25]和文献[47]
Figure 3. The universality of the spin excitations in doped high-Tc cuprates—the hourglass dispersion revealed by neutron scattering experiments. Figure (b) is shown for a comparison with the dispersion in the undoped system, which is represented schematically by the solid lines. The figures are reproduced from Ref. [25] and Ref. [47], respectively.
图 4 中子散射实验揭示的铜氧化物高温超导YaBa2Cu3O6.6自旋激发在动量空间的分布 (a)激发能
$ E = 24\;{\rm{meV}} $ ; (b)激发能$ E = 34\;{\rm{meV}} $ ; (c)激发能$ E = 75\;{\rm{meV}} $ . 其中$ 34\;{\rm{meV}} $ 对应于自旋共振模能量. 图来自文献[43]Figure 4. Images of spin excitations in the momentum space for YaBa2Cu3O6.6 revealed by neutron scattering experiments, at different excitation energies: (a)
$ E = 24\;{\rm{meV}} $ ; (b)$ 34\;{\rm{meV}} $ ; (c)$ 75\;{\rm{meV}} $ .$ 34\;{\rm{meV}} $ is the energy of the spin resonance. Figures are reproduced from Reference [43].
图 5 铁基高温超导母体BaFe2As2沿着
$ (\pi, q _{y}) $ 和$ (q_{x}, 0) $ 方向的自旋波色散. 图(a)中的实线表示利用各向异性海森伯模型计算的结果, 参数为$SJ_{1 {\rm a}} = 59.2 \pm 2.0,\; SJ_{1 {\rm b}} = -9.2 \pm 1.2,\; SJ_{2} = 13.6 \pm 1.0,\; SJ_{\rm c} = 1.8 \pm 0.3$ meV. 图(a)中的虚线代表利用各向同性海森伯模型计算的结果, 参数为$SJ_{1 {\rm a}} = SJ_{1 {\rm b}} = 18.3 \pm 1.4,\; SJ_{2} = 28.7 \pm 0.5, \;SJ_{\rm c} = 1.8$ meV. 其中S表示自旋. 图来自文献[55]Figure 5. Figures show spin wave dispersions in BaFe2As2 along the
$ (\pi, q_{y}) $ and$ (q_{x}, 0) $ directions, respectively. In the panel (a), the solid line is a Heisenberg model calculation using anisotropic exchange couplings$SJ_{1 {\rm a}} = 59.2 \pm 2.0,\; SJ_{1 {\rm b}} = -9.2 \pm 1.2,\; SJ_{2} = 13.6 \pm 1.0,\; SJ_{\rm c} = 1.8 \pm 0.3$ meV, and the dotted line is that assuming isotropic exchange coupling$SJ_{1 {\rm a}} = SJ_{1 {\rm b}} = 18.3 \pm 1.4,\; SJ_{2} = 28.7 \pm 0.5,\; SJ_{\rm c} = 1.8$ meV. S is the spin of the system. Figures are reproduced from Ref. [55].
图 6 哈伯德相互作用导致的粒子-粒子通道散射梯型费曼图. 其中的虚线表示哈伯德库仑互作用U (a)自旋单态通道的散射; (b)自旋三重态通道的散射. 图来自文献[67]
Figure 6. Ladder Feynman diagram in the particle-particle channel coming from the Hubbard interaction, where the dotted lines denote the Hubbard interaction U: (a) The spin-single channel; (b) the spin-triplet channel. Figures are reproduced from Ref. [67].
图 7 (a)由(3)式计算的有效电子间相互作用势
$ V_{\rm s}({ q}) $ 沿着四方晶格布里渊区高对称方向的分布. 计算参数为12%空穴掺杂, 并取次近邻跃迁参数$ t' = -0.3 t $ 和$ U = 2 t $ ; (b)$ V_{\rm s}({ q}) $ 经傅里叶变换后在实空间的分布. 图(b)来自文献[67]Figure 7. (a) Effective interaction
$ V_{\rm s}({ q}) $ between electrons arising from the exchanges of spin fluctuations along the high symmetry directions in the Brillouin zone, calculated by Eq.(3) with the next-nearest-neighbor hopping$ t' = -0.3 t $ ,$ U = 2 t $ and 12% hole doping; (b) Fourier transformation of$ V_{\rm s}({ q}) $ . Figure (b) is reproduced from Ref. [67].
图 8 利用哈伯徳模型的弱耦合计算方法(方程(3)和方程(5))得到的最有利能隙函数在布里渊区的分布. 其中的粗黑线表示电子费米面, 点线表示配对能隙函数的节点(能隙为零的点). 计算参数基于对铜氧化物高温超导的近似描述, 具体见图7的说明文字. 图来自文献[67]
Figure 8. The most favorable pairing function obtained from the weak-coupling approach to the Hubbard model Eqs.(3) and (5). The solid lines denote the Fermi surface and dotted lines denote the gap nodes. The parameters are the same as those given in the caption of Fig. 7, which are thought to describe approximately high-Tc cuprates. Figure is reproduced from Ref. [67].
图 9 (a)自旋激发率
$ \chi ({{q}}, \omega = 0) $ , (b)电子型能带和(c)空穴型能带上最有利配对函数在动量空间的分布. 这些结果基于对描写铁基超导最简单的两带哈伯徳模型[75]的弱耦合理论计算[76]. (d)费米面, 其中$ Q_{\rm{AF}} $ 表示围绕Γ的空穴费米面与围绕M的电子费米面之间的套叠波矢. 图来自文献[76]Figure 9. Momentum dependence of the spin susceptibility
$ \chi ({{q}}, \omega = 0) $ (a), the most favorable pairing functions on the electron Fermi pocket (b) and hole Fermi pocket (c). The results are obtained from the weak-coupling approach to the two-band Hubbard model [76], which is thought to be the simplest model describing iron iron pnictides [75]. (d) The Fermi surface in which$ Q_{\rm{AF}} $ denote the nesting wavevector between the hole pocket around Γ and electron pocket around M. Figures are reproduced from Ref. [76].
图 10 中子散射实验在(a)铜氧化物超导YBa2Cu3O6.92, (b)铁基超导BaFe1.85Co0.15As2和(c)重费米子超导CeCoIn5中测量的自旋极化率Im
$ \chi ({{q}}, \omega) $ 与能量的依赖关系. 其中不同符号点标志的曲线表示不同温度的结果. 图来自文献[18,59, 20]Figure 10. Spin susceptibility Im
$ \chi ({{q}}, \omega) $ measured by neutron scattering on (a) YBa2Cu3O6.92, (b) BaFe1.85Co0.15As2 and (c) CeCoIn5 for different temperatures below and above the superconducting transition temperature. Figures are reproduced from Refs. [18, 59, 20], respectively.
图 11 (a)铜氧化物超导YBa2Cu3O6.97, (b)铁基超导Ba0.6K0.4Fe2As2和(c)重费米子超导Nd0.05Ce0.95CoIn5中自旋共振峰强度随温度的依赖关系. 图分别来自文献[27, 19, 28]
Figure 11. Temperature evolution of the neutron intensity around the spin resonance in (a) the high-Tc cuprate YBa2Cu3O6.97, (b) iron-based superconductor Ba0.6K0.4Fe2As2 and (c) heavy fermion superconductor Nd0.05Ce0.95CoIn5. Figures are reproduced from References [27, 19, 28], respectively.
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