The ac Josephson effect phenomenon is studied in fractional quantum Hall states (FQHS). Usual quantum Hall devices have an electron layer of uniform thickness. We consider a new quantum Hall device. The device has a narrow part that is vertical to the direction of the current. The narrow part has a smaller electron layer thickness than in the other part. The ac Josephson effect might be observed when the magnetic field or the electric current is modulated by an oscillation with a constant frequency value f. Steps of the voltage appear when an electric current exceed a threshold value. The step value V is related to the transfer charge Q as V=(2 pi hbar f)/Q. We examine how the value of the transfer charge Q depends on the fractional filling factor nu. The nu - dependence of Q is affected by the wave function of the fractional quantum Hall state. We classify the value of Q according to the wave function type. Accordingly, observation of the ac Josephson effect in FQHS clarifies the structure of the fractional quantum Hall states.
Time-reversal invariance places strong constraints on the properties of the quantum spin Hall edge. One such restriction is the inevitability of dissipation in a Josephson junction between two superconductors formed on such an edge without the presence of interaction. Interactions and spin-conservation breaking are key ingredients for the realization of dissipationless ac Josephson effect on such quantum spin Hall edges. We present a simple quantum impurity model that allows to create a dissipationless fractional Josephson effect on a quantum spin Hall edge. We then use this model to substanti…
We study ring shaped (Corbino) devices made of bilayer two-dimensional electron gases in the total filling factor one quantized Hall phase which is considered to be a coherent BCS-like state of interlayer excitons. Identical Josephson currents are observed at the two edges while only a negligible conductance between them is found. The maximum Josephson current observed at either edge can be controlled by passing a second interlayer Josephson current at the other edge. Due to the large electric resistance between the two edges, the interaction between them can only be mediated by the neutral in…
We study the Josephson effect in a quantum spin Hall system coupled to a localized magnetic impurity. As a consequence of the fermion parity anomaly, the spin of the combined system of impurity and spin-Hall edge alternates between half-integer and integer values when the superconducting phase difference across the junction advances by $2π$. This leads to characteristic differences in the splittings of the spin multiplets by exchange coupling and single-ion anisotropy at phase differences, for which time-reserval symmetry is preserved. We discuss the resulting $8π$-periodic (or $\mathbb{Z}_4$)…
We show that a quantum Hall bilayer with the total filling $ν= 1$ should exhibit a dynamical regime similar to the flux-flow in large Josephson junctions. This analogy may explain a conspicuous peak in the interlayer tunneling conductance [\prl {\bf 84}, 5808 (2000)]. The flux-flow is likely to be spatiotemporally chaotic at low bias voltage, which will manifest itself through broad-band noise. The peak position can be controlled by an in-plane magnetic field.
An interlayer phase coherence develops spontaneously in the bilayer quantum Hall system at the filling factor $ν=1$. On the other hand, the spin and pseudospin degrees of freedom are entangled coherently in the canted antiferromagnetic phase of the bilayer quantum Hall system at the filling factor $ν=2$. There emerges a complex Nambu-Goldstone mode with a linear dispersion in the zero tunneling-interaction limit for both cases. Then its phase field provokes a Josephson supercurrent in each layer, which is dissipationless as in a superconductor. We study what kind of phase coherence the Nambu-G…
In the early days of superconductivity, Ivar Giaver discovered that it was possible to make a novel DC transformer by using one superconductor to drag vortices through another. An analogous effect was predicted to exist in quantum Hall bilayers and has recently been discovered experimentally by Eisenstein's group at Caltech. Similarly, new experiments from the Caltech group have demonstrated the existence of a Josephson-like `supercurrent' branch for electrons coherently tunnelling between the two layers.
We propose a way to realize a programmable quantum current standard (PQCS) from the Josephson voltage standard and the quantum Hall resistance standard (QHR) exploiting the multiple connection technique provided by the quantum Hall effect (QHE) and the exactness of the cryogenic current comparator. The PQCS could lead to breakthroughs in electrical metrology like the realization of a programmable quantum current source, a quantum ampere-meter and a simplified closure of the Quantum Metrological Triangle. Moreover, very accurate universality tests of the QHE could be performed by comparing PQCS…
We study the transport properties of a voltage-biased Josephson junction where the BCS superconducting leads are coupled via the edges of a quantum Hall sample. In this scenario, an out of equilibrium Josephson current develops, which is numerically studied within the Floquet-Keldysh Green's function formalism. We particularly focus on the time-averaged current as a function of both the bias voltage and the magnetic flux threading the sample and analyze the resonant multiple Andreev reflection processes that lead to an enhancement of the quasiparticle transmission. We find that a full tomograp…
I review recent novel experimental and theoretical advances in the physics of quantum Hall effect bilayers. Of particular interest is a broken symmetry state which optimizes correlations by putting the electrons into a coherent superposition of the two different layers.
We show that a spin-polarized Landau level in a two-dimensional electron gas can carry a spin-triplet supercurrent between two spin-singlet superconductors. The supercurrent results from the interplay of Andreev reflection and Rashba spin-orbit coupling at the normal--superconductor (NS) interface. We contrast the current-phase relationship and the Fraunhofer oscillations of the spin-triplet and spin-singlet Josephson effect in the lowest Landau level, and find qualitative differences.
At total filling factor $ν_T=1$, interlayer phase coherence in quantum Hall bilayers can result in a tunneling anomaly resembling the Josephson effect in the presence of strong fluctuations. The most robust experimental signature of this effect is a strong enhancement of the tunneling conductance at small voltages. The height and width of the conductance peak depend strongly on the area and tunneling amplitude of the samples, applied parallel magnetic field and temperature. We find that the tunneling experiments are in quantitative agreement with a theory which treats fluctuations due to meron…
We explore a spin Josephson effect in a system of two ferromagnets coupled by a tunnel junction formed of 2D time-reversal invariant topological insulators. In analogy with the more commonly studied instance of the Josephson effect for charge in superconductors, we investigate properties of the phase-coherent {\it spin} current resulting from the misalignment of the in-plane magnetization angles of the two ferromagnets. We show that the topological insulating barrier offers the exciting prospect of hosting a {\it fractional} spin Josephson effect mediated by bound states at the ferromagnet-top…
Coupling superconductors to quantum Hall edge states is the subject of intense investigation as part of the ongoing search for non-abelian excitations. Our group has previously observed supercurrents of hundreds of picoamperes in graphene Josephson junctions in the quantum Hall regime. One of the explanations of this phenomenon involves the coupling of an electron edge state on one side of the junction to a hole edge state on the opposite side. In our previous samples, these states are separated by several microns. Here, a narrow trench perpendicular to the contacts creates counterpropagating …
Josephson junctions based on three-dimensional topological insulators offer intriguing possibilities to realize unconventional $p$-wave pairing and Majorana modes. Here, we provide a detailed study of the effect of a uniform magnetization in the normal region: We show how the interplay between the spin-momentum locking of the topological insulator and an in-plane magnetization parallel to the direction of phase bias leads to an asymmetry of the Andreev spectrum with respect to transverse momenta. If sufficiently large, this asymmetry induces a transition from a regime of gapless, counterpropag…
In this study, we examine multiple encapsulated graphene Josephson junctions to determine which mechanisms may be responsible for the supercurrent observed in the quantum Hall (QH) regime. Rectangular junctions with various widths and lengths were studied to identify which parameters affect the occurrence of QH supercurrent. We also studied additional samples where the graphene region is extended beyond the contacts on one side, making that edge of the mesa significantly longer than the opposite edge. This is done in order to distinguish two potential mechanisms: a) supercurrents independently…
Quantum spin Hall-superconductor hybrids are promising sources of topological superconductivity and Majorana modes, particularly given recent progress on HgTe and InAs/GaSb. We propose a new method of revealing topological superconductivity in extended quantum spin Hall Josephson junctions supporting `fractional Josephson currents'. Specifically, we show that as one threads magnetic flux between the superconductors, the critical current traces an interference pattern featuring sharp fingerprints of topological superconductivity---even when noise spoils parity conservation.
Hybrid structures of quantum spin-Hall insulators (QSHIs) and superconductors (Ss) present a unique opportunity to access dissipationless topological states of matter, which, however, is frequently hindered by the lack of control over the spin polarization in QSHIs. We propose a very efficient spin-polarization mechanism based on the magnetoelectric (Edelstein) effect in superconducting QSHI structures. It acts akin to the Zeeman splitting in an external magnetic field, but with an effective $g$-factor of order of 1000, resulting in an unprecedented spin-splitting effect. It allows a magnetic …
We argue on the basis of phenomenological and microscopic considerations that there is no d.c. Josephson effect in ordered bilayer quantum Hall systems, even at T=0. Instead the tunnel conductance is strongly enhanced, approaching a finite value proportional to the square of the order parameter as the interlayer tunneling amplitude vanishes.
The Josephson tunneling through a 2D normal contact with the spin-orbit split conduction band has been studied in the diffusive regime. Linearized Usadel equations for triplet components of the pairing function revealed a striking similarity to the equations of spin diffusion driven by the electric field in normal metals. Consequently, we predict that the out-of-plane spin-Hall polarization accumulates towards lateral sample edges and the in-plane polarization is finite throughout the entire normal region. At the same time, the spin-Hall current is absent in the considered case of the stationa…
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 $θ$. The phase shift $φ$ appealing in the current-phase relationship $J\propto \sin(θ-φ$) 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 t…
Josephson phase shift and diode effect due to the inverse spin Hall effect
RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所新兴物质科学中心）；Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学先进理工学研究科）；CINaM, Aix-Marseille University（艾克斯-马赛大学中子与磁性材料中心）
We investigate anomalous Hall effect in a magnet coupled to a triplet superconductor under phase gradient. It is found that the anomalous Hall supercurrent arises from non-trivial structure of the magnetization. The magnetic structure manifested in the Hall supercurrent is characterized by even order terms of the exchange coupling, essentially different from that discussed in the context of anomalous Hall effect, reflecting the disspationless nature of supercurrent. We also discuss a possible candidate for magnetic structure to verify our prediction.
Periodic in time spin-Hall current and spin polarization induced by a dc electric bias has been calculated in a superconductor-normal 2DEG-superconductor (SNS) Josephson junction. We assumed that the band energies of electrons in the normal system are splitted due to Rashba spin-orbit coupling. The transport parameters have been calculated within the diffusion approximation and using perturbation expansion over a small SN contact transparency. We found out that in contrast to the stationary Josephson effect, the spin-Hall current does not turn to zero. Besides a direct proximity effect caused …
The anomalous $4π$-periodic ac Josephson effect, a hallmark of topological Josephson junctions, was experimentally observed in a quantum spin Hall insulator. This finding is unexpected due to time-reversal symmetry preventing the backscattering of the helical edge states and therefore suppressing the $4π$-periodic component of the Josephson current. Here we analyze the two-particle inelastic scattering as a possible explanation for this experimental finding. We show that a sufficiently strong inelastic scattering restores the $4π$-periodic component of the current beyond the short Josephson ju…
$\mathbb{Z}_4$ parafermions can be realized in a strongly interacting quantum spin Hall Josephson junction or in a spin Hall Josephson junction strongly coupled to an impurity spin. In this paper we study a system that has both features, but with weak (repulsive) interactions and a weakly coupled spin. We show that for a strongly anisotropic exchange interaction, at low temperatures the system enters a strong coupling limit in which it hosts two $\mathbb{Z}_4$ parafermions, characterizing a fourfold degeneracy of the ground state. We construct the parafermion operators explicitly, and show tha…
We investigate the interplay between ferromagnetic and superconducting order at the edge of a quantum spin Hall insulator. Using complementary analytical and self-consistent numerical approaches, we study a ferromagnetic Josephson junction and show how the direct coupling between magnetism and the superconducting U(1) phase gives rise to several unusual phenomena which distinguishes the present system from its non-topological equivalent. In particular, we demonstrate how the anomalous current-phase relation triggers supercurrent-induced magnetization dynamics and also study the spatial localiz…
We study junctions between superconductors mediated by the edge states of a quantum spin Hall insulator. We show that such junctions exhibit a fractional Josephson effect, in which the current phase relation has a 4π, rather than a 2πperiodicity. This effect is a consequence of the conservation of fermion parity - the number of electrons modulo 2 - in a superconducting junction, and is closely related to the Z_2 topological structure of the quantum spin Hall insulator. Inelastic processes, which violate the conservation of fermion parity, lead to telegraph noise in the equilibrium supercurrent…
We consider DC supercurrents in SNS junctions. Spin-orbit coupling in combination with Zeeman fields can induce an effective vector potential in the normal conductor. As a consequence, an out-of-plane spin-density varying along the transverse direction causes a longitudinal phase difference between the superconducting terminals. The resulting equilibrium phase coherent supercurrent is analogue to the non-equilibrium inverse spin Hall effect in normal conductors. We explicitly compute the effect for the Rashba spin orbit coupling in a disordered two-dimensional electron gas with an inhomogeneou…
The hydrodynamical phase-slippage geometry and a semi-classical Josephson-RCSJ formulism are combined to make an analogy between the periodically driven Josephson ac effect and the quantum Hall effect. Both of these two macroscopic quantum effects are postulated to be originated from the one-dimensional linear vortex-electron relative motion. The mechanisms of the quantized Hall resistance of IQHE and FQHE are unified under the phase-locking dynamics of a Josephson-like oscillation which is periodically driven by the electron wave of the coherent Hall current, with the time-averaged frequency …
We systematically study the coherent transport (Josephson tunneling and counterflow current) and its breakdown which leads to incoherent charge flow in in the excitonic BCS condensate formed in GaAs bilayers at $ν_{tot}=1/2+1/2$. The Josephson currents in samples with three different interlayer distances vary by four orders of magnitude. In contrast, the breakdown thresholds for the $ν_{tot}=1$ quantum Hall state are comparable. Furthermore, Coulomb drag in a Corbino ring reveals that the coherent counterflow current coexists with the dissipative charge transport.
We present a study of a graphene-based Josephson junction with dedicated side gates carved from the same sheet of graphene as the junction itself. These side gates are highly efficient, and allow us to modulate carrier density along either edge of the junction in a wide range. In particular, in magnetic fields in the $1-2$ Tesla range, we are able to populate the next Landau level, resulting in Hall plateaus with conductance that differs from the bulk filling factor. When counter-propagating quantum Hall edge states are introduced along either edge, we observe supercurrent localized along that…
Can Majorana zero modes in quantum Hall edges survive edge reconstruction?
Aix Marseille Univ, Université de Toulon, CNRS, CPT（艾克斯马赛大学，土伦大学，法国国家科学研究中心，理论物理中心）；Department of Physics, Indian Institute of Science Education and Research (IISER) Kolkata（印度科学与教育研究所加尔各答分校物理系）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究所国际理论科学中心）
We consider recent experiments on wide superconductor-quantum spin Hall insulator (QSHI)-superconductor Josephson junctions, which have shown preliminary evidence of proximity-induced superconductivity at the edge-modes of the QSHI system based on an approximate analysis of the observed Fraunhofer spectra of the Josephson critical current as a function of the applied magnetic field. Using a completely independent exact numerical method involving a non-linear constrained numerical optimization, we calculate the supercurrent profiles, comparing our results quantitatively with the experimental Fr…
Despite the robustness of the chiral edge modes of quantum Hall systems against the superconducting proximity effect, Cooper pairs can penetrate into the chiral edge channels and carry the Josephson current in an appropriate setup. In our work, the Josephson junction of a spin-polarized quantum anomalous Hall insulator (QAHI) with a Chern number $ν=1$ connecting conventional superconductors is studied from the perspective of pairing symmetry consistent with the chiral edge mode. Induced pairing states are equal-spin triplet, a combination of the even- and odd-frequency components, nonlocally e…
The zero-bias tunneling resonance in quantum Hall bilayer systems is investigated via numerical simulations of the classical two dimensional XY model with a symmetry-breaking field. Disorder is included in the model, and is shown to nucleate strings of overturned spins proliferated through the system, with unpaired vortices and antivortices at their endpoints. This string glass state supports low energy excitations which lead to anomalously large dissipation in tunneling, as observed in experiment. The effect of an in-plane magnetic field is discussed.
We construct a generalization of the Chalker-Coddington network model to the case of fractional quantum Hall effect, which describes the tunneling between multiple chiral edges. We derive exact local and global duality symmetries of this model, and show that its infrared properties are identical to those of disordered planar Josephson junction array (JJA) in a weak magnetic field, which implies the same universality class. The zero frequency Hall resistance of the system, which was expressed through exact correlators of the tunneling fields, is shown to be quantized both in the quantum Hall li…
Hall effect in SN and SNS junctions is considered. It is shown that at small temperature the Hall voltage is significantly suppressed as compared to its normal metal value. The time dependence of the Hall voltage in SNS junctions has a form of narrow pulses with the Josephson frequency.
We investigate transverse Josephson current in superconductor/normal metal/superconductor junctions where the normal metal has Rashba type spin-orbit coupling. It is shown that trans- verse current arises from the spin-orbit coupling in the normal metal. This effect is specific to superconducting current and the transverse current vanishes in the normal state. In addition, this transverse Josephson effect is purely stationary and applied magnetic field is unnecessary to realize this effect, in contrast to the Hall effect in the normal state. We also discuss physical interpretation of this effe…
We study the Josephson effect in spin-triplet superconductor$-$quantum anomalous Hall insulator$-$spin-triplet superconductor junctions using the nonequilibrium Green function method. The current-phase difference relations show strong dependence on the orientations of the $\bf{d}$-vectors in superconductors. We focus on two $\bf{d}$-vector configurations, the parallel one with the left and right ${\bf{d}}$-vectors being in the same direction, and the nonparallel one with the left ${\bf{d}}$-vector fixed at the $z$-axis. For the parallel configuration, the $0$-$π$ transition can be realized whe…
Hybrid superconductor/semiconductor devices constitute a powerful platform where intriguing topological properties can be investigated. Here we present fabrication methods and analysis of Josephson junctions formed by a high-mobility InAs quantum-well bridging two Nb superconducting contacts. We demonstrate supercurrent flow with transport measurements, critical temperature of 8.1 K, and critical fields of the order of 3 T. Modulation of supercurrent amplitude can be achieved by acting on two side gates lithographed close to the two-dimensional electron gas. Low-temperature measurements reveal…
Multi-terminal Josephson junctions were recently proposed as a versatile and tunable platform to emulate topological Bloch-like Hamiltonians in arbitrary dimensions. In this perspective article, we will give a brief overview of the subject and recognize these mesoscopic devices as realizations of topological flux networks as the ones envisioned by J. E. Avron and coworkers in their seminal works on the early days of the quantum Hall effect. We summarize the current state-of-the-art theoretical and experimental research regarding these Josephson devices, highlighting recent developments and giv…
We consider an annular superconductor-insulator-superconductor Josephson-junction, with the insulator being a double layer of electron and holes at Abelian fractional quantum Hall states of identical fillings. When the two superconductors gap out the edge modes, the system has a topological ground state degeneracy in the thermodynamic limit akin to the fractional quantum Hall degeneracy on a torus. In the quasi-one-dimensional limit, where the width of the insulator becomes small, the ground state energies are split. We discuss several implications of the topological degeneracy that survive th…
Spontaneous fractional Josephson current from parafermions
Aix Marseille Univ, Université de Toulon, CNRS, CPT（艾克斯-马赛大学、土伦大学、法国国家科学研究中心、理论物理中心）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心、塔塔基础研究所）；Department of Physics, Indian Institute of Science Education and Research (IISER) Kolkata（印度科学教育研究院加尔各答分校物理系）
We study the effect of a magnetic field on the current-phase relation of a topological Josephson junction formed by connecting two superconductors through the helical edge states of a quantum spin-Hall insulator. We predict that the Zeeman effect along the spin quantization axis of the helical edges results in an anomalous Josephson relation that allows for a supercurrent to flow in the absence of superconducting phase bias. We relate the associated field-tunable phase shift $ϕ_0$ in the Josephson relation of such a $ϕ_0$-junction to the existence of a so-called helical superconductivity, whic…
It has been proposed that band-inverted electron-hole bilayers support a phase transition from an insulating phase with spontaneously broken time-reversal symmetry to a quantum spin Hall insulator phase as a function of increasing electron and hole densities. Here, we show that in the presence of proximity-induced superconductivity it is possible to realize Majorana zero modes in the time-reversal symmetry broken phase in the absence of magnetic field. We develop an effective low-energy theory for the system in the presence of time-reversal symmetry breaking order parameter to obtain analytica…
Bilayer quantum Hall systems have a broken symmetry ground state at filling factor $ν=1$ which can be viewed either as an excitonic superfluid or as a pseudospin ferromagnet. We present a theory of inter-layer transport in quantum Hall bilayers that highlights remarkable similarities and critical differences between transport in Josephson junction and ferromagnetic metal spin-transfer devices. Our theory is able to explain the size of the large but finite low bias interlayer conductance and the voltage width of this collective transport anomaly.
The promise of fault-tolerant quantum computing has made topological superconductors the focus of intense research during the past decade. In this context, topological Josephson junctions based on nanowires or on topological insulators provide an alternative route for probing topological superconductivity. As a hallmark of their topological nature, such junctions exhibit a ground-state fermion parity that is $4π$-periodic in the superconducting phase difference $ϕ$. Finding unambiguous experimental evidence for this $4π$-periodicity still proves a difficult task, however. Here we propose a top…
