Impact of anisotropy on polar optical vibration in a wurtzite cylindrical nanowire with ring geometry

The quasi-confined (QC) phonon modes, surface optical (SO) phonon modes and corresponding Fröhlich-like Hamiltonian in a wurtzite cylindrical nanowire with ring geometry are investigated in the framework of the dielectric continuum model and Loudon’s uniaxial crystal model. Numerical calculations are focused on the dispersion relations of the SO phonons and the electron–SO phonon coupling strength. Results show that there are only two branches of SO phonon modes. The dispersions of the two branches of SO phonon modes are obvious when the phonon wave-number k_z or the azimuthal quantum number m is small. Typical degenerating behavior of the SO modes is evidenced due to the anisotropic effect of wurtzite crystal. Moreover, when k_z or m are large enough, the frequencies of the two branches of SO modes converge to a definite limiting frequency in single planar heterostructure. The calculations of the electron–SO phonon coupling strength reveal that the high-frequency SO modes (SO₊) play a more important role in the coupling strength than the low-frequency ones (SO₋). Furthermore, the long-wavelength SO phonons with small m are the main factor contributing to the electron–phonon interaction.     

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 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 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.     

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.     

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.     

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.     

Polar propagating optical phonon states and their dispersive spectra in wurtzite nitride superlattices: Quantum size effect

The polar optical phonon states of propagating (PR) modes in wurtzite GaN/Al_xGa_1−xN superlattices (SLs) are investigated within the dielectric continuum model framework. It is proved that the PR phonon modes only appear in wurtzite GaN/Al_xGa_1−xN SL with a small x$x$ Al mole ratio concentration (such as x<0.34$x<0.34$). The analytical phonon states of PR modes and their dispersive equation in the wurtzite GaN/Al_xGa_1−xN SL structures are obtained. Numerical calculations on the dispersive spectra of PR modes and their electrostatic potential properties as well as the quantum size effect are performed for a wurtzite GaN/Al_0.15Ga_0.85N SL. Results reveal that dispersive curves of PR modes in SLs form frequency bands. As the well width of GaN well-layer increases, the frequency bands of PR modes become steeper and narrower. The discussion of electrostatic potentials shows that the wavelength of PR phonon modes with a phase qL=π$qL=\pi$ (0) is 2L$2L$ (L$L$). With the increase of the SL well width, the wave-node number of the PR phonon modes in the barrier regions increases. The present theoretical scheme and numerical results are quite useful for analyzing the dispersive spectra of PR phonon modes and their polaronic effect in wurtzite GaN/AlGaN SL 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.     

Propagating Optical Phonon Modes and Their Electron-Phonon Interaction Hamiltonians in Asymmetric Wurtzite Nitride Semiconductor Quantum Wells

Within the framework of the dielectric continuum model and Loudon's uniaxial crystal model, the properties of frequency dispersion of the propagating (PR) optical phonon modes and the coupling functions of electron-PR phonons interaction in an asymmetrical wurtzite quantum well (QW) are deduced and analyzed via the method of electrostatic potential expanding. Numerical calculation on an asymmetrical Al_0.25Ga_0.75 N/GaN/Al_0.15Ga_0.85N wurtzite QW were performed. The results reveal that there are infinite branches of PR phonon modes in the systems. The behaviors of frequency forbidden of PR modes in the asymmetric QWs have been clearly observed. The mathematical and physical origins for these features have been analyzed in depth. The PR optical phonon branches have been distinguished and labelled reasonably in terms of the oscillating properties of the PR modes in the well-layer material. Moreover, the amplitudes and frequency properties of the electron-PR modes coupling functions in the barrier and well materials have also been analyzed from both of the mathematical and physical viewpoints.     

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.     

Optical phonon modes in a free-standing quantum wire with ring geometry

The confined longitudinal-optical (LO) phonon and surface-optical (SO) phonon modes of a free-standing quantum wire with ring geometry are discussed within the dielectric continuum (DC) approximation. Two branches of SO phonon modes have been investigated. The frequencies of the SO phonons are found to be dispersed and radius dependent for small size systems. When the wave vector qz→∞, the frequencies of each SO modes converge to the frequency values of the single planar heterostructure.     

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.     

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.     

Surface phonon–polaritons in rectangular quantum wires of polar ternary mixed crystals

In this paper, the dispersion relations of surface phonon–polaritons in freestanding rectangular quantum wire systems of polar ternary mixed crystals are derived. The numerical calculations for Al_xGa_1−xAs and Zn_xCd_1−xSe quantum wire systems are performed. The results reveal that the frequencies of surface phonon–polariton modes are sensitive to the geometric structures of the quantum wires, the wave-vectors in z-direction, and the compositions of the ternary mixed crystal materials. The effects of the “two-mode” and “one-mode” behaviors of the ternary mixed crystals on the surface phonon–polariton modes are also discussed.     

纤锌矿氮化物量子阱中光学声子模的三元混晶效应

本文先比较了几种常用方法(修正的无规元素等位移模型、虚晶近似和简化相干势近似等)对纤锌矿三元混晶体声子频率的拟合结果,再选用与实验数据接近的拟合方法,结合介电连续和单轴晶体模型导出含纤锌矿三元混晶In_xGa_1－xN和Al_xGa_1－xN单量子阱各类光学声子模的色散关系,进一步分析了声子模随组分的变化. 结果表明,修正的无规元素等位移模型对单模性纤锌矿     

Studies on the complex behavior of optical phonon modes in wurtzite (ZnO)1−x (Cr2O3) x

This paper reports on the use of phonon spectra obtained with laser Raman spectroscopy for the uncertainty concerned to the optical phonon modes in pure and composite ZnO_1?x (Cr₂O₃) _x . Particularly, in previous literature, the two modes at 514 and 640 cm^?1 have been assigned to ZnO are not found for pure ZnO in our present study. The systems investigated for the typical behavior of phonon modes with 442 nm as excitation wavelength are the representative semiconductor (ZnO)_1?x (Cr₂O₃) _x (x = 0, 5, 10 and 15 %). Room temperature Raman spectroscopy has been demonstrated polycrystalline wurtzite structure of ZnO with no structural transition from wurtzite to cubic with Cr₂O₃. The incorporation of Cr³⁺ at most likely on the Zn sub-lattice sites is confirmed. The uncertainty of complex phonon bands is explained by disorder-activated Raman scattering due to the relaxation of Raman selection rules produced by the breakdown of translational symmetry of the crystal lattice and dopant material. The energy of the E ₂ (high) peak located at energy 53.90 meV (435 cm^?1) due to phonon–phonon anharmonic interaction increases to 54.55 meV (441 cm^?1). A clear picture of the dopant-induced phonon modes along with the B ₁ silent mode of ZnO is presented and has been explained explicitly. Moreover, anharmonic line width and effect of dislocation density on these phonon modes have also been illustrated for the system. The study will have a significant impact on the application where thermal conductivity and electrical properties of the materials are more pronounced.     

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.