Boltzmann Transport Equation Calculations Research Articles

Defect scattering can lead to enhanced phonon transport at nanoscale

Defect scattering is well known to suppress thermal transport. In this study, however, we perform both molecular dynamics and Boltzmann transport equation calculations, to demonstrate that introducing defect scattering in nanoscale heating zone could surprisingly enhance thermal conductance of the system by up to 75%. We further reveal that the heating zone without defects yields directional nonequilibrium with overpopulated oblique-propagating phonons which suppress thermal transport, while introducing defects redirect phonons randomly to restore directional equilibrium, thereby enhancing thermal conductance. We demonstrate that defect scattering can enable such thermal transport enhancement in a wide range of temperatures, materials, and sizes, and offer an unconventional strategy for enhancing thermal transport via the manipulation of phonon directional nonequilibrium.

Monte Carlo simulation of phonon transport from ab-initio data with Nano-κ

Understanding of heat transport in nanodevices is a challenging issue for several new technologies. It requires specific modelling approaches and dedicated simulation tools. Yet the latter are not numerous, and are often adapted to some “classic” materials or restricted to simple geometries (i.e. thin films, nanowires). Looking to address this deficiency, the present work brings Nano-κ, a Python code to simulate the phonon transport in nano and micro scale devices from ab-initio phonon data. The code has personalizing capabilities that allow to simulate several geometries and materials, offering insights of temperature distribution, heat flux, thermal conductivity, and mode contribution to energy transport. In the present work, a detailed description of the physical model implementation is provided and several simulation test cases are discussed. Nano-κ provides reliable predictions of the thermal conductivity in different materials, and is able to correctly predict the relative effect of size, temperature, and surface roughness on thermal transport properties. Program summaryProgram Title: Nano-κCPC Library link to program files:https://doi.org/10.17632/tr29mhkjh8.1Developer's repository link:https://github.com/brunohs1993/NanokappaLicensing provisions: MITProgramming language: PythonNature of problem: The estimation of heat conduction at nanoscales depends on the estimation of phonon transport, since Fourier's law may not be valid in these situations. The phonon transport is described by the Boltzmann Transport Equation (BTE), that needs to be solved for each phonon mode separately, while estimating temperature distributions that depends on all modes at the same time. The phonon data can be retrieved from ab-initio simulations. Some BTE calculations have been already done through the years for simple geometries, but there is no publicly available code that allows enough flexibility to be used by the scientific community.Solution method: Some previous works [1-3] have had success in applying the Monte Carlo method to solve the BTE for phonon transport. In the more recent works, the ab-initio data is retrieved from density functional theory (DFT) simulations [4-5]. Nano-κ is a Python code built on the same theoretical basis of previous developments, but adds more flexibility so that it can be open for public research use.

Extraction of thermal conductivity using phonon Boltzmann Transport Equation based simulation of frequency domain thermo-reflectance experiments

The multidimensional phonon Boltzmann Transport Equation (BTE) was solved numerically in cylindrical coordinates and in time domain to simulate a Frequency Domain Thermo-Reflectance (FDTR) experimental setup. The phase lag between the pump and probe laser signals were computed for a pump laser modulation frequency ranging from 20 to 200 MHz. Results were obtained both with and without the inclusion of optical phonons, as well as with two different relaxation time-scale expressions (Holland versus Broido) for scattering, obtained from the literature for silicon. It was found that inclusion of optical phonons significantly improved the agreement between the measured and the computed phase lag. Subsequently, the thermal conductivity was extracted by fitting the Fourier heat conduction equation results—also solved numerically in time domain—to the measured and computed (using BTE) phase lag values. While both relaxation time-scales exhibited thermal conductivity suppression, a clear superiority of one time-scale expression over the other could not be established. With the Broido time-scale, the thermal conductivity extracted from BTE calculations overpredicted the value extracted from experiments regardless of whether optical phonons were included in the calculations. In contrast, with the Holland time-scale, the extracted thermal conductivity value overpredicted the value extracted from experiments when optical phonons were included, but underpredicted the value when optical phonons were excluded.

Comprehensive first-principles study of bulk, bilayer, and monolayer α-PtO2 properties

Abstract We calculated the structural stability, electronic, optical, and thermoelectric properties for α-PtO2 structures (bulk, bilayer, and monolayer) via first-principles density functional theory calculations. The results show that there is good agreement between the calculated lattice parameters and the available experimental data for the bulk structure. The in-plane Young's modulus and Poisson's ratio were calculated in the harmonic elastic strain range, and the results show that the monolayer is harder than the bulk. GW0 calculations on top of a semi-local generalised gradient approximation for the exchange-correlation energy predicted that the bilayer, monolayer, and the bulk α-PtO2 are all semiconductors with indirect band gaps. The results reveal that the band gap increases from bulk to monolayer. Calculations of optical properties show that the monolayer can absorb up to 8% of incident radiation in the visible range, which is higher than the 2.3% for graphene of the same thickness. In contrast, the bulk absorbs 1.77%. Phonon calculations confirm that all structures are dynamically stable. To examine the lattice thermal conductivity, the Boltzmann transport equation calculations in conjunction with density functional theory were implemented. The in-plane calculated lattice thermal conductivity were 8.47 × 10−8Wm−1K−1 for bulk, 4.59 × 10−8Wm−1K−1 for bilayer and 1.06 × 10−8Wm−1K−1 per layer at 300 K. The obtained thermoelectric figure of merit (ZT) per layer were 0.11 for bulk, 0.62 for bilayer and 0.74 for monolayer, respectively. The investigation shows that the n-type in monolayer has the most promise for thermoelectric applications.

Simulating transient heat transfer in graphene at finite Knudsen number via the Boltzmann transport equation and molecular dynamics

The phonon Boltzmann transport equation (BTE) with the relaxation time approximation (RTA) scattering model is used to calculate transient temperature profiles in graphene, and the results are compared to analogous molecular dynamics (MD) simulations. For the BTE calculations, the phonon dispersion relation and frequency-dependent scattering rates are obtained from a combination of MD data and semi-empirical power-law expressions for the normal and Umklapp phonon lifetimes. The dimensions and initial temperature conditions of graphene are varied to study the size and temperature dependence of thermal transport physics at the mesoscopic scale. Good quantitative agreement to within 5% is found between the BTE and MD results, over a wide range of temperatures and lengthscales of the temperature variation in the graphene sheet. Small differences are attributed to the inaccuracy of the RTA as applied to graphene, and to neglecting four-phonon scattering in the BTE simulations. The present results may further understanding in applications such as the transient heating of nanoelectronics.

Effect of dimensionality on thermoelectric powerfactor of molybdenum disulfide

We present that two-dimensional (2D) bilayer molybdenum disulfide (MoS2) exhibits an enhanced Seebeck coefficient over its three-dimensional (3D) counterpart arising from dimensionality confinement. It has been predicted that quantum confinement enhances thermoelectric performance but no studies have focused on a single material to present a theoretical and experimental comparison, which would illustrate the enhancement of thermoelectric performance. Layered MoS2 provides an opportunity to verify this hypothesis and in this work, we extensively study the Seebeck coefficient, S, the electrical conductivity, σ, and the thermoelectric powerfactor, S2σ of 2D monolayer and bilayer MoS2 using theoretical Boltzmann Transport Equation calculations and compare the results to well-characterized experimental data. We conclude that dimensional confinement indeed enhances the Seebeck coefficient by up to ∼50% in 2D bilayer MoS2 over 3D MoS2 under similar doping concentrations because of the discretization of density of states. We also consider electrical conductivity with various energy-dependent scattering rates considering charged-impurities and acoustic phonon mediated scattering, and comment on a theoretical comparison of the powerfactor to the best-case scenario for 3D MoS2.

Thermoelectric transport calculations using the Landauer approach, ballistic quantum transport simulations, and the Buttiker approximation

An implementation of the Landauer approach utilizing the ballistic quantum transport package Kwant to calculate thermoelectric transport properties is presented. Incoherent scattering is included by an approach suggested by Buttiker, where scattering is mimicked by virtual contacts (Buttiker probes). In this paper we present the implementation and provide a simple validation by comparing results to the Boltzmann transport equation (BTE) for bulk CdTe and a simple short period superlattice. The electronic structure, serving as a base for fitting a tight-binding model, is calculated from first-principles. From there, the electronic structure on a dense k-point grid is generated and passed to the BTE routines, while the transmission coefficients in the Landauer approach are calculated from Green’s functions based on the tight-binding model. Our results show good agreement between the results from the Buttiker and the BTE calculations, which indicates a successful initial implementation. We discover significant performance bottlenecks tied to the density of the Buttiker probes and as such the implementation needs additional optimization to be viable for production work.

Phonon transport in single-layerMo1−xWxS2alloy embedded withWS2nanodomains

Two-dimensional (2-D) transition metal dichalcogenides (TMDs) have shown numerous interesting physical and chemical properties, making them promising materials for electronic, optoelectronic, and energy applications. Tuning thermal conductivity of two-dimensional (2-D) materials could expand their applicability in many of these fields. In this paper, we propose a strategy of using alloying and nanodomains to suppress the thermal conductivity of 2-D materials. To predict the thermal conductivity of 2-D alloy embedded with nanodomains, we employ the Green's function approach to assess the phonon scattering strength due to alloying and nanodomain embedding. Our first-principles-driven phonon Boltzmann transport equation calculations show that the thermal conductivity of single-layer MoS2 can be reduced to less than one-tenth of its intrinsic thermal conductivity after alloying with W and introducing nanodomains due to the strong scattering for both high- and low-frequency phonons. The strategies to further reduce the thermal conductivity are also discussed.

Effects of stacking sequence on the electrical conductivity of InAs: A combination of density functional theory and Boltzmann transport equation calculations

The effects of the stacking sequence on the electrical conductivity of InAs are theoretically investigated using a combination of density functional theory and Boltzmann transport equation calculations. The calculations within the constant-relaxation-time approximation demonstrate that the electrical conductivity for n-type InAs with wurtzite segments is ∼47% larger than that with zinc blende structure. In contrast, the conductivity for p-type InAs with zinc blende structure is three times larger than those with wurtzite segments owing to the difference in the electronic states ∼0.1 eV below the valence band maximum depending on the staking sequence. The results within the constant-relaxation-time approximation suggest that differences in the stacking sequence along the growth direction could affect the electrical conductivity of InAs nanowires.

Anharmonic force constants extracted from first-principles molecular dynamics: applications to heat transfer simulations

A systematic method to calculate anharmonic force constants of crystals is presented. The method employs the direct-method approach, where anharmonic force constants are extracted from the trajectory of first-principles molecular dynamics simulations at high temperature. The method is applied to Si where accurate cubic and quartic force constants are obtained. We observe that higher-order correction is crucial to obtain accurate force constants from the trajectory with large atomic displacements. The calculated harmonic and anharmonic force constants are, then, combined with the Boltzmann transport equation (BTE) and non-equilibrium molecular dynamics (NEMD) methods in calculating the thermal conductivity. The BTE approach successfully predicts the lattice thermal conductivity of bulk Si, whereas NEMD shows considerable underestimates. To evaluate the linear extrapolation method employed in NEMD to estimate bulk values, we analyze the size dependence in NEMD based on BTE calculations. We observe strong nonlinearity in the size dependence of NEMD in Si, which can be ascribed to acoustic phonons having long mean-free-paths and carrying considerable heat. Subsequently, we also apply the whole method to a thermoelectric material Mg2Si and demonstrate the reliability of the NEMD method for systems with low thermal conductivities.

Atomistic Simulations of Heat Transport in Silicon Nanowires

We present a series of molecular dynamics, lattice dynamics, and Boltzmann transport equation calculations aimed at understanding heat transport in Silicon nanowires. In agreement with recent experiments, we find that the computed thermal conductivity strongly depends on the surface structure. It may be as high as that of bulk Si for crystalline wires, while wires with amorphous surfaces have the smallest thermal conductivity, about 100 times lower than the bulk. Two, combined effects are responsible for this dramatic difference: the presence, at disordered surfaces, of extended, nonpropagating modes analogous to heat carriers in amorphous Si, together with decreased lifetimes of propagating modes.

Modeling uphill diffusion of Mg implants in GaAs using suprem-i v

Transient, uphill diffusion of implanted Mg in GaAs during a 900 °C anneal is simulated using suprem-iv. The diffusion is believed to occur via the substitutional-interstitial-diffusion (SID) mechanism, with excess interstitials and vacancies produced by the implantation process causing this abnormal diffusion behavior. The SID mechanism is shown to be equivalent, in terms of the governing equations, to the interstitial-dopant pair diffusion model used in suprem-iv. This allows one to use suprem-iv, a silicon process simulator that includes dopant–point-defect interactions, to model uphill diffusion once the appropriate diffusivity and defect parameters are included. The profiles of excess interstitials and vacancies produced by the implantation process are obtained from Boltzmann transport equation calculations. The transient uphill diffusion phenomenon can be well simulated using the diffusion model in suprem-iv, with the dopant diffusing from the region of excess interstitials toward the surface and the region of excess vacancies. Once the defect concentrations return to their steady-state levels, either by diffusion, recombination, or capture by sinks, the normal concentration-dependent diffusion into the substrate occurs.

Displacement criterion for amorphization of silicon during ion implantation

Results of Boltzmann transport equation calculations are used to estimate what fraction of a crystalline silicon lattice must be displaced to cause a crystalline-to-amorphous transition during ion implantation. Comparison of these calculations with experimental MeV He channeling and backscattering results for 150-keV boron implantation at 77 °K indicates that the displacement of about 10% of the lattice will cause amorphization provided the substrate is at a temperature which inhibits self-annealing processes and defect diffusion. The calculations also indicate that the number of atoms displaced is proportional to the deposited energy density, one displacement occurring on average for each 200 eV of deposited energy. Experimental results for room-temperature silicon implantation confirm the fact that higher temperature substrates require a greater fractional displacement of the lattice before amorphization occurs.