Superconducting Order Parameter Research Articles

Zero energy bound states on nano atomic line defect in iron-based high temperature superconductors

Motivated by recent scanning tunneling microscopy experiments on Fe atomic line defect in iron-based high temperature superconductors, we explore the origin of the zero energy bound states near the endpoints of the line defect by employing the two-orbit four-band tight binding model. With increasing the strength of the Rashba spin–orbit coupling along the line defect, the zero energy resonance peaks move simultaneously forward to negative energy for s+− pairing symmetry, but split for s++ pairing symmetry. The superconducting order parameter correction due to As(Te, Se) atoms missing does not shift the zero energy resonance peaks. Such the zero energy bound states are induced by the weak magnetic order rather than the strong Rashba spin–orbit coupling on Fe atomic line defect.

Spontaneous time-reversal symmetry breaking by disorder in superconductors

A growing number of superconducting materials display evidence for spontaneous time-reversal symmetry breaking (TRSB) below their critical transition temperatures. Precisely what this implies for the nature of the superconducting ground state of such materials, however, is often not straightforward to infer. We review the experimental status and survey different theoretical mechanisms for the generation of TRSB in superconductors. In cases where a TRSB complex combination of two superconducting order parameter components is realized, defects, dislocations and sample edges may generate superflow patterns that can be picked up by magnetic probes. However, even single-component condensates that do not break time-reversal symmetry in their pure bulk phases can also support signatures of magnetism inside the superconducting state. This includes, for example, the generation of localized orbital current patterns or spin-polarization near atomic-scale impurities, twin boundaries and other defects. Signals of TRSB may also arise from a superconductivity-enhanced Ruderman-Kittel-Kasuya-Yosida exchange coupling between magnetic impurity moments present in the normal state. We discuss the relevance of these different mechanisms for TRSB in light of recent experiments on superconducting materials of current interest.

Dome structure in pressure dependence of superconducting transition temperature for HgBa2Ca2Cu3O8 : Studies by ab initio low-energy effective Hamiltonian

The superconducting (SC) cuprate HgBa2Ca2Cu3O8 (Hg1223) has the highest SC transition temperature Tc among cuprates at ambient pressure Pamb, namely, Tcopt≃138 K experimentally at the optimal hole doping concentration. Tcopt further increases under pressure P and reaches 164 K at optimal pressure Popt≃30 GPa, then Tcopt decreases with increasing P>Popt generating a dome structure [Gao , ]. This nontrivial and nonmonotonic P dependence of Tcopt calls for a theoretical understanding and mechanism. To answer this open question, we consider the low-energy effective Hamiltonian (LEH) for the antibonding (AB) Cu3dx2−y2/O2pσ band derived generally for the cuprates. In the AB LEH for cuprates with Nℓ≤2 laminated CuO2 planes between block layers, it was proposed that Tcopt is determined by a universal scaling Tcopt≃0.16|t1|FSC [Schmid , ], where t1 is the nearest-neighbor hopping, and the SC order parameter at optimal hole doping FSC mainly depends on the ratio u=U/|t1| where U is the onsite effective Coulomb repulsion: The u dependence of FSC has a peak at uopt≃8.5 and a steep decrease with decreasing u in the region u<uopt irrespective of materials dependent on other parameters. In this paper, we show that |t1| increases with P, whereas u decreases with P in the Hamiltonian of Hg1223. Based on these facts, we show that the domelike P dependence of Tcopt can emerge at least qualitatively if we assume Hg1223 with Nℓ=3 follows the same universal scaling for Tcopt, and Hg1223 is located at the slightly strong coupling region u≳uopt at Pamb and u≃uopt at Popt by taking account of expected corrections to our calculation. The consequence of these assumptions is the following: With increasing P within the range P<Popt, the increase in Tcopt is accounted for by the increase in |t1|, whereas FSC is insensitive to the decrease in u around ≃uopt and hence to P as well. At P>Popt, the decrease in Tcopt is accounted for by the decrease in u below uopt, which causes a rapid decrease in FSC dominating over the increase in |t1|. We further argue the appropriateness of these assumptions based on the insight from studies on other cuprate compounds in the literature. In addition, we discuss the dependencies of u and |t1| on each crystal parameter (CP), which provides hints for design of even higher Tcopt materials. Published by the American Physical Society 2024

Magnetic exchange interactions at the proximity of a superconductor

Interfacing magnetism with superconductivity gives rise to a wonderful playground for intertwining key degrees of freedom: Cooper pairs, spin, charge, and spin–orbit interaction, from which emerge a wealth of exciting phenomena, fundamental in the nascent field of superconducting spinorbitronics and topological quantum technologies. Magnetic exchange interactions (MEIs), being isotropic or chiral such as the Dzyaloshinskii–Moriya interactions, are vital in establishing the magnetic behavior at these interfaces as well as in dictating not only complex transport phenomena, but also the manifestation of topologically trivial or non-trivial objects. Here, we propose a methodology enabling the extraction of the tensor of MEI from electronic structure simulations accounting for superconductivity. We apply our scheme to the case of a Mn layer deposited on Nb(110) surface and explore proximity-induced impact on the MEI. The latter are weakly modified by a realistic electron-phonon coupling. However, tuning the superconducting order parameter, we unveil potential change of the magnetic order accompanied with chirality switching, as induced by the interplay of spin-orbit interaction and Cooper pairing. Owing to its simple formulation, our methodology can be readily implemented in state-of-the-art frameworks capable of tackling superconductivity and magnetism. We thus foresee implications in the simulations and prediction of topological superconducting bits as well as of cryogenic superconducting hybrid devices involving magnetic units.

Violation of emergent rotational symmetry in the hexagonal Kagome superconductor CsV3Sb5

Superconductivity is caused by electron pairs that are canonically isotropic, whereas some exotic superconductors are known to exhibit non-trivial anisotropy stemming from unconventional pairings. However, superconductors with hexagonal symmetry, the highest rotational symmetry allowed in crystals, exceptionally have strong constraint that is called emergent rotational symmetry (ERS): anisotropic properties should be very weak especially near the critical temperature Tc even for unconventional pairings such as d-wave states. Here, we investigate superconducting anisotropy of the recently-found hexagonal Kagome superconductor CsV3Sb5, which is known to exhibit various intriguing phenomena originating from its undistorted Kagome lattice formed by vanadium atoms. Based on calorimetry performed under accurate two-axis field-direction control, we discover a combination of six- and two-fold anisotropies in the in-plane upper critical field. Both anisotropies, robust up to very close to Tc, are beyond predictions of standard theories. We infer that this clear ERS violation with nematicity is best explained by multi-component nematic superconducting order parameter in CsV3Sb5 intertwined with symmetry breakings caused by the underlying charge-density-wave order.

Broadened quantum critical ground state in a disordered superconducting thin film.

A superconductor-insulator transition (SIT) in two dimensions is a prototypical quantum phase transition (QPT) with a clear quantum critical point (QCP) at zero temperature (T = 0). The SIT is induced by a field B and observed in disordered thin films. In some of weakly disordered or crystalline thin films, however, an anomalous metallic (AM) ground state emerges over a wide B range between the superconducting and insulating phases. It remains a fundamental open question how the QPT picture of the SIT is modified when the AM state appears. Here we present measurements of the Nernst effect N, which has great sensitivity to the fluctuations of the superconducting order parameter. From a thorough contour map of N in the B-T plane, we found a thermal-to-quantum crossover line of the superconducting fluctuations, a so-called ghost-temperature line associated with the QPT, as well as a ghost-field line associated with a thermal transition. The QCP is identified as a T = 0 intercept of the ghost-temperature line inside the AM state, which verifies that the AM state is a broadened critical state of the SIT.

Theoretical investigation of the Co-occurrence of superconductivity and antiferromagnetism in iron-based high-temperature superconductors

In this research work, the Co-occurrence of superconductivity and antiferromagnetism in Ca0.74 1 La0.26 1 Fe1−xCox As2 iron-based high-temperature superconductors have been investigated. Based on the multi-band nature of Ca0.74 1 La0.26 1 Fe1−xCox As2 iron-based superconductors, a two-band model Hamiltonian that contains intra- and inter-bands is developed. By employing the matrix form of the temperature-dependent Green’s function formalism with the two-band Hamiltonian, the mathematical expressions of superconducting transition temperature as functions of the superconducting order parameter and antiferromagnetism translational temperature as a function of the magnetic order parameter are obtained, respectively. The plotted graph of the superconducting and magnetic temperature as a function of the magnetic order parameter indicates a clear possibility of Co-occurrence of superconductivity and the antiferromagnetism order parameter in Ca0.74 1 La0.26 1 Fe1−xCox As2 iron-based superconductors in the range of magnetic order parameters between 2.3 meV and 6.07 meV, which is in good agreement with experimental observations. This research contributes to understanding the complex behavior of high-temperature superconductors and provides valuable technological applications for other fields.

Casimir Forces with Periodic Structures: Abrikosov Flux Lattices

We investigate the influence of the Abrikosov vortex lattice on the Casimir force in a setup constituted by high-temperature superconductors subject to an external magnetic field. The Abrikosov lattice is a property of type II superconductors in which normal and superconducting carriers coexist and the latter define a periodic pattern with square symmetry. We find that the optical properties determined by spatial redistribution of the superconducting order parameter induce Casimir forces with a periodic structure whose minimal strengths coincide with the vortex cores.

Effects of Rashba and Dresselhaus spin-orbit couplings on the critical temperature and paramagnetic limiting field of superconductors with broken inversion symmetry

Transition temperature (Tc) and paramagnetic limiting of a superconductor without spatial inversion symmetry in the presence of both Rashba and Dresselhaus antisymmetric spin-orbit couplings is studied. The critical temperature is derived for anisotropy of the superconducting order parameter, ranging from isotropic s-wave to any pairing state with nonzero angular momentum and mixed parity singlet-triplet states emerge due to spin-orbit coupling (SOC). It will be shown that for unprotected odd parity pairing, pure Rashba and Dresselhaus SOCs have similar effects on the reduction of Tc and for combined effects of the two spin-orbit couplings at fixed SOC, Tc reduced by increasing Dresselhaus component and decreased down to its minimum value in the equal-Rashba-Dresselhaus case. The paramagnetic limiting is also analyzed for spin-singlet pairing and it is shown that the low temperature divergence behavior of paramagnetic limiting is less affected by both Rashba and Dresselhaus SOCs but for fixed SOC strength by increasing Dresselhaus component the paramagnetic limiting field is increased.

Fully gapped pairing state in spin-triplet superconductor UTe2.

The recently discovered superconductor UTe2 is a promising candidate for spin-triplet superconductors, but the symmetry of the superconducting order parameter remains highly controversial. Here, we determine the superconducting gap structure by the thermal conductivity of ultraclean UTe2 single crystals. We find that the a-axis thermal conductivity divided by temperature κ/T in zero-temperature limit is vanishingly small for both magnetic field H‖a and H‖c axes up to H/Hc2 ∼ 0.2, demonstrating the absence of nodes around the a axis contrary to the previous belief. The present results, combined with the reduction of nuclear magnetic resonance Knight shift, indicate that the superconducting order parameter belongs to the isotropic Au representation with a fully gapped pairing state, analogous to the B phase of superfluid 3He. These findings reveal that UTe2 is likely to be a long-sought three-dimensional strong topological superconductor, hosting helical Majorana surface states on any crystal plane.

Superconducting Order Parameter Structure in the Nematic Phase of Iron-based Materials

We consider the effect of the nematic order on the formation of the superconducting state in iron pnictides and chalcogenides. Nematic order with the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{B}_{{2g}}}$$\\end{document} symmetry is modelled as the d-type Pomeranchuk instability and treated within the mean-field approach. Calculated nematic order parameter depends on the nematic interaction coefficient and abruptly changes with the coefficient’s increase. The superconducting solution is obtained within the spin-fluctuation pairing theory. We show that the leading solution in the nematic phase has a \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{{\\pi \\pm }}}$$\\end{document} structure. From the critical temperature \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{T}_{c}}$$\\end{document} estimations, we conclude that the nematic superconducting state of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{{\\pi \\pm }}}$$\\end{document} type is more favorable than the usual \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{s}_{ \\pm }}$$\\end{document} and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{d}_{{{{x}^{2}} - {{y}^{2}}}}}$$\\end{document} type states appearing in the absence of the nematicity.

Two-component nematic superconductivity in 4Hb-TaS2

Most superconductors have an isotropic, single component order parameter and are well described by the standard (BCS) theory for superconductivity. Unconventional, multiple-component superconductors are exceptionally rare and are much less understood. Here, we combine scanning tunneling microscopy and angle-resolved macroscopic transport for studying the candidate chiral superconductor, 4Hb-TaS2. We reveal quasi-periodic one-dimensional modulations in the tunneling conductance accompanied by two-fold symmetric superconducting critical field. The strong modulation of the in-plane critical field, Hc2, points to a nematic, unconventional order parameter. However, the imaged vortex core is isotropic at low temperatures. We suggest a model that reconciles this apparent discrepancy and takes into account previously observed spontaneous time-reversal symmetry breaking at low temperatures. The model describes a competition between a dominating chiral superconducting order parameter and a nematic one. The latter emerges close to the normal phase. Our results strongly support the existence of two-component superconductivity in 4Hb-TaS2 and can provide valuable insights into other systems with coexistent charge order and superconductivity.

Observing dynamical phases of BCS superconductors in a cavity QED simulator.

In conventional Bardeen-Cooper-Schrieffer superconductors1, electrons with opposite momenta bind into Cooper pairs due to an attractive interaction mediated by phonons in the material. Although superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system parameters are abruptly changed2-8. The resulting out-of-equilibrium phases are predicted to occur in real materials and ultracold fermionic atoms, but not all have yetbeen directly observed. Here we realize an alternative way to generate the proposed dynamical phases using cavity quantum electrodynamics (QED). Our system encodes the presence or absence of a Cooper pair in a long-lived electronic transition in 88Sr atoms coupled to an optical cavity and represents interactions between electrons as photon-mediated interactions through the cavity9,10. To fully explore the phase diagram, we manipulate the ratio between the single-particle dispersion and the interactions after a quench and perform real-time tracking of the subsequent dynamics of the superconducting order parameter using nondestructive measurements. We observe regimes in which the order parameter decays to zero (phase I)3,4, assumes a non-equilibrium steady-state value (phase II)2,3 or exhibits persistent oscillations (phase III)2,3. This opens up exciting prospects for quantum simulation, including the potential to engineer unconventional superconductors and to probe beyond mean-field effects like the spectral form factor11,12, and for increasing the coherence time for quantum sensing.

Quasi-2D Fermi surface in the anomalous superconductor UTe2

The heavy fermion paramagnet UTe2 exhibits numerous characteristics of spin-triplet superconductivity. Efforts to understand the microscopic details of this exotic superconductivity have been impeded by uncertainty regarding the underlying electronic structure. Here we directly probe the Fermi surface of UTe2 by measuring magnetic quantum oscillations in pristine quality crystals. We find an angular profile of quantum oscillatory frequency and amplitude that is characteristic of a quasi-2D Fermi surface, which we find is well described by two cylindrical Fermi sheets of electron- and hole-type respectively. Additionally, we find that both cylindrical Fermi sheets possess considerable undulation but negligible small-scale corrugation, which may allow for their near-nesting and therefore promote magnetic fluctuations that enhance the triplet pairing mechanism. Importantly, we find no evidence for the presence of any 3D Fermi surface sections. Our results place strong constraints on the possible symmetry of the superconducting order parameter in UTe2.

Exploring superconductivity in Ba3Ir4Ge16: Experimental and theoretical insights

We explore both experimental and theoretical aspects of the superconducting properties in the layered caged compound, Ba3Ir4Ge16. Our approach integrates muon spin rotation and relaxation (μSR) measurements with magnetization and heat capacity experiments, accompanied by first-principle calculations. The compound’s bulk superconductivity is unequivocally established through DC magnetization measurements, revealing a critical temperature (TC) of 5.7 K. A noteworthy characteristic observed in the low-temperature superfluid density is its saturating behavior, aligning with the features typical of conventional Bardeen-Cooper-Schrieffer (BCS) superconductors. The assessment of moderate electron-phonon coupling superconductivity is conducted through transverse field μSR measurements, yielding a superconducting gap to TC ratio (2Δ(0)∕kBTC) of 4.04, a value corroborated by heat capacity measurements. Crucially, zero field μSR measurements dismiss the possibility of any spontaneous magnetic field emergence below TC, highlighting the preservation of time-reversal symmetry. Our experimental results are reinforced by first-principles density functional calculations, underscoring the intricate interplay between crystal structure and superconducting order parameter symmetry in caged compounds. This comprehensive investigation enhances our understanding of the relationship between crystal structure and superconductivity in such unique compounds.

Origins of anomalies in the temperature dependences of specific heat and superfluid density in doped high-Tc cuprates: Signatures of Bose-liquid superconductivity

We show that the temperature-dependent superconducting order parameter and related superconducting properties (in particular, the temperature dependences of specific heat, superfluid density and related London penetration depth) of high-[Formula: see text] cuprates are fundamentally different from those of conventional superconductors and cannot be understood within the existing theories based on the Bardeen–Cooper–Schrieffer (BCS)-type condensation of weakly bound Cooper pairs into a superfluid Fermi-liquid and on the usual Bose–Einstein condensation (BEC) of bosonic Cooper pairs. We examine the validity of an alternative approach to the unconventional superconductivity in high-[Formula: see text] cuprates and establish that these materials exhibiting a [Formula: see text]-like superconducting transition at the critical temperature [Formula: see text] are similar to the superfluid 4He and are also superfluid Bose systems. We argue that the doped high-[Formula: see text] cuprates, from underdoped to overdoped regime, are unconventional (bosonic) superconductors and the tightly bound (polaronic) Cooper pairs in these polar materials behave like composite bosons just like 4He atoms and condense into a Bose superfluid at [Formula: see text]. We identify the superconducting order parameter [Formula: see text] in underdoped and optimally doped cuprates as the coherence parameter [Formula: see text] of bosonic Cooper pairs, which appears just below [Formula: see text] and has a kink-like temperature dependence near the characteristic temperature [Formula: see text]. We find that the [Formula: see text]-like specific heat anomaly in high-[Formula: see text] cuprates near [Formula: see text] predicted by the theory of Bose-liquid superconductivity is similar to that observed both in superfluid 4He near [Formula: see text] and in Hg-based cuprate superconductor HgBa2Ca2Cu3O8 near [Formula: see text][Formula: see text]K. We demonstrate that in high-[Formula: see text] superconductor YBa2Cu3[Formula: see text] the superfluid density [Formula: see text] exhibits distinctly different temperature dependences in the temperature ranges [Formula: see text] and [Formula: see text]. In this superconductor, a pronounced anomaly in [Formula: see text] exists near [Formula: see text] and the temperature dependence of [Formula: see text] below [Formula: see text] deviates downwards from the high-temperature behavior. Our results for the normalized superfluid density [Formula: see text] are in good agreement with the experimental data on the temperature dependence of [Formula: see text] in YBa2Cu3[Formula: see text]. The anomalous temperature dependences of specific heat and superfluid density observed in Hg- and Y-based high-[Formula: see text] superconductors are clear signatures of Bose-liquid superconductivity.

Theoretical study of the interplay of spin density wave and superconductivity in nickel substitution of the strontium–iron–arsenide (SrFe2−xNixAs2) superconductor in a two-band model

The main objective of this manuscript is to focus on the computational study of the interplay of spin density wave (SDW) and superconductivity using a two-band model for SrFe2−xNixAs2. We derived mathematical statements for the superconducting critical temperature, SDW critical temperature, superconductivity order parameter, and the SDW order parameter using the Hamiltonian model and Green’s function formalism for the SrFe2−xNixAs2 superconductor. A mathematical expression for the dependence of transition temperatures on the SDW order parameter was obtained for SrFe2−xNixAs2. Using these mathematical statements, transition temperatures versus the SDW order parameter phase diagrams were plotted to show the dependence of the SDW order parameter on transition temperatures. By merging these diagrams, we have depicted the intriguing possibility of the interplay of superconductivity and magnetism for the SrFe2−xNixAs2 superconductor. Phase diagrams of temperature versus superconducting order parameters and the SDW order parameter were also plotted to show the dependence of order parameters on temperature for the SrFe2−xNixAs2 superconductor.

Time-reversal symmetry breaking superconductivity between twisted cuprate superconductors.

Twisted interfaces between stacked van der Waals (vdW) cuprate crystals present a platform for engineering superconducting order parameters by adjusting stacking angles. Using a cryogenic assembly technique, we construct twisted vdW Josephson junctions (JJs) at atomically sharp interfaces between Bi2Sr2CaCu2O8+x crystals, with quality approaching the limit set by intrinsic JJs. Near 45° twist angle, we observe fractional Shapiro steps and Fraunhofer patterns, consistent with the existence of two degenerate Josephson ground states related by time-reversal symmetry (TRS). By programming the JJ current bias sequence, we controllably break TRS to place the JJ into either of the two ground states, realizing reversible Josephson diodes without external magnetic fields. Our results open a path to engineering topological devices at higher temperatures.

Emergent ferromagnetism with superconductivity in Fe(Te,Se) van der Waals Josephson junctions

Ferromagnetism and superconductivity are two key ingredients for topological superconductors, which can serve as building blocks of fault-tolerant quantum computers. Adversely, ferromagnetism and superconductivity are typically also two hostile orderings competing to align spins in different configurations, and thus making the material design and experimental implementation extremely challenging. A single material platform with concurrent ferromagnetism and superconductivity is actively pursued. In this paper, we fabricate van der Waals Josephson junctions made with iron-based superconductor Fe(Te,Se), and report the global device-level transport signatures of interfacial ferromagnetism emerging with superconducting states for the first time. Magnetic hysteresis in the junction resistance is observed only below the superconducting critical temperature, suggesting an inherent correlation between ferromagnetic and superconducting order parameters. The 0-π phase mixing in the Fraunhofer patterns pinpoints the ferromagnetism on the junction interface. More importantly, a stochastic field-free superconducting diode effect was observed in Josephson junction devices, with a significant diode efficiency up to 10%, which unambiguously confirms the spontaneous time-reversal symmetry breaking. Our work demonstrates a new way to search for topological superconductivity in iron-based superconductors for future high Tc fault-tolerant qubit implementations from a device perspective.

Synthetic complex Weyl superconductors, chiral Josephson effect and synthetic half-vortices

We show that the most generic form of spin-singlet superconducting order parameter for chiral fermions in systems with broken time reversal symmetry and inversion symmetry is of the Delta _s+igamma ^5Delta _5 where Delta _s is the usual order parameter and Delta _5 is the pseudo-scalar order parameter. After factoring out the U(1) phase e^{iphi }, this form of superconductivity admits yet additional complex structure in the plane of (Delta _s,Delta _5). The polar angle chi in this plane, which we call the chiral angle, can be controlled by the external flux bias. We present a synthetic setup based on stacking of topological insulators (TIs) and superconductors (SCs). Alternatively flux biasing the superconductors with a fluxes pm Phi leads to Delta _5=Delta _0 sin (chi ), where Delta _0 is the superconducting order parameter of the SC layers, and the chiral angle chi ={2pi }Phi /Phi _0 is directly given by the flux Phi in units of the flux quantum Phi _0=h/(2e). This can be used as a building block to construct a two-dimensional Josephson array. In this setup chi will be a background field defining a pseudoscalar Delta _5 that can be tuned to desired configuration. While in a uniform background field Delta _5 the dynamics of phi is given by standard XY model and its associated vortices, a staggered background pm Delta _5 (or equivalently chi and chi +pi in alternating lattice sites) creates a new set of minima for the phi field that will support half-vortex excitations. An isolated single engineered “half-vortex” in the chi field in an otherwise uniform background will bind a phi -half-vortex. This is similar to the way a p-wave superconducting vortex core binds a Majorana fermion.