Arbitrary Crystallographic Direction Research Articles

Semiconductor Epitaxy in Superconducting Templates.

Integration of high-quality semiconductor–superconductor devices into scalable and complementary metal-oxide-semiconductor compatible architectures remains an outstanding challenge, currently hindering their practical implementation. Here, we demonstrate growth of InAs nanowires monolithically integrated on Si inside lateral cavities containing superconducting TiN elements. This technique allows growth of hybrid devices characterized by sharp semiconductor–superconductor interfaces and with alignment along arbitrary crystallographic directions. Electrical characterization at low temperature reveals proximity induced superconductivity in InAs via a transparent interface.

Multiorbital edge and corner states in black phosphorene

We theoretically study emergent edge- and corner-localized states in monolayer black phosphorene. Using the tight-binding model based on the density functional theory, we find that the multiorbital band structure due to the nonplanar puckered geometry plays an essential role in the formation of the boundary localized modes. In particular, we demonstrate that edge states emerge at a boundary along an arbitrary crystallographic direction, and it can be understood from the fact that the Wannier orbitals associated with $3{p}_{x},3{p}_{y},3{p}_{z}$ orbitals occupy all the bond centers of phosphorene. At a corner where two edges intersect, we show that multiple corner-localized states appear due to hybridization of higher-order topological corner state and the edge states nearby. These characteristic properties of the edge and corner states can be intuitively explained by a simple topologically equivalent model where all the bond angles are deformed to ${90}^{\ensuremath{\circ}}$.

Study of the elastic properties of hexagonal metal single crystals

Hexagonal metals (e.g., Be, Zr, Ti) are widely used in the nuclear industry, space and aircraft engineering (in manufacturing of the structural elements operating under extreme conditions). A promising way to improve the quality of products made of them is to improve the physical properties of materials using the natural anisotropy of metal single crystals. The results of studying anisotropy and a comparative analysis of the technical characteristics of the elastic properties of single crystals of hexagonal metals are presented. The equations of the elastic compliance matrix components are derived in the explicit form for arbitrary crystallographic direction proceeding from transformations of the elastic compliance tensor in the principal axes to a new arbitrary coordinate system with a subsequent use of Euler angles. Analytical expressions are presented for the technical characteristics of the elastic properties (shear and Young's moduli, Poisson's ratio) of the single crystals of 10 hep metals for an arbitrary crystallographic direction. The axial symmetry of the characteristics about the hexagonal axis is revealed. The sums of the elastic compliance coefficients which determine the shear moduli and the Poisson's ratios in two mutually perpendicular directions are constant in any crystallographic plane of the single crystal. A comparative analysis of the anisotropy of the elastic properties of single crystals of the studied group of metals revealed auxetic properties of Zn and Be single crystals and the region of crystallographic directions of uniaxia tension, leading to an auxetic effect The auxetic effect in Zn was observed under tension in the directions of the plane perpendicular to the hexagonal axis of the single crystal. The planes of the auxetic effect manifestation in Be single crystals are perpendicular to the directions making an angle of 45° with the hexagonal axis.

Electrons in Quasi-1D Waveguides with the Spin–Orbit Interaction: Manifestations of Additional Spin Symmetry

We consider quasi-one-dimensional electron waveguides with spin–orbit interaction, which are formed in quantum wells grown along arbitrary crystallographic directions. An analytic solution to the Schrodinger equation is obtained for systems with Hamiltonians possessing additional spin symmetry. It is shown that the dispersion curves for electrons, which correspond to different size-quantization modes, can intersect only when such symmetry exists. We analyze the structure of dips appearing on the dependences of the conductance of an inhomogeneous waveguide on the energy of carriers. It is shown that the width of the dips substantially depends on the waveguide orientation in the plane of the quantum well. In particular, it vanishes when the waveguide is formed along the direction of the “magic” vector of the initial 2D system.

Persistent spin helices in 2D electron systems

We present a theoretical investigation of persistent spin helices in two-dimensional electron systems with spin-orbit coupling. For this purpose, we consider a single-particle effective mass Hamiltonian with a generalized linear-in-k spin-orbit coupling term corresponding to a quantum well grown in an arbitrary crystallographic direction, and derive the general condition for the formation of the persistent spin helix. This condition applied for the Hamiltonians describing quantum wells with different growth directions indicates the possibility of existence of the persistent spin helix in a wide class of 2D systems apart from the [001] model with equal Rashba and Dresselhaus spin-orbit coupling strengths and the [110] Dresselhaus model.

INVARIANTS IN ELASTIC PROPERTIES OF METAL SINGLE CRYSTALS OF CUBIC SYMMETRY

The equations of the elastic compliance matrix components are derived for arbitrary crystallographic directions determined by the Euler angles for cubic single crystals using the approach [4] developed by the authors which consists in transformation of the elastic compliance tensor in the principal axes into an arbitrary coordinate system with subsequent use of the Euler angles. Euler’s angles are applied in the following format: rotation around the hexagonal axis z (azimuth angle α), inclination of the hexagonal axis to an arbitrary position z ’ (polar angle β), rotation around the new position of z ’ axis (shear direction angle γ). Analysis of the equations derived for the components of elastic compliance matrix for cubic single crystals revealed invariant combinations of the components which, in turn, revealed relations between technical characteristics of the elastic properties of cubic single crystals: Young’s modulus along an arbitrary crystallographic direction, shear modulus and Poisson’s coefficients in a plane perpendicular to the selected crystallographic direction, shear modulus under torsion around this direction, and others. The validity of the obtained relations is verified in the calculations done for arbitrary crystallographic directions and various cubic single crystals. Calculations for single crystals of copper and nickel are presented.

Simulation and experimental study of maskless convex corner compensation in TMAH water solution

Maskless etching with convex corner compensation in the form of a 〈1 0 0〉 oriented beam is investigated using both experiments and simulations. The maskless convex corner compensation technique is defined as a combination of masked and maskless anisotropic etching of {1 0 0} silicon in 25 wt% TMAH water solution at a temperature of 80 °C. This technique enables fabrication of three-level micromachined silicon structures with compensated convex corners at the bottom of the etched structure. All crystallographic planes that appear during etching are determined and their etch rates are used to calculate the etch rate value in an arbitrary crystallographic direction necessary for simulation by an interpolation procedure. A 3D simulation of the profile evolution of the etched structure during masked and maskless etching of silicon based on the level set method is presented. All crystallographic planes of the etched silicon structures determined in the experiment are recognized in the corresponding simulated etching profiles obtained by the level set method.

Theoretical Study on Interferometric Illumination of Gold Colloid-Sphere Monolayers to Produce Complex Structures for Spectral Engineering

Theoretical study on the interferometric illumination of colloid-sphere monolayers (IICSM) is presented to produce complex plasmonic structures consisting of wavelength-scaled periodic arrays of nano-objects with arbitrary array symmetry and controllable nanoscaled substructure. The IICSM method is based on illumination of hexagonal colloidsphere monolayers by interference patterns synchronized with sphere arrays along arbitrary preselected crystallographic directions. This nanokaleidoscope method enables tuning four structure parameters independently: the symmetry and characteristic periodicity of the interference pattern might be varied by the number, wavelength, and angle of incidence of the interfering beams; the distance between the nano-objects is controllable by the relative orientation of the interference pattern with respect to the hexagonal lattice of colloid spheres; the size of individual nano-objects is determined by the colloid-spheres' diameter and by the light wavelength; and the substructure size-parameter sensitively depends on the polarization state and can be tuned with the nano-object size simultaneously. Finite element method is applied to demonstrate the capabilities of IICSM based on gold colloid spheres and the impact of the resulted complex plasmonic patterns on spectral properties of thin gold films. The possibility of realizing spectral engineering with predesigned rectangular arrays of hole doublets that may be produced uniquely by IICSM is shown.

Phonon deformation potentials for the corundum structure of sapphire

AbstractA theoretical Raman polarization analysis is proposed for the corundum structure of sapphire (α‐Al2O3) and validation experiments conducted with the purpose of retrieving the full set of phonon deformation potentials (PDPs). From the theoretical side, the change in force constants under stress/strain has been expressed in matrix form, and close‐form solutions were obtained for the eigenvalues that take into account the local dependence of oblique phonons on crystallographic orientation (i.e. uncoupling the effects of local crystal orientation and stress tensor from the shifts of Raman bands). From the experimental side, controlled (uniaxial) stress fields were applied to sapphire parallelepiped bars (along known crystallographic axes) while Raman spectra were systematically recorded along the bar thickness. An untextured alumina polycrystal with fine grain size was also investigated according to the same procedure. As a result of this set of experiments, PDPs for both A1g and Eg vibrational bands could be retrieved. Validation of PDP constants was obtained by measuring the steeply graded stress fields developed ahead of a surface crack propagated along an arbitrary crystallographic direction in the R‐plane of the sapphire crystal. Copyright © 2011 John Wiley & Sons, Ltd.

Strain in layered zinc blende and wurtzite semiconductor structures grown along arbitrary crystallographic directions

We present a simple approach to the evaluation of strain in zinc blende and in wurtzite layered semiconductor structures. These crystallographic structures are of particular interest because of their importance in optoelectronic device applications. The composite layered materials are currently grown pseudomorphically on substrates, which dictate the strain in the layers. Components of the strain are derived for arbitrary crystallographic growth directions. The strain in the layer determines the piezoelectric field in each layer in the structure. The strain and the strain-induced electric field are important in designing layered heterostructures with specific electronic energy levels for device applications. The methods presented are more generally applicable to other crystallographic structures and composite pseudomorphically grown materials. Illustrative problems and solutions are included.

Elasticity theory of pseudomorphic heterostructures grown on substrates of arbitrary thickness

A theoretical model for lattice mismatched pseudomorphically grown heterostructure, which is based on continuum elasticity theory is described. Two distinct types of coherently grown structures are considered, namely, those grown on a thick substrate and these grown on a freestanding one. Special cases of structures that are homogeneous or periodic in some directions are considered. The theory can be applied for an arbitrary crystallographic direction of the heterostructure growth. The model has been applied to AlN∕GaN nanocolumn and GaAs∕InAs heterostructure. Calculation of strain induced shape deformation is shown.

Wavefunction engineering of layered wurtzite semiconductors grown along arbitrary crystallographic directions

The electronic band structure of wurtzite semiconductor heterostructures is investigated theoretically using the envelope function k ⋅ P formalism. We use a Lagrangian formulation for the valence bands so that the order of the derivatives appearing in the multiband description is explicitly specified when Schrödinger’s equations for the envelope functions are generated through the application of the principle of least action. The issues of derivative operator ordering and boundary conditions at material interfaces are examined in detail. The theoretical results are presented for arbitrary growth directions and the spin–orbit interaction is taken into account. This is of interest, for example, in treating A -plane wurtzite heterostructures such as GaN/AlGaN quantum wells grown on R -plane sapphire. Strain effects are included using a general rotation method that diagonalizes strain and is appropriate for pseudomorphic wurtzite structures in any allowed orientation. It is shown that inversion asymmetry in the valence band leads to shifts in the band-edge energies in the [ 11 2 ¯ 0 ]-grown quantum wells. The finite element method is used with the Lagrangian for the composite layered semiconductor structure in order to obtain the energy eigenvalues for multi-quantum well systems. Calculations for quantum wells and superlattices are presented.

Vector magnetic properties in arbitrary direction of various kinds of magnetic materials

Two-dimensional (vector) magnetic properties under alternating magnetic flux in arbitrary crystallographic direction have been obtained in four typical kinds of magnetic materials. B—H characteristics in arbitrary direction are well derived numerically from two B—H loops of easy axis and hard axis.

Channeling measurements methodology

A complete set of methods for channeling measurements is presented. It is based on a full discussion of transformations between laboratory and crystal frames of reference. Specific methods include crystal orientation and goniometer calibration procedure, beam-goniometer misalignment analysis, beam alignment along an arbitrary crystallographic direction, calculations of the beam spot as a function of the goniometer XY translation, as well as two methods for scattering angle calibration. The methods are exemplified by ion scattering experiments.

Elasto-optical characterization of strained heterostructures

It is shown through a detailed calculation that in strained layers grown along arbitrary crystallographic directions, the strain components can be determined from oblique transmission experiments. The analysis is based on the elasto-optical effect. Electro-optical contributions due to piezoelectric fields are also considered. Specific examples are treated in detail.

Elastic distortions of strained layers grown epitaxially in arbitrary directions

We consider the elastic distortions which are due to lattice mismatch between epitaxial layers and substrates. The growth is assumed to take place in an arbitrary crystallographic direction. Explicit forms are obtained for the deformation of the unit cell and its two components, i.e., the angular distortions (symmetric strain) and the rigid-body rotation (antisymmetric strain). The correction factors which connect the relaxed lattice constants of the layer with observables from X-ray diffractometry are obtained in analytical forms. As shown through specific applications, the results agree with those obtained numerically in the literature.

Stress-optic law in a single crystal and its application to photo-anisotropic elasticity

A general stress-optic law in a plate made of a single crystal with birefringence is developed, the plate has an arbitrary crystallographic direction. From the general stressoptic law, a condition for obtaining stress distributions in the plate under a plane stress state is derived. Some optical and mechanical properties when the plate is used as a photoanisotropic model are also explained. Experiments on silicon beams in pure bending are performed by using an infrared photoelastic method. Experimental results show that the stress-optic law is valid. The optical and mechanical properties of silicon beams are shown in tables.

Theory of the intraband magneto-optics of cadmium arsenide

Using the band-structure model proposed by Bodnar (1977), the authors have calculated the dielectric function of Cd3As2 in the presence of a magnetic field. This calculation, which is carried out in full detail only for waves propagating along the direction of the magnetic field, which may, however, be in an arbitrary crystallographic direction, permits the analysis of cyclotron resonance as well as measurements of magnetoreflectivity or absorption. Numerical calculations are presented for propagation along or perpendicular to the c axis.

General matrix equations for diffusion drift velocity in a driving force

From a complete-path approach in which defects are followed from the time of their creation at defect sources to the time of their destruction at defect sinks, general matrix equations are developed which relate the atom drift velocity ${\ensuremath{\nu}}_{k}$ for diffusion in a driving force to simple matrices whose elements can be written by inspection. Previous equations for ${\ensuremath{\nu}}_{k}$ have been restricted either to crystals and defects having high symmetry or to cases where the $x$ components of any atom jumps were either \ifmmode\pm\else\textpm\fi{} $b$ or 0. By contrast, the present equations involve no such restrictions. They apply to diffusion by interstitialcy, divacancy, and other more complex mechanisms, independent of the presence of symmetry planes, and allow calculation of diffusion drift velocities even when the individual atom jumps provide a variety of jump distances along the diffusion direction. These equations apply, for example, even when jumps to both nearest- and next-nearest-neighbor sites are allowed and to diffusion in arbitrary crystallographic directions even in noncubic crystals. The present equations can be expressed in terms of a general matrix $S$. The relation of $S$ to the less general matrices ${S}_{+}$ and ${S}_{\ensuremath{-}}$ defined previously for crystals having mirror symmetry planes normal to the diffusion direction is discussed.

Optimum thermal compensation axes in YIG and GaYIG ferrimagnetic spheres

A new measurement technique using a bandpass filter structure, called a filter-orienter, allows determination of magnetic properties of single crystal ferrimagnetic samples at arbitrary crystallographic directions. Materials properties can be conveniently measured over several octaves in field and over a considerable temperature rang. This technique was utilized to examine the ferromagnetic resonance thermal compensation axes of YIG and GaYIG spheres. An optimum thermal compensation axis was found which for GaYIG is an order of magnitude more stable over a multi-octave field range than the thermal compensation axis usually identified in the {110} crystal plane. Experimental results were explained by ferromagnetic resonance theory when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</tex> -factor temperature dependence, propagation scattering, and other frequency perturbation effects were included.