Semi-infinite Crystal Research Articles

Accurate Model of Electromagnetic Wave Propagation in Unidimensional Photonic Crystals with Defects

Complete vectorial studies of electromagnetic wave propagation in complex media are difficult and computationally intensive. A mathematical continuous model can simplify some of such difficulties. Using this technique, we have modeled the propagation of electromagnetic plane waves through a semiinfinite photonic crystal with defects. The model is exact in the sense that the only approximations are the numerical accurate solution of ordinary differential equations and the simulation of defects by a high frequency sinusoidal continuous variation of the main crystal period.

Surface optical waves in semi-infinite one-dimensional photonic crystals with a thin nonlinear cap layer

An analytical direct matching procedure within the Kronig–Penney model was applied to analyse the dispersion behavior of the localized surface states supported in the surface of a semi-infinite one-dimensional photonic crystals truncated with a thin nonlinear cap layer. We find that the surface states are very sensitive to the nonlinearity of cap layer on the surface.

Symmetric and asymmetric domain structures in a semi-infinite single crystal

The distribution of magnetization in a semi-infinite single crystal of square cross section has been determined by means of micromagnetic modeling. It is established that both symmetric and asymmetric stable domain structures exist in the system in a broad range of variation of the magnetic anisotropy.

Micromagnetic simulation of the distribution of magnetization in semiinfinite single crystals

A method for calculating magnetization distribution in a single crystal in the form of a semiinfinite rod is suggested. The method is based on the division of the single crystal into two regions one of which, adjacent to the butt end of the rod, is finite and has a three-dimensional field of magnetization, and the remaining one (semiinfinite), in which the magnetization field is assumed to be two-dimensional. When computing the magnetization in the finite region, the effect of the magnetostatic field of the semiinfinite part of the rod is taken into account. An example of the calculation for a Co single crystal with a transverse orientation of the easy axis is given. This example shows that in the one and the same sample there are possible equilibrium states such that the magnetization distribution is virtually the same in any transverse section of the rod and differs qualitatively at the surface of the butt end of the rod from the distribution in the bulk of the rod.

Surface states of wurtzite semiconductor nanowires with identical lateral facets: A transfer-matrix approach

The focus of this paper is to investigate systematically surface states of semiconductor nanowires of wurtzite crystals with identical lateral surfaces. We first calculate the surface spectrum of a semi-infinite crystal. Then, using cyclic boundary conditions, we calculate the quantized spectrum of the surface states in nanowires. We find that the spectrum of the nanowire surfaces consists of a number of quantized levels inside the band gap. We further study absorption coefficients due to dipole transitions between the surface states in such nanowires. We demonstrate that such transitions lead to absorption above the fundamental band edge transition. Our calculations also show that the absorption coefficient induced by the transitions between the surface states depends weakly on the light polarization.

A plane-wave-based approach for complex photonic band structure and its applications to semi-infinite and finite system

A plane-wave-based approach for complex photonic band structure is developed and applied to analyse the behaviour of electromagnetic (EM) waves at the surface of semi-infinite and finite photonic crystals (PCs). The complex photonic band structures of two-dimensional and three-dimensional PCs have been calculated by the present approach and verified by conventional plane wave expansion method and plane-wave-based transfer matrix method. Based on the eigen modes of present approach, equations in matrix form are given to describe the boundary conditions of PCs, and consequently the behaviour of EM waves at the surface of semi-infinite and finite PCs is discussed. These results have also verified the validity and efficiency of the current approach.

Temperature dependence of surface magnetization in local-moment systems

We present a theory to study the temperature-dependent behavior of surface magnetization in a ferromagnetic semi-infinite crystal. Our approach is based on the single-site approximation for the s– f model. The effect of the semi-infinite nature of the crystal is taken into account by a localized perturbation method. Using the mean-field theory for the layer-dependent magnetization, the local density of states and the electron–spin polarization are investigated at different temperatures for ordinary and surface transition cases. The results show that the surface magnetic properties may differ strongly from those in the bulk and the coupling constant of atoms plays a decisive role in the degree of spin polarization. In particular, for the case in which the exchange coupling constant on the surface and between atoms in the first and second layer is higher than the corresponding in the bulk, an enhancement of surface Curie temperature and hence the spin polarization can be obtained.

A dissociated dislocation in an ultrathin silicon plate

Simplified explicit expressions are presented to describe the elastic displacement field of a periodic family of misfit dislocations running parallel to the two free surfaces of an elastically isotropic plate. In the situation where the period tends to infinity, the use of these expressions proves to be quite valuable for investigating the change of the separation distance, S, between two partial dislocations as a function of the position of one partial and the orientation of the fault plane. For the two 30° Shockley partials of a dissociated screw dislocation in an ultrathin silicon plate, numerical results indicate that S can change drastically. This property is confirmed in anisotropic elasticity for a dislocation located near the free surface of a semi-infinite crystal. The results emphasize that particular attention should be paid to precise measurement of the local thickness and positions of the partials in weak beam or high resolution transmission electron microscopy experiments.

Ab initiocalculations of the dispersion contribution to the physisorption potential: Application to theN2-BN system

We have developed an ab initio method to calculate the long-range dispersion coefficient for a molecule adsorbing on a surface modeled by a biperiodic slab. The variations of the dynamic polarizabilities versus imaginary frequencies of each system---considered separately---permit us to evaluate a ${C}_{4}$ dispersion coefficient, via the Casimir-Polder formula. The polarizability of the molecule is computed by using the time-dependent gauge-invariant variational method while the electric properties of the surface are calculated by using the periodic ab initio linear combination of atomic orbitals density functional theory method implemented in the CRYSTAL03 code. We have applied this method to the physisorption of the nitrogen molecule---in its ground and lowest-lying triplet states---on the (001) surface of hexagonal boron nitride. The semi-infinite crystal is simulated by slabs with different numbers of atomic layers.

Method for calculating electronic structures near surfaces of semi-infinite crystals

We present a stable scheme to calculate continum and bound electronic states in the vicinity of a surface of a semi-infinite crystal within the framework of density functional theory. The method is designed for solution of the Kohn-Sham equations in a pseudpotential formulation, including both local and separable nonlocal contributions. The method is based on the Numerov integration algorithm and uses singular value decomposition to control the exponentially growing contributions. The method has been successfully tested on the Li (110) surface with and without adsorbed H. For this model system, we are able to locate the energies of H-induced surface states relative to the corresponding energies of bulk continum states. Results encourage further development.

Semianalytic approach for analyzing coupling issues in photonic crystal structures

A semianalytic approach based on previously derived closed-form expressions for the transmission and reflection matrices between a dielectric waveguide and a semi-infinite photonic crystal (PhC) waveguide is proposed for analyzing coupling issues in PhC structures. The proposed approach is based on an eigenmode expansion technique and introduces several advantages with respect to other conventional numerical methods such as a shorter computation time and the possibility to calculate parameters, such as the reflection into PhC structures, difficult to obtain with others methods. Two different examples are analyzed and results compared to finite-difference time-domain simulations to prove the usefulness of the proposed approach: (i) An especially designed two-defects configuration placed within a PhC taper to improve the coupling efficiency and (ii) a coupled-cavity waveguide coupled to a single-line defect PhC waveguide.

Optical properties and photonic modes in patterned semiconductor systems

The optical response of a photonic crystal slab, where a dielectric grating is the elementary building block, is computed in the semiclassical framework, and compared with the dispersion curves of the bulk. The formalism adopted for the calculation employs the eigenmodes of a single layer as basis set for electromagnetic field expansion; it allows us to compute photonic crystal slab optical response as well as dispersion curves. Recently, the method was generalized to take into account spatial dispersion effects close to an exciton resonance of the system, and it was possible also to study the transition from dissipative to dispersive regime for a system under Bragg condition. The role of bulk and surface waves in the system is discussed as a function of the layer number N in the range N = 1 ÷ ∞. The “mirror effect”, observed in rectangular self-sustained gratings, due to the interplay among traveling, evanescent and guided waves, is recovered. This effect can be preserved also in the semi-infinite photonic crystal for selected values of the physical parameters. The role of light polarization in 1D and 2D systems is briefly discussed.

Light propagation in finite and infinite photonic crystals: The recursive Green’s function technique

We report a computational method based on the recursive Green's function technique for calculation of light propagation in photonic crystal structures. The advantage of this method in comparison to the conventional finite-difference time domain (FDTD) technique is that it computes Green's function of the photonic structure recursively by adding slice by slice on the basis of Dyson's equation. This eliminates the need for storage of the wave function in the whole structure, which obviously strongly relaxes the memory requirements and enhances the computational speed. The second advantage of this method is that it can easily account for the infinite extension of the structure both into air and into the space occupied by the photonic crystal by making use of the so-called ``surface Green's functions.'' This eliminates the spurious solutions (often present in the conventional FDTD methods) related to, e.g., waves reflected from the boundaries defining the computational domain. The developed method has been applied to study scattering and propagation of the electromagnetic waves in the photonic band-gap structures including cavities and waveguides. Particular attention has been paid to surface modes residing on a termination of a semi-infinite photonic crystal. We demonstrate that coupling of the surface states with incoming radiation may result in enhanced intensity of an electromagnetic field on the surface and very high $Q$ factor of the surface state. This effect can be employed as an operational principle for surface-mode lasers and sensors.

Electromagnetic modes in semi-infinite photonic crystals

In a real photonic crystal, there exist three modes: propagation mode, evanescence mode and surface mode. Using ideal modal, semi-infinite photonic crystals, we study their effects on the transmission spectrum of photonic crystals, respectively. Because there exists only one air–crystal interface in a semi-infinite photonic crystal, no multiple reflection occurs and no evanescent modes and backward-propagating modes exist in this structure. The effects of the evanescence modes are studied by comparing the transmission spectrum of a finite-thickness photonic crystal slab and that of a semi-infinite photonic crystal. In addition, the effects of the backward-propagating modes are investigated using a coated semi-infinite photonic crystal structure. Finally, we study the effects of the surface modes, and find that the transmission spectrum of a semi-infinite photonic crystal is strongly dependent on its termination.

Time saving techniques for electronic structure calculations of infinite and semi-infinite crystals, interfaces, and slabs of arbitrary thickness

Hard confined functions (HCF) are proposed as a basis set for electronic structure calculations. The basis functions have enough number of continuous derivatives to perform space integrations numerically with desired accuracy. Replacing unconfined basis functions in ADF-BAND package by HCF with cut-off radiuses of the order of the nearest neighbor distance leads to the reduction of the time of computations without losing the accuracy. For the crystals with surfaces, special techniques based on representation of the wave function as a linear combination of a finite number of HCF and Bloch waves, were elaborated. Exact finite sets of equations for coefficients of HCF and Bloch waves have been developed. The order of the sets depends only on the thickness of a perturbed region, but not on the size of the whole system. For integration of the density of states over perpendicular to the surface wave vector component and energy, the residue theorem and a shift of the energy path into the complex plane are used.

Behavior of light at photonic crystal interfaces

Band structures and Bloch modes give a generalized description of light in infinite photonic crystals. We show that the band structure and Bloch modes also contain the information necessary to find the amplitude and phase of light reflected and transmitted from interfaces in systems made using finite and semi-infinite photonic crystals. We obtain the equivalent of the Fresnel coefficients for photonic crystals. We use these coefficients to derive the reflection of light from a photonic crystal of finite size and the resonant modes of photonic crystal cavities and line defects. Results are given for ideal two-dimensional crystals, as well as crystals etched in semiconductor slab waveguides.

Efficient mode coupling technique between photonic crystal heterostructure waveguide and silica waveguides

We report an efficient mode coupling technique between silica waveguides (SWGs) and a planar photonic crystal heterostructure waveguide (PPCHWG) composed of two semi-infinite two-dimensional photonic crystals (PCs) with different filling fractions. By calculating the transmittance at output end of SWG, we find that the designed structures possess the merits of high coupling efficiency and relaxing fabricating tolerance. The maximal transmittance exceeds 90% at a wavelength near 1.65 μm. The high transmission efficiency of PPCHWGs can be achieved without introducing any defect structures for optimal mode matching. Besides, the width of PPCHWGs can be flexibly adjusted via changing the separation of two component PCs in PC heterostructures.

On the optical properties of anisotropic media in the presence of negative components of dielectric and (or) magnetic tensors

The studies of the media with an open surface of wave vectors (SWV) undertaken by the author in the 1970s are continued. A number of new specific features of these media are revealed. It is shown, in particular, that the focusing upon refraction at a plane boundary (a phenomenon that has recently attracted much attention and is related to the media whose permittivity and permeability are simultaneously negative) can also be realized in a nonmagnetic medium with an open SWV. The boundary problem for a plate and a semi-infinite crystal with an open SWV is solved, and the qualitative differences from the media with a closed SWV are revealed. Biaxial nonmagnetic media and nonmagnetic media in the presence of negative components of the dielectric and magnetic tensors are briefly considered.

Surface motion induced by laser action on opaque anisotropic crystals

Analytical solutions for the acoustic wave equations obtained by temporal Fourier and spatial Laplace transformations directly provide a description of the motion of the crystal surface caused by the spatially distributed laser heating of a semi-infinite crystal. Evaluation of the acoustic field in the bulk of the material is not needed here. In general, all three acoustic modes are excited due to the laser-induced thermoelastic effect and contribute to each of the three components of the transient surface displacement. Numerical simulations of the surface displacement as a function of time and crystal surface orientation are performed with the use of the analytical formulae derived in the case of a hexagonal crystal, for which only two modes are excited. The formulae obtained make it possible to optimize the orientation of the surface of the crystal in order to improve the efficiency of the excitation of the in-plane motion of the surface.

Augmented-plane-wave approach to scattering of Bloch electrons by an interface

A full-potential augmented-plane-wave-based variational method is proposed to construct the scattering wave function for a system consisting of two semi-infinite crystals separated by an interface region. The two half spaces are represented by their complex band structures, and a basis set expansion is used to represent the wave function in the embedded region. The method is based on solving the equation $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\gamma}}(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}E)\ensuremath{\Psi}=0$ in the scattering region, which is equivalent to the original Schr\"odinger equation, and the presence of the operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\gamma}}$ makes it possible to formulate the variational problem in terms of plane waves. Current conservation considerations are drawn on to include the requirement of the smooth continuity of the wave function into the variational functional. The problem of the overcompleteness of embedding basis sets is discussed and a solution is presented. The method is verified by calculating the electron diffraction and surface states on (100) and (111) surfaces of Al and Cu. The accuracy and convergence properties of the computational scheme are analyzed.