The resistive state of a superconducting thin film channel (bridge) of intermediate width W(~,~ W ... A kinematic vortex with infinite velocity is a phase slip line. 1.
Physica C 213 (1993) 193-199 North-Holland
Kinematic vortices and phase slip lines in the dynamics of the resistive state of narrow superconductive thin film channels A. A n d r o n o v , I. G o r d i o n , V. K u r i n , I. N e f e d o v a n d I. S h e r e s h e v s k y Institute of Applied Physics, Russian Academy of Sciences, 46 Ulyanov St., Nizhnj~Novgorod 603600, Russian Federation Received 15 April 1993
The resistive state of a superconducting thin film channel (bridge) of intermediate width W(~,~ W,~2~r, ~ is the coherence length and A~r is the effective magnetic penetration depth) - this case is relevant to the nanoscale high-Tosuperconductor thin film bridges - is studied by simulations of two-dimensional time-dependent Ginzburg-Landau equations. It is found that in the uniform narrow channel at high dimensionless conductivity Z the dynamical behavior is due to appearance of the phase slip lines, but at lower r vortices appear in the resistive state. In inhomogeneous channels vortices appear at any ,r. We call these vortices ldnematic vorticesbecause, unlike the Abrikosov vortices, they can not be treated as quasiparticles. They move with any velocity depending on the current distribution over the bridge (the bridge inhomogeneity). The more uniform the current distribution, the higher the vortex velocity. A kinematic vortex with infinite velocity is a phase slip line.
1. Introduction The resistive state o f thin superconducting film channels with widths substantially larger than the effective magnetic penetration depth 2err (2eff=22/d, where d is the film thickness and 2 is L o n d o n penetration depth) results from m o v e m e n t o f the Abrikosov (Pearl) vortices due to transport current. Slowly moving vortices retain their structure and, consequently, can be treated as quasiparticles with characteristic size 2~ff. Their velocity is determined by balance o f the Lorentz force produced by the current and by the viscous force. On the other hand, resistive state o f narrow superconducting channels with widths being below the lengths ~ ("whiskers") is one-dimensional in nature and results from a pair-breaking process, which produces the phase slip centers (PSC) [ 1 ]. The intermediate case when the width o f a channel lies between coherent length ~ and 2¢ff requires special consideration, which is currently quite important because this case corresponds to the high-To superconductor thin film nanobridges. Here because o f W>> ~ it is possible that the two-dimensional dynamics o f the current j and o f the order parameter ~¢takes place, but, on the other hand, the width is too
small for the Abrikosov (Pearl) vortices to be placed directly inside the channel. In this paper we discuss the resistive state of such superconducting channels on the basis of simulation o f a version o f the time-dependent Ginzburg-Landau equations ( T D G L E ) . In section 2 we present the T D G L E used, discuss appropriate boundary conditions and the applicability of the T D G L E for high-To superconductors. Section 3 contains the simulation results for uniform and nonuniform bridges. Creation and annihilation o f vortices and the vortex velocity dependence on current distribution are considered. In section 4 we give a discussion o f the results o f section 3 and present also the reasons for the term kinematic vortex (which in fact corresponds to crossing o f the bridge by a running PSC). The results are summarized in section 5.
2. Model and basic equations The well known simplest equation for describing the dynamics o f a resistive state is the time-dependent G i n z b u r g - L a n d a u equation ( T D G L E ) .
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A. Andronov et al. / Kinematic vortices and phase slip lines
(O e ) rGL + 2 i ~ ¢ 7,+7,(17,12--1)
(1)
+¢2(__iV__ 2eA~ 2 hc
]7,=0,
where ~ is the appropriately normalized superconducting complex order parameters, ~ and A the scalar and vector electromagnetic potentials, rGL is the characteristic time of Ginzburg-Landau (GL) theory, which is connected to diffusion constant D in the normal state: D=~2/ZGL=IVF/3, where ~ is the temperature-dependent coherent length of GL theory, l and VF are the free path and Fermi velocity of electrons, respectively. Equation (1) is to be supplemented by the Maxwell equation with the electrical current density j = - trVq~+ ( c / 4 n 2 2 ) Im{ V* ( ~ o / 2 n V - iA ) q/).
(2) Here the first term is the normal state current, tr is the normal state conductivity and ~ is the scalar potential. The second part is the supercurrent; here q~o is the quantum of magnetic flux and 2 is the temperature-dependent magnetic penetration depth. Equation (1) is derived from the quasiparticle kinetic equation only for gapless superconductors, and strictly speaking, should not describe superconductors with finite gap. The difficulties in describing the dynamics of the resistive state of finite-gap superconductors arise owing to a complicated (nonlocal) manner by which the quasiparticle distribution function is connected with the electric field and the rate of the order parameter change. Sometimes these difficulties can be overcome. For example, if the energy relaxation length IE and the energy relaxation time rE of quasiparticles are short enough, while the characteristic time and length scales of the order parameter change are large enough so that the ansatz of local equilibrium may be used one can derive the equation for finite-gap superconductors [ 2 ] : ZGL( 1 +/-21 7,12) - l / 2 ( O / O t + 2 i e / h ( ~
(3)
+½F 2 Ol 7,12/0t) 7, =~2(V-2ieA)27,+ (I - 1 7 ' 1 2 ) 7 . Here F=2rEAo>> 1 (~u=A/Ao), where Ao is the equi-
librium gap value. In the high-T¢ superconductors at T ~ Tc rE is quite short (rE < 10 -13 s). This short rE provides in turn a short energy relaxation length IE - about several lattice cells. Since in the high-To superconductors ~ is of about 1-3 lattice cells long, one may consider that for these superconductors the structures of nanometerscale for the time-scale corresponding to microwave frequencies may be well described by eq. (3). Equations (1) and (3) are qualitatively alike and differ by the characteristic times only: rGL in ( 1 ) and refr= "['GL/( 1 +F2I qJ[ 2) in (3). Since in a resistive state I 7,12~ const, nearly always throughout the bridge, one can use the more simple eq. ( 1 ) with ZGLexchanged to zeff=const, instead ofeq. (3). This consideration is also supported by results of parallel simulation of eqs. ( 1 ) and (3) in some particular cases [2,3 ]. In general for a more detailed description of the resistive state dynamics one should solve the quasiparticle kinetic equation to find dynamics of order parameter and quasiparticle distribution function. However, as the first step for consideration of the complicated two-dimensional phenomena in the resistive state of narrow superconducting channels the approach based on eq. ( 1 ) with rGL changed to reff is quite appropriate. In this paper we use this approach to study the resistive state of thin superconducting film bridges of intermediate width (~ 0 , in normal state regions ( N ) a ( ( x , y ) < 0 ; to simplify the simulation we assume the normal state region to be finite. Boundary conditions for the order parameter and the scalar potential are now as follows: at the ends of the normal regions at x = 0 and x = L the order parameter ~u=0, and the input-output currents are uniform, i.e. O~/OX=jo/a, where Jo is the input-output current density. At the lateral boundaries, at y = 0 and y= W, O~/Oy=O, and O~o/Oy=O. Discussions of the resistive state in the framework of the one-dimensional version of eq. ( 1, 2, 3) were performed in lots of publications starting from the pioneering works of Kramer and co-workers [4,5 ]. The two-dimensional simulation of these equations have been started only recently [6,7 ]. And in particular in ref. [6] it was found that in a homogeneous bridge the appearance of PSL's or vortices depends on the value of the normal state conductivity - the result which is also found in the present work.
,j
>
£
"t..
L=26~ Fig. 1. Geometry ofa NSN bridge with additional N-type region.
195
However, it should be emphasized that in most of the papers including ref. [6 ] dealing with simulations of different versions of TDGLE, the periodic boundary conditions along channels are usually used. The results of simulation even in the one-dimensional case for NSN channels studied with boundary conditions used in the present paper differ often qualitatively from the results obtained for periodic conditions (cf. discussion in ref. [5] where the onedimensional consideration of a resistive state in the N S ' S S ' N channel is also discussed).
3. Dynamics of resistive state Equations ( 1 ) and (2) describe the evolution of the two functions - the order parameter ¥ and the electrical potential ~. In the present paper transverse inhomogeneity of the bridge is introduced only by considering transversely inhomogeneous distributions of the function a(x, y) while the width of the channel W is supposed to be constant. The above boundary conditions remain appropriate of course for this case also. For W=const. one may decompose the order parameter ~ and the electric potential to a discrete set of transverse spatial modes. The modes are coupled due to nonlinearity of eq. (4a, 4c). We use the standard grid method to solve the equations describing evolution of every mode. To find amplitudes of these modes the fast Fourier transformation is employed.
3.1. Phase slip lines and vortices in uniform bridges First we discuss the dynamics of the order parameter for a case of a uniform bridge, where function a(x, y) depends on x only. The simulations show that the bridge resistive state dynamics depends crucially on 2:." if27 is larger than some critical value 27¢r (which is a function of W and Jo the system evolves to onedimensional behavior under any initial conditions, i.e. a~u/ay--)o, a~/o~u--)o as t--,og, and the bridge resistivity results from periodical appearance of PSL's. This process takes place for the transport current density exceeding some critical value near the pairbreaking current JGL =a3/2(4/27) 1/2 (small differences are due to boundary effects). Atj¢ >J~r the PSL's arise and their number grows with the current while
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A. Andronov et al. / Kinematic vorticesand phase slip lines
their behavior becomes complicated. Reference [ 5 ] is entirely devoted to the one-dimensional simulation o f the resistive state o f superconducting N S N channels. In fig. 2 regions with different numbers and behavior o f PSL's are shown (for more details see ref. [ 5 ] ) . If the dimensionless conductivity is small enough, the one-dimensional state becomes unstable, transverse perturbations grow, and resistivity results from the m o v e m e n t o f the vortex across the channel. For other boundary conditions similar results have been obtained earlier in ref. [ 6 ]. In the present paper we find that these vortices can be anisotropic. The anisotropy o f the vortex and its velocity grow when parameters X or W a r e getting close to the region where the resistive state is determined by the one-dimensional PSL dynamics. There also exists a narrow region o f X and j where the character of one-dimensional behavior depends on initial conditions so that near the boundary "~cr( W, jo) there is a hysteresis. This section may be summarized as follows. In homogeneous bridges the resistive state appears nearly at the pair-breaking current density. The resistivity is caused by PSL's in narrow bridges or by the movement o f anisotropic vortices in more wider bridges. The value o f the bridge width Wet separating these two regions with different character o f the resistive state depends on normal state conductivity and on the transport current.
3.2. Kinematic vortices in nonuniform bridges
If a bridge is not transversely uniform, i.e. function a ( x , y ) depends also on y, the distribution o f the order parameter is not one-dimensional even for a stationary nonresistive case - i.e. if the superconducting current is below its critical value. As the transport current grows the current density in some points exceeds the critical value Jcr = (4 / 2 7 ) ~/2a 3/.2 and at these points the pair-breaking process has to begin leading to a decrease o f the order parameter and to a formation o f a vortex which moves across the bridge. Figure 3 shows the creation of a vortex at the local inhomogeneity situated at the edge of the channel at x = L / 2 . All results for the inhomogeneous bridges presented below in this section are for the region o f X and W, where in a corresponding uniform bridge the one-dimensional dynamics (PSL's) occurs. In the inhomogeneous case the creation and the m o v e m e n t o f the vortex look as follows. At first the dynamics o f the order parameter is almost onedimensional. Then in the same place, where the PSL should arise in a corresponding uniform bridge, the order parameter decreases and a "ditch" appears in the order parameter distribution. Then a vortex "runs" along this ditch, the vortex is strongly anisotropic as is seen from fig. 3.
I
(7 w=18
r.=lO
0
:0.55
0.50
N 000
/ ......... 0.00
,. ,.' ......
~o.oo
¥=Yv
J Fig. 2. Regions with a different number of PSL's in the resistive state of one-dimensional current-carrying superconductors.
, .....
20.00
I,,
,1:).00
3o.oo o.oo
X
~o.oo
X=Xv
Y
Fig. 3. Structure of a vortex created at the edge inhomogeneity: isolines of I~l, longitudinal and transverse cross sections of the vortex.
A. Andronov et al. / Kinematic vortices and phase slip lines
197
fact these kinematic vortices are the PSC's moving across the bridge.
Figure 4 illustrates how two PSL's that simultaneously appeared in the homogeneous channel (at x = x t and x = x 2 ) transform to vortices by an inhomogeneity at the channel edge. The vortices move almost along the PSL ditches; however they appear at different times, move in different directions and give rise to a change in time evolution of the voltage across the bridge compared with the homogeneous case with PSL's. Figure 5 shows the creation of vortex-antivortex at the inhomogeneity in the center of the channel. These vortices also are anisotropic and move along the "former" PSL ditch. The velocity of a kinematic vortex can be any up to infinity as in the case of a PSL in a uniform bridge. For fixed transport current and bridge width the value of the inhomogeneity determines the velocity of the vortex and its anisotropy: the vortex velocity grows with its size along the y-axes. Figure 6 shows the velocity of the vortex versus inhomogeneity of the bridge. This unusual dependence - for the Abrikosov vortices the dependence would be just the opposite - supports our consideration that these vortices are kinematic phenomena and that they cannot be treated as quasiparticles. In
4 . Origin of kinematic vortices
To understand the cause of breaking up of PSL's to kinematic vortices and to evaluate their velocity let us discuss the following consideration. A vortex is an isolated point where I~1 =0. This point can be determined as a crossing of two lines with Re ~ ( x , y ) = 0 and Im ~ ( x , y ) =0. In a uniform superconducting bridge at high enough 27 the order parameter does not depend on the transverse coordinate, so these lines are the straight ones parallel to the y-axes. At some time t* these lines coincide and PSL's appear. In a nonuniform bridge and in a uniform bridge at low ~ lines with Re ~ ( x , y ) = 0 and Im ~F(x,y)=0 they become curved and are not parallel; they can cross at isolated points only. It is those points that are the centers of vortices. The angle between these lines is determined by the nonuniformity of the distribution of the order parameter (which may be due to the inhomogeneity of function a ( x , y) ) and by the electric potential nonuniformity
50.00
.i=0"7
X2 . . . . .
0.00 140.00
~ r
. . . . . . ',, . . . . . . . . . . . . . . . . 160.00 180.00
2
0.60
i], , ,v ! = xf t _ _
x2,
N 0
0.20 t4.0,00
160.00
t80.00
2
X
volt ooo
0.00
~lwl,~,,,
14o.oo
,,III,,D,
160.oo
,
180.00
,,,
2
time Fig. 4. Coordinatesof vortices (--) and PSL's ( - , - ) and voltageacrossthe structure shown with (--) and without (-*-) inhomogeneity; directions of the vortex movementare also schematicallydepicted.
A. Andronov et al. I Kinematic vortices and phase slip lines
198
'xSl
pends on the velocities of these lines vR and ~. These are velocities perpendicular to the corresponding lines. They depend on O~w/Otwhich is of course limited. The vortex velocity depends also on the angle c~ between these lines at their crossing. Then the vortex velocity is
s
Vvor-
I'I)
IVR--EI sin a
'\
"x..'
J
/ \ 1 ,m ((**I- ° _ _ _
ooo.,
Fig. 5. Order parameter distributions just before and just after creation of a kinematic vortex-antivortex pair for the structure shown. At
Since the value of this angle can be arbitrarily small, the velocity of vortex can be arbitrarily high. On the other hand, it is clear that the degree of the vortex anisotropy should be determined by the same angle: say, at small a the ratio Lx/Ly ~ sin a, where Lx and Ly are the longitudinal and transverse characteristic sizes of a vortex; consequently Vvor~L J L x . From this interpretation one can understand that the movement of such a vortex is a kinematic phenomenon which is similar to the movement of the crossing point of two rulers (or two sheets of paper) in a well-known physical school demonstration. The movement of a vortex may have nothing to do with transfer of information across the film but may be caused just by the appearance of PSL's in separated current lines. Therefore, a vortex velocity should be determined by the current distribution over the whole bridge.
4 5
5. Summary
30.00
L
20.00
10.00
0.00 0.00
s.o0
lo.oo time
15.00
2o.o0
Fig. 6. Vortex m o v e m e n t versus bridge i n h o m o g e n e i t i e s .
(which depends on normal conductivity: the potential is the more uniform at high X). The velocity of the points where the lines with Re ~ ( x , y ) = 0 and Im ~U(x,y)=0 cross, i.e. the vortex velocity, de-
We discussed the dynamics of a resistive state in a narrow superconducting thin film bridge of intermediate width ~
