The goal of this paper is to review in brief the basic physics of single-election devices, as well as their-current and prospective applications. These devices based on the controllable transfer of single electrons between small conducting "islands", have already enabled several important scientific experiments. Several other applications of analog single-election devices in unique scientific instrumentation and metrology seem quite feasible. On the other hand, the prospect of silicon transistors being replaced by single-electron devices in integrated digital circuits faces tough challenges and remains uncertain. Nevertheless, even if this replacement does not happen, single electronics will continue to play an important role by shedding light on the fundamental size limitations of new electronic devices. Moreover, recent research in this field has generated some by-product ideas which may revolutionize random-access-memory and digital-data-storage technologies.

Single-electron devices and their applications

Within the past 20 years or so, there has occurred an explosion of interest in the magnetic behavior of pyrochlore oxides of the type $A_{2}^{3+}$$B_{2}^{4+}$O$_{7}$ where $A$ is a rare-earth ion and $B$ is usually a transition metal. Both the $A$ and $B$ sites form a network of corner-sharing tetrahedra which is the quintessential framework for a geometrically frustrated magnet. In these systems the natural tendency to form long range ordered ground states in accord with the Third Law is frustrated, resulting in some novel short range ordered alternatives such as spin glasses, spin ices and spin liquids and much new physics. This article attempts to review the myriad of properties found in pyrochlore oxides, mainly from a materials perspective, but with an appropriate theoretical context.

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Magnetic Pyrochlore Oxides

Magnetic flux can penetrate a type-II superconductor in the form of Abrikosov vortices (also called flux lines, flux tubes, or fluxons) each carrying a quantum of magnetic flux phi 0=h/2e. These tiny vortices of supercurrent tend to arrange themselves in a triangular flux-line lattice (FLL), which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-Tc superconductors (HTSCs) also by thermal fluctuations. Many properties of the FLL are well described by the phenomenological Ginzburg-Landau theory or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb alloys and HTSCs the FLL is very soft mainly because of the large magnetic penetration depth lambda . The shear modulus of the FLL is c66~1/ lambda 2, and the tilt modulus c44(k)~(1+k2 lambda 2)-1 is dispersive and becomes very small for short distortion wavelengths 2 pi /k<< lambda . This softness is enhanced further by the pronounced anisotropy and layered structure of HTSCs, which strongly increases the penetration depth for currents along the c axis of these (nearly uniaxial) crystals and may even cause a decoupling of two-dimensional vortex lattices in the Cu-O layers. Thermal fluctuations and softening may `melt` the FLL and cause thermally activated depinning of the flux lines or ofthe two-dimensional `pancake vortices` in the layers. Various phase transitions are predicted for the FLL in layered HTSCs. Although large pinning forces and high critical currents have been achieved, the small depinning energy so far prevents the application of HTSCs as conductors at high temperatures except in cases when the applied current and the surrounding magnetic field are small.

https://iopscience.iop.org/article/10.1088/0034-4885/58/11/003/pdf

The flux-line lattice in superconductors

One-dimensional nanostructures: Chemistry, physics & applications

Magnetic flux can penetrate a type-II superconductor in form of Abrikosov vortices. These tend to arrange in a triangular flux-line lattice (FLL) which is more or less perturbed by material inhomogeneities that pin the flux lines, and in high-$T_c$ supercon- ductors (HTSC's) also by thermal fluctuations. Many properties of the FLL are well described by the phenomenological Ginzburg-Landau theory or by the electromagnetic London theory, which treats the vortex core as a singularity. In Nb alloys and HTSC's the FLL is very soft mainly because of the large magnetic penetration depth: The shear modulus of the FLL is thus small and the tilt modulus is dispersive and becomes very small for short distortion wavelength. This softness of the FLL is enhanced further by the pronounced anisotropy and layered structure of HTSC's, which strongly increases the penetration depth for currents along the c-axis of these uniaxial crystals and may even cause a decoupling of two-dimensional vortex lattices in the Cu-O layers. Thermal fluctuations and softening may melt the FLL and cause thermally activated depinning of the flux lines or of the 2D pancake vortices in the layers. Various phase transitions are predicted for the FLL in layered HTSC's. The linear and nonlinear magnetic response of HTSC's gives rise to interesting effects which strongly depend on the geometry of the experiment.

The Flux-Line Lattice in Superconductors

Etude faite par un oscillateur mecanique au silicium a haut Q. La fusion du reseau de flux dans les deux supraconducteurs varie en fonction de l'orientation du champ magnetique

Evidence from mechanical measurements for flux-lattice melting in single crystals YBa2Cu3O7 and Bi2.2Sr2Ca0.8Cu2O8.

We report on current-voltage measurements in clean, untwinned ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ single crystals with picovolt voltage sensitivity and millikelvin temperature resolution in magnetic fields ranging up to 7 T. We find evidence for a melting transition in the vortex lattice which is hysteretic in both temperature and magnetic field. The measured thermal and magnetic hysteresis widths are related by the local slope of the phase boundary. This strongly supports the picture that, in the clean limit, the melting transition of the Abrikosov vortex lattice is a first-order phase transition.

Experimental evidence for a first-order vortex-lattice-melting transition in untwinned, single crystal YBa2Cu3O7.

The high-resolution Bitter pattern technique has been used to reveal the magnetic structure of single-crystal samples of high-${T}_{c}$ superconductor Y${\mathrm{Ba}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ at 42 K Typical patterns consist of hexagonally correlated, singly quantized vortices of flux $\frac{\mathrm{hc}}{2e}$ That is, the structures are comparable to those that would be observed in conventional type-II superconductors under similar conditions

Observation of Hexagonally Correlated Flux Quanta In YBa2Cu3O7

tion of unperturbed OP and of its behavior in the vicinity of the transition. We have used a Corbino disk geometry (inset of Fig. 1), in which the vortices circulate in the bulk without crossing the edges, and compared it with the regular strip configuration in the same crystals. If the instability effects are a bulk phenomenon, no significant difference between the two geometries is expected; whereas if the sample edges do play a dominant role, qualitatively different behavior should be observed. This paper demonstrates that the two geometries yield a strikingly different response and, significantly, the instability effects are eliminated in the Corbino geometry. Furthermore, by preventing the edge contamination, the true bulk dynamics in the vicinity of the transition can be probed for the first time. The results suggest that the disorder-driven transition TDT is possibly a thermodynamic first-order phase transition, which shows, in addition, sharp reentrant behavior.

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Instabilities and disorder-driven first-order transition of the vortex lattice

We report the observation of a novel flux-lattice structure, a commensurate array of flux-line chains. Our experiments consist of the magnetic decoration of the flux lattices in single crystals of Ba-Sr-Ca-Cu-O where the magnetic field is applied at an angle with respect to the conducting planes. For a narrow range of angles, the equilibrium structure is one with uniformly spaced chains with a higher line density of vortices than the background lattice. Our observations are in qualitative agreement with theories which suggest that, in strongly anisotropic materials the vortices develop an attractive interaction in tilted magnetic fields.

P. L. Gammel

Papers

Evidence from mechanical measurements for flux-lattice melting in single crystals YBa2Cu3O7 and Bi2.2Sr2Ca0.8Cu2O8.

Experimental evidence for a first-order vortex-lattice-melting transition in untwinned, single crystal YBa2Cu3O7.

Observation of Hexagonally Correlated Flux Quanta In YBa2Cu3O7

Instabilities and disorder-driven first-order transition of the vortex lattice

Observation of a commensurate array of flux chains in tilted flux lattices in Bi-Sr-Ca-Cu-O single crystals.