Abstract
Planar Josephson junctions provide a versatile platform, alternative to the nanowirebased geometry, for the generation of the Majorana bound states, due to the additional phase tunability of the topological superconductivity. The proximity induction of chiral magnetism and superconductivity in a twodimensional electron gas showed remarkable promise to manipulate topological superconductivity. Here, we consider a Josephson junction involving a skyrmion crystal and show that the chiral magnetism of the skyrmions can create and control the Majorana bound states without the requirement of an intrinsic Rashba spinorbit coupling. Interestingly, the Majorana bound states in our geometry are realized robustly at zero phase difference at the junction. The skyrmion radius, being externally tunable by a magnetic field or a magnetic anisotropy, brings a unique control feature for the Majorana bound states.
Introduction
The unification of nontrivial spin texture and superconductivity via advanced interface engineering is a futuristic approach to create and manipulate nonAbelian Majorana bound states (MBS) for their controlled usage in faulttolerant topological quantum computing^{1,2,3,4,5}. The nanoscale control of magnetism not only relaxes the need for a specific form of Rashba spinorbit coupling, but also motivates for a magnetic fieldfree platform for the braiding of the MBS^{6,7,8,9,10,11,12}. Despite numerous successes in the search for the MBS in onedimensional geometries, the associated limitations such as the intrinsic instabilities of onedimensional systems, the need for fine tuning of parameters, and the technological obstacles in physical implementation, suggest to look for a twodimensional platform^{13,14}. The discovery of topological superconductivity in phasecontrolled planar Josephson junctions is, therefore, a major step towards the realization of a twodimensional array of MBS for designing scalable braiding protocols^{15,16,17,18}. The Josephson junction geometry provides additional control to tune the MBS by changing the shape of the junction, strain, and unconventional spinorbit coupling^{19,20,21,22,23}. A timereversal invariant topological superconductivity can also be induced by placing the Josephson junction on top of a strong topological insulator^{24}. Previous works on the Josephson junctionbased platforms, however, revealed the requirements of a strong intrinsic Rashba spinorbit coupling and πphase biasing of the Josephson junction. These constraints pose serious challenges in the detection and manipulation of the MBS under realistic conditions. Chiral magnetism in proximity to an swave superconductor generates exotic effects including the appearance of the Majorana modes^{25,26,27,28,29,30,31,32}; however, the location and stability of the Majorana states in these platforms are difficult to anticipate due to their nonlocalized nature.
In our considered geometry, the planar Josephson junction, composed of a twodimensional electron gas and an swave superconductor, is placed on top of a Néeltype skyrmion crystal (SkX) in such a way that the twodimensional electron gas experiences the spatially varying magnetic field from the bottom SkX and it is also proximitized to the electron pairing from the top superconductors, as described in Fig. 1a. The interplay between the SkX spin texture and the proximityinduced superconductivity leads to topological superconductivity near the middle quasionedimensional channel of the Josephson junction with localized MBS at its two ends. The advantages of using the SkX are: (i) the chiral magnetism generates a robust fictitious gauge field, that can also be visualized as a spinorbit coupling, and a local Zeeman field which remove the stringent criteria of a strong Rashbatype spinorbit coupling and, therefore, essentially expands the region of parameter space to realize the MBS, (ii) the existence of the MBS can be further controlled externally by tuning the skyrmion radius, and (iii) the current SkXbased Josephson junction is not required to be phase biased and the MBS can be found robustly at a phase difference φ = 0, between the two superconducting regions, unlike the usual planar Josephson junctions that are required to be phase biased with φ = π, to minimize the critical magnetic field for the topological transition and to maximize the chemicalpotential range within which the MBS appear^{15}. Using the zeroenergy feature of the quasiparticle states with a topological energy gap, sharp localization of these states, chargeneutrality condition, two order parameters, viz. Majorana polarization and curvature of the density of states, we confirm the existence of the MBS in our set up. The tunable phase difference and the skyrmion radius together provide broad, flexible control of the MBS which is indispensable to achieve the longsoughtafter goal of the braiding.
Results
Theoretical set up
For the generation of the SkX, we consider a heterointerface of a thinlayer ferromagnet and a heavy compound (metal or insulator). The advantage of the heavy compound is that it helps to generate a large DzyaloshinskiiMoriya interaction (DMI) at the interface between the ferromagnet and the heavy compound. The cooperation between the DMI and the ferromagnetic exchange interaction of the ferromagnet produces a triangular SkX, in the presence of a magnetic field or an anisotropy. Our Monte Carlo simulations reveal that columns of skyrmions, arranged in a triangular array, appear spontaneously within a sixlayer ferromagnet, although the DMI exists predominantly at the interface between the ferromagnet and the heavy compound, as shown in Fig. 1b. We perform simulated annealing using the Metropolis energyminimization algorithm, formulated with the following Hamiltonian
where J is the nearestneighbor ferromagnetic exchange interaction strength in the ferromagnet, D is the DMI strength at the bottom ferromagnet layer that interfaces with the heavy compound, H_{z} is the perpendicular magnetic field, A is the easyplane magnetic anisotropy at the bottom ferromagnet layer, i,j are the site indices in the entire ferromagnet, and a,b are the twodimensional site indices at the bottom ferromagnet layer. The DMI, present dominantly at the interface between the ferromagnet and the heavy compound, generates a Néeltype SkX^{33,34,35}. Besides the engineered interfaces, the SkX naturally appears in a wide variety of materials^{36,37,38,39} that can also be utilized in the proposed device geometry, instead of the combination of the ferromagnet and the heavy compound. Also, the SkX can be artificially created without the need for any external magnetic field by nanopatternization^{40}.
The spin texture B_{i} on the top layer of the ferromagnet, obtained from the Monte Carlo simulations, is used to obtain the lowenergy spectrum of the planar Josephson junction by solving selfconsistently the Bogoliubovde Gennes equations. Since the SkX lies underneath the twodimensional electron gas without a finite separation between them, it is reasonable to assume that the deviation in the magnetic fringing field in the twodimensional electron gas from the original SkX texture is negligible. The proximityinduced superconductivity in the twodimensional electron gas, which is subject to the SkX spin texture B_{i}, is described by the Hamiltonian
where t = ℏ^{2}/(2m^{*}a^{2}) is the hopping energy, m^{*} is the effective mass of electrons, a is the unit spacing of the lattice grid, μ is the chemical potential, and Δ_{i} is the induced local swave pairing amplitude on the two sides of the Josephson junction that are attached to the top Al layer. The pairing amplitude Δ_{i} = −U_{i}〈c_{i↑}c_{i↓}〉 is calculated selfconsistently using the onsite attractive interaction strength U_{i} of the induced superconducting states in the twodimensional electron gas. U_{i} = U in the twodimensional electron gas below the Al superconductors and zero in the middle metallic channel. The value U = 2 meV is determined by setting Δ_{i} = 0.2 meV, the estimated proximityinduced gap magnitude for a twodimensional electron gas with an SC interface^{41,42}, without any spin texture. The g factor and the effective mass are set to g = 50 and m^{*} = 0.017m_{0} for InSb^{43,44}. The lattice grid spacing used is a = 10 nm (a is the lattice grid spacing used to discretize the kinetic energy term \(\frac{{p}^{2}}{2m}\) within finitedifference approximation. It is immaterial as long as the lattice lengths remain fixed) with which the hopping energy becomes t = 22.44 meV. The amplitude of the spin texture B_{i} is set to B_{0} = 0.3 T, compatible with the saturation magnetization M_{s} = 1.7 × 10^{6} A/m for CoFe^{10,11}. HM/ferromagnet interfaces with Pt, Pd, Ag, Ir, and Au as the HM have been developed together with Co, Fe, and their alloys (see e.g., ref. ^{45}). We present results for a planar Josephson junction with length L_{y} = 2 μm, transverse length of the SC leads L_{x} = 200 nm, and width of the quasionedimensional metallic channel W = 50 nm.
Emergence of the MBS
The lowenergy spectrum, shown in Fig. 2a, reveals that there exist multiple ranges of the chemical potential within which the zeroenergy MBS appear. To determine the Majorana character of the quasiparticle states, we compute the Majorana polarization, defined as^{46,47}
where \({u}_{i\uparrow }^{n}\) and \({v}_{i\uparrow }^{n}\) are the Bogoliubovde Gennes quasiparticle and quasihole amplitudes, respectively, corresponding to the n^{th} eigenstate, spin ↑, and site i. As evident from Fig. 2a, \( {{\mathcal{P}}}_{{{\mathcal{M}}}_{,n}} \approx 1\) indicates the occurrence of a pair of robust MBS with a finite topological energy gap. The Majorana polarization \( {{\mathcal{P}}}_{{\mathcal{M}},1}\) of the first positive eigenstate, plotted with μ in Fig. 2b, acquires finite values within the range of μ, in which the MBS emerge. The deltafunctionlike peaks in \( {{\mathcal{P}}}_{{\mathcal{M}},1}\) are the signatures of the Majorana oscillations, which is also clearly seen in the lowenergy spectrum in Fig. 2a, originating due to the overlap of the MBS wave functions at the two ends of the finitelength quasionedimensional channel. The Majorana oscillations in \( {{\mathcal{P}}}_{{\mathcal{M}},1}\) have also been confirmed from the calculations of a onedimensional wire (for results in the wire geometry, see Supplementary Note 4). The Majorana polarization, with a modification in the expression used in Eq. (3), was proposed to be probed in this planar Josephson junction geometry using the spinselective Andreev reflection technique^{48}. To further characterize the evolution of the topological superconductivity with changing a parameter, such as μ, we look at the curvature of the density of states at zero energy \(\frac{{\partial }^{2}D}{\partial {E}^{2}}\), where D(E) is defined as^{49}
and δ(E − E_{n}) is modeled using a Gaussian with broadening 0.001 meV (≪t). The second derivative is computed using the secondorder finitedifference method. These two quantities, \( {{\mathcal{P}}}_{{\mathcal{M}},1}\) and \(\frac{{\partial }^{2}D}{\partial {E}^{2}}\), may provide additional insight in the experimental detection of the MBS, besides the conventional zerobias conductance peak^{50}, which often leads to ambiguity due to other possible zerobias states in a superconductor^{51}.
As shown in Fig. 2c, \(\frac{{\partial }^{2}D}{\partial {E}^{2}}\) takes finite values in the same ranges of μ as that of the Majorana polarization \( {{\mathcal{P}}}_{{\mathcal{M}},1}\). The Majorana oscillations, in the form of deltafunctionlike peaks, is also noticeable in \(\frac{{\partial }^{2}D}{\partial {E}^{2}}\), albeit with changes in the sign. To visualize the location of the zeroenergy MBS, we show, in Fig. 2d, the profile of the local density of states \({\rho }_{{}_{{\rm{LDOS}}}}^{i}\ =\ {\sum }_{\sigma }( {u}_{i\sigma }{ }^{2}+ {v}_{i\sigma }{ }^{2})\), corresponding to the lowest positiveenergy eigenstate at μ = 0.5 meV where the Josephson junction is in the topological superconducting regime. The sharp peaks in \({\rho }_{{}_{{\rm{LDOS}}}}^{i}\) indicate that the MBS are localized predominantly near the two ends of the quasionedimensional channel. Figure 2e shows the profile of the charge density of states \({\rho }_{{}_{{\rm{CDOS}}}}^{i}\ =\ {\sum }_{\sigma }( {u}_{i\sigma }{ }^{2} {v}_{i\sigma }{ }^{2})\) corresponding to the lowest positiveenergy eigenstate at μ = 0.5 meV. The profiles of \({\rho }_{{}_{{\rm{LDOS}}}}^{i}\) and \({\rho }_{{}_{{\rm{CDOS}}}}^{i}\), at μ = −0.5 meV where the Josephson junction is in the topologically trivial superconducting regime, are shown in Fig. 2f, g, respectively. In this case, both the quasiparticle state and the charge density are distributed near the middle of the quasionedimensional channel. Interestingly, a comparison of Fig. 2e, g, implies an orderofmagnitude suppression in \({\rho }_{{}_{{\rm{CDOS}}}}^{i}\), which is reminiscent of the local chargeneutrality signature of the MBS and is another confirmation of the Majorana character of this state. The above results establish that the spinorbit coupling, generated by the SkX, alone can lead to the emergence of the MBS in the planar Josephson junction devices.
Skyrmion control
The skyrmion size in a SkX is tunable, with remarkable precession, by an external magnetic field, magnetic anisotropy and advanced symmetry protocol at heterointerfaces^{52,53,54}. In our Monte Carlo simulations, the skyrmion size was varied by tuning the magnetic field and the DMI, as shown in Fig. 3ac. The Bogoliubovde Gennes quasiparticle spectra at different skyrmion sizes, shown in Fig. 3df, imply that the presence of the zeroenergy MBS at a given chemical potential can be turned ON or OFF by only changing the skyrmion properties. The minimum diameter of the skyrmions, realized in our Monte Carlo simulations, is 10 lattice spacings for which the MBS appear in the discussed planar Josephson junction setting. With increasing the skyrmion size, we find that the range of the chemical potential within which the MBS appear decreases effectively, however, the oscillation amplitude of the MBS is suppressed gradually. Therefore, the skyrmions offer a unique ability to manipulate the localization length of the MBS in the planar Josephson junctions. The strongly localized MBS in the skyrmiontuned planar Josephson junctions can, therefore, have advantageous over other platforms for MBS realization in faulttolerant topological quantum computing.
The broken inversion symmetry at the interface between the twodimensional electron gas and the superconductor, often leads to a sizable intrinsic Rashba spinorbit coupling, which is usually considered as the primary mechanism for modifying the pairing symmetry of the induced superconductivity^{55,56}, leading to the desired topological superconductivity. We find that the MBS remain robust in the presence of the intrinsic Rashba spinorbit coupling (for details on the effect of Rashba spinorbit coupling, see Supplementary Note 2). By placing the SkX texture only below the middle quasi1D channel, we didn’t find robust MBS formation (for details, see Supplementary Note 3) and hence it suggests the significance of placing the SkX texture below the entire Josephson junction.
Phase control
Another important control parameter, that sets the planar Josephson junctions apart from other platforms hosting the MBS, is the phase difference φ between the two superconducting regions of a Josephson junction. The theoretical prediction^{15} and the subsequent experimental discoveries^{16,17} suggest that the Josephson junction needs to be biased by a phase difference φ = π to minimize the critical Zeeman field, required for inducing the topological superconductivity. Remarkably, in the current Josephson junction set up with the SkX, the topological superconductivity is induced at φ = 0, as depicted by the quasiparticle spectrum with varying φ in Fig. 4a. With increasing φ, the MBS move gradually from zero to higher energies, indicating an enhancement in the localization length of the MBS. The MBS appear again at zero energy above φ ≈ 3π/2.
This dephasing effect of the MBS can be understood from the Majorana oscillations—as we find that the oscillation increases with increasing φ in the range 0 < φ ≤ π (for details, see Supplementary Note 1). The finite length of the quasionedimensional metallic channel gives rise to the oscillations of the zeroenergy MBS with varying chemical potential μ. Furthermore, the finite width of the metallic channel provides extra room for delocalization of the MBS at the two ends, contributing additively to the Majorana oscillation. In the previous works on this geometry, a magnetic field is applied in the plane of the planar Josephson junction, which locks the phases of the MBS to φ = π. In our set up, there is no magnetic field applied to a particular direction, but a chiral magnetism exists throughout the junction and, therefore, a π phase biasing is not required in this case. The middle metallic region of the Josephson junction can be perceived as a quasionedimensional void region surrounded by the superconducting twodimensional electron gas. Additional phase difference between the two superconducting sides, therefore, only causes disruption to the induced topological superconductivity. This phenomenon generically takes place at several values of the chemical potential, as shown in Fig. 4b, where we plot the Majorana polarization \( {{\mathcal{P}}}_{{\mathcal{M}},1}\) of the lowest positiveenergy eigenstate in the parameter space spanned by the phase difference φ and the chemical potential μ. For the chosen range of μ values, the Majorana polarization decreases substantially within the range π/2 ≾ φ ≾ 3π/2. The Majorana oscillation in \( {{\mathcal{P}}}_{{\mathcal{M}},1}\), however, survives up to φ ≈ π. At φ = π, \( {{\mathcal{P}}}_{{\mathcal{M}},1}\) vanishes completely, indicating the disappearance of the MBS. Therefore, φ = 0 is the most favorable condition to realize the MBS in our Josephson junction set up and the phase difference can be further tuned to control the presence of the MBS. To check the consistency of our assignment of MBS, we also compute the Z_{2} topological index Q at several points of the above phase diagram using an effective onedimensional Hamiltonian of the planar Josephson junction with the SkX. In Fig. 4b, we show the topological invariant which confirms the phase diagram.
Conclusion
The skyrmions bring outstanding control functionalities to the planar Josephson junctions for the creation and manipulation of the zeroenergy MBS and their localization properties. The SkXs, being realized in an abundance of magnetic materials and also artificially created in patterned magnetic materials, offer a feasible approach for advanced manipulation of the zeroenergy MBS. The proposed planar Josephson junction, combined with a SkX, has the major advantages that there is no need for a strong intrinsic Rashbatype spinorbit coupling and phasebiasing constraint for the realization of the zeroenergy MBS. The enhanced tunability of the MBS in the proposed twodimensional platform opens up opportunities for designing feasible MBS braiding protocols for the faulttolerant topological quantum computing, and investigating Majorana spectroscopy using the multiterminal superconducting quantum interference devices.
Methods
Monte Carlo simulations
The SkX spin configurations were obtained using a L_{x} × L_{y} × L_{z} lattice with periodic boundary conditions along the x and y directions and open boundary conditions along the z direction. A biasfree sampling method, that provides a full and uniform coverage of the phase space spanned by the spin angles, was used for generating the completely random spin configurations. The calculation was started at a high temperature T = 10J with a random spin configuration and the temperature was lowered slowly down to a low value T = 0.001J in 2000 steps. At each temperature step, 10^{10} Monte Carlo spin updates were performed. In each spin update step, a new spin direction was chosen randomly within a small cone spanned around the initial spin direction. The new spin configuration was accepted or rejected according to the Metropolis energyminimization algorithm by comparing the total energies of the previous and the new trial spin configurations, calculated using the Hamiltonian Eq. (1).
Selfconsistent Bogoliubovde Gennes formalism
The Hamiltonian Eq. (2), which is quadratic in the fermionic operators \({\hat{c}}_{i\sigma }\), can be solved by exact diagonalization via a unitary transformation \({\hat{c}}_{i\sigma }={\sum }_{n}{u}_{i\sigma }^{n}{\hat{\gamma }}_{n}+{v}_{i\sigma }^{n* }{\hat{\gamma }}_{n}^{\dagger }\), where \({\hat{\gamma }}_{n}^{\dagger }\) (\({\hat{\gamma }}_{n}\)) is a fermionic creation (annihilation) operator of the quasiparticle/quasiphole state in the n^{th} energy eigenstate. The quasiparticle amplitudes \({u}_{i\sigma }^{n}\) and the quasihole amplitudes \({u}_{i\sigma }^{n}\) are determined by solving the Bogoliubovde Gennes equations: \({\sum }_{j}{{\mathcal{H}}}_{ij}{\psi }_{j}^{n}={\epsilon }_{n}{\psi }_{n}^{i}\), where \({\psi }_{i}^{n}={[{u}_{i\uparrow }^{n},{u}_{i\downarrow }^{n},{v}_{i\uparrow }^{n},{v}_{i\downarrow }^{n}]}^{T}\) is the basis wave function and ϵ_{n} is the n^{th} energy eigenvalue. The Hamiltonian \({{\mathcal{H}}}_{ij}\) is expressed in the following matrix form
where \({{\mathcal{H}}}_{\uparrow \uparrow ,\downarrow \downarrow }=t(1{\delta }_{ij})+(4t\mu ){\delta }_{ij}1/2\sigma g{\mu }_{B}{B}_{z}\), where σ = ± for ↑↑, ↓↓, and \({{\mathcal{H}}}_{\uparrow \downarrow }=1/2g{\mu }_{B}({B}_{x}+i{B}_{y})\). The swave pairing amplitude Δ_{i} = −U_{i}〈c_{i↑}c_{i↓}〉 is computed using \({{{\Delta }}}_{i}={U}_{i}{\sum }_{n}\left[\right.{u}_{i\uparrow }^{n}{v}_{i\downarrow }^{n* }(1f({\epsilon }_{n}))+{u}_{i\downarrow }^{n}{v}_{i\uparrow }^{n* }f({\epsilon }_{n})\left]\right.\), where U_{i} = U inside the two superconducting regions and zero in the middle quasionedimensional metallic channel, \(f({\epsilon }_{n})=1/(1+{e}^{{\epsilon }_{n}/{k}_{B}T})\) is the FermiDirac distribution function at temperature T. The selfconsistency iterations were performed until the pairing amplitudes Δ_{i} were converged at every lattice sites.
Calculation of topological invariant
A triangular SkX can be described by the following spin structure
where S is the spin amplitude and the spin angles θ_{i} and ϕ_{i} are defined as
R_{i} = (x_{i}, y_{i}) is the skyrmion center near position r = (x, y) and R is the skyrmion radius. To obtain an effective Hamiltonian for the planar JJ, we perform the following unitary transformation that rotates the local spin S_{i} with respect to the SkX plane normal, the z direction
where
The Hamiltonian (2) in the main text, in the transformed basis, becomes
where the new hopping integral is given by
α being the strength of the generated SOC.
To obtain an effective onedimensional Hamiltonian, we consider an infinitely long junction (i.e., L_{y} → ∞) and perform a partial Fourier transform \({d}_{i,{k}_{y},\sigma }={\sum }_{j}{e}^{i{k}_{y}y}{d}_{i,j,\sigma }\). The resulting Hamiltonian \({\mathcal{H}}(x,{k}_{y})\) is then used to compute the topological invariant Q, the Z_{2} topological index associated with the broken chiral symmetry, given by^{57}
where ‘Pf’ denotes the Pfaffian. A topologically nontrivial phase is determined by Q = −1, while Q = 1 represents the trivial phase.
Data availability
The data presented in this paper are available from the corresponding author upon reasonable request.
Code availability
Codes used in this paper are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences, and Engineering Division.
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N.M. and E.D. planned the project. N.M. performed the calculations. S.O. provided theoretical inputs. All authors discussed the results, wrote, and reviewed the manuscript.
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Mohanta, N., Okamoto, S. & Dagotto, E. Skyrmion control of Majorana states in planar Josephson junctions. Commun Phys 4, 163 (2021). https://doi.org/10.1038/s42005021006665
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DOI: https://doi.org/10.1038/s42005021006665
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