Physical Review B

(Condensed Matter and Materials Physics - 15 (II))

November 2009

Volume 80, Number 20 , partial issue

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RAPID COMMUNICATIONS

Semiconductors II: surfaces, interfaces, microstructures, and related topics

Surface physics, nanoscale physics, low-dimensional systems

ARTICLES

Electronic structure: wide-band, narrow-band, and strongly correlated systems

Semiconductors I: bulk

Semiconductors II: surfaces, interfaces, microstructures, and related topics

Surface physics, nanoscale physics, low-dimensional systems

RAPID COMMUNICATIONS

Semiconductors II: surfaces, interfaces, microstructures, and related topics

Hyperfine mediated triplet-singlet transition probability in a double-quantum-dot system: Analogy with the double-slit experiment

Fernando Domínguez and Gloria Platero

Published 2 November 2009 (4 pages)
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We apply an elementary measurement scheme to calculate the electronic triplet-singlet transition mediated by hyperfine interaction in a double-quantum dot. We show how the local character of the hyperfine interaction and the nuclear back-action process (flip-flop) are crucial to cancel destructive interferences of the triplet-singlet transition probability. It is precisely this cancellation that differentiates the hyperfine interaction from an anisotropic magnetic field which mixes the triplet and the singlet eigenstates.

Localization in an inhomogeneous quantum wire

A. D. Güçlü, C. J. Umrigar, Hong Jiang, and Harold U. Baranger

Published 2 November 2009 (4 pages)
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We study interaction-induced localization of electrons in an inhomogeneous quasi-one-dimensional system—a wire with two regions, one at low density and the other high. Quantum Monte Carlo techniques are used to treat the strong Coulomb interactions in the low-density region, where localization of electrons occurs. The nature of the transition from high to low density depends on the density gradient—if it is steep, a barrier develops between the two regions, causing Coulomb blockade effects. Ferromagnetic spin polarization does not appear for any parameters studied. The picture emerging is in good agreement with measurements of tunneling between wires.

Interaction-tuned compressible-to-incompressible phase transitions in quantum Hall systems

Z. Papić, N. Regnault, and S. Das Sarma

Published 2 November 2009 (4 pages)
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We analyze transitions between quantum Hall ground states at prominent filling factors in the spherical geometry by tuning the width parameter of the Zhang-Das Sarma interaction potential. We find that incompressible ground states evolve adiabatically under this tuning, whereas the compressible ones are driven through a first-order phase transition. Overlap calculations show that the resulting phase is increasingly well described by appropriate analytic model wave functions (Laughlin, Moore-Read, Read-Rezayi). This scenario is shared by both odd (=1/3,1/5,3/5,7/3,11/5,13/5) and even denominator states (=1/2,1/4,5/2,9/4). In particular, the Fermi-liquid-like state at =1/2 gives way, at large enough value of the width parameter, to an incompressible state identified as the Moore-Read Pfaffian on the basis of its entanglement spectrum.

Surface-induced polarity inversion in ZnO nanowires

Giancarlo Cicero, Andrea Ferretti, and Alessandra Catellani

Published 2 November 2009 (4 pages)
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In this work we use ab initio calculations to address the polar behavior of ZnO nanowires. Moving from a description based on Wannier functions, we employ a computational approach that allows one to express the polarization of a nanostructure in terms of local contributions. In particular, we discuss the changes in the nanostructure polarity in terms of two contributions, one related to changes in the equilibrium lattice parameters and the other related to surface effects. The former contribution is also interpreted on the basis of piezoelectric constants. Surprisingly, we find that for the smallest nanostructures, the average dipole is opposite to that of an infinite bulk structure.

Surface physics, nanoscale physics, low-dimensional systems

Surface modes in plasmonic crystals induced by diffractive coupling of nanoantennas

G. Vecchi, V. Giannini, and J. Gómez Rivas

Published 4 November 2009 (4 pages)
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Lattice surface modes in two-dimensional plasmonic crystals of metallic nanoantennas are sustained by the diffractive coupling between localized plasmon resonances. We have investigated experimentally the dispersion of these modes by combining variable-angle transmittance and reflectance spectroscopy. Transmittance spectra through plasmonic crystals reveal quality factors for these modes exceeding 150 in the near infrared, 30 times higher than the quality factor associated to localized plasmon resonances. We have derived the characteristic lengths of the lattice surface modes, distinguishing a regime in which the group velocity is reduced while the mode intensity is strongly confined near the nanoantennas and a regime in which the surface mode propagates over several unit cells of the plasmonic crystal.

Low-temperature specific heat of one-dimensional multicomponent systems at the commensurate-incommensurate phase transition point

T. Vekua

Published 6 November 2009 (4 pages)
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Low-temperature dependence of specific heat of one-dimensional multicomponent systems at the commensurate-incommensurate phase transition point is studied. It is found that for canonical systems, with a fixed total number of particles, low-temperature specific heat linearly depends on temperature with a diverging prefactor.

ARTICLES

Electronic structure: wide-band, narrow-band, and strongly correlated systems

Quantum Hall effect in bilayer graphene: Disorder effect and quantum phase transition

R. Ma, L. Sheng, R. Shen, M. Liu, and D. N. Sheng

Published 2 November 2009 (5 pages)
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We numerically study the quantum Hall effect (QHE) in bilayer graphene based on tight-binding model in the presence of disorder. Two distinct QHE regimes are identified in the full energy band separated by a critical region with nonquantized Hall Effect. The Hall conductivity around the band center (Dirac point) shows an anomalous quantization proportional to the valley degeneracy, but the =0 plateau is markedly absent, which is in agreement with experimental observation. In the presence of disorder, the Hall plateaus can be destroyed through the float-up of extended levels toward the band center and higher plateaus disappear first. The central two plateaus around the band center are most robust against disorder scattering, which is separated by a small critical region in between near the Dirac point. The longitudinal conductance around the Dirac point is shown to be nearly a constant in a range of disorder strength, until the last two QHE plateaus completely collapse.

Generalized Slater-Jastrow trial function: Application to the electron gas at high density

James C. Porter

Published 3 November 2009 (14 pages)
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The function u(i,j)=u(|ri−rj|), in terms of which variational trial functions of the Slater-Jastrow type are expressed, is generalized to include dependence on the electron momenta as well as on the electron-pair separations. The resulting Euler equations are used to evaluate the ground-state coulomb correlation energy giving c=0.0622 log(rs)−0.1441  Ry, where rs is the ratio of the radius of a sphere, containing, on average one electron, to the Bohr radius. This correlation energy is 0.050 Ry lower than the corresponding result c=0.0622 log(rs)−0.094  Ry found by Gell-Mann and Brueckner. Also evaluated are the corresponding correlation corrections to the Hartree-Fock (HF) exchange potential energy and to the average exchange charge density as a function of distance. The correction to the HF exchange potential energy is shown to cancel the divergence of the HF exchange energy at the Fermi energy so that the electron effective mass remains finite.

Optimizing the thermoelectric efficiency of icosahedral quasicrystals and related complex alloys

Enrique Maciá

Published 3 November 2009 (7 pages)
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In this work we analyze the potential role of quasicrystals and related alloys in thermoelectric material research. Relatively large figure of merit values are expected for those samples exhibiting two properly located narrow features in the density of states close to the Fermi level. It is expected that optimized quasicrystals will perform better at relatively low temperatures, whereas the ZT curve of complex metallic alloys reaches its maximum at high temperatures. Among state-of-the-art quasicrystals most promising samples for thermoelectric applications are found in the AlPd(Mn,Re) system. Quasicrystalline and related approximants in the ScMgCuGa and CaAuIn systems, synthesized on the basis of pseudogap tuning concepts, appear as promising candidates as well.

Highly efficient method for Kohn-Sham density functional calculations of 500–10 000 atom systems

M. J. Rayson and P. R. Briddon

Published 4 November 2009 (11 pages)
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A method for the solution of the self-consistent Kohn-Sham equations using Gaussian-type orbitals is presented. Accurate relative energies and forces are demonstrated to be achievable at a fraction of the computational expense for large systems. With this approach calculations involving around 1000 atoms can easily be performed with a serial desktop computer and ~10 000 atom systems are within reach of relatively modest parallel computational resources. The method is applicable to arbitrary systems including metals. The approach generates a minimal basis on the fly while retaining the accuracy of the large underpinning basis set. Convergence of energies and forces are given for clusters as well as cubic cells of silicon and aluminum, for which the formation energies of defects are calculated in systems up to 8000 and 4000 atoms, respectively. For these systems the method exhibits linear scaling with the number of atoms in the presently important size range of ~500–3000 atoms.

Electronic stopping of low-energy H and He in Cu and Au investigated by time-of-flight low-energy ion scattering

S. N. Markin, D. Primetzhofer, M. Spitz, and P. Bauer

Published 4 November 2009 (4 pages)
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Electronic stopping has been investigated experimentally for slow H⁺ and He⁺ ions (v<0.6  a.u.) in polycrystalline Au and Cu, by means of time-of-flight low-energy ion scattering. Measurements were performed in backscattering geometry using polycrystalline films of several nanometers thickness; high-resolution Rutherford backscattering spectrometry was used for precise thickness calibration (<4%). Absolute stopping cross-section data for Cu and Au were obtained down to v~0.07  a.u. Both Au and Cu exhibit a rather sharp, distinctive threshold velocity of ~0.18  a.u. for excitation of d electrons by H⁺ and He⁺. Below this threshold, projectiles interact exclusively with s electrons.

Ligand isotope structure of the optical ⁷F₀⁵D₀ transition in EuCl₃·6H₂O

R. L. Ahlefeldt, A. Smith, and M. J. Sellars

Published 5 November 2009 (5 pages)
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The structure of the ⁷F0⁵D0 transition of Eu³⁺ in EuCl3·6H2O is investigated in crystals with three different isotopic compositions [natural abundance (0.2% ¹⁸O, 0.01%²H), 1.9% ¹⁸O, and 0.06% ²H]. Raman-heterodyne-detected NMR is used to identify shifts caused by isotopes of the ligands O and H in sites neighboring the Eu site in the lattice. The structure of the ⁷F0⁵D0 transition is explained as a combination of the hyperfine structure of ¹⁵¹Eu and ¹⁵³Eu and isotope shifts caused by Cl, O, and H.

Effective Coulomb interactions within BEDT-TTF dimers

Edan Scriven and B. J. Powell

Published 6 November 2009 (9 pages)
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We calculate the parameters for Hubbard models of -(BEDT-TTF)2X and -(BEDT-TTF)2X. We use density-functional theory (DFT) to calculate the interactions between holes in dimers of the organic molecule bis(ethylenedithio)tetrathiafulvalene (BEDT-TTF) for 23 experimental geometries taken from a range of materials in both the and polymorphs. We find that the effective Coulomb interactions are essentially the same for all of the compounds studied. We highlight the disagreement between our parametrization of the Hubbard model and previous results from both DFT and Hückel methods. We show that this is caused by the failure of an assumption made in previous calculations (which estimate the effective Coulomb interaction from the intradimer hopping integral). We discuss the implications of our calculations for theories of the BEDT-TTF salts based on the Hubbard model and use our calculated parameters to explain a number of phenomena caused by conformational disorder in these materials.

Semiconductors I: bulk

Ferroelectric nature and real-space observations of domain motions in the organic charge-transfer compound tetrathiafulvalene-p-chloranil

Hideo Kishida, Hisashi Takamatsu, Ken Fujinuma, and Hiroshi Okamoto

Published 6 November 2009 (7 pages)
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Ferroelectricity in an organic charge-transfer compound, tetrathiafulvalene-p-chloranil (TTF-CA), originating from the one-dimensional valence and lattice instabilities, has been investigated by an electroreflectance (ER) method. Microscopic ER spectroscopy in the visible region enables real-space observations of both ferroelectric domain structures with a few hundred micrometers in size and depinning of the domain walls under strong electric fields. In addition, from ER spectroscopy in the infrared molecular-vibration region, we demonstrate that field-induced changes in the dimeric molecular displacement as well as charge transfer between TTF and CA molecules play an important role in the large dielectric response in TTF-CA.

Temperature-dependent resistivity of ferromagnetic Ga_1−xMn_xAs: Interplay between impurity scattering and many-body effects

F. V. Kyrychenko and C. A. Ullrich

Published 6 November 2009 (6 pages)
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The static conductivity of the diluted magnetic semiconductor Ga1−xMnxAs is calculated using an equation of motion approach for the current response combined with time-dependent density-functional theory to account for Hartree and exchange interactions within the hole gas. We find that the Coulomb scattering off the charged impurities alone is not sufficient to explain the experimentally observed drop in resistivity below the ferromagnetic transition temperature: the often overlooked scattering off the fluctuations of localized spins is shown to play a significant role.

Semiconductors II: surfaces, interfaces, microstructures, and related topics

Change in the character of quasiparticles without gap collapse in a model of fractional quantum Hall effect

Csaba Tőke and Jainendra K. Jain

Published 4 November 2009 (11 pages)
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It is commonly assumed in the studies of the fractional quantum Hall effect that the physics of a fractional quantum Hall state, in particular the character of its excitations, is invariant under a continuous deformation of the Hamiltonian during which the gap does not close. We show in this article that, at least for finite systems, as the interaction is changed from a model three body interaction to Coulomb, the ground state at filling factor =2/5 evolves continuously from the so-called Gaffnian wave function to the composite fermion wave function, but the quasiholes alter their character in a nonperturbative manner. This is attributed to the fact that the Coulomb interaction opens a gap in the Gaffnian quasihole sector, pushing many of the states to very high energies. Interestingly, the states below the gap are found to have a one-to-one correspondence with the composite fermion theory, suggesting that the Gaffnian model contains composite fermions, and that the Gaffnian quasiholes are unstable to the formation of composite fermions when a two-body interaction term is switched on. General implications of this study are discussed.

Realizing singlet-triplet qubits in multivalley Si quantum dots

Dimitrie Culcer, ukasz Cywiński, Qiuzi Li, Xuedong Hu, and S. Das Sarma

Published 5 November 2009 (6 pages)
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There has been significant progress in the implementation and manipulation of singlet-triplet qubits in GaAs quantum dots. Given the considerably longer spin coherence times measured in Si, considerable interest has been generated recently in Si quantum dots. The physics of these systems is considerably more complex than the physics of GaAs quantum dots owing to the presence of the valley degree of freedom, which constitutes the focus of this work. In this paper we investigate the physics of Si quantum dots and focus on the feasibility of quantum coherent singlet-triplet qubit experiments analogous to those performed in GaAs. This additional degree of freedom greatly increases the complexity of the ground state and gives rise to highly nontrivial and interesting physics in the processes of qubit initialization, coherent manipulation and readout. We discuss the operational definition of a qubit in Si-based quantum dots. We find that in the presence of valley degeneracy a singlet-triplet qubit cannot be constructed, whereas for large valley splitting (kBT) the experiment is similar to GaAs. We show that experiments on singlet-triplet qubits analogous to those in GaAs would provide a method for estimating the valley splitting in Si. A Zeeman field distinguishes between different initialized states for any valley splitting and provides a tool to determine the size of this splitting.

Measurement of the Knight field and local nuclear dipole-dipole field in an InGaAs/GaAs quantum dot ensemble

T. Auer, R. Oulton, A. Bauschulte, D. R. Yakovlev, M. Bayer, S. Yu. Verbin, R. V. Cherbunin, D. Reuter, and A. D. Wieck

Published 5 November 2009 (15 pages)
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We present a comprehensive investigation of the electron-nuclear system of negatively charged InGaAs/GaAs self-assembled quantum dots (QDs) under the influence of weak external magnetic fields (up to 3 mT). We demonstrate that, in contrast to conventional semiconductor systems, these small fields have a profound influence on the electron spin dynamics, via the hyperfine interaction. QDs, with their comparatively limited number of nuclei, present electron-nuclear behavior that is unique to low-dimensional systems. We show that the conventional Hanle effect used to measure electron-spin relaxation times, for example, cannot be used in these systems when the spin lifetimes are long. An individual nucleus in the QD is subject to milli-Tesla effective fields, arising from the interaction with its nearest neighbors and with the electronic Knight field. The alignment of each nucleus is influenced by application of external fields of the same magnitude. A polarized nuclear system, which may have an effective field strength of several Tesla, may easily be influenced by these milli-Tesla fields. This in turn has a dramatic effect on the electron-spin dynamics and we use this technique to gain a measure of both the dipole-dipole field and the maximum Knight field in our system thus allowing us to estimate the maximum Overhauser field that may be generated at zero external magnetic field. We also show that one may fine tune the angle which the Overhauser field makes with the optical axis.

Tunnel coupling in an ensemble of vertically aligned quantum dots at room temperature

V. V. Nikolaev, N. S. Averkiev, M. M. Sobolev, I. M. Gadzhiyev, I. O. Bakshaev, M. S. Buyalo, and E. L. Portnoi

Published 5 November 2009 (10 pages)
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We report unambiguous observation of the formation of mixed electronic states in an ensemble of self-assembled vertically aligned quantum dots at room temperature. Three closely spaced layers containing stacked In(Ga)As/GaAs quantum dots are placed in the active region of a two-section semiconductor device, and investigations of the quantum-dot optical properties at different applied electric fields are carried out by means of differential-absorption spectroscopy. A simple semianalytical model, which describes absorption of two layers of coupled quantum dots with an account of the size dispersion, is developed. A comparison between our experimental and theoretical results allows clear attribution of the observed low-photon-energy field-dependent spectral features to the four mixed optical transitions due to the two upper quantum-dot layers. Interpretation of the experimental results reveals an anticrossing of spatially direct and indirect transitions characterized by the energy splitting of approximately 30 meV.

Role of nonlinear effects in nanowire growth and crystal phase

V. G. Dubrovskii, N. V. Sibirev, G. E. Cirlin, A. D. Bouravleuv, Yu. B. Samsonenko, D. L. Dheeraj, H. L. Zhou, C. Sartel, J. C. Harmand, G. Patriarche, and F. Glas

Published 6 November 2009 (8 pages)
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We study theoretically and experimentally nonlinear effects during the “vapor-liquid-solid” growth of semiconductor nanowires. Nonlinear growth equation considered contains kinetic coefficients from the surface and sidewall diffusion which can be of either signs. We predict four possible growth scenarios: (I) infinite growth; (II) decomposition; (III) averaging growth to a finite length; and (IV) continuing growth such that nanowires of small initial length decay and longer nanowires grow infinitely. We present the experimental evidence of nontrivial scenarios (II) and (IV) during the Au-assisted molecular-beam epitaxy of GaAs nanowires. Scenario (II) corresponds to the evaporation of GaAs nanowires during the annealing. Scenario (IV) is observed during two-step growth procedure where a low-temperature growth is followed by deposition at 630 °C. We show that the growth via scenario (IV) enables to control the crystal phase and to obtain the stacking-fault-free sections of zinc-blende GaAs NWs having only 15–20 nm in radius.

Surface physics, nanoscale physics, low-dimensional systems

Anomalous finite size effects on surface states in the topological insulator Bi₂Se₃

Jacob Linder, Takehito Yokoyama, and Asle Sudbø

Published 2 November 2009 (5 pages)
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We study how the surface states in the strong topological insulator Bi2Se3 are influenced by finite size effects and compare our results with those recently obtained for two-dimensional topological insulator HgTe. We demonstrate two important distinctions: (i) contrary to HgTe, the surface states in Bi2Se3 display a remarkable robustness towards decreasing the width L down to a few nm thus ensuring that the topological surface states remain intact and (ii) the gapping due to the hybridization of the surface states features an oscillating exponential decay as a function of L in Bi2Se3 in sharp contrast to HgTe. Our findings suggest that Bi2Se3 is suitable for nanoscale applications in quantum computing or spintronics. Also, we propose a way to experimentally detect both of the predicted effects.

Capacitance of graphene nanoribbons

A. A. Shylau, J. W. Kos, and I. V. Zozoulenko

Published 4 November 2009 (9 pages)
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We present an analytical theory for the gate electrostatics and the classical and quantum capacitance of the graphene nanoribbons (GNRs) and compare it with the exact self-consistent numerical calculations based on the tight-binding p-orbital Hamiltonian within the Hartree approximation. We demonstrate that the analytical theory is in a good qualitative (and in some aspects quantitative) agreement with the exact calculations. There are however some important discrepancies. In order to understand the origin of these discrepancies we investigate the self-consistent electronic structure and charge density distribution in the nanoribbons and relate the above discrepancy to the inability of the simple electrostatic model to capture the classical gate electrostatics of the GNRs. In turn, the failure of the classical electrostatics is traced to the quantum mechanical effects leading to the significant modification of the self-consistent charge distribution in comparison to the noninteracting electron description. The role of electron-electron interaction in the electronic structure and the capacitance of the GNRs is discussed. Our exact numerical calculations show that the density distribution and the potential profile in the GNRs are qualitatively different from those in conventional split-gate quantum wires; at the same time, the electron distribution and the potential profile in the GNRs show qualitatively similar features to those in the cleaved-edge overgrown quantum wires. Finally, we discuss an experimental extraction of the quantum capacitance from experimental data.

Charge transfer satellite in Pr@C₈₂ metallofullerene observed using resonant x-ray emission spectroscopy

H. Yamaoka, H. Sugiyama, Y. Kubozono, A. Kotani, R. Nouchi, A. M. Vlaicu, H. Oohashi, T. Tochio, Y. Ito, and H. Yoshikawa

Published 4 November 2009 (5 pages)
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Resonant x-ray emission spectroscopy (RXES) was performed on the metallofullerene Pr@C82 at the Pr L3 absorption edge. We verify not only nearly three-electron charge transfers from the metal to the cage but also back-electron transfer observed as a charge transfer satellite. The results are compared to theoretical calculations with a single-impurity Anderson model. Theory shows that the electronic structure of endohedral atom in the cage is atomiclike. The satellite structure originates from the charge transfer, i.e., dynamical screening effect, induced by the core-hole potential in the final state rather than from the valence fluctuation of the rare-earth metal in the ground state. We also performed the RXES measurement of Pr2O3 for comparison.

Infrared optical properties of chromium nanoscale films with a phase transition

Robert Lovrinčić and Annemarie Pucci

Published 5 November 2009 (6 pages)
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Thickness-dependent infrared spectra measured during growth of Cr on diamond C(100) under ultrahigh vacuum conditions reveal a structural phase transition from a discontinuous phase to the crystalline bulk one with Drude-type optical properties. The thickness of 2.5 nm of the observed phase transition well agrees with literature data on the theoretically predicted maximum size for stable fcc nanoclusters the formation of which should be supported on C(100) owing to a good lattice match. Below 2.5 nm, the spectral behavior surprisingly well corresponds to a Drude-Smith type dielectric function, which allows one to determine the conductivity for the discontinuous phase and to get quantitative information on the effect of coherent backscattering.

Plasmon modes of spatially separated double-layer graphene

E. H. Hwang and S. Das Sarma

Published 6 November 2009 (5 pages)
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We derive the plasmon dispersion in doped double-layer graphene (DLG), made of two parallel graphene monolayers with carrier densities n1 and n2, respectively, and an interlayer separation of d. The linear chiral gapless single-particle energy dispersion of graphene leads to DLG plasmon properties with several unexpected experimentally observable characteristic features such as a nontrivial influence of an undoped (n2=0) layer on the DLG plasmon dispersion and a strange influence of the second layer even in the weak-coupling d limit. At long wavelengths (q0), the density dependence of the plasma frequencies is different from the usual two-dimensional (2D) electron system with quadratic energy dispersion. Our predicted DLG plasmon properties clearly distinguish graphene from the extensively studied usual parabolic 2D electron systems.

Atomic configuration, conductance, and tensile force of platinum wires of single-atom width

Tokushi Kizuka and Kosuke Monna

Published 6 November 2009 (9 pages)
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Platinum (Pt) wires of single-atom width were produced by the retraction of a Pt nanotip from contact with a Pt plate at room temperature inside a transmission electron microscope. The distance between the nanotip and the plate was controlled using a conductance feedback system, as a result of which wires showing certain conductance values were observed continuously by in situ lattice imaging. Simultaneously, the force acting on the wires was measured using a function of atomic force microscopy. The tip-plate distance was also increased with a constant speed, and the atomic configuration, force, and conductance were similarly investigated. The single-atom-width Pt wires were found to exhibit straight shapes with an interatomic distance of 0.28±0.03  nm. The wires were stable at a tensile force of approximately 1 nN; the observed interatomic distance resulted from elastic expansion. The present study demonstrated experimental evidence for the relationship between wire length and conductance; the wires extend from a three-atom length to a five-atom length as the selected feedback conductance decreases from 2.0 to 0.5G0 (where G0=2e²/h, e being the charge of an electron and h Planck's constant). Contacts exhibiting a conductance of 3.0G0 were two-atom-width contacts. In a conductance histogram constructed from the simple retraction, only one peak was observed at 1.3G0. Thus, it was found that the conductance of single-atom-width Pt wires is less than 3.0G0, with 1.3G0 being that of the most-stable state.