Corresponding Author

Dirk Manske

Max Planck Institute for Solid State Research

References

1.
Brydon, P.; Asano, Y.; Timm, C.
Spin Josephson effect with a single superconductor
2.
Kuboki, K.; Takahashi, H.
Spontaneous spin current near the interface between unconventional superconductors and ferromagnets
3.
Gentile, P.; Cuoco, M.; Romano, A.; Noce, C.; Manske, D.; Brydon, P.
Spin-Orbital Coupling in a Triplet Superconductor-Ferromagnet Junction

In collaboration with:

P. Gentile, M. Cuoco, A. Romano, C. Noce (SPIN-CNR and Department of Physics, University of Salerno)

P. Brydon (Institute for Theoretical Physics, TU Dresden)

Department "Quantum Many-Body Theory"

Role of orbital states at the interface of a triplet superconductor and a ferromagnet

Authors

D. Manske

Departments

Quantum Many-Body Theory (Walter Metzner)

In order to study the interplay between spin and orbital degrees of freedom in a Triplet Superconductor (TSC) − Ferromagnet (FM) junction we solve a lattice model of a TSC-FM heterostructure using a self-consistent Bogoliubov–de Gennes theory. We show that increasing the angle between the ferromagnetic moment and the triplet vector order parameter enhances or suppresses the p-wave gap close to the interface, according to whether the gap antinodes are parallel or perpendicular to the boundary, respectively. On the contrary, for a px or py TSC, we find that the variation of the gap controls the orientation of the FM’s moment via the change in condensation energy. As a result, the stable configuration is either parallel or perpendicular to the TSC vector order parameter, depending on the alignment of the TSC gap with respect to the interface, thus evidencing a unique form of spin-orbital coupling. The stable configuration for the chiral px+ ipy state reveals competition between the different orbital components, with a first-order transition from the perpendicular to the parallel configuration as the FM exchange field is increased.

In general, the phases of a singlet superconductor and a FM are incompatible, since the exchange field of the FM destroys the superconductivity by aligning the antiparallel spins of the electrons in singlet Cooper pairs. This pair-breaking effect makes the homogeneous coexistence of both phases almost impossible. On the other hand, the coexistence of FM and TSC states is more favorable, as the exchange field is only pair breaking when it is perpendicular to the Cooper pair spins. Therefore, one can expect interesting effects, since the orientation of the FM moment relative to the TSC vector order parameter is now a crucial variable. In addition to the pair breaking, spin-flip reflection processes at the interface with the FM scatter the triplet Cooper pairs between the spin ↑ ­ and ↓ condensates, setting up a Josephson-like coupling between them [1]. The resulting "spin Josephson effect" is manifested as a spontaneous spin current in the TSC normal to the TSC/FM interface. Both the pair breaking and spin Josephson coupling make significant contributions to the free energy of a TSC-FM junction through the proximity effect, interface electronic reconstruction, and the variation of the TSC gap. Although these contributions depend upon the direction of the FM exchange field, the two effects do not necessarily act constructively: while pair breaking is always absent for a moment perpendicular to the TSC vector order parameter, the effective Josephson phase difference can vanish for parallel and perpendicular configurations, depending on the orbital pairing state.

We examine a lattice model of the TSC-FM junction shown in Fig. 1. The lattice size is (L+1)×(L+1) with periodic boundary conditions imposed along the direction parallel to the interface. Indicating each site by a vector i=(ix,iy) with ix and iy being integers ranging from –L/2 to L/2, we consider the Hamiltonian [2]

H = Σ<ij>,σ tij (ciσ+ cjσ + h.c.) − μ Σiσ niσ − Σ<ij>∈TSC V(ni↑­nj↓ + ni↓nj­↑) − ΣiFM h·si

<p><strong>Fig.1:</strong> Schematic diagram of the two-dimensional TSC-FM junction. The FM region is located at x&lt;0, while the TSC is realized for x&gt; 0. The magnetization <strong>M</strong> of the FM is collinear to the exchange field <strong>h</strong> and forms an angle&nbsp;&Phi; with the <strong>d</strong>-vector of the TSC, which defines the z-axis. We study TSC states with <em>p<sub>x</sub></em>, <em>p<sub>y</sub></em>, and <em>p<sub>x</sub>+</em>i<em>p<sub>y</sub></em> symmetry.</p> Zoom Image

Fig.1: Schematic diagram of the two-dimensional TSC-FM junction. The FM region is located at x<0, while the TSC is realized for x> 0. The magnetization M of the FM is collinear to the exchange field h and forms an angle Φ with the d-vector of the TSC, which defines the z-axis. We study TSC states with px, py, and px+ipy symmetry.

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where ciσ is the annihilation operator of an electron with spin σ at the site i, niσ=c+c is the spin-σ number operator, and siss’ cis+sss’ cis’ is the local spin density. The lattice is divided into three regions: the FM subsystem for ix<0, the TSC subsystem for ix>0, and the interface at ix=0. The chemical potential m is the same across the lattice. The order parameter of the TSC, the so-called d-vector, encodes the spin structure of the Cooper pairs: it is defined as d=(1/2)×(Δ1−Δ-1)x−(i/2)(Δ1-1)y0z, where ΔSz is the gap for triplet pairing with the z component of the spin Sz=−1, 0, 1. Since the TSC state is invariant under spin rotations about the d-vector, h can be restricted to the x-z plane, i.e., h=h(sin(Φ),0,cos(Φ)). We obtain a single-particle Hamiltonian HMF from Eq.(1) by decoupling the interaction term and solving self-consistently for the mean-field amplitudes Δij=<ci↑ci↓>. We then calculate the condensation energy EΔ of the TSC and the Gibbs free energy F of the junction.

<strong>Fig. 2:</strong> Zero temperature pairing amplitude normalized to its bulk value as a function of the distance <em>i</em><sub>x</sub> from the interface for several different angles&nbsp;&Phi; between <strong>d</strong> and <strong>M</strong>. The spin-triplet orbital symmetry is of (a) <em>p</em><sub>x</sub>, (b) <em>p</em><sub>y</sub>, and chiral type with (c) a real <em>p</em><sub>x</sub> and (d) an imaginary <em>p</em><sub>y</sub> component, respectively. Zoom Image
Fig. 2: Zero temperature pairing amplitude normalized to its bulk value as a function of the distance ix from the interface for several different angles Φ between d and M. The spin-triplet orbital symmetry is of (a) px, (b) py, and chiral type with (c) a real px and (d) an imaginary py component, respectively. [less]

In Fig. 2 we present the pairing amplitude profile near the interface for h=1.5, tint=1, and several different values of 0≤Φ≤π/2. This is determined by minimizing the Gibbs energy functional with respect to the pairing amplitudes for a fixed angle. Independent of the orbital symmetry, the proximity effect in the FM smoothly evolves from a monotonic decay at Φ=π/2 to a damped oscillating behavior at Φ=0. On the other hand, the pairing amplitude in the TSC side of the interface strongly depends upon both the angle Φ and the orbital symmetry of the TSC. For a TSC with px orbital symmetry, the pairing amplitude near the interface is reduced as the exchange field is tilted from parallel to perpendicular with respect to the d-vector (Fig. 2(a)); the opposite behavior is observed for a py TSC, although the effect is less pronounced (Fig. 2(b)). The chiral px±ipy TSC evidences both trends: decreasing Φ from π/2 to 0 enhances the real (px) part of the gap (Fig. 2(c)), but suppresses the imaginary (py) part (Fig. 2(d)). Competition between the two gap components enhances their variation with f compared to the time-reversal symmetric states.

The pair breaking due the spin-spin coupling alone cannot explain the different f dependence of the py and px gap profiles. This instead originates from the spin-flip reflection of triplet Cooper pairs at the interface with the FM, which is crucial for the spin Josephson effect. During this scattering process an incident Cooper pair acquires a spin- and orbital-dependent phase shift [3].

<p><strong>Fig. 3:</strong> Sketch of the most favorable magnetic (small red arrow) configurations with respect to the orbital symmetry and the d-vector (large blue arrow) of the TSC as well as the character of interface transparency.</p> Zoom Image

Fig. 3: Sketch of the most favorable magnetic (small red arrow) configurations with respect to the orbital symmetry and the d-vector (large blue arrow) of the TSC as well as the character of interface transparency.

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Finally, we have calculated Gibbs free energy F directly from HMF. The minimum of F fixes the stable moment orientation. For the px orbital symmetry, the profile exhibits a single minimum at Φ=0 and a maximum at Φ=π/2, and vice versa for the py TSC. The stable magnetic orientation is therefore parallel (perpendicular) to the d-vector if the antinodes of the p-wave TSC gap are perpendicular (parallel) to the interface. Thus, the orbital pairing state in the bulk TSC plays a critical role in fixing the stable orientation of the magnetization in the FM, which is summarized by the sketch in Fig. 3. For the time-reversal symmetric gaps, the easy axis in the FM originates from the maximization of the TSC’s condensation energy. On the other hand, the orbital frustration of the condensation energy in a chiral TSC leads to a magnetic configuration with a first-order transition between the perpendicular and parallel configurations as a function of the exchange field [3].


 
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