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.     

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.     

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.     

Vibration Spectra of Quasi-confined Optical Phonon Modes in an Asymmetric Wurtzite AlxGa1-xN/GaN/AlyGa1-yN Quantum Well

Based on the dielectric continuum model and Loudon's uniaxial crystal model,the properties of the quasiconfined (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 naturally reduce to the results in symmetric wurtzite QW once a set of symmetric structural parameters are chosen.Numerical calculations on an asymmetric A1N/GaN/Al0.15Ga0.85N wurtzite QW are performed.A detailed comparison with the symmetric wurtzite QW was also performed.The results show that the structural asymmetry of wurtzite QW changes greatly the dispersion frequencies and the electrostatic potential distributions of the QC optical phonon modes.     

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.     

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.     

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.     

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.     

Electron-LO-phonon interaction in wurtzite GaN quantum wells under a magnetic field

We calculate the electron-LO-phonon relaxation rates in wurtzite GaN quantum wells in the presence of a magnetic field parallel to the growth direction. Using the dielectric continuum model (DCM), we are able to include contributions from both the interface and the quasi-confined phonon modes. The relaxation rate expression takes the phonon dispersion into account, and is applicable to all phonon modes. We find that the relaxation rates show strong oscillations as a function of the applied magnetic field. In relatively wide (8 nm) quantum wells, the inclusion of interface phonon mode decreases this oscillation amplitude. But in thin wells (5 nm), the interface phonon mode is of the same importance as the quasi-confined mode, and it strongly modifies the oscillation behavior.     

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.     

Electron quasi-confined-optical-phonon interactions in wurtzite GaN/AlN quantum wells

The equation of motion for the p-polarization field in a wurtzite GaN/AlN multilayer heterostructure is solved for the quasi-confined-optical-phonon modes based on the dielectric-continuum model and Loudons uniaxial crystal model. The polarization eigenvector, the dispersion relation of the quasi-confined-optical-phonon modes and the electron-quasi-confined-phonon interaction Fröhlich-like Hamiltonian are derived. The analytical formulas can be directly applied to single/multiple quantum wells (QWs) and superlattices. The electron-quasi-confined-phonon coupling functions are investigated for a given AlN/GaN/AlN single QW with full account of the strains of the QW structures and the anisotropy effect of wurtzite crystals. We find that there are two kinds of quasi-confined-optical-phonon modes in the GaN/AlN QWs: the GaN-layer quasi-confined-optical-phonon modes and the AlN-layer quasi-confined-optical-phonon modes. There are infinite quasi-confined-optical-phonon branches, labelled by a quantum number n (n=1,2,...), with definite symmetry with respect to the center of the AlN/GaN/AlN single QW for a given phonon wave number q. The dispersions of the quasi-confined-optical-phonon modes with smaller n are more obvious than the ones with larger n. Moreover, the modes with smaller n are much more important for their electron-quasi-confined-phonon interactions than those with larger n. In most cases, it is enough to consider the modes with n 8 for the electron-quasi-confined-phonon interactions in a single GaN/AlN QW. The higher frequency modes are more significant than the lower ones. The long-wavelength quasi-confined-optical-phonon modes are much more important for the electron-quasi-confined-phonon interactions. The GaN-layer quasi-confined-optical-phonon energies and their electron-quasi-confined-phonon interaction strength are markedly increased due to the strains of the QW structures. The influence of the strains on the the AlN-layer electron-quasi-confined-phonon interactions can be ignored.     

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.     

Vibrational properties of single-wall BC3 nanotubes

The phonon dispersions of single-wall BC₃ nanotubes with any chirality are calculated within a symmetry-based force constant model of the lattice dynamics. Based on the non-symmorphic symmetry group of the BC₃ tubes, the symmetries and number of the Raman- and infrared-active modes at Γ point of the one-dimensional Brillouin zone are given. The neighbor atom-atomic interaction force constants are recalculated by fitting them to the experimental phonon energy-dispersion curves of honeycomb BC₃ sheet. The frequencies of the optically active modes are presented as the function of diameters and chiralities for BC₃ tubes. The obtained phonon density-of-states spectra, phonon dispersion relations, and vibrational patterns of the zone-center phonons are presented and discussed in detail. The calculated frequencies of infrared-active modes are compared with the experimental values reported in the literature. The results provide comprehensive information about the vibrational properties of the BC₃ tubes and shed light on the interpretation of Raman scattering and infrared spectroscopies.     

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.     

Thermal and quantum fluctuations induced additional gap in single-particle spectrum of d-p model

The possibility of thermal and quantum fluctuations induced attractive interaction leading to a pairing gap Δ_tq in the single-particle spectrum of d-p model in the limit of a large N of fermion flavor is investigated analytically. This is an anomalous pairing gap in addition to the one with d-wave symmetry originating from partially screened, inter-site coulomb interaction. The motivation was to search for a hierarchy of multiple many-body interaction scales in high-T_c superconductor as suggested by recent experimental findings. The pairing gap anisotropy stems from more than one source, namely, nearest neighbor hoppings and the p-d hybridization, but not the coupling of the effective interaction. The temperature at which Δ_tq vanishes may be driven to zero by using a tuning parameter to have access to quantum criticality (QC) only when N?1. For the physical case N=2, the usual coherent quasi-particle feature surfaces in the spectral weight everywhere in the momentum below the pairing gap Δ_tq. Thus it appears that the reduction in spin degeneracy has the effect of masking quantum criticality.     

Energy of a Polaron in a Wurtzite Nitride Finite Parabolic Quantum Well

The effects of electron-phonon interaction on energy levels of a polaron in a wurtzite nitride finite parabolic quantum well （PQW） are studied by using a modified Lee-Low-Pines variational method. The ground state, first excited state, and transition energy of the polaron in the GaN/Al0.3Ga0.7N wurtzite PQW are calculated by taking account of the influence of confined LO（TO）-like phonon modes and the half-spaee LO（TO）-like phonon modes and considering the anisotropy of all kinds of phonon modes. The numerical results are given and discussed. The results show that the electron-phonon interaction strongly affects the energy levels of the polaron, and the contributions from phonons to the energy of a polaron in a wurtzite nitride PQW are greater than that in an A1GaAs PQW. This indicates that ehe electron-phonon interaction in a wurtzite nitride PQW is not negligible.     

Electron-phonon interaction effects on the surface states in wurtzite nitride semiconductors

A variational approach is used to study the surface states of an electron in a semi-infinite wurtzite nitride semiconductor. The surface-state energy of the electron is calculated, by taking the effects of the electron-surface optical phonon interaction and structure anisotropy into account. The numerical computation has been performed for the energies of the electronic surface states as a function of the surface potential V₀ for wurtzite GaN, AlN, and InN, respectively. The results show that the electron-phonon interaction lowers the surface state energy. It is also found that the energies of the electronic surface-state in wurtzite structures are lower than that in the zinc-blende structures by hundreds of meV for the materials calculated. The influence of e-p-interactions on the surface state of electron cannot be neglected.     

Exciton effects on the refractive index and optical absorption in wurtzite quantum wells via a fractional-dimensional space approach

The optical refractive index changes and absorption coefficients of quantum wells (QWs) are theoretically investigated with considering exciton effects within the framework of the fractional-dimensional space approach. The exciton wave functions and bound energies are obtained as a function of spatial dimensionality, and the dimension increases with the well width increasing. Then optical properties are obtained by using the compact-density matrix approach and an iterative method. Numerical results are presented for wurtzite ZnO/Mg_xZn_1−xO QWs. The calculated results show that the changes of refractive index and absorption coefficients are greatly enhanced due to the quantum confinement of exciton. And the smaller the QW width (dimension) is, the larger influence of exciton on the optical properties will be. Furthermore, the exciton effects make the resonant peaks move to a lower energy. In addition, the optical properties are related to the QW width, the incident optical intensity and carrier density.