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Heavy fermion superconductors belong to a special class of strongly correlated systems and unconventional superconductors. The emergence of superconductivity in these materials is closely associated with the presence of quantum critical fluctuations. Heavy fermion superconductors of different structures often exhibit distinct competing orders and superconducting phase diagrams, implying sensitive dependence of their electronic structures and pairing mechanism on the crystal symmetry. Here we give a brief introduction on recent theoretical and experimental progress in several different material families. We develop a new phenomenological framework of superconductivity combining the Eliashberg theory, a phenomenological form of quantum critical fluctuations, and strongly correlated band structure calculations for real materials. Our theory provides a unified way for systematic understanding of various heavy fermion superconductors.
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Keywords:
- heavy fermion superconductivity /
- competing order /
- quantum critical fluctuation /
- pairing symmetry
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图 1 稀磁合金和常规超导体的电阻率随温度演化示意图. 稀磁合金中, 由于Kondo效应, 电阻率会在一定温度之下呈现
$ -{\rm{log}} T $ 的行为, 而在$ T\rightarrow 0 $ 时以$ -T^2 $ 的方式趋于饱和; 超导中, 电阻率在$ T_{\rm{c}} $ 之下变为零Figure 1. Characteristic evolution of resistivity as a function of temperature for dilute magnetic alloys and superconductors. In dilute magnetic alloys, the resistivity shows
$ -{\rm{log}} T $ behavior within a certain range of temperature due to the Kondo effect and eventually saturates as$ -T^2 $ when$T\rightarrow 0$ . In superconductors, resistivity becomes zero below$ T_{\rm{c}} $ .图 5 重费米子二流体模型基本相图. 其中T *,
$ T_{\rm{L}} $ 分别表示相干温度和退局域化温度,$ f_0 $ 表示f电子与导带电子之间集体杂化的效率Figure 5. The basic phase diagram of the two-fluid model for heavy fermion systems. T * and
$ T_{\rm{L}} $ are the coherence temperature and the delocalization temperature, respectively. And$ f_0 $ represents the effectiveness of the collective hybridization between f electrons and conduction electrons.图 7 重费米子超导体的典型相图 (a) CeIn3和CeRhIn5的温度-压力相图[69]; (b) UGe2的温度- 压力相图[69]; (c) CeCu2Si2和CeCu2Ge2的温度-压力相图[69]; (d) URu2Si2的温度-压力相图[70], 其中HO, SC, AF分别代表隐藏序(Hidden order)、超导和反铁磁; (e) CeCoIn5的磁场-温度相图[71]; (f) UPt3的磁场-温度相图[72], 其中A, B, C表示三种不同的超导序参量; (g) U1–xThxBe13的掺杂浓度-温度相图[73]; (h) PrOs4Sb12的磁场-温度相图[74], 其中FIOP表示磁场诱导的电四极矩相
Figure 7. Typical phase diagrams of heavy fermion superconductors. The temperature-pressure phase diagrams for: (a) CeIn3 and CeRhIn5[69]; (b) UGe2[69]; (c) CeCu2Si2 and CeCu2Ge2[69]; (d) URu2Si2, in which HO, SC, AF refer to the hidden order, superconducting and antiferromagnetic phases[70]. The magnetic field-temperature phase diagrams for: (e) CeCoIn5[71]; (f) UPt3 (A, B, C denote three different superconducting states)[72]; (h) PrOs4Sb12 (FIOP is a field-induced quadrupole phase)[74]. (g) The phase diagram of U1–xThxBe13 as a function of Th doping[73].
图 12 (a) CeRh1–xIrxIn5和CeCoIn5中轨道各向异性
$ \alpha^2 $ 与体系基态的关系, 其中C (IC)表示公度(非公度)反铁磁[172];(b) CeRhIn5的磁场-压力-温度相图[176]Figure 12. (a) Relation between the ground states of CeRh1–xIrxIn5 and CeCoIn5 and the orbital anisotropy
$ \alpha^2 $ , where C (IC) denote commensurate (incommensurate) antiferromagnetism[172]; (b) the magnetic field-pressure-temperature phase diagram of CeRhIn5[176].图 15 (a)理论计算的YbRh2Si2超导随反铁磁波矢
$ {{Q}}=(h, h, l) $ 变化的相图[126], 其中${{Q}}^{\rm{EXPT}}=(0.14\pm0.04, 0.14\pm 0.04, 0)$ 为中子散射实验得到的反铁磁波矢[202]; (b)理论预言的两种磁场-温度相图[126]Figure 15. (a) The theoretical superconducting phase diagram of YbRh2Si2 depending on the antiferromagnetic wave vector
$ {{Q}}=(h, h, l) $ [126], where${{Q}}^{\rm{EXPT}}=(0.14\pm0.04, 0.14\pm 0.04, 0)$ is the wave vector obtained from neutron scattering experiments[202]; (b) two candidate scenarios for the magnetic field-temperature phase diagram[126].图 16 (a) β-YbAlB4的磁化强度M对温度导数的
$ T/B $ 标度行为, 其中左下方的内插图为β-YbAlB4的磁场-温度相图, 右上方的内插图为Pearson关联系数R (反映两个变量之间关联强度)的拟合值[206]; (b) α-YbAlB4和β-YbAlB4的晶体结构图比较[210]Figure 16. (a)
$ T/B $ -scaling of the temperature derivatives of the magnetization M in β-YbAlB4. The insets in the left-bottom and right-upper figures show the magnetic field-temperature phase diagram and the fitted Pearson coefficient (R), respectively. (b) comparison of the crystal structures of α-YbAlB4 and β-YbAlB4[210].图 17 UTe2的(a)晶体结构和(b)四种可能的磁构型; (c)U离子的磁矩和四种磁构型与基态的能量差值随库仑相互作用U的变化; (d)计算得到的磁交换系数
$ J_i $ ($ i=1, 2, 3 $ )随U的变化[127]Figure 17. (a) Crystal structures and (b) four candidate magnetic configurations of UTe2; (c) magnetic moments of U ion and the energy difference between the four magnetic orders and the ground state as a function of the Coulomb interaction U; (d) calculated magnetic exchange interactions
$ J_i $ ($ i=1, 2, 3 $ ) as a function of U [127].图 18 (a) DFT + U和(b) DFT + DMFT计算得到的UTe2能带结构; (c) UTe2的费米面结构及费米速度分布; (d) 三种超导不可约表示下节点在费米面上的分布
Figure 18. Electronic band structures of UTe2 obtained from (a) DFT + U and (b) DFT + DMFT calculations; (c) Fermi surface topology with colored Fermi velocities; (d) node distributions on the Fermi surfaces for three irreducible representations of superconductivity[127].
表 1 重费米子超导材料及基本性质
Table 1. Heavy fermion superconductors and their basic properties
类别 材料 晶系(空间群) $ T_{\rm{c}} $/K $ \gamma $/mJ·mol–1·K2 节点 特殊性质 Ce基 CeCu2Si2 四方($ I4/mmm $) 0.7 1000 无 超导与SDW相分离; 加压诱导第二个超导 CeCu2Ge2 四方($ I4/mmm $) 0.64 (10.1 GPa) 200 — 反铁磁竞争序; 加压诱导第二个超导 CePd2Si2 四方($ I4/mmm $) 0.43 (3 GPa) 65 — 反铁磁竞争序 CeRh2Si2 四方($ I4/mmm $) 0.42 (1.06 GPa) 23 — 反铁磁竞争序 CeAg2Si2[61] 四方($ I4/mmm $) 1.25 (16 GPa) — — 反铁磁竞争序 CeAu2Si2 四方($ I4/mmm $) 2.5 (22.5 GPa) — — 反铁磁竞争序 CeNi2Ge2 四方($ I4/mmm $) 0.3 350 — 非费米液体正常态 CeIn3 立方($ Pm3 m $) 0.23 (2.45 GPa) 140 线 反铁磁竞争序 CeIrIn5 四方($ P4/mmm $) 0.4 750 线 非费米液体正常态 CeCoIn5 四方($ P4/mmm $) 2.3 250 线 自旋单态配对; 强磁场诱导Q相 CeRhIn5 四方($ P4/mmm $) 2.4 (2.3 GPa) 430 — 压力和磁场诱导费米面突变; 强磁场诱导向列序 CePt2In7 四方($ I4/mmm $) 2.3 (3.1 GPa) 340 — 反铁磁竞争序 Ce2RhIn8 四方($ P4/mmm $) 2.0 (2.3 GPa) 400 — 反铁磁竞争序 Ce2PdIn8 四方($ P4/mmm $) 0.68 550 线 非费米液体正常态 Ce2CoIn8 四方($ P4/mmm $) 0.4 500 — 非费米液体正常态 Ce3PdIn11 四方($ P4/mmm $) 0.42 290 — 两个反铁磁序 CePt3Si 四方($ P4 mm $) 0.75 390 线 反铁磁竞争序; 破缺中心反演; 混合宇称配对? CeIrSi3 四方($ I4 mm $) 1.65 (2.5 GPa) 120 — 反铁磁竞争序; 破缺中心反演; 混合宇称配对? CeRhSi3 四方($ I4 mm $) 1.0 (2.6 GPa) 120 — 反铁磁竞争序; 破缺中心反演; 混合宇称配对? CeCoGe3 四方($ I4 mm $) 0.69 (6.5 GPa) 32 — 反铁磁竞争序; 破缺中心反演; 混合宇称配对? CeRhGe3[62] 四方($ I4 mm $) 1.3 (21.5 GPa) — — 反铁磁竞争序; 破缺中心反演; 混合宇称配对? CeIrGe3 四方($ I4 mm $) 1.6 (24 GPa) 80 — 反铁磁竞争序; 破缺中心反演; 混合宇称配对? CeNiGe3 正交($ Cmmm $) 0.43 (6.8 GPa) 45 — 反铁磁竞争序; 加压诱导第二个超导 Ce2Rh3Ge5 正交($ Ibam $) 0.26 (4.0 GPa) 90 — 反铁磁竞争序 CePd5Al2 四方($ I4/mmm $) 0.57 (10.8 GPa) 56 — 反铁磁竞争序 Yb基 YbRh2Si2 四方($ I4/mmm $) 0.002 — — 磁场诱导非常规量子临界点 β-YbAlB4 正交($ Cmmm $) 0.08 150 — $ T/B $标度行为; 磁场诱导拓扑金属正常态? U基 UGe2 正交($ Cmmm $) 0.8 (1.2 GPa) 34 线 铁磁竞争序; 等自旋三重态配对; 超导态破缺时间反演对称性 UTe2[63,64] 正交 ($ Immm $) 1.6 110 点 铁磁涨落; 自旋三重态配对; 磁场诱导
多个超导相URhGe 正交 ($ Pnma $) 0.25 163 — 铁磁竞争序; 等自旋三重态配对; 磁场
诱导两个超导相UCoGe 正交 ($ Pnma $) 0.8 57 点? 线? 等自旋三重态配对; 铁磁竞争序; 磁场
诱导两个超导相UIr 单斜 ($ P2_1 $) 0.15 (2.6 GPa) 49 — 多个铁磁相; 破缺中心反演; 混合宇称配对? U2PtC2 四方($ I4/mmm $) 1.47 150 — 无磁有序; 自旋三重态配对; 铁磁涨落 UPd2Al3 六方($ P6/mmm $) 2.0 200 线 反铁磁竞争序; 自旋单态配对;
磁场调制FFLO?UNi2Al3 六方($ P6/mmm $) 1.1 120 — 反铁磁竞争序; 自旋三重态配对; 超导
与反铁磁共存UBe13 立方($ Fm\bar{3}c $) 0.95 1000 无 非费米液体正常态; 自旋三重态配对;
Th掺杂诱导多个超导相UPt3 六方($ P6_3/mmc $) 0.530, 0.480 440 线+点 自旋三重态配对; 多个超导相; 低温超
导破缺时间反演对称性U6Fe 四方($ I4/mcm $) 3.8 157 — 电荷密度波竞争序; 磁场调制FFLO? URu2Si2 四方($ I4/mmm $) 1.53 70 线 隐藏序正常态; 自旋单态配对; 破缺
时间反演对称性Pr基 PrOs4Sb12 立方($ Im\bar3 $) 1.82, 1.74 500 点? 无? 磁场诱导反铁电四极矩序; 两个超导相; 低温超导破缺时间反演对称性 PrIr2Zn20 立方($ Fd\bar3 m $) 0.05 — — 反铁电四极矩竞争序 PrRh2Zn20 立方($ Fd\bar3 m $) 0.06 — — 反铁电四极矩竞争序 PrV2Al20 立方($ Fd\bar3 m $) 0.05 900 — 反铁电四极矩竞争序 PrTi2Al20 立方($ Fd\bar3 m $) 0.2 100 — 铁电四极矩竞争序 Pu基 PuCoGa5 四方($ P4/mmm $) 18.5 77 线 混合价态; 价态涨落机制? 自旋涨落机制? PuCoIn5 四方($ P4/mmm $) 2.5 200 线 混合价态; 自旋涨落机制 PuRhGa5 四方($ P4/mmm $) 8.7 70 线 混合价态; 自旋涨落机制 PuRhIn5 四方($ P4/mmm $) 1.6 350 线 混合价态; 价态涨落机制? 自旋涨落机制? Np基 NpPd5Al2 四方($ I4/mmm $) 4.9 200 点 非费米液体正常态 *表格中$ T_{\rm{c}}$后有括号表明为压力下超导, “—”表示尚无相关实验, “?”表示还不确定或存在争议. 表中主要数据及特殊性质可参考文献[65-68]. 表 2 超导能隙函数的对称性变换
Table 2. Symmetry transformation of the superconducting gap functions
对称性变换 自旋单态 自旋三重态 费米子交换$ P $ $ P\psi({{k}})=\psi(-{{k}})=\psi({{k}}) $ $ P{{d}}({{k}})={{d}}(-{{k}})=-{{d}}({{k}}) $ 空间旋转$ g $ $ g\psi({{k}})=\psi(D(g){{k}}) $ $ g{{d}}({{k}})={{d}}(D(g){{k}}) $ 自旋旋转$ g_s $ $ g_s\psi({{k}})=\psi({{k}}) $ $ g_s{{d}}({{k}})=\bar D(g_s){{d}}({{k}}) $1 时间反演$ \theta $ $ \theta\psi({{k}})=\psi^*(-{{k}}) $ $ \theta{{d}}({{k}})=-{{d}}^*(-{{k}}) $ 空间反演$ I $ $ I\psi({{k}})=\psi(-{{k}}) $ $ I{{d}}({{k}})={{d}}(-{{k}}) $ $ U(1) $规范$\varPhi$ $\varPhi\psi({{k} })={\rm e}^{{\rm i}\phi}\psi({{k} })$ $\varPhi{{d} }({{k} })={\rm e}^{{\rm i}\phi}{{d} }({{k} })$ *其中$D(g)$为晶体点群G的表示矩阵, $\bar D(g_s)$为SU(2)群的表示矩阵.
1存在自旋-轨道耦合时, 自旋的旋转与${{k}}$ 的旋转不再独立, 即$g_s{{d}}({{k}})=\bar D(g_s){{d}}(\bar D(g_s){{k}})$. -
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