Theoretical Researches On The Phonon Dissipation Mechanisms Of Quartz Crystals And Carbon Nanotubes

With the great developments of the mesoscopic electromechanical,the optomechanical,the hybrid mechanical systems,etc.,especially in quantum regime,the origins of mechanical dissipation have been under intensive study.In particular,as a kind of piezoelectric materials,the quartz crystals can be used to manufacture the low-loss mechanical resonator.On the other hand,the unique mechanical properties of carbon nanotubes have been under intense investigations,and the phonon damping of these materials is thus worthy of studying due to its close relevance to physical properties and phenomenon of carbon nanotubes.In this thesis,thus,we study the phonon(mechanical excitation)dissipation and quality factor of both these materials.In chapter two,a phonon dissipation mechanism is proposed to interpret the measured results of the mechanical quality factors in a recent experiment[M.Goryachev et al.,Phys.Rev.Lett.111,085502(2013)]where the cryogenic resonant sound waves are produced by a kind of piezoelectric bulk acoustic cavities made from quartz crystals.The frequency range for measurements is from several MHz to near 1 GHz.As a result,the dependence of the measured quality factors on frequency displays Rayleigh phonon scattering in these acoustic cavities at liquid helium or even lower temperature.Moreover,the overall trend of the quality factor surprisedly gets smaller as temperature declines.Obviously,these experimental results demonstrate a new mechanism of phonon dissipation.The anharmonic effect,described by phonon-phonon coupling,is suppressed and not responsible for phonon dissipation at such low temperature.Therefore,the main limitation on quality is due to the material itself.The two-level systems(TLSs)are distributed in the disordered part of quartz crystals,and thus we suggest that the phonon-TLS elastic scattering leads to the dissipative phenomenon based on the well-known phonon-TLS coupling.Then,according to the second-order perturbation theory,the four elastic scattering processes are used to derive the formulae of the phonon scattering rate and quality factor Q.Finally,the results manifest that the quality factor obeys the law Q ×f3=const,where f denotes the resonant frequency of the acoustic cavity,and becomes smaller with the reduction of temperature.The coincidence of the calculated and the experimental results means that the phonon-TLS elastic scattering is reasonable to interpret the phenomenon observed in the experiment.Besides,the transition frequencies of the TLS ensemble widely are implied to lie in the region of several gigahertz or even higher,and therefore the phonon-TLS elastic scattering sufficiently happens for the phonons at 1 GHz or lower which are in the frequency range of the experiment.While the phonon-TLS elastic scattering leads to a kind of phonon dissipation mentioned in the previous chapter,the phonon-static point defect elastic scattering can be considered as another source for the phonon dissipation.In chapter three,therefore,the static scattering is introduced to improve the dissipation mechanism based on the phonon elastic scattering.The derived results indicate that,if the static point defect scattering dominates the phonon dissipation,the quality factor is independent of temperature though presenting Rayleigh-type scattering.On the contrary,if the phonon-TLS elastic scattering plays a considerable role on the phonon dissipation,the quality factor conforms to the Rayleigh-type scattering and declines with temperature reducing,which agrees with the experiment.Besides,we also show how the energy splitting distribution of the TLS ensemble affects the total Q factor,and analyze the contribution of the static defect scattering.Therefore,the study shows that,as the anharmonic effect is limited at ultra-low temperature,the phonon attenuation originates from phonon elastic scattering by static and the defects,including TLSs and static point defects,in crystals.Apart from the vibrational damping of quartz crystals,the damping mechanism of carbon nanotubes(CNTs)attracts considerable interest and has been intensively investigated.In chapter four,we employ the continuum elasticity model to calculate the phonon damping rate and quality factor of C.mode,i.e.,the lowest-lying phonon branch with |l|=2,where l denotes the integer angle momentum quantum number,while phonon modes with |l|=0,1 have been studied previously.This model is available in long-wavelength regime because the carbon nanotubes can be considered as a kind of one-dimensional continuous medium through which the long-wavelength phonons propagate freely until their dissipation.In this model,the elastic potential energy is described as the quadratic form of the deformation tensors containing the linear and nonlinear parts.Hence,except the harmonic terms,the coupling terms of a linear strain tensor with a nonlinear strain tensor are also contained in this elastic model and responsible for the three-phonon processes which lead to a kind of phonon damping.Besides,since the C mode is a kind of curvature phonon mode,the derivation and calculation for C mode damping should take account of the curvature effect.Based on the model mentioned above,the dispersive relation and polarization vector of long-wavelength C mode are obtained,and analytic formulae of three-coupling terms involved with C mode can also be derived.Since the main contributions of C mode damping are only from a decay channel C?F+F and a combination channel C+C?T in long-wavelength regime,there are only two coupling coefficients given in this chapter through the derivations.We estimate the upper bounds of quality factors at zero and finite temperature,respectively,and depict their dependence on tube radius R.A comparison displays that the results change a lot without the consideration of curvature energy term,and thus it is necessary to consider this term.We also show the relaxation time of C mode as a function of wave vector k,and calculate the thermal conductivity including the C mode.In some cases,the C mode damping makes a considerable contribution to the thermal conductivity.Finally,in chapter five,we give a brief summary and some outlooks on working directions in the future.