Free-standing Quantum Wire Research Articles

An efficient finite-difference scheme for computation of electron states in free-standing and core–shell quantum wires

The electron states in axially symmetric quantum wires are computed by means of the effective-mass Schrödinger equation, which is written in cylindrical coordinates φ, ρ, and z. We show that a direct discretization of the Schrödinger equation by central finite differences leads to a non-symmetric Hamiltonian matrix. Because diagonalization of such matrices is more complex it is advantageous to transform it in a symmetric form. This can be done by the Liouville-like transformation proposed by Rizea et al. (2008), which replaces the wave function ψ(ρ) with the function F(ρ)=ψ(ρ)ρ and transforms the Hamiltonian accordingly. Even though a symmetric Hamiltonian matrix is produced by this procedure, the computed wave functions are found to be inaccurate near the origin, and the accuracy of the energy levels is not very high. In order to improve on this, we devised a finite-difference scheme which discretizes the Schrödinger equation in the first step, and then applies the Liouville-like transformation to the difference equation. Such a procedure gives a symmetric Hamiltonian matrix, resulting in an accuracy comparable to the one obtained with the finite element method. The superior efficiency of the new finite-difference scheme (FDM) is demonstrated for a few ρ-dependent one-dimensional potentials which are usually employed to model the electron states in free-standing and core–shell quantum wires. The new scheme is compared with the other FDM schemes for solving the effective-mass Schrödinger equation, and is found to deliver energy levels with much smaller numerical error for all the analyzed potentials. It also gives more accurate results than the scheme of Rizea et al., except for the ground state of an infinite rectangular potential in freestanding quantum wires. Moreover, the PT symmetry is invoked to explain similarities and differences between the considered FDM schemes.

Surface optical phonon-assisted electron Raman scattering in a free-standing quantum wire with ring geometry

We present a theoretical calculation of the differential cross section for the electron Raman scattering process associated with surface optical phonon modes in semiconductor quantum wires (QWs) with ring geometry. Electron states are considered to be confined within the QWs. We consider the Fröhlich electron–phonon interaction in the framework of the dielectric continuum approach. Some singularities for the ring with various sizes in the Raman spectra are found and interpreted. A discussion of the phonon behavior for the QWs with large and small sizes is presented. The numerical results are also compared with those of experiments.

EFFECTS OF CONFINED LO AND SO PHONON MODES ON POLARON IN FREESTANDING CYLINDRICAL QUANTUM WIRE WITH PARABOLIC CONFINEMENT

The electron self-energy and correction to the electron effective mass in a freestanding quantum wire with parabolic confining potential was investigated by the perturbation approach. Both the electron-confined longitudinal optical (LO) phonon and surface optical (SO) phonon interactions were considered. Results shows that, for small wire radius, the contributions of electron–LO phonon interaction to the electron self-energy and the correction to the electron effective mass are relatively small in compare with those of the electron–SO phonon interaction.

Markovian and Non-Markovian Light-Emission Channels in Strained Quantum Wires

We have achieved conditions to obtain optical memory effects in semiconductor nanostructures. The system is based on strained InP quantum wires where the tuning of the heavy-light valence band splitting has allowed the existence of two independent optical channels with correlated and uncorrelated excitation and light-emission processes. The presence of an optical channel that preserves the excitation memory is unambiguously corroborated by photoluminescence measurements of free-standing quantum wires under different configurations of the incoming and outgoing light polarizations in various samples. High-resolution transmission electron microscopy and electron diffraction indicate the presence of strain effects in the optical response. By using this effect and under certain growth conditions, we have shown that the optical recombination is mediated by relaxation processes with different natures: one a Markov and another with a non-Markovian signature. Resonance intersubband light-heavy hole transitions assisted by optical phonons provide the desired mechanism for the correlated non-Markovian carrier relaxation process. A multiband calculation for strained InP quantum wires was developed to account for the description of the character of the valence band states and gives quantitative support for light hole-heavy hole transitions assisted by optical phonons.

Grading effects in semiconductor nanowires with longitudinal heterostructures

The role of graded interfaces between materials in a cylindrical free-standing quantum wire with longitudinal heterostructures is theoretically investigated, by solving the Schr\"odinger equation within the effective mass approximation. Previous work on such wires with abrupt interfaces have predicted that, as the wire radius is reduced, the effective potential along the growth direction is altered and might lead to a carrier confinement at the barriers, as in a type-II system. Our results show that when graded interfaces are considered, such potential acquires a peculiar form, which presents cusps at the interfacial regions, yielding to electron confinement at interfaces. Numerical results also show that, in some special cases, interfacial confinement and type-I to type-II transitions can also be induced by applying a magnetic field parallel to the wire axis.

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 q z → ∞ , the frequencies of each SO modes converge to the frequency values of the single planar heterostructure.

ONE-PHONON-ASSISTED ELECTRON RESONANT RAMAN SCATTERING IN FREE-STANDING QUANTUM WIRES

The scattering intensity (SI) for an electron resonant Raman scattering (ERRS) process in a free-standing semiconductor quantum wire of cylindrical geometry associated with bulk longitudinal optical (LO) phonon modes or the surface optical (SO) phonon modes is calculated for T=0 K . The Fröhlich interaction is considered to illustrate the theory for a GaAs system. Electron states are confined within a free-standing quantum wire (FSW). Single parabolic conduction and valence bands are assumed. The selection rules are studied. Numerical results and a discussion are also presented for various radii of the cylindrical quantum wires.

One phonon resonant Raman scattering in free-standing quantum wires

Abstract The scattering intensity (SI) of a free-standing cylindrical semiconductor quantum wire for an electron resonant Raman scattering (ERRS) process associated with bulk longitudinal optical (LO) phonon modes and surface optical (SO) phonon modes is calculated separately for T = 0 K . The Frohlich interaction is considered to illustrate the theory for GaAs and CdS systems. Electron states are confined within a free-standing quantum wire (FSW). Single parabolic conduction and valence bands are assumed. The selection rules are studied. Numerical results and a discussion are also presented for various radii of the cylindrical.

One-phonon resonant Raman scattering in cylindrical quantum wires

The scattering intensity (SI) for an electron resonant Raman scattering (ERRS) process associated with bulk longitudinal optical (LO) phonon modes and surface optical (SO) phonon modes is calculated separately for T=0K in a free-standing semiconductor quantum wire of cylindrical geometry. We consider the Fröhlich interaction to illustrate the theory for a AlAs system. Electron states are considered to be confined within a free-standing quantum wire (FSW). We also assume single parabolic conduction and valence bands. The selection rules are studied. Numerical results and a discussion are also presented for various cylindrical radii.

Electron Raman scattering in cylindrical quantum wires

Electron Raman scattering (ERS) is investigated in a free-standing semiconductor quantum wire of cylindrical geometry for two classes of materials CdS and GaAs. The differential cross section (DCS) involved in this process is calculated as a function of a scattering frequency and the radius of the cylinder. Electron states are considered to be confined within a free-standing quantum wire (FSW). Single parabolic conduction and valence bands are assumed. The selection rules are studied. Singularities in the spectra are found and interpreted for various radii of the cylinder.

Oscillating spectra of polar IO and SO phonons and electron–IO and SO phonon couplings in a freestanding wurtzite quantum wire

AbstractBy employing the transfer matrix method, the polar interface optical (IO) and surface optical (SO) phonon modes and the corresponding Fröhlich electron–phonon interaction Hamiltonians in a freestanding multishell wurtzite cylindrical quantum wire (QWR) are derived and studied under the dielectric continuum approximation and Loudon's uniaxial crystal model. The present theoretical scheme can be looked on as a generalization of the previous studies regarding the IO phonon modes in quasi‐one‐dimensional (Q1D) wurtzite QWRs with a certain layer‐number. Numerical calculations on a freestanding wurtzite GaN/AlN QWR are performed. The degenerating behaviors of the IO and SO phonon modes in the freestanding wurtzite QWR seem prominent in contrast to the situation of Q1D wurtzite QWR with infinite polar matrix. The calculations of the electron–phonon coupling function show that the high‐frequency IO phonon branch and SO mode play a more important role in the electron–phonon interaction. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A general treatment of deformation effects in Hamiltonians for inhomogeneous crystalline materials

In this paper, a general method of treating Hamiltonians of deformed nanoscale systems is proposed. This method is used to derive a second-order approximation both for the strong and weak formulations of the eigenvalue problem. The weak formulation is needed in order to allow deformations that have discontinuous first derivatives at interfaces between different materials. It is shown that, as long as the deformation is twice differentiable away from interfaces, the weak formulation is equivalent to the strong formulation with appropriate interface boundary conditions. It is also shown that, because the Jacobian of the deformation appears in the weak formulation, the approximations of the weak formulation is not equivalent to the approximations of the strong formulation with interface boundary conditions. The method is applied to two one-dimensional examples (a sinusoidal and a quantum-well potential) and one two-dimensional example (a freestanding quantum wire), where it is shown that the energy eigenvalues of the second-order approximations lie within 1% of the exact energy eigenvalues for a linear strain of up to 9.8%, whereas the first-order approximation has an error of less than 1% for a linear strain of up to 5.5%.

Polaron self-energy and effective mass in a freestanding cylindrical quantum wire

The polaron self-energy and correction to the electron effective mass in a freestanding quantum wire is investigated by the perturbation approach.The polaron effect of the electron-confined longitudinal optical (LO) phonon and surface optical (SO) phonon interactions are separately worked out. Numerical calculation on a GaAs quantum wire shows that the confined LO phonon contribution to the polaron self-energy is relatively small for a narrow wire and gradually approach that of the bulk material when the radius of the wire increases. While the contribution of the SO phonon modes is big for small wire radius and then decreases as the radius increases.

Theory of thermoelectric power factor in quantum well and quantum wire superlattices

Calculations are presented for thermoelectric transport in quantum well and quantum wire superlattices, using (i), the full superlattice electronic band structure in (ii) a multisubband inelastic Boltzmann equation for carrier-phonon scattering. The transport direction is taken to be in the quantum well planes and along quantum wires. It is demonstrated that these two features are needed to give a quantitative treatment of the power factor P in superlattice systems. Results are given for PbTe and for GaAs quantum well and quantum wire superlattices, including the dependence of P on growth direction and on potential offset. For both quantum well and quantum wire superlattices, the dependence of P on potential offset ${V}_{0}$ is found to be qualitatively weaker than in previous work based on the constant relaxation time approximation for carrier scattering. These weaker dependences on ${V}_{0}$ are traced mainly to the enhancement of the electron-phonon scattering rates upon confinement. These results give a different picture of the effects of confinement on P suggesting, for example, that increased confinement in superlattices does not lead to significantly higher P and that free-standing structures, such as free-standing quantum wires, may be particularly attractive for thermoelectric applications.

Phase instability and electron-assisted renormalization of acoustic phonon spectrum in free-standing quantum wires

In this paper we consider the possibility of a Peierls phase transition for a system of one-dimensional electrons and acoustic phonons confined in a free-standing quantum wire (FSQWI). We assume that electrons and phonons are coupled via the deformation potential interaction. The dispersion relation for this system is derived taking into account the role of electrostatic effects. The Peierls phase transition to the state characterized by periodic lattice deformation (PLD) is obtained in the high-permittivity limit. The formalism developed in this paper facilitates investigation of the renormalized acoustic phonon spectrum and predicts the possibility of the Peierls transition in FSQWIs fabricated from semiconductor materials with large permittivity.

Velocity fluctuations and Johnson noise in quantum wires:the effect of phonon confinement

We have theoretically studied electron velocity fluctuations and the resulting Johnson (thermal) noise in a free-standing GaAs quantum wire at low and intermediate driving electric fields. One-dimensional confinements of electrons and phonons have been taken into account. Acoustic phonon confinement introduces infrared frequency peaks in the noise power spectrum which are an unmistakable signature of phonon confinement and provide an experimental `handle' to use in assessing the importance of such confinement. Phonon confinement also suppresses the dc component of the noise spectral density (and the hot-carrier diffusivity) by several orders of magnitude. When a transverse magnetic field is applied to the quantum wire, it introduces three remarkable features: (i) it reduces the temporal decay rate of the velocity autocorrelation function and increases the dc component of diffusivity, (ii) it promotes prolonged and persistent oscillations in the velocity autocorrelation function which is indicative of a long memory of the electron ensemble, and finally (iii) it red-shifts the peaks in the noise power spectrum by increasing the length of an electron's trajectory in momentum space between two successive phonon scattering events.

Renormalization of acoustic phonon spectra and rudiments of the Peierls transition in free-standing quantum wires

We have analyzed a quasi-one-dimensional electron-phonon system coupled through the deformation potential interaction. In a free-standing quantum wire, we calculated the renormalization of acoustic phonon spectra. The results illustrate remarkable cusplike features in the vicinity of the phonon wave vector that is twice the electron Fermi wave vector. These cusps are rudiments of the Peierls transition in the quasi-one-dimensional system. The transition is suppressed in conventional materials because of the electrostatic effects. For such materials, however, the renormalization of acoustic phonon spectra and the cusps can be observed in various acoustical and optical experiments. Furthermore, our study demonstrates that the Peierls transition can occur in free-standing quantum wires fabricated from semiconductor materials with high permittivity.

Current-voltage instability in free-standing semiconductor quantum wires

Current-voltage instability in free-standing semiconductor quantum wires

The self-consistent electronic structure of rectangular free-standing quantum wires: Fourier expansion approach

A method for the self-consistent calculation of rectangular free-standing quantum wires is presented. The method is also used for the electronic structure calculation of cladded quantum wires. It relies on the wavefunction's Fourier expansion, and uses the structure symmetry to save computation time. Calculations are performed for these two types of quantum wire at a number of dopant and surface state densities, and the influence of various parameters is analyzed. At low temperatures, the occurrence of Friedel oscillations is noticed, arising from a very limited number of well separated electronic states.

Acoustic phonon modes of rectangular quantum wires

Acoustic phonon modes of a free-standing rectangular quantum wire composed of cubic crystals are theoretically investigated using an algorithm developed to analyse data from resonant ultrasound spectroscopy. The normal phonon modes are classified according to their spatial symmetries into a compressional mode termed the dilatational mode and non-compressional modes referred to as the flexural, torsional, and shear modes. The formalism that we present is quite general and can be applied to wires of any cubic material. As an example, the dispersion relations are obtained for square and rectangular wires of GaAs, taking into account the anisotropic elasticity of the material. The dispersion curves for a rectangular wire are compared with those of the approximate hybrid modes referred to as the thickness and width modes, and the validity of the modes is discussed. The existence of edge modes is confirmed by examining the spatial distribution of displacement vectors.