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 report a remarkable symmetry between the resistivity and conductivity on opposite sides of the B=0 metal-insulator transition in a two-dimensional electron gas in high-mobility silicon metal-oxide-semiconductor field-effect transistors. This symmetry implies that the transport mechanisms on the two sides are related. {copyright} {ital 1997} {ital The American Physical Society}

https://arxiv.org/pdf/cond-mat/9611147.pdf

Reflection symmetry at a B=0 metal-insulator transition in two dimensions

Measurements of the susceptibility between 1.25 and 300 K and of the magnetization from 0 to 50 kG are reported for a series of boron-doped silicon samples spanning the metal-nonmetal transition. As for Si:P, the magnetic properties change gradually in this field and temperature range as the boron concentration is varied across the transition. The susceptibility consists of (at least) two contributions: a component associated with the randomly distributed localized holes interacting via short-range antiferromagnetic exchange, and a component which maps the evolution of extended holes from a degenerate to a nondegenerate gas. The latter is a positive term which decreases in magnitude as the number of delocalized holes increases with increasing boron concentration. The impurity magnetization is not saturated at 1.25 K and 50 kG, and is small compared to the magnetization of an equivalent number of uncoupled boron impurities. Results for both the susceptibility and magnetization indicate that exchange plays a more important role in Si:B than it does in Si:P.

Magnetic properties of boron-doped silicon.

We report measurements of the resistance of silicon metal-oxide-semiconductor field-effect transistors as a function of temperature in high parallel magnetic fields where the two-dimensional system of electrons has been shown to be fully spin polarized. In a field of 10.8 T, insulating behavior is found for densities up to n{sub s}{approx}1.35x10{sup 11}cm{sup -2}{approx}1.5n{sub c}; above this density the resistance is a very weak function of temperature, varying less than 10% between 0.25 and 1.90 K. At low densities {rho}{yields}{infinity} more rapidly as the temperature is reduced than in zero field and the magnetoresistance {Delta}{rho}/{rho} diverges as T{yields}0.

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Temperature dependence of the resistivity of a dilute two-dimensional electron system in high parallel magnetic field

In three-dimensional n-CdSe samples that obey Mott variable-range hopping, \ensuremath{\rho}=${\mathrm{\ensuremath{\rho}}}_{0}$exp(${\mathit{T}}_{0}$/T${)}^{1/4}$, in the absence of a magnetic field, we report a field-induced crossover at low temperatures to a resistivity that exhibits a weak power-law divergence, \ensuremath{\rho}=${\mathrm{\ensuremath{\rho}}}_{0}$${\mathit{T}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, with an exponent \ensuremath{\alpha} that decreases slowly with increasing field. \textcopyright{} 1996 The American Physical Society.

Magnetic-field-induced crossover from Mott variable-range hopping to weakly insulating behavior

For an electron density near the H=0 insulator-to-conductor transition, the magnetoconductivity of the low-temperature conducting phase in high-mobility silicon MOSFETs is consistent with the form {Delta}{sigma}(H{sub {vert_bar}{vert_bar}},T){equivalent_to}{sigma}(H{sub {vert_bar}{vert_bar}},T){minus}{sigma}(0,T)=f(H{sub {vert_bar}{vert_bar}}/T) for magnetic fields H{sub {vert_bar}{vert_bar}} applied parallel to the plane of the electron system. This sets a valuable constraint on theory and provides further evidence that the electron spin is central to the anomalous H=0 conducting phase in two dimensions. {copyright} {ital 1998} {ital The American Physical Society}

Myriam P. Sarachik

Papers

Reflection symmetry at a B=0 metal-insulator transition in two dimensions

Magnetic properties of boron-doped silicon.

Temperature dependence of the resistivity of a dilute two-dimensional electron system in high parallel magnetic field

Magnetic-field-induced crossover from Mott variable-range hopping to weakly insulating behavior

H/t scaling of the magnetoconductance near the conductor-insulator transition in two dimensions