Theoretical progress and material studies of heavy fermion superconductors

Li Yu; Sheng Yu-Tao; Yang Yi-Feng

doi:10.7498/aps.70.20201418

Abstract

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.

Keywords:

Authors and contacts

1.
Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China
2.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
3.
University of Chinese Academy of Sciences, Beijing 100049, China
4.
Songshan Lake Materials Laboratory, Guangdong 523808, China

Corresponding author: Yang Yi-Feng, yifeng@iphy.ac.cn

Funds: Project supported by the National Basic Research Program of China (Grant No. 2017YFA0303103), the National Natural Science Foundation of China (Grant Nos. 11774401, 11974397), the Strategic Priority Research Program of Chinese Academy of Sciences, China (Grant No. XDB33010100), and the China Postdoctoral Science Foundation (Grant No. 2020M670422)

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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}} $.

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图 2 (a) Kondo屏蔽和 (b) RKKY相互作用示意图

Figure 2. Sketch of the (a) Kondo screening and (b) RKKY interaction.

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图 3 Doniach相图. 其中AFM表示反铁磁, $ T_{\rm{N}} $为反铁磁转变温度

Figure 3. The Doniach phase diagram, where AFM denotes the antiferromagnetic phase and $ T_{\rm{N}} $ is the AFM transition temperature.

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图 4 重费米子的平均场能带杂化图像及“小”费米面到“大”费米面的转变

Figure 4. The mean-field hybridized band picture for heavy fermions and the associated transition from “small” to “large” Fermi surface.

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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.

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图 6 CeCu₂Si₂中比热随温度的演化. 内插图为两个不同CeCu₂Si₂样品在$ T_{\rm{c}} $附近的比热系数$ C/T $^[57]

Figure 6. The specific heat (C) as a function of temperature (T) in CeCu₂Si₂. The inset compares $ C/T $ near $ T_{\rm{c}} $ in two different samples^[57].

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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) U_1–xTh_xBe₁₃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 U_1–xTh_xBe₁₃ as a function of Th doping^[73].

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图 8 不同超导材料中$ T_{\rm{c}} $与自旋涨落特征温度$ T_{\rm{0}} $的关系^[67]

Figure 8. $ T_{\rm{c}} $ versus the characteristic spin-fluctuation temperature $ T_{\rm{0}} $ in different superconductors^[67].

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图 9 重费米子超导的唯象理论框架

Figure 9. A phenomenological framework for heavy fermion superconductivity.

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图 10 CeCu₂Si₂的能带结构和费米面^[125]. 费米面的颜色标记了费米速度的大小

Figure 10. Band structures and Fermi surfaces of CeCu₂Si₂^[125]. The colors of the Fermi surfaces represent the Fermi velocity.

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图 11 CeCu₂Si₂的超导相图^[125]

Figure 11. The superconducting phase diagram of CeCu₂Si₂^[125]

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图 12 (a) CeRh_1–xIr_xIn₅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 CeRh_1–xIr_xIn₅ 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].

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图 13 (a)${\rm{Ce}} MX_{3} $超导体在加压下的最高$ T_{\rm{c}} $与相应原胞体积的关系图^[62]; (b) CeRhSi₃的磁场-温度相图^[187]

Figure 13. (a) Relation between the highest $ T_{\rm{c}} $ under pressure and the relative unit-cell volume of ${\rm{Ce}} MX_{3} $^[62]; (b) the magnetic field-temperature phase diagram of CeRhSi₃^[187].

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图 14 YbRh₂Si₂的能带结构和费米面^[126]

Figure 14. Band structures and Fermi surfaces of YbRh₂Si₂^[126].

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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].

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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].

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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].

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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].

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图 19 UTe₂超导态的磁场-转角相图. 其中SC_PM, SC_RE, SC_FP表示三种不同的超导相, FP表示磁场极化相^[64]

Figure 19. The magnetic field-azimuthal angle phase diagram for superconducting UTe₂, where SC_PM, SC_RE, SC_FP are three different superconducting phases, and FP denotes the field-polarized phase^[64].

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图 20 UBe₁₃的(a)温度-压力相图和(b)磁场-温度相图^[246]

Figure 20. (a) Temperature-pressure phase diagram and (b) magnetic field-temperature phase diagram of UBe₁₃^[246].

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图 21 PuCoGa₅超导机理的两种可能图像:　(a)价态涨落机制; (b)自旋涨落机制^[322]

Figure 21. Two possible scenarios for the pairing mechanism of PuCoGa₅: (a) The valence-fluctuation mechanism; (b) the spin-fluctuation mechanism^[322].

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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 $)	0.002	—	—	磁场诱导非常规量子临界点
Yb基	β-YbAlB₄	正交($ 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].

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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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