Study of thermal conductivity and thermal rectification in exponential mass graded lattices

Concept of exponential mass variation of oscillators along the chain length of N oscillators is proposed in the present Letter. The temperature profile and thermal conductivity of one-dimensional (1D) exponential mass graded harmonic and anharmonic lattices are studied on the basis of Fermi-Pasta-Ulam (FPU) β model. Present findings conclude that the exponential mass graded chain provide higher conductivity than that of linear mass graded chain. The exponential mass graded anharmonic chain generates the thermal rectification of 70-75% which is better than linear mass graded materials, so far. Thus instead of using linear mass graded material, the use of exponential mass graded material will be a better and genuine choice for controlling the heat flow at nano-scale.     

Thermal rectification and thermal resistive phase cross over in exponential mass graded materials

Concept of the functional graded materials (FGMs) has been explored by consideringexponential mass variation along the chain of anharmonic oscillators in the study of heattransport at low dimensions. This exponential distribution of mass along the space invokesthe diffusion of phonons transport which results to temperature gradient, asymmetric heatflow, thermal rectification and cross over between positive differential thermalresistance (PDTR) and negative differential thermal resistance (NDTR) in one-dimensional(1D) exponential mass graded chain. The temperature dependence thermal rectificationachieved is 4?74% and also predicted that the thermal rectification can be controlled bytuning the higher and lower average temperature limits of two thermal reservoirs. It isalso seen that in FGMs, the thermal conductivity does not change drastically against theaverage temperature of two heat baths. The cross over between PDTR and NDTR can be tunedeither by mass ratio of one dimensional (1D) exponential mass graded anharmonic chainand/or by temperature difference between two heat baths. The figure of merit of the 1Dstructure can also be tuned by mass gradient, the higher mass gradient material will workas the potential candidate for better thermoelectric material.     

The stationary state and the heat equation for a variant of Davies' model of heat conduction

We study a variant of Davies' model of heat conduction, consisting of a chain of (classical or quantum) harmonic oscillators, whose ends are coupled to thermal reservoirs at different temperatures, and where neighboring oscillators interact via intermediate reservoirs. In the weak coupling limit, we show that a unique stationary state exists, and that a discretized heat equation holds. We give an explicit expression of the stationary state in the case of two classical oscillators. The heat equation is obtained in the hydrodynamic limit, and it is proved that it completely describes the macroscopic behavior of the model.     

Normal heat conduction in a chain with a weak interparticle anharmonic potential

We analytically study heat conduction in a chain with an interparticle interaction V(x)= lambda[1-cos(x)] and harmonic on-site potential. We start with each site of the system connected to a Langevin heat bath, and investigate the case of small coupling for the interior sites in order to understand the behavior of the system with thermal reservoirs at the boundaries only. We study, in a perturbative analysis, the heat current in the steady state of the one-dimensional system with a weak interparticle potential. We obtain an expression for the thermal conductivity, compare the low and high temperature regimes, and show that, as we turn off the couplings with the interior heat baths, there is a "phase transition": Fourier's law holds only at high temperatures.     

Thermal Conductivity for a Momentum Conservative Model

We introduce a model whose thermal conductivity diverges in dimension 1 and 2, while it remains finite in dimension 3. We consider a system of oscillators perturbed by a stochastic dynamics conserving momentum and energy. We compute thermal conductivity via Green-Kubo formula. In the harmonic case we compute the current-current time correlation function, that decay like t ^−d/2 in the unpinned case and like t ^−d/2–1 if an on-site harmonic potential is present. This implies a finite conductivity in d ≥ 3 or in pinned cases, and we compute it explicitly. For general anharmonic strictly convex interactions we prove some upper bounds for the conductivity that behave qualitatively as in the harmonic cases.     

Strange Heat Flux in (An)Harmonic Networks

We study the heat transport in systems of coupled oscillators driven out of equilibrium by Gaussian heat baths. We illustrate with a few examples that such systems can exhibit “strange” transport phenomena. In particular, circulation of heat flux may appear in the steady state of a system of three oscillators only. This indicates that the direction of the heat fluxes can in general not be “guessed” from the temperatures of the heat baths. Although we primarily consider harmonic couplings between the oscillators, we explain why this strange behavior persists under weak anharmonic perturbations.     

Path integral computation of phonon anharmonicity

The partition function of an oscillator disturbed by a set of electron particle paths has been computed by a path integral formalism which permits to evaluate at any temperature the relevant cumulant terms in the series expansion. The low temperature cutoffs in the anharmonic cumulant series are determined fulfilling the constraint of the third law of thermodynamics. The general method here proposed has been applied to the semiclassical Su-Schrieffer-Heeger model whose time dependent source current is linear in the oscillator displacement field. We find that this peculiar current induces large electron-phonon anharmonicities on the phonon subsystem. As a signature of anharmonicity the phonon heat capacity shows a peak whose temperature location strongly varies with the strength of the e-ph coupling. Since the electron hopping potential provides a sizeable background in the low and intermediate temperature range, such a peak is partly smeared in the total heat capacity. Low energy oscillators are more sensitive to anharmonic perturbations.Received: 7 January 2004, Published online: 3 August 2004PACS: 71.20.Rv Polymers and organic compounds - 31.15.Kb Path-integral methods - 63.20.Kr Phonon-electron and phonon-phonon interactions     

Heat Transport in Ordered Harmonic Lattices

We consider heat conduction across an ordered oscillator chain with harmonic interparticle interactions and also onsite harmonic potentials. The onsite spring constant is the same for all sites excepting the boundary sites. The chain is connected to Ohmic heat reservoirs at different temperatures. We use an approach following from a direct solution of the Langevin equations of motion. This works both in the classical and quantum regimes. In the classical case we obtain an exact formula for the heat current in the limit of system size N→∞. In special cases this reduces to earlier results obtained by Rieder, Lebowitz and Lieb and by Nakazawa. We also obtain results for the quantum mechanical case where we study the temperature dependence of the heat current. We briefly discuss results in higher dimensions.     

Nonequilibrium invariant measure under heat flow

We provide an explicit representation of the nonequilibrium invariant measure for a chain of harmonic oscillators with conservative noise in the presence of stationary heat flow. By first determining the covariance matrix, we are able to express the measure as the product of Gaussian distributions aligned along some collective modes that are spatially localized with power-law tails. Numerical studies show that such a representation applies also to a purely deterministic model, the quartic Fermi-Pasta-Ulam chain.     

New equipartition results for normal mode energies of anharmonic chains

The canonical and microcanonical distributions of energy among the normal modes of an anharmonic chain with nearest-neighbor interactions and free ends are examined. If the interparticle potential is an even function, then energy is distributed uniformly among the normal modes at all energy densities. If the interparticle potential is not an even function but includes quadratic, cubic, and quartic terms, then the energy sharing among the normal modes is also uniform in both the small- and large-energy density limits. At large energies, in this latter case the energy per normal mode scales as the square root of the energy density. Thus we find equipartition of energy among the normal modes of an anharmonic chain. The sum of the normal mode energies is less than the total energy of the chain.     

Heat Transport in Harmonic Lattices

We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green’s function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.     

Single mode heat rectifier: controlling energy flow between electronic conductors

We study heat transfer between conductors, mediated by the excitation of a monomodal harmonic oscillator. Using a simple model, we show that the onset of rectification in the system is directly related to the nonlinearity of the electron gas dispersion relation. When the metals have a strictly linear dispersion relation, a Landauer-type expression for the thermal current holds, symmetric with respect to the temperature difference. Rectification becomes prominent when deviations from linear dispersion exist, and the fermionic model cannot be mapped into a harmonic bosonized representation. The effects described here are relevant for understanding radiative heat transfer and vibrational energy flow in electrically insulating molecular junctions.     

Normal thermal conduction in lattice models with asymmetric harmonic interparticle interactions

We study the thermal conduction behaviors of one-dimensional lattice models with asymmetric harmonic interparticle interactions. Normal thermal conductivity that is independent of system size is observed when the lattice chains are long enough. Because only the harmonic interactions are involved, the result confirms, without ambiguity, that asymmetry plays a key role in normal thermal conduction in one-dimensional momentum conserving lattices. Both equilibrium and nonequilibrium simulations are performed to support the conclusion.     

Can Disorder Induce a Finite Thermal Conductivity in 1D Lattices?

We study heat conduction in one-dimensional mass-disordered harmonic and anharmonic lattices. It is found that the thermal conductivity kappa of the disordered anharmonic lattice is finite at low temperature, whereas it diverges as kappa approximately N0.43 at high temperature. Moreover, we demonstrate that a unique nonequilibrium stationary state in the disordered harmonic lattice does not exist at all.     

Theory of Non-Equilibrium Heat Transport in Anharmonic Multiprobe Systems at High Temperatures

We consider the problem of heat transport by vibrational modes between Langevin thermostats connected by a central device. The latter is anharmonic and can be subject to large temperature difference and thus be out of equilibrium. We develop a classical formalism based on the equation of motion method, the fluctuation–dissipation theorem and the Novikov theorem to describe heat flow in a multi-terminal geometry. We show that it is imperative to include a quartic term in the potential energy to insure stability and to properly describe thermal expansion. The latter also contributes to leading order in the thermal resistance, while the usually adopted cubic term appears in the second order. This formalism paves the way for accurate modeling of thermal transport across interfaces in highly non-equilibrium situations beyond perturbation theory.     

Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation

We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the ?x ⁴-potential, which serves as a toy model for the quantum field theoretical ?⁴-theory.     

Periodic, Quasiperiodic and Chaotic Discrete Breathers in a Parametrical Driven Two-Dimensional Discrete Klein-Gordon Lattice

We study a two-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the two-dimensional Klein-Gordon lattice with hard on-site potential. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.     

Different kinds of discrete breathers in a Sine-Gordon lattice

We study a one-dimensional Sine-Gordon lattice of anharmonic oscillators with cubic and quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the one-dimensional Sine-Gordon lattice no matter whether the nonlinear interaction is cubic or quartic. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.     

Symmetry properties of the heat current in non-ballistic asymmetric junctions: A case study

We consider quantum heat flow in two-terminal junctions and inquire on the connection between the transport mechanism and the junction functionality. Using simple models, we demonstrate that the violation of the Landauer behavior in asymmetric junctions does not necessarily imply the onset of thermal rectification. We also demonstrate through a simple example that a spatial inhomogeneity of the energy spectra is not a necessary condition for thermal rectification.