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Difficulties of supercurrents in narrow pores of 3He-A
Volume 86A, number 1
PHYSICS LETTERS
26 October 1981
DIFFICULTIES OF SUPERCURRENTS IN NARROW PORES OF 3He-A E.V. THUNEBERG a,i and J. KURKIJARVI b a Research Institute for Theoretical Physics, SF-001 70 Helsinki 17, Finland b Department of TechnicalPhysics, SF.02150 Espoo 15, Finland Received 29 July 1980
We consider resistanceless suf,ercurrents through narrow pores and find such currents to vanish in most cases because of end effects at the entries and exits of the pores. Under pressure dc supercurrents are found to arise.
There has been a great deal of discussion in the literature on the effects of supercurrents on 3He-A textures in narrow pores [1—3].Currents in pores as a result of phase gradients are also interesting in view of any eventual Josephson experiments through “weak links” as a truly short orifice may seem hard to make with a small enough diameter to compete with the short coherence length ~(T) ~ 1000 A. In this letter we wish to emphasize the point that supercurrents through long pores are probably hard to produce and will in most cases be controlled by effects at the ends of the pores rather than by the largest possible super. fluid velocity within the tubes. The end effects belong to the well-known family of textural rearrangements such as to eliminate phase gradients [4]. The prototype of such an effect is the early observation of Ambegaokar et al. [5] that supercurrents would vanish for tunneling junctions with 1 vectors opposite on the two sides of the junction. One can establish their result with a rather general geometrical argument, an argument which works for pores as well: If one rotates the order parameter around the axis of the junction, which costs no energy at all, one winds away a phase difference across the junction as the left and right hand sides of the junction have their phases turned in opposite directions on account of the opposite directions of 1. A similar argument can be constructed for parallel I’s at the ends of a pore as well if the order parameter has rotaPresent address: Nordita, 2100 Copenhagen
0, Denmark.
0031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
tion symmetry around the axis of the pore. One now
turns the order parameter on just one side of the pore, say left of the half way mark through the pore, and thereby winds away a phase difference at no cost of energy because the rotationally symmetrical order parameter in the pore notices nothing. The latter case applies, in particular, to the disgyration texture [6] (see fig. 1) which therefore accepts no resistanceless current at all in a long circular pore. The pore has to be long on account of a small imperfec. tion in the axial symmetry of the disgyration texture order parameter, the singular lines that have to emerge from its ends [7] into the bulk fluid and go somewhere such as a wall. The interaction between these lines will lead to a tiny “Josephson energy” E(i.~.p)as a function of the phase difference t~p.E(~p)will, however, be an exponentially decreasing function of the length of the pore and vanishingly small in practice. It is clearest at first confine oneself, in addition
to
______
______
Fig. 1. Sketch of the disgyration texture in a pore with paral-
lel i’s. The disgyration is assumed to go over to a vortex at the ends. The phase difference acrossthe pore is unwound by rotating the vortices around the pore axis.
35
Volume 86A, number 1
PHYSICS LETTERS
26 October 1981
to the disgyration texture, to just one case where the order parameter has no axial symmetry, the Pan-Am texture in a pore [6]. This means limiting the discus. sion to pore radii R in the range dipolar length ~6 pm ~‘ R ~ temperature dependent coherence length. With the 1 vector parallel at its ends, the Pan-Am texture behaves according to one s expectations and accepts a phase gradient and a maximum resistanceless super-
of the Cross—Anderson [101 orbital viscosity leading to a dc supercurrent driven by the chemical potential difference. This current again depends on the texture. For the Pan-Am texture it is on the order of
current limited by some Landau critical velocity on the order of up to say several cm/s (if one takes the critical velocity to be larger the smaller the pore [8]). All other combinations, Pan-Am texture with the I vectors opposite to each other at the ends of the pore and disgyration texture no matter how the 1 vectors, will lead to maximum resistanceless supercurrents deter.
with the length of the pore 1 = 1 pm, p denoting the mass density of the fluid, ~ superfluid density and L~pthe pressure head in Pa. This is large enough to dominate over the small pinning effect of the Pan.Am texture at pressures of only a few Pa but is limited to the maximum resistanceless supercurrent. In the case of the disgyration texture, the Cross—Anderson current is probably small because it vanishes for axially symmetric 1 vectors and all contribution to it would come from just the ends of the pore. One should note that
mined by much lower energies than that required to
distort the texture. Such energies arise from pinning to irregularities of the pores and the type of effect described above between the end singularities of the disgyration texture. One expects only small pinning gularities run along the walls of the pore. Assuming
that the disgyration singularity turns into vortex lines at the ends of the pore, the pinning of these can lead to currents on the order of v (fl/m)(4R)~ lnFd/~(T)] I
‘~
‘
‘
/
where m is the bare 3He mass, d the size of the irregularity, d ~ ~(T) and R the radius of the pore. Taking R = 1 pm, d = 100 nm, ~(T) = 30 nm we get ~ 6.3
mm/s. The interaction of the end singularities with each other, as mentioned above, leads to negligible currents in any pore long enough to house the disgyration texture in the first place. If a chemical potential difference, a pressure head for instance, is applied at the ends of the pore, things get more complicated. In principle the “Josephson energies” E(i~p)from both sources, phase gradient or pinning, are periodic in ~p and could therefore lead to ac Josephson effects. The Pan-Am texture with a phase gradient in a long pore, however, suffers from the complication, well known from superconductivity [9], of phase winding up in the pore and slipping in an irregular fashion. If the maximum Josephson current again depends on one of the weaker mechanisms, a relatively large “excess” supercurrent will appear with finite chemical potential difference. The unwinding of the phase will lag behind the drive as a result
36
/T (L~p/Pa)i \
T~\1/2 j 140 mm/s, T~ I
—
(2)
the normal current velocity would be completely negligible in all these cases as a result of the huge viscosity of 3He at superfluid temperatures, u~ 10 pm/s per Pa for the pore R = 1 pm, 1 = 1 pm.
effects in the case of the Pan-Am texture whose sin-
C
2pl
m
As a complementary tool to NMR the above information can be used to distinguish between the Pan-Am and the disgyration textures in pores. A pore supporting a resistanceless supercurrent in the proper range of diameters certainly houses a Pan-Am texture. Even if the I vectors should align unfavorably the Pan-Am texture will admit a large constant supercurrent under a
pressure head. The disgyration texture admits no resistanceless supercurrent and next to no current even under pressure with a tendency to small amplitude ac Josephson effect from the pinning. Our considerations may have some relevance also in the discussion relating to two NMR experiments in narrow pores reported in the literature [5,11]. The possibility [1] that a transition from a Pan-Am texture to a disgyration texture at constant supercurrent mass flow rate would explain differences between the two NIMR experiments seems out of the question to us. In pores of the size of the coherence length, the equilibrium phase is other than the bulk A-phase. This phase is called the polar phase and its order parameter has the axial symmetry, same as the disgyration texture. The kind of analysis applied to the disgyration texture above suggests that the maximum supercurrent through such a pore would be extremely small although its axial symmetry has to be broken at the transition
Volume 86A, number 1
PHYSICS LETTERS
regions between the A-phase and the polar phase. It is therefore important in experiments looking for weak link effects in the A-phase to make the orifices shorter
26 October 1981
two sides independent of the size of the orifice. It should be noted that although such a texture has cy-
[4] N.D. Mermin, in: Quantum fluids and solids, eds. S.B. Trickey, E.D. Adams and J.W. Dufty (Plenum, New York, 1977) p. 3. [5] V. Ambegaokar, PG. deGennes and D. Rainer, Phys. Rev. A9 (1974) 2676. [61 CM. Gould and D.M. Lee, Phys. Rev. Lett. 41(1978) 967. [7] G.E. Volovik and V.P. Mineev, Soy. Phys. JETP 45 1186. [8] (1977) J.M. Parpia and J.D. Reppy, Phys. Rev. Lett. 43 (1979)
lindrical symmetry the associated order parameter has not, and a Josephson current can flow.
1332. [9] J.S. Langer and V. Ambegaokar, Phys. Rev. 164 (1967)
than or of the order of their radii. One then expects a Mermin—Ho texture resembling arrangement going through the hole if the 1 vectors are parallel on the
References [1] W.S. Truscott, Phys. Lett. 74A (1979) 80. [2] R. Bruinsma and K. Maki, J. Low Temp. Phys. 37 (1979)
607. [3] L.J. Buchholtz and A.L. Fetter, Phys. Rev. B15 (1977) 5225.
498. [10] M.C. Cross and P.W. Anderson, in: Proc. 14th Intern. Conf. on Low Temp. Physics (Otaniemi, Finland, 1975), eds. M. Krusius and M. Vuorio (North-Holland, Amsterdam, 1975) p. 29. [11] J. Saunders, D.S. Betts, D.F. Brewer, SJ. Swithenby and W.S. Truscott, Phys. Rev. Lett. 40 (1978) 1278.
PHYSICS LETTERS
26 October 1981
DIFFICULTIES OF SUPERCURRENTS IN NARROW PORES OF 3He-A E.V. THUNEBERG a,i and J. KURKIJARVI b a Research Institute for Theoretical Physics, SF-001 70 Helsinki 17, Finland b Department of TechnicalPhysics, SF.02150 Espoo 15, Finland Received 29 July 1980
We consider resistanceless suf,ercurrents through narrow pores and find such currents to vanish in most cases because of end effects at the entries and exits of the pores. Under pressure dc supercurrents are found to arise.
There has been a great deal of discussion in the literature on the effects of supercurrents on 3He-A textures in narrow pores [1—3].Currents in pores as a result of phase gradients are also interesting in view of any eventual Josephson experiments through “weak links” as a truly short orifice may seem hard to make with a small enough diameter to compete with the short coherence length ~(T) ~ 1000 A. In this letter we wish to emphasize the point that supercurrents through long pores are probably hard to produce and will in most cases be controlled by effects at the ends of the pores rather than by the largest possible super. fluid velocity within the tubes. The end effects belong to the well-known family of textural rearrangements such as to eliminate phase gradients [4]. The prototype of such an effect is the early observation of Ambegaokar et al. [5] that supercurrents would vanish for tunneling junctions with 1 vectors opposite on the two sides of the junction. One can establish their result with a rather general geometrical argument, an argument which works for pores as well: If one rotates the order parameter around the axis of the junction, which costs no energy at all, one winds away a phase difference across the junction as the left and right hand sides of the junction have their phases turned in opposite directions on account of the opposite directions of 1. A similar argument can be constructed for parallel I’s at the ends of a pore as well if the order parameter has rotaPresent address: Nordita, 2100 Copenhagen
0, Denmark.
0031-9163/81/0000—0000/s 02.75 © 1981 North-Holland
tion symmetry around the axis of the pore. One now
turns the order parameter on just one side of the pore, say left of the half way mark through the pore, and thereby winds away a phase difference at no cost of energy because the rotationally symmetrical order parameter in the pore notices nothing. The latter case applies, in particular, to the disgyration texture [6] (see fig. 1) which therefore accepts no resistanceless current at all in a long circular pore. The pore has to be long on account of a small imperfec. tion in the axial symmetry of the disgyration texture order parameter, the singular lines that have to emerge from its ends [7] into the bulk fluid and go somewhere such as a wall. The interaction between these lines will lead to a tiny “Josephson energy” E(i.~.p)as a function of the phase difference t~p.E(~p)will, however, be an exponentially decreasing function of the length of the pore and vanishingly small in practice. It is clearest at first confine oneself, in addition
to
______
______
Fig. 1. Sketch of the disgyration texture in a pore with paral-
lel i’s. The disgyration is assumed to go over to a vortex at the ends. The phase difference acrossthe pore is unwound by rotating the vortices around the pore axis.
35
Volume 86A, number 1
PHYSICS LETTERS
26 October 1981
to the disgyration texture, to just one case where the order parameter has no axial symmetry, the Pan-Am texture in a pore [6]. This means limiting the discus. sion to pore radii R in the range dipolar length ~6 pm ~‘ R ~ temperature dependent coherence length. With the 1 vector parallel at its ends, the Pan-Am texture behaves according to one s expectations and accepts a phase gradient and a maximum resistanceless super-
of the Cross—Anderson [101 orbital viscosity leading to a dc supercurrent driven by the chemical potential difference. This current again depends on the texture. For the Pan-Am texture it is on the order of
current limited by some Landau critical velocity on the order of up to say several cm/s (if one takes the critical velocity to be larger the smaller the pore [8]). All other combinations, Pan-Am texture with the I vectors opposite to each other at the ends of the pore and disgyration texture no matter how the 1 vectors, will lead to maximum resistanceless supercurrents deter.
with the length of the pore 1 = 1 pm, p denoting the mass density of the fluid, ~ superfluid density and L~pthe pressure head in Pa. This is large enough to dominate over the small pinning effect of the Pan.Am texture at pressures of only a few Pa but is limited to the maximum resistanceless supercurrent. In the case of the disgyration texture, the Cross—Anderson current is probably small because it vanishes for axially symmetric 1 vectors and all contribution to it would come from just the ends of the pore. One should note that
mined by much lower energies than that required to
distort the texture. Such energies arise from pinning to irregularities of the pores and the type of effect described above between the end singularities of the disgyration texture. One expects only small pinning gularities run along the walls of the pore. Assuming
that the disgyration singularity turns into vortex lines at the ends of the pore, the pinning of these can lead to currents on the order of v (fl/m)(4R)~ lnFd/~(T)] I
‘~
‘
‘
/
where m is the bare 3He mass, d the size of the irregularity, d ~ ~(T) and R the radius of the pore. Taking R = 1 pm, d = 100 nm, ~(T) = 30 nm we get ~ 6.3
mm/s. The interaction of the end singularities with each other, as mentioned above, leads to negligible currents in any pore long enough to house the disgyration texture in the first place. If a chemical potential difference, a pressure head for instance, is applied at the ends of the pore, things get more complicated. In principle the “Josephson energies” E(i~p)from both sources, phase gradient or pinning, are periodic in ~p and could therefore lead to ac Josephson effects. The Pan-Am texture with a phase gradient in a long pore, however, suffers from the complication, well known from superconductivity [9], of phase winding up in the pore and slipping in an irregular fashion. If the maximum Josephson current again depends on one of the weaker mechanisms, a relatively large “excess” supercurrent will appear with finite chemical potential difference. The unwinding of the phase will lag behind the drive as a result
36
/T (L~p/Pa)i \
T~\1/2 j 140 mm/s, T~ I
—
(2)
the normal current velocity would be completely negligible in all these cases as a result of the huge viscosity of 3He at superfluid temperatures, u~ 10 pm/s per Pa for the pore R = 1 pm, 1 = 1 pm.
effects in the case of the Pan-Am texture whose sin-
C
2pl
m
As a complementary tool to NMR the above information can be used to distinguish between the Pan-Am and the disgyration textures in pores. A pore supporting a resistanceless supercurrent in the proper range of diameters certainly houses a Pan-Am texture. Even if the I vectors should align unfavorably the Pan-Am texture will admit a large constant supercurrent under a
pressure head. The disgyration texture admits no resistanceless supercurrent and next to no current even under pressure with a tendency to small amplitude ac Josephson effect from the pinning. Our considerations may have some relevance also in the discussion relating to two NMR experiments in narrow pores reported in the literature [5,11]. The possibility [1] that a transition from a Pan-Am texture to a disgyration texture at constant supercurrent mass flow rate would explain differences between the two NIMR experiments seems out of the question to us. In pores of the size of the coherence length, the equilibrium phase is other than the bulk A-phase. This phase is called the polar phase and its order parameter has the axial symmetry, same as the disgyration texture. The kind of analysis applied to the disgyration texture above suggests that the maximum supercurrent through such a pore would be extremely small although its axial symmetry has to be broken at the transition
Volume 86A, number 1
PHYSICS LETTERS
regions between the A-phase and the polar phase. It is therefore important in experiments looking for weak link effects in the A-phase to make the orifices shorter
26 October 1981
two sides independent of the size of the orifice. It should be noted that although such a texture has cy-
[4] N.D. Mermin, in: Quantum fluids and solids, eds. S.B. Trickey, E.D. Adams and J.W. Dufty (Plenum, New York, 1977) p. 3. [5] V. Ambegaokar, PG. deGennes and D. Rainer, Phys. Rev. A9 (1974) 2676. [61 CM. Gould and D.M. Lee, Phys. Rev. Lett. 41(1978) 967. [7] G.E. Volovik and V.P. Mineev, Soy. Phys. JETP 45 1186. [8] (1977) J.M. Parpia and J.D. Reppy, Phys. Rev. Lett. 43 (1979)
lindrical symmetry the associated order parameter has not, and a Josephson current can flow.
1332. [9] J.S. Langer and V. Ambegaokar, Phys. Rev. 164 (1967)
than or of the order of their radii. One then expects a Mermin—Ho texture resembling arrangement going through the hole if the 1 vectors are parallel on the
References [1] W.S. Truscott, Phys. Lett. 74A (1979) 80. [2] R. Bruinsma and K. Maki, J. Low Temp. Phys. 37 (1979)
607. [3] L.J. Buchholtz and A.L. Fetter, Phys. Rev. B15 (1977) 5225.
498. [10] M.C. Cross and P.W. Anderson, in: Proc. 14th Intern. Conf. on Low Temp. Physics (Otaniemi, Finland, 1975), eds. M. Krusius and M. Vuorio (North-Holland, Amsterdam, 1975) p. 29. [11] J. Saunders, D.S. Betts, D.F. Brewer, SJ. Swithenby and W.S. Truscott, Phys. Rev. Lett. 40 (1978) 1278.