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Theoretical progress and material studies of heavy fermion superconductors

Li Yu ,  Sheng Yu-Tao ,  Yang Yi-Feng

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Theoretical progress and material studies of heavy fermion superconductors

Li Yu, Sheng Yu-Tao, Yang Yi-Feng
Acta Phys. Sin., 2021, 70(1): 017402.
doi: 10.7498/aps.70.20201418
科大讯飞翻译 (iFLYTEK Translation)
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  • 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.

    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}} $.

    DownLoad: Full-Size Img PowerPoint

    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.

    DownLoad: Full-Size Img PowerPoint

    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.

    DownLoad: Full-Size Img PowerPoint

    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.

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    图 9  重费米子超导的唯象理论框架

    Figure 9.  A phenomenological framework for heavy fermion superconductivity.

    DownLoad: Full-Size Img PowerPoint

    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.

    DownLoad: Full-Size Img PowerPoint

    图 11  CeCu2Si2的超导相图[125]

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

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    图 19  UTe2超导态的磁场-转角相图. 其中SCPM, SCRE, SCFP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 SCPM, SCRE, SCFP are three different superconducting phases, and FP denotes the field-polarized phase[64].

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    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].

    DownLoad: Full-Size Img PowerPoint

    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].
    DownLoad: CSV

    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$g_s{{d}}({{k}})=\bar D(g_s){{d}}(\bar D(g_s){{k}})$.
    DownLoad: CSV
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  • Abstract views:  19968
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Publishing process
  • Received Date:  27 August 2020
  • Accepted Date:  22 September 2020
  • Available Online:  24 December 2020
  • Published Online:  05 January 2021

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