Vortex dynamics studied by nonreciprocal terahertz responses
*Sachiko Nakamura1,2

Nonreciprocal charge transport, where the electrical conductivity depends on the direction of the current, is a great tool to detect intrinsic symmetry breaking. With an AC input, the nonreciprocal transport can be detected with high sensitivity as rectification or second harmonic generation (SHG). Because it becomes prominent when both space-inversion (P) and time-reversal (T) symmetries are broken, the nonreciprocal transport has been studied on noncentrosymmetric materials whose structure breaks P symmetry, in a magnetic field or with spontaneous magnetization which break T symmetry, and has illuminated various concepts related to applications of topological materials, e.g. Berry phase and toroidal moment [1].

The nonreciprocal transport can also be induced in centrosymmetric materials as long as both P and T symmetries are broken. In this talk, I will present our recent experiments on a film of an s-wave centrosymmetric superconductor NbN, where P and T symmetries are simultaneously broken by DC supercurrent injection [2]. Although most currents (or flows) break those symmetries, supercurrent is advantageous because it does not induce additional breaking of P symmetry by an electric field, or macroscopic irreversibility due to the dissipation which is a sort of T symmetry breaking. Under the supercurrent injection, giant nonreciprocal SHG is observed in the terahertz (THz) frequency range.

THz electromagnetic waves are in between microwaves and infrared light. Its photon energy is a few meV as much as the superconducting gap energy 2Δ of the NbN film. Because the frequency is high enough, the conductivity does not diverge below the superconducting critical temperature Tc, which makes it possible to observe the second-harmonic (SH) electric field below Tc. The electric field of the high-intensity THz pulses used in the experiments is a few kV/cm and induces a large supercurrent about 1 MA/cm2 which is comparable to the critical current density of the film (≈3 MA/cm2).

The microscopic origin of the nonreciprocal THz SHG is found to be magnetic flux quanta (=vortices) pinned in the type-II superconductor NbN. The magnetic flux originates from the geomagnetic field (≈0.5 Oe). A vortex in an anharmonic pinning potential tilted by the DC supercurrent is oscillated by a sinusoidal supercurrent induced by the THz electric field, and radiates an electromagnetic wave proportional to its non-sinusoidal velocity including the SH component. Although the directions of movement are orthogonal, the equation of motion (EOM) of the vortex is quite similar to that of an electron in a non-symmetric crystal oscillated by an electric field of light which is generally used to explain optical SHG. The EOM of the vortex not only explains the SHG but also suggests a resonant behavior at a frequency determined by the curvature at the bottom of the effective pinning potential, which explains the resonance peak observed in the temperature dependence of the SH intensity.

The resonance frequency can be written as a function of the DC supercurrent density and the vortex mass. The vortex mass determined from the supercurrent density dependence of the resonant frequency is about the electron mass, which is consistent with the vortex core mass also known as Suhl’s mass [3] estimated for the 25-nm-thick film of dirty-limit superconductor. This suggests that we can observe the motion of the vortex cores only accompanied by superconducting vortex current. Although the vortex mass has little influence on low-frequency or small-amplitude electromagnetic responses of vortices, e.g., critical current, flux-flow resistivity, and depinning frequency, there are a few experiments to determine the vortex mass using nanosecond dynamics where the vortex mass is 1,000 to 10,000 times larger than the core mass possibly due to tangling of vortex lines and/or normal components around the core. Considering that, observation of the tiny vortex core of a few nanometers becomes successful in our experiments by studying faster picosecond dynamics.

Electromagnetic response of vortices also appear at the fundamental frequency (FH) directly and indirectly: the former is very small and the latter is depletion due to the SHG. However, the FH response is difficult to distinguish from other responses of the Higgs (amplitude) mode of the superconducting order parameter and quasiparticle excitations emerging at the same frequency. By contrast, the SHG can be extracted as a difference between the THz waveforms with positive and negative DC supercurrents as it has nonreciprocal properties. The DC supercurrent convert the anharmonicity of the pinning potential to the nonreciprocity in this work, but it can also convert other nonlinearities in superconductors, e.g., nonlinear viscosity or vortices and nonlinear light-Higgs coupling. The demonstrated scheme of current-induced nonreciprocal response will be a new probe to study various nonlinear phenomena in superconductors.

This work was done in collaboration with Ryo Shimano, Kota Katsumi (Univ. Tokyo), and Hirotaka Terai (NICT).

[1] Y. Tokura and N. Nagaosa, Nat. Commun. 9, 3740 (2018).
[2] S. Nakamura, K. Katsumi, H. Terai, and R. Shimano, Phys. Rev. Lett. 125, 097004 (2020).
[3] H. Suhl, Phys. Rev. Lett. 14, 226 (1965).

Keywords: nonreciprocal transport, vortex, terahertz, type-II superconductor