In-plane Zeeman Field Research Articles

Supercurrent vortices and Majorana zero modes induced by an in-plane Zeeman field on the surface of a three-dimensional topological insulator

A nonuniform in-plane Zeeman field can induce spontaneous supercurrents of spin-orbit-coupled electrons in superconducting two-dimensional systems and thin films. In this work it is shown that current vortices can be created at the ends of a long homogeneously magnetized strip of a ferromagnetic insulator, which is deposited on the surface of a three-dimensional topological insulator. The $s$-wave superconductivity on its surface is assumed to have an intrinsic origin, or to be induced by the proximity effect. It is shown that vortices with the odd vorticity can localize Majorana zero modes. The latter may also be induced by sufficiently narrow domain walls inside the strip, which opens a way for manipulating these modes by moving the walls. It is shown that the vorticity can be tuned by varying the magnetization and width of the strip. A stability of the strip magnetization with respect to the Berezinsky-Kosterlitz-Thouless transition has been analyzed.

Critical supercurrent and φ0 state for probing a persistent spin helix

We theoretically study the profile of a supercurrent in two-dimensional Josephson junctions with Rashba-Dresselhaus spin-orbit interaction (RDSOI) in the presence of a Zeeman field. Through investigating self-biased supercurrent (so called $\varphi_0$-Josephson state), we obtain explicit expressions for the functionality of the $\varphi_0$ state with respect to RDSOI parameters ($\alpha,\beta$) and in-plane Zeeman field components ($h_x,h_y$). Our findings reveal that, when the chemical potential ($\mu$) is high enough compared to the energy gap ($\Delta$) in superconducting electrodes, i.e., $\mu \gg \Delta$, RSOI and DSOI with equal strengths ($|\alpha|=|\beta|$) cause vanishing $\varphi_0$ state independent of magnetization and the type of RDSOI. A Zeeman field with unequal components, i.e., $|h_x|\neq |h_y|$, however, can counteract and nullify the destructive impact of equal-strength RDSOIs (for one type only), where $\mu\sim\Delta$, although $|h_x|= |h_y|$ can still eliminate the $\varphi_0$ state. Remarkably, in the $\mu\sim\Delta$ limit, the $\varphi_0$ state is proportional to the multiplication of both components of an in-plane Zeeman field, i.e., $h_xh_y$, which is absent in the $\mu \gg \Delta$ limit. Furthermore, our results of critical supercurrents demonstrate that the persistent spin helices can be revealed in a high enough chemical potential regime $\mu\gg \Delta$, while an opposite regime, i.e., $\mu\sim\Delta$, introduces an adverse effect. In the ballistic regime, the "maximum" of the critical supercurrent occurs at $|\alpha|=|\beta|$ and the Zeeman field can boost this feature. The presence of disorder and nonmagnetic impurities change this picture drastically so the "minimum" of the critical supercurrent occurs at and around the symmetry lines $|\alpha|=|\beta|$.

Majorana corner modes in an s-wave second order topological superfluid

In sharp contrast to conventional topological superfluids, higher order (order r > 1) topological superfluids in n dimensions do not host n − 1 dimensional Majorana boundary states, instead host n − r dimensional Majorana excitations. In this paper, we propose Majorana corner modes can emerge in a second order superfluid with s-wave pairing, instead of unconventional pairings such as d-wave and -wave pairings in most of previous proposals. There are three key ingredients in this scheme consisting of a topological insulator, an in-plane Zeeman field, and an s-wave pairing. Based on the low energy theory for edge states, where the effective Dirac mass sign changes at the corner, we unveil the emergence of Majorana corner modes. Our proposal provides a promising platform for implementing 2D second order topological superfluids and Majorana corner modes.

Stark and Zeeman effects on the topological phase and transport properties of topological crystalline insulator thin films.

The fundamental investigation of topological crystalline insulator (TCI) thin films is essential for observing interesting phenomena. In practice, a promising pathway involves the application of electric and magnetic fields to tune the topological phases of TCI thin films. To achieve this, we applied a perpendicular electric field and an in-plane magnetic field to not only tune the Dirac gap of a SnTe(001) thin film and find the phase transition but also to directly connect them with their effects on the group velocity of both massless and massive surface Dirac fermions. The TCI thin film is an inherent insulator due to the hybridization between the front and back surfaces, and it transitions to a semimetal phase at a critical perpendicular electric field due to the Stark effect. Correspondingly, the anisotropic group velocity of the upper (lower) conduction (valence) band decreases (increases) with the electric field at certain momenta. We found that when one of the in-plane Zeeman field components becomes stronger than the intrinsic hybridization potential, the anisotropic Weyl cones with opposite chiralities retrieve at the critical momenta and the corresponding group velocities become zero. Further, the isotropic in-plane Zeeman field leads to rotation of the band structure, as expected, resulting in non-zero group velocities along all directions. Finally, for the sake of completeness, the combined Stark and Zeeman effects are tracked and the results show that the system is an insulator at all fields and the group velocities are altered more than when the individual Stark and Zeeman effects are applied. Our findings may provide interesting physical insights for practical applications in nanoelectronics and spintronics.

Geometry-Assisted Topological Transitions in Spin Interferometry.

We identify a series of topological transitions occurring in electronic spin transport when manipulating spin-guiding fields controlled by the geometric shape of mesoscopic interferometers. They manifest as distinct inversions of the interference pattern in quantum conductance experiments. We establish that Rashba square loops develop weak-(anti)localization transitions (absent in other geometries as Rashba ring loops) as an in-plane Zeeman field is applied. These transitions, boosted by nonadiabatic spin scattering, prove to have a topological interpretation in terms of winding numbers characterizing the structure of spin modes in the Bloch sphere.

Helicoidal excitonic phase in an electron-hole double-layer system

We propose helicoidal excitonic phase in a Coulomb-coupled two-dimensional electron-hole double layer (EHDL) system with relativistic spin-orbit interaction. Previously, it was demonstrated that layered InAs/AlSb/GaInSb heterostructure is an ideal experimental platform for searching excitonic condensate phases, while its electron layer has non-negligible Rashba interaction. We clarify that due to the Rashba term, the spin-triplet (spin-1) exciton field in the EHDL system forms a helicoidal structure and the helicoid plane can be controlled by an in-plane Zeeman field. We show that due to small but finite Dirac term in the heavy hole layer the helicoidal structure of the excitonic field under the in-plane field results in a helicoidal {\it magnetic} order in the electron layer. Based on linearization analyses, we further calculate momentum-energy dispersions of low-energy Goldstone modes in the helicoidal excitonic phase. We discuss possible experimental probes of the excitonic phase in the EHDL system.

Gap-filling states induced by disorder and Zeeman coupling in the nodeless chiral superconducting Bi/Ni bilayer system

Motivated by the recently discovered time-reversal symmetry-breaking superconductivity in epitaxial Bi/Ni bilayer system with transition temperature $T_c\approx 4.2$K and the observation of zero-bias anomaly in tunneling measurements, we show that gap-filling states can appear in the fully gapped $d_{xy}\pm id_{x^2-y^2}$ superconducting states. We consider a model of helical electron states with d-wave pairing. In particular, we show that both magnetic and non-magnetic impurities can create states within the superconducting gap. Alternatively, we also show that the coupling of the electron spins to the in-plane Zeeman field provided by nickel can also create gap-filling states by producing Bogoliubov Fermi surfaces. Our findings may explain the origin of zero-bias anomaly observed in the point-contact tunneling measurements.

Magnetic-field induced s+if pairing in Ising superconductors

We show that an in-plane Zeeman field applied to non-centrosymmetric Ising superconductors converts singlet $s$-wave Cooper pairs to equal-spin triplet $if$ pairs, leading to an enhancement of the critical transition line beyond expected from Ising spin-orbit coupling. Singlet to triplet conversion relates to a phase transformation due to spin rotation by the Zeeman field and has a geometric origin. The discussion is especially relevant, but not limited to monolayer transition metal dichalcogenides.

Weak localization in transition metal dichalcogenide monolayers and their heterostructures with graphene

We calculate the interference correction to the conductivity of doped transition metal dichalcogenide (TMDC) monolayers. Because of the interplay between valley structure and intrinsic spin-orbit coupling (SOC), these materials exhibit a rich weak localization (WL) behavior that is qualitatively different from conventional metals or similar two-dimensional materials such as graphene. Our results can also be used to describe graphene/TMDC heterostructures, where the SOC is induced in the graphene sheet. We discuss new parameter regimes that go beyond existing theories, and can be used to interpret recent experiments in order to assess the strength of SOC and disorder. Furthermore, we show that an in-plane Zeeman field can be used to distinguish the contributions of different kinds of SOC to the WL magnetoconductance.

Second-Order Topological Superconductivity in π-Junction Rashba Layers.

We consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom s-wave superconductor with phase difference ϕ close to π. We show that, in the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If ϕ=π, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference ϕ deviates from π by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference ϕ increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling ϕ by magnetic flux, the other involves an additional layer of randomly oriented magnetic impurities responsible for the phase shift of π in the proximity-induced superconducting pairing.

Majorana fermions in semiconducting nanowire and Fulde–Ferrell superconductor hybrid structures

The novel idea that spin-orbit coupling (SOC) and $s$-wave pairing can lead to induced $p$-wave pairing at strong magnetic limit has stimulated widespread interests in the whole community for the searching of Majorana Fermions (MFs), a self-hermitian particle, in semiconductor-superconductor hybrid structures. However, this system has several inherent limitations that prohibit the realization and identification of MFs with the major advances of semiconductor nanotechnology. We show that these limitations can be resolved by replacing the $s$-wave superconductor with type-II Fulde-Ferrell (FF) superconductor, in which the Cooper pair center-of-mass momentum plays the role of renormalizing the in-plane Zeeman field and chemical potential. As a result, the MFs can be realized for semiconductor nanowires with small Land\'e $g$ factor and high carrier density. The SOC strength directly influences the topological boundary, thus the topological phase transition and associated MFs can be engineered by an external electric field. Almost all the semiconductor nanowires can be used to realize MFs in this new platform. In particular, we find that InP nanowire, in some aspects, is more suitable for the realization of MFs than InAs and InSb nanowires. This new platform therefore can integrate the advances of semiconductor nanotechnology to the realization and identification of MFs in this hybrid structure.

Superfluid weight and Berezinskii–Kosterlitz–Thouless temperature of spin-imbalanced and spin–orbit-coupled Fulde–Ferrell phases in lattice systems

We study the superfluid weight Ds and Berezinskii–Kosterlitz–Thouless (BKT) transition temperatures TBKT in case of exotic Fulde–Ferrell (FF) superfluid states in lattice systems. We consider spin-imbalanced systems with and without spin–orbit coupling (SOC) accompanied with in-plane Zeeman field. By applying mean-field theory, we derive general equations for Ds and TBKT in the presence of SOC and the Zeeman fields for 2D Fermi–Hubbard lattice models, and apply our results to a 2D square lattice. We show that conventional spin-imbalanced FF states without SOC can be observed at finite temperatures and that FF phases are further stabilized against thermal fluctuations by introducing SOC. We also propose how topologically non-trivial SOC-induced FF phases could be identified experimentally by studying the total density profiles. Furthermore, the relative behavior of transverse and longitudinal superfluid weight components and the role of the geometric superfluid contribution are discussed.

Momentum space Aharonov-Bohm interferometry in Rashba spin-orbit coupled Bose-Einstein condensates

The recent experimental realization of synthetic Rashba spin-orbit coupling (SOC) paves a new avenue for exploring topological phases in ultracold atoms. The unequivocal characterization of such topological physics requires a simple scheme for measuring the Berry phase originating from the SOC. Here we propose a scheme to realize momentum space Aharonov-Bohm interferometry in a Rashba spin-orbit–coupled Bose-Einstein condensate through a sudden change of the in-plane Zeeman field. We find that the π Berry phase for the Dirac point of the Rashba SOC is directly revealed by a robust dark interference fringe in the momentum space. An external perpendicular Zeeman field opens a band gap at the Dirac point, which reduces the Berry phase along the Rashba ring, leading to lower brightness of the interference fringe. We develop a variational model with semiclassical equations of motion of essential dynamical quantities for describing the interference process, yielding real and momentum space trajectories and geometric phases agreeing with the real-time simulation of the Gross-Pitaevskii equation. Our study may pave the way for the experimental detection of Berry phases in ultracold atomic systems and further exploration of momentum space interference dynamics.

Rashba sandwiches with topological superconducting phases

We introduce a versatile heterostructure harboring various topological superconducting phases characterized by the presence of helical, chiral, or unidirectional edge states. Changing parameters, such as an effective Zeeman field or chemical potential, one can tune between these three topological phases in the same setup. Our model relies only on conventional non-topological ingredients. The bilayer setup consists of an s-wave superconductor sandwiched between two two-dimensional electron gas layers with strong Rashba spin-orbit interaction. The interplay between two different pairing mechanisms, proximity induced direct and crossed Andreev superconducting pairings, gives rise to multiple topological phases. In particular, helical edge states occur if crossed Andreev superconducting pairing is dominant. In addition, an in-plane Zeeman field leads to a 2D gapless topological phase with unidirectional edge states, which were previously predicted to exist only in non-centrosymmetric superconductors. If the Zeeman field is tilted out of the plane, the system is in a topological phase hosting chiral edge states.

Spin interferometry in anisotropic spin-orbit fields

Electron spins in a two-dimensional electron gas (2DEG) can be manipulated by spin-orbit (SO) fields originating from either Rashba or Dresselhaus interactions with independent isotropic characteristics. Together, though, they produce anisotropic SO fields with consequences on quantum transport through spin interference. Here we study the transport properties of modelled mesoscopic rings subject to Rashba and Dresselhaus [001] SO couplings in the presence of an additional in-plane Zeeman field acting as a probe. By means of 1D and 2D quantum transport simulations we show that this setting presents anisotropies in the quantum resistance as a function of the Zeeman field direction. Moreover, the anisotropic resistance can be tuned by the Rashba strength up to the point to invert its response to the Zeeman field. We also find that a topological transition in the field texture that is associated with a geometric phase switching is imprinted in the anisotropy pattern. We conclude that resistance anisotropy measurements can reveal signatures of SO textures and geometric phases in spin carriers.

Tunable- φ Josephson junction with a quantum anomalous Hall insulator

We theoretically study the Josephson current in a superconductor/quantum anomalous Hall insulator/superconductor junction by using the lattice Green function technique. When an in-plane external Zeeman field is applied to the quantum anomalous Hall insulator, the Josephson current $J$ flows without a phase difference across the junction $\theta$. The phase shift $\varphi$ appealing in the current-phase relationship $J\propto \sin(\theta-\varphi$) is proportional to the amplitude of Zeeman fields and depends on the direction of Zeeman fields. A phenomenological analysis of the Andreev reflection processes explains the physical origin of $\varphi$. A quantum anomalous Hall insulator breaks time-reversal symmetry and mirror reflection symmetry simultaneously. However it preserves magnetic mirror reflection symmetry. Such characteristic symmetry property enable us to have a tunable $\varphi$-junction with a quantum Hall insulator.

Topological Fulde-Ferrell and Larkin-Ovchinnikov states in spin-orbit-coupled lattice system

The spin-orbit coupled lattice system under Zeeman fields provides an ideal platform to realize exotic pairing states. Notable examples range from the topological superfluid/superconducting (tSC) state, which is gapped in the bulk but metallic at the edge, to the Fulde-Ferrell (FF) state (having a phase-modulated order parameter with a uniform amplitude) and the Larkin-Ovchinnikov (LO) state (having a spatially varying order parameter amplitude). Here, we show that the topological FF state with Chern number ($\mathcal{C}=-1$) (tFF$_{1}$) and topological LO state with $\mathcal{C}=2$ (tLO$_{2}$) can be stabilized in Rashba spin-orbit coupled lattice systems in the presence of both in-plane and out-of-plane Zeeman fields. Besides the inhomogeneous tSC states, in the presence of a weak in-plane Zeeman field, two topological BCS phases may emerge with $\mathcal{C}=-1$ (tBCS$_{1}$) far from half filling and $\mathcal{C}=2$(tBCS$_{2}$) near half filling. We show intriguing effects such as different spatial profiles of order parameters for FF and LO states, the topological evolution among inhomogeneous tSC states, and different non-trivial Chern numbers for the tFF$_{1}$ and tLO$_{1,2}$ states, which are peculiar to the lattice system. Global phase diagrams for various topological phases are presented for both half-filling and doped cases. The edge states as well as local density of states spectra are calculated for tSC states in a 2D strip.

Topological Superconductivity in a Planar Josephson Junction

We consider a two-dimensional electron gas with strong spin-orbit coupling contacted by two superconducting leads, forming a Josephson junction. We show that in the presence of an in-plane Zeeman field the quasi-one-dimensional region between the two superconductors can support a topological superconducting phase hosting Majorana bound states at its ends. We study the phase diagram of the system as a function of the Zeeman field and the phase difference between the two superconductors (treated as an externally controlled parameter). Remarkably, at a phase difference of $\pi$, the topological phase is obtained for almost any value of the Zeeman field and chemical potential. In a setup where the phase is not controlled externally, we find that the system undergoes a first-order topological phase transition when the Zeeman field is varied. At the transition, the phase difference in the ground state changes abruptly from a value close to zero, at which the system is trivial, to a value close to $\pi$, at which the system is topological. The critical current through the junction exhibits a sharp minimum at the critical Zeeman field, and is therefore a natural diagnostic of the transition. We point out that in presence of a symmetry under a modified mirror reflection followed by time reversal, the system belongs to a higher symmetry class and the phase diagram as a function of the phase difference and the Zeeman field becomes richer.

A Realistic Model for Observing Spin-Balanced Fulde–Ferrell Superfluid in Honeycomb Lattices**Supported by the Natural Science Foundation of Jiangsu Province under Grant No BK20130424, and the National Natural Science Foundation of China under Grant No 11547047.

The combination of spin-orbit coupling (SOC) and in-plane Zeeman field breaks time-reversal and inversion symmetries of Fermi gases and becomes a popular way to produce single plane wave Fulde–Ferrell (FF) superfluid. However, atom loss and heating related to SOC have impeded the successful observation of FF state until now. In this work, we propose the realization of spin-balanced FF superfluid in a honeycomb lattice without SOC and the Zeeman field. A key ingredient of our scheme is generating complex hopping terms in original honeycomb lattices by periodical driving. In our model the ground state is always the FF state, thus the experimental observation has no need of fine tuning. The other advantages of our scheme are its simplicity and feasibility, and thus may open a new route for observing FF superfluids.

SU(3) Quantum Hall Ferromagnetism in SnTe.

The (111) surface of SnTe hosts one isotropic Γ[over ¯]-centered and three degenerate anisotropic M[over ¯]-centered Dirac surface states. We predict that a nematic phase with spontaneously broken C_{3} symmetry will occur in the presence of a perpendicular magnetic field when the N=0 M[over ¯] Landau levels are 1/3 or 2/3 filled. The nematic state phase boundary is controlled by a competition between intravalley Coulomb interactions that favor a valley-polarized state and weaker intervalley scattering processes that increase in relative strength with magnetic field. An in-plane Zeeman field alters the phase diagram by lifting the threefold M[over ¯] Landau-level degeneracy, yielding a ground state energy with 2π/3 periodicity as a function of Zeeman-field orientation angle.