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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

抛物量子线中强耦合极化子的有效质量

采用改进的线性组合算符法、Lagrange乘子和变分法,在考虑电子与LO声子相互作用情况下,研究了抛物量子线中强耦合极化子的有效质量和光学声子平均数。通过数值计算,讨论了约束强度ω₀和拉格朗日乘子u对极化子的有效质量m^*和光学声子平均数N及极化子振动频率λ的影响。计算结果表明:有效质量m^*和光学声子平均数N及极化子振动频率λ都随着约束强度ω₀和拉格朗日乘子u的增加而增大。     

Electron–phonon interaction in fan-shaped quantum dot and quantum wire

The optical phonon modes and electron–optical-phonon interaction in fan-shaped quantum dot and quantum wire are studied with the dielectric continuum (DC) model and separation of variables. The explicit expressions for the longitudinal optical (LO) and interface optical (IO) phonon eigenmodes are deduced. It is found that there exist two types of IO phonon modes: top interface optical (TIO) phonon mode and arc interface optical (AIO) phonon mode, in a fan-shaped quantum dot. 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 potential applications of these results are also discussed.     

Uniform Descriptions of Electron-IO Phonon Interaction in Structures of Multi-layer Coupling Low-dimensional Systems

By using the transfer matrix method, within the framework of the dielectric continuum approximation,uniform forms for the interface optical (IO) phonon modes as well as the corresponding electron-IO phonon interaction Hamiltonians in n-layer coupling low-dimensional systems (including the coupling quantum well (CQ W), coupling quantum-well wire (CQWW), and coupling quantum dot (CQD)) have been presented. Numerical calculations on the three-layer asymmetrical AIGaAs/GaAs systems are performed, and the analogous characteristics for limited frequencies of IO phonon in the three types of systems (CQW, CQWW, and CQD) when the wave-vector and the quantum number approach zero or infinity are analyzed and specified.     

Uniform Descriptions of Electron-IO Phonon Interaction in Structures of Multi-layer Coupling Low-dimensional Systems

By using the transfer matrix method, within the framework of the dielectric continuum approximation, uniform forms for the interface optical (IO) phonon modes as well as the corresponding electron-IO phonon interaction Hamiltonians in n-layer coupling low-dimensional systems (including the coupling quantum well (CQW), coupling quantum-well wire (CQWW), and coupling quantum dot (CQD)) have been presented. Numerical calculations on the three-layer asymmetrical AlGaAs/GaAs systems are performed, and the analogous characteristics for limited frequencies of IO phonon in the three types of systems (CQW, CQWW, and CQD) when the wave-vector and the quantum number approach zero or infinity are analyzed and specified.     

抛物量子线中弱耦合极化子的有效质量和光学声子平均数

讨论电子与体纵光学(LO)声子弱耦合时对抛物量子线中极化子性质的影响.采用Tokuda改进的线性组合算符法、Lagrange乘子和变分法,导出了抛物量子线中弱耦合极化子的有效质量和光学声子平均数随拉格朗日乘子变化的规律及极化子振动频率随量子线约束强度的变化规律.并以ZnS量子线为例进行了数值计算,结果表明:抛物量子线中弱耦合极化子的有效质量m^*和光学声子平均数N随着拉格朗日乘子u的增加而增大;该结论与体材料中结论基本一致,但量子线中的效应比体材料更明显,表明量子线对电子约束的增强,使极化子效应更明显.同时,极化子振动频率λ随约束强度ω₀的增强而增大.     

Vibration Spectrums of Polar Interface Optical Phonons in GaAs/AlAs Cylindrical Quantum Dots

The dispersions of the top interface optical phonons and the side interface optical phonons in cylindrical quantum dots are solved by using the dielectric continuum model. Our calculation mainly focuses on the frequency dependence of the IO phonon modes on the wave-vector and quantum number in the cylindrical quantum dot system. Results reveal that the frequency of top interface optical phonon sensitively depends on the discrete wave-vector in z direction and the azimuthal quantum number, while that of the side interface optical phonon mode depends on the radial and azimuthal quantum numbers. These features are obviously different from those in quantum well, quantum well wire, and spherical quantum dot systems. The limited frequencies of interface optical modes for the large wave-vector or quantum number approach two certain constant values, and the math and physical reasons for this feature have been explained reasonably.