拓扑材料中的超导

顾开元; 罗天创; 葛军; 王健

doi:10.7498/aps.69.20191627

摘要

近来, 人们在凝聚态体系中发现了由拓扑不变量定义的物相, 其中最重要的有拓扑绝缘体、拓扑半金属和拓扑超导体等. 这些物相的拓扑性质由非平凡的拓扑数描述, 相应的材料被称为拓扑材料, 具有诸多新奇的物理特性. 其中拓扑超导体由于边界上有满足非阿贝尔统计的Majorana零能模, 成为实现拓扑量子计算的主要候选材料. 除了探索本征的拓扑超导体外, 由于拓扑性质上的相似性, 在不超导的拓扑材料中调制出超导自然成为了实现拓扑超导的重要手段. 目前, 人们发展了栅极调制、掺杂、高压、近邻效应调制和硬针尖点接触等多种技术, 已经成功地在许多拓扑绝缘体和半金属中诱导出了超导, 并对超导的拓扑性和Majorana零能模进行了研究. 本文回顾了本征拓扑超导候选材料, 以及拓扑绝缘体和半金属中诱导出超导的代表性工作, 评述了不同实验手段的优势和缺陷、分析了其超导拓扑性的证据, 并提出展望.

关键词:

拓扑绝缘体 /
拓扑半金属 /
非常规超导 /
拓扑超导 /
拓扑材料

Abstract

In recent years, by introducing topological invariants into condensed matter systems, new phases of mater are revealed. Of these new phases, the topological insulator, topological semimetal and topological superconductor are the most important. They are called topological materials due to nontrivial topological parameters. Topological superconductors hold Majorana zero modes at the edges, satisfying non-abelian statistics, which makes them major candidate for realizing topological quantum computation. Besides exploring intrinsic topological superconductor, a promising way to realize topological superconductor is to induce superconductivity into other kinds of topological materials. Up to now, experimentalists have developed some techniques, such as gating, doping, high pressure, interface effect and hard point contact to introduce superconductivity into various topological materials, and also they have studied the topological properties of the induced superconductivity. In this review, we summarize the representative researches on intrinsic topological superconductor candidates and induced superconductivities in topological insulators and semimetals. The advantages and disadvantages of different techniques are discussed. Besides, the potential evidences of topological superconductors are analyzed. In the end, the outlook of this actively pursued research field is given.

Keywords:

topological insulator /
topological semimetal /
unconventional superconductivity /
topological superconductivity /
topological material

作者及机构信息

顾开元^1,,
罗天创^1,,
葛军¹,
王健^1,2,3,4

1.
北京大学物理学院, 量子材料科学中心, 北京　100871

2.
量子物质科学协同创新中心, 北京　100871

3.
中国科学院大学, 中国科学院拓扑量子计算卓越创新中心, 北京　100190

4.
北京量子信息科学研究院, 北京　100193

通信作者: 王健, jianwangphysics@pku.edu.cn

基金项目: 国家重点研发计划(批准号: 2018YFA0305604, 2017YFA0303302)、国家自然科学基金(批准号: 11888101, 11774008)、中国科学院战略性先导科技专项(B类)(批准号: XDB28000000)和北京市自然科学基金(批准号: Z180010)资助的课题

Authors and contacts

Gu Kai-Yuan^1,,
Luo Tian-Chuang^1,,
Ge Jun¹,
Wang Jian^1,2,3,4

1.
International Center for Quantum Material, School of Physics, Peking University, Beijing 100871, China

2.
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China

3.
CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China

4.
Beijing Academy of Quantum Information Sciences, Beijing 100193, China

Corresponding author: Wang Jian, jianwangphysics@pku.edu.cn

Funds: Project supported by the National Key Research and Development Program of China (Grant Nos. 2018YFA0305604, 2017YFA0303302), the National Natural Science Foundation of China (Grant Nos. 11888101, 11774008), the Strategic Priority Research Program of Chinese Academy of Sciences, China (Grant No. XDB28000000), and the Natural Science Foundation of Beijing, China (Grant No. Z180010)

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施引文献

图 1 Au₂Pb的输运测量　(a) Au₂Pb单晶的ρ(T)曲线, 插图为30—60 K的实验数据; (b) 温度对磁化率χ的依赖, 体现了低温下的Meissner效应, 插图为不同温度下的M(H)曲线; (c) 不同垂直场下的ρ(T)曲线; (d) 不同温度下的ρ(H_⊥)曲线, 插图为归一化的上临界场与温度的关系, 红色虚线为p波超导拟合, 而黑色虚线为WHH理论所描述的s波超导拟合. 引自文献[ 52]

Fig. 1. Transport measurements of Au₂Pb: (a) ρ(T) curves of Au₂Pb single crystal. Inset: close-up of the same data from 30 to 60 K; (b) temperature dependence of χ, shows the Meissner effect: sharp diamagnetic drops at 1.15 K. The inset presents low-field M(H) curves at various temperatures from 0.5 to 1.1 K; (c) ρ(T) characteristics at various H_⊥ up to 0.05 T. The ρ(T) obtained in zero field shows $ T_{\rm c}^{\rm onset} = 1.3~{\rm K},~T_{\rm c}^{\rm zero} = 1.18~{\rm K} $ ; (d) magnetoresistance of Au₂Pb crystal at various temperatures under H_⊥. Inset: normalised upper critical field ${h^*} = {H_{{\rm{c}}2}}/{T_{\rm{c}}}\left( {{\rm{d}}{H_{{\rm{c}}2}}/{\rm{d}}{T_{T{\rm{}} = T_{\rm{c}}}}} \right)$ as a function of normalised temperature t = T/T_c, with the red dashed line indicating the expectation for a polar p-wave state. The black dashed line indicates the WHH theory for s-wave superconductor. From Ref. [52].

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图 2 Fatemi等^{[ 62]}的WTe₂栅压实验　(a)单层WTe₂晶体结构的示意图; (b) 实验样品的光学图像; (c) 底压为4 V、顶压为5 V时的R-T曲线, 插图表示60 mK时电阻随底压和顶压变化; (d)不同温度下的V-I特性曲线; (e)不同外磁场下的微分电导随电流变化曲线. 引自文献[ 62]

Fig. 2. Device schematic and superconductivity characteristics^{[ 62]}: (a) Cartoon illustration of the device structure and the crystal structure of monolayer WTe₂; (b) optical microscopy image of device 1; (c) temperature dependence of the resistance for V_bg = 4 V and V_tg = 5 V. The inset shows the resistance as a function of both gate voltages, at a base temperature of 60 mK; (d) V-I characteristics from base temperature (black) up to 940 mK (red); (e) nonlinear V-I behavior, captured by differential resistance curves, at base temperature for different perpendicular magnetic fields. From Ref. [ 62].

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图 3 Sajadi等^{[ 63]}的WTe₂栅压实验　(a)零外磁场下, 不同载流子浓度下的lnR_xx-T曲线, 其中出现明显的中间态平台. 插图为扫描线在相图中的位置; (b)载流子浓度n_e = 20 × 10¹² cm^–2时, 不同垂直场下的lnR_xx–T曲线. 插图: 扫温得到的特征温度 T_1/2和扫场得到特征磁场B_1/2; (c)载流子浓度n_e = 18 × 10¹² cm^–2时, 不同平行场下的lnR_xx–T曲线(B_// = 0 数据对应载流子浓度 n_e = 19 × 10¹² cm^–2; 剩余数据对应载流子浓度 n_e = 18 × 10¹² cm^–2). 插图: 特征温度T_1/2随平行场B_//的变化关系, 其中泡利极限B_P由灰色虚线表示; (d)不同垂直磁场下的lnR_xx–1/T曲线, 体现低温下电阻的饱和; (e)载流子浓度n_e = 15 × 10¹² cm^–2时, 不同温度下的R_xx–B_⊥曲线体现极低垂直磁场下电阻的升高; (f)和(e)相同载流子浓度下, 不同温度下的R_xx–B_//曲线, 体现相比(e)更为尖锐的电阻跳变. 插图: 线性坐标下(c)中的数据. 引自文献[ 63]

Fig. 3. Resistance characterization of WTe₂ device in the superconducting regime: (a) R_xx on log scale versus temperature T at a series of positive-gate doping levels n_e showing a drop of several orders of magnitude at low T for larger n_e. Inset: Locations of sweeps on the phase diagram; (b) effect of perpendicular magnetic field B_⊥ on resistance at the highest n_e value in (a). Inset: Chara-cteristic temperatures T_1/2 obtained from these temperature sweeps, as well as characteristic fields B_1/2 measured from field sweeps under similar conditions; (c) same as (b) but for the in-plane magnetic field B_// (the B_// = 0 data are for n_e = 19 × 10¹² cm^–2; the remaining data are for n_e = 18 × 10¹² cm^–2). Inset: Reduction of T_1/2 with B_//, fit to the expected form for materials with strong spin-orbit scattering (solid line). The Pauli limit B_P, assuming g = 2, is indicated by the dashed line; (d) data from (b) replotted to highlight the saturation of R_xx at low T; (e) sweeps of B_⊥ showing rise of resistance beginning at very low field; (f) sweep of B_// showing sharper onset of resistance comparing to (e). Inset: Data from (c) on a linear scale. From Ref. [ 63].

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图 4 高压诱导的Bi₂Te₃和Bi₂Se₃中的超导　(a) 不同压强下Bi₂Te₃中的超导相变^{[ 70]}; (b) Bi₂Te₃的相图和霍尔系数与压强的关系^{[ 70]}; (c) 不同压强下Bi₂Se₃中的超导相变^{[ 68]}; (d) Bi₂Se₃的相图和载流子浓度与压强的关系^{[ 68]}

Fig. 4. SC in Bi₂Te₃ and Bi₂Se₃ induced by pressure: (a) SC transition in Bi₂Te₃ at various pressures^{[ 70]}; (b) SC phase diagram and Hall coefficient as a function of pressure^{[ 70]} in Bi₂Te₃; (c) SC transition in Bi₂Se₃ at various pressures^{[ 68]}; (d) phase diagram and carrier density as a function of pressure in Bi₂Se₃^{[ 68]}.

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图 5 高压诱导的Cd₃As₂中的超导　(a), (b)不同压强下Cd₃As₂中的超导相变; (c) Cd₃As₂在高压下的相图. 引自文献[ 72]

Fig. 5. Superconductivity in Cd₃As₂ induced by pressure: (a), (b) SC transition in Cd₃As₂ at various pressures; (c) phase diagram of Cd₃As₂ under pressure. From Ref. [ 72].

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图 6 在第二类Weyl半金属WTe₂中高压诱导的超导和MoTe₂中高压增强的超导　(a) DFT计算得到的WTe₂的晶格常数和各向异性随压强的变化^{[ 65]}; (b)费米面在k_ak_b面内的轮廓线, 随着压强的施加, 费米面显著变大, 费米面附近态密度增大, 从而有利于超导配对^{[ 65]}; (c)上半图为根据实验结果绘制出的WTe₂相图, 绿色区域是高磁阻区, 红色区域是超导相; 下半图为1 T, 10 K下的霍尔系数(R_H)和压强的关系, 插图是其二阶导数, 阴影部分指出R_H反号和超导出现的压强区域^{[ 75]}; (d) 高压增强的MoTe₂超导上临界场和温度的关系表现出多带超导的特征^{[ 66]}; (e) MoTe₂在高压下的相图, 其中黑色和绿色符号分别是用电阻-温度关系和XRD测得的从1T' 相到Td相的相变温度^{[ 66]}

Fig. 6. Pressure induced SC in WTe₂ and pressure enhanced SC in MoTe₂: (a) Lattice parameters and c/a as a function of applied pressure calculated by density functional theory (DFT)^{[ 65]}; (b) calculated evolution of Fermi surface contour in ab plane at various pressures. Fermi surface enlarges substantially with the application of pressure, which is favorable for the formation of Cooper pairs^{[ 65]}; (c) upper panel: measured phase diagram of WTe₂. Green and red region respectively correspond to large magnetoresistance (LMR) and superconductivity (SC). Lower panel: Hall coefficient R_H at 1 T and 10 K as a function of pressure. Inset is its second order derivative. The shaded area indicates where R_H sign changes and SC takes place^{[ 75]}; (d) upper critical field as a function of temperature of pressure enhanced SC in MoTe₂, whose behavior is reminiscent of multi-band SC^{[ 66]}; (e) phase diagram of MoTe₂ under high pressure. Black and green symbols represent 1T' to Td structural phase transition temperature measured by resistivity and XRD^{[ 66]}.

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图 7 Cu_xBi₂Se₃ 中的非常规超导　(a) 能够发生超导相变的层间Cu掺杂Bi₂Se₃结构示意图^{[ 81]}; (b) 输运测得的Cu_0.12Bi₂Se₃超导在电阻率-温度)(ρ-T)关系中的反映, 左下角插图是超导转变区的放大, 上方插图为超导转变中电阻率随垂直于面的磁场强度的关系, 右下方插图是不同掺杂方式和比例的样品Seebeck系数随温度的关系, 说明了填隙方式掺入Cu有更好的掺杂效果^{[ 81]}; (c)在点接触谱中观察到的Cu_xBi₂Se₃超导的ZBCP^{[ 90]}; (d) STM观测到的Cu_0.31Bi₂Se₃在电子温度310 mK和0.2 T的垂直磁场下磁通涡旋中的ZBCP^{[ 84]}

Fig. 7. Unconventional superconductivity in Cu_xBi₂Se₃: (a) Structure of Cu intercalated Bi₂Se₃ where superconductivity is possible^{[ 81]}; (b) resistivity-temperature) relation(ρ-T) relation of Cu_0.12Bi₂Se₃. Lower left inset magnifies the superconductivity transition region. Upper inset shows resistivity as a function of perpendicular magnetic field. A third inset shows Seebeck coefficients of differently doped materials, where we can see doping effect is stronger for intercalated Bi₂Se₃^{[ 81]}; (c) ZBCP in point contact spectrum (PCS) of superconducting Cu_xBi₂Se₃^{[ 90]}; (d) ZBCP observed in vortex cores of Cu_0.31Bi₂Se₃ surface under 0.2 T perpendicular magnetic field at effective electron tempearature 310 mK by STM^{[ 84]}.

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图 8 掺S的MoTe₂(MoTe_1.8S_0.2)中的双带超导　(a) 不同温度下电阻率和磁场的关系, 在低温和低磁场下有明显的超导零电阻; (b) 超导的临界磁场和温度的关系, 可以用双带模型很好地拟合, 拟合参数指出很可能是非常规的s^+– 配对超导; (c) 超导电子热容和温度的关系, 可以用双带模型很好地拟合; (d) STM 在MoTe_1.8S_0.2的表面观测到的远大于体态的超导带隙, 谱形不能用常规的s波超导拟合. 引自文献[ 77]

Fig. 8. Superconductivity in S doped MoTe₂ (MoTe_1.8S_0.2): (a) ρ-B relation of MoTe₂ under various temperatures. Superconductivi-ty with zero resistance is clearly seen; (b) critical magnetic field as a function of temperature, which can be well fitted by a two-band mode. The fitting parameter supports unconventional s^+– pairing superconductivity; (c) superconducting electron heat capacity as a function of temperature can be well fitted by two-band mode; (d) superconductivity gap observed at the surface of MoTe_1.8S_0.2 by STM. The spectrum differs from what is expected for a conventional s wave superconductor. From Ref. [ 77].

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图 9 自旋分辨针尖对Bi₂Se₃薄膜上磁通探测的示意图　(a)在涡旋中心自旋向上极化的Andreev反射具有高电导值, 因为一个自旋向上入射的零能电子被自旋向上Majorana零能模反射为空穴; (b)在涡旋中心自旋向下极化的安德略夫反射具有低电导值, 因为入射电子和Majorana零能模的自旋无法配对. 引自文献[ 104]

Fig. 9. Schematic diagram of a spin selective tip on a vortex: (a) Illustration of spin selective Andreev reflection in spin polarized (M↑) STM/STS on a vortex center r = 0 in an interface of a topological insulator and s-wave superconductor. An incoming spin-up electron of zero energy is reflected as an outgoing spin-up hole induced by Majorana zero mode with spin-up at r = 0, which gives out a higher tunneling conductance; (b) an incoming spin-down electron of zero energy is reflected directly because of the mismatch of the spins of the electron and the Majorana zero mode, which results in a lower tunneling conductance. From Ref. [ 104].

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图 10 Bi(111)双层台阶边界态的Majorana零能模　(a) Bi(111)面上双层台阶的卡通示意图, 其中Nb (110)衬底提供超导近邻效应, 铁原子簇提供质量壁垒; (b) 紧束缚模型计算得到的空间分辨低能局域态密度图, 其中各颜色箭头与图(a)中对应; (c) 三种颜色箭头所在位置的点谱; (d), (e) 由紧束缚模型计算的能带结构, (d)中磁场与边缘A平行, 而(e)中磁场包含垂直边缘A的分量, 其中Δ_H为铁磁杂化的能隙, 而Δ_M为高简并点的Zeeman劈裂. 引自文献[ 109]

Fig. 10. Topological superconductivity and Majorana zero modes in the topological edge state of a Bi(111) bilayer: (a) Schematic representation of a hexagonal Bi bilayer island sitting on the surface of a Bi(111) thin film and exhibiting topological helical states on every other edge. Topological superconductivity DSC is induced into these helical states by superconducting proximity from the underlying Nb(110) substrate. Attaching a ferromagnetic cluster to the bilayer edge can open a magnetic hybridization gap. An MZM is localized at the mass domain wall, which is realized at the cluster-helical edge state interface, and can be detected in STM experiments; (b) spatially resolved low-energy local density of states (LDOS) calculated from a tight binding model for the edge state cluster arrangement shown in (a). The LDOS is a spectroscopic line cut taken along the A edge in (a); (c) point spectra extracted from the calculated spectroscopic line cut shown in (b) (positions indicated by the colored triangles); (d), (e) calculated band structure along the G-M direction from a tight-binding model of a Bi(111) bilayer, for which the A edge is coupled to the spin-polarized d-bands of a ferromagnetic cluster, resulting in a magnetic hybridization gap and a Zeeman gap. In (d), the cluster magnetization is parallel to the A edge M = (M_x, 0, 0); in (e), it has an additional component of the same amplitude perpendicular to the A edge M = (M_x, M_y, 0); The wave function weight on the Bi(111) edge in contact with the cluster is represented by symbol size and position on color scale. The magnetic hybridization gap, spanning the entire Brillouin zone, and the Zeeman gap at the high-symmetry point are indicated. From Ref. [ 109].

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图 11 三种点接触结构的示意图　(a) 用纳米微加工技术实现的隧穿结; (b) 使用银胶的软点接触; (c) 针尖硬点接触. 引自文献[ 118]

Fig. 11. Schematics of three common point contact confi-guration: (a) Tunneling junction fabricated by nanolithography; (b) soft point contact using silver paint; (c) hard point contact configuration, also called needle-anvil confi-guration. From Ref. [ 118].

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图 12 针尖点接触在拓扑半金属Cd₃As₂和TaAs中诱导的超导　(a) Cd₃As₂中零偏压点接触微分电导与温度的关系, 电阻的下降可以被磁场抑制, 这意味着它代表超导相变, 插图是针尖点接触实验装置示意图^{[ 121]}; (b) 不同温度下归一化的钨针尖Cd₃As₂点接触微分电导谱^{[ 121]}; (c), (d) PtIr针尖点接触在TaAs中测得的超导^{[ 123]}

Fig. 12. Tip induced superconductivity in topological semimetal Cd₃As₂ and TaAs: (a) Zero-bias point contact differential resistance of Cd₃As₂ as a function of temperature with W tip, magnetic field suppresses resistance drop, which signals a SC transition^{[ 121]}; (b) normalized point contact differential conductance spectrum of Cd₃As₂ with W tip^{[ 121]}; (c), (d) superconductivity in TaAs induced by PtIr tip^{[ 123]}.

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