Phonon Gas Model Research Articles

Bond Strain and Rotation Behaviors of Anharmonic Thermal Carriers in ‐RDX

AbstractIn this paper, we examine how bond strain and rotation carry heat in the molecular crystal α‐RDX. We calculate anharmonic lifetimes and then rank order the phonons and their distortions to the lattice by their importance as carriers of thermal energy. The motions of the atoms that constitute the strain and rotation are determined using the phonon mode energy and mode shapes under the harmonic approximation. To draw the distinction between propagating and diffusive carriers, we additionally compare the thermal conductivity estimates from three microscale models: phonon gas model, Cahill‐Watson‐Pohl formula and Allen‐Feldman harmonic theory. We found that the modes that contribute substantially to the thermal conductivity are also the dominant contributors to the strain and rotation of all bonds in ‐RDX, among which N−N and N−O bonds were found to exhibit the largest strains and rotations. We also found that on average, the N−N bonds exhibit the largest deformation, more strain and more rotation than N−O bonds. Our calculations confirm that large straining of the N−N and N−O bonds results from relatively larger displacement of the nitrogen atom in the nitro group −NO2.

(Invited) Limits to Phonon Thermal Boundary Resistance across GaN Interfaces

As material length scales decrease below typical vibrational heat carrier (e.g., phonon) mean free paths, the thermal boundary resistances (TBR) at heterogeneous material interfaces become the dominant thermal resistances that dictate temperature rises, thermal runaway, and power density at failure in a wide array of devices. In this work, I will discuss the phonon scattering processes that dictate the thermal boundary conductance (TBC, the inverse of TBR, so TBC = 1/TBR) across GaN-based interfaces, including the maximum limits to TBC at GaN interfaces and the roles of point defect scattering and interfacial layers on GaN TBC. I will first present experimental measurements of the TBC from 78−500 K across isolated heteroepitaxially grown ZnO films on GaN substrates. This data provides an assessment of the underlying assumptions driving phonon gas-based models, such as the diffuse mismatch model (DMM), and atomistic Green’s function (AGF) formalisms used to predict TBC. Our measurements, when compared to previous experimental data, suggest that TBC can be influenced by long wavelength, zone center modes in a material on one side of the interface as opposed to the “vibrational mismatch” concept assumed in the DMM; this disagreement is pronounced at high temperatures. At room temperature, we measure the ZnO/GaN TBC as 490[+150,−110] MW m-2 K-1. The disagreement among the DMM and AGF, and the experimental data at elevated temperatures, suggests a non-negligible contribution from other types of modes that are not accounted for in the fundamental assumptions of these harmonic based formalisms, which may rely on anharmonicity. Given the high quality of these ZnO/GaN interfaces, these results provide an invaluable, critical, and quantitative assessment of the accuracy of assumptions in the current state of the art computational approaches used to predict phonon TBC across interfaces. I will then turn my discussion to focus on the role that interfacial imperfections, such as point defects and continuous films at the interface, have on GaN TBC. These results demonstrate the large variability that can occur in TBC at GaN-based interfaces based on interfacial non-idealities. 1. J. T. Gaskins, et al. "Thermal boundary conductance across heteroepitaxial zno/gan interfaces: Assessment of the phonon gas model," Nano Letters, 18, 7469–7477 (2018). 2. B. F. Donovan, et al. "Thermal boundary conductance across metal-gallium nitride interfaces from 80 - 450 K," Applied Physics Letters, 105, 203502 (2014).

Lattice anharmonicity, phonon dispersion, and thermal conductivity of PbTe studied by the phonon quasiparticle approach

We investigated the vibrational property of lead telluride (PbTe) with a focus on lattice anharmonicity at moderate temperatures $(300lTl800$ K) using the phonon quasiparticle approach which combines first-principles molecular dynamics and lattice dynamics. The calculated anharmonic phonon dispersions are strongly temperature dependent and some phonon modes adopt giant frequency shifts, e.g., transverse optical modes in the long-wavlength regime. As a result, we witness the avoided crossing between transverse optical modes and longitudinal acoustic modes at elevated temperature, in good agreement with experimentation and available theoretical studies. These results, together with the large root-mean-square displacements of atoms, reveal a strong anharmonic effect in PbTe. The obtained phonon lifetimes allow studies of transport properties. For considered temperatures, the phonon mean free paths can be shorter than lattice constants at relatively high temperature, especially for optical modes. This finding goes against the widely employed minimal phonon mean free path concept. As such, the calculated lattice thermal conductivity of PbTe, which is indeed relatively small, does not have the prescribed minima at high temperature, showcasing the breakdown of the minimal mean free path theory. Our study provides a basis for delineating vibrational and transport properties of PbTe and other thermoelectric materials within the framework of the phonon gas model.

The Importance of Phonons with Negative Phase Quotient in Disordered Solids

Current understanding of phonons is based on the phonon gas model (PGM), which is best rationalized for crystalline materials. However, most of the phonons/modes in disordered materials have a different character and thus may contribute to heat conduction in a fundamentally different way than is described by PGM. For the modes in crystals, which have sinusoidal character, one can separate the modes into two primary categories, namely acoustic and optical modes. However, for the modes in disordered materials, such designations may no longer rigorously apply. Nonetheless, the phase quotient (PQ) is a quantity that can be used to evaluate whether a mode more so shares a distinguishing property of acoustic vibrations manifested as a positive PQ, or a distinguishing property of an optical vibrations manifested as negative PQ. In thinking about this characteristic, there is essentially no intuition regarding the role of positive vs. negative PQ vibrational modes in disordered solids. Given this gap in understanding, herein we studied the respective contributions to thermal conductivity for several disordered solids as a function of PQ. The analysis sheds light on the importance of optical like/negative PQ modes in structurally/compositionally disordered solids, whereas in crystalline materials, the contributions of optical modes are usually small.

Temperature dependence and cation effects in the thermal conductivity of glassy and molten alkali borates

The thermal conductivity of Li2O-B2O3, Na2O-B2O3 and K2O-B2O3 glass systems was measured as a function of the temperature. As the temperature increases, the thermal conductivity of the glass phase initially increases and then reaches a plateau. Afterwards, in the liquid phases, a further increase in the temperature leads to a decrease in the thermal conductivity. The thermal conduction phenomenon can be better described by considering the glass and molten oxide systems as a one-dimensional continuum. It was found that the temperature corresponding to the highest thermal conductivity lies close to the one-dimensional Debye temperature (ΘD1). According to phonon gas model, the variables affecting the thermal conductivity were evaluated. Below ΘD1, the increase in heat capacity with the temperature leads to a corresponding increase in the thermal conductivity. The heat capacity then becomes constant above ΘD1 leading to the observed plateau in the thermal conductivity of the glass phase. After melting the glass, the decrease in thermal conductivity with increasing temperature is due to changes in sound velocity and mean free path. The relative content of 3- and 4-coordinated boron was analyzed by 11B MAS-NMR. The cation effect on the thermal conductivity of alkali borate glasses was evaluated through their ionization potentials.

Phonon transport at interfaces between different phases of silicon and germanium

Current knowledge and understanding of phonon transport at interfaces are wholly based on the phonon gas model (PGM). However, it is difficult to rationalize the usage of the PGM for disordered materials, such as amorphous materials. Thus, there is essentially no intuition regarding interfaces with amorphous materials. Given this gap in understanding, herein we investigated heat conduction at different crystalline and amorphous Si/Ge interfaces using the recently developed interface conductance modal analysis method, which does not rely on the PGM and can therefore treat an interface with a disordered material. The results show that contrary to arguments based on lower mean free paths in amorphous materials, the interface conductances are quite high. The results also show that the interfacial modes of vibration in the frequency region of 12–13 THz are so important that perturbing the natural vibrations with velocity rescaling heat baths (i.e., in non-equilibrium molecular dynamics simulations) affects the conductance even when the heat baths are >60 nm away from the interface. The results suggest that it may be possible to affect interfacial heat transfer by perturbations very far away from the interface, which is an effect that cannot be explained or even rationalized by the traditional paradigm that stems from the Landauer formalism.

Examining the Validity of the Phonon Gas Model in Amorphous Materials.

The idea of treating phonon transport as equivalent to transport through a gas of particles is termed the phonon gas model (PGM), and it has been used almost ubiquitously to try and understand heat conduction in all solids. However, most of the modes in disordered materials do not propagate and thus may contribute to heat conduction in a fundamentally different way than is described by the PGM. From a practical perspective, the problem with trying to apply the PGM to amorphous materials is the fact that one cannot rigorously define the phonon velocities for non-propagating modes, since there is no periodicity. Here, we tested the validity of the PGM for amorphous materials by assuming the PGM is applicable, and then, using a combination of lattice dynamics, molecular dynamics (MD) and experimental thermal conductivity data, we back-calculated the phonon velocities for the vibrational modes. The results of this approach show that if the PGM was valid, a large number of the mid and high frequency modes would have to have either imaginary or extremely high velocities to reproduce the experimental thermal conductivity data. Furthermore, the results of MD based relaxation time calculations suggest that in amorphous materials there is little, if any, connection between relaxation times and thermal conductivity. This then strongly suggests that the PGM is inapplicable to amorphous solids.

Non-negligible Contributions to Thermal Conductivity From Localized Modes in Amorphous Silicon Dioxide.

Thermal conductivity is important for almost all applications involving heat transfer. The theory and modeling of crystalline materials is in some sense a solved problem, where one can now calculate their thermal conductivity from first principles using expressions based on the phonon gas model (PGM). However, modeling of amorphous materials still has many open questions, because the PGM itself becomes questionable when one cannot rigorously define the phonon velocities. In this report, we used our recently developed Green-Kubo modal analysis (GKMA) method to study amorphous silicon dioxide (a-SiO2). The predicted thermal conductivities exhibit excellent agreement with experiments and anharmonic effects are included in the thermal conductivity calculation for all the modes in a-SiO2 for the first time. Previously, localized modes (locons) have been thought to have a negligible contribution to thermal conductivity, due to their highly localized nature. However, in a-SiO2 our results indicate that locons contribute more than 10% to the total thermal conductivity from 400 K to 800 K and they are largely responsible for the increase in thermal conductivity of a-SiO2 above room temperature. This is an effect that cannot be explained by previous methods and therefore offers new insight into the nature of phonon transport in amorphous/glassy materials.

Phonon Quasiparticles and Anharmonic Free Energy in Complex Systems

We use a hybrid strategy to obtain anharmonic frequency shifts and lifetimes of phonon quasiparticles from first principles molecular dynamics simulations in modest size supercells. This approach is effective irrespective of crystal structure complexity and facilitates calculation of full anharmonic phonon dispersions, as long as phonon quasiparticles are well defined. We validate this approach to obtain anharmonic effects with calculations in MgSiO3 perovskite, the major Earth forming mineral phase. First, we reproduce irregular thermal frequency shifts of well characterized Raman modes. Second, we combine the phonon gas model (PGM) with quasiparticle frequencies and reproduce free energies obtained using thermodynamic integration. Combining thoroughly sampled quasiparticle dispersions with the PGM we then obtain first-principles anharmonic free energy in the thermodynamic limit (N→∞).

A transient ballistic–diffusive heat conduction model for heat pulse propagation in nonmetallic crystals

A hybrid phonon gas model was developed to describe heat pulse propagation in dielectric crystals. In this model, heat energy is described as a mixture of longitudinal and transverse phonon gases. The novel aspects of this model include: (1) the distinction between longitudinal and transverse phonon excitations is taken into account; (2) transitions between ballistic and second sound transport, ballistic and diffusive transport, and second sound and diffusive transport are successfully captured. For the first time, benchmark cases of heat-pulse experiments in NaF at low temperatures in ballistic, ballistic–diffusive, and diffusive regions have been completely reconstructed in numerical simulations, which demonstrate the validity of the new model. It was elucidated how heat pulses are transmitted by longitudinal, ballistic transverse and dispersive transverse phonons at low temperatures. The numerical results not only yield new insight in physics of transient ballistic–diffusive heat conduction with effects from different phonon scattering processes, but also provide numerical tools to study similar transient ballistic–diffusive heat conduction in nanoelectronics and modern optoelectronics.

Phonon transport analysis of silicon germanium alloys using molecular dynamics simulations

The phonon transport properties and the lattice thermal conductivity of silicon germanium alloy crystals have been investigated based on phonon gas model by using classical molecular dynamics simulations. The attenuation of the mode-dependent phonon relaxation time due to alloying and its dependence on the alloy fraction were quantified by projecting the molecular dynamics phase space trajectory onto the normal mode of the alloyed crystal. By empirically approximating the group velocities from the extended dispersion relations, the lattice thermal conductivity was calculated based on the phonon gas model under relaxation time approximation. The obtained reduction in the lattice thermal conductivity caused by alloying agrees well with that of the experiment and direct non-equilibrium molecular dynamics calculations. The phonon-mean-free-path dependent contribution to thermal conductivity suggests that the effect of nanostructuring can have non-monotonic dependence on the alloy fraction.

A Hybrid Phonon Gas Model for Transient Ballistic-Diffusive Heat Transport

We present a continuum hybrid phonon gas model to describe transient ballistic-diffusive heat transport. In this model, heat energy is carried by a mixture of longitudinal and transverse phonon gases so that the distinction between longitudinal and transverse phonon excitations is taken into account. This new model is validated by the successful reconstruction of benchmark cases of heat-pulse experiments in NaF, which have never been completely reconstructed before. It is elucidated how thermal pulses are transmitted by longitudinal and transverse phonon gases. This model not only helps us yield new insight in transient ballistic-diffusive heat conduction mechanisms but also provides numerical tools to study transient ballistic-diffusive heat conduction in nanoelectronic and modern optoelectronics.

Thermal diffusivity of alkali and silver halide crystals as a function of temperature

The phonon component of thermal diffusivity (D) for ten synthetic single-crystals (LiF, NaCl, NaI, NaI:Tl, KCl, KBr, CsI, CsI:Tl, AgCl, and AgBr) with the B1 and B2 structures was measured from ambient temperature (T) up to ∼1093 K using contact-free, laser-flash analysis, from which effects of ballistic radiative transfer were removed. We investigated optical flats from different manufacturers as well as pellets made from compressed powders of most of the above chemical compositions plus LiI, NaBr, KI, RbCl, RbBr, RbI, CsCl, CsBr, and AgI. Impurities were characterized using various spectroscopic methods. With increasing T,D decreases such that near melting the derivatives ∂D/∂T are low, −0.0006±0.0004 mm2 s−1 K−1. Our results are ∼16% lower than D298 previously obtained with contact methods, which are elevated by ballistic radiative transfer for these infrared (IR) windows, and are well described by either D−1 following a low order polynomial in T, or by D−1∝T+n, where n ranges from 1.0294 to 1.9429. Inverse correlations were found between D298 and both density and thermal expansivity (α). Primitive lattice constant times compressional velocity correlates directly with D but changes much more slowly with temperature. Instead, D(T) is proportional to (TαL)−1 from ∼0 K up to the limit of measurements, in accord with these physical properties being anharmonic. On average, the damped harmonic oscillator–phonon gas model reproduces D298 based on two physical properties: compressional velocity and the damping coefficient (Γ) from analysis of IR reflectivity data. Given large uncertainties in Γ(T), D−1(T) is reproduced for LiF, NaCl, MgO, and the silver halides, for which IR reflectivity data are available. Our correlations show that optical phonons largely govern heat transport of insulators, and permit prediction of D and thus thermal conductivity for simple, diatomic solids.

Heat pulse propagation by second sound in dielectric crystals

A non-linear thermodynamic model describing heat pulse propagation in dielectric crystals at low temperature is proposed. This work is a generalization of those of Cattaneo and of Guyer and Krumhansl, and is complementary to an earlier paper (Lebon G, Torrisi M and Valenti A 1995 J. Phys.: Condens. Matter 7 1461), which was mainly devoted to a linear approach. The model is based on extended irreversible thermodynamics, and uses as field variables the temperature, the heat flux, and the flux of the heat flux. Unlike the simple phonon gas model, the present formalism is compatible with the experimental observation that second sound is temperature dependent. In this paper, explicit expressions for the internal energy and velocity of propagation of weak discontinuities are determined. By making use of experimental data for NaF and Bi, the values of the relevant parameters have been evaluated.