A theoretical study on discrete vortex bound states is carried out near a vortex core in the presence of a van Hove singularity(VHS) near the Fermi level by solving Bogoliubov–de Gennes(Bd G) equations. When the VHS ...A theoretical study on discrete vortex bound states is carried out near a vortex core in the presence of a van Hove singularity(VHS) near the Fermi level by solving Bogoliubov–de Gennes(Bd G) equations. When the VHS lies exactly at the Fermi level and also at the middle of the band, a zero-energy state and other higher-energy states whose energy ratios follow integer numbers emerge. These discrete vortex bound state peaks undergo a splitting behavior when the VHS or Fermi level moves away from the middle of the band. Such splitting behavior will eventually lead to a new arrangement of quantized vortex core states whose energy ratios follow half-odd-integer numbers.展开更多
The bound states around a vortex in anisotropic superconductors is a longstanding yet important issue.In this work,we develop a variational theory on the basis of the Andreev approximation to obtain the energy levels ...The bound states around a vortex in anisotropic superconductors is a longstanding yet important issue.In this work,we develop a variational theory on the basis of the Andreev approximation to obtain the energy levels and wave functions of the low-energy quantized bound states in superconductors with anisotropic pairing on arbitrary Fermi surface.In the case of circular Fermi surface,the effective Schr¨odinger equation yielding the bound state energies gets back to the theory proposed by Volovik and Kopnin many years ago.Our generalization here enables us to prove the equidistant energy spectrum inside a vortex in a broader class of superconductors.More importantly,we are now able to obtain the wave functions of these bound states by projecting the quasiclassical wave function on the eigenmodes of the effective Schr¨odinger equation,going beyond the quasiclassical Eilenberger results,which,as we find,are sensitive to the scattering rate.For the case of isotropic Fermi surface,the spatial profile of the low-energy local density of states is dominated near the vortex center and elongates along the gap antinode directions,in addition to the ubiquitous Friedel oscillation arising from the quantum inteference neglected in the Eilenberger theory.Moreover,as a consequence of the pairing anisotropy,the quantized wave functions develop a peculiar distribution of winding number,which reduces stepwise towards the vortex center.Our work provides a flexible way to study the vortex bound states in the future.展开更多
Topological crystalline insulators are known to support multiple Majorana zero modes(MZMs)at a single vortex,their hybridization is forbidden by a magnetic mirror symmetry MT.Due to the limited energy resolution of sc...Topological crystalline insulators are known to support multiple Majorana zero modes(MZMs)at a single vortex,their hybridization is forbidden by a magnetic mirror symmetry MT.Due to the limited energy resolution of scanning tunneling microscopes and the very small energy spacing of trivial bound states,it remains challenging to directly probe and demonstrate the existence of multiple MZMs.In this work,we propose to demonstrate the existence of MZMs by studying the hybridization of multiple MZMs in a symmetry breaking field.The different responses of trivial bound states and MZMs can be inferred from their spatial distribution in a vortex.However,the theoretical simulations are very demanding since it requires an extremely large system in real space.By utilizing the kernel polynomial method,we can efficiently simulate large lattices with over 108 orbitals to compute the local density of states which bridges the gap between theoretical studies based on minimal models and experimental measurements.We show that the spatial distribution of MZMs and trivial vortex bound states differs drastically in tilted magnetic fields.The zero-bias peak elongates when the magnetic field preserves MT,while it splits when MT is broken,giving rise to an anisotropic magnetic response.Since the bulk of SnTe are metallic,we also study the robustness of MZMs against the bulk states,and clarify when can the MZMs produce a pronounced anisotropic magnetic response.展开更多
Vortices and bound states offer an effective means of comprehending the electronic properties of superconductors.Recently,surface-dependent vortex core states have been observed in the newly discovered kagome supercon...Vortices and bound states offer an effective means of comprehending the electronic properties of superconductors.Recently,surface-dependent vortex core states have been observed in the newly discovered kagome superconductors CsV_(3)Sb_(5).Although the spatial distribution of the sharp zero energy conductance peak appears similar to Majorana bound states arising from the superconducting Dirac surface states,its origin remains elusive.In this study,we present observations of tunable vortex bound states(VBSs)in two chemically-doped kagome superconductors Cs(V_(1-x)Tr_(x))_(3)Sb_(5)(Tr=Ta or Ti),using low-temperature scanning tunneling microscopy/spectroscopy.The CsV_(3)Sb_(5)-derived kagome superconductors exhibit full-gap-pairing superconductivity accompanied by the absence of long-range charge orders,in contrast to pristine CsV_(3)Sb_(5).Zero-energy conductance maps demonstrate a field-driven continuous reorientation transition of the vortex lattice,suggesting multiband superconductivity.The Ta-doped CsV_(3)Sb_(5)displays the conventional cross-shaped spatial evolution of Caroli-de Gennes-Matricon bound states,while the Tidoped CsV_(3)Sb_(5)exhibits a sharp,non-split zero-bias conductance peak(ZBCP)that persists over a long distance across the vortex.The spatial evolution of the non-split ZBCP is robust against surface effects and external magnetic field but is related to the doping concentrations.Our study reveals the tunable VBSs in multiband chemically-doped CsV_(3)Sb_(5)system and offers fresh insights into previously reported Y-shaped ZBCP in a non-quantum-limit condition at the surface of kagome superconductor.展开更多
基金the National Natural Science Foundation of China (Grant No. 11804154)the Scientific Research Foundation of NJIT (Grant Nos. YKJ201853 and CKJA201807)。
文摘A theoretical study on discrete vortex bound states is carried out near a vortex core in the presence of a van Hove singularity(VHS) near the Fermi level by solving Bogoliubov–de Gennes(Bd G) equations. When the VHS lies exactly at the Fermi level and also at the middle of the band, a zero-energy state and other higher-energy states whose energy ratios follow integer numbers emerge. These discrete vortex bound state peaks undergo a splitting behavior when the VHS or Fermi level moves away from the middle of the band. Such splitting behavior will eventually lead to a new arrangement of quantized vortex core states whose energy ratios follow half-odd-integer numbers.
基金supported by the National Key R&D Program of China(Grant No.2022YFA1403201)the National Natural Science Foundation of China(Grant Nos.12274205,12374147,92365203,and 11874205)。
文摘The bound states around a vortex in anisotropic superconductors is a longstanding yet important issue.In this work,we develop a variational theory on the basis of the Andreev approximation to obtain the energy levels and wave functions of the low-energy quantized bound states in superconductors with anisotropic pairing on arbitrary Fermi surface.In the case of circular Fermi surface,the effective Schr¨odinger equation yielding the bound state energies gets back to the theory proposed by Volovik and Kopnin many years ago.Our generalization here enables us to prove the equidistant energy spectrum inside a vortex in a broader class of superconductors.More importantly,we are now able to obtain the wave functions of these bound states by projecting the quasiclassical wave function on the eigenmodes of the effective Schr¨odinger equation,going beyond the quasiclassical Eilenberger results,which,as we find,are sensitive to the scattering rate.For the case of isotropic Fermi surface,the spatial profile of the low-energy local density of states is dominated near the vortex center and elongates along the gap antinode directions,in addition to the ubiquitous Friedel oscillation arising from the quantum inteference neglected in the Eilenberger theory.Moreover,as a consequence of the pairing anisotropy,the quantized wave functions develop a peculiar distribution of winding number,which reduces stepwise towards the vortex center.Our work provides a flexible way to study the vortex bound states in the future.
基金support from the National Key R&D Program of China(Grants No.2021YFA1401500)the Hong Kong Research Grants Council(Project No.16306220).
文摘Topological crystalline insulators are known to support multiple Majorana zero modes(MZMs)at a single vortex,their hybridization is forbidden by a magnetic mirror symmetry MT.Due to the limited energy resolution of scanning tunneling microscopes and the very small energy spacing of trivial bound states,it remains challenging to directly probe and demonstrate the existence of multiple MZMs.In this work,we propose to demonstrate the existence of MZMs by studying the hybridization of multiple MZMs in a symmetry breaking field.The different responses of trivial bound states and MZMs can be inferred from their spatial distribution in a vortex.However,the theoretical simulations are very demanding since it requires an extremely large system in real space.By utilizing the kernel polynomial method,we can efficiently simulate large lattices with over 108 orbitals to compute the local density of states which bridges the gap between theoretical studies based on minimal models and experimental measurements.We show that the spatial distribution of MZMs and trivial vortex bound states differs drastically in tilted magnetic fields.The zero-bias peak elongates when the magnetic field preserves MT,while it splits when MT is broken,giving rise to an anisotropic magnetic response.Since the bulk of SnTe are metallic,we also study the robustness of MZMs against the bulk states,and clarify when can the MZMs produce a pronounced anisotropic magnetic response.
基金supported by the National Natural Science Foundation of China(61888102,52022105,92065109,and 12174428)the National Key Research and Development Projects of China(2022YFA1204100,2018YFA0305800,2019YFA0308500,2020YFA0308800,and 2022YFA1403400)+4 种基金the CAS Project for Young Scientists in Basic Research(YSBR-003 and 2022YSBR-048)the Innovation Program of Quantum Science and Technology(2021ZD0302700)the financial support from the European Research Council(ERC Consolidator Grant “Nonlinear Topo”,No.815869)ISF-Singapore-Israel Research Grant(3520/20)supported by the US DOE,Basic Energy Sciences(DE-FG02-99ER45747)。
文摘Vortices and bound states offer an effective means of comprehending the electronic properties of superconductors.Recently,surface-dependent vortex core states have been observed in the newly discovered kagome superconductors CsV_(3)Sb_(5).Although the spatial distribution of the sharp zero energy conductance peak appears similar to Majorana bound states arising from the superconducting Dirac surface states,its origin remains elusive.In this study,we present observations of tunable vortex bound states(VBSs)in two chemically-doped kagome superconductors Cs(V_(1-x)Tr_(x))_(3)Sb_(5)(Tr=Ta or Ti),using low-temperature scanning tunneling microscopy/spectroscopy.The CsV_(3)Sb_(5)-derived kagome superconductors exhibit full-gap-pairing superconductivity accompanied by the absence of long-range charge orders,in contrast to pristine CsV_(3)Sb_(5).Zero-energy conductance maps demonstrate a field-driven continuous reorientation transition of the vortex lattice,suggesting multiband superconductivity.The Ta-doped CsV_(3)Sb_(5)displays the conventional cross-shaped spatial evolution of Caroli-de Gennes-Matricon bound states,while the Tidoped CsV_(3)Sb_(5)exhibits a sharp,non-split zero-bias conductance peak(ZBCP)that persists over a long distance across the vortex.The spatial evolution of the non-split ZBCP is robust against surface effects and external magnetic field but is related to the doping concentrations.Our study reveals the tunable VBSs in multiband chemically-doped CsV_(3)Sb_(5)system and offers fresh insights into previously reported Y-shaped ZBCP in a non-quantum-limit condition at the surface of kagome superconductor.
