Polar optical phonon states and dispersive spectra of wurtzite ZnO nanocrystals embedded in zinc-blende MgO matrix

Based on the macroscopic dielectric continuum model and Loudon’s uniaxial crystal model, the polar optical phonon modes of a quasi-0-dimensional (Q0D) wurtzite spherical nanocrystal embedded in zinc-blende dielectric matrix are derived and studied. It is found that there are two types of polar phonon modes, i.e. interface optical (IO) phonon modes and the quasi-confined (QC) phonon modes coexisting in Q0D wurtzite ZnO nanocrystal embedded in zinc-blende MgO matrix. Via solving Laplace equations under spheroidal and spherical coordinates, the unified and analytical phonon states and dispersive equations of IO and QC modes are derived. Numerical calculations on a wurtzite/zinc-blende ZnO/MgO nanocrystal are performed. The frequency ranges of the IO and QC phonon modes of the ZnO/MgO nanocrystals are analyzed and discussed. It is found that the IO modes only exist in one frequency range, while QC modes may appear in three frequency ranges. The dispersive frequencies of IO and QC modes are the discrete functions of orbital quantum numbers l and azimuthal quantum numbers m. Moreover, a pair of given l and m corresponds to one IO mode, but to more than one branches of QC. The analytical phonon states and dispersive equations obtained here are quite useful for further investigating Raman spectra of phonons and other relative properties of wurtzite/zinc-blende Q0D nanocrystal structures.     

Polar optical phonon states and their electron-phonon coupling properties in a wurtzite nitride quantum dot

Within the framework of the macroscopic dielectric continuum model, the interface-optical-propagating (IO-PR) mixing phonon modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) structure are derived and studied. The approximative analytical-phonon-states of IO-PR mixing modes are given. It is found that there are two types of IO-PR mixing phonon modes, i.e. ρ-IO/z-PR mixing modes and the z-IO/ρ-PR mixing modesexisting in Q0D wurtzite QDs. And each IO-PR mixing modes also have symmetrical and antisymmetrical forms. Via a standard procedure of field quantization, the Fröhlich Hamiltonians of electron-(IO-PR) mixing phonons interaction are obtained. And the orthogonal relations of polarization eigenvectors for these IO-PR mixing modes are also deduced. Numerical calculations of dispersive relation and electron-phonon coupling properties on a wurtzite GaN cylindrical QD are carried out. The behaviors that the IO-PR mixing phonon modes in wurtzite QDs reduce to the IO modes and PR modes in wurtzite QW and QWR systems are analyzed deeply from both of the viewpoints of physics and mathematics. The result shows that the present theories of polar mixing phonon modes in wurtzite cylindrical QDs are consistent with the phonon modes theories in wurtzite QWs and QWR systems. The coupling properties of electron-(IO-PR) mixing modes interactions are studied and analyzed in detail. An abnormal increase of electron-phonon coupling strength are observed as the azimuthal quantum numbers and order of phonon modes increase, which is ascribed to the modulation effect of different dielectric functions of wurtzite crystals in radius- and axial-directions. The analytical electron-phonon interaction Hamiltonians obtained here are useful for further investigating phonon influence on optoelectronics properties of wurtzite Q0D QD structures.     

Fröhlich electron-interface and -propagating optical phonon interactions in a wurtzite multi-shell cylindrical heterostructure

Under the dielectric continuum model and Loudon's uniaxial crystal model, the polar optical phonon modes in a wurtzite multi-shell cylindrical heterostructure are analyzed and discussed. The analytical electrostatic potential functions are presented for all the five types of polar optical phonon modes including the interface optical (IO) modes, the propagating (PR) modes, the quasi-confined (QC) modes, the half-space-like (HSL) modes and the exactly confined (EC) modes. By adopting a transfer matrix method, the free IO and PR phonon fields and corresponding Fröhlich electron -IO and -PR interaction Hamiltonians are obtained via the method of electrostatic potential expansion. The analytical formulas are universal and can be applied to single, double and some complex cylindrical wurtzite quantum systems.     

Anisotropy Effects on Polar Optical Vibrations in a Freestanding Wurtzite Cylindrical Quantum Dot

The properties of polar optical phonon vibrations in a quasi-zero- dimensional (Q0D) anisotropic wurtzite cylindrical quantum dot (QD) are analyzed based on the dielectric continuum model and Loudon's uniaxial crystal model.The analytical electrostatic potentials of the phonon vibrations in the systems are deduced and solved exactly. The result shows that there exist four types of polar mixing optical phonon modes in the Q0D wurtzite cylindrical QD systems. The dispersive equations and electron-phonon coupling function for the quasi-confined-half-space (QC-HS) mixing modes are derived and discussed. It is found that once the radius or the height of the QD approach infinity, the dispersive equations of the QC-HS mixing modes in the Q0D cylindrical QD can naturally reduce to those of the QC and HS modes in Q2D QWs or Q1D QWWs systems. This has been analyzed reasonably from both of physical and mathematical viewpoints.     

Anisotropy Effects on Polar Optical Vibrations in a Freestanding Wurtzite Cylindrical Quantum Dot

The properties of polar optical phonon vibrations in a quasi-zero- dimensional （QOD） anisotropic wurtzite cylindrical quantum dot （QD） are analyzed based on the dielectric continuum model and Loudon＇s uniaxial crystal model. The analytical electrostatic potentials of the phonon vibrations in the systems are deduced and solved exactly. The result shows that there exist four types of polar mixing optical phonon modes in the QOD wurtzite cylindrical QD systems. The dispersive equations and electron-phonon coupling function for the quasi-confined-half-space （QC-HS） mixing modes are derived and discussed. It is found that once the radius or the height of the QD approach infinity, the dispersive equations of the QC-HS mixing modes in the QOD cylindrical QD can naturally reduce to those of the QC and HS modes in Q2D QWs or Q1D QWWs systems. This has been analyzed reasonably from both of physicM and mathematical viewpoints.     

Interface and propagating optical phonon mixing modes in a wurtzite GaN-based quantum dot

The polar optical phonon vibrating modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) are solved exactly based on the dielectric continuum model and Loudon’s uniaxial crystal model. The result shows that there exist four types of polar mixing optical phonon modes in the Q0D wurtzite cylindrical QD systems, which is obviously different from the situation in blende cylindrical QDs. The dispersive equations for the interface-optical-propagating (IO-PR) mixing modes are deduced and discussed. It is found that the dispersive frequency of IO-PR mixing modes in wurtzite QD just take a series of discrete values due to the three-dimensional confined properties. Moreover, once the radius or the height of the QD approach infinity, the dispersive equations of the IO-PR mixing modes in the wurtzite Q0D cylindrical QD can naturally reduce to those of the IO and PR modes in Q2D QWs or Q1D QWWs systems. This has been analyzed reasonably from both physical and mathematical viewpoints. The analytical expressions obtained in the paper are useful for further investigating phonon influence on physical properties of the wurtzite Q0D QD systems.     

Confinement of polar optical phonons in quasi-one-dimensional Wurtzite GaN-based quantum well wires: propagating and half-space phonon modes

With the aid of the macroscopic dielectric continuum and Loudon’s uniaxial crystal models, the propagating (PR) and half-space (HS) optical phonon modes and corresponding Fröhlich-like electron-phonon interaction Hamiltonians in a quasi-one-dimensionality (Q1D) wurtzite quantum well wire (QWW) structure are derived and studied. Numerical calculations on a wurtzite GaN/A_l0.15Ga_0.85N QWW are performed, and discussion is focused mainly on the dependence of the frequency dispersions of PR and HS modes on the free wave-number k _z in the z-direction and on the azimuthal quantum number m. The calculated results show that, for given k _z and m, there usually exist infinite branches of PR and HS modes in the high-frequency range, and only finite branches of HS modes in the low-frequency range in wurtzite QWW systems. The reducing behaviors of the PR modes to HS modes, and of the HS mode to interface phonon mode have been observed clearly in Q1D wurtzite heterostructures. Moreover, the dispersive properties of the PR and HS modes in Q1D QWWs have been compared with those in Q2D quantum well structures. The underlying physical reasons for these features have also been analyzed in depth.     

Polaronic effects due to surface optical vibration modes in a freestanding cylindrical wurtzite nanowire

The ground-state polaron self-trapped energy and effective mass due to the surface optical (SO) phonon modes in a freestanding wurtzite GaN nanowire (NW) were studied by means of the Lee–Low–Pines variational approach. Based on the dielectric continuum and Loudon’s uniaxial crystal models, the polar optical phonon modes in the one-dimensional (1D) systems are analyzed, and the vibrating spectra of SO modes and electron–SO phonon coupling functions are discussed and analyzed. The calculations on the ground-state polaron self-trapped energy and correction of effective mass due to the SO phonon modes in the 1D GaN NWs reveal that the polaron self-trapped energy and correction of effective mass are far larger than those in 1D GaAs NW systems. The reasons resulting in this obvious difference in the two 1D structures are mainly due to the different electron–phonon coupling constants and electron effective masses of bulk materials constituting the two types of 1D confined system. Finally, the polaronic properties of the wurtzite 1D GaN NWs have been compared with those of the wurtzite GaN-based two-dimensional quantum wells. The physical origination of these characteristics and their distinction in the different-dimensionality systems has been analyzed in depth.     

Polar interface and surface optical vibration spectra in multi-layer wurtzite quantum wires： transfer matrix method

The polar interface optical （IO） and surface optical （SO） phonon modes and the corresponding Froehlich electron phonon-interaction Hamiltonian in a freestanding multi-layer wurtzite cylindrical quantum wire （QWR） are derived and studied by employing the transfer matrix method in the dielectric continuum approximation and Loudon＇s uniaxial crystal model. A numerical calculation of a freestanding wurtzite GaN/AlN QWR is performed. The results reveal that for a relatively large azimuthal quantum number m or wave-number kz in the free z-direction, there exist two branches of IO phonon modes localized at the interface, and only one branch of SO mode localized at the surface in the system. The degenerating behaviours of the IO and SO phonon modes in the wurtzite QWR have also been clearly observed for a small kz or m. The limiting frequency properties of the IO and SO modes for large kz and m have been explained reasonably from the mathematical and physical viewpoints. The calculations of electron-phonon coupling functions show that the high-frequency IO phonon branch and SO mode play a more important role in the electron phonon interaction.     

Polar Oscillation of Interface and Surface Optical Phonons in Free-Standing Cylindrical Quantum-Well Wires

Under dielectric continuum approximation, interface optical (IO) and surface optical (SO) phonon modes as well as the corresponding Fro^ehlich electron-phonon interaction Hamiltonian in a free-standing cylindrical quantum-well wire (QWW) are derived and studied. Numerical calculations on GaAs/AlzGa1-x As cylindrical QWW are performed. Results reveal that there are two branches of IO phonon modes and one branch of SO phonon mode, and the dispersion frequencies of IO or SO phonon modes sensitively depend on the Al mole fraction x in AlzGa1-x As material and the wavevector in z direction, kz. With the increasing of kz and quantum number m, the frequency of each IO mode approaches one of the two frequency values of the single GaAs/AlxGa1-x As heterostructure, and the electrostatic potential distribution of the phonon mode tends to be more and more localized at a certain interface or surface, meanwhile, the coupling between the electron-IO and -SO phonons becomes weaker.     

外电场作用下纤锌矿氮化物抛物量子阱中极化子能级

The energy levels of a polaron in a wurtzite nitride finite parabolic quantum well (PQW)are studied by a modified Lee-Low-Pines variational method. The ground state of the polaron, the transition energy from first exited state to the ground state and the 关键词： 氮化物抛物量子阱 电子-声子相互作用 极化子     

Interface-Optical-Phonon Modes in Quasi-one-dimensional Wurtzite Rectangular Quantum Wires

By employing the dielectric continuum model and Loudon's uniaxial crystal model, the interface optical (IO) phonon modes in a freestanding quasi-one-dimensional (Q1D) wurtzite rectangular quantum wire are derived and analyzed. Numerical calculation on a freestanding wurtzite GaN quantum wire is performed. The results reveal that the dispersion frequencies of IO modes sensitively depend on the geometric structures of the Q1D wurtzite rectangular quantum wires, the free wave-number k_z in z-direction and the dielectric constant of the nonpolar matrix. The degenerating behavior of the IO modes in Q1D wurtzite rectangular quantum wire has been clearly observed in the case of small wave-number k_z and large ratio of length to width of the rectangular crossing profile. The limited frequency behaviors of IO modes have been analyzed deeply, and detailed comparisons with those in wurtzite planar quantum wells and cylindrical quantum wires are also done. The present theories can be looked on as a generalization of that in isotropic rectangular quantum wires, and it can naturally reduce to the case of Q1D isotropic quantum wires once the anisotropy of the wurtzite material is ignored.     

Electron-phonon interaction in hemispherical quantum dot

Within the framework of the dielectric continuum (DC) model, the optical phonon modes and electron-optical-phonon interaction in hemispherical quantum dot are investigated. The proper eigenfunctions for longitudinal optical (LO) and interface optical (IO) phonon modes are constructed. After having quantized the eigenmodes, we derive the Hamiltonian operators describing the LO and IO phonon modes as well as the corresponding Fröhlich electron-phonon interaction. The dispersion relation of IO phonon modes is size independent. The potential applications of these results are also discussed.     

The quasi-confined optical phonons in wurtzite GaN/AlN multiple quantum wells

Within the framework of the dielectric-continuum model and Loudon's uniaxial crystal model, the equation of motion for p-polarization field in arbitrary wurtzite multilayer heterostructures are solved for the quasi-confined phonon (QC) modes. The polarization eigenvector, the dispersion relation, and the electron-QC interaction Fröhlich-like Hamiltonian are derived by using the transfer-matrix method. The dispersion relations and the electron-QC coupling strength are investigated for a wurtzite GaN/AlN single QW. The results show that there are infinite branches of dispersion curve with definite symmetry with respect to the center of the QW structure. The confinement of the quasi-confined phonons in the QW leads to a quantization of q_z,j characterized by an integer m that defines the order of corresponding quasi-confined modes. The QC modes are more dispersive for decreasing m. The QC modes display an interface behavior in the barrier and a confined behavior in the well. The symmetric modes have more contribution to electron-QC interaction than the antisymmetric modes. The strains have more effect on symmetry modes, and can be ignored for symmetry modes.     

Polar oscillation and dispersion properties of quasi-confined optical phonon modes in a wurtzite GaN/AlxGa1−xN nanowire

Under the dielectric continuum model and Loudon’s uniaxial crystal model, the properties of the quasi-confined (QC) optical phonon dispersions and the electron-QC phonons coupling functions in a cylindrical wurtzite nanowire are deduced via the method of electrostatic potential expanding. Numerical computations on a GaN/Al_0.15Ga_0.85N wurtzite nanowire are performed. Results reveal that, for a definite axial wave number k_z and a certain azimuthal quantum number m, there are infinite branches of QC modes. The frequencies of these QC modes fall into two regions, i.e. a high frequency region and a low frequency region. The dispersion of the QC modes are quite apparant only when k_z and m are small. The lower-order QC modes in the higher frequency region play more important role in the electron-QC phonon interactions. Moreover, for the higher-order QC modes in the high frequency region, the electrostatic potentials “escaping” out of the well-layer material nearly could be ignored.     

Electron-interface-optical-phonon interaction in rectangular quantum wire and quantum dot

Under the dielectric continuum model and separation of variables, the interface optical (IO) phonon modes and electron-optical-phonon interaction in rectangular quantum wire and quantum dot embedded in a nonpolar matrix are studied. We found that there exist various types of IO phonon modes in rectangular nanostructures. The IO phonon modes in rectangular quantum wire include IO-propagating (IO-PR) and IO-IO hybrid phonon modes, while the IO phonon modes in rectangular quantum dot contain IO-IO-PR and IO-PR-PR hybrid phonon modes. The results of numerical calculation show that these hybrid phonon modes contain corner optical (CO) phonon modes and edge optical (EO) phonon modes. The potential applications of these results are also discussed.     

Interface Optical Phonon Modes and Frohlich Electron-Phonon Interaction Hamiltonian in a Multi-shell Spherical Nanoheterosystem

Under dielectric continuum approximation, interface optical (IO) phonon modes and the Frohlich electron-IO phonon interaction Hamiltonian in a multi-shell spherical nanoheterosystem were derived and studied. Numericalcalculations on three-layer and four-layer CdS/HgS spherical nanoheterosystems have been performed. Results revealthat there are four IO phonon modes for the three-layer system and six IO phonon modes for the four-layer system.On each interface, there are two IO phonon modes, the frequency of one is between WTO,CdS and WLO,CdS, and that ofthe other is between WTO,HgS and WLO,HgS. With the increasing of quantum number l, the frequency of each IO modeapproaches one of the two frequency values of the single CdS/HgS heterostructure, and the potential for each IO modeis more and more localized at a certain interface, furthermore, the coupling between the electron-lO phonons becomes weaker.     

Temperature and layer number dependence of the G and 2D phonon energy and damping in graphene

We have studied the temperature and size dependence of the G and 2D phonon modes in graphene. It is shown that in a graphene monolayer the phonon energy decreases whereas the phonon damping increases with increasing temperature. The electron-phonon interaction leads to hardening whereas the fourth-order anharmonic phonon-phonon processes lead to softening of the phonon energy with increasing temperature. We have shown that the electron-phonon interaction plays an important role also by the dispersion dependence of the phonon G mode, by the observation of the Kohn anomaly. The G mode frequency decreases and damping increases, whereas the 2D phonon frequency and damping increase with increasing layer number. The temperature and size effects of the 2D mode are much stronger than those of the G mode.     

Polar Interface Optical Phonon Modes and Fr(o*)hlich Electron-Phonon Interaction Hamiltonians in an Arbitrary Layer-Number Quantum Well System

By using determinant method as in our recent work, the IO phonon modes, the orthogonal relation forpolarization vector, electron-IO phonon Frohlich interaction Hamiltonian, the dispersion relation, and the electron-phonon coupling function in an arbitrary layer-number quantum well system have been derived and investigated withinthe framework of dielectric continuum approximation. Numerical calculation on seven-layer Alx Ga1-x As/GaAs systemshave been performed. Via the numerical results in this work and previous works, the general characters of the IO phononmodes in an n-layer coupling quantum well system were concluded and summarized. This work can be regarded as ageneralization of previous works on IO phonon modes in some fixed layer-number quantum well systems, and it providesa uniform method to investigate the effects of IO phonons on the multi-layer coupling quantum well systems.