### Book chapter

## Quantum fluctuations in low-dimensional superconductors

### In book

With rapid development of nanotechnology it became realistic to fabricate artificial nanostructures with dimensions in sub-50 nm scales. The physics of quasi-one-dimensional superconductors of corresponding dimensions is rather interesting [1]. The particular manifestation of size-dependent quantum fluctuations of superconducting order parameter - the quantum phase slip (QPS) – appeared capable to suppress such ‘text-book’ properties of superconductivity as zero resistivity [2] and persistent currents [3].

Here we demonstrate that one can build a superconducting analogue of a single-electron transistor (Cooper pair transistor) without any tunnel junctions. Instead a pair of thin superconducting wires in QPS regime - the quantum phase slip junctions (QPSJ) - can be used (Fig. 1). At sufficiently low temperatures, well below the critical temperature of the superconductor, the clear Coulomb blockade develops at the I-V characteristic of such a system [4,5]. Application of static gate potential efficiently modulates the amplitude of the Coulomb gap. The same device can be considered as the potential candidate for building a quantum standard of electric current [6].

We have employed noise thermometry for investigations of thermal relaxation between the electrons and the substrate in nanowires patterned from 40-nm-thick titanium film on top of silicon wafers covered by a native oxide. By controlling the electronic temperature Te by Joule heating at the base temperature of a dilution refrigerator, we probe the electron–phonon coupling and the thermal boundary resistance at temperatures Te = 0.5–3 K. Using a regular T 5-dependent electron–phonon coupling of clean metals and a T 4-dependent interfacial heat flow, we deduce a small contribution for the direct energy transfer from the titanium electrons to the substrate phonons due to inelastic electron-boundary scattering.

We consider the optical conductivity of a clean two-dimensional metal near a quantum spin-density-wave transition. Critical magnetic fluctuations are known to destroy fermionic coherence at “hot spots” of the Fermi surface but coherent quasiparticles survive in the rest of the Fermi surface. A large part of the Fermi surface is not really “cold” but rather “lukewarm” in a sense that coherent quasiparticles in that part survive but are strongly renormalized compared to the noninteracting case. We discuss the self-energy of lukewarm fermions and their contribution to the optical conductivity σ(), focusing specifically on scattering off composite bosons made of two critical magnetic fluctuations. Recent study [S. A. Hartnoll et al., Phys. Rev. B 84, 125115 (2011)] found that composite scattering gives the strongest contribution to the self-energy of lukewarm fermions and suggested that this may give rise to a non-Fermi-liquid behavior of the optical conductivity at the lowest frequencies. We show that the most singular term in the conductivity coming from self-energy insertions into the conductivity bubble σ(Omega) ∝ ln^3(Omega) /(Omega)^(1/3) is canceled out by the vertex-correction and Aslamazov-Larkin diagrams. However, the cancellation does not hold beyond logarithmic accuracy, and the remaining conductivity still diverges as 1/(Omega)^(1/3). We further argue that the 1/(Omega)^(1/3) behavior holds only at asymptotically low frequencies, well inside the frequency range affected by superconductivity. At larger Omega, up to frequencies above the Fermi energy, σ(Omega) scales as 1/(Omega), which is reminiscent of the behavior observed in the superconducting cuprates.

Continuing miniaturization of electronic devices, together with the quickly growing number of nanotechnological applications, demands a profound understanding of the underlying physics. Most of the fundamental problems of modern condensed matter physics involve various aspects of quantum transport and fluctuation phenomena at the nanoscale. In nanostructures, electrons are usually confined to a limited volume and interact with each other and lattice ions, simultaneously suffering multiple scattering events on impurities, barriers, surface imperfections, and other defects. Electron interaction with other degrees of freedom generally yields two major consequences, quantum dissipation and quantum decoherence. In other words, electrons can lose their energy and ability for quantum interference even at very low temperatures. These two different, but related, processes are at the heart of all quantum phenomena discussed in this book.This book presents copious details to facilitate the understanding of the basic physics behind a result and the learning to technically reproduce the result without delving into extra literature. The book subtly balances the description of theoretical methods and techniques and the display of the rich landscape of the physical phenomena that can be accessed by these methods. It is useful for a broad readership ranging from master's and PhD students to postdocs and senior researchers.

Book of abstracts

The topic of superconductivity in strongly disordered materials has attractedsignificant attention. These materials appear to be rather promising for fabrication of various nanoscale devices such as bolometers and transition edge sensors of electromagnetic radiation. The vividly debated subject of intrinsic spatial inhomogeneity responsible for thenon-Bardeen–Cooper–Schrieffer relation between the superconducting gap and the pairing potential is crucial both for understanding the fundamental issues of superconductivity in highly disordered superconductors, and for theoperation of corresponding nanoelectronic devices. Here we report an experimental study of theelectron transport properties of narrow NbN nanowires with effective cross sections of the order of the debated inhomogeneity scales. The temperature dependence of the critical current follows the textbook Ginzburg–Landau prediction for thequasi-one-dimensional superconducting channel Ic∼(1-T/Tc)3/2. We find that conventional models based on the thephase slip mechanism provide reasonable fits for the shape of R(T) transitions. Better agreement with R(T) data can be achieved assuming theexistence of short ‘weak links’ with slightly reduced local critical temperature Tc. Hence, one may conclude that an ‘exotic’ intrinsic electronic inhomogeneity either does not exist in our structures, or, if it doesexist, itdoes not affect their resistive state properties, or does not provide any specific impact distinguishablefrom conventional weak links.

Superconducting properties of metallic nanowires can be entirely different from those of bulk superconductors because of the dominating role played by thermal and quantum fluctuations of the order parameter. For superconducting channels with diameters below ∼ 50 nm fluctuations of the phase of the complex order parameter - the phase slippage - lead to non-zero resistance below the critical temperature. Fluctuations of the modulus of the complex order parameter broaden the gap edge of the quasiparticle energy spectrum and modify the density of states. In extreme case of very narrow channels imbedded in high-impedance environment (which fix the charge and, hence, enable strong fluctuations of the quantum-conjugated variable, the phase) the superconductor can be driven to insulating state – the Coulomb blockade. We review recent experimental activities in the field demonstrating rather unusual phenomena.