Imaging anisotropic vortex dynamics in FeSe

Strong vortex pinning in FeSe could be useful for technological applications and could provide clues about the coexistence of superconductivity and nematicity. To characterize the pinning of individual, isolated vortices, we simultaneously apply a local magnetic field and image the vortex motion with scanning superconducting quantum interference devices susceptibility. We find that the pinning is highly anisotropic: the vortices move easily along directions that are parallel to the orientations of twin domain walls and pin strongly in a perpendicular direction. These results are consistent with a scenario in which the anisotropy arises from vortex pinning on twin domain walls and quantify the dynamics of individual vortex pinning in FeSe.

Figure 1

Magnetization and resistivity (b) curves used to determine $T_c$ and $T_s$, respectively. In (a), the magnetization $M$ is normalized by the applied field $H$ and mass, and does not take into account demagnetization effects, hence, the low-temperature value of $<-1$. In (b), the in-plane resistivity ${\rho }_{\parallel }$ is plotted, normalized by its value at room temperature.

Figure 2

Images of vortices and vortex motion in FeSe. (a) Optical microscope image of two samples imaged for this paper. The black mark with red lines indicates the approximate tetragonal [100] directions for sample 1 (right) and the black mark with green lines indicates the same for sample 2 (left). (b) Layout of the pickup-loop/field-coil geometry for the SQUID susceptometer used. (c) Large-area magnetometry image of the surface of FeSe. The square outlines the area imaged in (d) and (e). (d) Magnetometry image of a single vortex. The full-scale variation of the false color lookup table corresponds to 31 $\mathrm{m}{\mathrm{\Phi }}_0$ magnetic flux through the SQUID pickup loop. (e) Susceptibility image taken simultaneously with the magnetometry image in (d), showing a butterfly in one of two types of domains. The full-scale variation here is 4.3 ${\mathrm{\Phi }}_0$/A. The white arrows indicate the FeSe tetragonal $a$ crystal axes directions. (f) Susceptibility image of a second vortex in the second TB direction in the same sample as (c)–(e). Full-scale variation 0.57 ${\mathrm{\Phi }}_0/\mathrm{A}$.

Figure 3

Simultaneous large-area magnetometry (a) and susceptometry (b) images showing vortices pinned on TBs and corresponding butterflies. Dashed lines mark the inferred domain boundaries. The butterfly lobes are oriented parallel to the TBs.

Figure 4

Temperature series of susceptometry images for two butterflies in sample 1 [(a) and (b)] and one in sample 2 [(c)]. The long-wavelength modulation in (c) is due to height variation in the scan plane. (d) Temperature series in sample 3 showing striped features in susceptometry along the TB direction. The tetragonal [100] direction in each series is marked for clarity.

Figure 5

Vortex susceptibility images calculated using the model described in the text, the susceptometer layout of Fig. 2, and scan height $z_0=2\mu \text{m}$. Unlike in the SSM data, the axes of anisotropy are chosen to lie along the image axes. The spring constants along the $x$ and $y$ axes are varied from $1\times {10}^{-9}$ N/m to $20\times {10}^{-9}$ N/m. In the cases where there is anisotropy, the weak axis $k$ sets the scale of the signal while the strong axis $k$ changes the shape of the lobes. The full scale color variation is 180 ${\mathrm{\Phi }}_0/\mathrm{A}$.

Figure 6

Fits of the model to experiment using the spring constants $k_s$ and $k_w$ as fitting parameters. (a) The dependence of the ${\chi }^2$ difference between model and experiment along the weak axis (dashed line in inset) on $k_w$ using a fixed value of $k_s=1$ N/m. The $\widehat{s}$ axis is assumed to be rotated by ${28}^{\circ }$ relative to the scanned $x$ axis, with $z_0=2 \mu \mathrm{m}$. (b) The dependence of ${\chi }^2$ on $k_s$ computed for cross sections through the strong axis (dashed line in inset) for a fixed value of $k_w=4.1\times {10}^{-9}$ N/m. The black dashed line indicates the value of $k_s$ at which ${\chi }^2$ is doubled from its minimum value. (c), (d) Experimental (blue dashed line) and best-fit model (red solid line) cross section along the weak (c) and strong (d) axes.

Figure 7

Susceptibility data (a), best-fit model (b), and difference (c) for the butterfly shown in 2. Data (d), model (e), and difference (f) for the butterfly shown in 2. The color maps are in units of ${\mathrm{\Phi }}_{\mathit{0}}/\mathrm{A}$.

Figure 8

(a), (b) Cross sections of vortex displacement $dr$ (blue solid line) extracted from susceptibility data and simulated susceptometer force (red dashed line), for a 1-mA field-coil current, along the weak (a) and strong (b) axes of the butterfly in Fig. 7. The displacement tracks with the strength of the force from the susceptometer along the weak axis. For the strong axis, a linear background was subtracted from the susceptibility cross section before calculating vortex displacements. A small nonzero background leads to a finite $dr$ which does not track with $F_{\mathrm{SQUID}}$. (c) Susceptometer force versus vortex displacement along the weak axis plotted separately for the brighter (blue circles) and dimmer (red squares) lobes. A linear model (green dashed line) was used to fit the data to calculate an effective $k_w$. (d) Susceptometer force plotted against vortex displacement for the strong axis cross section. We do not see any apparent correlation between applied force and calculated displacement.