Ultrasmall Metallic Grains Research Articles

The specific heat of superconducting metallic grains in magnetic field

More and more studies indicate that the effects of quantum size and energy level statistics play a crucial role in the thermodynamic properties of ultrasmall metallic grains. This paper aims to investigate how they affect the specific heat of ultrasmall metallic grains in magnetic field. As the particle size decreases, fluctuation effects and the impact of energy level separation are becoming more and more important. The method of static path approximation (SPA) is adopted to handle the fluctuation effect. Random matrix theory (RMT) is adopted due to its successful description of the energy level of metal nanoparticles. The normalized specific heat of several typical temperatures and electron spins were taken in the calculation, and the results were analyzed. It was found that spin and the spin-orbit coupling affect the specific heat very obviously, and the suppressed high spin weakens the contribution of electrons to the heat capacity.

Statistical odd-even staggering in few fermions systems

The odd-even staggering (OES) in nuclear binding energies is a well-known fact. A rather similar effect can be found in other finite fermion systems. For instance, ultra small metallic grains and metal clusters. The staggering in nuclei and grains is attributed primarily to pairing correlations. In clusters, it is originated by the Jahn–Teller effect [Phys. Rev. Lett. 81, 3599 (1998)]. Here, we work with a simple, Lipkin-like, exactly solvable two-level fermion model. A statistical mechanics’ treatment of it shows that OES effects also emerge here, as revealed by theoretical tools connected with the so-called statistical complexity.

Spectral Explanation for Statistical Odd-Even Staggering in Few Fermions Systems

Odd-even statistical staggering in a Lipkin-like few fermions model has been recently encountered. Of course, staggering in nuclear binding energies is a well established fact. Similar effects are detected in other finite fermion systems as well, as for example, ultra small metallic grains and metal clusters. We work in this effort with the above-mentioned Lipkin-like, two-level fermion model and show that statistical staggering effects can be detailedly explained by recourse to a straightforward analysis of the associated energy-spectra.

Influence of Disorder on Superconducting Correlations in Nanoparticles

We investigate how the interplay of quantum confinement and level broadening caused by disorder affects superconducting correlations in ultra-small metallic grains. We use the electron-phonon interaction-induced electron mass renormalization and the reduced static-path approximation of the BCS formalism to calculate the critical temperature as a function of the grain size. We show how the strong electron-impurity scattering additionally smears the peak structure in the electronic density of states of a metallic grain and imposes additional limits on the critical temperature under strong quantum confinement.

Phonon limited superconducting correlations in metallic nanograins.

Conventional superconductivity is inevitably suppressed in ultra-small metallic grains for characteristic sizes smaller than the Anderson limit. Experiments have shown that above the Anderson limit the critical temperature may be either enhanced or reduced when decreasing the particle size, depending on the superconducting material. In addition, there is experimental evidence that whether an enhancement or a reduction is found depends on the strength of the electron-phonon interaction in the bulk. We reveal how the strength of the e-ph interaction interplays with the quantum-size effect and theoretically obtain the critical temperature of the superconducting nanograins in excellent agreement with experimental data. We demonstrate that strong e-ph scattering smears the peak structure in the electronic density-of-states of a metallic grain and enhances the electron mass, and thereby limits the highest Tc achievable by quantum confinement.

Magnetic response of energy levels of superconducting nanoparticles with spin-orbit scattering

Discrete energy levels of ultrasmall metallic grains are extracted in single-electron-tunneling-spectroscopy experiments. We study the response of these energy levels to an external magnetic field in the presence of both spin-orbit scattering and pairing correlations. In particular, we investigate $g$-factors and level curvatures that parametrize, respectively, the linear and quadratic terms in the magnetic-field dependence of the many-particle energy levels of the grain. Both of these quantities exhibit level-to-level fluctuations in the presence of spin-orbit scattering. We show that the distribution of $g$-factors is not affected by the pairing interaction and that the distribution of level curvatures is sensitive to pairing correlations even in the smallest grains in which the pairing gap is smaller than the mean single-particle level spacing. We propose the level curvature in a magnetic field as a tool to probe pairing correlations in tunneling spectroscopy experiments.

Relative phase and Josephson dynamics between weakly coupled Richardson models

We consider two weakly coupled Richardson models to study the formation of a relative phase and the Josephson dynamics between two mesoscopic attractively interacting fermionic systems. Our results apply to superconducting properties of coupled ultrasmall metallic grains and to Cooper-pairing superfluidity in neutral systems with a finite number of fermions. We discuss how a definite relative phase between the two systems emerges and how it can be conveniently extracted from the many-body wave function, finding that a definite relative phase difference emerges even for very small numbers of pairs ($\ensuremath{\sim}$10). The Josephson dynamics and the current-phase characteristics are then investigated, showing that the critical current has a maximum at the BCS-BEC crossover. For the considered initial conditions a two-state model gives a good description of the dynamics and of the current-phase characteristics.

Thermodynamics of ultrasmall metallic grains in the presence of pairing and exchange correlations: Mesoscopic fluctuations

We study the mesoscopic fluctuations of thermodynamic observables in a nanosized metallic grain in which the single-particle dynamics are chaotic and the dimensionless Thouless conductance is large. Such a grain is modeled by the universal Hamiltonian describing the competition between exchange and pairing correlations. The exchange term is taken into account exactly by a spin-projection method, and the pairing term is treated in the static-path approximation together with small-amplitude quantal fluctuations around each static fluctuation of the pairing field. Odd-even particle-number effects induced by pairing correlations are included using a number-parity projection. We find that the exchange interaction shifts the number-parity effects in the heat capacity and spin susceptibility to lower temperatures. In the regime where the pairing gap is similar to or smaller than the single-particle mean level spacing, these number-parity effects are suppressed by exchange correlations, and the fluctuations of the spin susceptibility may be particularly large. However, for larger values of the pairing gap, the number-parity effects may be enhanced by exchange correlations.

Ultra-small metallic grains: effect of statistical fluctuations of the chemical potential on superconducting correlations and vice versa

Superconducting correlations in an isolated metallic grain are governed by the interplay between two energy scales: the mean level spacing δ and the bulk pairing gap Δ0, which are strongly influenced by the position of the chemical potential with respect to the closest single-electron level. In turn superconducting correlations affect the position of the chemical potential. Within the parity projected BCS model we investigate the probability distribution of the chemical potential in a superconducting grain with randomly distributed single-electron levels. Taking into account statistical fluctuations of the chemical potential due to the pairing interaction, we find that such fluctuations have a significant impact on the critical level spacing δc at which the superconducting correlations cease: the critical ratio δc/Δ0 at which superconductivity disappears is found to be increased.

Parity-fluctuation induced enlargement of the ratioΔE/kBTcin metallic grains

We investigate how the interplay of quantum confinement and particle number-parity fluctuations affects superconducting correlations in ultra-small metallic grains. Using the number-parity projected BCS formalism we calculate the critical temperature and the excitation gap as a function of the grain size for grains with even and odd number of confined carriers. We show that the experimentally observed anomalous increase of the coupling ratio ${\ensuremath{\Delta}}_{E}/{k}_{B}{T}_{c}$ with decreasing superconducting grain size can be attributed to an enhancement of the number-parity fluctuations in ultra-small grains.

Mesoscopic Competition of Superconductivity and Ferromagnetism: Conductance Peak Statistics for Metallic Grains

We investigate the competition between superconductivity and ferromagnetism in chaotic ultrasmall metallic grains in a regime where both phases can coexist. We use an effective Hamiltonian that combines a BCS-like pairing term and a ferromagnetic Stoner-like spin exchange term. We study the transport properties of the grain in the Coulomb-blockade regime and identify signatures of the coexistence of pairing and exchange correlations in the mesoscopic fluctuations of the conductance peak spacings and peak heights.

Thermal Signatures of Pairing Correlations in Nuclei and Nanoparticles

The Bardeen-Cooper-Schrieffer (BCS) mean-field theory of the pairing interaction breaks down for nuclei and ultra-small metallic grains (nanoparticles). Finite-temperature pairing correlations in such finite-size systems can be calculated beyond the BCS theory in an auxiliary-field Monte Carlo approach. We identify thermal signatures of pairing correlations in both nuclei and nanoparticles that depend on the particle-number parity.

Polynomial-time simulation of pairing models on a quantum computer.

We propose a polynomial-time algorithm for simulation of the class of pairing Hamiltonians, e.g., the BCS Hamiltonian, on an NMR quantum computer. The algorithm adiabatically finds the low-lying spectrum in the vicinity of the gap between the ground and the first excited states and provides a test of the applicability of the BCS Hamiltonian to mesoscopic superconducting systems, such as ultrasmall metallic grains.

Mesoscopic fluctuations in superconducting dots at finite temperatures

We study the thermodynamics of ultrasmall metallic grains with the mean level spacing comparable or larger than the pairing correlation energy in the whole range of temperatures. A complete picture of the thermodynamics in such systems is given taking into account the effects of disorder, parity and classical and quantum fluctuations. Both spin susceptibility and specific heat turn out to be sensitive probes to detect superconducting correlations in such samples.

Superconducting Correlations in Small Metallic Grains**The project partially supported by National Natural Science Foundation of China under Grant No. 19874004

By applying the path-integral formulation and the Feynman theorem, we calculate the off-diagonal superconducting correlation in an ultrasmall metallic grain. Unlike its behavior in the bulk limit, we find that this correlation function is an intensive quantity in this case. Therefore, superconductivity is indeed completely suppressed by quantum fluctuations as the diameter of the grain shrinks to the nanometer scale. This conclusion confirms the previous results by numerical calculations. Furthermore, it also imposes a strong constraint on the delocalizing amplitude of the pair-mixing function, which was recently proposed to characterize superconductivity in the canonical ensemble.

Pairing Correlation Effect on the Zero-Bias Tunneling Spectra of Ultrasmall Superconducting Grains

We study the superconducting pairing correlation (SPC) effect on the resonant tunneling through an ultrasmall metallic grain with a gate electrode. The resonant conductance peaks appearing with an increase or decrease of the gate voltage are accompanied by the transition of the total electron number N on the grain, such as N ↔ N ±1. We focus our attention on the tunneling resonance associated with an odd- N state. In the absence of the SPC effect, the structure of the zero-bias resonant conductance peak for N ↔ N - 1 is equivalent to that for N ↔ N + 1. It is shown that due to the interplay between the SPC effect and the nonuniform spacings of single-particle energy levels, we expect a notable difference between the two peak structures.

Superconductivity in ultrasmall metallic grains

AbstractWe review recent experimental and theoretical work on superconductivity in ultrasmall metallic grains, i.e. grains sufficiently small that the conduction electron energy spectrum becomes discrete. The discrete excitation spectrum of an individual grain can be measured by the technique of single‐electron tunneling spectroscopy, and reveals parity effects indicative of pairing correlations in the grain. After introducing the discrete BCS model that has been used to model such grains, we review a phenomenological, grand‐canonical, variational BCS theory describing the paramagnetic breakdown of these pairing correlations with increasing magnetic field. We also review recent canonical theories that have been developed to describe how pairing correlations change during the crossover, with decreasing grain size, from the bulk limit to the limit of few electrons, and compare their results to those obtained using Richardson's exact solution of the discrete BCS model.

Pair-mixing superconducting correlation in ultrasmall metallic grains

The superconducting order in an ultrasmall metallic grain with a fixed number of electrons is analyzed. By exploiting the intrinsic symmetries of the BCS Hamiltonian, we establish some general properties of a recently proposed pair-mixing correlation function on a mathematically rigorous basis. We also prove a relation between this function and the conventional off-diagonal correlation function. By variationally estimating both correlation functions, we show that the superconducting crossover in ultrasmall metallic grains is smooth, a conclusion reached previously by numerical calculations.

Crossover from bulk to few-electron limit in ultrasmall metallic grains

We study the properties of ultrasmall metallic grains with sizes in the range 20--400 electrons. Using a particle-hole version of the density-matrix renormalization-group (DMRG) method we compute condensation energies, spectroscopic gaps, pairing parameters, and particle-hole probabilities of the ground-state wave function. The results presented in this paper confirm that the bulk superconducting regime (large grains) and the fluctuation dominated regime (small grains) are qualitatively different, but show that the crossover between them is very smooth with no signs of critical level spacings separating them. We compare our DMRG results with the exact ones obtained with the Richardson solution finding complete agreement. We also propose a simplified version of the DMRG wave function, called the particle-hole BCS Ansatz, which agrees qualitatively with the DMRG solution and illustrates what is lacking in the projected BCS (PBCS) wave function in order to describe correctly the crossover. Finally we present a recursive method to compute norms and expectation values with the PBCS wave function.

Models exhibiting order by projection

We present calculations on a simple spin model which shows the phenomenon of order by projection. We discuss a two-level atomic system coupled to the radiation field to explore the possible relevance of order by projection in quantum optics. We point out the similarity between the effective Hamiltonian for the two-level atoms and the reduced BCS Hamiltonian used in the study of the ultrasmall metallic grains, and we present some calculations to show the transition from a nonsuperconducting to a superconducting ground state in the grains. We present a sum rule for the interacting electron systems on a lattice, exhibiting order by projection. This relates a nonextensive change in the ``kinetic energy'' due to a projection term to the extensive expectation value of the interaction in the ground state without projection.