Showing papers on "Spin wave published in 2005"
TL;DR: Graphene is converted from an ideal two-dimensional semimetallic state to a quantum spin Hall insulator and the spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder, and symmetry breaking fields are discussed.
Abstract: We study the effects of spin orbit interactions on the low energy electronic structure of a single plane of graphene. We find that in an experimentally accessible low temperature regime the symmetry allowed spin orbit potential converts graphene from an ideal two-dimensional semimetallic state to a quantum spin Hall insulator. This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are nonchiral, but they are insensitive to disorder because their directionality is correlated with spin. The spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder, and symmetry breaking fields are discussed.
6,058 citations
01 Jan 2005
TL;DR: Nagai et al. as discussed by the authors investigated the phase transition in frustrated ising spin systems and proposed two-dimensional spin liquid antiferromagnets and spin glasses, respectively.
Abstract: Frustration - Exactly Solved Frustrated Models (H T Diep & H Giacomini) Properties and Phase Transitions in Frustrated Ising Spin Systems (O Nagai et al.) Renormalization Group Approaches to Frustrated Magnets in D = 3 (B Delamotte et al.) Phase Transitions in Frustrated Vector Spin Systems: Numerical Studies (D Loison) Two-Dimensional Quantum Antiferromagnets (G Misguich & C Lhuillier) One-Dimensional Spin Liquids (P Lecheminant) Spin Ice (S T Bramwell et al.) Experimental Studies of Frustrated Pyrochlore Antiferromagnets (B D Gaulin & J S Gardner) Recent Progress in Spin Glasses (N Kawashima & H Rieger).
737 citations
TL;DR: It is shown that electron spin flips are dominated by nuclear interactions and are slowed by several orders of magnitude when a magnetic field of a few millitesla is applied, having significant implications for spin-based information processing.
Abstract: The GaAs double quantum dot is the classic spin qubit widely studied for its potential as information carrier in quantum computers. The discovery that electron spin flips in this system are governed by nuclear interactions, and slowed dramatically by a weak magnetic field, is promising in terms of the control and manipulation of spin-based memory. The spin of a confined electron, when oriented originally in some direction, will lose memory of that orientation after some time. Physical mechanisms leading to this relaxation of spin memory typically involve either coupling of the electron spin to its orbital motion or to nuclear spins1,2,3,4,5,6,7. Relaxation of confined electron spin has been previously measured only for Zeeman or exchange split spin states, where spin-orbit effects dominate relaxation8,9,10; spin flips due to nuclei have been observed in optical spectroscopy studies11. Using an isolated GaAs double quantum dot defined by electrostatic gates and direct time domain measurements, we investigate in detail spin relaxation for arbitrary splitting of spin states. Here we show that electron spin flips are dominated by nuclear interactions and are slowed by several orders of magnitude when a magnetic field of a few millitesla is applied. These results have significant implications for spin-based information processing12.
436 citations
TL;DR: It is shown that the Raman spin coherence in the quantum beats is caused not only by the usual stimulated Raman interaction but also by simultaneous spontaneous radiative decay of either excited trion state to a coherent combination of the two spin states.
Abstract: We report on the coherent optical excitation of electron spin polarization in the ground state of charged GaAs quantum dots via an intermediate charged exciton (trion) state. Coherent optical fields are used for the creation and detection of the Raman spin coherence between the spin ground states of the charged quantum dot. The measured spin decoherence time, which is likely limited by the nature of the spin ensemble, approaches 10 ns at zero field. We also show that the Raman spin coherence in the quantum beats is caused not only by the usual stimulated Raman interaction but also by simultaneous spontaneous radiative decay of either excited trion state to a coherent combination of the two spin states.
232 citations
TL;DR: It is demonstrated that the lowest threshold of spin-wave excitation in an in-plane magnetized magnetic nanocontact driven by spin-polarized current is achieved for a nonlinear self-localizedspin-wave mode-standing spin- wave bullet--stabilized by current-induced nonlinear dissipation.
Abstract: We demonstrate that the lowest threshold of spin-wave excitation in an in-plane magnetized magnetic nanocontact driven by spin-polarized current is achieved for a nonlinear self-localized spin-wave mode-standing spin-wave bullet--stabilized by current-induced nonlinear dissipation. This nonlinear mode has a nonpropagating evanescent character, is localized in the region comparable with the contact radius, and has a frequency that is lower than the frequency of the linear ferromagnetic resonance. The threshold current and generated frequency at the threshold theoretically calculated for this mode are in quantitative agreement with experiment.
228 citations
TL;DR: In this article, the authors investigated the dc voltage simultaneously generated with the excitation of the uniform mode of magnetization precession in ferromagnetic trilayers with different spacer-layer materials.
Abstract: Direct current voltage appears across and in plane of a ferromagnetic multilayer experiencing ferromagnetic resonance. We have investigated the dc voltage simultaneously generated with the excitation of the uniform mode of magnetization precession in ferromagnetic∕normal-metal∕ferromagnetic trilayers with different spacer-layer materials. The generated voltage strongly depends on the chemical nature and the thickness of the normal-metal spacer as well as on the microwave incident power. This dc voltage might be correlated with the spin-pumping effect recently predicted.
210 citations
TL;DR: In this paper, the eigenfrequencies and precession patterns of some of the lowest-frequency magnetic normal modes of submicron patterned elements were determined for a Permalloy-like ellipse, 350nm×160nm×5nm.
Abstract: Micromagnetic calculations are used to determine the eigenfrequencies and precession patterns of some of the lowest-frequency magnetic normal modes of submicron patterned elements. Two examples are presented. For a Permalloy-like ellipse, 350nm×160nm×5nm thick in zero field, the lowest frequency normal mode at 4GHz corresponds to precession in the “ends” of the ellipse. Other resonant frequencies are compared with the frequencies of spinwaves with discrete wave vectors. For a normally magnetized 50nmdiameter×15nm thick cobalt disk, the calculated eigenfrequencies increase linearly with applied field, mimicking the behavior of the experimental critical current for spin transfer instabilities in an experimental realization of this disk.
208 citations
TL;DR: In this paper, the spin density distribution in transition metal compounds is discussed, the characteristic trends involving spin delocalization and spin polarization mechanisms are summarized, and the characteristic shapes of spin density distributions around a transition metal atom are presented.
163 citations
TL;DR: The excitation spectrum of six micron magnetic squares is measured using x-ray magnetic circular dichroism and precessional motion of the magnetization within the domains as well as a domain wall mode and vortex motion is observed.
Abstract: We have measured the excitation spectrum of six micron magnetic squares using x-ray magnetic circular dichroism. We observe all three excitations expected in a Landau flux-closure pattern. High temporal and spatial resolution allows quantitative analysis of the excitations. A short magnetic in plane pulse excites the magnetic element and we observe precessional motion of the magnetization within the domains as well as a domain wall mode and vortex motion. The vortex moves perpendicular to the excitation field and relaxes without showing a circulating orbit.
144 citations
TL;DR: In this paper, the spin-orbital superexchange models for a series of representative orbital-degenerate transition metal oxides with different multiplet structure were analyzed.
Abstract: The temperature dependence and anisotropy of optical spectral weights associated with different multiplet transitions is determined by the spin and orbital correlations. To provide a systematic basis to exploit this close relationship between magnetism and optical spectra, we present and analyze the spin-orbital superexchange models for a series of representative orbital-degenerate transition metal oxides with different multiplet structure. For each case we derive the magnetic exchange constants, which determine the spin wave dispersions, as well as the partial optical sum rules. The magnetic and optical properties of early transition metal oxides with degenerate t2g orbitals titanates and vanadates with perovskite structure are shown to depend only on two parameters, viz. the superexchange energy J and the ratio of Hund’s exchange to the intraorbital Coulomb interaction, and on the actual orbital state. In eg systems important corrections follow from charge transfer excitations, and we show that KCuF3 can be classified as a charge transfer insulator, while LaMnO3 is a Mott insulator with moderate charge transfer contributions. In some cases orbital fluctuations are quenched and decoupling of spin and orbital degrees of freedom with static orbital order gives satisfactory results for the optical weights. On the example of cubic vanadates we describe a case where the full quantum spin-orbital physics must be considered. Thus information on optical excitations, their energies, temperature dependence, and anisotropy, combined with the results of magnetic neutron scattering experiments, provides an important consistency test of the spin-orbital models, and indicates whether orbital and/or spin fluctuations are important in a given compound.
134 citations
TL;DR: In this article, it was shown that the angular spin current (or spin torque) can also induce an electric field, similar to the linear spin current, and the formula for the induced electric field was derived, playing the role of ''Biot-Savart law'' or ''Ampere law''.
Abstract: We find that in order to completely describe the spin transport, apart from spin current (or linear spin current), one has to introduce the angular spin current. The two spin currents, respectively, describe the translational and rotational motion (precession) of a spin. The definitions of these spin current densities are given and their physical properties are discussed. Both spin current densities appear naturally in the spin continuity equation. In particular, we predict that the angular spin current (or the spin torque as called in previous works), similar to the linear spin current, can also induce an electric field $\stackrel{P\vec}{E}$. The formula for the induced electric field $\stackrel{P\vec}{E}$ by the angular spin current element is derived, playing the role of ``Biot-Savart law'' or ``Ampere law.'' When at large distance $r$, this induced electric field $\stackrel{P\vec}{E}$ scales as $1∕{r}^{2}$, whereas the $\stackrel{P\vec}{E}$ field generated from the linear spin current goes as $1∕{r}^{3}$.
TL;DR: In this article, the authors investigated ultrafast photo-induced spin dynamics by time-resolved magneto-optical Kerr spectroscopy for various ferromagnetic and ferrimagnetic compounds.
Abstract: Ultrafast photoinduced spin dynamics has been investigated by time-resolved magneto-optical Kerr spectroscopy for various ferromagnetic and ferrimagnetic compounds: FeCr2S4, CoCr2S4, CuCr2Se4, CdCr2Se4, La0.6Sr0.4MnO3, and SrRuO3. The temporal demagnetization process, which is observed commonly for all the compounds, essentially consists of two components: One is an instantaneous change which originates perhaps from multiple emissions of magnetic excitations during nonradiative decay of photoexcited carriers, and the other is a delayed response due to thermalization of the spin system. The time constant of the delayed change depends strongly on materials and is scaled with the magnetocrystalline anisotropy, indicating that spin-orbit coupling is a dominant interaction for this process.
TL;DR: Simulations show that as the field decreases from saturation, the rings switch from a "bamboo" to a novel "twisted bamboo" state at a certain critical field, and predict a corresponding dip in the dependence of the spin wave frequency on the magnetic field.
Abstract: The spin dynamics of high-aspect-ratio nickel nanorings in a longitudinal magnetic field have been investigated by Brillouin spectroscopy and the results are compared with a macroscopic theory and three-dimensional micromagnetic simulations. Good agreement is found between the measured and calculated magnetic field dependence of the spin wave frequency. Simulations show that as the field decreases from saturation, the rings switch from a ``bamboo'' to a novel ``twisted bamboo'' state at a certain critical field, and predict a corresponding dip in the dependence of the spin wave frequency on the magnetic field.
TL;DR: In this article, a Brillouin light scattering study and theoretical interpretation of spin-wave modes in arrays of in-plane magnetized micron-size rectangular elements are reported, and the frequency spectra and spatial profiles of these eigenmodes are calculated both analytically and numerically.
Abstract: A Brillouin light scattering study and theoretical interpretation of spin-wave modes in arrays of in-plane magnetized micron-size rectangular ${\mathrm{Ni}}_{80}{\mathrm{Fe}}_{20}$ elements are reported. It is shown that two-dimensional spin-wave eigenmodes of these elements can be approximately described as products of one-dimensional spin-wave eigenmodes of longitudinally and transversely magnetized long finite-width permalloy stripes. The lowest eigenmodes of rectangular elements are of dipole-exchange nature and are localized near the element edges, while the higher eigenmodes are of a mostly dipolar nature and are weakly localized near the element center. The frequency spectra and spatial profiles of these eigenmodes are calculated both analytically and numerically, and are compared with the results of the Brillouin light scattering experiment.
TL;DR: Spatially resolved phase and amplitude spectra of ferromagnetic disks reveal that the lowest order azimuthal spin-wave mode splits into a doublet as the disk size decreases, and it is demonstrated that the splitting is due to the coupling between spin waves and the gyrotropic motion of the vortex core.
Abstract: We have investigated azimuthal spin-wave modes in magnetic vortex structures using time-resolved Kerr microscopy. Spatially resolved phase and amplitude spectra of ferromagnetic disks with diameters from 5 microm down to 500 nm reveal that the lowest order azimuthal spin-wave mode splits into a doublet as the disk size decreases. We demonstrate that the splitting is due to the coupling between spin waves and the gyrotropic motion of the vortex core.
TL;DR: The magnetic structures and phase diagram have been determined, and reveal additional transitions below 8 K where the ferroelectrically displaced Ho3+ ions are ordered magnetically.
Abstract: Hexagonal HoMnO3 is a frustrated antiferromagnet (T(N)=72 K) ferroelectric (T(C)=875 K) in which these two order parameters are coupled. Our neutron measurements of the spin-wave dispersion for the S=2 Mn3+ on the layered triangular lattice are well described by a two-dimensional nearest-neighbor Heisenberg exchange J=2.44 meV, and an anisotropy D that is 0.28 meV above the spin-reorientation transition at 40 K and 0.38 meV below. For H parallel c the magnetic structures and phase diagram have been determined, and reveal additional transitions below 8 K where the ferroelectrically displaced Ho3+ ions are ordered magnetically.
TL;DR: In this paper, a new scheme of realizing both spin filtering and spin pumping by using ac-driven double quantum dots in the Coulomb blockade regime was proposed and analyzed, where the spin polarization of the current can be controlled by tuning the parameters of the ac field.
Abstract: We propose and analyze a new scheme of realizing both spin filtering and spin pumping by using ac-driven double quantum dots in the Coulomb blockade regime. By calculating the current through the system in the sequential tunneling regime, we demonstrate that the spin polarization of the current can be controlled by tuning the parameters (amplitude and frequency) of the ac field. We also discuss spin relaxation and decoherence effects in the pumped current.
TL;DR: Quantized spin-wave eigenmodes in single, 16 nm thick and 0.75 to 4 mum wide square permalloy islands with a fourfold closure domain structure have been investigated by microfocus Brillouin light scattering spectroscopy and time resolved scanning magneto-optical Kerr microscopy.
Abstract: Quantized spin-wave eigenmodes in single, 16 nm thick and 0.75 to $4\text{ }\ensuremath{\mu}\mathrm{m}$ wide square permalloy islands with a fourfold closure domain structure have been investigated by microfocus Brillouin light scattering spectroscopy and time resolved scanning magneto-optical Kerr microscopy. Up to six eigenmodes were detected and classified. The main direction of the spin-wave quantization in the domains was found to be perpendicular to the local static magnetization. An additional less pronounced quantization along the direction parallel to the static magnetization was also observed.
TL;DR: In this article, the authors studied magnetic vortex oscillations in soft submicron cylindrical dots with variable thickness and diameter by an analytical approach and micromagnetic simulations, and they considered two kinds of modes of the vortex magnetization oscillations: (1) low-frequency translation mode, corresponding to the movement of a vortex as a whole near its equilibrium position and (2) high-frequency vortex modes, which correspond to radially symmetric oscillations of the magnetization, mainly outside the vortex core.
Abstract: We have studied magnetic vortex oscillations in soft submicron cylindrical dots with variable thickness and diameter by an analytical approach and micromagnetic simulations. We have considered two kinds of modes of the vortex magnetization oscillations: (1) low-frequency translation mode, corresponding to the movement of the vortex as a whole near its equilibrium position and (2) high-frequency vortex modes, which correspond to radially symmetric oscillations of the vortex magnetization, mainly outside the vortex core. The vortex translational eigenmode was calculated numerically in frequency and time domains for different dot aspect ratios. To describe the discrete set of vortex high-frequency modes we applied the linearized equation of motion of dynamic magnetization over the vortex ground state. We considered only radially symmetric magnetization oscillations modes. The eigenfrequencies of both kinds of the excitation modes are determined by magnetostatic interactions. They are proportional to the thickness/diameter ratio and lie in the GHz range for typical dot sizes.
TL;DR: Using elastic and inelastic neutron scattering it is shown that a cubic spinel, CdCr2O4, undergoes an elongation along the c axis (c > a = b) at its spin-Peierls-like phase transition at T(N) = 7.8 K.
Abstract: Using elastic and inelastic neutron scattering we show that a cubic spinel, CdCr2O4, undergoes an elongation along the c axis (c > a = b) at its spin-Peierls-like phase transition at T(N) = 7.8 K. The Neel phase (T < T(N)) has an incommensurate spin structure with a characteristic wave vector Q(M) = (0, delta,1) with delta approximately 0.09 and with spins lying on the ac plane. This is in stark contrast to another well-known Cr-based spinel, ZnCr2O4, that undergoes a c-axis contraction and a commensurate spin order. The magnetic excitation of the incommensurate Neel state has a weak anisotropy gap of 0.6 meV and it consists of at least three bands extending up to 5 meV.
TL;DR: It is shown that a new resonant magnetic excitation at incommensurate momenta, observed recently by inelastic neutron scattering experiments on YBa2Cu3O6.85 and YBa 2Cu3 O6.6, is a spin exciton, located in the Brillouin zone and determined by the momentum dependence of the particle-hole continuum.
Abstract: We show that a new resonant magnetic excitation at incommensurate momenta, observed recently by inelastic neutron scattering experiments on YBa2Cu3O6.85 and YBa2Cu3O6.6, is a spin exciton. Its location in the Brillouin zone and its frequency are determined by the momentum dependence of the particle-hole continuum. We identify several features that distinguish this novel mode from the previous resonance mode observed near Q=(pi,pi).
TL;DR: A one-dimensional (iso)spin 1/2 Bose gas with repulsive delta-function interaction by the Bethe Ansatz method is studied and the excitations above the polarized ground state are discussed.
Abstract: We study a one-dimensional (iso)spin 1/2 Bose gas with repulsive delta-function interaction by the Bethe Ansatz method and discuss the excitations above the polarized ground state. In addition to phonons the system features spin waves with a quadratic dispersion. We compute analytically and numerically the effective mass of the spin wave and show that the spin transport is greatly suppressed in the strong coupling regime, where the isospin-density (or ``spin-charge\'\') separation is maximal. Using a hydrodynamic approach, we study spin excitations in a harmonically trapped system and discuss prospects for future studies of two-component ultracold atomic gases.
TL;DR: In this article, the authors used the dynamical matrix method to calculate spin excitations in magnetic particles and found that the mode frequencies strongly depend on the eccentricity of the dots and on the direction of the applied magnetic field.
Abstract: Brillouin light scattering (BLS) spectra have been measured in arrays of cylindrical Permalloy dots with elliptical cross section, $200\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ wide, $15\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ thick, and eccentricities from 1 to 3. Several spin modes are observed and their frequencies tracked as a function of the direction of the applied $1.5\phantom{\rule{0.3em}{0ex}}\mathrm{kOe}$ magnetic field $\mathbit{H}$. The experimental data are interpreted within the framework of the recently introduced dynamical matrix method to calculate spin excitations in magnetic particles. We find that the mode frequencies strongly depend on the eccentricity of the dots and on the direction of the applied field. For fields along the principal axes the solutions can be classified into: (i) modes localized near the particle ends, (ii) modes with nodal lines perpendicular to $\mathbit{H}$ (backwardlike modes), (iii) modes with nodal lines parallel to $\mathbit{H}$ (Damon-Eshbach-like modes) and (iv) modes with both parallel and perpendicular nodal lines. In cases where the frequencies of two modes in different families are similar, some hybridization between the modes is observed. For $\mathbit{H}$ along nonsymmetry directions only the modes of type (i) remain reasonably well defined, other modes can at best be described as hybrids of modes in the above categories. Determining which of the modes is active in BLS experiments leads to excellent agreement with the experimental results.
TL;DR: A spin-wave theory is presented for the magnetization dynamics in a ferromagnetic film that is traversed by spin-polarized carriers at high direct-current densities and explains quantitatively recent experimental results in nanometric point contacts onto magnetic multilayers.
Abstract: A spin-wave theory is presented for the magnetization dynamics in a ferromagnetic film that is traversed by spin-polarized carriers at high direct-current densities. It is shown that nonlinear effects due to four-magnon interactions arising from dipolar and surface anisotropy energies limit the growth of the driven spin wave and produce shifts in the microwave frequency oscillations. The theory explains quantitatively recent experimental results in nanometric point contacts onto magnetic multilayers showing downward frequency shifts (redshifts) with increasing current, if the external field is on the film plane, and upward shifts (blueshifts), if the field is perpendicular to the film.
TL;DR: In this paper, the authors studied spin pumping in Pd∕Fe(001) ultrathin crystalline films prepared on GaAs(001), by ferromagnetic resonance (FMR).
Abstract: We have studied spin pumping in Pd∕Fe(001) ultrathin crystalline films prepared on GaAs(001) by ferromagnetic resonance (FMR). FMR measurements show that the Pd(001) overlayers lead to an appreciable attenuation of the spin current, which was generated by the precessing magnetization of Fe. Pd overlayers thicker than about 10 nm act as perfect spin sinks. It is argued that the loss of spin coherence in Pd is caused by scattering with spin fluctuations.
TL;DR: The time-resolved magnetic response of ultrathin epitaxial Fe(001) films grown on GaAs (001) and covered by Au, Pd, and Cr capping layers was investigated by time and spatially resolved Kerr effect measurements.
Abstract: The time-resolved magnetic response of ultrathin epitaxial Fe(001) films grown on GaAs(001) and covered by Au, Pd, and Cr capping layers was investigated by time and spatially resolved Kerr effect measurements. The magnetization was excited by an in-plane magnetic field pulse using the transient internal field generated at a Schottky barrier while the wavelength of the excitation (resonant mode) was roughly 4 mu m. Each of the three cap layers affected the spin relaxation in a unique way. Au cap layers resulted in the bulk Gilbert damping of the Fe film. Pd cap layers caused an additional Gilbert damping due to spin-pump or spin-sink effects. Cr cap layers lead to a strong extrinsic damping which can be described by two-magnon scattering. In this case the strength of the extrinsic damping can be controlled by a field induced shift of the spin wave manifold with respect to the excited k vector.
TL;DR: It is shown that spin-wave instability is absent in a state containing a domain wall, which indicates that nucleation of magnetic domains occurs above a certain critical spin current.
Abstract: A large spin current applied to a uniform ferromagnet leads to a spin-wave instability as pointed out recently. In this Letter, it is shown that such spin-wave instability is absent in a state containing a domain wall, which indicates that nucleation of magnetic domains occurs above a certain critical spin current. This scenario is supported also by an explicit energy comparison of the two states under spin current.
TL;DR: In this paper, the spin polarization near the edge induced by the spin Hall effect was shown to be an intrinsic property independent of the strength of disorder scattering, and it becomes larger and more strongly localized as the spin-orbit coupling strength increases.
Abstract: The controlled generation of localized spin densities is a key enabler of semiconductor spintronics In this work, we study spin Hall effect induced edge-spin accumulation in a two-dimensional hole gas with strong spin orbit interactions. We argue that it is an intrinsic property, in the sense that it is independent of the strength of disorder scattering. We show numerically that the spin polarization near the edge induced by this mechanism can be large, and that it becomes larger and more strongly localized as the spin-orbit coupling strength increases, and is independent of the width of the conducting strip once this exceeds the elastic scattering mean-free-path. Our experiments in two-dimensional hole gas microdevices confirm this remarkable spin Hall effect phenomenology. Achieving comparable levels of spin polarization by external magnetic fields would require laboratory equipment whose physical dimensions and operating electrical currents are a million times larger than those of our spin Hall effect devices.
TL;DR: In this article, a fast-rise-time magnetic field pulse perpendicular to the plane is observed in magneto-optic Kerr microscopy and Fourier transformation and two types of modes are observed: modes with circular nodes and modes with diametric nodes.
Abstract: Micron-sized ferromagnetic permalloy disks having an in-plane vortexlike configuration are excited by a fast-rise-time magnetic-field pulse perpendicular to the plane. The excited modes are imaged using time-resolved magneto-optic Kerr microscopy and Fourier transformation. Two types of modes are observed: modes with circular nodes and modes with diametric nodes. The frequency of the modes with circular nodes increases with the number of nodes. In contrast, the frequency of the modes with diametric nodes decreases with the number of nodes. This behavior is explained accurately by an analytical model.