Physica I 0 7B (1981J 407-408 GD $ North.Holland Publishing Company

BOUNDARY AND SCREENING EFFECTS IN ~He FILM DYNAMICS

YU Lu

Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China

and

Department of Physics, Harvard University, Cambridge, MA 02138

and

Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106

Experimental results of Ekholm and Hallock on persistent current decay in ~He films are interpreted quantitatively by considering both boundary effects and vortex density relaxation. The screening due to free vortices in nonequilibritun states is discussed self-consistently and is applied to explain the nonlinear effects observed in Bishop and Reppy's oscillating substrate experiments at the Kosterlitz-Thouless transition.

i. Very careful measurements of the persistent current decay made by Ekholm and Hallock on unsaturated He films(l) have shown significant deviations from the logarithmic law predicted by lordanskii-Langer-Fisher theory as summarized in Ref. 2. Several attempts (3-5) have been made towards the theoretical understanding, but the whole variety of experimentally observed behavior remains unexplained. In the present communica- tion the experimental results of Ref. 1 are interpreted quantitatively by taking into account both the relaxation and boundary effects. Details will be published elsewhere.(6)

2. The rate equations for the superfluid veloc- ity V s and the free vortex density n have been derived by AHNS.(7) In dimensionless variables they can be written as (4,6)

dV/dT = -NV, (i)

dN/dT = VA-N ~ + GV ~/2 + I-GNV. (2)

The first equation describes the decay of current velocity V E Vs/Vs(0) , where Vs(0) is the velo- city at the beginning of the free decay. The second equation describes the time evQlution of free vortex density N ~ (n/no)(rco/a)A/2, where n o is proportional to the density of bound pairs, a being the core radius and rco ~/mVs(0) being the escape radius at the initial moment. The dimensionless coupling constant ~ E 2~p~2/m2kBT>4__ below transition according to the Kosterlitz- Thouless theory,(8) where Ps denotes the areal density of the superfluid. The first two terms of Eq.(2) describe the bulk nucleation and anni- hilation of vQrtices in 2D films, while the last two terms correspond to the boundary creation and recombination process with parameter G -I ~ 2~rcoWno(a/rco )~/2, where W is the film width. The time t ~s scaled as T E t/to, where

/2, D being to-1 ~ 2~Dlno(a/rco) the diffusion constant of the vortex.

The system of nonlinear differential equations (1) and (2) has been solved numerically for dif- ferent values of I and G, the solution being compared with the experimental data of Ref. I at

1.4K for films with thickness varying from 6.1 to 9 layers. Theoretical curves and experimen- tally observed values for the persistent current decay are shown in Figs. 1 and 2. For thin films, i.e. G< 1 case, we use the values of A, determined by the third sound velocity measure- ~ents.(1) For certain range of parameter G, i.e. G< ;2 for the thinnest film, we can fit the data by adjusting only one parameter t o . For thick films, i.e. when G > A/2, theoretical curves almost do not change with G, the only relevant parameters being ~ and to, which are still correlated with each other. The discrep- ancy for films of intermediate thickness (d = 7.2 and 7.6) is interpreted in terms of pinning cen- ter effects which are important when the density of bulk and wall created vortices is small.(5,6)

t 4

O4 •

Fig. i. Comparison of theoretical and experi- mental results(l) for persistent current decay of thin films. Parameters used are ~ = 13, 15, 16; G = .01, .I, .4 and t o = 9.5, 75, 240 sec. for d = 6.1, 6.4, 6.6 correspondingly.

3. According to Kosterlitz-Thouless theory,(8) there are no free vortices below transition tem- perature in the thermal equilibrium. In non- equilibrium states, however, the free vortices can exist and therefore the screening due to these vortices might be important even below the transition temperature. In the simplest Debye- Huckel approximation, the logarithmic interaction

2 2 22 between vortices 2q £n(r/a) where q ~ Wps ~ /m ,

into -2q2Ko(<r) , where K o is the zeroth changes order modified Bessel function and ~ = (2~An) ~ is the inverse of the Debye screening length.

0378-4363/81/0000-0000/$02.50 © Noah-Holland Publ~l~lng Company 407

408

oe

oe

O.4

02

L i o oloi lO = )03 io*

tl,,d

72 7.6 SO e.S

9.0

"t o o o

io s

Fig. 2. Comparison of theory with experiment(l) for persistent current decay of thick films. Parameters used are ~ = 30, 26, 18, i0, 8; G = 30, 30, 30, 20, 20 and t o = 3000, 1200, 380, 380, 240 sec. for d = 9, 8.6, 8.0, 7.6, 7.2 corre-

spondingly.

The system of differential equations (1), (2) with modifications to take the screening into account has also been solved numerically. Since Ekholm and Hallock's experiments have been done far away from the onset region, the screening effect only contributes minor corrections.

On the other hand, Bishop and Reppy's oscillat- ing substrate experiments(9) have been done exactly in the onset region. The theoretical interpretation of these experiments was sug- gested by Huberman et al.(10) and Ambegaokar et al.(ll). AHNS(7,11) worked out a detailed linear response theory for it and Teitel and Ambegaokar(12) got quite good fits to the data at low driving level. But, the nonlinear effects, i.e. the shift of the dissipation peak towards lower temperatures and the broadening of the peak itself with the increase of drive(9) remain un- explained until now. There are two characteris- tic lengths in the problem, one is the escape radius r c, which is greater than 2xl0-3cm in the experiments,(9) the other one is the diffusion length over one period r D = /8D/w = 10-4cm. Since r c >> r D, the static picture(7,10,11) is

not valid.

We start from the Langevin equation for the vortex pair with the Screened binding potential in a periodic external field of frequency w:

d~/dt =-aKl(r) r/r +Eb cos(wt+ ~) + ~(t), (3)

where

4D~Ps~2<2/kB T ~ 8DK 2 (4) a

b H 4D~Ps~2K/kB T r c ~ 8D</r c, (5)

<~i(t ) ~j(t~)> = 4DK26ij6(t-t~), (6)

and E is the unit vector of the external field. Considering b as a formal small parameter, ex- panding r to the secon~ order of it, averaging over the random noise ~ and the random phase ~, integrating over one period At = 2~/~, we obtain the effective d~iving velocity due to the exter-

nal field. If r ± E,

Ar = d 2 + h--t 2 b2/(c2 + [0 )r, (7)

and if r II E,

A~ _£ b2/(g2 A-~ = + ~2)r ' (8)

where c,d,g,i are some combinations of modified Bessel functions.(13) As described by Eq. (7), a vortex pair gains additional net repulsions which may cause the unbinding beyond a certain threshold value of driving force. One can cal- culate the escape rate using the equivalent Fokker-Planck equation. It turns out that in the experimental situation of Ref. 9 a free vortex density of order 106/cm 2 is enough to explain the extra dissipation at higher driving level. This number can be obtained from the above sketched self-consistent calculation. Details will be given later. (13)

4. The author is especially grateful to B.I. Halperin for the numerous enlightening discus- sions, and also to R. B. Hallock and many other people for helpful conversations. The work is supported by NSF through Grants Nos. DMR-77- 10210, DMR-77-23420 and PHY-77-27084.

References

[i] Ekholm, D.T. and Hallock, R.B., Phys. Rev. B21, 3902(1980); Phys. Rev. Lett. 42, 449

(1979). [2] Langer, J.S. and Reppy, J.D., Prg~reSs in

Low Temperature Physics, edited by C.J. Gorter (North-Holland, Amsterdam, 1970), Vol.6, p.l.

[3] Donnelly, R.J., Hills, R.N., and Roberts, P.H., Phys. Rev. Lett. 42, 725 (1979).

[4] McCauley,Jr., J.L., Phys, Rev. Lett. 45, 467 (1980).

[5] Browne, D.A. and Doniach, S., in Inhomogene- ous Superconductors, 1979 (West Virginia), AIP Proceedings, No. 58, P.304.

[6] Yu, L., Phys. Rev. B23, April 1 (1981) and to be published.

[7] Ambegaokar, V., Halperin, B.I., Nelson, D.R. and Siggia, E.D., Phys. Rev. B21,1806(1980).

[8] Kosterlitz, J.M. and Thouless, D.J., J.Phys. C6, 1181(1973); Nelson, D.R. and Kosterlitz, J.M., Phys. Rev. Lett. 39, 1201(1977).

[9] Bishop, D.J. and Reppy, J.D., Phys. Rev.B22, 5171(1980); ibid, Phys. Rev. Lett. 4_O0, 1727

(1978). [i0] Huberman, B.A., Myerson, R.J., and Doniach,

S., Phys. Rev. Lett. 40, 780(1978). [ii] Ambegaokar, V., Halperin, B.I., Nelson, D.R.,

and Siggia, E.D., Phys.Rev.Lett.40,783(1978). [12] Teitel, S. and Ambegaokar, V., Appendix A

in Ref. 9. [13] Yu, L. (to be published).