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

Thermally activated paramagnetism in carbon chars

We report the observation of a nonlinear (quadratic) dependence of the Hall voltage {ital V}{sub {ital H}} on magnetic field at small fields in the hopping regime. With increasing field the Hall coefficient, {ital R}{sub {ital H}}={ital V}{sub {ital H}}/{ital IH}, of insulating compensated {ital n}-type CdSe increases approximately linearly, then remains roughly constant at a plateau value, and increases again at higher fields. No theory currently exists with which to compare this unanticipated behavior.

Nonlinear Hall voltage in the hopping regime.

The low-temperature dc conductivities of barely metallic samples of $p$-type Si:B are compared for a series of samples with different dopant concentrations, $n,$ in the absence of stress (cubic symmetry), and for a single sample driven from the metallic into the insulating phase by uniaxial compression, $S.$ For all values of temperature and stress, the conductivity of the stressed sample collapses onto a single universal scaling curve, $\ensuremath{\sigma}(S,T)={\ensuremath{\sigma}}_{0}(\ensuremath{\Delta}{S/S}_{c}{)}^{\ensuremath{\mu}}G[{T/T}^{*}(S)],$ with ${T}^{*}\ensuremath{\propto}(\ensuremath{\Delta}{S)}^{z\ensuremath{\nu}}.$ The scaling fit indicates that the conductivity of Si:B is $\ensuremath{\propto}{T}^{1/2}$ in the critical range. Our data yield a critical conductivity exponent $\ensuremath{\mu}=1.6,$ considerably larger than the value reported in earlier experiments where the transition was crossed by varying the dopant concentration. The larger exponent is based on data in a narrow range of stress near the critical value within which scaling holds. We show explicitly that the temperature dependences of the conductivity of stressed and unstressed Si:B are different, suggesting that a direct comparison of the critical behavior and critical exponents for stress-tuned and concentration-tuned transitions may not be warranted.

Conductivity of metallic Si:B near the metal-insulator transition: Comparison between unstressed and uniaxially stressed samples

Based on data for insulating [ital n]-type CdSe at temperatures between 0.05 and 4.2 K and magnetic fields above 3 T (up to the highest measured field of 9 T), we demonstrate an empirical relation between the longitudinal and transverse conductivities given by [sigma][sub [ital x][ital y]]=[ital scrAH][sigma][sub [ital x][ital x]][sup 3/2].

Myriam P. Sarachik

Papers

Thermally activated paramagnetism in carbon chars

Nonlinear Hall voltage in the hopping regime.

Conductivity of metallic Si:B near the metal-insulator transition: Comparison between unstressed and uniaxially stressed samples

Empirical relation between longitudinal and transverse transport in insulating n-CdSe.

PROBING THE COULOMB GAP IN INSULATING n-TYPE CdSe