P.M.R. Brydon and D. Manske Since the discovery of the high-Tc cuprates,
much attention has been directed towards the fabrication of phase-sensitive devices using un-conventional superconducting materials. Such devices are not only of technological impor-tance, but they also provide insight into funda-mental questions such as the symmetry of the pairing wavefunction [1]. Focus has recently
shifted to the properties of junctions involving tripletsuperconductors, motivated in large part by the confirmation of triplet superconductiv-ity in Sr2RuO4[2]. The triplet spin-state of the Cooper pairs implies an intimate relationship to magnetism, leading to novel tunneling effects in structures involving both triplet superconduc-tors and magnetic phases.
Together with B. Kastening (RWTH Aachen) and D.K. Morr (University of Illinois), we have theoretically studied a device combining triplet superconductivity with magnetism, the so-called triplet-superconductor – ferromagnet – triplet-superconductor (TFT) junction [3,4]. In this junction, a thin ferromagnetic barrier is sandwiched between two bulk pz-wave super-conductors, as shown schematically in Fig. 63.
The triplet-superconductor vector order param-eters, the so-called d-vectors, lie in the x–y plane, giving Cooper pairs with Sz=±1. The d-vector in the superconductor on the left de-fines the x-axis; the d-vector on the right is inclined to this by the angleθ. The two super-conductors both have a gap of magnitude Δ0, but there is a phase differenceφbetween them.
Within the barrier, there is a component of the magnetizationM⊥in thex–yplane, and a com-ponentMalong thez-axis; charge scattering is included by the potentialUP.
Figure 63: Schematic diagram of the TFT junction studied in this work. Thed-vectors in the two super-conductors are misaligned by an angleθ, and there is a phase differenceφbetween the two condensates.
We decompose the magnetic moment of the barrier into a component along thez-axis, M, an a com-ponent in thex–yplane,M⊥, parameterized by the angleα. Charge scattering is described by the poten-tialUP.
We obtain the current through the barrier by solving the Bogoliubov-de Gennes equations for the junction. This yields the energies of the so-called Andreev bound states (subgap states localized at the barrier) from which the Josephson currentIJ may be deduced. In what follows, we express the current in units of
ekFΔ0/ and the scattering potentials in units of2kF/m, wherekFandmare respectively the Fermi wavevector and effective mass in the bulk superconductors. All results are for zero tem-perature.
Figure 64: Dependence of the Josephson current on the angle α for several values of φ. M=UP= 0, M⊥= 1, andθ= 0.
In the case of a ferromagnetic barrier between two singlet superconductors, only the magni-tude of the ferromagnetic moment is relevant to the transport. In contrast, the current through the TFT junction also depends crucially upon the orientation of the moment with respect to the d-vectors. In the special case when the d-vectors are parallel, the current is strongly de-pendent uponM⊥: Rotating the moment about the z-axis, the current at fixed φ shows a pe-riodic dependence upon the angle α between M⊥ and the x-axis, with a maximum value when M⊥⊥dL,R (Fig. 64). The maximum be-comes very sharply peaked as the phase dif-ference approaches φ=π; in this limit, small changes in the alignment of M⊥ produce very large changes in the current. This effect can be exploited to build a ‘Josephson current switch’
in which the current is turned ‘on’ (IJ=0) or
‘off’ (IJ≈0) by rotation of the barrier moment.
This ‘switch’ has possible relevance to quan-tum computing, as the behavior of the junction is similar to other two-level systems proposed as qubits. Importantly from this perspective, we find that the effect is robust to a component of the magnetization along the z-axis and charge scattering at the barrier.
As another example of the non-trivial depen-dence of the current on the orientation of the magnetic moment, we consider the case when
|M⊥|= 0. Here the absence of spin-flip scat-tering at the barrier decouples the two spin channels. Misalignment of the d-vectors then producesspin-dependenteffective phase differ-ences,φσ=φ–σθ. Atφ= 0, therefore, each spin channel sees a finiteeffective phase difference
−σθ; although the current through each spin sector isfinite, the total current is nevertheless vanishing (see Fig. 65). This is because the bar-rier transparency is the same for each spin chan-nel, and the two contributions therefore can-cel. When there is both charge scattering and scattering off a longitudinal moment, however, the effective scattering potential at the barrier is also spin-dependent and given by UP–σ|M|. This reduces the transmission through the spin↓ channel more than that through the spin↑ chan-nel, and so as shown in Fig. 65 the current through the two channels does not cancel at φ= 0. The appearance of afinite current without a phase difference between the two condensates is a very unusual result.
Figure 65: Current vs. phase relations for the junc-tion showing the appearance of a finite current at φ= 0 for misalignedd-vectors (θ=π/3).
The critical current of the junction is defined as the maximum current with respect to φfor given barrier potentials andd-vector alignment.
Wefind that the critical current of the TFT junc-tion shows pronounced non-analyticities as a function of the barrier parameters. These are
shown for example in Fig. 66, where the criti-cal current is plotted as a function of UP and
|M⊥|. As marked in white, there is a line of first-order non-analyticity (solid) and also third-order non-analyticity (broken). These lines de-fine the boundaries of different ‘phases’ of the junction, A, B and C. Within each phase, the critical current occurs at qualitatively different points along theIJvs.φcurve: In region A it oc-curs atφ=π±θ±0+, on one side of jump dis-continuities in IJ (e.g., as in Fig. 65); in region B it occurs at a stationary point; and in region C it occurs atφ=π±θ∓0+, on the opposite side of the jump discontinuities compared to region A. As the scattering at the barrier is changed, or the alignment of thed-vectors altered, the junc-tion undergoes a ‘transijunc-tion’ from one critical current state to another. We predict that these transitions will produce an observable signal in the critical current, especially so when the ac-companying non-analyticity is offirst-order.
Figure 66: ‘Phase diagram’ for the critical current as a function of UP and |M⊥|, for α= 0.2π and θ= 0.35π. The solid white line indicates afirst-order non-analyticity, while the broken white line denotes a third-order non-analyticity. The ‘phases’ A, B and C are defined in the text.
To summarize, we have extensively character-ized the Josephson current through the TFT junction. We have found that the magnetic structure of the triplet pairing state significantly expands the phase space of the junction com-pared to a singlet system. In particular, the ori-entation of the magnetic moment strongly influ-ences the current, leading to a number of exotic Josephson effects.
We have also constructed a phase diagram for the junction, which provides a novel classifi-cation scheme for the different critical current states. Our study of the TFT junction is ongo-ing: In addition to the charge Josephson effect discussed here, it is likely that there will also be a spontaneousspincurrent through the TFT junction. The occurrence of such a highly un-conventional effect would expand the potential applications for this device.
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