Depinning and dynamics of vortex matter in mesoscopic channels R. Besseling, N. Kokubo, P.H. Kes

Part of the research in the superconductivity/vortex matter section of the MSM-group focuses on the behavior of the vortex lattice (VL) in mesoscopic flow channels in thin film superconductors. In this peculiar system vortices are driven (under the influence of a transport current J) through narrow channels which are a few vortex lattice spacings wide (see Fig.1). The samples are fabricated from a-NbGe(weak pinning)/NbN(strong pinning) double layers using submicron lithography, resulting in channel widths ranging from 200 to 1000 nm. Vortex motion in the channel is impeded by the (shear) interaction with strongly pinned vortex lattices forming the channel edges. By changing the applied field one can tune the commensurability between the lattice constants a0,b0 (the latter is the equilibrium rowspacing) and the channel width w. The system offers a unique chance to study the microscopic effects of plasticity, dislocation dynamics and lattice tearing as a function of the microstructure inside the channels.

                    

Fig. 1 Double layer device with weak pinning channels.

The investigations are twofold: Various transport experiments are carried out (e.g. IV and IcB measurements to obtain the stress-velocity relation and the critical flow stress of the vortices) and numerical simulations combined with analytical approaches provide the background for understanding the experimental results. 

A typical measurement of shear force density Fs = JcB versus field of a channel with etched width of 290 nm is shown in Fig. 2 (field sweep down, T=1.7K). 

                   

Fig. 2.  Shear force density Fs versus field. The red line expresses the result expected from continuum elasticity theory where Fs=0.17 c66/w  where the prefactor A follows from Frenkels considerations. 

The envelope of the experimental data follows the field dependence of the shearmodulus c66 (red line). This stems from the phenomenological continuum relation Fs=2Ac66/w where Ac66 is the flow stress at the channel edge. The factor A depends on the lattice potential and microscopic structural properties both inside the channel and in the edge. The observed oscillations in the experiment reflect the structural changes inside the channel on changing from n to n+1 rows. 

                    

Numerical and analytical work

Simulations are performed using  molecular dynamics with a modified London interaction between vortices and periodic boundary conditions. Our attention focusses specially on the influence of positional disorder of the vortex configurations in the channel edges on the flow stress and dynamics of matching and mismatching configurations inside the channel. Three cases, namely perfect, weakly disordered and strongly disordered, can be distinguished. These regimes give rise to different, but quite typical flow characteristics, ranging from 1D charge density wave(CDW)-like motion to quasi-2D disordered flow.    

ORDERED CASE: the commensurate (i.e. when w=nb0) state in the channel is simply a continuation of the perfect lattice in the edge. The flow stress coincides with Frenkels historical result  for slip between perfect lattice planes: Fsideal=0.34 c66/w (red line in Fig. 2, phenomenological constant A=0.17). At incommensurability pairs of misfit dislocations with Burgers vector in the channel direction occur at the channel edge, see Fig. 3. Their core size ~(c11/c66)1/2a0  (c11 and c66 are the compression and shear modulus of the VL, respectively) considerably exceeds a0 and their Peierls barrier (stress required to move them) is vanishing small (A~0). As a result, sharp peaks in Fs as function of commensurability parameter should occur.

                                                         

Fig.3 Triangulation of an incommensurate channel w~3.6 b0. Pairs of misfit dislocations occur at the channel edges, which are marked grey. 

WEAK DISORDER:  for weakly disordered, correlated arrays in the channel edge the commensurate flow stress is reduced due to driven nucleation of defect pairs at the channel edge. The dynamic displacement field of this quasi-1D motion shows interesting nucleation dynamics (a MOVIE, F~0.9 Fsideal  is available), having a direct and formal analogy with depinning of the phase field of a weakly disordered CDW or an elastic string in a tilted disordered washboard potential  (in case of slow connection save THIS target, ~600kB .gif file and a GIF PLAYER using right mouse button). The incommensurate flow stress is enhanced due to pinning of the existing edge dislocations by random strain in the pinned arrays (For a movie of such behavior in a single chain just above threshold, click Dislocation/soliton motion, F~0.2Fsideal).  Thus, due to weak disorder the sharp peaks in flow stress are lowered and broadened. The pinning and flow mechanisms remain quasi-1D, i.e. determined by dislocations with Burgers vector (glide direction) along the channel. These mechanisms, found for weak disorder confirm the intuitive guess that defects (incommensurability) lead to a reduction of the flow stress. 

STRONG DISORDER: recently, we have found very surprising and counterintuitive results in case the arrays in the channel edge are strongly disordered.

     

Fig. 4. (a)  flow stress versus channel width for RMS- strain in the pinned arrays ~10% . (b) Flow stress in units of the ideal lattice value. Right hand side figure are normalized experimental data.

The disorder strength is such that the limit of elastic deformations in the edge is considerably exceeded. Accordingly disorder induced edge dislocations are always present, also at commensurability. Fig. 2 shows the disorder averaged  flow stress of vortices inside the channel versus the commensurability parameter w/b0. The data are in good correspondence with the experimentally observed smooth oscillations in the Jc-Ha curves, shown in normalized form in the right figure. However, the maxima in flow stress are seen to occur at incommensurate channel widths, while minima occur close to commensurate widths ! The origin of this surprising observation is the following: around matching situations w~ nb0, the (moving) structure inside the channel consists of n rows, see Fig. 5.

Fig. 5.  Snapshot of the flow for w~4.1b0 and f~fFrenkel~4fc. 

The flow stress is roughly determined by the (collective) pinning of edge dislocations (Burgers vector along the channel) due to disorder at the channel walls. A movie is available HERE. The four vortices which are marked red (at  ~1/4 from the LHS of the figure) show that over a large part motion inside the channel remains elastic.

Moving to mismatching channel widths (w/b0~n+1/2), drastically different phenomena occur. Due to disorder fluctuations in the pinned arrays in the edge, it can become energetically favorable to form n+1 rows in certain parts of the channel and n rows in other parts, see Fig. 6. 

Fig. 6.  Snapshot of the flow for w~4.6b0 and f~fFrenkel~4fc. The structure appears broken up in domains.

The most pronounced effect in terms of dislocations is that regions develop in which dislocations have a Burgers vector misaligned with the channel direction. These regions can roughly be located in between those with n and n+1 rows. Ideally such dislocations have a misorientation with the channel direction of 60 degrees. Since these fault zones are determined by quenched disorder from the edge, they remain during the flow. It are these regions with misoriented dislocations (which block each others flow) that lead to the enhanced flow stress at mismatching channel widths. The resulting traffic jam like vortex motion can be observed in the MOVIE: plastic motion inside the channel, appearing most pronounced as transverse jumps of vortices in the fault zones, dominates the flow. 

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