Three-phonon Umklapp Processes Research Articles

Characterization of thermal transport and laser absorption properties of an individual pristine SWCNT

Determination of thermal transport properties and optical absorption cross-section of nanowires is of central importance for designing nano-optoelectronics and nano-photovoltaics. Experimental study of these properties is challenging because of the very small size of a typical sample. In this study, we present a steady-state Raman method for determining the temperature (T)-dependent thermal conductivity (λ) and laser absorption cross-section (Cabs) for an individual nanowire. This method was verified and then applied to determine λ and Cabs of an individual pristine single-walled carbon nanotube (SWCNT, L = 22 μm, d = 1.29 nm). The results show that λ decreases with increasing T in this experimental temperature range, indicating that higher temperature induces a larger scattering rate of three-phonon Umklapp processes. By measuring the temperature increase at the sample center induced by a 0.284 mW focused polarized Gaussian laser spot (514.5 nm, re = 1.0 μm), we derived the absorbed laser power (Pα) and Cabs to be 3.98 × 10−5 mW and 5.46 × 10−15 m2 for this individual SWCNT. The laser absorption cross-section per carbon atom (σc) was determined to 1.64 × 10−17 cm2 with an uncertainty lower than 10.6%.

An ab initio method for the prediction of the lattice thermal transport properties of oxide systems: Case study of Li2O and K2O

A method for the prediction of the thermal transport properties of macroscopic and isotropic oxides systems, above the standard temperature of T=298.15 K, is presented. This method combines: (i) the kinetic theory, (ii) a thermodynamically self consistent method for the density of the lattice vibration energy, and (iii) the three-phonon umklapp processes for the description of the phonon-phonon scattering. The proposed approach is purely predictive, as no experimental data are required for the model parameterization; they are derived from ground sate electronic structure calculations. Case studies on Li2O and K2O are presented and discussed. The predicted thermal transport properties are found to be in excellent agreement with available experimental data. The thermal conductivity of K2O is found to differ from the thermal conductivity of Li2O by an order of magnitude. This difference is explained in terms of electron localization within the crystal.

Model calculation of thermal conductivity in antiferromagnets

A theoretical study is given of thermal conductivity in antiferromagnetic materials. The study has the advantage that the three-phonon interactions as well as the magnon phonon interactions have been represented by model operators that preserve the important properties of the exact collision operators. A new expression for thermal conductivity has been derived that involves the same terms obtained in our previous work in addition to two new terms. These two terms represent the conservation and quasi-conservation of wavevector that occur in the three-phonon Normal and Umklapp processes respectively. They gave appreciable contributions to the thermal conductivity and have led to an excellent quantitative agreement with the experimental measurements of the antiferromagnet FeCl2.

Phonon Scatterings in the Lattice Thermal Conductivity of Si_(1-x) Ge_x Alloy Nanowires: Theoretical Study

Theoretical investigation of the alloy concentration and temperature dependences of the lattice thermal conductivity of silicon-germanium nanowires is performed using the Steigmeier and Abeles model. Phonon scattering processes are represented by frequency-dependent relaxation time approximation. In addition to the commonly considered acoustic three-phonon umklapp processes, phonon-boundary and point-defect scattering mechanisms are assumed. No distinction is made between longitudinal and transverse phonons. The importance of all the mechanisms involved in the model is clearly demonstrated. Analysis of the results shows that: (1) alloy scattering is the dominant scattering mechanism at intermediate and high temperatures; (2) thermal conductivity is mainly depends on the alloy concentration across the full range of temperatures; (3) weak diameter dependence of thermal conductivity is observed in Si_(1-x) Ge_x alloy nanowires; (4) the roughness of nanowires depends on the alloy concentration and has a major role in decreasing thermal conductivity at low temperatures; (5) the anharmonicity parameter is not size-dependent, as compared to Si and Ge nanowires. These findings provide new insights into the fundamental understanding of high-performance nanostructural semiconductors of relevance to optoelectronic and thermoelectric devices.

Thermophysical properties of multi-wall carbon nanotube bundles at elevated temperatures up to 830 K

Thermal transport measurements in multi-wall carbon nanotube (MWCNT) bundles at elevated temperatures up to 830 K are reported using a novel generalized electrothermal technique. Compared with individual CNTs, the thermal conductivity ( k) of MWCNT bundles is two to three orders of magnitude lower, suggesting the thermal transport in MWCNT bundles is dominated by the tube-to-tube thermal contact resistance. The effective density for the two MWCNT bundles, which is difficult to measure using other techniques, is determined at 116 kg/m 3 and 234 kg/m 3. The thermal diffusivity slightly decreases with temperature while k exhibits a small increase with temperature up to 500 K and then decreases. For the first time, the behavior of specific heat for MWCNTs above room temperature is determined. The specific heat is close to graphite at 300–400 K but is lower than that for graphite above 400 K, indicating that the behavior of phonons in MWCNT bundles is dominated by boundary scattering rather than by the three-phonon Umklapp process. The analysis of the radiation heat loss suggests that it needs to be considered when measuring the thermophysical properties of micro/nano wires of high aspect ratios at elevated temperatures, especially for individual MWCNTs due to their extremely small diameters.

Phonon thermal conduction in graphene: Role of Umklapp and edge roughness scattering

We investigated theoretically the phonon thermal conductivity of single-layer graphene. The phonon dispersion for all polarizations and crystallographic directions in graphene lattice was obtained using the valence-force field method. The three-phonon Umklapp processes were treated exactly using an accurate phonon dispersion and Brillouin zone, and accounting for all phonon relaxation channels allowed by the momentum and energy conservation laws. The uniqueness of graphene was reflected in the two-dimensional phonon density of states and restrictions on the phonon Umklapp scattering phase-space. The phonon scattering on defects and graphene edges has also been included in the model. The calculations were performed for the Gruneisen parameter, which was determined from the ab initio theory as a function of the phonon wave vector and polarization branch, and for a range of values from experiments. It was found that the near room-temperature thermal conductivity of single-layer graphene, calculated with a realistic Gruneisen parameter, is in the range $\ensuremath{\sim}2000--5000\text{ }\text{W}/\text{mK}$ depending on the flake width, defect concentration and roughness of the edges. Owing to the long phonon mean free path the graphene edges produce strong effect on thermal conductivity even at room temperature. The obtained results are in good agreement with the recent measurements of the thermal conductivity of suspended graphene.

Phonon intrinsic mean free path in zincblende AlN

We have carried out calculations of phonon anharmonic inverse relaxation time in the cubic phase of AlN by using phonon dispersion results from a full lattice dynamical model and applying Fermi's golden rule formula. The cubic anharmonicity in crystal potential has been modelled within the anharmonic elastic continuum model. The relative roles of three-phonon Normal and Umklapp processes have been examined at different temperatures. Increase in the relaxation rate of acoustic phonons due to their interaction with optical phonons has also been quantified. The calculated results for the relaxation times of the acoustic phonon modes show appreciable differences from the results obtained from the application of the isotropic continuum model, both at low and high temperatures. The calculated results for the mean free path over a large temperature range have been compared with available estimates obtained from thermal conductivity measurements.

Thermal conductivity of GaAs/AlAs superlattices: The Umklapp process

The thermal conductivity of GaAs/AlAs superlattices limited by the three-phonon Umklapp process and boundary scattering has been studied theoretically based on the model of lattice dynamics with force constant matrix. It was found that the Umklapp relaxation rate approximates BTw(2)/n with a fitting parameter B. The thermal conductivity increases with the increase of temperature at low temperatures, and would show a peak behaviour at about 60K before falling off at high temperatures. In addition, the thermal conductivity increases with the increase of period thickness of the superlattices.

Thermal properties of zeolites: effective thermal conductivity of dehydrated powdered zeolite 4A

The determination of the subambient effective thermal conductivity of evacuated powdered zeolite 4A is described. The results are modelled successfully in terms of boundary scattering, point defects and three-phonon Umklapp processes.

Modification of the three-phonon Umklapp process in a quantum wire

We have investigated the modification of the three-phonon Umklapp process in a quantum wire caused by the effect of phonon confinement. Our formalism rigorously takes into account variations in phonon group velocity as well as the presence of an energy gap between the phonon modes due to the quantization of the radial phonon wave vector. Numerical calculations of the Umklapp relaxation rate were carried out for Si wires. Based on the obtained results we predict the suppression of the acoustic phonon modes decay at low temperatures that can lead to the unusual thermal conductivity increase. We also present a simplified empiric formula for the Umklapp relaxation rate calculation in a quantum wire.

OXYGEN-DOPING DEPENDENCE OF THERMOELECTRIC POWER AND THERMAL CONDUCTIVITY MEASUREMENTS IN EuBa2Cu3O7-δ

The temperature variation of thermoelectric power, S (T) , and thermal conductivity, k (T) , in EuBa 2 Cu 3 O 7-δ polycrystalline samples with various degrees of oxygenation is reported. The temperature dependence in the range of highly oxygenated to deoxygenated samples displays a gradual transformation from metallic-like to semiconductor-like features in the normal state. For δ values near zero; the critical temperature, indicated by S (T) , shows a lowering which is attributed to a possible raising in electron carriers due to oxygen overdoping. Above T c, thermal conductivity is found to decrease with increase in temperature, pointing towards a contribution to thermal resistivity from three-phonon Umklapp processes. Below T c, thermal conductivity increases rapidly before reaching a maximum, attributed this effect to electron-phonon enhancement combined with a temperature dependent defect scattering limitation in the lifetime of the phonons. A nonmonotonic doping dependence on k (T) is indicated in data, suggesting sensitivity to both oxygen content and oxygen order. The experimental set up is described and results discussed.

Intrinsic thermal-resistive process of crystals: Umklapp processes at low and high temperatures.

Three-phonon umklapp processes are considered as the main intrinsic thermal-resistive processes in the crystals. The splitting umklapp processes are considered as the important process in thermal-resistive processes at low and high temperatures. Based upon a two-dimensional model, the simplest possible model is constructed for umklapp process. Real computational expressions are provided using the data of a phonon-dispersion relation of the direction of the principal reciprocal-lattice vectors and the given lattice characteristics. To test the model, the calculations are compared with the existing data of the purest crystals of LiF and NaF. \textcopyright{} 1996 The American Physical Society.

Anharmonic thermal resistivity of dielectric crystals at low temperatures.

The intrinsic thermal resistance of dielectric crystals at low temperatures due to three-phonon Umklapp processes has been treated theoretically, taking account of the restrictions imposed by wave-vector selection and frequency conservation rules, as well as the known phonon-dispersion curves. Since low-frequency phonons cannot interact directly with zone-boundary phonons, the transfer of excess momentum takes place in two steps: a normal process with modes of intermediate frequency, followed by an Umklapp process of intermediate-frequency phonons with phonons near the zone boundary of two different polarization branches with different frequencies. The intermediate frequency is given by that frequency difference. The second step can also be scattering by point defects, so that the apparent Umklapp resistance can be modified by point defects, which scatter the intermediate-frequency phonons much more strongly than the thermal phonons. The important processes in different frequency ranges depend on the dispersion curves and on the residual point defects; various cases are discussed. The theory has only one adjustable parameter, the Gr\uneisen constant. The point-defect scattering strength is deduced from the departures of the thermal conductivity in the boundary scattering regime at very low temperatures from a ${\mathit{T}}^{3}$ dependence. The theory has been compared to measurements on LiF, NaF, diamond, Si, and enriched $^{74}\mathrm{Ge}$. Reasonably good agreement with experimental data can be obtained without invoking an adjustable exponent in the Umklapp resistance.

Thermal conductivity of YBa2Cu3O7 from 7 K to 260 K

Thermal conductivity of YBa2Cu3O7 has been measured on samples of a few mm thickness. AboveT c thermal conductivity is found to decrease with increase in temperature, pointing towards a contribution to thermal resistivity from three-phonon Umklapp processes. BelowT c thermal conductivity increases rapidly before reaching a maximum at about 50 K and then falls towards zero at lower temperatures. The experimental set up is described and results discussed.

Phonon conductivity of InSb in the temperature range 2–800 K

The lattice thermal conductivity of InSb has been calculated in the temperature range 2–800 K in the frame of Dubey’s model, which makes use of Guthrie’s classification of three-phonon scattering events. Excellent agreement has been found between the theoretical and experimental values of the phonon conductivity over a wide range of temperatures. The temperature dependence of three-phonon scattering relaxation rate has been calculated for both class I and class II events in the study of phonon conductivity of InSb. The percentage contributions of transverse and longitudinal phonons have been studied separately in the temperature range of investigation. The role of four-phonon processes is also included at high temperatures. The percentage contributions of the three-phonon normal and umklapp processes towards τ 3 −1 have also been investigated.

Analysis of the lattice thermal conductivity and phonon-phonon scattering relaxation rate: Application to Mg2Ge and Mg2Si

The lattice thermal conductivities of Mg2Ge and Mg2Si have been analysed in the entire temperature range 2–1000 K in the frame of a new expression for the phonon-phonon scattering relaxation rate proposed by Dubey as $$\tau _{3ph}^{ - 1} = (B_{N,I} + B_{U,I} e^{ - \theta /\alpha T} )g(\omega )T^{m_I (T)} + (B_{N,II} + B_{U,II} e^{ - \theta /\alpha T} )g(\omega )T^{m_{II} (T)}$$ based on the Guthrie classification of the phonon-phonon scattering events, and a very good agreement has been obtained between the calculated and experimental values of the lattice thermal conductivity for both samples in the entire temperature range of the study. The separate percentage contributions due to three-phonon normal and umklapp processes towards the three-phonon scattering relaxation rate have also been studied. The role of the four-phonon processes has been included in the present analysis.

Thermal Conductivity, Heat Capacity and Phase Diagram of Tetrahydrofuran Under Pressure

Abstract The thermal conductivity, Λ, and the heat capacity per unit volume, pcp , have been measured for tetrahydrofuran (THF) using the transient hot-wire method in the temperature range 130-300 K and at pressures up to 2 GPa. Most of the data relate to the single solid phase which was found to exist over these ranges of the variables. The liquidus curve was determined roughly up to the maximum pressure, and we estimated the density of the solid phase at the normal melting point using the Clausius-Clapeyron relation. We inferred from the measurements that solid THF was a normal crystal phase in which three-phonon umklapp processes provided the only significant source of thermal resistivity. There was evidence for the effect of thermal expansion on Λ, and this effect decreased under pressure.

Thermoelectric properties of Ag2Se

To obtain information on the temperature and concentration dependence of thermoelectric quality factor Z and to clarify the nature of scattering in Ag2Se a study was made of electrical conductivity σ, thermo-emf α, and thermal conductivity ϰ in silver selenide over the temperature range T=80–450°K at concentration levels of 2–43·1024 m−3. It is shown that in Ag2Se the Lorentz number L, determined experimentally from the electronic fraction of the thermal conductivity, is less than the Sommerfeld number Lo. Calculation of L/Lo(n) performed with a theory considering inelastic scattering of carriers, shows that the inelasticity is produced by electron interaction. Comparison of experimental data on the temperature dependence of lattice thermal conductivity with theory permits the conclusion that in Ag2Se in range 80–300°K the basic role in phonon scattering is played by three-phonon Umklapp processes. It is shown that with increase in T and decrease in n the thermoelectric quality factor of silver selenide increases. The highest value of Z was achieved in a specimen with electron concentration n=2·1024 m−3 at T=320°K. The rapid decrease in Z upon phase transition is related to discontinuous decrease in α and σ at this point.

Analysis of lattice thermal conductivity and phonon-phonon scattering relaxation rate: Application to Ge

The three-phonon scattering relaxation rates and their temperature exponents have been analysed in the frame of Guthrie's classification of the phonon-phonon scattering events as class I and class II events and as a result of this, a new expressionτ -13ph =(B N,I+B U,Ie-θ/αT) g(w)T m I (T)+(B N,II+B U,IIe-θ/αT)g(w)T mII(T) for the three phonon scattering relaxation rates has been proposed for the first time to calculate the lattice thermal conductivity of a sample. Using the expression proposed above, the lattice thermal conductivity of Ge has been analysed in the temperature range 2–1000K and result obtained shows a very good agreement with the experimental data. The percentage contributions due to three-phonon normal and umklapp processes are also reported. The role of four phonon processes is also included at high temperatures. To estimate an approximate value of the scattering strength and the phonon conductivity, the analytical expression is also obtained in the frame of the expression proposed above forτ -13ph .

Analysis of the lattice thermal conductivities of Mg2Si and Mg2Sn in the temperature range 2–1000 K in the frame of the two-mode conduction of phonons

The lattice thermal conductivities of Mg2Si and Mg2Sn have been analyzed in the entire temperature range 2–1000 K in the frame of the expression for the three-phonon scattering relaxation rate recently proposed by Dubey and Misho, and a very good agreement is found between the calculated and the experimental values of the lattice thermal conductivity in the entire temperature range of the study. The separate contributions due to transverse and longitudinal phonons towards the total lattice thermal conductivity have also been studied by calculating their percentage contributions. The percentage contributions of the three-phonon normal and umklapp process scattering relaxation rates towards the three-phonon scattering relaxation rate have been studied for both Mg2Si and Mg2Sn. The percentage contribution of the three-phonon scattering relaxation rate towards the combined scattering relaxation rate has also been studied for transverse as well as for longitudinal phonons, for four different values of the phonon frequencies. The role of the four-phonon processes is. also included in the present analysis.