PHYSICAL REVIEW B VOLUME 45, NUMBER 5 1 FEBRUARY 1992-I

Some phenomenological theoretical aspects of intrinsic superfluid critical velocities

Richard E. PackardUniversity of California, Berkeley, California 94720

Stefano VitaleUniversity of Trento and Istituto Nazionale di Fisica Nucleare, I 3805-0 Povo, Trento, Italy

(Received 28 May 1991;revised manuscript received 9 September 1991)

We use a phenomenological theory of superfluid critical velocities in small orifices to explain threephenomena: (1) The low-temperature intrinsic critical velocity U, in dc experiments should depend loga-rithmically on the driving pressure head; the logarithmic increment in U, is the width of the velocity dis-

tribution function for phase slips. (2) The critical velocity measured in dc experiments is related to thatseen in ac experiments through the quantum number of the phase slips involved. (3) It is possible thatquantum tunneling processes may be significant at temperatures as high as 0.3 K.

For over half a century physicists have recognized thatdissipation-free superflow can exist only below some criti-cal velocity, v, . Measurements of this critical velocityhave been made in many laboratories' exploring theeffects of various parameters on the apparent breakdownof superfluidity. Out of this wealth of observational data,several theories have emerged to explain the observedcritical velocity.

A prediction of what that velocity should be was madeby Landau, who demonstrated that, above about 60 m/s,superflow would decay by means of creation of elementa-ry excitations. Although the Landau velocity is apresumed upper limit to critical velocities, it is now un-derstood that motion of quantized vortex lines can leadto dissipation mechanisms at velocities considerably lessthan the Landau limit. For example, the complex motionof a tangle of quantum turbulence can lead to dissipationprocesses in long channels. Also, near the transitiontemperature T&, thermal activation of small quantizedvortex rings can lead to finite dissipation.

In 1965 Anderson described the fundamental process-es that might lead to flow dissipation when a puresuperfluid (e.g. , He below l K) passes through a smallorifice of area s in a very thin wall. He suggested that, atthe critical velocity v„an instability could occur in theflow which would lead to the motion of a quantized vor-tex line across the orifice, crossing all the enclosedstreamlines. In such a process, the order-parameterphase difference across the hole would change by 2m anda fixed amount of energy would be removed from the flowfield. The energy decrement ' due to a 2~ phase slip isAE =~p, v, s, where ~ is the quantum of circulation and p,is the Auid's mass density.

A confirmation of the Anderson phase-slip picturecame recently in the experiments of Avenel and Varo-quaux (AV). They studied the time evolution of asuperfluid Helmholtz oscillator which contained a sub-microrneter orifice within the superfluid flow path. Whenthe velocity in the orifice reaches a critical value, the os-cillation amplitude abruptly drops by an amount corre-sponding to the energy removed by an Anderson 2'

phase slip. This observation has recently been confirmedelsewhere.

For several decades before the AV experiment, thepredominant method for studying the critical velocityproblem involved trying to measure the largest flow ve-

locity at which fluid could pass through an orificewithout any accompanying pressure drop. ' Typically aninitial pressure difference, AP, is applied across the orificeand, as b,P relaxes to zero, the correlation between aver-age mass current and pressure head is recorded. In theAnderson picture, the ideal current versus bP charac-teristic would exhibit a vertical line at EP=O extendingto a critical current. The curve would then follow an al-most horizontal line, thus demonstrating that the flow issaturated at the critical value. For finite AP, the numberof 2m phase slips per second would be given by theJosephson-Anderson frequency, fJA

=hP Ip, K.

Actual data differs in several ways from the above idealpicture. The current versus AP curve exhibits a rapid butsmooth transition to the dissipative regime. In addition,the mass current does not seem to totally saturate evenwhen the pressure head is substantially higher thanzero. ' Commonly the observed current is noticeablydifferent for the two directions of flow through thehole. ' Finally, there appears to be two different criticalcurrent regimes: one being large and temperature depen-dent and a second smaller current which is temperatureindependent. How can such seemingly complex behaviorbe reconciled with the phase-slip picture that seems soclearly demonstrated in the AV experiment?

The central clues to answer this question come fromseveral experimental observations. The first fact is thatthe phase-slip critical velocity, measured below 1 K inthe small orifice experiments, is a linearly decreasingfunction of increasing temperature:

v =v p( 1 T/Tp)

Here v, o is approximately 10 m/s and T, =2.5 K. Thesecond observation is that the velocity at which a phaseslip occurs has a small random variation about the meanvalue. ' '

45 2512 1992 The American Physical Society

45 BRIEF REPORTS 2513

A final point comes from a recent experiment using anAV-type oscillator with one single orifice. It is observedthat, although the critical velocity is the same for flow inboth directions through the hole, slips may be all of size2m for one direction yet be 2mn (where n is integer) forthe other direction.

The first point we wish to address is the explanation ofwhy dc intrinsic critical velocities (i.e., the temperaturedependent v, ) depend on the magnitude of the drivingpressure head. We will derive the form of the v, versushP characteristic curve and show that its departure fromthe simplified picture mentioned above is due to the sta-tistical nature of the of the phase-slip events.

It has been pointed out that, " in ac experiments, theobserved linear temperature dependence of v„given byEq. (1), can be deduced by assuming that the phase slip isinitiated by a thermally activated process. The lineartemperature dependence of v, will follow if the activationenergy E, is a linearly decreasing function of superfluidvelocity:

E =Ep(1 U/U p) (2)

A(t) =1 exp I—(PEp )[1—v(t)/U, p]], (3)

where r is an attempt frequency and P=(ks T)The second factor is the time the system spends near a

given velocity, a time that is determined by the accelera-tion of the fluid. We consider the situation where thefluid in the orifice with effective length, L, experiences aconstant pressure-driven acceleration a =AP/p, L. Theprobability density, g(U), for having a phase slip at veloci-ty v if the fluid is accelerated from an initial velocity v; is

where the energy Eo and the velocity v,o are phenomeno-logical constants connected to the fluctuation process.

The statistical nature of the thermal activation processleads to a finite width in the observed distribution func-tion characterizing the critical velocity. The authors ofRef. 12 show that, if thermal activation is the sole sourceof phase-slip nucleation, then the observed values of thephase-slip critical velocity, v„should have a width, Av„which is proportional to temperature. The actual datadisplayed in Ref. 12 suggest that thermal process are(especially at low temperatures) not the only mechanismsinvolved in the phase-slip problem. Nevertheless, a fit tothe data, in a temperature ra'age where the data appearlinear, yields the value' of Ep/ks =106 K. Although itis not known whether this is a universal value character-izing the energy barrier, we will use it to make numericalestimates in what follows.

To derive the v, versus hP curve for dc flow, we startfrom the same statistical assumptions and calculate theobserved critical velocity seen in a pressure-driven flowexperiment on a hole characterized by 2m phase slips.The important point in the derivation is that the totalprobability for a phase slip to occur near a velocity v,

g(v)dv, depends on two factors. The first is the condi-tional probability rate, A,(t), that the system experiencesthe slip at time t, provided that it did not occur previous-ly. This function depends on the velocity v (t). Using theactivation energy given in Eq. (2),

given by' 'A(U) V A(U')

g(U)= exp — dv'a a

(4)

Performing the integral gives, for v & v,-,

g(v) = Ae ' 'exp[ —( Av, p/PEp)

tiEp v/Vp PEp V&' /VpX e —e (5)

with A =(I /a )e ').For the kind of orifices that are used in real experi-

ments, a phase slip always makes the fluid velocity dropwell below the values for which the probability density issignificantly larger than zero. Thus, U, in Eq. (5) can bereplaced by zero without affecting the result of the fol-lowing calculations.

By multiplying Eq. (5) by v and integrating over all

possible velocities, we calculate the mean velocity atwhich a phase slip occurs, (U, ). After some effort onefinds

(v, ) =U,&[1—(k&T/Eo)ln(a, p, L /KP)],

where the number

(6)

Qi = 1.78(ks T/Ep)I v p

and the numerical factor comes from evaluating a definiteintegral.

To the extent that the temperature dependence of thelogarithmic term is very weak, Eq. (6) recovers theobserved linear temperature dependence with

T& =(E&/ks )/1n(a, p, L /bP).One feature, very relevant for dc Bow experiments, is

the explicit dependence of ( v, ) on driving pressure head.The logarithmic increment in critical velocity is given by

(v, ) 'd(v, )/d(lnbP)=(ksT/Eo)/(1 —T/Tp) . (8)

U~sing, in Eq. (8), the values of Ep and Tp given by ex-

periment, we predict that the magnitude of the logarith-mic increment, at 1.39 K, is 0.029. In a dc orifice-flowexperiment, Hess reported that, in the temperaturerange where Eq. (1) is obeyed, the logarithmic incrementof the average velocity is 0.026+0.003 at T=1.39 K.Though this velocity is not exactly a measurement of( v, ) (see below), the agreement with the prediction ofEq. (8) is suggestive although possibly coincidental.

A second prediction from Eq. (6) is that, in a dc experi-ment, the observed temperature intercept To will dependlogarithmically on hP. In the AV type of oscillator ex-periment one can directly measure the distribution func-tion of phase slips and its associated width Av, . Theanalysis in Ref. 12 predicts this width to be

bu, /(U, ) =(2/in2)(ks T/Ep)/(1 —T/Tp),

which, except for the numerical factor, is precisely thesame as the expression for the logarithmic increment Eq.(8). This explicitly shows that the increase in averagecritical velocity due to pressure head is due to the statisti-cal width of the phase-slip distribution function.

2514 BRIEF REPORTS 45

At a sufticiently low temperature, quantum tunnelingprocesses can dominate thermal fluctuations to produce atemperature-independent distribution width. ' By thesimplest argument one might estimate the crossover tem-perature to the quantum limit. By equating the charac-teristic Boltzmann factor exponent Eo/k&T to the ex-ponent for tunneling through a square barrier of width w

and energy height Eo, we find the characteristic quantumtemperature TQ.

T =(hE' )/(21rwm' )Q

It is not obvious what the value of the mass m orlength w should be in this equation. In Ref. 1 it is sug-gested that the characteristic length scale in the vortexnucleation process is on the order of 10 m. If we usethis length and let the mass be the bare He mass, onefinds a value of TQ =0.35 K. At such a temperature onewould expect that the fractional width in the distributionfunction would be determined approximately half bythermal fluctuations and half by quantum fluctuations.

In actual measurements of the distribution width, Ref.12 reports a linear temperature dependence superimposedon a temperature-independent part. The two parts arecomparable at about 0.5 K. The similarity between thistemperature and the estimate of TQ given above may becoincidental. Clearly additional experiments and moredetailed theory are needed to determine the role of quan-tum tunneling in the vortex nucleation problem.

Another point we will discuss is the relation betweenthe average critical velocity measured in a dc experimentand the critical velocity characteristic of 2~ phase slips.In the simplest dissipation event, a small vortex segmentis created through some thermal fluctuation. ' Aftercreation the segment evolves in the flow field, crossing allthe flow lines and annihilating at a boundary. It is alsopossible for the thermally created vortex to evolve in amore complicated manner, perhaps undergoing a fewtwists and reconnections before finally annihilating. Thiswill lead to a 2m.n phase slip and the energy dissipatedwill be n times greater than the 2~ event.

The magnitude of the temperature-dependent phase-slip critical velocity (u, ) is symmetric with respect toflow direction. This is because, for pure potential flow inthe absence of vortices, the flow pattern depends only onthe boundaries and must be independent of direction.However, once a vortex is created, its motion can be verydependent on direction of flow and geometry.

An oscillator experiment of the AV type measuresdirectly the velocity which initiates the phase slip. Thiscritical velocity, (u, ), is independent of the n value ofthe slip. In contrast, a pressure-driven dc flow experi-ment measures the average mass current through theorifice. The average velocity (u), which characterizesthe mass current, is not simply the critical velocity ( u, ) .

It is straightforward to calculate the relation between(u) and (u, ). We assume that the average currentthrough an aperture is characterized by phase slips ofsize 2~n. The applied pressure head hP causes the in-stantaneous velocity in the hole to increase linearly withtime with acceleration a =hP /p, L, where L is the

effective length of the orifice. ' When the velocityreaches the critical velocity, the phase slip of 2mn causesthe velocity to drop from the ( u, ) to ( u, ) —n~/L(Here z enters from the expression for the energy dissi-pated in a 2' phase slip. ) The instantaneous velocity thusconsists of periods of constant acceleration to ( u, ) fol-lowed by abrupt drops. The process repeats at the fre-quency fJA/n.

The time average of such a pattern is easily shown tobe

( u ) = ( u, ) —n a/2L -.

Thus, the current through the orifice, which is propor-tional to ( u ) is not an indicator of ( u, ) unless the multi-

ple number n is known. If the slip number, n is a randomvariable, depending on T and hP, the dependence of ( u )on AP would not necessarily reflect the simple tempera-ture dependence given in Eq. (8). The disparity betweensome dc flow experiments may be contained in Eq. (11).For a typical effective length, L =10 m, and a typical( u, ) =5 m/s, the second term in Eq. (11) is about an n%change in ( u ) relative to ( u, ) .

In recent experiments using a submicron orifice, it wasobserved that, using the oscillator technique of AV, flowthrough the hole in one direction always produced single2m. phase slips, whereas flow in the opposite direction ex-hibited slips of n ) 1 at the same ( u, ). In a dc flow mea-surement on the same hole, the flow in the directioncharacterized by single phase slips exhibited an averagevelocity very close to the (u, ) in the oscillator experi-ment. However, dc flow in the opposite directiondisplays an average velocity substantially less than that ofthe former case. In the latter case, expansion of the sen-sitivity of the flow detector actually permits observationof the individual large deceleration events. It is foundthat, when the current is small, the fluid is undergoingdeceleration events characterized by large-n values.

The discussion above considers dissipation arising fromdiscrete phase-slip events which are thermally initiatedand finite in time. However, it is possible for dissipationto occur in quite a different manner. If the hole issufficiently large, it is possible for a single, thermally ac-tivated vortex to grow indefinitely into a tangle of twistedvorticity. ' In such an instance, the vortex tangle re-moves large amounts of energy continuously from thepressure drive. The average velocity in the orifice willthen be small and temperature independent. This is thetype of dissipation typically seen in holes larger than10 m. The limiting velocity of such flow has no con-nection with the thermal activation ( u, ) and can only becharacterized by complex numerical simulations.

Some experiments have reported seeing spontaneousand uncontrollable switching between a largetemperature-dependent flow to a sma11 temperature-independent flow. ' This would fit with the discussionabove. '

It is interesting to note that, even in an oscillator ex-periment with a hole larger than 10 rn, if the actual on-set event leading to turbulence is detected, the critical ve-

locity for this event is the large temperature-dependentvalue, even though subsequent motion of the oscillator is

BRIEF REPORTS 2515

heavily damped due to the sustained vortex tangle nearthe orifice. '

We summarize the above comments by pointing outthat we have explained several important points relatedto the understanding of the superAuid critical velocity.We have shown that the average velocity where phaseslips occur is a function of the pressure head across theorifice. The effect arises due to the finite width of thephase-slip distribution function. We showed that thecurrent through an orifice is characterized by both thecritical velocity and the size of the phase slip. Differingdc measurements can be reconciled by assuming one criti-cal velocity but differing multiple numbers for the slips.We point out that the existence of two different criticalcurrent regimes is consistent with the picture that thetemperature-independent regime is characterized by sus-tained vortex evolution. '

The important remaining questions dealing with the

onset of dissipation at low temperatures concern the (pos-sibly tractable) problem to determine the size of U,c and

Fo as well as the probably more complex question of try-ing to predict the time evolution of nascent vorticitywithin an orifice. The role of possible quantum Auctua-tions is also of interest.

It is a great pleasure to acknowledge several peoplewhose conversations stimulated some of the ideasdeveloped herein: L. Pitaevskii, M. Cerdonio, A. Amar,Y. Sasaki, S. Davis, M. Bonaldi, R. Dolesi, K. W.Schwarz, E. Varoquaux, O. Avenel, and Wrn. Zimmer-mann. One of us (R.E.P.) thanks the University of Tren-to and The Bronzini Institute for gracious hospitalityduring his stay in Italy. This work was partially support-ed by the National Science Foundation Grant No.DMR-88-19110.

'A recent review of the critical velocity problem, relevant tosmall orifices, is given in E. Varoquaux, W. Zimmermann,and O. Avenel, in Excitations in 2-Dimensional and 3-Dimensional Quantum Fluids, edited by A. F. G. Wyatt andH. J. Lauter (Plenum, New York, 1991). This paper has themost up-to-date list of references.

2J. T. Tough, in Progress in Lou Temperatures Physics, edited byD. F. Brewer (North-Holland, Amsterdam, 1981),Vol. 8.

J. S. Langer and J. D. Reppy, in Progress in Low TemperaturePhysics, edited by C. J. Gorter (North-Holland, Amsterdam,1970), Vol. 6.

4P. W. Anderson, Rev. Mod. Phys. 38, 298 (1966).~E. R. Huggins, Phys. Rev. A 1, 332 (1970).60. Avenel and E. Varoquaux, Phys. Rev. Lett. 55, 2704 (1985).7A. Amar, Y. Sasaki, J. C. Davis, and R. E. Packard (unpub-

lished); preliminary information on this experiment has beengiven in A. Amar, J. C. Davis, R. E. Packard, and R. L.Lozes, Physica B 1654166, 753 (1990).

G. B.Hess, Phys. Rev. Lett. 27, 977 (1971).J.P. Hulin et a/. , Phys. Rev. A 9, 885 (1973).

' G. L. Schofield, Jr., Ph. D. thesis, University of Michigan,1971~

E. Varoquaux, M. W. Meisel, and O. Avenel, Phys. Rev. Lett.57, 229 (1986);B.P. Beeken and W. Zimmermann, Phys. Rev.B 35, 1630 (1987). The linear temperature dependence is alsoseen for dc ffow described in Ref. 7.W. Zimmermann, Jr., O. Avenel, and E. Varoquaux, Physica

B 165)%166, 749 (1990); see also, Excitations in 2-Dimensionaland 3 Dimen-sional Quantum Fluids (Ref. 1).Our calculation differs from those in Ref. 3 by involving avariable velocity in the velocity dependent rate, A.. In singleorifice experiments a single 2~ phase slip lowers the velocitybelow that which further slips can occur. However, inpersistent-current experiments, single slips do not appreciablyalter the velocity, and the rate A, may be treated as constant intime.A. Papoulis, Probability, Random Variables and StochasticProcesses (McGraw-Hill, New York, 1965), p. 109.

~sp. M. Morse and H. Feshbach, Methods of Theoretica!Physics(McGraw-Hill, New York, 1953), p. 1294.

~6K. W. Schwarz, Phys. Rev. Lett. 64, 1130(1990).There is some evidence in Berkeley (Ref. 7) and in Trento, Ita-ly [Bonaldi et al. (unpublished)] that, at temperatures above 1

K, there are two distinct temperature-dependent critical ve-

locities. Additional measurements are needed to determinewhether these observations are due to some artifact under-standable within the framework of this paper.

M. Bonaldi, S. Vitale, and M. Cerdonio, Phys. Rev. B 42, 9865(1990).

t9Quantum nucleation of vortices near moving electron bubbleshas been reported by P. C. Hendry, N. S. Lawson, P. V. E.McClintock, C. D. H. Williams, and R. M. Bowley, Phys.Rev. Lett. 60, 604 (1988).