A new numerical method for quasi-particle spectroscopy around a pair of vortices in triplet superconductors

Triplet superconductors such as Sr₂RuO₄ and Na_xCoO₂·yH₂O are now found to be p-wave (p_x±ip_y) or f-wave ((p_x±ip_y)coscp_z) superconductors. In conventional singlet superconductors, vortices are quantized because phase of order parameter must rotate by 2π around a vortex. But triplet superconductors have a degree of freedom of spin, which is described by d-vector. The d-vector and phase can rotate by π around a vortex, separately. Therefore appearance of HQVs is predicted. Theoretically, it is found that a pair of HQVs is more stable than a singly quantized vortex, for several parameter regions.In this study, in order to investigate quasi-particle bound states around two vortices in s-wave superconductors, we have developed a new numerical method to solve the BdG equation for two vortices state, using Mathieu functions. We confirmed the validity of this method for two vortices state and applied it in case of a pair of vortices. And we solved it.     

Quasi-particle excitations around a half-quantum vortex in p- and f- wave superconductors

Triplet superconductors such as Sr₂RuO₄ and Na_xCoO₂·yH₂O are now found to be p-wave (k_x ± ik_y) or f-wave ((k_x ± ik_y)cos ck_z) superconductors. Kee phenomenologically suggested that in these p-wave or f-wave superconductors, two half-quantum vortices (HQVs) become stable. Using Bougoliubov–de Gennes equation with the Fourier-Bessel expansion, we analyze quasi-particle excitations around an HQV at one end of the d-soliton for both p-wave and f-wave superconductors. We find that the bound state peak in the total local density of states around the HQV in f-wave superconductors becomes rather low compared to that around a singly quantized vortex. This is because, when flux and spin of the Cooper pairs are parallel, local density of states of quasi-particles shows bound state at zero energy. On the other hand, when flux and spin are anti-parallel, there is no phase singularity in the order parameter.     

Electronic properties of nano-structured anisotropic superconductors

Using the Bogoliubov-de Gennes equation on the tight-binding electron model, we have investigated nano-structured anisotropic superconductors. For a π/4-rotated square d-wave superconducting plate, the superconducting symmetry becomes s+id, not only at the surfaces, but over the whole superconductor. Also we have investigated quasi-particle structures of this superconducting state.     

Vortex structure in Sr2RuO4

The vortex structure in p-wave superconductors is investigated by the Bogoliubov–de Gennes theory on a tight-binding model. We calculate the temperature dependence of the electronic state at each site in the vortex lattice state, and show the difference between sin p_x+i sin p_y-wave and sin p_x−i sin p_y-wave superconducting state. Furthermore the relation of the electronic structure and the site-dependence of the nuclear magnetic relaxation time is also discussed.     

Induced p-wave superfluidity in two dimensions: Brane world in cold atoms and nonrelativistic defect CFTs

We propose to use a two-species Fermi gas with the interspecies s-wave Feshbach resonance to realize p-wave superfluidity in two dimensions. By confining one species of fermions in a two-dimensional plane immersed in the background three-dimensional Fermi sea of the other species, an attractive interaction is induced between two-dimensional fermions. We compute the pairing gap in the weak-coupling regime and show that it has the symmetry of p_x+ip_y. Because the magnitude of the pairing gap increases toward the unitarity limit, it is possible that the critical temperature for the p_x+ip_y-wave superfluidity becomes within experimental reach. The resulting system has a potential application to topological quantum computation using vortices with non-Abelian statistics. We also discuss aspects of our system in the unitarity limit as a “nonrelativistic defect conformal field theory (CFT)”. The reduced Schrödinger algebra, operator-state correspondence, scaling dimensions of composite operators, and operator product expansions are investigated.     

Mass flows and angular momentum density for px + ipy paired fermions in a harmonic trap

We present a simple two-dimensional model of a p_x + ip_y superfluid in which the mass flow that gives rise to the intrinsic angular momentum is easily calculated by numerical diagonalization of the Bogoliubov-de Gennes operator. We find that, at zero temperature and for constant director l, the mass flow closely follows the Ishikawa-Mermin-Muzikar formula .     

Superconducting slab in contact with thin superconducting layer at higher critical temperature

Within the framework of nonlinear time dependent Ginzburg–Landau equations (TDGL) we study the properties of a mesoscopic superconducting film with both surfaces in contact with a thin superconducting layer at a higher critical temperature. The properties of the layer are taken into account by the de Gennes boundary conditions via the extrapolation length b. We assume that the magnetic field is parallel to the multilayer interfaces. We obtain magnetization curves and calculate the spatial distribution of the superconducting electron density using a numerical method based on the technique of gauge invariant variables. This work tests both the rectangular cross-section size and b limit for the occurrence of vortices in a mesoscopic sample of area d_xxd_y where d_y = 80ξ(0) and dx varies discretely from 20ξ(0) to 3ξ(0). Our data also show a linear behavior of the magnetization curve and a power-law of order parameter modulus in limit b → 0_-.     

Two dimensional vortex structures in a superconductor slab at low temperatures

The dynamics of a two dimensional chain like structure of vortices is studied in the model of nonlinear time dependent Ginzburg–Landau equations (TDGL). The transition between different linear chains of vortices in a superconducting homogeneous slab with both surfaces in contact with a thin layer of metallic material is analyzed. The magnetization curve, vortex number, vortex configurations and modulus of the order parameter are studied as a function of the external magnetic field. We show how these vortex configurations are affected by the extrapolation length b (de Gennes boundary conditions), Ψ due to the proximity effects in a mesoscopic sample of area d_x × d_y, where d_y = 60ξ(0) and d_x varies discretely from 30ξ(0) to 12ξ(0). Possible connection with recent theoretical results in a two dimensional system of charged particles is discussed.     

铁磁超导态/绝缘层/自旋三重态p波超导体结的直流Josephson电流

由Bogoliubov-de Gennes方程得到铁磁超导共存态(FS)的自洽方程，利用推广的Furusaki-Tsukada的电流公式计算了铁磁超导态/绝缘层/自旋三重态p波超导体(FS/I/p)结的直流Josephson电流随结的温度、相位差以及FS中磁交换能、结界面的势垒散射强度的变化关系.研究表明：FS中磁交换能、结界面的势垒散射均抑制FS/I/p结的直流Josephson电流.当自旋三重态超导体具有p_x波配对势时，自旋三重态超导体结的直流Josephson电流随结两侧相位差的振荡周期是π. 关键词： 铁磁超导态 自旋三重态超导体 p波超导体 直流Josephson电流     

Quasi-particle transport in vortex state of nodal superconductors

The quasi-classical approach has proved very useful for studying the vortex state of nodal superconductors like d-wave superconductors in high T_c cuprates. After a brief introduction, we review our work on the thermal conductivity tensor in d-wave superconductors and f-wave superconductors.     

Exotic superconducting state embedded in the hidden order state of URu2Si2

The heavy-fermion compound URu₂Si₂ has mystified researchers since the superconducting state (T_c = 1.45 K) is embedded within the enigmatic ‘‘hidden order” phase (T_h = 17.5 K). Here, we report charge and thermal transport measurements on ultraclean single crystals of URu₂Si₂ with very large residual-resistivity-ratio down to 30 m K (∼T_c/50), which reveal a number of unprecedented superconducting properties. The results provide strong evidence for a new type of unconventional superconductivity with two distinct gaps having different nodal topology. We propose a gap function with chiral d-wave form Δ(k) = Δ₀k_z(k_x + ik_y). We also demonstrate that a distinct flux line lattice melting transition with outstanding characters occurs well below the upper critical fields even at sub-Kelvin temperature. The intriguing superconducting state of URu₂Si₂ adds a unique and exciting example to the list of unconventional superconductors.     

Nuclear spin-lattice relaxation rate in electron-doped Pr0.91LaCe0.09CuO4 − y: Constraint on the gap symmetry

We present numerical calculations of the nuclear spin-lattice relaxation (Rs) rate in the superconducting state of pure indium and slightly underdoped n-type cuprate Pr_0.91LaCe_0.09CuO_4 − y. By properly taking into account electron-phonon coupling, our calculated Rs for the conventional s-wave superconductor, indium, is in quantitative agreement the experimental data with a clear Hebel-Slichter peak. In contrast, the absence of the Hebel-Slichter peak in the Rs data of Pr_0.91LaCe_0.09CuO_4 − y can be explained by either d-wave or highly anisotropic s-wave gap symmetry. Thus, the absence of the Hebel-Slichter peak does not necessarily argue against an s-wave gap symmetry in this electron-doped cuprate.     

Quantum interference effects induced by multiple magnetic impurities on a triangular lattice f-wave superconductor

The effects of multi-impurity quantum interference on triangular lattice f-wave superconductors are studied by self-consistently solving Bogoliubov-de Gennes equations within the t?t′?J?V model. An overall phase diagram is presented, which shows that f-wave superconductivity dominates near 0.3 doping. Rich phenomena are induced by quantum interference effects, such as periodic modulations in charge orders, pyramid frustum structures, and a magnetic moment reverse transition, which are qualitatively different from the single-impurity case. We also examine the local density of states to show how localized quasiparticle states are created at or near the impurity sites, which can be directly measured by scanning tunneling microscopy experiments.     

Iron isotope effect on Tc in optimally-doped (Ba,K)Fe2As2 (Tc = 38 K) and SmFeAsO1−y (Tc = 54 K) superconductors

We report the iron isotope effect on a transition temperature (T_c) in an optimally-doped (Ba,K)Fe₂As₂ (T_c = 38 K) and SmFeAsO_1−y (T_c = 54 K) superconductors. In order to obtain the reliable isotope shift in T_c, twin samples with different iron isotope mass are synthesized in the same conditions (simultaneously) under high-pressure. We have found that (Ba,K)Fe₂As₂ shows an inverse iron isotope effect α_Fe = −0.18 ± 0.03 while SmFeAsO_1−y shows a small iron isotope effect α_Fe = −0.02 ± 0.01, where the isotope exponent α is defined by T_c ∼ M^−α (M is the isotopic mass). The results show that α_Fe changes in the iron-based superconductors depending on the system. The distinct iron isotope effects imply the exotic coupling mechanism in the iron-based superconductors.     

欠掺杂高温超导体中的涡旋电荷结构相变

利用平均场t-t′-U-V-V_c模型,通过自洽求解Bogoliubov-de Gennes方程,研究了高温超导体中涡旋结构的相变.发现增大原位排斥势U,自旋密度波、电荷密度波以及d波序参量由棋盘结构转变为条纹结构.模型哈密顿量中引入合适强度的长程库仑势后,欠掺杂高温超导体样品中也可以出现二维或者棋盘结构,结果与文献报道的扫描隧道显微镜实验结果一致. 关键词： 高温超导 涡旋结构 长程库仑势     

Quasi-particle structure around a single vortex in high-Tc superconductors

For analyzing the checker-board like modulation of the local density of states (LDOS) around a vortex observed in the slightly overdoped Bi₂Sr₂CaCu₂O_x, we examined the effect of pseudogap state of high-T_c superconductors to the LDOS around the vortex. We first derived the Bogoliubov-de Gennes equation for d-wave superconductivity (d-SC) in the presence of d-spin density wave (d-SDW). Using the Fourier–Bessel expansion, we solved this equation for a single vortex state, numerically. We found that the peak of the bound states around E = 0 becomes small and modulation of the LDOS is observed for larger d-SDW order parameter.     

Nanostructuring of Fe films by oblique incidence deposition on a FeSi2 template onto Si(1 1 1): Growth, morphology, structure and faceting

The growth of thin Fe films deposited at oblique incidence on an iron silicide template onto Si(1 1 1) single crystal has been investigated as a function of Fe thickness (0 < t_Fe ? 180 monolayers (MLs)) and incidence angle (0 ? θ ? 80°). The growth mode is determined in situ by means of scanning tunnelling microscopy (STM) and low energy electron diffraction (LEED). Stripes oriented perpendicularly to the incident atomic flux are formed for θ ? 30°. Self-correlation functions are used to extract characteristic lengths from STM images. The correlation lengths in the direction of the incident flux (ξ_x) and perpendicular to the atomic flux (ξ_y) grow with different powers versus time (ξ_x∝t^σ and ξ_y∝t^ρ, with σ = 0.34 ± 0.03 and ρ = 0.67 ± 0.03) following the exact solution of the (1 + 1) dimensional Kardar-Parisi-Zhang (KPZ) equation. The root mean square roughness follows also a scaling law for t_Fe < 120 ML leading to a growth exponent β = 0.73 ± 0.02. Shadowing and steering effects are discussed on the basis of our STM data.     

Ferromagnetic insulator effects in ferromagnetic semiconductor/ferromagnetic insulator/p-wave superconductor junctions

The effect of the ferromagnetic insulator on tunneling conductance in ferromagnetic semiconductor/ferromagnetic insulator/p-wave superconductor (FS/FI/P) junctions is studied based on a scattering theory. Three kinds of pairings for the P side are chosen: p_x, p_{y ,}p_x+ip_y waves. It is shown that the spin filtering effect originating from the exchange field in the FI strongly modifies the normalization conductance. Many novel features including the zero-bias conductance dip and splitting are exhibited for fixed spin polarization in the FS. The tunneling spectrum for the heavy holes is much different from that for the light holes due to the different mismatches in the effective mass and Fermi velocity between FS and P.     

Nonlinear generation of vorticity in thin smectic films

The energy levels of the fermions bound to the vortex are considered for vortices in the superfluid/superconducting systems that contain the symmetry protected plane of zeroes in the gap function in bulk. The Caroli–de Gennes–Matricon branches with different approach zero energy level at p_z → 0. The density of states of the bound fermions diverges at zero energy giving rise to the \(\sqrt \Omega \) dependence of the density of states in the polar phase of superfluid ³He rotating with the angular velocity Ω and to the \(\sqrt B \) dependence of the density of states for superconductors in the (d_xz + id_yz)-wave pairing state.     

Scanning Hall probe microscopy of vortex matter

Scanning Hall probe microscopy (SHPM) is a novel scanned probe magnetic imaging technique whereby the stray fields at the surface of a sample are mapped with a sub-micron semiconductor heterostructure Hall probe. In addition an integrated scanning tunnelling microscope (STM) or atomic force microscope (AFM) tip allows the simultaneous measurement of the sample topography, which can then be correlated with magnetic images. SHPM has several advantages over alternative methods; it is almost completely non-invasive, can be used over a very wide range of temperatures (0.3–300 K) and magnetic fields (0–7 T) and yields quantitative maps of the z-component of magnetic induction. The approach is particularly well suited to low temperature imaging of vortices in type II superconductors with very high signal:noise ratios and relatively high spatial resolution (>100 nm). This paper will introduce the design principles of SHPM including the choice of semiconductor heterostructure for different measurement conditions as well as surface tracking and scanning mechanisms. The full potential of the technique will be illustrated with results of vortex imaging studies of three distinct superconducting systems: (i) vortex chains in the “crossing lattices” regime of highly anisotropic cuprate superconductors, (ii) vortex–antivortex pairs spontaneously nucleated in ferromagnetic-superconductor hybrid structures, and (iii) vortices in the exotic p-wave superconductor Sr₂RuO₄ at milliKelvin temperatures.