One-dimensional Superconductor Research Articles

Analytical investigation of omnidirectional photonic band gaps in one-dimensional superconductor–dielectric photonic crystals

In this paper, the first two total omnidirectional photonic band gaps (OBGs) of one-dimensional (1D) superconductor–dielectric photonic crystals (SDPCs) which composed of dielectric–dielectric and superconductor–dielectric periodic structures have been investigated theoretically by using the transfer matrix method (TMM). Those OBGs originate from Bragg gaps. From the analysis, it is found that this kind of heterostructure has wider frequency ranges of total OBGs for both polarizations in comparison to the dielectric photonic crystals (DPCs) or SDPCs, and the bandwidths of OBGs can be notably enlarged by increasing the thickness of superconductor layer and decreasing the ambient temperature.

Stability of Majorana fermions in proximity-coupled topological insulator nanowires

It has been shown previously that a finite-length topological insulator nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and subject to a longitudinal applied magnetic field, realizes a one-dimensional topological superconductor with an unpaired Majorana fermion (MF) localized at each end of the nanowire. Here, we study the stability of these MFs with respect to various perturbations that are likely to occur in a physical realization of the proposed device. We show that the unpaired Majorana fermions persist in this system for any value of the chemical potential inside the bulk band gap of order 300 meV in Bi$_2$Se$_3$ by computing the Majorana number. From this calculation, we also show that the unpaired Majorana fermions persist when the magnetic flux through the nanowire cross-section deviates significantly from half flux quantum. Lastly, we demonstrate that the unpaired Majorana fermions persist in strongly disordered wires with fluctuations in the on-site potential ranging in magnitude up to several times the size of the bulk band gap. These results suggest this solid-state system should exhibit unpaired Majorana fermions under accessible conditions likely important for experimental study or future applications.

Interplay of Disorder and Interaction in Majorana Quantum Wires

We study the interplay between disorder and interaction in one-dimensional topological superconductors which carry localized Majorana zero-energy states. Using Abelian bosonization and the perturbative renormalization group approach, we obtain the renormalization group flow and the associated scaling dimensions of the parameters and identify the critical points of the low-energy theory. We predict a quantum phase transition from a topological superconducting phase to a nontopological localized phase, and obtain the phase boundary between these two phases as a function of the electron-electron interaction and the disorder strength in the nanowire. Based on an instanton analysis which incorporates the effect of disorder, we also identify a large regime of stability of the Majorana-carrying topological phase in the parameter space of the model.

Omnidirectional photonic band gaps enlarged by Fibonacci quasi-periodic one-dimensional ternary superconductor photonic crystals

First two omnidirectional photonic band gaps (OBGs) realized by one-dimensional (1D) Fibonacci quasi-periodic structure which is composed of superconductor and two kinds of isotropic dielectric are theoretically studied by the transfer matrix method (TMM). From the numerical results, it has been shown that such OBGs are insensitive to the incident angle and the polarization of electromagnetic wave (EM wave), and the frequency ranges and central frequencies of OBGs cease to change with increasing Fibonacci order, but vary with the ambient temperature of system and the thickness of the superconductor, respectively. The bandwidths of OBGs can be notably enlarged. The damping coefficient of superconductor layers has no effect on the frequency ranges of OBGs. Those OBGs originate from a Bragg gap in contrast to zero-n˜ gap or single negative (negative permittivity or negative permeability) gap. It has been proved that Fibonacci quasi-periodic 1D ternary superconductor dielectric photonic crystals (SDPCs) have a superior feature in the enhancement of OBGs frequency ranges compared with the conventional binary dielectric photonic crystals (DPCs).

Unwinding of a one-dimensional topological superconductor

We show that a topological superconductor made of four chains of superconducting spinless fermions characterized by four Majorana edge states can adiabatically be deformed into a trivial band insulator. To unwind this time-reversal invariant topological superconductor, interactions to spinful fermions are switched on along an adiabatic path. Thereby, we couple modes which belong to two different representations of the time-reversal symmetry operator $\mathcal{T}$ with ${\mathcal{T}}^{2}=1$ and ${\mathcal{T}}^{2}=\ensuremath{-}1$, respectively. This observation can be understood by investigating how the relevant symmetries act on the entanglement spectrum giving rise to four instead of eight different topological phases with Majorana edge modes. We also show that a simple level crossing of doubly and singly degenerate states occurs in the entanglement spectrum on deforming the quantum state.

Andreev Bound States in a One-Dimensional Topological Superconductor

We study the charge character of the Andreev bound states (ABSs) in one-dimensional topological superconductors with spatial inversion symmetry (SIS) breaking. Despite the absence of the SIS, we show a hidden symmetry for the Bogoliubov-de Gennes equations around Fermi points in addition to the particle-hole symmetry. This hidden symmetry protects that the charge of the ABSs is solely dependent on the corresponding Fermi velocities. On the other hand, if the SIS is present, the ABSs are charge neutral, similar to Majorana fermions. We also propose that the charge of the ABSs can be experimentally measured in the tunneling transport spectroscopy from the resonant differential tunneling conductance.

Microscopic theory for a ferromagnetic nanowire/superconductor heterostructure: Transport, fluctuations, and topological superconductivity

Motivated by the recent experiment of Wang et al. [Nature Physics 6, 389 (2010)], who observed a highly unusual transport behavior of ferromagnetic Cobalt nanowires proximity-coupled to superconducting electrodes, we study proximity effect and temperature-dependent transport in such a mesoscopic hybrid structure. It is assumed that the asymmetry in the tunneling barrier gives rise to the Rashba spin-orbit-coupling in the barrier that enables induced p-wave superconductivity in the ferromagnet to exist. We first develop a microscopic theory of Andreev scattering at the spin-orbit-coupled interface, derive a set of self-consistent boundary conditions, and find an expression for the p-wave minigap in terms of the microscopic parameters of the contact. Second, we study temperature-dependence of the resistance near the superconducting transition and find that it should generally feature a fluctuation-induced peak. The upturn in resistance is related to the suppression of the single-particle density of states due to the formation of fluctuating pairs, whose tunneling is suppressed. In conclusion, we discuss this and related setups involving ferromagnetic nanowires in the context of one-dimensional topological superconductors. It is argued that to realize unpaired end Majorana modes, one does not necessarily need a half-metallic state, but a partial spin polarization may suffice. Finally, we propose yet another related class of material systems -- ferromagnetic semiconductor wires coupled to ferromagnetic superconductors -- where direct realization of Kitaev-Majorana model should be especially straightforward.

Interacting One-Dimensional Fermionic Symmetry-Protected Topological Phases

In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, 0, Z2, or Z. For example, noninteracting one-dimensional fermionic superconducting phases with S(z) spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z4. Using group cohomology, one can additionally show that there are only four distinct phases for such one-dimensional superconductors even with strong interactions. Comparing their projective representations, we find that all these four symmetry-protected topological phases can be realized with free fermions. Further, we show that one-dimensional fermionic superconducting phases with Z(n) discrete S(z) spin rotation and time-reversal symmetries are classified by Z4 when n is even and Z2 when n is odd; again, all these strongly interacting topological phases can be realized by noninteracting fermions. Our approach can be applied to systems with other symmetries to see which one-dimensional topological phases can be realized with free fermions.

Nonlocal noise cross correlation mediated by entangled Majorana fermions

Due to their nonlocality, qubits nested in Majorana bound states may be the key to realize decoherence-free quantum computation. Majorana bound states could be achieved at the ends of a one-dimensional topological superconductor. However, when the bound states couple directly to electron reservoirs their nonlocal correlation is quenched by local Andreev reflections. Here we propose a scheme to generate nonlocal noise cross correlation between two well-separated quantum dots, mediated by a pair of Majorana bound states. Both positive and negative cross correlations can be obtained by tuning the gate voltages applied to the dots. Within a limited range of finite temperatures, the cross correlation is not suppressed by thermal fluctuations. Furthermore, we show how the local Andreev reflections suppress the noise cross correlation when multiple dot energy levels are coupled to the Majorana bound states. The measurable cross correlation is expected to serve as a sensitive indicator for the generation of Majorana fermions.

Decoherence of Majorana qubits by noisy gates

We propose and study a realistic model for the decoherence of topological qubits, based on Majorana fermions in one-dimensional topological superconductors. The source of decoherence is the fluctuating charge on a capacitively coupled gate, modeled by noninteracting electrons. In this context, we clarify the role of quantum fluctuations and thermal fluctuations and find that quantum fluctuations do not lead to decoherence, while thermal fluctuations do. We explicitly calculate decay times due to thermal noise and give conditions for the gap size in the topological superconductor and the gate temperature. Based on this result, we provide simple rules for gate geometries and materials optimized for reducing the negative effect of thermal charge fluctuations on the gate.

Enhancement of Omnidirectional Photonic Bandgaps in One-Dimensional Superconductor–Dielectric Photonic Crystals with a Staggered Structure

In this paper, the properties of the omnidirectional photonic bandgap (OBG) realized by one-dimensional (1D) photonic crystals with a staggered structure which is composed of superconductor and isotropic dielectric have been theoretically investigated by the transfer matrix method (TMM). From the numerical results, it has been shown that such OBG is insensitive to the incident angle and the polarization of electromagnetic wave (EM wave), and the frequency range and central frequency of OBG can be tuned by the ambient temperature of system, the average thickness of superconductor layer, the average thickness of dielectric layer, and staggered parameters, respectively. The bandwidth of OBG can be notably enlarged with increasing average thickness and staggered parameter of superconductor layer. Moreover, the frequency range of OBG can be narrowed with increasing the average thickness, staggered parameter of dielectric layer, and ambient temperature, respectively. The damping coefficient of superconductor layer has no effect on the bandwidth of OBG under low-temperature conditions. It is shown that 1D superconductor–dielectric photonic crystals (SDPCs) have a superior feature in the enhancement of frequency range of OBG. This kind of OBG has potential applications in filters, microcavities, and fibers, etc.

Phase slippage driven dissipation and high-field Little-Parks effect in superconducting MoGe nanowire networks formed on nanoporous substrates

Superconducting MoGe nanowires with extremely small transverse dimensions have served as unique platforms for exploring physical phenomena in one-dimensional (1D) superconductors. Here we report on networks of superconducting MoGe nanowires with transverse dimensions down to a few nanometers, fabricated by sputter-deposition of MoGe onto commercially available filtration membranes containing dense nanopores. These networks of nanowires exhibit physical properties of individual MoGe nanowires, such as thermally activated phase slips---a dissipation mechanism expected for 1D superconducting nanowires. Furthermore, anomalies in the magnetic field versus temperature phase diagram around 1.2--1.5 Tesla and in the magnetoresistance curves can be understood with the Little--Parks effect, consistent with the multiply connected nature of a network of 1D nanowires.

Low-energy subgap states in multichannelp-wave superconducting wires

One-dimensional $p$-wave superconductors are known to harbor Majorana bound states at their ends. Superconducting wires with a finite width $W$ may have fermionic subgap states in addition to possible Majorana end states. While they do not necessarily inhibit the use of Majorana end states for topological computation, these subgap states can obscure the identification of a topological phase through a density-of-states measurement. We present two simple models to describe low-energy fermionic subgap states. If the wire's width $W$ is much smaller than the superconductor coherence length $\ensuremath{\xi}$, the relevant subgap states are localized near the ends of the wire and cluster near zero energy, whereas the lowest-energy subgap states are delocalized if $W\ensuremath{\gtrsim}\ensuremath{\xi}$. Notably, the energy of the lowest-lying fermionic subgap state (if present at all) has a maximum for $W\ensuremath{\sim}\ensuremath{\xi}$.

NOTE ON ANALYTICAL STUDIES OF ONE-DIMENSIONAL HOLOGRAPHIC SUPERCONDUCTORS

We employ the variational method for the Sturm–Liouville eigenvalue problem to analytically study phase transition of one-dimensional holographic superconductors. It is shown that this method is not a very powerful method to analytically calculate the properties of holographic superconductors. From the analytical treatment of scalar operator condensate at critical temperature, we also show that the mean-field critical exponent 1/2 results from the coupling term between scalar field and vector field, which may be an universal property of holographic superconductors with a similar coupling term in their equations of motion.

Topological properties of superconducting junctions

Motivated by recent developments in the field of one-dimensional topological superconductors, we investigate the topological properties of s-matrix of generic superconducting junctions where dimension should not play any role. We argue that for a finite junction the s-matrix is always topologically trivial. We resolve an apparent contradiction with the previous results by taking into account the low-energy resonant poles of s-matrix. Thus no common topological transition occur in a finite junction. We reveal a transition of a different kind that concerns the configuration of the resonant poles.

Crossover from Peierls distortion to one-dimensional superconductivity in arrays of (5,0) carbon nanotubes

By performing second-order renormalization group analysis on thin arrays of 4-Angstrom (5,0) carbon nanotubes (CNTs) embedded in aluminophosphate--five (AFI) zeolite crystals, we identify singlet superconductivity instability to be dominant at low temperatures, attributable to the screening of the electron-electron Coulomb interaction in the array configuration. Our analysis also shows that there is a crossover as the system scales to lower energy/temperatures, whereby one-dimensional (1D) superconductivity is the ground state, but the response function of the Peierls distortion/charge density wave (CDW) order dominates at the higher energy regime. This crossover behavior indicates that for a thin array of (5,0) CNTs the CDW order may represent an excited state of the array, so that the CDW characteristics can appear at finite temperatures, in conjunction with 1D superconductivity. Experimental results are presented to support this interpretation.

Detecting Majorana bound states by nanomechanics

We propose a nanomechanical detection scheme for Majorana bound states, which have been predicted to exist at the edges of a one-dimensional topological superconductor, implemented, for instance, using a semiconducting wire placed on top of an s-wave superconductor. The detector makes use of an oscillating electrode, which can be realized using a doubly clamped metallic beam, tunnel coupled to one edge of the topological superconductor. We find that a measurement of the nonlinear differential conductance provides the necessary information to uniquely identify Majorana bound states.

Majorana fermions in a topological-insulator nanowire proximity-coupled to ans-wave superconductor

A topological insulator nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and subject to a longitudinal applied magnetic field, is shown to realize a one-dimensional topological superconductor with unpaired Majorana fermions localized at both ends. This situation occurs under a wide range of conditions and constitutes what is possibly the most easily accessible physical realization of the elusive Majorana particles in a solid-state system.

Probability Distribution of Majorana End-State Energies in Disordered Wires

One-dimensional topological superconductors harbor Majorana bound states at their ends. For superconducting wires of finite length L, these Majorana states combine into fermionic excitations with an energy ε(0) that is exponentially small in L. Weak disorder leaves the energy splitting exponentially small, but affects its typical value and causes large sample-to-sample fluctuations. We show that the probability distribution of ε(0) is log normal in the limit of large L, whereas the distribution of the lowest-lying bulk energy level ε(1) has an algebraic tail at small ε(1). Our findings have implications for the speed at which a topological quantum computer can be operated.

Majorana edge states in interacting two-chain ladders of fermions

In this work we study interacting spinless fermions on a two-chain ladder with inter-chain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a $\mathbb{Z}_2$ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground state degeneracy against local perturbations. Lattice fermion models realizing such effective field theory are discussed.