Observing Majorana bound states of Josephson vortices in topological superconductors

Abstract

In recent years there has been an intensive search for Majorana fermion states in condensed matter systems. Predicted to be localized on cores of vortices in certain nonconventional superconductors, their presence is known to render the exchange statistics of bulk vortices non-Abelian. Here we study the equations governing the dynamics of phase solitons (fluxons) in a Josephson junction in a topological superconductor. We show that the fluxon will bind a localized zero energy Majorana mode and will consequently behave as a non-Abelian anyon. The low mass of the fluxon, as well as its experimentally observed quantum mechanical wave-like nature, will make it a suitable candidate for vortex interferometry experiments demonstrating non-Abelian statistics. We suggest two experiments that may reveal the presence of the zero mode carried by the fluxon. Specific experimental realizations will be discussed as well.

Non-Abelian statistics (1–3) has recently been the subject of intensive research driven both by its possibly profound impact on the field of quantum computation (4–6) and by the search for its manifestations (7, 8). Among all mechanisms giving rise to such statistics, the route via spin-polarized p-wave superfluidity may be the simplest one. It was previously argued (9–11) that an Abrikosov vortex in a p-wave superfluid can trap a zero energy Majorana fermion, being a self-conjugate “half” fermion. A pair of Majorana modes constitute a regular fermion, and the resulting nonlocal occupancies label a set of degenerate ground states. Braiding of vortices results in mixing of these ground states, sometimes in a noncommutative fashion: it matters in which order multiple braidings are performed. The search for an explicit experimental signal of the resulting vortex exchange statistics, as well as for the presence of Majorana modes on their cores, is currently on its way.

In this paper we propose an experiment that probes Majorana fermions in Josephson vortices (fluxons). Josephson vortices are trapped in insulating regions between superconductors. For conventional superconductors, they are described as solitonic solutions of the sine-Gordon equation moving with small inertial mass (estimated to be smaller than the electron mass). In the case of topological superconductors, we find that such vortices bind a localized Majorana zero mode and would therefore behave as non-Abelian anyons, despite the fact that they lack a normal core. We show that the non-Abelian nature of these vortices manifests itself in measurable transport properties of the Josephson junctions that house them.

We start by showing that Josephson vortices traveling in a Josephson junction in a topological superconductor bind a single Majorana zero energy state (see Eq. 6). We then discuss two experiments that can be used to measure the presence of these Majorana fermions. The first probes a thermodynamical property of a circular charge biased Jospephson junction (Fig. 1), by measuring the nonlinear capacitance induced by the persistent motion of the vortex trapped in the junction (12–15). The second is an interference experiment of fluxons demonstrating Aharonov–Casher (16) oscillations (Fig. 2), similar in spirit to the one proposed in ref. 17.

Fig. 1.

Aharonov–Casher effect in a long circular Josephson junction. The junction traps a single fluxon that is traveling around the ring propelled by a bias charge Q induced between the two ring-shaped superconductors. The energy spectrum of the junction is periodic in Q with periodicity e when Φ is increased to nucleate a vortex within the interior hole. Copper wires act as reservoirs of unpaired electrons.

Fig. 2.

Vortex interferometry experiment based on the Aharonov–Casher effect (17) adapted to Josephson vortices. A superconducting wire creates a circulating magnetic field acting as a source for the entrance of Josephson vortices into the sample. An applied supercurrent drives the vortex along one of two paths circumventing an island toward the top of the sample. A charge Q enclosed in the island controls the interference term via the Aharonov–Casher effect. When the flux Φ nucleates a vortex in the central region, the interference term would be obliterated.

Results

Hamiltonian of a Circular Josephson Junction.

We start by considering a circular Josephson junction, made of two concentric superconducting annuli, separated by a thin insulator. We assume that the hole at the center of the inner superconductor is of a size comparable to the superconducting coherence length, and encloses N_v vortices. The Hamiltonian governing the junction would be composed of three parts, H = H^ϕ + H^ψ + H_tun. The first, H^ϕ is related to the dynamics of the phase across the junction. For a Josephson junction of height h_z, this part of the dynamics is derived from the following Hamiltonian (see, for example, refs. 14 and 18)

[1]

where ϕ is the phase difference across the junction, n is the two dimensional (2D) density of Cooper pairs on, say, the inner plate, and σ is the 2D density of the externally induced charge. The first part of the Hamiltonian is the capacitive energy, the second the magnetic energy, and the third is the Josephson energy. The resulting equation of motion is the sine-Gordon equation, with the typical speed of light reduced to (here d is the width of the insulating barrier and λ_L is the London penetration length). The Josephson vortex is a soliton described by this equation, with its typical size set by the Josephson penetration length λ_J. We shall assume throughout that the circumference of the junction, L, is much larger than λ_J, and that λ_L ≪ h_z ≪ λ_J. The parameter .

The second part of the Hamiltonian, H^ψ, originates from the neutral protected edge modes of the topological superconductor, which give rise to its quantized thermal Hall conductance. The Hamiltonian governing these neutral modes is with

[2]

where 1 and 2 refer to the two counterpropagating Majorana edge modes. Here ψ_i(x) is a Majorana field, , x is the coordinate running along the Josephson junction, and v_ψ is the velocity along the edge. In terms of the electron’s creation and annihilation operators c_i(x), and the phase fields of the two superconductors on the two sides of the junctions ϕ_i(x) the Majorana fields may be expressed as

[3]

In this explicit form, the equation holds true for a spin-polarized p-wave superconductor. For a general topological superconductor, it is still the case that the Majorana field will acquire a minus sign going around the edge, in addition to one minus sign per each vortex enclosed in its path (i.e., one minus sign for every 2π winding of the phase ϕ_i). The boundary conditions of the fields ψ₁(x), ψ₂(x) therefore depend on N_v, the number of vortices enclosed by the two superconducting annuli.

When the two superconducting islands are brought into close proximity, tunneling terms of the form translate to

[4]

with ϕ(x) ≡ ϕ₂(x) - ϕ₁(x) and m being a tunneling amplitude. Writing Eqs. 2 and 4 compactly as a matrix equation, the Hamiltonian becomes

[5]

where Ψ = (ψ₁,ψ₂)^T is a spinor composed of the two counterpropagating Majorana modes. The Hamiltonian possesses a symmetry under ϕ → ϕ + 2π and Ψ → σ_zΨ.

Bound Majorana Mode on the Background of a Soliton.

A solitonic solution of Eq. 1, also known as a fluxon or a Josephson vortex, is a finite energy solution that interpolates between two minima of the periodic potential described by the Josephson term. For a long Josephson junction (L≫λ_J) it acquires the form where x₀ is the position of the soliton (see, for example, refs. 14 and 18). In the following we solve Eq. 5 in the background of a single soliton, explicitly plugging ϕ_s into ϕ, and using . This would in turn result in a tunneling amplitude whose sign is different on the two sides of the soliton. In light of the Jackiw–Rebbi mechanism, Eq. 5 will now bind a zero energy mode at the position of the soliton x₀,

[6]

In the limit of a long junction, L≫λ_J and L≫v_ψ/m, the shape of the Majorana mode is described by the localized function , with N a normalization factor and x₀ the center of the Josephson vortex. The operator γ_J is a localized Majorana fermion and the subscript J indicates that this mode is bound to a Josephson vortex. This mode satisfies . Indeed, the entire low energy spectrum of bound states can be extracted. Plugging the solitonic solution into Eq. 5 and rotating the spinors according to

the Hamiltonian can be written conveniently as

[7]

The spectrum of this Hamiltonian possesses a supersymmetry with a superpotential . In particular, the zero mode is annihilated by either A or A^† depending on the sign of the mass, whereas the other operator will not have a normalizable eigenfunction at zero energy. The rest of spectrum is doubly degenerate with A and A^† connecting the states of the doublet. Squaring the Hamiltonian in Eq. 7 and linearizing the potential we get a shifted harmonic oscillator, and the spectrum of excitations above the zero energy mode is

[8]

Majorana fermions have to come in pairs, due to the properties of the BdG equations. The second Majorana fermion will be on one of the superconductor edges, and will be denoted by γ_e, with e standing for edge. Its position will depend on the number of vortices N_v in the inner hole. A zero mode is found on every edge that encloses an odd number of vortices. Hence, when N_v is odd, γ_e will be localized on the edge separating the inner superconductor from the vacuum. When N_v is even, γ_e will be localized on the edge separating the outer superconductor from the vacuum. The two Majorana fermions, γ_e and γ_J, cannot shift from zero energy without hybridizing, and the superconductor that separates them prevents that from happening. Note that in the absence of electron tunneling across the junction, the presence of these two Majorana modes is protected by an index theorem. Their wavefunction will be spread evenly around the two edges of the superconductor. When we next turn on electron tunneling, the Majorana mode γ_J cannot disappear without hybridizing with the distant Majorana mode on the edge of the sample. Therefore it necessarily persists in the junction. Due to the effect of tunneling it gets localized around the center of the soliton as described by Eq. 6.

Due to the presence of the soliton in the junction, the boundary conditions are different for the two Majorana fields and depend on N_v: and . A rotation of the soliton around the junction shifts x by L and ϕ₁(x) by 2π. These shifts have two effects. First, they multiply ψ₁ by and ψ₂ by due to the boundary conditions. Second, they multiply the off-diagonal term of [5] by -1 and hence multiply ψ₁(x) in Eq. 6 by -1. Combining these two effects together, we see that γ_J is multiplied by . In view of this, the transformation may be understood as a consequence of the winding of one Majorana fermion around another for odd N_v, and the absence of such winding for even N_v. Remarkably, we find below that the capacitance of the junction in the two cases is different.

Manifestation of the Majorana Mode in Thermodynamics and Transport.

We now consider the energy of the long circular Josephson junction when it is biased by an external charge Q = 2eσLh_z. For a “conventional” Josephson junction made of s-wave superconductors this problem was studied in ref. 14. In that case the Hamiltonian consists of Eq. 1 only. The magnetic and Josephson energies constitute the rest mass of the vortex. The charging energy constitutes its kinetic energy, and is the only component that depends on Q. This kinetic/charging energy may be understood in two ways. First, by viewing the vortex as a particle of mass in a one dimensional ring, subjected to a vector potential , with a set of energy eigenstates

[9]

with n being the quantum number that quantifies the momentum of the vortex. Second, by writing this energy as a capacitive energy of the form, where now C is the effective capacitance (which for a short junction coincides with the geometric capacitance). The quantum number n is now identified as the number of Cooper pairs charging the Josephson junction, and the capacitance is C = M(2eL)²/(2πℏ)².

The spectrum of [9] is described by a set of parabolas, each characterized by a different integer value of n. When Q is increased so that the spectrum reaches a crossing point of two parabolas of n and n + 1, it becomes energetically favorable to tunnel a Cooper pair to decrease the charging energy, thus crossing to the next parabola. The matrix element required for this tunneling is generated by a small amount of disorder. The velocity of the moving fluxon is proportional to the derivative of the ground state energy with respect to Q. In turn, the moving vortex generates a measurable voltage via the Josephson relation between the inner ring and outer ring of the device, which will be oscillating as function of Q with a periodicity of 2e. The periodicity of 2e is quite expected, given the energy cost needed to break a Cooper pair. None of these results depend on the value of N_v.

The same spectrum shows a strikingly different behavior when the junction is made of two topological superconductors. In particular, this difference manifests itself in the dependence of the spectrum on the charge Q and N_v. The fermionic part of the Hamiltonian now has two degenerate eigenstates, which are two eigenstates of the operator iγ_Jγ_e, with eigenvalues ± 1. When it comes to the spectrum of the Josepshon vortex, there is a crucial difference between the case where γ_e is localized at the edge of the outer superconductor (even N_v) and that in which it is localized at the center of the inner superconductor (odd N_v). As we saw before, in the latter case the winding of the soliton around the junction multiplies γ_J by -1. It is easy to see that it multiplies γ_e by -1 as well. These two operator transformations are implemented by the unitary transformation U = e^iαiγ_Jγ_e on the state of the system (2). The phase α cannot be determined by this consideration, and is not important for what follows. The two ground states are eigenstates of U, and thus the application of U multiplies them by two phases, that differ by π. The phase accumulated by the soliton encircling the ring affects its spectrum. When the energy [9] is viewed as the energy of a charged particle on a ring threaded by a magnetic flux, a π phase shift corresponds to the introduction of half a quantum of flux. The spectrum [9] is then changed to be

[10]

where the f =  ± 1 correspond to the two eigenstates of U for α = 0. The charge Q^′ may be shifted with respect to the induced charge Q by a nonuniversal number that depends on α. The spectrum is now periodic with a period of e, rather than the period of 2e as was the case for the nontopological superconductor, as well as for the topological superconductor with an even N_v.

For the two sets of parabolas to be observed in an experiment when N_v is odd, the system has to be able to switch between the two values of f, namely the two states iγ_Jγ_e =  ± 1. This switch requires the total number of electrons to change by one (7, 19, 20). To allow for that to happen, the junction should be weakly coupled to a reservoir of single electrons; i.e., to a metal. The ground state energy would be . The end result is that the voltage measured between the inner and outer edges of the superconductor would be periodic as function of Q (or V₀) with the periodicity now being e for odd N_v and 2e for even N_v.

Whereas this experiment probes a thermodynamical property, the next one is an interference experiment that probes electronic transport. The fluxons are generated near the bottom of the interferometer (Fig. 2), and driven by a supercurrent driven from left to right. The fluxon beam is split into two partial waves enclosing an island with externally imposed charge Q. Due to the Aharonov–Casher effect, the magnitude of the vortex current oscillates according to

[11]

where ζ is the visibility of the oscillations and J_v0 the average current. By the Josephson relation, a measurable voltage difference is created between the left and right ends of the superconductor, which would oscillate with changing Q. If the flux Φ on the central island is increased so that another non-Abelian vortex is nucleated in the central hole, the interference pattern would be zero, ζ = 0. These results are sensitive to effects of decoherence, which would be reflected in a reduction of the value of ζ as the temperature is increased. However, the fluxons are quite light, and it was predicted (21) and experimentally demonstrated (15, 22) that they can exhibit quantum behavior (23, 24). Also, the gap to the next fermionic state on the background of the soliton has different parametric dependence as compared to Abrikosov vortices.

For a soliton in a topological superconductor there are two main energy scales that control decoherence effects. The first is related to the bosonic part H^ϕ, whereas the second stems from the fermionic part H^ψ.

The spectrum of the bosonic Hamiltonian H^ϕ contains nontopological excitations with which the fluxon can interact. These are quantized phase oscillations, or plasmons. Written in terms of collective coordinates for the fluxon, the Hamiltonian would be similar to the Caldeira–Leggett Hamiltonian of a quantum particle interacting with a bath. However, unlike the Caldeira–Leggett mechanism, here the plasmon spectrum is gapped, with the plasmon gap given by . At energy scales below this gap (estimated to be a few Kelvin) the coupling to plasmons is exponentially suppressed, and the internal dephasing thus mostly avoided (25) over lengths much larger than the Josephson penetration length. This lies at the origin of the prediction of quantum behavior of Josephson vortices in long Josephson junctions. In particular, scattering of the fluxon on static inhomogeneities in the junction would be mostly elastic due to the presence of this plasmon gap (21). Tunneling of a fluxon through barriers was predicted to be enhanced by the presence of the plasmons (21) and experimentally observed (22).

The second energy scale is unique to topological superconductors, and is related to the presence of higher energy fermionic states above the zero energy mode carried by the soliton, as described by H^ψ and the fermionic tunneling terms, see Eq. 8. The presence of these states would result in decoherence of the non-Abelian mechanism. The gap to the first state would be . To avoid these effects, the temperature should be lower than this gap as well.

Proposed Realizations.

There are several systems in which the ideas presented above may be implemented in practice. The most relevant for our purpose is a topological insulator whose surface states become superconducting by proximity to an s-wave superconductor, and are driven into an effective p-wave pairing for a single fermionic species. The second is a hybrid structure of three materials: a semiconductor with a magnetic material and a superconductor layer placed on top of it, as discussed in ref. 26. The third is the perovskite material Sr₂RuO₄ (SRO) that becomes superconducting at temperatures below 1.5 K, and for which there is growing evidence that it realizes an unconventional pairing of a p_x ± ip_y (spin-triplet) form.

The surface state of a topological insulator is described by the Hamiltonian (where ψ = (ψ_↑,ψ_↓)^T is a spinor composed of the electronic operators associated with the spin components). The chemical potential is tuned to the Dirac point. Superconductivity is induced on the surface state by the proximity effect (27), represented by an extra term to the Hamiltonian. The superconducting layer of height h_z is deposited as described in Fig. 1. At the thin insulating layer between the two annuli, as well as in the inner and outer holes, an insulating magnetic material of constant magnetization should be deposited, resulting in a Zeeman term of the form Mψ^†σ_zψ in the surface Hamiltonian. This breaks the time-reversal symmetry at the edges and by that chooses a single chirality for the flow of the neutral chiral edge modes. The dynamics of the soliton will be largely determined by the s-wave superconducting layer, whereas a zero energy Majorana mode will be trapped by the soliton on the surface state of the topological insulator. Other than these material specific details, the rest of the arguments in the paper can be applied with no further changes.

To estimate the experimental parameters we take typical values λ_L = 0.1 μm and λ_J = 30 μm for an s-wave superconducting layer with h_z = 0.5 μm and an insulating region of width d = 20 Å. For these parameters, the dimensionless parameter β² is approximately 0.01, and the velocity of light is . We take the neutral edge velocity to be v_ψ = 10⁵ m/s (27) (which coincides with the surface state Fermi velocity close to the Dirac point), and the tunneling between Majorana edge states to be 0.025 meV. The plasmon gap E_p is approximately 5 K (25), whereas the intracore gap E_n given by direct substitution to Eq. 8, is approximately 120 mK. The junction charging energy E_c = (2e)²/2C is approximately 250 (λ_J/L)² mK. To observe quantum phenomena, we need the temperature to be smaller than all these energy scales, T < min{E_n,E_c,E_p}. The operating temperatures are therefore in the range of 10–100 mK.

The implementation of the Josephson junction in a hybrid structure of semiconductor quantum well coupled to an s-wave superconductor and a ferromagnetic insulator (26) is similarly straightforward. The superconducting layer is deposited as in Fig. 1, whereas the ferromagnetic insulator layer will be deposited throughout with no restrictions, breaking the time-reversal symmetry for the quantum well. Another option is to use a quantum well with both Rashba and Dresselhaus spin-orbit coupling, and deposit the superconducting layer as before, with an external magnetic field applied in the in-plane direction (28). Both cases will result in a superconductor-normal-superconductor type Josephson junction.

The implementation using SRO is different in several respects. First, SRO is not spin-polarized. As a consequence, in bulk SRO Majorana modes are carried by half-vortices, that are made of a π rotation of the pairing d-vector glued together with a π phase winding of the order parameter. These half-vortices were recently observed (29) in mesoscopic samples of the type that may be useful for the present context. Similarly, Majorana modes in a long Josephson junction would require half-fluxons. Second, SRO is a three dimensional material, made of two dimensional layers. The physics described above for the Majorana modes carried by semifluxons decouples into different layers, one Majorana edge per layer. The multitude of Majorana modes does not affect the even N_v case. In the case of odd N_v, the factor γ_eγ_J in U is replaced by γ_e,i γ_J,i, with the index i numbering the layers, and L_z being the number of layers. The unitary transformation still has two eigenvalues that differ by a minus sign, and a transfer of a single electron facilitates a transition between the two states. Thus, as long as the zero energy states do not split, the multilayer case will not be different from the single layer one.

Tunnel coupling between the layers may split the Majorana zero energy modes. However, a tunneling term between the i, j layers is of the form

[12]

with i, j being the layers involved in the tunneling. This tunneling is suppressed when the phase difference between the layers vanishes. Fluctuations of ϕ are massive due to the presence of the interlayer Josephson coupling, and thus we may expect the Majorana modes not to split (see also ref. 30).

Third, SRO may take two possible forms of p-wave pairing, commonly denoted by p_x ± ip_y. The analysis above assumes that both superconductors are of the same pairing form. If the converse is true, the shape of the Majorana zero mode is different, but the rest of the analysis is unaffected. We analyze this case in SI Text.

We note that SRO suffers from the presence of a small minigap approximately Δ²/E_F for quasi-particle excitations in the core of bulk vortices and in edge states. This sets a constraint of about 10 mK on the maximum temperature. For the case of a topological insulator tuned to the Dirac point this problem is avoided.

Conclusion

In summary, we predict that phase solitons in a long Josephson junction embedded in a topological superconductor would carry a localized Majorana zero mode. Consequently, these solitons would constitute anyons with non-Abelian exchange statistics. Exploiting the quantum nature of these solitons we suggest two experiments that can reveal the presence of the Majorana modes. One experiment involves voltage oscillations of a topological superconducting capacitor, realized by a circular Josephson junction hosting a single fluxon. The other experiment involves interference effects of a fluxon beam.