Umklapp Interaction Research Articles

Size and dimensionality dependent phonon conductivity in nanocomposites

We have studied size and dimensionality dependent phonon conductivity of PbTe-PbSe nanocomposites by considering three configurations: superlattice, embedded nanowire and embedded nanodot. Calculations have been performed in the framework of an effective medium theory. The required bulk thermal conductivities of PbTe and PbSe are evaluated by using Callaway’s effective relaxation-time theory, and by accounting for relevant scattering mechanism including three-phonon Normal and Umklapp interactions involving acoustic as well as optical branches. The thermal interface resistance is computed using the diffuse mismatch theory. It is found that the size (thickness) and volume fraction of PbSe are the two main factors that control the effective thermal conductivity in these nanocomposites. In particular, for PbSe size d = 10 nm and volume fraction , our results predict significant reductions over the weighted average of room-temperature bulk results of 9%, 17% and 15% in the conductivity across the interfaces for the superlattice, embedded nanowire, and nanosphere structures, respectively. For a given , an increase in d reduces the interface density and the effective conductivity varies approximately as . It is shown that nanocompositing in any of the three configurations can beat the alloy limit for lattice thermal conductivity.

Conductivity of a generic helical liquid

A quantum spin Hall insulator is a two-dimensional state of matter consisting of an insulating bulk and one-dimensional helical edge states. While these edge states are topologically protected against elastic backscattering in the presence of disorder, interaction-induced inelastic terms may yield a finite conductivity. By using a kinetic equation approach, we find the backscattering rate $\tau^{-1}$ and the semiclassical conductivity in the regimes of high ($\omega \gg \tau^{-1}$) and low ($\omega \ll \tau^{-1}$) frequency. By comparing the two limits, we find that the parametric dependence of conductivity is described by the Drude formula for the case of a disordered edge. On the other hand, in the clean case where the resistance originates from umklapp interactions, the conductivity takes a non-Drude form with a parametric suppression of scattering in the dc limit as compared to the ac case. This behavior is due to the peculiarity of umklapp scattering processes involving necessarily the state at the "Dirac point". In order to take into account Luttinger liquid effects, we complement the kinetic equation analysis by treating interactions exactly in bosonization and calculating conductivity using the Kubo formula. In this way, we obtain the frequency and temperature dependence of conductivity over a wide range of parameters. We find the temperature and frequency dependence of the transport scattering time in a disordered system as $\tau \sim [\max{(\omega,T)}]^{-2K-2}$, for $K>2/3$ and $\tau \sim [\max{(\omega,T)}]^{-8K+2}$ for $K <2/3$.

1D FFLO state in absence of time reversal symmetry breaking

We propose a route to a one-dimensional Fulde-Ferrell-Larkin-Ovchinnikov state in the absence of broken time-reversal symmetry. At present such a state may be encouraged in a clean (no disorder) AlAs quantum wire fabricated using the cleaved edge overgrowth technique. The fabrication technique captures two degenerate nonoverlapping bands separated in momentum-space by half an umklapp vector which leads to four Fermi points. Using field theoretic methods such as abelian bosonization and the renormalization group scheme we treat the important low energy long wavelength fermionic interaction terms for this one dimensional system. Due to the specific bandstructure arrangement of the quantum wire there is a new class of unique umklapp assisted interactions. These umklapp interactions are present at all electronic densities and are not related to the commensurability of the electron gas with the underlying lattice. We show that in the presence of the umklapp interactions and without any external perturbations such as a magnetic or electric field a singlet superconducting ground state is preferred with non-zero center-of-mass momentum for the Cooper pairs. The finite pairing momentum of the Cooper pairs is an indication of a Fulde-Ferrell-Larkin-Ovchinnikov state which is known to lead to inhomogeneous superconductivity.

Phonon transport in silicon, influence of the dispersion properties choice on the description of the anharmonic resistive mechanisms

Thermal conductivity in semiconductors is investigated through the phonon transport description. From the former studies based on the gas kinetic theory to the recent ones using the Monte Carlo methods or the radiative multifux models, each technique involves the dispersion properties of the crystal lattice. Besides, most of these studies have been done in the frame of the relaxation time approximation of collision processes. In this study, the importance of the choice of the dispersion curves and time relaxation parameters are pointed out. A complete description of the transverse umklapp interaction is provided without any adjustment parameters. New insights of the energy conservation surface calculation, involved in phonon-phonon time relaxation expression, are provided. Using simple and more complex models, we have determined the heat transport properties, in the temperature range from 50 K to 1000 K in bulk silicon. We especially focused our attention on the thermal conductivity as a function of the phonon anharmonic resistive mechanisms for two models of dispersion: “linear” and “quadratic”. We demonstrate that very different thermal conductivity properties can be calculated according to the chosen description of lattice vibrations.

Interaction induced dimerization in zigzag single wall carbon nanotubes

We derive a low-energy effective model of metallic zigzag carbon nanotubes at half filling. We show that there are three important features characterizing the low-energy properties of these systems: the long-range Coulomb interaction, umklapp scattering, and an explicit dimerization generated by interactions. The ratio of the dimerization induced gap and the Mott gap induced by the umklapp interactions is dependent on the radius of the nanotube and can drive the system upon increasing dimerization strength from a Haldane spin-liquid phase through a quantum phase transition with $\mathrm{SU}{(2)}_{1}$ quantum symmetry to a dimerized phase. We consider the physical properties of the phases on either side of this transition, which should be relevant for realistic nanotubes.

Cubic magnets with Dzyaloshinskii–Moriya interaction

Ground-state energy and spin-wave spectrum are studied theoretically using conventional exchange, Dzyaloshinskii–Moriya interaction, anisotropic exchange and cubic anisotropy. The spin-wave spectrum is strongly anisotropic: excitations with momentum q along and perpendicular to the helix wave vector k have linear and quadratic dispersion, respectively, if q ⪡ k . It is a result of the umklapp interaction connecting the spin-waves with q and q ± k . The classical ground-state energy depends on the magnetic field component along the vector k only. Transition to the ferromagnetic state holds at H > H c where g μ B H c = Ak 2 and A is the spin-wave stiffness at q ⪢ k . For low perpendicular field the helical order is stabilized by the spin-wave gap Δ . For g μ B H ⊥ < Δ 2 there is Bose condensation of the spin-waves with momenta k and zero. The perpendicular susceptibility and the second harmonic of the spin rotation appear. For larger field the vector k establishes along the field and the condensation disappears. The theory is in agreement with the existing experimental data.

Non-local Thirring model with backward and umklapp interactions

We extend a non-local and non-covariant version of the Thirring model in order to describe a many-body system with backward and umklapp scattering processes. We express the vacuum to vacuum functional in terms of a non-trivial fermionic determinant. Using path-integral methods we find a bosonic representation for this determinant which allows us to obtain an effective action for the collective excitations of the system. By introducing a non-local version of the self-consistent harmonic approximation, we get an expression for the gap of the charge-density excitations as functional of arbitrary electron–electron potentials. As an example we also consider the case of a non-contact umklapp interaction.

Phonon scattering in diamond films

Abstract We have studied the thermal conductivity κ of free standing diamond films with different quality and sizes of the crystallines in bulk. The measurements were performed in the temperature range 10–350 K by steady-state heat flux technique. We have estimated the value and temperature dependence of the phonon mean free path lp using the simple gas kinetic equation for thermal conductivity: κ≈1/3 l p C p v. The values of heat capacity Cp and mean sound velocity v are known from literature for massive crystals. At high (room) temperatures the mean free path of phonons is limited by the phonon–phonon Umklapp interaction and phonon scattering on point defect in bulk. Both these processes lead to increasing of lp(T) with cooling the sample but they have different temperature dependencies. So from lp(T) dependence one can judge the quality (purity) of the sample. At low temperatures the phonon-boundary scattering dominates and lp is limited by the crystalline sizes. The mean free path of phonons in the best of samples has reached the value of a few microns at temperatures 20–70 K, which agrees with the middle sizes of crystallites estimated from X-ray measurements and from microphotos. At lower temperatures the value of lp is increasing again. It can be attributed to the phonon penetration through the boundaries between crystallines. The temperature dependence of λ in this process follows l(T)∼T−(1−2).

Competition between the Mott transition and Anderson localization in one-dimensional disordered interacting electron systems.

The competition between the Mott transition and the Anderson localization in one dimensional electron systems is studied based upon the bosonization and the renormalization group method. The beta function is calculated up to the second order in the strength of diagonal disorder by using a replica trick. It is found that the sufficiently strong forward scattering by random impurities destroys the Mott-Hubbard gap, and the backward scattering gives rise to the Anderson localization for the resulting gapless state. On the other hand, if the Umklapp interaction is strong enough, the Mott insulating state still overwhelms the Anderson localization.

Anharmonic interactions in beryllium oxide.

BeO is an insulator with an unusually high thermal conductivity (92% that of Cu at 25 \ifmmode^\circ\else\textdegree\fi{}C) and a high Debye temperature (1280 K). We measured its first-order Raman scattering as a function of temperature from 300 down to 30 K. The temperature dependence of the energy and lifetime of its normal modes were analyzed in light of the theory of anharmonic phonon-phonon interactions, revealing that three-phonon up-conversion and down-conversion processes are present. We also measured the second-order Raman spectra of BeO, finding a number of bands that correspond well to previously reported second-order infrared-absorption measurements. In this way an old inconsistency was solved, since group theory predicts the infrared second-orders to be Raman-active as well, but they were never observed before. In addition, the thermal conductivity data of BeO were reexamined in order to explain their temperature dependence. We found that they have a quadratic dependence on the inverse absolute temperature after the conductivity maximum, all the way from 60 to 2000 K. Terms accounting for isotope and impurity scattering cannot be invoked in this material which is practically isotopically pure and with a low concentration of impurities. Umklapp interactions should be playing a dominant role, but the corresponding $\mathrm{exp}(\frac{{\ensuremath{\Theta}}_{D}}{\ensuremath{\alpha}T})$ dependence is not observed. By taking into account the combined relaxation times of umklapp and normal processes it was possible to simulate these data, thus evidencing that the indirect effect of normal processes on thermal resistivity is significant and need to be taken into account.

“Hot spots” in quasi-one-dimensional organic conductors

Abstract The distribution of the electron scattering rate on the Fermi surface of a quasi-one-dimensional conductor is calculated for the electron-electron umklapp interaction. We find that in certain regions on the Fermi surface the scattering rate is anomalously high. The reason for the existence of these “hot spots” is analogous to the appearance of the van Hove singularities in the density of states. We employ a generalized τ-approximation (where the scattering integral in the Boltzmann equation is replaced by the scattering time which depends on the position at the Fermi surface) to study the dependence of the electric resistance on the amplitude and the orientation of a magnetic field. We find that the “hot spots” do not produce a considerable magnetoresistance or commensurability effects at the “magic angles”.

Thermal conductivity, heat capacity and phase stability of solid sodium chlorate (NaClO3) under pressure

The thermal conductivity, lambda , and the specific heat capacity at constant pressure, cp, have been measured for solid sodium chlorate (NaClO3) in the temperature range 100-300K. Differential scanning calorimetry (DSC) was used to measure cp at atmospheric pressure, whilst the transient hot-wire method was used to measure simultaneously both lambda and the heat capacity per unit volume rho cp up to a pressure of 2 GPa. The authors found that measured DSC values of cp exceeded those previously predicted from a lattice dynamical model, consistent with a previous prediction of disorder in Cl atom positions at high temperatures. However, such disorder, if it exists, did not affect the transport of heat, since they found that lambda (T) under isochoric conditions appeared to be determined entirely by three-phonon umklapp interactions. They found no evidence for a phase transition in NaClO3 up to 2 GPa at 300K, contrary to some previous results. However, they did observe an explosive chemical instability to occur at about 1 GPa and 130K.

Excited states of double donors in silicon

A calculation of the binding energy of excited states of an electron bound to a substitutional +2e point charge in Si is presented. Direct and umklapp intervalley interactions are included. Very large valley-orbit splittings for 1s and 2s levels are obtained, whereas p-like and ns levels (n≥3) are well described by one-valley effective-mass theory. The spectrum is in fair agreement with recent experiments on chalcogen donors.

Renormalization group for one-dimensional Fermi systems with long-range potential, exchange and umklapp interactions

Renormalization group for one-dimensional Fermi systems with long-range potential, exchange and umklapp interactions

Anisotropy in the Electron-Phonon Umklapp Interaction and Deviations from Matthiessen's Rule

Detailed calculations and experimental evidence show that significant deviations from Matthiessen's rule arise naturally as a result of anistropy in the low-temperature electron-scattering probability. Such anisotropy can be considerable in the pure metal as a result of the strong orientational dependence of the electron-phonon umklapp interaction at low temperatures. It is shown that elastic electron-impurity scattering can produce a significant impurity-dependent change in the intrinsic resistivity simply through the reduction of these angular distortions. As a consequence calculations made in the usual relaxation-time approximation are more appropriately compared with resistivity data dominated by such isotropic-impurity scattering than with ideal resistivity data obtained in the pure limit (where umklapp-generated anisotropy is usually substantial). Using the variational method, various model pseudopotentials, and a realistic phonon spectrum, the mechanism's effect has been accurately evaluated for the case of potassium in the temperature range below $\frac{{\ensuremath{\Theta}}_{D}}{5}$. A thorough analysis of the role played by the various scattering mechanisms indicates that the effect is strongly dependent on the relative amounts of umklapp versus normal scattering present. The variational results are found to account for a substantial portion of the observed impurity dependence of the intrinsic resistivity of potassium. The predicted deviations in polyvalent metals are expected to be significantly larger and agree, in general form, with observations of such effects in aluminum.

Theoretical Calculation of the Electronic Thermal Expansion of Simple Metals

At low temperatures the excitation of conduction electrons in metals gives rise to an electronic contribution C=Γ̃T to the heat capacity, and also a contribution β∝T to the thermal expansion coefficient, where T is the absolute temperature. The electronic Grüneisen parameter is γ=VβB/C=(d lnΓ̃/d lnV), where V and B are the crystal volume and bulk modulus, respectively, at T=0. We have used a local pseudopotential model to calculate all the contributions to Γ̃ and its volume derivative, including the electron-phonon interactions in first-order perturbation. The calculated γ is 1.18 for Na, 1.01 for K, and 1.63 for Al, while the measured γ is 1.8 for Al. The free electron model predicts γ=⅔. From our calculations it appears that the Umklapp interactions of electrons with transverse phonons give important contributions to Γ̃.

Thermal Conductivity of SolidHe3

Measurements are reported of the thermal conductivity above 1.1 deg K for the two solid phases of He/sup 3/. The lowerdensity lower-pressure phase ( alpha ) is body-centered cubic;, the higher-density higher-pressure phase ( beta ) has a hexagonal close-packed structure as does solid He/sup 4/. The thermal conductivity in the beta phase is similar to that of solid He/sup 4/ and is in reasonable agreement with that expected from Peierls' theory for umklapp scattering. The thermal conductivity as a function of reciprocal temperature for the density is given, which clearly indicates that there is a discontinuity and a change in the slope of the curve that occurs at the alpha - beta phase transition. In the alpha phase a change in the slope of the curve occurs at 1.3 to 1.6 deg K. The results in the beta phase are consistent with the assumption that the thermal resistance of the crystal arises principally from phonon-phonon umklapp interaction. Several mechanisms are considered to explain the behavior in the alpha phase. (B.O.G.)