Closer look at the low-frequency dynamics of vortex matter using scanning susceptibility microscopy

Using scanning susceptibility microscopy, we shed light on the dynamics of individual superconducting vortices and examine the hypotheses of the phenomenological models traditionally used to explain the macroscopic ac electromagnetic properties of superconductors. The measurements, carried out on a ${\text{2H-NbSe}}_2$ single crystal at relatively high temperature ($T=6.8$ K), show a linear amplitude dependence of the global ac susceptibility for excitation amplitudes between 0.3 and 2.6 Oe. We observe that the low amplitude response, typically attributed to the oscillation of vortices in a potential well defined by a single, relaxing, Labusch constant, actually corresponds to strongly nonuniform vortex shaking. This is particularly pronounced in the field-cooled disordered phase, which undergoes a dynamic reorganization above 0.8 Oe as evidenced by the healing of lattice defects and a more uniform oscillation of vortices. These observations are corroborated by molecular dynamics simulations when choosing the microscopic input parameters from the experiments. The theoretical simulations allow us to reconstruct the vortex trajectories, providing deeper insight into the thermally induced hopping dynamics and the vortex lattice reordering.

Figure 1

(a) Schematic overview of the scanning susceptibility microscopy setup. A superconducting sample is placed in a dc magnetic field, $H$, generated by a superconducting coil surrounding a collinear copper coil generating an ac field $h_{\mathrm{ac}}\left(t\right)$. The time-averaged magnetic field profile due to the present vortices and the screening currents is schematically shown by the black lines. The magnifying glass provides a closer look at the induced ac vortex motion. When the drive is small, the ac magnetic field induces a periodic force on the vortices, shaking them back and forth. A Hall sensor picks up locally the associated time-dependent Hall voltage, $V_{\mathrm{Hall}}$. (b) A lock-in amplifier, provided with both $h_{\mathrm{ac}}\left(t\right)$ as a reference and $V_{\mathrm{Hall}}$, allows one to extract both the in-phase, $b_1^{\prime }(x,y)$, and the out-of-phase, $b_1^{\prime \prime }(x,y)$, components of the local magnetic response.

Figure 2

(a) Vortex configuration at 4.2 K after a field-cooling experiment in $H=1$ Oe applied perpendicularly to the sample surface. The Voronoi construction is plotted on top of the image. (b) A sequence of two different field-cooling experiments in 0.9 Oe, imaged at 4.2 K. (c) Difference image between the two aforementioned experiments at 0.9 Oe. (d) Typical vortex configuration obtained from molecular dynamics simulations on a weakly disordered two-dimensional (2D) superconducting system (cf. Sec. 4).

Figure 3

In-phase (circles) and out-of-phase (squares) components of the background signal averaged over the scan area as functions of the ac amplitude $H_{\mathrm{ac}}$ at $T=6.8$ K and after a $H=1.0$ Oe FC procedure. The lines are linear fits to the data.

Figure 4

Top row: Scanning Hall probe microscopy images of the local induction, $b_z(x,y)$, acquired at a temperature of $T=6.8$ K and a dc magnetic field $H=1.0$ Oe while shaking with an external applied ac field of frequency $f=77.123$ Hz and increasing amplitude (left to right) $H_{\mathrm{ac}}=0.3$, 0.8, 1.0, 1.2, and 2.6 Oe. The Voronoi construction is plotted on top of the image. The numbers in the center of some Voronoi polygons indicate the coordination number $\nu$ of the respective vortex. Middle and bottom rows: Simultaneously acquired maps of (center) in-phase, $b_v^{\prime }(x,y)$, and (bottom) out-of-phase, $b_v^{\prime \prime }(x,y)$, response components. To unify the color map scale, we normalized both $b_v^{\prime }$ and $b_v^{\prime \prime }$ by $1\times {10}^{-3}{H}_{\mathrm{ac}}$ (in Gaussian units). The black arrow in the bottom panel of $H_{\mathrm{ac}}=2.6$ Oe indicates schematically the direction of the ac screening currents.

Figure 5

The $H_{\mathrm{ac}}$ dependence of the real part of the average sixfold bond-angle order parameter as obtained from the data presented in Fig. 4. Higher values of ${\psi }_6$ indicate closer proximity to a perfect triangular lattice. The error bars are standard deviations from the mean.

Figure 6

Amplitude dependence of (a) the real part of the time-averaged sixfold bond-angle order parameter and (b) the time-averaged number of defects normalized by the total number of vortices. The curves are plotted for three different temperature values as labeled in (a). (c–e) Snapshots of vortex configurations and Voronoi constructions for the points indicated in (b). Defects of positive (negative) topological charge are depicted in light (dark) gray. The black arrows indicate schematically the direction of the ac driving force.

Figure 7

Real ($u_1^{\prime }$) and imaginary ($u_1^{\prime \prime }$) parts of the first harmonic of the vortex displacements averaged over all vortices for (a) $T=0$ and (b) $T=0.36U_0/k$. The shaded regions correspond to the amplitude range where the defect density effectively decreases below its static value, reaching a minimum value. For comparison, we show the response predicted for the flux-flow regime, $u=iA/\eta \omega$ (red dashed curve). (c) Root-mean-square values of the in-phase (circles) and out-of-phase (squares) components of vortex response within the scan area as functions of $h_{\mathrm{ac}}$.

Figure 8

Density plots of the (a) real and (b) imaginary components of the first harmonic of the vortex flux distribution, $b_v$, in a $26\times 26{\lambda }^2$ area of the sample for a driving force amplitude $A=0.5\times {10}^{-3}\epsilon /\lambda$. (d–f) Contour plots of the absolute value of $b_v$ for drive amplitudes $A=0.5$, 1.0, and 3.0 (in units of $1\times {10}^{-3}\epsilon /\lambda$), respectively. Individual vortex trajectories during one forcing period are also shown. For better visualization, all vortex trajectories with respect to their mean position were magnified by a factor (d) $M=4$ and (e) $M=3$. Panel (c) is a zoom-in of the $2.6\times 2.6{\lambda }^2$ region depicted in (f) and shows one vortex trajectory and the pinning landscape nearby. The black arrows indicate schematically the direction of the ac driving force.