Third Sound in Superfluid 3He
(materials prepared by graduate student Andrew M.R. Schechter in 1998)
Third Sound in Superfluid 3He
(materials prepared by
graduate student Andrew M.R. Schechter
in 1998)
Unconventional Superfluidity in 3He
Basic Facts about 3He
Basic Facts about
3HeLiquid at T=0
As you cool a material, like water, it goes through a series of abrupt phase transitions, first condensing from a gas to a liquid, and then changing from a liquid to a solid. Helium is one of the few exceptions to this rule. Both 4He gas and 3He gas condense into liquids (at 4.2 K and 3.2 K, respectively), but unless you apply a significant amount of pressure, both 4He and 3He remain liquid down to T=0.
The reason for this is two-fold. First, the forces between helium atoms, which would tend to bind them together, are extremely weak. Second, the mass of helium atoms is very small, so the quantum-mechanical uncertainty principle ordains a large "zero-point" motion, or low density. Quantum mechanical effects strongly influence the properties of both 4He and 3He; for this reason, they are referred to as quantum fluids.
Superfluid phase transition
Both of these quantum fluids have the amazing property that, as they cool down, they go through another phase transition to become a superfluid. This means that they are able to flow without viscosity, and without dissipating energy.
The mechanisms for this phase transition, and the details of the superfluidity in 4He and 3He, are very different. 4He has an even number of constituent particles (protons, electrons, and neutrons), which makes it a boson, meaning it is governed by Bose-Einstein statistics. At low temperatures, all the bosons in a sample will want to occupy the same quantum-mechanical ground state, forming a Bose-Einstein condensate. This condensate is responsible for the superfluid behavior of 4He below Tl = 2.17 K.
In contrast, 3He is a fermion, since it has one less neutron than 4He. Fermions obey the Pauli exclusion principle, which says that in a sample of many identical fermions, no two can occupy the same quantum-mechanical state. Bose-Einstein condensation is ruled out for 3He, so another mechanism is needed to explain its superfluid behavior. That mechanism is provided by the Bardeen-Cooper-Schrieffer (BCS) theory.1
BCS superconductivity
The BCS theory was first developed1 to account for the superconductivity of electrons in some metals at temperatures below 20 K. Electrons are fermions, so they are unable to Bose-condense as 4He does; yet, in a superconductor they are able to flow without electrical resistance or dissipation. The basic idea of BCS theory is that the electrons in a metal interact with the ions in the metal, in such a way that pairs of electrons will be attracted to each other. These Cooper pairs become the ("boson-like") constituent particles of the fluid. The Cooper pairs are able to occupy lower energy-levels than the electrons did, which the system likes, and crucially, the spectrum of energy levels contains a gap, D(T), between the highest occupied state and the next higher available state. The Cooper pairs and the energy gap are responsible for superconducting behavior in metals.
Conventional Cooper pairs: s-wave spin singlets
Cooper pairs in conventional superconductors are made out of 2 electrons. The combination has no orbital angular momentum (L=0), so it is a symmetric, "s-wave" object, and by the fundamental requirement that all combinations of fermions must be anti-symmetric, we see that the spin angular momentum must be anti-symmetric. (Anti-symmetric means that when you interchange any two fermions, the overall wave-function describing the combination is just multiplied by -1.) There is only one anti-symmetric way to combine the spins of two spin-1/2 fermions Y = |+-> - |-+>, which is called the spin singlet state. The wavefunction for this state can be described by a complex function Y = De if, which is called the order parameter, where D is the energy gap, which is zero above the transition temperature.
3He Cooper pairs: p-wave spin triplets
The BCS theory can be applied to 3He, as well as to electrons. However, the Cooper pairs for 3He are much more complex creatures than the those in a conventional superconductor. Due to the hard-core repulsion of the helium nuclei, the two 3He atoms in the Cooper pair feel a greater need to keep away from each other than the electrons do; so while the electrons bind together in a tight, symmetric, s-wave package, the 3He atoms bind together in a loose, anti-symmetric, p-wave package. "P-wave" just means that the pair has angular orbital momentum L=1. An L=1 system has three different quantum-mechanical states, denoted by ml = 1, 0, or -1.
Since all combinations of fermions must be anti-symmetric, the spin angular momentum in this case must be symmetric. There are three symmetric ways to combine the spins of two spin-1/2 fermions, Y = |++>, |-->, or |+-> + |-+>. (Read this as both spin up, both spin down, or the symmetric combination of the two.)
So a 3He Cooper pair has three different possible orbital states, and three different possible spin states. This gives a total of nine different combinations, each of which is weighted in the order parameter Y by a complex number, giving 18 degrees of freedom. This allows the 3He superfluid to behave in much more complex ways than the conventional superconductor, with its two degrees of freedom.
Other unconventional superconductors
A great part of the motivation for studying 3He comes from complexities which are allowed its Cooper pairs. Apart from the purely academic interest in the material for its own sake, it is now known that several of the "high-Tc superconductors," materials which become superconducting at 100 K rather than 4K, have complex Cooper pair symmetries, such as d-wave for YBCO. The prospect for practical applications arising from such materials encourages us to examine closely superfluids like 3He which have similarly unconventional Cooper pairs.
Bulk sample of 3He
Due to the many degrees of freedom in its order parameter, 3He has not just one, but several, stable superfluid phases. These phases have qualitatively different Cooper pairs, order parameters, and energy gaps, and the phase 3He prefers depends on the temperature T, the pressure P, and the applied magnetic field H.
Here we show a sketch of the phase diagram for 3He, found from experiment, as a function of temperature and pressure. There are 4 different regions of phase space:
Different physical states of bulk 3He:
- Solid 3He
- 3He-A superfluid
- 3He-B superfluid
- Normal (Fermi) liquid
Normal (Fermi) liquid: From 3.2 K down to about 2 mK, 3He is a normal liquid, which is well-described by the theory for Fermi liquids.
3He-B superfluid: This superfluid state has equal components of each of the three possible spin states, and each of the three L=1 spatial states. This results in an isotropic superfluid; for example, the energy gap D has the same magnitude for any excitation, independent of the direction in momentum-space (k-space) of that excitation.
3He-A superfluid: In contrast to 3He-B, this state is decidedly anisotropic. It's Cooper pairs have only two
of the spin components, and only two of the spatial components. This causes the energy gap
for excitations to depend strongly on the direction in momentum-space (k-space), as shown
below. In fact, the gap completely vanishes along one axis, which has important
consequences for the superfluid's properties affecting, for example, the heat capacity,
thermal conductivity, and superfluid density.
Films or slabs of 3He
The exact phase diagram of films, or slabs, of 3He is not experimentally known. However, there has been some theoretical and experimental investigations, and they have suggested the phase diagram drawn below. (Note: the details of the diagram are expected to vary somewhat depending on the exact interaction between the superfluid and the substrate, the boundary condition.)
Different states for
slabs of He-3:
Solid 3He
3He-A-like superfluid
3He-B-like superfluid
Normal (Fermi) liquid
As for the bulk liquid, we have a solid phase under pressure, and a normal liquid above 2 mK.However, below the superfluid transition temperature, we that the A- and B-phase regions have shifted. An A-like phase extends all the way down to zero pressure, only giving way to a B-like phase at lower temperatures. This is particularly important for our films, since the free surface means that we are always operating at zero pressure.
We can understand the shift in the phase diagram with the aid of the picture below.
When the slab or film is near the same size as a single Cooper pair, the L=1 component perpendicular to the substrate gets suppressed, creating an A-like phase. However, the size of a Cooper pair, x (also called the coherence length), is strongly dependent on the temperature. As the temperature falls, the Cooper pair gets smaller, so the confinement of the film is less important and a B-like phase becomes stable.
History of superfluid 3He films
Chronology
1972: Osheroff, Richardson, and Lee found the superfluid transition in bulk 3He (for which they won the 1996 Nobel Prize).2
1985: Sachrajda, Harris-Lowe, Harrison, Turkington, and Daunt showed that 3He films go superfluid, using beaker-emptying experiment. They watched the rate at which liquid flowed through the film from the beaker into the surrounding bath, as they cooled down the sample. 3
1988: Davis, Amar, Pekola, and Packard created stable films in a parallel plate geometry. A film forms on the surface of a disk partially submerged in a liquid helium bath, and the film's thickness can be controlled by raising or lowering the bath's level. Placing electrodes above the film allow it to be measured and manipulated. 4
1988: Freeman, Germain, Thuneberg, and Richardson made two important advances. First, they used NMR to identify the phase of thick slabs of 3He as A-like, filling in part of the phase diagram we saw above. Second, they showed that covering the substrate with an extremely thin blanket of 4He enhances the superfluidity of the 3He slabs greatly. 5
What's to come
The 3He film is a relatively unexplored system which has the potential for some very interesting physics, with relevance beyond its own sub-field. It is a 2-D superfluid with an unconventional, p-wave symmetry. The role that 2-D systems play in layered superconductors has just begun to be appreciated; for instance, the Kosterlitz-Thouless 2-D phase transition (which was first seen in 4He films) has just been observed in a d-wave superconductor.6 The 3He film is another system in which the same types of questions can be asked, and similar phenomena observed and understood.
D. R. Tilley and J. Tilley, Superfluidity and Superconductivity, Adam Hilger Ltd., Bristol, 1986.
F. Pobell, Matter and Methods at Low Temperatures, Springer-Verlag, Berlin, 1992.
Nobel Prize Addresses:
David M. Lee, "The extraordinary phases of liquid 3He." Rev. Mod. Phys. 69, p.645-665, 1997.
Douglas. D. Osheroff, "Superfluidity in 3He: discovery and understanding." Rev. Mod. Phys. 69, p.667-681, 1997.
Robert C. Richardson, "The Pomeranchuk effect." Rev. Mod. Phys. 69, p.683-690, 1997.
1. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of Superconductivity." Phys. Rev. 108, p.1175-1204, 1957.
2. D. D. Osheroff, R. C. Richardson, and D .M. Lee, "Evidence for a new phase of solid 3He." Phys. Rev. Lett. 28, p.885-888, 1972. (The authors initially thought their observations were due to the solid 3He in their cell; their later experiments demonstrated that the effect was in fact a new (superfluid) phase of liquid 3He.)
3. A. Sachrajda, R. F. Harris-Lowe, J. P. Harrison, R. R. Turkington, and J. G. Daunt, "3He film flow: two-dimensional superfluidity." Phys. Rev. Lett. 55, p.1602-1605, 1985.
4. J. C. Davis, A. Amar, J. P. Pekola, and R. E. Packard, "Superfluidity of 3He films." Phys. Rev. Lett. 60, p.302-304, 1988.
5. M. R. Freeman, R. S. Germain, E. V. Thuneberg, and R. C. Richardson, "Size effects in thin films of superfluid 3He." Phys. Rev. Lett. 60, p.596-599, 1988.
6. J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, "Vanishing of phase coherence in underdoped Bi2Sr2CaCu208+d"to be published.
Comments: packard@socrates.berkeley.edu
