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.     

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 (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.     

Phonon States and Dispersive Spectra of Polar Optical Phonons in Quasi-One-Dimensional Nanowires of Wurtzite ZnO and Zinc-Blend MgO Semiconductors

Within the framework of the macroscopic dielectric continuum model and Loudon's uniaxial crystal model, the phonon modes of a wurtzite/zinc-blende one-dimensional (1D) cylindrical nanowire (NW) are derived and studied. The analytical phonon states of phonon modes are given. It is found that there exist two types of polar phonon modes, i.e. interface optical (IO) phonon modes and the quasi-confined (QC) phonon modes existing in 1D wurtzite/zinc-blende NWs. Via the standard procedure of field quantization, the Fröhlich electron-phonon interaction Hamiltonians are obtained. Numerical calculations of dispersive behavior of these phonon modes on a wurtzite/zinc-blende ZnO/MgO NW are performed. The frequency ranges of the IO and QC phonon modes of the ZnO/MgO NWs 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 properties of the IO and QC modes on the free wave-number k_z and the azimuthal quantum number m arediscussed. The analytical Hamiltonians of electron-phonon interaction obtained here are quite useful for further investigating phonon influence on optoelectronics properties of wurtzite/zinc-blende 1D NW structures.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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     

First-order Raman scattering in cylindrical wurtzite nanowire

We have presented a theoretical study on electron resonant Raman scattering (ERRS) process associated with the bulk longitudinal optical (LO), surface optical (SO) and quasi-confined (QC) phonon modes in a free-standing wurtzite nanowire (NW). We consider the Fröhlich electron–phonon interaction in the framework of the dielectric continuum model. Numerical calculations on the GaN material reveal that differential cross-section (DCS) is sensitive to the wire size. The bulk LO and high-frequency quasi-confined (QC₊) phonons make main contributions to the DCS and the impact of the SO phonon can be negligible in the ERRS process. Moreover, scattering intensity of the bulk LO phonon is strongly enhanced as the incident photon energy approaches the energy band-gap of the GaN.     

Dispersions of quasi-confined optical phonon modes and their electron–phonon interactions in an asymmetric wurtzite AlxGa1−xN / GaN / Aly Ga1−yN quantum well

Within the framework of 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 an asymmetric wurtzite quantum well (QW) are deduced via the method of electrostatic potential expanding. The present theoretical scheme can be treated as a generalization of the QC optical phonons in an ordinary wurtzite double heterostructures QWs, and it can be reduced naturally to the situation of the symmetrical wurtzite QW once a suite of symmetrical parameters are adopted. Numerical computation on an asymmetric AlN/ GaN/ Al_0.15Ga_0.85N wurtzite QW are performed, and a detailed comparison with the case in symmetric wurtzite QW is carried out. The calculated results show that the structural asymmetry of wurtzite QW changes greatly the dispersion behaviors and the electrostatic potential distributions of the QC optical phonon modes.     

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.     

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.     

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.     

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.