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THEORY OF THE PHASE-SLIP STATE IN
SUPERCONDUCTING FILAMENTS
R. Watts-Tobin, L. Kramer
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 8, Tome 39, ao6t 1978, page C6-554
THEORY OF THE PHASE-SLIP STATE IN SUPERCONDUCTING FILAMENTS
.F
R.J. Watts-Tobin and L. Kramer
Department of Physics, University of Lancaster, Lancaster LA1 4YB, EngZcmd
*~nstitut diir Theoretische Physik der Universit2t Bayreuth, 8580 Bayreuth, West Gemany
RSsum6.- Une gdn6ralisation de l'dquation de Ginzburg-Landau dependante du temps, qui 6tait ddrivde autrefois pour des supraconducteurs sales prSs de T a des solutions oscillantes avec des glissse- ments de phase au-dessous et au-dessus du courant c:itique pour des fils fins et homogznes. Nous discutons la stabilit6 des diffdrentes solutions. Les rdsultats sont compares avec les caractdris- tiques courant-tension trouvdes dans des whiskers et micro-ponts et avec la structure correspondant
5 des centres de glissements de phase observde rdcemment.
Abstract.- A previously derived generalized time-dependent Ginzburg-Landau equation valid for dirty superconductors in the vicinity of Tc exhibits for thin homogeneous filaments oscillatory phase-slip solutions below and above the critical depairing current. The stability of the various solutions is discussed. The results are compared with I-V curves found experimentally in whiskers and microbrid- ges and with the structure of a phase-slip center observed recently.
Starting from a set of equations which describe dirty in the presence of time-dependent electromagne- tic fields A, (I /1,2/, we have recently derived the following simple generalized time-dependent Ginzburg- Landau (TDGL) equation /3/
(U = n4/145(3) ; y = 2TE
A ~ ;
T~ = inelastic colli- sion ;ime ;=
:
A
8n2 :T &/75(3) ; E= (Tc-T)/Tc ; J, = A/& ; p = 2e(I to ; to = S ~ / D U ;E2=
T D/8Tc&;t
D = vFR/3). Time t and current density J are measu- red in units of t and jo = uo/2e to€, (ao = normal conductivity).
Equations (l) and (2) are valid as long as the various quantities vary in space slowly over the inelastic diffusion length LE = (DrE)l/' and in time slowly over The condition LE<< €,
,
i.e.is in general sufficient (but sometimes not necessa- ry) for this to be the case. For y < < l , i.e. (T T )2
c E
E < < 1, (which is a much stronger condition than eq.(3)) eq. ( 1 ) goes over into the simple TDGL equa- tion which was investigated previously /4/.
We consider solutions of (15) and (16) for long, thin current-carrying filaments where A can be neglected and all quantities depend on X only. There exist two simple stationary solutions : the
fully superconducting (S) state $ = f m exp iqx, p EO, which exists and is locally stable for all j < jmx = 0.385, and the normal (N) state jE0,
p' = -j/€, which can be shown to be locally stable for all j # 0 and y # m .
Let us consider global stability of these states, i.e. stability with respect to all fluctua- tions with nonvanishing probability (i.e., finite energy) to occur. We use the following principle /4/ : a necessary condition for a locally stable state to be globally unstable is the existence of an unstable threshold solution localized about that state which neither decreases nor increases as t + m. Stationary threshold solutions localized
about the normal state do indeed exist for O<j<jc where the current jc is a function of y (jc=0.335,
m for
Y
= 0, 5.5, m ) . Thus, below jc the J, x 9
normal state is globally unstable; we expect it to be globally stable above je.
Using the same arguments as in Ref. /4/ the S state appears to be globally stable for j < min (jc, jmax). Thus, as long as j < jmax holds (i.e. fory< 5.5) a reversible S-N transition can take place
.
For jc > jmx, on the other hand, no such transition can occur and there must exist some stable solution besides the N and S states.Consequently we looked for oscillatory phase
slip solutions which for y = 0 exist in a very nar- row range of currents jmin < j < j with jmin= 0.326.
References
As y increases this range actually increases to hi- /l/ Larkin, A.I. and Ovchinnikov, Yu N., Zh. Eksp. Teor. Fiz.
12
(1977) 299 (To be translated in gher as well as to lower currents. Clearly "isolatedSov. Phvs. JETP) ;
phase-slip solutions" where the S state is approa- ~vchinnikov, Y U N:
,
J. Low Temp .Phys.2
(1 977) ched asymptotically on both sides can exist only 4 3/2/ Schmid, A. and Schijn, G., J. Low Temp.Phys.
g
below j For these solutions the electric fieldmax ' (1975) 207 decays asymptotically over the "quasiparticle diffu-
/3/ Kramer, L. and Watts-Tobin, R.J., Phys.Rev.Lett. sion length" (in press)
h = (I *y2f2) '/*I hfa(+(4~c~E~/~dofD)1 for y>>l)
Q
Wwhich may become much larger than the coherence len- gth. We find that solutions with a periodic array of phase-slip centers exists above j and can carry
m-ax
an average supercurrent larger than j The maximum max '
current up to which the solutions exist first incre- ases with decreasing perSodic length d and then de- creases.
The I-V curves of such phase-slip states exhi- bit significant almost-linear portions which may be parametrized approximately by
V = (j
-
Bjmax)1\15
with A-
7.55 (independent of d and y) andB
= 0.6 and 0.46 for y = 10 and 20 (in- dependent of d). Thus each phase-slip center acts like a piece of filament of length A through which a time-averaged normal current of magnitude j-B jmax flows. Notethat I\ and
X
have different temperature-Q
dependence.
It follows that the I-V curves of filaments of finite length have a step-structure, each voltage jump corresponding to the appearance of anadditional phase-slip center. This kind of behavior is well established experimentally /5,6/. Recent measure- ments by Dolan and Jackel /7/ have revealed to the local structure of a phase-slip center. Their mea- surements of
X
on Sn agree with eq. (17) whenQ
taking T~ = 2 X 10-l0 S. which is consistent with a theoretical estimate 161. This restricts the validity of our theory to c< 0.4 X 10-' (see Eq.
(3)) and here the temperature dependence of the measured differential resistance appears to be
consistent with our results whereas for E >0.7 X I O - ~
the temperature-dependence becomes weaker.
/4/ Kramer, L. and Baratoff, A., Phys.Rev.Lett.
38
(1977) 518 and to be published/5/ Meyer, J.D., Appl. Phys.
2
(1973) 303 ;Meyer, J. and Tidecks, R., Solid State Commun. 24 (1977) 639 and 643
-
/6/ Skocpol, W.J., Beasley, M.R. and Tinkam, M,,
J. Low Temp.Phys.
16
(1974) 145/7/ Dolan, G.J. and Jackel, L.D., Phys. Rev.Lett. 39 (1977) 1628
-
We wish to thank Pr. A. Baratoff and Pr. A.
