#
Nematic Superconductivity in Doped Bi_{2}Se_{3} Topological Superconductors

## Abstract

**:**

## 1. Introduction

## 2. Nematic Superconductivity: Rotational Symmetry Breaking in the Gap Amplitude

#### 2.1. Symmetry Breaking in Superconductivity

#### 2.2. Gap-Nematic and Spin-Nematic Superconductivity

## 3. Superconductivity in Doped Bi${}_{2}$Se${}_{3}$

#### 3.1. Crystal Structure of the Mother Compound Bi${}_{2}$Se${}_{3}$

#### 3.2. Basic Properties of Doped Bi${}_{2}$Se${}_{3}$ Superconductors

#### 3.3. Possible Superconducting States

#### 3.4. Early Experiments on the Superconducting State in Doped Bi${}_{2}$Se${}_{3}$

## 4. Recent Experiments on Nematic Superconducting Behavior

#### 4.1. Beginning of the Story: Nuclear Magnetic Resonance

#### 4.2. Pioneering Reports of Bulk Properties

#### 4.3. Recent Reports

#### 4.4. Direct Visualization

## 5. Known Issues

#### 5.1. Which of ${\Delta}_{4x}$ or ${\Delta}_{4y}$ Is Realized?

#### 5.2. Normal-State and Superconducting-State Nematicities

#### 5.3. Nematic Domains

#### 5.4. Possible Nematic Superconductivity in Other Systems

## 6. Summary and Perspectives

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AMR | Angular magnetoresistance |

ARPES | Angle-resolved photoemission spectroscopy |

BCS | Bardeen–Cooper–Schrieffer |

BG | Bridgman method |

BW | Balian–Werthamer |

ECI | Electrochemical intercalation |

MBE | Molecular-beam epitaxy |

MG | Melt growth |

NMR | Nuclear magnetic resonance |

QL | Quintuple layer |

SC | Superconducting |

STM | Scanning tunneling microscope |

vdW | van der Waals |

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**Figure 1.**Schematic comparison of various known superconductivity and gap/spin nematic superconductivity, for the case of a tetragonal-lattice system.

**Figure 2.**Schematic description of the crystal structure of the mother compound Bi${}_{2}$Se${}_{3}$ [42]. The purple spheres are the Bi atoms, and the green and light-blue spheres are the Se(1) and (2) atoms, respectively. The colors of the spheres are modified depending on the depth along the view direction: atoms closer to the view point have thicker colors. (

**a**) View from the a axis direction. The intercalated metallic ions most likely sit in the van der Waals (vdW) gap between the quintuple layers (QL). (

**b**) View from the c axis direction. The crystal structure figures were made using the software VESTA-3 [44].

**Figure 3.**Representative experiments on the nematic superconductivity in ${A}_{x}$Bi${}_{2}$Se${}_{3}$. (

**a**) In-plane field-angle dependence of the NMR Knight shift of Cu${}_{x}$Bi${}_{2}$Se${}_{3}$ [8]. (

**b**) In-plane field-angle dependence of the specific heat and ${H}_{\mathrm{c}2}$ of Cu${}_{x}$Bi${}_{2}$Se${}_{3}$ [9]. (

**c**) In-plane angular dependence of ${H}_{\mathrm{c}2}$ evaluated from magnetoresistance measurements on Sr${}_{x}$Bi${}_{2}$Se${}_{3}$ [10]. (

**d**) In-plane field-angle dependence of the irreversible component of the magnetic torque of Nb${}_{x}$Bi${}_{2}$Se${}_{3}$ [11]. We should be careful for the definition of the field angle: in (

**a**,

**b**), 0${}^{\xb0}$ corresponds to $H\parallel a$; whereas in (

**c**,

**d**), it corresponds to $H\parallel {a}^{\ast}$. The panels (

**a**,

**b**) are respectively quoted from [8,9] with the permission of Springer Nature; (

**c**,

**d**) are respectively from [10,11] under the Creative Commons License.

**Figure 4.**Schematic figure of the coupling between the nematic order parameter and the in-plane symmetry breaking field.

**Table 1.**Proposed superconducting states for doped Bi${}_{2}$Se${}_{3}$ [13,14,15,56,57]. The $\mathit{d}$ vector structures in the band bases are from [56,57]. Here, $\lambda $ represents the strength of the spin-orbit coupling; and $\epsilon $ represents the gap minima for the ${\Delta}_{4y}$ state (see the text). In the bottom row, schematic gap and d-vector structures of each state are shown, together with various cut views. The $\lambda $ value is chosen to be 0.5, and the $\epsilon $ value is just set to be 0.1. The sphere at the center of a cut view is the Fermi surface. The gap structure is expressed with colored surfaces, whose distance from the Fermi surface corresponds to the SC gap amplitude $\left|\mathit{d}\right|$ normalized by its maximal value ${d}_{0}$. The color of this surface also depicts the gap value, as well as the d-vector direction, as explained in the left bottom cell: The hue and lightness of the color indicate the azimuthal and polar angles of the d vector, ${\varphi}_{\mathit{d}}$ and ${\theta}_{\mathit{d}}$, respectively; whereas the grayness of the color depicts the normalized gap, with 50% gray corresponding to $\left|\mathit{d}\right|=0$.

${\mathsf{\Delta}}_{1\mathit{a}}$ | ${\mathsf{\Delta}}_{1\mathit{b}}$ | ${\mathsf{\Delta}}_{2}$ | ${\mathsf{\Delta}}_{3}$ | ${\mathsf{\Delta}}_{4\mathit{x}}$ | ${\mathsf{\Delta}}_{4\mathit{y}}$ | |
---|---|---|---|---|---|---|

Irreducible representation | ${A}_{1g}$ | ${A}_{1g}$ | ${A}_{1u}$ | ${A}_{2u}$ | ${E}_{u}$ | |

Pairing potential | ${\sigma}_{0}$ | ${\sigma}_{x}$ | ${\sigma}_{y}{s}_{z}$ | ${\sigma}_{z}$ | ${\sigma}_{y}{s}_{x}$ | ${\sigma}_{y}{s}_{y}$ |

$\mathit{d}$-vector | - | - | $\sim (\lambda {k}_{x},\lambda {k}_{y},{k}_{z})$ | $\sim (-\lambda {k}_{y},\lambda {k}_{x},0)$ | $\sim ({k}_{z},0,-\lambda {k}_{x})$ | $\sim (\epsilon {k}_{x},-{k}_{z},\lambda {k}_{y})$ |

Parity | even | even | odd | odd | odd | |

Topo.SC | no | no | yes | yes | yes | |

Nematic SC | no | no | no | no | yes | |

Schematic $\mathit{d}$-vector/gap structures |

**Table 2.**Comparison of experimental reports on nematic superconductivity in doped Bi${}_{2}$Se${}_{3}$ and related systems.

Material | Reference | Growth Method ${}^{\mathbf{i}}$ | Doping Level x | Probe | Large ${\mathit{H}}_{c2}$ | Suggested State |
---|---|---|---|---|---|---|

Cu${}_{x}$Bi${}_{2}$Se${}_{3}$ | Matano 2016 [8] | MG + ECI | 0.29–0.31 | NMR | y | ${\Delta}_{4x}$ |

Yonezawa 2017 [9] | MG + ECI | 0.3 | C | x | ${\Delta}_{4y}$ | |

Tao 2018 [70] | MG + ECI | 0.31 | STM | - | ${\Delta}_{4x}$ | |

Sr${}_{x}$Bi${}_{2}$Se${}_{3}$ | Pan 2016 [10] | MG | 0.10, 0.15 | ${\rho}_{ab}$ | x | |

Nikitin 2016 [71] | MG | 0.15 | ${\rho}_{ab}$ in P | x | ||

Du 2017 [72] | MG | NA | ${\rho}_{c}$ | x (#1, #2) | ||

y (#3) | ||||||

Smylie 2018 [73] | MG | 0.1 | ${\rho}_{ab}$, M | x | ||

Kuntsevich 2018 [74] | BG | 0.10–0.20 | ${\rho}_{ab}$ | x (some) | ||

y (others) | ||||||

Willa 2018 [75] | MG | 0.1 | C | y | ||

Nb${}_{x}$Bi${}_{2}$Se${}_{3}$ | Asaba 2017 [11] | MG | NA | torque | - | |

Shen 2017 [53] | MG | 0.25 | ${\rho}_{ab}$, M | y | ||

Cu${}_{x}$(PbSe)${}_{5}$(Bi${}_{2}$Se${}_{3}$)${}_{6}$ | Andersen 2018 [76] | BG + ECI | 1.5 | ${\rho}_{ab}$, ${\rho}_{c}$, C | x | ${\Delta}_{4x}$ |

Bi${}_{2}$Te${}_{3}$/Fe(Se, Te) | Chen 2018 [77] | MBE | - | STM | - | ${\Delta}_{4y}$ |

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**MDPI and ACS Style**

Yonezawa, S.
Nematic Superconductivity in Doped Bi_{2}Se_{3} Topological Superconductors. *Condens. Matter* **2019**, *4*, 2.
https://doi.org/10.3390/condmat4010002

**AMA Style**

Yonezawa S.
Nematic Superconductivity in Doped Bi_{2}Se_{3} Topological Superconductors. *Condensed Matter*. 2019; 4(1):2.
https://doi.org/10.3390/condmat4010002

**Chicago/Turabian Style**

Yonezawa, Shingo.
2019. "Nematic Superconductivity in Doped Bi_{2}Se_{3} Topological Superconductors" *Condensed Matter* 4, no. 1: 2.
https://doi.org/10.3390/condmat4010002