Nonequilibrium Phonon Distribution Research Articles

Thermal resistance across Si–SiGe alloy interface from phonon distribution mismatch

Interfacial thermal resistance has often been attributed to the mismatch of phonon spectra between two materials and resulting phonon-interface scattering. However, we use the solution of Peierls–Boltzmann transport equation to reveal a substantial nonequilibrium thermal resistance across the interfaces of Si and SiGe alloys at room temperature, despite their nearly identical phonon dispersion and negligible phonon-interface scattering. The Kapitza length of the Si–Si0.99Ge0.01 interface is approximately 600 nm of Si. This originates from the mismatch in phonon distribution between Si and SiGe alloys due to their distinct scattering rates. The mismatch is relaxed by phonon scattering over a region of 1 μm around the interface, corresponding to the upper bound of mean free path Λx of heat-carrying phonons. The relaxation process leads to the significant entropy generation and increased thermal resistance. Introducing a gradual variation in Ge concentration near the interface markedly reduces thermal resistance when implemented over the 1 μm period. Our finding demonstrates that the interfacial thermal resistance can be significant due to the nonequilibrium phonon distribution, even in the absence of phonon-interface scattering. In addition, among various phonon modes with a wide range of Λx, the relaxation of the nonequilibrium is predominantly governed by the phonons with long Λx.

Non-equilibrium phonon distribution caused by an electrical current

In an attempt to explain flash sintering experiments, it had been proposed that the electron–phonon coupling leads to a proliferation of short wave-length lattice vibrations in an electric field. In this paper we investigate this by solving two coupled Boltzmann equations, describing a free electron gas in an electric field scattering from a crystal lattice coupled to a heat bath. The electric field imposes cylindrical symmetry and drives the electrons and the phonons into a non-equilibrium steady state. We find that the phonon distribution shows a strong excess population at the Brillouin zone edge in the direction of the electric field. We argue analytically, that this can be traced back to the shifted Fermi sphere for the electrons. Furthermore, not only energy but also momentum is exchanged in the electron–phonon system, which defies any attempt at describing the system by a two-temperature model.

Hot and cold phonons in electrically biased graphene

Based on first-principles calculations and semiclassical transport theory, we study the nonequilibrium phonon distribution in graphene in the presence of electrical current. In addition to the hot-phonon effect, we observe a cooling of phonon modes at the same time. Interestingly, the presence of electric current along the direction connecting $K$ and ${K}^{\ensuremath{'}}$ valleys induces an opposite dipolelike temperature distribution in the two valleys and at the $\mathrm{\ensuremath{\Gamma}}$ point. This leads to a ``high-order'' valley polarization of phonon distribution between $K$ and ${K}^{\ensuremath{'}}$. Based on the nonequilibrium phonon distribution, we furthermore study their effect on the lattice parameters and find a negative current-induced expansion coefficient. Similar to graphene's negative thermal expansion, this is rooted in the dominant contribution of out-of-plane acoustic phonons, which do not couple directly to electrons, but gain energy from in-plane phonons through anharmonic scattering.

Derivation of a hydrodynamic heat equation from the phonon Boltzmann equation for general semiconductors

We present a formalism to solve the phonon Boltzmann transport equation (BTE) for finite Knudsen numbers that supplies a hydrodynamic heat transport equation similar to the Navier-Stokes equation for general semiconductors. This generalization of Fourier's law applies in general cases, from systems dominated by momentum-preserving normal collisions, as is well known, to kinetic materials dominated by resistive collisions, where it captures nonlocal effects. The key feature of our framework is that the macrostate is described in terms of the heat flux and its first derivatives. We obtain explicit expressions for the nonequilibrium phonon distribution and for the geometry-independent macroscopic parameters as a function of phonon properties that can be calculated from first principles. Ab initio model predictions are found to agree with a wide range of experiments in silicon. In contrast to approaches directly based on the BTE, the hydrodynamic equation can be solved in arbitrary geometries, thus providing a powerful tool for nanoscale heat modeling at a low computational cost.

Suppressing quantum effects by optically driven nonequilibrium phonons

Optically-generated nonequilibrium phonon-distribution is used for exploring the origin of a nonlocal adiabatic response in an interacting Anderson insulator. Exposing the system to weak infrared radiation is shown to effectively suppress a long-range effect observed in field-effect experiments while producing little heating and barely changing the system conductance. These effects are shown to be consistent with the quantum nature of the effect and therefore are peculiar to disordered systems that are quantum-coherent.

Observation of Anomalous Non-Ohmic Transport in Current-Driven Nanostructures

Sufficiently large electric current applied to metallic nanostructures can bring them far out-of-equilibrium, resulting in non-Ohmic behaviors characterized by current-dependent resistance. We experimentally demonstrate a linear dependence of resistance on current in microscopic thin-film metallic wires at cryogenic temperatures, and show that our results are inconsistent with common non-Ohmic mechanisms such as Joule heating. As the temperature is increased, the linear dependence becomes smoothed out, resulting in the crossover to behaviors consistent with Joule heating. A plausible explanation for the observed behaviors is the strongly non-equilibrium distribution of phonons generated by the current. Analysis based on this interpretation suggests that the observed anomalous current-dependent resistance can provide information about phonon transport and electron-phonon interaction at nanoscale. The ability to control the properties of phonons generated by current can lead to new routes for the optimization of thermal properties of electronic nanodevices.

Nonequilibrium kinetics of the electron–phonon subsystem can give rise to electric- and magnetic-plasticity effects in crystals in alternating electric and/or magnetic fields

Kinetic processes in magnetic crystals in an alternating magnetic field and/or a pulsed electric field are studied theoretically, experimentally, and numerically to establish the prime mechanisms by which they influence the structure and the mechanical, dissipative, and magnetic characteristics of crystals. The specific materials studied are severly strained ferritic pearlite steel 15Kh2NMFA and nickel. The paper presents a systematic kinetic analysis of the nonequilibrium dynamics of the electron–phonon subsystem of a magnetic crystal in an electric field. Our proposed method that underlies the analysis solves the system of Boltzmann equations for the electron and phonon distribution functions numerically without expanding the electron distribution function in a power series of the phonon energy. It is shown that an electronic subsystem excited by an electric field transfers energy to the phonon subsystem and thereby massively produces short-wave phonons which act strongly on lattice defects (such as point and linear ones and phase boundaries) and thus redistribute and decrease their density, as well as providing damage healing, decreasing local peak stresses, and reducing the degradation level of construction materials properties. It is found that, under the action of an induction electric field, the electron distribution function becomes nonequilibrium near the Fermi level energy and, as a result of electron–phonon collisions, transfers significant energy to the phonon subsystem, resulting in forming a nonequilibrium phonon distribution function. Based on modified Granato–Lücke and Landau–Gofman models, it is shown, using the calculated phonon distribution function, that the action of phonons on dislocations is much stronger than it would be in the case of thermodynamic equilibrium at the experimentally measured sample heating by 12 K.

Relaxation of a nonequilibrium phonon distribution induced by femtosecond laser irradiation

Ultrafast laser irradiation of solids leads to a thermodynamic nonequilibrium within and between the electron and phonon subsystems of the material. Due to electron-electron and phonon-phonon collisions, both subsystems relax into respective new thermodynamic equilibria within a characteristic thermalization time, which is different for each one of them. Moreover, they equilibrate their temperatures by electron-phonon coupling. The relaxation of the electronic nonequilibrium and its effect on the electron-phonon coupling was the subject of a number of studies, and it is comparably well understood, while the nonequilibrium within the phononic subsystem is usually neglected and the influence of the nonequilibrium phonons on other relaxation processes is unclear. Our calculations show significant differences in the energy transfer rate between the electrons and the phonons depending on whether a nonequilibrium distribution is assumed for the phonons or not. Here, we present a model to study the relaxation of the nonequilibrium phonon subsystem. Collisions between phonons are described within the frame of Boltzmann integrals. From this, an energy-dependent relaxation time can be extracted and inserted into a relaxation-time approach. Within the frame of this model, we study the thermalization of a phonon distribution induced by ultrafast laser irradiation. We show that the thermalization time of such a distribution is of the order of some hundreds of picoseconds. Moreover, we discuss the energy transfer between Fermi-distributed electrons and nonequilibrium phonons and compare it to the energy transfer for equilibrium distributions in both subsystems.

Enhanced quasiparticle lifetime in a superconductor by selective blocking of recombination phonons with a phononic crystal

When quasiparticles in a BCS superconductor recombine into Cooper pairs, phonons are emitted within a narrow band of energies above the pairing energy at 2$\Delta$. We show that a phonon bandgap restricting the escape of recombination phonons from a superconductor can increase the quasiparticle recombination lifetime by more than an order of magnitude. A phonon bandgap can be realized and matched to the recombination energy with a phononic crystal, a periodically-patterned dielectric membrane. We discuss in detail the non-equilibrium quasiparticle and phonon distributions that arise in a superconductor due to a phonon bandgap and a pair-breaking photon signal. Although intrinsically a non-equilibrium effect, the lifetime enhancement in the small-signal regime is remarkably similar to an estimate from an equilibrium formulation. The equilibrium estimate closely follows $\exp(\Omega_{bg}/k_BT_b)$, where $\Omega_{bg}$ is the phonon bandgap energy bandwidth above 2$\Delta$, and $T_b$ is the phonon bath temperature of the coupled electron-phonon system. We discuss the impact of a phononic bandgap on the performance of superconducting devices, and propose a superconducting microwave resonator circuit to measure the enhancement in the quasiparticle lifetime.

Simulation of Phonon Transport in Semiconductors Using a Population-Dependent Many-Body Cellular Monte Carlo Approach

A Monte Carlo rejection technique for numerically solving the complete, nonlinear phonon Boltzmann transport equation (BTE) is presented in this work, including three particles interactions. The technique has been developed to explicitly model population-dependent scattering within a full-band cellular Monte Carlo (CMC) framework, to simulate phonon transport in semiconductors, while ensuring conservation of energy and momentum for each scattering event within gridding error. The scattering algorithm directly solves the many-body problem accounting for the instantaneous distribution of the phonons. Our general approach is capable of simulating any nonequilibrium phase space distribution of phonons using the full phonon dispersion without the need of approximations used in previous Monte Carlo simulations. In particular, no assumptions are made on the dominant modes responsible for anharmonic decay, while normal and umklapp scattering are treated on the same footing. In this work, we discuss details of the algorithmic implementation of both the three-particle scattering for the treatment of the anharmonic interactions between phonons, as well as treating isotope and impurity scattering within the same framework. The simulation code was validated by comparison with both analytical and experimental results; in particular, the simulation results show close agreement with a wide range of experimental data such as thermal conductivity as function of the isotopic composition, the temperature, and the thin-film thickness.

A Finite Difference Lattice Boltzmann Method to Simulate Multidimensional Subcontinuum Heat Conduction

In this paper, a lattice Boltzmann method (LBM)-based model is developed to simulate the subcontinuum behavior of multidimensional heat conduction in solids. Based on a previous study (Chen et al., 2014, “Sub-Continuum Thermal Modeling Using Diffusion in the Lattice Boltzmann Transport Equation,” Int. J. Heat Mass Transfer, 79, pp. 666–675), phonon energy transport is separated to a ballistic part and a diffusive part, with phonon equilibrium assumed at boundaries. Steady-state temperature/total energy density solutions from continuum scales to ballistic scales are considered. A refined LBM-based numerical approach is applied to a two-dimensional simplified transistor model proposed by (Sinha et al. 2006, “Non-Equilibrium Phonon Distributions in Sub-100 nm Silicon Transistors,” ASME J. Heat Transfer, 128(7), pp. 638–647), and the results are compared with the Fourier-based heat conduction model. The three-dimensional (3D) LBM model is also developed and verified at both the ballistic and continuous limits. The impact of film thickness on the cross-plane and in-plane thermal conductivities is analyzed, and a new model of the supplementary diffusion term is proposed. Predictions based on the finalized model are compared with the existing in-plane thermal conductivity measurements and cross-plane thermal conductivity molecular dynamics (MD) results.

Field evaporation of semiconductors assisted by nonequilibrium phonon excitations

Laser-assisted field evaporation of materials is often interpreted as an equilibrium phenomenon that can be described in terms of a thermally assisted process due to the increase of the lattice temperature defined for equilibrium phonon distribution. Our simulations, taking into account nonequilibrium phonon distribution, produced by femtosecond laser radiation, and coupled carrier-phonon kinetics in the sample, show that the evaporation process is assisted by hot phonons. An out-of-thermal-equilibrium phenomenon on the picosecond time scale is thus demonstrated in the case of femtosecond-laser-assisted field evaporation of semiconductors. It is experimentally confirmed by atom-probe measurements using an original double-pulse correlation setup.

Nonequilibrium superconducting thin films with sub-gap and pair-breaking photon illumination

We calculate nonequilibrium quasiparticle and phonon distributions for a number of widely-used low transition temperature thin-film superconductors under constant, uniform illumination by sub-gap probe and pair-breaking signal photons simultaneously. From these distributions we calculate material-characteristic parameters that allow rapid evaluation of an effective quasiparticle temperature using a simple analytical expression, for all materials studied (Mo, Al, Ta, Nb, and NbN) for all photon energies. We also explore the temperature and energy-dependence of the low-energy quasiparticle generation efficiency η by pair-breaking signal photons finding in the limit of thick films at low bath temperatures that is material-independent. Taking the energy distribution of excess quasiparticles into account, we find as the bath temperature approaches the transition temperature in agreement with the assumption of the two-temperature model of the nonequilibrium response that is appropriate in that regime. The behaviour of η with signal frequency scaled by the superconducting energy gap is also shown to be material-independent, and is in qualitative agreement with recent experimental results. An enhancement of η in the presence of sub-gap (probe) photons is shown to be most significant at signal frequencies near the superconducting gap frequency and arises due to multiple photon absorption events that increase the average energy of excess quasiparticles above that in the absence of the probe.

Full-counting statistics for molecular junctions: Fluctuation theorem and singularities

We study the full-counting statistics of charges transmitted through a single-level quantum dot weakly coupled to a local Einstein phonon which causes fluctuations in the dot energy. An analytic expression for the cumulant generating function, accurate up to second order in the electron-phonon coupling and valid for finite voltages and temperatures, is obtained in the extended wide-band limit. The result accounts for nonequilibrium phonon distributions induced by the source-drain bias voltage, and concomitantly satisfies the fluctuation theorem. Extending the counting field to the complex plane, we investigate the locations of possible singularities of the cumulant generating function, and exploit them to identify regimes in which the electron transfer is affected differently by the coupling to the phonons. Within a large-deviation analysis, we find a kink in the probability distribution, analogous to a first-order phase transition in thermodynamics, which would be a unique hallmark of the electron-phonon correlations. This kink reflects the fact that although inelastic scattering by the phonons once the voltage exceeds their frequency can scatter electrons opposite to the bias, this will never generate current flowing against the bias at zero temperature, in accordance with the fluctuation theorem.

Bath dynamics in an exactly solvable qubit model with initial qubit-environment correlations

We study the bath dynamics in the dephasing model of a two-state quantum system (qubit) coupled to an environment of harmonic oscillators. This model was shown [Morozov et al., Phys. Rev. A, 2012, 85, 022101] to admit the analytic solution for the qubit and environment dynamics. Using this solution, we derive the exact expression for the bath reduced density matrix in the presence of initial qubit-environment correlations. We obtain the non-equilibrium phonon distribution function and discuss in detail the time behavior of the bath energy. It is shown that only the inclusion of dynamic correlations between the qubit and the bath ensures the proper time behavior of the quantity which may be interpreted as the "environment energy".

Non-equilibrium superconductivity in quantum-sensing superconducting resonators

Low temperature microwave superconducting resonators (SRs) are attractive candidates for producing quantum-sensitive, arrayable energy or power detectors for astrophysical and other precision measurement applications. Their readout uses a microwave probe signal with quanta of energy well below the threshold for pair-breaking in the superconductor. We have calculated the non-equilibrium quasiparticle and phonon distributions generated by the photons of the probe signal of a resonator operating well below its superconducting transition temperature Tc as the absorbed probe power was changed using the coupled kinetic equations described by Chang and Scalapino. The calculations give insight into a rate equation estimate which suggests that the quasiparticle distributions can be driven far from their thermal equilibrium value for typical readout powers. From the driven quasiparticle distribution functions, the driven quasiparticle number densities and lifetimes were calculated. An effective temperature to describe the driven quasiparticles was defined. The non-equilibrium lifetimes were compared to the distribution-averaged thermal lifetimes at the effective temperature and good agreement was found typically within a few per cent. We used the non-equilibrium quasiparticle distribution to model a representative SR. The complex conductivity and hence the frequency dependence of the experimentally measured forward scattering parameter S21 of the SR as a function of absorbed power were found. The non-equilibrium S21 cannot be accurately modeled by a thermal distribution even at its own elevated temperature, having a higher quality factor in all cases studied, although for low absorbed powers the two effective temperatures are similar. From the non-equilibrium lifetimes and number densities we determined the achievable noise equivalent power (NEP) of the resonator used as a power detector as a function of absorbed microwave power. Simpler expressions to evaluate the effective quasiparticle temperature as a function of absorbed power have also been derived. We conclude that multiple photon absorption from the microwave probe increases the quasiparticle number above the thermal background and ultimately limits the achievable NEP of the resonator at temperatures well below Tc.

Phonon thermal conductivity of GaN nanotubes

We theoretically investigated the phonon thermal conductivity of gallium nitride (GaN) nanotubes with diameters ranging from a few nanometers to 120 nanometers using the Boltzmann transport equation and took into account the phonon dispersion relations of the nanotubes and the influence of boundary scattering on the non-equilibrium phonon distribution. The calculation results show that the phonon thermal conductivity of GaN nanotubes is much lower than that of the bulk counterpart and it depends on the thickness, inner and outer diameters, and surface roughness of the nanotubes. A small thickness or a large surface roughness leads to a small thermal conductivity. The reduction of the phonon thermal conductivity of the nanotubes is mainly due to the decrease of the phonon group velocity, change of the phonon relaxation rate, and enhancement of phonon boundary scattering. The understanding and results on the thermal conductivity obtained in this work are important for the optoelectronic devices based on GaN nanotubes and nanowires, and the developed calculation method on the phonon thermal conductivity is generally applicable and can be used for other nanotube systems.

Impact of Self-Heating Effect on the Electrical Characteristics of Nanoscale Devices

Hot phonon generation and its impact on the current conduction in a nanoscale Si-device are investigated using a Monte Carlo simulation technique. In the quasi-ballistic transport regime, electrons injected from the source lose their energies mainly by emitting optical phonons in the drain. Due to the slow group velocity of the optical phonons, the efficiency of the heat dissipation is so poor that a region with a nonequilibrium phonon distribution, i.e., a hot spot, is created. In this study, we have implemented the hot phonon effect in an ensemble Monte Carlo simulator for the electron transport, and carried out the steady state simulations. Although it is confirmed that the optical phonon temperature in the hot spot is larger than that of acoustic phonons by > 100 K, the electron current density is not significantly affected. The local heating would degrade the hot electron cooling efficiency and the parasitic resistance in the drain, but they have a minor impact on the quasi-ballistic electron transport from the source to the drain.

Effects of dislocations on small signal high frequency hot electron mobility in n-GaN at low and high temperatures under high magnetic fields including hot phonon effect

The small signal high-frequency ac mobility of hot electrons in n-GaN in the extreme quantum limit at low- and high-temperatures has been calculated considering the non-equilibrium phonon distribution as well as the thermal phonon distributions. The energy loss rate has been calculated considering the dominance of the piezo electric coupling scattering and the polar optical phonon scattering while the momentum loss rate has been calculated considering the acoustic phonon scattering via deformation potential and the piezo electric coupling and the dislocation scattering.

Nonlinear effects of phonon fluctuations on transport through nanoscale junctions

We analyze the effect of electron-phonon coupling on the full counting statistics of a molecular junction beyond the lowest order perturbation theory. Our approach allows to take into account analytically the feedback between the nonequilibrium phonon and electronic distributions in the quantum regime. We show that for junctions with high transmission and relatively weak electron-phonon coupling this feedback gives rise to increasingly higher nonlinearities in the voltage dependence of the cumulants of the transmitted charges distribution.