Consequences of Confinement on Vortex Ordering and Dynamics in Superconducting Nanostripes

The scientific field of superconductivity has presented both theoretical and experimental physicists with unexpected findings since its initial discovery. Currently, nanoscale super- conductivity is a rapidly developing research area, as scientists endeavor to gain insight into how fluctuations at this scale significantly impact the behavior of superconductors. Super- conducting nanostripes (SNs) are a critical component in superconducting electronics, with widespread applications in quantum technology. For example, superconducting nanostripe single-photon detectors (SNSPDs) have been proposed for use in quantum communication, as well as in fields such as astronomy and spectroscopy [1, 2, 3, 4]. These developments have the potential to revolutionize technology and our understanding of the natural world. Super- conducting nanostripes are particularly suitable for the detection of individual photons due to their ability to tune the applied biased current in a manner that maximizes their intrinsic detection efficiency [5]. Using the same detection mechanism, SNSPDs can also serve as a bolometer, capable of detecting particle collisions [6, 7]. In addition to single-photon detec- tors, other examples of superconducting electronics include prototype logic devices [8, 9, 10]. Research has shown that the strength of interactions between vortices and strip edges can be controlled by changing the magnetization orientation in the strips using a small in-plane magnetic field. As a result of the controlled acceleration or deceleration of superconducting vortices, strong tunable anisotropy is introduced into the vortex dynamics, which is highly favorable for the development of logic devices [8]. Reference [10] discusses a study of a super- conducting H-bar nanostripe, which revealed that a single row of vortices was the optimal configuration for generating a giant non-local electrical resistance. In this configuration, the vortices were able to move several microns away from the local current drive without be- ing altered by the passing current, enabling feasible long-range information transfer by the vortices. Another important addition to the family of superconducting electronics is the flux qubit [11, 12, 13], which is a crucial component in the development of quantum computers. Flux qubits can be magnetically coupled, creating a system that is analogous to an Ising spin model [14, 15]. The magnetic coupling of flux qubits is particularly useful because optimiza- tion problems can often be transformed into a quadratic unconstrained binary optimization (QUBO) [16, 17]. By allowing the system to relax into the ground state via quantum an- nealing [18, 19], flux qubits can be utilized to solve a wide range of optimization problems. Superconducting diodes have garnered significant interest for their ability to transport cur- rent in a single direction with minimal energy loss, making them highly attractive for numerous applications. The first instance of a magnetically tunable superconducting diode effect was demonstrated in an artificial superlattice [20], which further highlights the potential of superconducting electronics for a wide range of practical applications. The discovery of the magnetically tunable superconducting diode effect in artificial superlattices has generated significant interest in the field, leading to recent studies on the intrinsic mechanisms under- lying this effect [21]. Interestingly, studies have also shown that superconducting vortices in fluxon pumps and lenses exhibit diode-like behavior when subjected to alternating current [22, 23]. Specifically, vortices are pumped out of a restricted region of the superconductor, and when AC is applied, the vortices become jammed, leading to the observed diode-like behavior. Narrow superconducting nanostripes have been shown to exhibit an enhancement in critical parameters. For instance, in an artificial heterostructure consisting of such stripes, the critical temperature was observed to increase from 23 K to 150 K [24]. Furthermore, when the dimensions of stripes are reduced from 3D to 2D, an amplification in the critical current density can occur, as observed for niobium film when the thickness is reduced [25]. The strong confinement forces present at the edges of narrow SNs can significantly impact the behavior of the condensate, particularly that of the vortex lattice due to its topology [26]. The edge barrier [27] creates a strong vortex confinement [28], resulting in effects such as the enhancement of critical current density, as observed experimentally for a NbN nanostripe. Confinement in narrow SNs can also result in significant magnetoresistance oscillations under specific experimental conditions. For instance, time-averaged voltage/resistance as a function of the applied magnetic field displays distinct peaks during transitions between static and dynamic vortex phases [29, 30, 31]. At higher applied fields with multiple rows of vortices or high currents, continuous motion of vortices leads to a monotonic background, on top of which resistance oscillations due to entries of additional vortices are superimposed [29, 32]. Observations of commensurate effects between the width of a superconductor and the number of rows of vortices have revealed a corresponding relationship with the critical cur- rent as a function of the magnetic field perpendicular to the plane (with a fixed SN width w), as well as a relationship with the critical current as a function of width (with a fixed magnetic field H), representing a manifestation of confinement effects [33, 34]. Over the years, research has explored the criteria that govern the entry of vortices from the edge of a thin superconductor, with these studies addressing this subject [35, 36, 37, 38, 39]. While it was initially proposed that the current density at the edge of the superconductor exceeding the depairing current (JDP ) was the driving mechanism [40], subsequent observations have shown that the edge current actually surpasses JDP , thereby challenging the previous hy- pothesis. According to Vodolazov’s research [39], the criteria governing vortex entry into a superconductor at its edge is a threshold value of the supervelocity (kinematic momentum) at the edge. Electromagnetic resonators represent a compelling use case for utilizing superconducting nanostripes or films, as demonstrated by several studies [41, 42, 43]. Under the influence of a sufficiently strong transport current, the crossing of vortices through a superconductor- normal metal system results in a dynamic electromagnetic field around the stripe. The highest intensity of the induced magnetic field is concentrated at the edges of the stripe. As a vortex crosses this field, the interaction between the magnetic fields triggers the emission of photons [42]. A solitary vortex crossing through the narrow structure results in the emission of a single photon that necessitates the use of superconducting nanowire single- photon detectors for detection. On the other hand, when a synchronized row of vortices passes through the structure, a much stronger signal is produced, which can be detected experimentally [43]. In the context of narrow SNs, rows of vortices can cross the stripe either asynchronously or synchronously, as reported in several studies [44, 29, 45]. The nature of the crossing is dependent on the competing forces such as confinement, vortex-vortex interaction, and Lorentzian forces, which together dictate the dynamics of the vortex motion. However, de- spite the extensive research in this area, the criteria governing synchronous crossings remain poorly understood. To gain a comprehensive understanding of the behavior of vortices in superconducting nanowires with small widths, it is important to investigate the favorable geometry of vortex configurations under static conditions (i.e., in the absence of an applied current) and their relationship to the dynamic case (i.e., when an applied current is present). Understanding how a vortex lattice is affected by the interaction with the edge confining force and other dynamic forces is crucial when considering SNs for applications mentioned above, especially EM radiation emitters. Investigating the dynamic dissipative states under strong confinement in the 1D-2D crossover regime can provide insights into the behavior of vortices as they cross superconducting nanostripes under various conditions, such as the width of the stripe, the magnetic field, and the intensity of the applied current. This information is critical to both the fast information transfer and the frequency of emitted radiation by moving vortices. Such studies can reveal how vortices move across the stripe under different conditions and provide additional information on the possible vortex velocity under con- finement [46, 47]. Therefore, it is crucial to scrutinize how confinement in superconducting nanostripes affects the vortex configurations and dynamics. In Chapter I, a chronological overview of superconductivity will be presented, including summaries of the theoretical descriptions for superconductivity, such as the London model, Ginzburg-Landau theory, and BCS theory. The chapter will also cover different types of superconducting materials and provide an overview of their applications in electronics and other fields. In Chapter II, the generalized time-dependent Ginzburg-Landau (gTDGL) framework will be described, including its uses and validity in nanoscale superconductors. This theory serves as the basis for the results obtained in this thesis and remains an essential tool for the study of superconductors today. In Chapter III, the confinement of vortices in thin, narrow superconducting nanostripes under static equilibrium conditions will be discussed, using the static case of the generalized time-dependent Ginzburg-Landau equations. The chapter will cover the basics of vortices, including an overview of the criteria for vortex entry, with a focus on the supervelocity. The organization of vortex rows will also be examined, and a vortex row phase diagram as a function of the applied magnetic field, H, for a given width w of the nanostripe will be presented. The critical supervelocity at the edge of the stripe associated with vortex entry will be investigated and shown to be larger for narrower stripes, which corresponds to values found in [39] for wider superconducting nanostripes. The magnetic field dependence of the average number of present vortices will also be analyzed to reveal strong confinement effects. Finally, the reconfiguration from the vortex rows to the vortex lattice with increasing width will be discussed, offering a criterion to define the quasi 1D-to-2D transition, where the nanostripe effectively becomes a nanofilm. Chapter IV delves into the effects of confinement in the dynamic state. Simulations of voltage-current (V-J) characteristics in SNs demonstrate transitions among different resistive regimes, including Meissner, flux-flow, flux-flow instability, and normal regimes. However, such V-J curves exhibiting similar features to simulations have only been experimentally measured for wider structures, as reported in [48, 49]. The commensurate behavior of the critical current with changes in the magnetic field is also explored. It is shown that the local minima values in Jc1(H) (the onset of dissipation) are related to the row transitions displayed in the vortex-row phase diagram. Additionally, the amplification of critical current values through the pinning of vortices is discussed. For stripes with an average vortex area of less than 80ξ 2 (ξ being the coherence length), vortices cross the SN periodically and continuously under the flux-flow regime, causing modulations of the voltage drop that can be detected experimentally. This process results in the emission of electromagnetic radiation [42], and the power spectra of this radiation can be obtained by performing a fast Fourier transformation of the voltage drop as a function of time [50]. The average vortex velocity obtained from these measurements can be used to discuss the washboard frequencies [43, 42], which can reach up to tens of kilometers per second for thin niobium SNs [49]. Assuming a sufficient vortex density, the transitions of vortex row crossings can vary between quasi- synchronous, synchronous, and asynchronous regimes, which is determined by the interplay of forces acting on the vortices. Synchronised crossings are particularly desirable for small- band electromagnetic emitters in the GHz or THz range. For typical thin niobium SNs [25], modulation frequencies range in the microwave regime, between 1-100GHz. The chapter also covers the dynamics of vortex crossings in viscous condensates, where the rearrangement of the vortex lattice occurs in the presence of defects and high viscosity. At low viscosity, vortex channels can facilitate synchronised vortex crossings, but in the presence of defects that lead to highly viscous condensates, strong distortions can occur in the vortex channels. Chapter V focuses on the response of superconductors to thermal fluctuations. It includes a brief discussion on how the dynamic behavior can be affected by changes in the material’s thermal properties. The chapter then demonstrates how heating can drive the superconduc- tor into the normal state through the creation of a hot belt. Hot superconducting bands can be useful for controlling synchronous vortex crossings, but it is important for the bands to remain narrow in relation to characteristic lengths, such as the coherence length. The mechanism of photon detection in single photon detectors is discussed, including simulations performed on realistic detectors coupled to a shunt resistor. The response of a detector is then examined in terms of delay time, timing jitter, and current crowding caused by the me- andering geometry of the typical detector. The intrinsic detection efficiency is summarized as the probability of a detection event on the main component, the superconducting trans- port current-carrying meandering nanostripe. Finally, the thesis concludes with a summary of the main points.