Imaging Unconventional Vortices in Superconductors

Scanning SQUID captures the magnetic flux $\phi$ over an isolated vortex and its magnetic dynamics $\chi$, estimating the symmetry of the pinning potential $U$. Reprinted figure from [2]. Copyright 2021, by American Physical Society.

A spoon can create large or small vortices in classical fluids, such as in a coffee cup. Similarly, in classical gases in the atmosphere, small vortices or giant vortices (tornadoes) can form. However, the quantum world is very different.

A magnetic field is created at the center when electrons form vortices in a material. Under certain conditions, superconductors allow the magnetic field to pass through as a thin bundle (magnetic flux) through the crystal (superconducting vortex). The magnetic flux trapped within isolated vortex lines in the single crystal of a type-II superconductor takes discrete values given by the equation $n\Phi_0 = nh/2e$. Here, $n$ is an integer, $h$ is Planck's constant, and $e$ is the charge of an electron. This discreteness occurs because the Cooper pairs around the vortex line encompass a phase of $2n\pi$. However, vortex lines with magnetic flux less than $\Phi_0$ can be stabilized under some conditions.

The scanning SQUID microscope can observe local magnetic flux in absolute units, allowing us to evaluate unconventional superconducting vortex and dynamic properties.

[1] I.P. Zheng, Y. Iguchi, et al., Physical Review B 100, 024514 (2019).
[2] Y. Iguchi et al., Physical Review B 103, L220503 (2021). [Letter]
[3] Y. Iguchi et al., Science 380, 1244-1247 (2023). (Open access link)

Imaging anisotropic dynamics of isolated vortex pinned in twin boundaries

Iron-based superconductor FeSe shows the nematic phase transition at 90 K and the superconducting phase coexisting with the nematic phase below $T_c$ = 9 K. We directly measured the vortex dynamics in FeSe and visualized the anisotropic vortex pinning potential along the twin boundaries.[1] The measurement of isolated vortex dynamics provides local vortex pinning potentials by simulations with a simple quadratic pinning potential $U(x,y) = 1/2( k_xx^2 + k_yy^2)$, where $k_x$ and $k_y$ are the spring constants. We observed the anisotropic susceptibility images over isolated vortices, which were consistent with the vortex dynamics with anisotropic pinning potential.

Observed anisotropic vortex dynamics on FeSe (a) is captured by our simulation with anisotropic pinning potential (b). (c) is the difference of (a) and (b). Reprinted figure from [1]. Copyright 2019, by American Physical Society.

Vortices were pinned along the twin boundaries on FeSe(a). The local pinning potential is anisotropic along the twin boundaries(b). Reprinted figure from [1]. Copyright 2019, by American Physical Society.

Inhomogeneous vortex pinning potential in single crystal URu$_2$Si$_2$

We also measured the local vortex dynamics on a heavy fermion superconductor URu$_2$Si$_2$, which has twin domains under the strain, and observed the isotropic and anisotropic pinning potential at different locations of the same crystal.[2]

The observed vortex dynamics on URu$_2$Si$_2$ (a),(b) are captured by our simulation with anisotropic (c) and isotropic (d) pinning potential, respectively. (e) The observed local anisotropy of pinning potential on URu$_2$Si$_2$ was not ordered. Reprinted figure from [2]. Copyright 2021, by American Physical Society.

The observed anisotropic pinning potentials on URu$_2$Si$_2$ didn't have a long range order in contrast to them on FeSe. Thus the local strain was dominant to break the rotational symmetry of the pinning potentials rather than electron nematic order (or the twin boundary if the sample forms it) on URu$_2$Si$_2$.

The temperature dependence of the pinning force constant $k$ which was estimated from the vortex dynamics at three different locations on URu$_2$Si$_2$ . Reprinted figure from [2]. Copyright 2021, by American Physical Society.

In addition, we first estimated the temperature dependence of the local pinning force, $(1-(T/T_c)^2)^2$, by fitting the isotropic vortex dynamics on URu$_2$Si$_2$. This result suggested that the size of impurity in this sample is of the same order as the coherence length.

Discovery of temperature-dependent un-quantized vortex in superconductor

The total magnetic flux in an isolated vortex in a single crystal of the type-II superconductors has discrete values, $n\Phi_0=nh/2e (n=1,2...)$. On the other hand, the half-integer ($n=1/2$) vortex has been observed at the intrinsic Josephson Junction formed on the grain boundaries of d-wave superconductors because of the $\pi$ phase winding.

Let's consider the effect of multiband in a single crystal of the type-II multiband superconductors. We theoretically expect that the quantum phase winding is only in one of the bands at the level of Ginzburg-Landau theory, inducing fractional vortices, whose total flux may be a part of $\Phi_0$ and temperature dependent. However, this vortex type had never been observed experimentally since its first theoretical discovery in 2002.

Experimentally observed magnetic flux scan over (a) the un-quantized vortex and (b) the quantized vortex. Simulation results with (c) $0.3\Phi_0$ and (d) $\Phi_0$ point sources for (a) and (b), respectively. Reprinted figure from [3]. Copyright 2023, by , The American Association for the Advancement of Science

The total flux trapped in the un-quantized vortices are smaller than $\Phi_0$ and temperature dependent, but almost independent on locations. Reprinted figure from [3]. Copyright 2023, by , The American Association for the Advancement of Science

We first report similar vortices in a multiband superconductor K$_x$Ba$_{1-x}$Fe$_2$As$_2$, which carries only a part of $\Phi_0$.[3] By manipulating vortices in K$_x$Ba$_{1-x}$Fe$_2$As$_2$ with the scanning SQUID microscope, we demonstrate that some vortices carry temperature-dependent unquantized fractions of a flux quantum, which is universal at the same temperature over millimeter-scale range. These are different phenomena from the half-integer vortex at the intrinsic Josephson junctions.