The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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Statistical mechanics of complex networks

We provide a broad review of fundamental electronic properties of two-dimensional graphene with the emphasis on density and temperature dependent carrier transport in doped or gated graphene structures. A salient feature of our review is a critical comparison between carrier transport in graphene and in two-dimensional semiconductor systems (e.g. heterostructures, quantum wells, inversion layers) so that the unique features of graphene electronic properties arising from its gap- less, massless, chiral Dirac spectrum are highlighted. Experiment and theory as well as quantum and semi-classical transport are discussed in a synergistic manner in order to provide a unified and comprehensive perspective. Although the emphasis of the review is on those aspects of graphene transport where reasonable consensus exists in the literature, open questions are discussed as well. Various physical mechanisms controlling transport are described in depth including long- range charged impurity scattering, screening, short-range defect scattering, phonon scattering, many-body effects, Klein tunneling, minimum conductivity at the Dirac point, electron-hole puddle formation, p-n junctions, localization, percolation, quantum-classical crossover, midgap states, quantum Hall effects, and other phenomena.

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Electronic transport in two-dimensional graphene

A revolution in electronics is in view, with the contemporary evolution of the two novel disciplines of spintronics and molecular electronics. A fundamental link between these two fields can be established using molecular magnetic materials and, in particular, single-molecule magnets. Here, we review the first progress in the resulting field, molecular spintronics, which will enable the manipulation of spin and charges in electronic devices containing one or more molecules. We discuss the advantages over more conventional materials, and the potential applications in information storage and processing. We also outline current challenges in the field, and propose convenient schemes to overcome them.

Molecular spintronics using single-molecule magnets

Shor and Grover demonstrated that a quantum computer can outperform any classical computer in factoring numbers1 and in searching a database2 by exploiting the parallelism of quantum mechanics. Whereas Shor's algorithm requires both superposition and entanglement of a many-particle system3, the superposition of single-particle quantum states is sufficient for Grover's algorithm4. Recently, the latter has been successfully implemented5 using Rydberg atoms. Here we propose an implementation of Grover's algorithm that uses molecular magnets6,7,8,9,10, which are solid-state systems with a large spin; their spin eigenstates make them natural candidates for single-particle systems. We show theoretically that molecular magnets can be used to build dense and efficient memory devices based on the Grover algorithm. In particular, one single crystal can serve as a storage unit of a dynamic random access memory device. Fast electron spin resonance pulses can be used to decode and read out stored numbers of up to 105, with access times as short as 10-10 seconds. We show that our proposal should be feasible using the molecular magnets Fe8 and Mn12.

Quantum computing in molecular magnets

Double-decker phthalocyanine complexes with Tb3+ or Dy3+ showed slow magnetization relaxation as a single-molecular property. The temperature ranges in which the behavior was observed were far higher than that of the transition-metal-cluster single-molecule magnets (SMMs). The significant temperature rise results from a mechanism in the relaxation process different from that in the transition-metal-cluster SMMs. The effective energy barrier for reversal of the magnetic moment is determined by the ligand field around a lanthanide ion, which gives the lowest degenerate substate a large |Jz| value and large energy separations from the rest of the substates in the ground-state multiplets.

Lanthanide Double-Decker Complexes Functioning as Magnets at the Single-Molecular Level

We present a systematic experimental investigation of the photoluminescence of heavily doped, compensated Si:P,B. The spectra broaden as the doping levels are increased, as expected. The spectral weights of the no-phonon peak and the phonon-assisted transitions vary with compensation and with excitation intensity. In contrast with the simple superposition of high-level and low-level spectra found for uncompensated material, increasing the input power for heavily doped, compensated Si produces a strong, continuous shift in energy of the entire spectrum in an essentially rigid way. A logarithmic power dependence found for the two most heavily doped samples is consistent with an exponential band tail. Deviations from this behavior are found for two samples with lower concentration and compensation. We suggest that these deviations may signal the onset of a minimum in the density of states that occurs when the acceptor (donor) band begins to separate from the host valence (conduction) band as the dopant concentration is reduced.

Photoluminescence of heavily doped, compensated Si:P,B.

For donor concentrations on both sides of the metal-insulator transition, the paramagnetic component of the susceptibility of Si:P at low temperature is described by a power law approx.T/sup -//sup ..cap alpha../, modified by a term which represents thermal activation to higher-energy states which are magnetically inert. For insulating samples, the power law is associated with the random Heisenberg antiferromagnet. The smooth continuation of the same behavior onto the metallic side of the transition suggests that local moments exist in metallic Si:P. We further suggest that this may imply that the exchange interaction between the magnetic moments does not undergo any abrupt or major change in character as the transition is crossed, in spite of the appearance of delocalized electrons.

Susceptibility of Si:P across the metal-insulator transition. II. Evidence for local moments in the metallic phase.

The van der Pauw geometry has been widely used for the measurement of resistivities and Hall coefficients. Although the measurement of a Hall coefficient requires a finite magnetic field, it should be noted that van der Pauw’s expression is valid only in the limit of zero field; in addition to the Hall contribution, measurements in a finite magnetic field generally include a term associated with field‐induced changes in the longitudinal resistivity. Although a simple solution to this problem entails taking the difference between readings in opposite field directions, there are circumstances where this may be impractical. In this note we present a straightforward extension of the van der Pauw calculation which allows a determination of the Hall coefficient from quantities measured in one field direction only.

Measurement of the Hall coefficient using van der Pauw method without magnetic field reversal

We present the results of a detailed study of the thermally-assisted-resonant-tunneling relaxation rate of Mn12 acetate as a function of an external, longitudinal magnetic field and find that the data can be fit extremely well to a Lorentzian function. No hint of inhomogeneous broadening is found, even though some is expected from the Mn nuclear hyperfine interaction. This inconsistency implies that the tunneling mechanism cannot be described simply in terms of a random hyperfine field.

http://arxiv.org/pdf/cond-mat/9807411.pdf

Resonant magnetization tunneling in Mn 12 acetate: The absence of inhomogeneous hyperfine broadening

The electrical resistivity of the shallow double-donor system Si:P,Bi, prepared by ion implantation, was investigated in the temperature range from 1.7 to 300 K. Good agreement was obtained between the measured resistivities and resistivities calculated by a generalized Drude approach for the same temperatures and dopant concentrations. The critical impurity concentration for the metal-nonmetal transition for the double-doped Si:P,Bi system was found to lie between the critical concentrations of the two single-doped systems, Si:P and Si:Bi. @S0163-1829~99!11747-8#

Myriam P. Sarachik

Papers

Photoluminescence of heavily doped, compensated Si:P,B.

Susceptibility of Si:P across the metal-insulator transition. II. Evidence for local moments in the metallic phase.

Measurement of the Hall coefficient using van der Pauw method without magnetic field reversal

Resonant magnetization tunneling in Mn 12 acetate: The absence of inhomogeneous hyperfine broadening

Impurity resistivity of the double-donor system Si:P,Bi