SUPERFLUID 3He, A TWO-FLUID SYSTEM, WITH THE NORMAL-FLUID DYNAMICS DOMINATED BY ANDREEV REFLECTION
G. R. Pickett*
Department of Physics, Lancaster University, Lancaster, LAI 4YB, UK Received May 26, 2014
As a specific offering towards his festschrift, we present a review the various properties of the excitation gas in superfluid JHo, which depend on Andreev reflection. This phenomenon dominates many of the properties of the normal fluid, especially at the lowest temperatures. We outline the ideas behind this dominance and describe a sample of the many experiments in this system which the operation of Andreev reflection has made possible, from temperature measurement, particle detection, vortex imaging to cosmological analogues.
Contribution for the JETP special issue in honor of A. F. Andreev's 75th birthday
DOI: 10.7868/S0044451014120062
1. INTRODUCTION
This article provides a brief review of the influence of Andreev reflection on the behavior of super-fluid 3He. Since, as we see below, the dispersion curve of the quasiparticlc/quasihole excitations depends on the relative motions of the fluid, any mechanical disturbance of the liquid gives rise to a disordered effective gap for the excitations which are no longer free to move through the liquid unconstrained. In this landscape of varying gaps, excitations are constantly being subjected to Andreev processes to the extent that the whole mechanical behavior of the normal fluid of excitations is completely dominated by such processes. This is the system where the Andreev process indeed conies into its own. However, this behavior is not a drawback. On the contrary, we are able to exploit these processes which allow us to undertake a range of experimental investigations which are otherwise experimentally inaccessible.
2. THE ANALOGY BETWEEN ANDREEV REFLECTION AT A BOUNDARY SEPARATING DIFFERENT
SUPERCONDUCTORS, AND IN A REGION
SEPARATING STATIONARY AND MOVING SUPERFLUID 3He
The original Andreev reflection process was proposed by Aleksandr Fedorovich for describing the be-
* E-mail: g.pickett'ffllancaster.ac.uk
havior of quasiparticles in superfluids in regions where the energy gap is changing spatially, as for example at a superconducting normal interface or at an interface between two dissimilar superconductors with different energy gaps [1]. Of course, the unique aspect of this process is the "flavor" change made by excitations as they approach a region where the gap rises above the total energy of the excitation. In this situation an approaching excitation finds itself arriving at a minimum in the dispersion curve at which its group velocity falls to zero and then it retraces its trajectory with almost no change in its momentum but with the opposite group velocity. Thus an incoming quasipar-ticle is retro-reflected as a quasihole and an incoming quasihole will be reflected as a quasiparticle.
From this new understanding of the process, a whole spectrum of unique phenomena can be recognized, quasiparticle quasihole bound states and many more. However, that is the picture we recognize from static systems to which the concept was originally applied. (The rigid metallic lattices of superconducting/ normal interfaces are clearly static.)
It is the purpose of this paper to recapitulate those aspects of this behavior which we have studied in non-static contexts, i.e., in superfluids. Here the phenomenon, while operating in a similar way, has a much richer and more complex behavior. In a moving BCS system the effective energy gaps are not static but governed by the flow of the fluid adding a whole new spectrum of properties with often very counter-intuitive effects.
Suporfluid 3He has a complex and anisotropic order parameter, but for the purposes of the present treatment we will consider only the B phase which has an energy gap in low magnetic fields with the same magnitude around the Fermi surface. Therefore, the straightforward ideas of Andreev reflection developed for simple superconductors can be transferred to superfluid 3He-U directly. It is true that, while the magnitude of the U-phase gap is isotropic, the pairing is in fact anisotropic since the directional properties of the Cooper pairs vary around the Fermi sphere. However, for the arguments used here this does not really come into play until we start applying magnetic fields.
To begin, let us look at the dispersion curve of the excitations in superfluid 3He (with energy gap A) as shown in Fig. 1A. For simplicity the curve only covers the single x-dimension and is drawn for the rest frame of the liquid. We note that there are four classes of excitations. On the right-hand side of the figure quasi-particles move in the positive x direction and quasi-holes move in the — x direction, both with momenta very close to the Fermi momentum value, pp. On the left-hand side of the curve there are quasiholes moving in the x direction and quasiparticles moving in the — x direction, both with momenta close to —pp-
Let us assume a neighboring region of superfluid with a larger (static) gap, A', with the dispersion curve as shown in Fig. IB. If we then imagine a low-energy quasiparticle from region A traveling in the x direction towards region B, then as the excitation moves into the region of increasing energy gap, it decelerates as its group velocity decreases (with the decreasing slope of the dispersion curve) and finally reaches the curve minimum, having slowed to zero velocity. It can then penetrate no further into the region of increasing gap but will retrace its trajectory with increasing velocity in the —x direction, but with more or less the same total momentum as it had originally. Hence, it now has its group velocity and momentum oppositely directed and it has become a quasihole. We can use similar arguments for an incoming quasihole. This is the classical Andreev scenario.
The new aspect, which we have to take into account in the superfluid, is the fact that we can set various parts of the liquid into relative motion. Since the dispersion curve is tied to the rest frame of the liquid, then for comparing excitation energies we have to apply the correct Galilean transformation to make the various curves consistent. This reduces to the classical argument that if we observe liquid approaching us with a (small) velocity v then a fermion in that liquid approaching us with velocity w in that rest frame
will appear to us to have a velocity of v + w and thus the corresponding kinetic energy will be m(v + w)2 /2 rather than the mw2 /2 in its own liquid rest frame. Near the Fermi energy, the fermions have a velocity ±vp. Thus in our rest frame, we will see the energy of the dispersion curve skewed by the transformation E' —¥ E+vpp. The curve for superfluid at rest is shown in Fig. 2A. The equivalent curve for liquid moving to the right with velocity v is shown in Fig. 2B. A quasiparticle on the right-hand side of Fig. 2A moving in the x direction could not penetrate into the moving liquid of Fig. 2B but would be Andreev-reflected and emerge traveling in the — x direction but as a quasihole. Unlike in Fig. 1 the behavior of quasiparticles and quasiholes is not symmetrical. A quasihole on the left-hand side of Fig. 2A traveling in the x direction would be able to penetrate into the moving region. Similar arguments can be used for excitations traveling from the moving to the stationary regions.
3. THE EFFECT OF THE ANDREEV
REFLECTION OF QUASIPARTICLES AND QUASIHOLES ON THE DAMPING OF A
MOVING OBJECT IN SUPERFLUID 3He-B
The above section concludes all the introduction to Andreev reflection in the superfluid in relative motion that we need initially to understand the basic ideas of the dynamics of the excitation gas in the superfluid. Throughout this work we assume that we are in the low-temperature limit T -C Tc. Here the mean free path of the excitations is invariably much greater than any reasonable experimental dimension and we can consider the excitation motion to be entirely ballistic.
The first topic we discuss is the mechanical damping of an object moving in the superfluid, since this serves as a good introduction to the dynamics and also provides us with some important experimental tools for probing the superfluid.
Let us take again a one-dimensional toy model of a moving object where we consider the bulk liquid to be stationary but the liquid near the moving object to be at rest with respect to it. The scenario is shown in Fig. 3. The important aspect of this process is the fact that the flow field associated with the motion of the wire provides a sort of Maxwell demon which leads to different responses to the moving object from quasiparticles and quasiholes. From the figure it is apparent that quasiparticles approaching the leading side of the object will be able reach the surface but quasiholes approaching from the front cannot reach the region of liquid moving with the wire and will be Andreev-reflected,
Fig. 1. The dispersion curves for two superconductors (or superfluids) with different energy gaps. A quasiparticle or quasihole in region A moving right would not be able to penetrate into the superconductor (superfluid) of region B but would undergo
Andreev reflection and retrace its incoming trajectory
Liquid stationary Liquid in motion
Fig. 2. A) The dispersion curve for excitations in superfluid JHoi? at rest. B) The dispersion curve for the superfluid when moving with velocity v in the x direction skewed by the Galilean transformation term E' E + vpF- A quasiparticle on the right-hand side of the curve in region A moving to the right would not be able to penetrate into the moving superfluid of region B, but would undergo Andreev reflection and retrace its incoming trajectory. Conversely, a quasihole on the left-hand
side of region A moving to the right is free to enter region B
and thus cannot exchange momentum with the wire. That means that particle hole symmetry is broken an
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