The superconductivity of a kagome superconductor CsV3Sb5 is studied by scanning tunneling microscopy / spectroscopy at an ultralow temperature with high resolution. Two kinds of superconducting gaps with multiple sets of coherent peaks and residual zero-energy density of states are observed on both half-Cs and Sb surfaces, implying multiband superconductivity with gap nodes. Sixfold star-shaped magnetic vortex is observed with conventional Caroli-de Gennes-Matricon bound states inside. Magnetic impurities suppress the superconductivity, while nonmagnetic impurities do not, suggesting the absence of sign-change in the superconducting order parameter. Moreover, the interplay between charge density waves and superconductivity differs on various bands, resulting in different density of state distributions. Our study provides critical clues for further understanding the superconductivity and its relation to charge density waves in CsV3Sb5.
We studied the effects of multiband electronic structure on the thermoelectric and electrical transport properties in the normal state of kagome superconductors $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs). In all three members, the multiband nature is manifested by sign changes in the temperature dependence of the Seebeck and Hall resistivity, together with sublinear response of the isothermal Nernst and Hall effects to external magnetic fields in the charge ordered state. Moreover, ambipolar transport effects appear ubiquitously in all three systems, giving rise to sizable Nernst signal. Finally, possib…
Classification of Lifshitz invariant in multiband superconductors: an application to Leggett modes in the linear response regime in Kagome lattice models
We investigate the temperature dependence of the upper critical field $H_{c 2} $ as a tool to probe the possible presence of multiband superconductivity at the interface of LAO/STO. The behaviour of $H_{c 2} $ can clearly indicate two-band superconductivity through its nontrivial temperature dependence. For the disorder scattering dominated two-dimensional LAO/STO interface we find a characteristic non-monotonic curvature of the $H_{c 2} (T) $. We also analyse the $H_{c 2} $ for multiband bulk STO and find similar behaviour.
We report a self-consistent microscopic theory of characteristic length scales, vortex structure and type-1.5 superconducting state in two-band systems using two-band Eilenberger formalism.
The thermal conductivity of the heavy fermion superconductor PrOs4Sb12 was measured down to Tc/40 throughout the vortex state. At lowest temperatures and for magnetic fields H ~ 0.07Hc2, already 40% of the normal state thermal conductivity is restored. This behaviour (similar to that observed in MgB2) is a clear signature of multiband superconductivity in this compound.
A theoretical overview of the classes of superconductors encompassing (a) high-Tc cuprate, (b) iron-based and (c) aromatic superconductors is given. Emphasis is put on the multiband natures of all the three classes, where the differences in the multiorbits and their manifestations in the electronic structures and pairing are clarified. From these, future directions and prospects are discussed.
We show that a clean multiband superconductor may display one or several phase transitions with increasing temperature from or to frustrated configurations of the relative phases of the superconducting order parameters. These transitions may occur when more than two bands are involved in the formation of the superconducting phase and when the number of repulsive interband interactions is odd. These transitions are signalled by slope changes in the temperature dependence of the superconducting gaps.
We study the superconductivity in the three-dimensional multiband d-p model, in which a Cu$_2$O$_3$-ladder layer and a CuO$_2$-chain layer are alternately stacked, as a model of the superconducting spin-chain ladder cuprate. $p_z$-Wave-like triplet superconductivity is found to be the most stable, and its dependence on interlayer coupling can explain the superconducting transition temperature dependence on pressure in real superconducting spin-chain ladder cuprates. The superconductivity may be enhanced if hole transfer from the chain layer to the ladder layer can be promoted beyond the typica…
In strontium titanate, the Froehlich electron - LO-phonon interaction dominates the electron response and can also provide superconductivity. Because of high LO-phonon frequencies in SrTiO3, the superconducting system is non-adiabatic. We demonstrate that the dielectric function approach is an adequate theoretical method for superconductivity in SrTiO3 and on the SrTiO3-LaAlO3 interface. The critical temperatures are calculated using realistic material parameters. The obtained critical temperatures are in line with experimental data both for bulk and interface superconductivity. The present me…
The upper critical field Hc2(T) of the multiband superconductor KFe2As2 has been studied via low-temperature thermal expansion and magnetostriction measurements. We present compelling evidence for Pauli-limiting effects dominating Hc2(T) for H || a, as revealed by a crossover from second- to first-order phase transitions to the superconducting state in the magnetostriction measurements down to 50 mK. Corresponding features were absent for H || c. To our knowledge, this crossover constitutes the first confirmation of Pauli limiting of the Hc2(T) of a multiband superconductor. The results are su…
We present a method to produce superconducting tips to be used in Scanning Tunneling Microscopy/Spectroscopy experiments. We use these tips to investigate the evolution of the electronic density of states of NbSe2 from 0.3K up to its critical temperature (7.2K). The use of a superconducting tip (Pb) as ounterelectrode provides an enhancement of the different features related to the DOS of NbSe2 in the tunneling conductance curves, along all the studied thermal range. The analysis of the experimental results gives evidence of the presence of multiband superconductivity in NbSe2.
Ta$_4$Pd$_3$Te$_{16}$ is a recently discovered superconductor that has been suggested to be an unconventional superconductor near magnetism. We report electronic structure calculations showing that in spite of the layered crystal structure the material is an anisotropic three dimensional metal. The Fermi surface contains prominent one and two dimensional features, including nested 1D sheets, a 2D cylindrical section and a 3D sheet. The electronic states that make up the Fermi surface are mostly derived from Te $p$ states, with small Ta $d$ and Pd $d$ contributions. This places the compound far…
A mean-field treatment is presented of a square lattice two-orbital-model for ${\rm BiS_2}$ taking into account intra- and inter-orbital superconductivity. A rich phase diagram involving both types of superconductivity is presented as a function of the ratio between the couplings of electrons in the same and different orbitals (${\rm η= V_{XX}/V_{XY}}$) and electron doping $x$. With the help of a quantity we call orbital-mixing ratio, denoted as $R(ϕ)$, the phase diagram is analyzed using a simple and intuitive picture based on how $R(ϕ)$ varies as electron doping increases. The predictive pow…
We report point contact Andreev Reflection (PCAR) measurements on a high-quality single crystal of the non-centrosymmetric superconductor Re6Zr. We observe that the PCAR spectra can be fitted by taking two isotropic superconducting gaps with Delta_1 ~ 0.79 meV and Delta_2 ~ 0.22 meV respectively, suggesting that there are at least two bands which contribute to superconductivity. Combined with the observation of time reversal symmetry breaking at the superconducting transition from muon spin relaxation measurements (Phys. Rev. Lett. 112, 107002 (2014)), our results imply an unconventional super…
The pseudogap (PG) excitations in the framework of multiband superconductivity are analysed. Calculations for the doping phase diagram of electron-doped cuprate superconductors have been made. A nonrigid multiband model has been used. The interband pairing acts between the upper Hubbard band (UHB) and midgap states created (and shifted upwards) by doping. The appearance of two PG and two phase diagram critical points (CP) is suggested. The larger antinodal PG reflects the destruction of the long-range antiferromagnetic (AF) order. The second, smaller PG is a near-nodal event with exposed dopin…
Superconductivity in Pb, H under extreme pressure and CaBeSi, in the framework of the density functional theory for superconductors, is discussed. A detailed analysis on how the electron-phonon and electron-electron interactions combine together to determine the superconducting gap and critical temperature of these systems is presented. Pb, H under pressure and CaBeSi are multigap superconductors. We will address the question under which conditions does a system exhibits this phenomenon. The presented results contribute to the understanding of multiband and anisotropic superconductivity, which…
We study three-dimensional time-reversal-invariant topological superconductivity in noncentrosymmetric materials such as RhSi, CoSi, and AlPt which host coupled multifold nodes energetically split by the spin-orbit coupling at the same time-reversal-invariant momentum (TRIM). The topological superconductivity arises from the $s_{+} \oplus s_{-}$ gap function, which is $\boldsymbol{k}$ independent, but with opposite signs for the two nodes split at the same TRIM. We consider various electron-electron interactions in the tight-binding model for RhSi and find that the topological superconducting …
I solved the Eliashberg equations for multiband non-phononic $s\pm$ wave spin glass superconductor and I calculated the temperature dependence of the gaps and superfluid density that reveals unusual behavior as non monotonic temperature dependence and reentrant superconductivity. For particular values of input parameters the phase diagram is still more complex with two different ranges of temperature where the superconductivity appears.
We report measurements of the London penetration depth $Δλ(T)$ and the electronic specific heat $C_e(T)$ on high-quality single crystals of the filled-skutterudite superconductor PrPt$_4$Ge$_{12}$ ($T_c\simeq$8K). Both quantities show a weak temperature dependence at $T\ll T_c$, following $Δλ\sim T^n$ ($n\simeq 3.2$) and $C_e/T\sim T^{2.8}$. Such temperature dependences deviate from both conventional s-wave type and nodal superconductivity. A detailed analysis indicates that the superfluid density $ρ_s(T)$, derived from the penetration depth, as well as the electronic specific heat can be cons…
Point-contact spectroscopy was originally developed for the determination of the electron-phonon spectral function in normal metals. However, in the past 20 years it has become an important tool in the investigation of superconductors. As a matter of fact, point contacts between a normal metal and a superconductor can provide information on the amplitude and symmetry of the energy gap that, in the superconducting state, opens up at the Fermi level. In this paper we review the experimental and theoretical aspects of point-contact spectroscopy in superconductors, and we give an experimental surv…
We carried out the measurements of magnetoresistance, magnetic susceptibility and specific heat on crystals of the low-dimensional transition metal telluride Ta$_4$Pd$_3$Te$_{16}$. Our results indicate that Ta$_4$Pd$_3$Te$_{16}$ is an anisotropic type-II superconductor with the extracted Ginzburg-Landau parameter $κ_{\text{GL}}=$ 84. The upper critical field $H_{c2}$($T$) shows a linear dependence at low temperature and the anisotropy of $H_{c2}$($T$) is strongly $T$-dependent, both of which indicate a multiband scenario. A detailed analysis reveals that the electronic specific heat $C_{\text{…
Recently it has been demonstrated that the pair-exchange coupling of quasi-one-dimensional (Q1D) bands with conventional higher-dimensional bands in one multiband superconducting material can result in the formation of robust aggregate pair condensate. In particular, it has been found that the Q1D thermal pair fluctuations are suppressed in the presence of deep conventional band(s), where the Fermi level is much larger than the characteristic cut-off energy. Here we report that impact of the Q1D fluctuations is significantly weakened even in the presence of nearly shallow higher-dimensional ba…
We report 31P-NMR and specific heat measurements on an iron (Fe)-based superconductor SrFe2(As0.65P0.35)2 with Tc=26 K, which have revealed the development of antiferromagnetic correlations in the normal state and the unconventional superconductivity(SC) with nodal gap dominated by the gapless low-lying quasiparticle excitations. The results are consistently argued with an unconventional multiband SC state with the gap-size ratio of different bands being significantly large; the large full gaps in s\pm-wave state keep Tc high, whereas a small gap with a nodal-structure causes gapless feature u…
We have investigated the specific heat of optimally-doped iron chalcogenide superconductor Fe(Te0.57Se0.43) with a high-quality single crystal sample. The electronic specific heat Ce of this sample has been successfully separated from the phonon contribution using the specific heat of a non-superconducting sample (Fe0.90Cu0.10)(Te0.57Se0.43) as a reference. The normal state Sommerfeld coefficient gamma_n of the superconducting sample is found to be ~ 26.6 mJ/mol K^2, indicating intermediate electronic correlation. The temperature dependence of Ce in the superconducting state can be best fitted…
We study the intrinsic anisotropy of the superconducting state in KFe$_2$As$_2$, using small-angle neutron scattering to image the vortex lattice as the applied magnetic field is rotated towards the FeAs crystalline planes. The anisotropy is found to be strongly field dependent, indicating multiband superconductivity. Furthermore, the high field anisotropy significantly exceeds that of the upper critical field, providing further support for Pauli limiting in KFe$_2$As$_2$ for field applied in the basal plane. The effect of Pauli paramagnetism on the unpaired quasiparticles in the vortex cores …
Emergence of superconductivity in a two-band system coupled with breathing and Jahn-Teller phonons is discussed in a weak-coupling limit. With the use of a standard quantum mechanical procedure, the phonon-mediated attraction is derived. From the analysis of the model including such attraction, a BCS-like formula for a superconducting transition temperature $T_{\rm c}$ is obtained. When only the breathing phonon is considered, $T_{\rm c}$ is the same as that of the one-band model. On the other hand, when Jahn-Teller phonons are active, $T_{\rm c}$ is significantly enhanced by the interband att…
Exotic quantum phase transitions in metals, such as the nematic and smectic states, were discovered one after another and found to be universal now. The emergence of unconventional density-wave order in frustrated kagome metal AV$_3$Sb$_5$ and its interplay with exotic superconductivity attract increasing attention. We reveal that the smectic bond-density-wave is naturally caused by the paramagnon interference mechanism, because strong scatterings among different van-Hove singularity points are induced. In addition, the fluctuations of the bond-order induce sizable "beyond-Migdal" pairing glue…
Measurements of the London penetration depth and tunneling conductance in single crystals of the recently discovered stoicheometric, iron - based superconductor, CaKFe$_4$As$_4$ (CaK1144) show nodeless, two effective gap superconductivity with a larger gap of about 6-9 meV and a smaller gap of about 1-4 meV. Having a critical temperature, $T_{c,onset}\approx$35.8 K, this material behaves similar to slightly overdoped Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ (e.g. $x=$0.54, $T_c \approx$ 34 K)---a known multigap $s_{\pm}$ superconductor. We conclude that the superconducting behavior of stoichiometric CaK11…
We examine the impact of interfacial phonons on the superconducting state of FeSe/SrTiO$_3$ developing a material's specific multiband, full bandwidth, and anisotropic Eliashberg theory for this system. Our self-consistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak-coupling constant λ$_m$ = 0.4, in mediating the high T$_c$, and explain other puzzling experimental observations, such as the s-wave symmetry and replica bands. We discover that the formation of replica bands has a T$_c$ decreasing effect that is ne…
We report muon spin rotation ($μ$SR) experiments together with first-principles calculations on microscopic properties of superconductivity in the kagome superconductor LaRu$_3$Si$_2$ with $T_{\rm c}$ ${\simeq}$ 7K. We find that the calculated normal state band structure features a kagome flat band and Dirac as well as van Hove points formed by the Ru-$dz^2$ orbitals near the Fermi level. Below $T_{\rm c}$, $μ$SR reveals isotropic type-II superconductivity, which is robust against hydrostatic pressure up to 2 GPa. Intriguingly, the ratio 2$Δ/k_{\rm B}T_{\rm c}$ ${\simeq}$ 4.3 (where $Δ$ is the…
A group of newly discovered non-magnetic metal kagome structures AV3Sb5 (A=K, Rb, Cs) have aroused widespread interest in experiment and theory due to their unusual charge density wave (CDW) and intertwined superconductivity. However, they all possess weak electron-phonon coupling (EPC) and low superconducting transition temperature. Here, we performed high-throughput first-principles calculations on novel kagome candidates with AV3Sb5 prototype structure, and proposed 24 dynamically stable novel kagome metals. The calculation based on Bardeen-Cooper-Schrieffer theory shows that most of these …
Prediction of 1:1 kagome metals with superconductivity and band topology
School of Physics and Physical Engineering, Qufu Normal University（曲阜师范大学物理与光电工程学院）；Hunan Provincial Key laboratory of Thin Film Materials and Devices, School of Materials Science and Engineering, Xiangtan University（湘潭大学材料科学与工程学院薄膜材料与器件湖南省重点实验室）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
Time-Reversal Symmetry Breaking Superconducting State and Collective Modes in Kagome Superconductors
Capital Normal University（首都师范大学）；Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学理论科学研究所）；Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所北京凝聚态物理国家研究中心）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
Original definition of the pseudogap [11,12] allowes to connect it conceptually with a topological insulator. It concerns the multiband superconductivity background with variable chemical potential position.
The interplay of superconductivity with electronic and structural instabilities on the kagome lattice provides a fertile ground for the emergence of unusual phenomena. The vanadium-based kagome metals $A$V$_3$Sb$_5$ ($A=$ K, Rb, Cs) exhibit superconductivity on an almost ideal kagome lattice, with the superconducting transition temperature $T_{\rm c}$ forming two domes upon pressure-tuning. The first dome arises from the competition between superconductivity and a charge-density-wave, whereas the origin for the second dome remains unclear. Herein, we show that the appearance of the second supe…
Superconductivity and Electron Correlations in Kagome Metal LuOs3B2
Southwest Jiaotong University（西南交通大学）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交大李政道研究所）；Shanghai Jiao Tong University（上海交通大学）；Zhejiang University（浙江大学）；Fujian Provincial Collaborative Innovation Center for Advanced High-Field Superconducting Materials and Engineering（福建省先进强磁场超导材料与工程协同创新中心）；Fujian Normal University（福建师范大学）
Materials with transition metals in triangular lattices are of great interest for their potential combination of strong correlation, exotic magnetism and electronic topology. Kagome nets are of particular importance since the discovery of geometrically frustrated magnetism and topological band structures in crystals like Herbertsmithite and Fe3Sn2, respectively. KV3Sb5 was discovered to be a layered topological metal with a Kagome net of vanadium. Here, we fabricated Josephson Junctions (JJ) of K1-xV3Sb5 and induced superconductivity over long junction lengths. Through magnetoresistance and cu…
Optical conductivity signatures of strong correlations and multiband superconductivity in infinite-layer nickelates
Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory（斯坦福材料能源科学研究所，SLAC国家加速器实验室）；Department of Applied Physics, Stanford University（斯坦福大学应用物理系）；Department of Material Science Engineering, Pusan National University（釜山国立大学材料科学与工程系）；Department of Physics, Stanford University（斯坦福大学物理系）；Center for Correlated Electron Systems, Institute for Basic Science（基础科学研究院相关电子系统中心）；Department of Physics and Astronomy, Seoul National University（首尔国立大学物理与天文学系）；State Key Laboratory of Surface Physics and Department of Physics, Fudan University（复旦大学表面物理国家重点实验室及物理系）；Department of Physics, Hanyang University（汉阳大学物理系）
The entanglement of charge density wave (CDW), superconductivity, and topologically nontrivial electronic structure has recently been discovered in the kagome metal $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs) family. With high-resolution angle-resolved photoemission spectroscopy, we study the electronic properties of CDW and superconductivity in CsV$_3$Sb$_5$. The spectra around $\bar{K}$ is found to exhibit a peak-dip-hump structure associated with two separate branches of dispersion, demonstrating the isotropic CDW gap opening below $E_{\rm F}$. The peak-dip-hump lineshape is contributed by linearly di…
Topological superconductors (TSC) become a focus of research due to the accompanying Majorana fermions. However, the experimentally reported TSC are extremely rare. The recent experiments reported the kagome TSC AV$_{3}$Sb$_{5}$ (A=K, Rb, Cs), which exhibit unique superconductivity, topological surface states (TSS), and Majorana bound states. More recently, the first titanium-based kagome superconductor CsTi$ _{3} $Bi$ _{5} $ with nontrivial topology was successfully synthesized as a perspective TSC. Given that Cs contributes little to the electronic structures of CsTi$ _{3} $Bi$ _{5} $ and bi…
The newly discovered Kagome superconductors $A$V$_3$Sb$_5$ ($A$ = K, Rb, Cs) exhibit superconductivity, charge order, and band topology simultaneously. To explore the intricate interplay between the superconducting and charge orders, we investigate the doping evolution of superconductivity, charge-density-wave (CDW) order, and band topology in doped topological kagome superconductors Cs(V$_{1-x}$Ti$_x$)$_3$Sb$_5$ where the Ti-dopant introduces hole-like charge carriers. Despite the absence of the CDW phase transition in doped compounds even for the lowest doping level of $x=0.047$, the superco…
Scanning tunneling experiments on single crystals of superconducting FeTe$_{0.55}$Se$_{0.45}$ have recently provided evidence for discrete energy levels inside vortices. Although predicted long ago, such levels are seldom resolved due to extrinsic (temperature, instrumentation) and intrinsic (quasiparticle scattering) limitations. We study a microscopic multiband model with parameters appropriate for FeTe$_{0.55}$Se$_{0.45}$. We confirm the existence of well-separated bound states and show that the chemical disorder due to random occupation of the chalcogen site does not affect significantly t…
We study the emergence of multiband superconductivity with $s$- and $d-$wave symmetry on the background of spin density wave (SDW). We show that the SDW coherence factors renormalize the momentum dependence of the superconducting (SC) gap, yielding a SC state with an \emph{unconventional} s-wave symmetry. Interband Cooper pair scattering stabilizes superconductivity in both symmetries. With increasing SDW order, the s-wave state is more strongly suppressed than the d-wave state. Our results are universally applicable to two-dimensional systems with a commensurate SDW.
A recent Little-Parks experiment on Kagome-structured superconductor CsV_3Sb_5 demonstrated remarkable resistance oscillations with period \phi0/3=hc/6e. Here, we perform analysis based on a theory involving three 2e superconductivity (SC) order parameters associated with the three reciprocal lattice vectors which connect M points of the hexagonal Brillouin zone. In a ring geometry we unveil that, as a series of metastable states, phase of one SC order parameter winds 2πmore or less than the other two ones around the ring, which yields local free energy minima at integer multiples of \phi0/3. …
Recent discovery of high T_c superconductivity in Fe-based compounds may have opened a new pathway to the room temperature superconductivity. The new materials feature FeAs layers instead of the signature CuO_2 planes of much-studied cuprates. A model Hamiltonian describing FeAs layers is introduced, highlighting the crucial role of puckering of As atoms in promoting d-electron itinerancy and warding off large local-moment magnetism of Fe ions, the main enemy of superconductivity. Quantum many-particle effects in charge, spin and multiband channels are explored and a nesting-induced spin densi…
Superconductivity has been first observed in TlNi$_2$Se$_2$ at T$_C$=3.7 K and appears to involve heavy electrons with an effective mass $m^*$=14$\sim$20 $m_b$, as inferred from the normal state electronic specific heat and the upper critical field, H_${C2}$(T). Although the zero-field electronic specific heat data, $C_{es}(T)$, in low temperatures (T < 1/4 T$_C$) can be fitted with a gap BCS model, indicating that TlNi$_2$Se$_2$ is a fully gapped superconductor, the two-gap BCS model presents the best fit to all the $C_{es}(T)$ data below $T_C$. It is also found that the electronic specific h…
The recently discovered kagome metal series $A$V$_3$Sb$_5$ ($A$=K, Rb, Cs) exhibits topologically nontrivial band structures, chiral charge order and superconductivity, presenting a unique platform for realizing exotic electronic states. The nature of the superconducting state and the corresponding pairing symmetry are key questions that demand experimental clarification. Here, using a technique based on the tunneling diode oscillator, the magnetic penetration depth $Δλ(T)$ of CsV$_3$Sb$_5$ was measured down to 0.07 K. A clear exponential behavior in $Δλ(T)$ with marked deviations from a $T$ o…
The kagome metal KV$_3$Sb$_5$ hosts charge order, topologically nontrivial Dirac band crossings, and a superconducting ground state with unconventional characteristics, providing an ideal platform to investigate the interplay between different electronic states on the kagome lattice. Here we study the evolution of charge order and superconductivity in KV$_3$Sb$_5$ under hydrostatic pressure using electrical resistivity measurements. With the application of pressure, the superconducting transition temperature $T_{\rm c}=0.9$ K under ambient pressure quickly increases to 3.1 K at $p=0.4$ GPa, as…
Cavity Tuning of the CDW--Superconductivity Interplay in a Kagome Metal
Tsientang Institute for advaced study（钱塘高等研究院）；Songshan Lake Materials Laboratory（松山湖材料实验室）；Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science（马普物质结构与动力学研究所及自由电子激光科学中心）；Department of Materials Science and Engineering, City University of Hong Kong（香港城市大学材料科学与工程系）；Department of Physics and Photon Science, Gwangju Institute of Science and Technology (GIST)（光州科学技术院物理与光子科学系）；The Flatiron Institute（平顿研究所）
