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Last lecture:Last lecture:

We discussed the peculiar properties of superconducting weak links. The Josephson phase relation Is = IJ sinφ led to remarkable non-linear I-V phenomena with important applications.

We have now covered some of the major consequences of the Ginzburg-Landau theory of the superconducting state. Recall that the starting point was quite general, and did not involve microscopic details.

It may come as no surprise therefore that we can apply much of what we have learned to the related phenomena of superfluidity.

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Lecture 6:Lecture 6:

• Phenomenology of superfluidity• The superfluid wavefunction• Superflow in response to pressure and temperature gradients –

the Josephson-Anderson phase relation and the Euler equation• Two fluid model and the fountain effect• Flow quantisation and vortices• Excitations: phonons, rotons and second sound• Landau’s critical velocity

• Literature: Annett ch. 2, Pethick & Smith chs. 9 & 10

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SuperfluiditySuperfluidity

Examples of superfluidity:• 4He below 2.17 K• 3He below 2 mK• Cold atoms in the nK range• neutron stars (probably)• hydrogen under extreme pressures (possibly)

In this lecture we will concentrate on 4He – we’ll discuss the much more intricate case of 3He later.

What’s so special about 4He? Zero point energy considerations tell us 4He can’t crystallize:

d

2

22

23

dmEZP

πh×=

44

For liquid 4He, d = 3.8 Å →→→→ EZP = 1.2 meV in this approximation, close to the depth E0 of the attractive potential well between two 4He atoms. [It is possible to quantify this further using a Lennard-Jones potential which gives an even higher estimate for EZP.] This means 4He is a quantum fluid which is prevented from solidifying by zero point motion.

4He liquefies at 4.2 K and first becomes a normal quantum fluid known as He-I. The first signature of superfluidity is the so-called λtransition visible in the specific heat at 2.17 K:

Below this transition, the liquid can flow through capillaries without an applied pressure difference: the viscosity is zero, η = 0. This new superfluid state of matter is known as He-II.

Tλ = 2.17 K

Picture credits: after Buckingham

& Fairban

k

p1 p2 = p1

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The Superfluid WavefunctionThe Superfluid Wavefunction

The specific heat anomaly indicates that He-II is an ordered state as is the superconducting state and the vanishing viscosity in He-II is similar to the vanishing resistivity in a superconductor. In analogy with superconductivity, we therefore postulate a superfluid macroscopic wavefunction ψ(r, t).

It is tempting to write down the usual the Ginzburg-Landau free energy. However, its quantitative applicability is limited since:(i) the effective coherence length ξ reaches atomic length scales

at low T and(ii) the region near Tλ is strongly affected by fluctuations.

Some features do survive such as the current-phase relation:

θρψψψψψ is e

imJ *)*(

2 s=∇−∇=h

θρ ∇⋅=m

sh

vs

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For superconductivity, we took the curl of this to arrive at thescreening and supercurrent acceleration (1st London) equations, respectively. Let’s try the same here: taking the curl we get

i.e., the superflow is irrotational. This is a rather bizarre property – imagine a liquid you cannot stir.

As regards the time evolution of the phase, we have to replace the electric potential energy qV with the chemical potential µ:

) (where He4mm

mvs =∇= θ

h

0=×∇ sv

)2( :ctorSupercondu eqqVt

−=−=∂

∂θh

µθ

−=∂

∂

th :Superfluid

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This is the Anderson-Josephson phase evolution relation, which yields the time evolution of vs as

The gradient of µ can be obtained from the Gibbs-Duhem relation for an equilibrium system

together with the Bernoulli effect for a system in dynamic flow

dTN

Sdp

N

Vd −=µ

2

21

svpp ρ−→

µθ

θ −∇=∂

∂∇=∇

∂

∂= )()(

tmtmvm s h

h&

N is the number of particles, V is the volume, S is the entropy, p is the pressure and T is the temperature.

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where σ is the entropy per unit mass. This is the superfluid equivalent of the 1st London equation. It is known as Euler’s equation and will be recognised as essentially the Navier-Stokes equation without the viscosity term.

Note that the superfluid has no entropy (the superfluid is described by a macroscopic wavefunction and so there is no disorder) so σ must arise from the residual normal fluid.

Tpv

t

v ss ∇+∇

−∇

−=∂

∂σ

ρ2)( 2

This leads to the relation:

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Two Fluid ModelTwo Fluid Model

A key experiment shows that a torsionaloscillator in He-II has a T-dependent frequency, and hence a T-dependent effective rotating mass. This is because it drags a boundary layer of normal fluid with it, showing that it is necessary to describe He-II in a two-fluid model:

Picture credits: Andronikashvili/S. Yuan

ViscosityEntropy*VelocityDensity

ηnσρvnρnNormal fluid

00vsρsSuperfluid

*per unit volume

This model has some peculiar thermo-mechanical effects. We consider one example below.

1010

Both the superfluid and normal fluid components carry particle currents, but only the normal component carries heat current.

What happens if we connect two reservoirs as shown below?

Assume steady state:

So that

T1 T2

r

)(or

0

1212 TTppTp

Tp

t

vs

−=−∇=∇

∇+∇

−==∂

∂

ρσρσ

σρ

p1 p2

1111

Fountain EffectFountain Effect

Two extremes: 1. A single large vessel of He-II: r → ∞→ ∞→ ∞→ ∞. Bulk He-II is a very good thermal conductor and effectively acts as a thermal short so that ∇∇∇∇pnever has a chance to build up.

2. A nearly blocked capillary: r →→→→ 0. Only the superfluid can flow while the heat flow is blocked. This is enough to lift the superfluid against gravity and force it out at the top of the vessel, as seen in the spectacular fountain effect.

Picture credits: J. Allen / Virtual Helium W

eb Site

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Flow QuantisationFlow Quantisation

Recall the problem of flux quantization in a superconducting ring. The corresponding problem in a superfluidleads to quantization of circulation κ. Since the superfluid is neutral, the vector potential does not enter and using an analysis similar to that in lecture 2, we find:

Integrationpath

vs

∫ ∫ =⋅∇=⋅ θ∆θm

dsm

dsvshh

circulation κ 2πn

mn

h=

quantum of circulation κ0

s/m 100.1 2700

−×===m

nh

κκκ

n is an integer

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Superfluid VorticesSuperfluid Vortices

Next, what about vortices? Let’s consider superflow around a vortex core with circulation equal to one quantum κ0. The superfluid velocity vs is then given by

There is no screening (effectively λ → ∞) and the superfluid vortex core is of the order of an Å and thus very small comparedwith that of conventional superconductors.

vs

rrvdsv ss πκ 2)( 0 ∫ =⋅=

rrvs π

κ

2)( 0=

1414

We can induce vortices by rotating the cryostat. But how many vortices do we get in equilibrium? Rotating a cylinder of radius rat ω induces a circulation

round the outside which must correspond to the number N of vortices in the fluid: N = 2πr2ω/κ0, each with a single quantum of circulation. (Of course, N must be an integer so there will in general be a slight mismatch between the cylinder wall motion and the superflow.) The superposition of vortices “simulates” bulk rotation, and we get a parabolic meniscus just like in a rotating bucket of water.

Picture credits: E. Thuneb

erg

In turbulent superflow it is also possible to generate vortex rings and tangles. The study of their dynamics is an active field of research.

∫ =⋅=⋅= 222 rrrdsvs πωωπκ

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Elementary Excitations of Liquid Elementary Excitations of Liquid 44HeHe

The energy spectrum of elementary excitations observed by neutron scattering is shown on the right:

rotonsphononsc

cmin

There are three main features which we’ll discuss in turn: the linear phonon regime whereω ���� ck, the peculiar rotonregion, and the Landau critical velocity cmin.

Phonons are density fluctuations of the fluid as a whole, with sound velocity c = 238 m/s (in the low-T limit). Phonons are the only low energy excitations – the single particle ω = �k2/2mparabola is absent (the reason for this will be explained in thenext lecture).

Picture credits: H. W. Jackson

ωk

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Second Sound and RotonsSecond Sound and RotonsBesides conventional phonons we also have in He-II:• Second sound: the motion of the normal fluid and superfluidyields a system with two coupled degrees of freedom. In ordinary sound ρn and ρs oscillate in phase. In second sound ρnand ρs oscillate in antiphase, and with a velocity different from that or ordinary sound.

• Rotons: the spectrum ω vs. k bends over and goes through a pronounced minimum at k ���� 2π/a (recall that in the extended zone scheme of a periodic solid ω →→→→ 0 at k= 2π/a). Detailed calculations show that the atomic motion has some parallels to that of a smoke ring (figure on right).

ρ

superfluid: ρs

normal fluid: ρn

First sound: phonons

ρ

superfluid: ρs

normal fluid: ρn

Second sound: thermal waves

excess entropy transport

Picture credits: hunter@

Monke

yView

r

1717

Now that we know the spectrum of the low lying excitations, let’s see whether we can excite them in the simplest possible way: by dragging an object through the superfluid at velocity v. This induces a disturbance potential U(r, t) ≡≡≡≡ U(r–vt). The Fourier transform in 1D gives:

Picture credits: H. W. Jackson

cmin

This means that the dragged object can only create excitations withω = vk, i.e., on the lower dashed line in the figure. If this line does not intersect the dispersion curve, in particular for v < cmin, no excitations can be created and the flow is dissipationless.

v

∫∫−−= ikxti

k eevtxUdxdtU ωω )( ,

dxds

vtxs

=

−=

)(

)(

vk

eeesUdsdt ikvtiksti

−∝

= ∫∫−

ωδ

ω

ωk

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Landau Critical VelocityLandau Critical Velocity

Picture credits: B. Samuelson

The existence of a finite cmin – called the Landau critical velocity –is therefore necessary for superfluidity. Note that the absence ofthe free particle parabola ω = �k2/2m is also crucial since for a quadratic spectrum ω ∝ k2, cmin is always 0. For He-II, cmin is less than the sound velocity c due to the roton minimum. A further reduction of cmin comes from the creation of vortices under certain flow conditions. Galilean invariance implies that cmin is also the critical flow velocity of the superfluid beyond which finite viscosity is observed.

This may sound unfamiliar, but similar ideas arise in other contexts from the propagation of an electron in an electromagnetic vacuum to subsonic / supersonic flight.

v < c – silent v ≥≥≥≥ c – BOOM!