π State in ³He Josephson Junctions

The Josephson effect in superfluid ³He was studied in the 1980's by Avenel and Varoquaux [1]. More recently, the flow of superfluid ³He-B through a 65x65 array of nanometer size apertures has been measured by Backhaus et al at Berkeley [2,3]. They find in the current-phase relation a new branch, so-called π state (pi state). The π state has subsequently been observed also in a single narrow slit [4].

In order to explain the π state, we have made calculations in two limiting cases. The first case is one large aperture, where the 18-component order parameter is solved numerically in and around the aperture using the Ginzburg-Landau theory of ³He. We find that the calculations have to be made in three dimensions because the π state has broken symmetry (m'm2') even in a circularly symmetric orifice. The current-phase relationship (figure) shows the existence of the π state for sufficiently large apertures. The physical interpretation is that the π state corresponds to a phase slip by a half-quantum vortex. The figure correnspons to the case with equal spin-orbit rotation matrices on the two sides of the weak link, where the current-phase relation is strongly hysteretic. The hysteresis is not usually present when more general boundary conditions are used, and the π states should therefore be experimentally observable.

As the second case we have made calculations for an aperture that is small compared to the superfluid coherence length. The presence of the π state in such a "pinhole" was first demonstrated by Yip using a simplified model [5]. We have done the pinhole calculation selfconsistently and find that the π state occurs in a single pinhole only at very low temperatures, approximately 0.2 T_c. The situation is different in an array of pinholes. There the π state can appear at higher temperatures because of anisotextural Josephson effect: the texture changes as a function of the phase difference. We find that this mechanism gives a good quantitative explanation for the π states and bistability observed in Berkeley [2,3]. For alternative theories trying to explain the π state see Refs. [5-8].

The next step is to understand the dynamics of ³He Josephson junctions.

1. O. Avenel and E. Varoquaux, Phys. Rev. Lett. 60, 416 (1988).
2. S. Backhaus, S. Pereverzev, R.W. Simmonds, A. Loshak, J.C. Davis, and R.E. Packard, Nature 392, 687 (1998).
3. A. Marchenkov, R.W. Simmonds, S. Backhaus, A. Loshak, J.C. Davis, and R.E. Packard, Phys. Rev. Lett. 83, 3860 (1999).
4. O. Avenel, Yu. Mukharsky, and E. Varoquaux, Physica 280, 130 (2000).
5. S.-K. Yip, Phys. Rev. Lett. 83, 3864 (1999), cond-mat 9907096.
6. N. Hatakenaka, Phys. Rev. Lett. 81, 3753 (1998); J. Phys. Soc. Japan 67, 3672 (1998).
7. O. Avenel, Y. Mukharsky, and E. Varoquaux, Nature 397, 484 (1999).
8. A. Smerzi, S. Raghavan, S. Fantoni, and S.R. Shenoy, cond-mat 0011298.

Publications

• J.K. Viljas and E.V. Thuneberg: Theory of the π state in ³He Josephson junctions, Phys. Rev. Lett. 83, 3868 (1999), cond-mat 9907181.
• J.K. Viljas and E.V. Thuneberg: Pinhole calculations of the Josephson effect in ³He-B, Phys. Rev. B 65, 64530 (2002), cond-mat 0107052.
• J.K. Viljas and E.V. Thuneberg: Equilibrium simulations of 2D weak links in p-wave superfluids, J. Low Temp. Phys. 129, 423 (2002), cond-mat/0210052.
• J. Viljas and E. Thuneberg, Stability of A-B phase boundary in a constriction, Physica B 329-333, 86-87 (2003), preprint.
• E. V. Thuneberg, Theory of Josephson Phenomena in Superfluid ³He, AIP Conference Proceedings 850, 103 (2006), cond-mat/0509504.
• Transparencies of a talk: π states and textural effects in superfluid ³He weak links (Sept. 2002, J. Viljas, pdf file, 684 kb).

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