Callaway Theory Research Articles

Thermal conductivity of pink CVD diamond: Influence of nitrogen-related centers

Thermal conductivity κ(T) of single-crystal CVD diamond lightly doped (about 3 ppm) with nitrogen has been measured at temperatures from 5.7 to 410 K. The sample was carefully characterized by optical absorption and photoluminescence spectroscopy for the presence of impurities. Nine different optically active defects related with nitrogen, hydrogen, and silicon impurities have been identified and quantified. This pink-tint crystal showed a high thermal conductivity of 24.0±0.5 W cm−1 K−1 at room temperature, which is very close to the highest value ever measured at about 25 W cm−1 K−1 for diamonds of natural isotopic composition. At the same time, the κ(T) of the crystal showed strong suppression >10% at temperatures 6<T<120 K with a maximum decrease of 2.7 times at ≈40 K compared to high purity diamonds. This behavior of the conductivity is attributed to a phonon scattering by charge carriers bound to nitrogen-related impurity centers, which is ineffective, however, at room and higher temperatures. The κ(T) has been calculated within the model based on the Callaway theory taking into account the elastic phonon scattering off charge carriers (holes and electrons) in the ground states of doping centers, and a very good agreement between the measured and theoretical data has been achieved. The model also gives a good approximation to the experimental data for κ(T) given in the literature for synthetic and natural single-crystal diamonds.

AICON: A program for calculating thermal conductivity quickly and accurately

AICON (Ab Initio Conductivities) is a program written in Python for computing lattice thermal conductivity of crystalline bulk materials using the modified Debye–Callaway model. Building upon the traditional Debye–Callaway theory, the modified model obtains the lattice thermal conductivity by averaging the contributions from acoustic and optical branches based on their specific heat. The only inputs of this program are the phonon spectrum, phonon velocity and Grüneisen parameter, all of which can be calculated using third-party ab initio packages, making the method fully parameter-free. This leads to a fast and accurate evaluation and enables high-throughput calculations of lattice thermal conductivity even in large and complex systems. In addition, this program calculates the specific heat and phonon relaxation times for different scattering processes, which will be beneficial for understanding the phonon transfer behavior. Program summaryProgram Title: AICONProgram Files doi:http://dx.doi.org/10.17632/s9b8y8t92c.1Licensing provisions: GNU General public license 3Programming language: Python3External routines/libraries: Numpy, Scipy, spglib, pymatgenNature of problem: The calculation of lattice thermal conductivity from first principles with an anharmonic approximation requires a large number of calculations to construct the third-order force constants matrix, which could be prohibitively long time.Solution method: Modified Debye–Callaway model, where only the phonon spectrum, phonon velocity and Grüneisen parameter are needed. The acoustic branch and optic branch are both considered to obtain the final lattice thermal conductivity.

Three-phonon scattering processes and thermal conductivity in IV-chalocogenides

We present a systematic study of allowed three-phonon scattering processes, involving acoustic and optical branches, and their relative roles in explaining the low thermal conductivity of IV-chalcogenide thermoelectric materials PbTe, PbSe, PbS, and SnTe. Using numerical results for , computed by employing the isotropic continuum scheme, we have examined the extent of the additional contribution the Callaway theory and the Allen theory provide over the single-mode relaxation time theory. Within the Callaway theory, for all these materials the acoustic (, ) and transverse optical () phonons contribute between 10–25% towards at and above room temperature, with . The longitudinal optical () phonons contribute negligibly (<5%) in Pb-chalogenides, but their contribution is larger (22%) than that of phonons (18%) in SnTe. Due to the presence of high defect concentration in these materials, the high temperature conductivity varies less strongly than T−1. In confirmation with experimental measurements, our study finds that below the Debye temperature the resistivity of SnTe varies as the square-root of the point defect concentration.

Effect of AlN addition on the thermal conductivity of pressureless sintered SiC ceramics

Dense silicon carbide ceramics with 0.5–10wt% AlN additions were pressureless densified by two steps sintering using B4C and C as sintering additives. The effects of AlN content on density, microstructure and thermal conductivity of SiC–AlN composites were investigated. SiC ceramics can be well densified in the presence of AlN. The phonon scattering mechanism and the Klemens–Callaway theory were used to calculate the thermal conductivity evolution of SiC ceramics result from the dissolved Al and N atoms. The main reason for the degradation of thermal conductivities is the point defect scattering dominated by the differences in size and interatomic forces between impurity and host atoms.

Effect of phonon focusing on the temperature dependence of thermal conductivity of silicon

The effect of phonon focusing on the temperature dependence of thermal conductivity of single crystalline silicon is studied at temperatures from 3 to 40 K in the frame of generalized Callaway theory. Thermal conductivity has been calculated for silicon samples in the form of finite length rectangular rods with different orientations of long axis and side faces. To evaluate the phonon scattering rate due to diffuse boundary scattering, the analytical expressions derived recently by us were utilized. Theoretical results representing the dependence of on temperature and sample orientation are in agreement with experimental data. Contributions of different phonon polarizations to the thermal conductivity have been analyzed. The transverse phonon branches are most important as expected, the slow transverse mode being the most prominent in the formation of conductivity anisotropy. The phonon dispersion was found to have a weak effect on the calculated at these temperatures.

Electron-phonon processes in semiconductors at low temperatures

The thermal conductivity and thermopower are simultaneously studied in the semiconductors HgSe, HgTe, InP, GaAs, Cd x H1 − x Te, and Ag2S in which the effect of electron drag by phonons was detected at low temperatures T. It is found that x = 1–3 in the dependences K ph ∝ T −x . It is shown that the underestimated values of x are caused by phonon scattering at certain defects. It is also found that the maxima of K ph(T) and αph(T) coincide in all studied crystals, except for Ag2S. For the first time, it is found that the strong effect αph ∝ T −3 is observed in Ag2S, the αph maximum is at T = 16 K, and the K ph maximum is at 27 K. The results obtained are in agreement with Callaway and Herring theory. The effect of magnetic field on the drag thermopower αph(T) in n-Cd x H1−x Te (in which the strong dependence αph ∝ T −3 and the strong effect of the magnetic field on it are detected) is considered for the first time. The results are compared with the Askerov theory. The force parameter A ph(ɛ) of the drag effect and its dependence on magnetic field are determined.

Effect of Parameters on Lattice Thermal Conductivity in Germanium Nanowires

Modified Callaway's theory was used to calculate lattice thermal conductivity (LTC) of Germanium nanowires. Results are compared to those of experimental values of the temperature dependence of LTC for nanowire diameters of 62, 19, and 15nm. In this calculation, both longitudinal and transverse modes are taken into account. Scattering of phonons is assumed to be by nanowire boundaries, imperfections, dislocations, electrons, and other phonons via both normal and Umklapp processes. Effect of parameters, phonon confinement and imperfections in limiting thermal conductivity for the nanowires under considerations are investigated. The suppression in thermal conductivity of these nanowires is arise from electron-phonon scattering and phonon-boundary scattering at low temperatures, while at high temperatures is due to imperfections and intrinsic properties.

Application of the Callaway theory to analysis of thermal transport by phonons in ceramic and biomorphic composites

By applying the Klemens–Callaway expression for the thermal conductivity in the framework of the Debye's model of phonons we present an analysis of the thermal transport by phonons in ceramic composites Si3N4/BN of various fibre structure (including plaited fibre configurations denoted [0/45] and [0/90]) and biomorphic composites (or ecocomposites) SiC/Si based on beech matrices filled with Si of two different volume percentage (2% and 26%). We obtained a good agreement with the Debye–Klemens–Callaway theory of the thermal conductivity at low temperatures for the ceramic composites Si3N4/BN with the plaited fibre configurations. A good fitting of the Callaway expression for the thermal conductivity coefficient in the whole temperature range can be made to our experimental results for the biomorphic composites SiC/Si. The latter fitting allowed us to assess the magnitude of the mean free path of phonons scattered by the boundaries of grains. We also estimated that the contribution of the conduction electrons to the heat transfer at low temperatures is 5–7% of the effective thermal conductivity in case of SiC/Si.

Thermal diffusivity and conductivity of thorium–plutonium mixed oxides

The thermal diffusivity of (Th, Pu)O 2 (with 3, 8 and 30 wt.% PuO 2) mixed oxides and of the single oxides PuO 2 and ThO 2 was measured using the laser flash technique. In several samples, a further degradation in the thermal diffusivity due to auto-irradiation damage was noted, and a recovery could be achieved by thermal treatment. The ThO 2 and (Th, Pu)O 2 samples with 3 and 8 wt.% PuO 2 were found to be semitransparent and had to be coated with C for the thermal diffusivity measurement. The experimental thermal conductivity of the fuels was calculated and analysed. The introduction of PuO 2 in the ThO 2 was found to have an effect similar to that observed in (U, Pu)O 2 mixed oxides: small decrease in the range 3–8 wt.% PuO 2 and higher decrease at 30 wt.% PuO 2 content. Using the classical phonon transport model in crystal structures, a new correlation for the thermal conductivity of (Th, Pu)O 2 mixed oxides is proposed and is compared to the results of another correlation built after the Klemens–Callaway theory.

Quantitative study of the enhancement of the thermal conductivity of AlN ceramics bynanoscale processing

We have theoretically studied and quantitatively analysed the enhancement of thethermal conductivity of AlN microceramics (i.e. commercially available powder)by the insertion of AlN nanosized particles of high-specific surface area, and ofY2O3 and CaO as sintering additives. We have also studied the enhancementof the thermal conductivity of AlN nanoceramics by usingY2O3 as the sintering additive. Thermal conductivity calculations have been carried out byapplying the Callaway theory in its full form and by incorporating a detailed and accurateaccount of three-phonon processes. The role of densification of the AlN ceramic samples inenhancing the thermal conductivity has been quantified at low, intermediate and hightemperatures. In addition to explaining the experimentally observed room-temperatureconductivity results, the conductivity variation has been predicted over a large temperaturerange.

Thermal conductivity of polycrystalline CVD diamond: effect of annealing‐induced transformations of defects and grain boundaries

AbstractPolycrystalline CVD diamond has been annealed in vacuum at temperatures tan = 1550 °C–1690 °C to investigate an influence of modifications of grain boundaries (GB) and defects on thermal conductivity and optical properties of originally optical quality material. While the sample was converted by this heat treatment to “black diamond”, excitons still could be observed in cathodoluminescence spectra. Photoluminescence spectra revealed an aggregation of nitrogen (formation of H3 optical center) after annealing at 1690 °C. In‐plane thermal conductivity k (T) measured by a heated bar method in the temperature range T = 5–410 K shows a strong (up to 5 times) and monotonic decrease at T &lt; 45 K upon annealing, while room temperature k (298 K) value remains almost constant (ca. 19.0 W/cm K) till tan = 1650 °C, rapidly degrading at higher tan. The experimental data on thermal conductivity are analyzed in the framework of full Callaway theory taking into account three‐phonon scattering processes, phonon scattering on point defects, sample boundaries, and GB. The proposed model reproduces well the observed k (T) behavior, the phonon scattering on the modified GB playing an important role. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Thermal conductivity of polycrystalline CVD diamond: Experiment and theory

The temperature dependences of thermal conductivity κ of polycrystalline CVD diamond are measured in the temperature range from 5 to 410 K. The diamond sample is annealed at temperatures sequentially increasing from 1550 to 1690°C to modify the properties of the intercrystallite contacts in it. As a result of annealing, the thermal conductivity decreases strongly at temperatures below 45 K, and its temperature dependence changes from approximately quadratic to cubic. At T > 45 K, the thermal conductivity remains almost unchanged upon annealing at temperatures up to 1650°C and decreases substantially at higher annealing temperatures. The experimental data are analyzed in terms of the Callaway theory of thermal conductivity [9], which takes into account the specific role of normal phonon-phonon scattering processes. The thermal conductivity is calculated with allowance for three-phonon scattering processes, the diffuse scattering by sample boundaries, the scattering by point and extended defects, the specular scattering by crystallite boundaries, and the scattering by intercrystallite contacts. A model that reproduces the main specific features of the thermal conductivity of CVD diamond is proposed. The phonon scattering by intercrystallite contacts plays a key role in this model.

Role of additives in enhancing the thermal conductivity of AlN ceramics

The thermal conductivity of AlN ceramics with different amounts of Y2O3 as sintering additive has been calculated at low, intermediate and high temperatures by applying the Callaway theory and using a detailed account of three-phonon scattering processes. By making a quantitative analysis of the results, it is found that the increase in the room-temperature conductivity due to the addition of Y2O3 is largely attributed to the decrease in the oxygen impurities level in the AlN crystal. In addition to matching the experimental measurements (available in a limited temperature range) our theoretical work predicts the resulting thermal conductivity over a large temperature range.

Role of optical phonon in Ge thermal conductivity

We report on the contribution of optical phonon decay into acoustic phonon to the thermal conductivity [k(T)] of semiconductors. We have considered this mechanism as an acoustic phonon generation rate within the full modified Callaway theory to accurately describe experimental data of k(T) for Ge as a function of isotopic composition. The proposed model requires only one adjustable parameter for each mode. Our approach to describe k(T) in semiconductors is found to provide much better agreement with the experimental results than previously reported models, which did not account for the role of optical phonon.

Thermal conductivity of single crystal and ceramic AlN

We have applied the Callaway theory and used a detailed account of three-phonon scattering processes to calculate the thermal conductivity of three AlN single crystal samples containing different amounts of oxygen and two AlN ceramic samples with different grain sizes and oxygen contamination levels. The N-drift contribution to the total conductivity has been quantified. The influence on the thermal conductivity of oxygen-related defects, and grain boundaries in ceramic samples, has been investigated. The theoretical results obtained from this work are in good agreement with available experimental data. Our calculations suggest that the “effective” boundary length is greater than the reported grain size for each of the two ceramic samples studied by Watari et al. [J. Mater. Res. 17, 2940 (2002)].

Quantitative estimate of phonon scattering rates in different forms of diamond

We have obtained a quantitative estimate of phonon scattering rates in natural type IIa, neutron-irradiated, microwave plasma assisted chemical vapor deposition and hot filament chemical vapor deposition diamond samples studied in the 1990s by Morelli and co-workers. This is done by successfully explaining the experimental thermal conductivity results with the employment of Callaway's theory in its full form and Srivastava's scheme for Normal and Umklapp three-phonon relaxation times involving Gr\uneisen's constant as the only (semi) adjustable parameter.

Disorder scattering effect on the high-temperature lattice thermal conductivity of TiCoSb-based half-Heusler compounds

The lattice thermal conductivities of TiCoSb-based half-Heusler alloys are presented in the temperature range between 300 and 900K. A phenomenological model calculation of the high-temperature lattice thermal conductivities of these alloys was derived based on the Klemens–Callaway theory [Phys. Rev. 119, 507 (1960); ibid. 113, 1046 (1959)]. Good agreement was obtained between the calculated and the experimental data for TiCoSb, TiCo0.5Rh0.5Sb, and Ti0.5Zr0.5CoSb. Furthermore, the model predicts that simultaneously isoelectronic alloying on both Ti and Co sublattices could reduce the lattice thermal conductivity, and a κL value of 0.3W∕mK is predicted for Ti0.5Zr0.5Co0.5Rh0.5Sb at 900K.

Thermal conductivity of germanium crystals with different isotopic compositions

We have measured the thermal conductivity of seven germanium crystals with different isotopic compositions in the temperature range between 2 K and 300 K. These samples, including one made of highly enriched ${}^{70}\mathrm{Ge}(99.99%)$, show intrinsic behavior at room temperature with the exception of a $p$-type sample with $|{N}_{d}\ensuremath{-}{N}_{a}|\ensuremath{\cong}2\ifmmode\times\else\texttimes\fi{}{10}^{16} {\mathrm{cm}}^{\ensuremath{-}3}$. The ``undoped'' samples exhibit a ${T}^{3}$ dependence at low temperatures, basically determined by boundary scattering. The maximum value of $\ensuremath{\kappa}$ (which falls in the range between 13 K and 23 K) is found to be a monotonically decreasing function of the isotopic mass variance parameter $g$. The maximum ${\ensuremath{\kappa}}_{m}$ measured for the most highly enriched ${}^{70}\mathrm{Ge}(99.99%)$ sample is 10.5 kW/mK, one order of magnitude higher than for natural germanium. The experimental data have been fitted with the full Callaway theory, modified by treating transverse and longitudinal modes separately, using three free adjustable parameters for each set of modes to represent anharmonic effects plus the calculated contributions from isotopic and boundary scattering. For the isotopically purest ${}^{70}\mathrm{Ge}(99.99%)$ sample, dislocation scattering, or a similar mechanism, must be added in order to fit the data. We have also checked the effect of various surface treatments on the thermal conductivity in the low temperature region. The highest values of $\ensuremath{\kappa}$ are found after polish etching with a SYTON suspension.

Thermal conductivity of CVD diamond films: high-precision, temperature-resolved measurements

We describe a high-precision measuring system for the determination of the in-plane thermal conductivity of CVD diamond samples at temperatures between 77 and 900 K. The thermal conductivity is determined by establishing a well-defined steady-state heat flux through a diamond bar and by measuring the temperature gradient, using an array of eight thermocouples. The heat is supplied by a meander-shaped microheater located at one edge of the bar. By measuring the temperature distribution without heat production in the microheater, radiation losses are detected and corrected for. At room temperature the in-plane thermal conductivity of polished CVD diamond samples is determined with an accuracy better than 1%. A variety of CVD diamond samples deposited by microwave plasma CVD under different growth conditions was investigated. Some of the samples showed extraordinarily high thermal conductivities of up to 54 W cm −1K −1 at −150 °C. To our knowledge, this is the highest value reported so far for CVD diamond. The temperature dependence of the thermal conductivity is well described by Callaway's theory. By fitting theoretical curves to the experimental values, information about defect densities and grain boundaries is derived.

The effective free path length of phonons in diamond

The influence of normal (N) processes of phonon-phonon interactions on the temperature dependence of the effective free path length l eff of phonons in diamond was considered in the limits of Callaway's theory. It was shown that in perfect crystals at low temperatures N processes prevail over all other types of scattering and that even in the range of room temperature they play a significant role. In connection with this the possibility of second sound spreading in diamond was considered together with the specific influence of crystal boundaries on l eff. It was shown that in diamond with a reduced content of 13C isotope it is possible to reveal a state corresponding both to the Knudsen minimum and to phonon flow close to the Poiseuille one.