Electrons In Crystalline Solids Research Articles

Multiple scattering of primary electrons in crystalline solids: Numerical calculations for Pt/Cu(1 1 1) and Cu/Pt(1 1 1) systems

The multiple scattering formalism used to describe the primary electron beam scattering events in crystalline solids results in theoretical distributions of elastically backscattered electrons and Auger electrons. In calculations different locations of the monolayer and the resulting intensity distributions are shown. The theoretical formalism concerns the final form of the wave field in a solid and the interference of the primary plane and the scattered spherical electron waves. The explicit form of the wave function at the emitter site, the formula used to calculate the signal of elastically backscattered electrons and Auger electrons as well as parameters involved in simulations are discussed. The results of numerical calculations presented as stereographic distributions reveal characteristic intensity maxima and bands associated with crystalline directions and planes, respectively. The data are discussed in the context of the sample structure within the surface near region, scattering properties of individual atoms, lattice parameters, location of the layer as well as the sample chemical composition. Moreover, a detailed analysis of intensities associated with different atomic directions in the face centred cubic crystals, as well as averaged intensities of whole distributions are shown. The short range order structural information, which concerns the nearest and next nearest neighbours in a three dimensional unit cell, can be applied in the modelling of heterostructures and the calculation of signal distributions with the use of the multiple scattering approach. This can be helpful in the detailed analysis of intensity features observed experimentally.

Emulating tightly bound electrons in crystalline solids using mechanical waves

Solid state physics deals with systems composed of atoms with strongly bound electrons. The tunneling probability of each electron is determined by interactions that typically extend to neighboring sites, as their corresponding wave amplitudes decay rapidly away from an isolated atomic core. This kind of description is essential in condensed-matter physics, and it rules the electronic transport properties of metals, insulators and many other solid-state systems. The corresponding phenomenology is well captured by tight-binding models, where the electronic band structure emerges from atomic orbitals of isolated atoms plus their coupling to neighboring sites in a crystal. In this work, a mechanical system that emulates dynamically a quantum tightly bound electron is built. This is done by connecting mechanical resonators via locally periodic aluminum bars acting as couplers. When the frequency of a particular resonator lies within the frequency gap of a coupler, the vibrational wave amplitude imitates a bound electron orbital. The localization of the wave at the resonator site and its exponential decay along the coupler are experimentally verified. The quantum dynamical tight-binding model and frequency measurements in mechanical structures show an excellent agreement. Some applications in atomic and condensed matter physics are suggested.

Wave and Uncertainty Properties of Electrons in Crystalline Solids

Herein, relations between particle and wave properties for charge carriers in periodic potentials of crystalline metals and semiconductors are derived. The particle aspects of electrons in periodic potentials are considered using properties of quasimomentum (QM), while the wave aspects are described using wave packets of Bloch waves. The two aspects are combined in the derivation of QM–wavelength relations for energy bands of arbitrary nonparabolicity and nonsphericity. An effective mass relating electron QM to its average velocity for spherical energy bands is defined and used to calculate energy dependences of wavelengths for electrons in narrow‐gap semiconductors, graphene, and surface states of topological insulators. An uncertainty relation between electron QM and its spatial coordinate in periodic potentials is derived. It is emphasized that the described properties apply to the average (not instantaneous) electron behavior. Analogies and differences between the wave and uncertainty properties of electrons in crystalline solids and those in vacuum are traced. An interference experiment with semiconductor nanostructures is proposed to verify the theoretical predictions and demonstrate to what extent the wave properties of electrons in solids are distinct from those in vacuum.

Tight-binding tunneling amplitude of an optical lattice

The particle in a periodic potential is an important topic in an undergraduate quantum mechanics curriculum and a stepping stone on the way to more advanced topics, such as courses on interacting electrons in crystalline solids, and graduate-level research in solid-state and condensed matter physics. The interacting many-body phenomena are usually described in terms of the second quantized lattice Hamiltonians which treat single-particle physics on the level of tight-binding approximation and add interactions on top of it. The aim of this paper is to show how the tight-binding tunneling amplitude can be related to the strength of the periodic potential for the case of a cosine potential used in the burgeoning field of ultracold atoms. We show how to approach the problem of computing the tunneling amplitude of a deep lattice using the JWKB (Jeffreys–Wentzel–Kramers–Brillouin, also known as semiclassical) approximation. We also point out that care should be taken when applying the method of the linear combination of atomic orbitals (LCAO) in an optical lattice context. A summary of the exact solution in terms of Mathieu functions is also given.

Bloch Waves in Minimal Landau Gauge and the Infinite-Volume Limit of Lattice Gauge Theory.

By exploiting the similarity between Bloch's theorem for electrons in crystalline solids and the problem of Landau gauge fixing in Yang-Mills theory on a "replicated" lattice, we show that large-volume results can be reproduced by simulations performed on much smaller lattices. This approach, proposed by Zwanziger [Nucl. Phys. B412, 657 (1994)NUPBBO0550-321310.1016/0550-3213(94)90396-4], corresponds to taking the infinite-volume limit for Landau-gauge field configurations in two steps: first for the gauge transformation alone, while keeping the lattice volume finite, and second for the gauge-field configuration itself. The solutions to the gauge-fixing condition are then given in terms of Bloch waves. Applying the method to data from MonteCarlo simulations of pure SU(2) gauge theory in two and three space-time dimensions, we are able to evaluate the Landau-gauge gluon propagator for lattices of linear extent up to 16 times larger than that of the simulated lattice. This approach is reminiscent of the Fisher-Ruelle construction of the thermodynamic limit in classical statistical mechanics.

Nonlinear dynamics and the nano-mechanical control of electrons in crystalline solids

Under the umbrella of nano-mechanical control of electrons in crystalline solids, provided here are i) a discussion of aspects of the influence of static piezoelectricity on semiconductors, ii) a description of electron surfing on traveling piezopotentials/surface acoustic waves, iii) comments on the role of solitons in (dopable) polymer conductors/synthetic metals, and iv) the major component of these notes, a discussion of basic aspects of lattice solitons and discrete breathers permitting to understand genuine electron surfing on nanosolitons. This surfing offers a form of long range, fast and robust transport process in crystalline solids. The particular case of the undopable highly crystalline polydiacetylene polymer serves to illustrate the invention of a novel solectron field effect transistor (SFET).

The Quantum Oscillatory Modulated Potential—Electric Field Wave Packets Produced by Electrons

In this work we are formulating a new theory for describing the waving nature of a microscopic electric particle. Based on the predictions of the Quantum Oscillatory Modulated Potential—QOMP, for describing the interaction between two microscopic electric particles, electron-electron, for instance, we are postulating that the waving behavior of these particles may be an attribute of the charges of the particles and not their masses as currently accepted. For a microscopic electric charge, we are presenting new arguments showing that the electric field in the vicinity of a microscopic charge is spatially waving and can be determined as the gradient per unit of charge of this new quantum interaction potential, with use of an appropriated phase factor to account for the behavior of an unbound electron. Differently of what is predicted by the classical Coulomb electric field, when a charged particle is moving under the action of a potential of V volts, the new electric field existing around the charge has the form of a wave packet. For typical values of the potential V, the wavelength of the waving electric field is in very good agreement with those experimentally observed with diffraction of electrons in crystalline solids.

Nature of electron Zitterbewegung in crystalline solids

We demonstrate both classically and quantum mechanically that the Zitterbewegung (ZB, the trembling motion) of electrons in crystalline solids is nothing else, but oscillations of velocity assuring the energy conservation when the electron moves in a periodic potential. We show that the two-band k.p model of electronic band structure, formally similar to the Dirac equation for electrons in a vacuum, gives a very good description of ZB in solids. Our results unambiguously indicate that the trembling motion is the basic way of electron propagation in periodic potentials and, as such, it is certainly observable. A recent experimental simulation of ZB with the use of trapped ions is in good agreement with our calculations.

Zitterbewegung of nearly-free and tightly-bound electrons in semiconductors

We show theoretically that non-relativistic nearly-free electrons in solids should experiencea trembling motion (Zitterbewegung, ZB) in the absence of external fields, similarly torelativistic electrons in a vacuum. The ZB is directly related to the influence of the periodicpotential on the free electron motion. The frequency of the ZB is , where Eg is the energy gap. The amplitude of the ZB is determined by the strength of periodicpotential and the lattice period, and it can be of the order of nanometres. We show thatthe amplitude of the ZB does not depend much on the width of the wavepacketrepresenting an electron in real space. An analogue of the Foldy–Wouthuysentransformation, known from relativistic quantum mechanics, is introduced in order todecouple electron states in various bands. We demonstrate that after the bands aredecoupled electrons should be treated as particles of a finite size. In contrast to nearly-freeelectrons we consider a two-band model of tightly-bound electrons. We show that in thiscase also the electrons should experience the trembling motion. It is concluded that thephenomenon of ZB of electrons in crystalline solids is the rule rather than the exception.

Maximally localized Wannier functions for photonic lattices

A method is described for the computation of the generalized Wannier functions corresponding to a composite group of electromagnetic bands in a photonic crystal. The Wannier functions are optimally localized, in that they have minimum real-space spread, defined as the average over the set of the second moment of the functions. The mathematical approach follows that developed by N. Marzari and D. Vanderbilt [Phys. Rev. B 56, 12 847 (1997)] to obtain similarly well-localized Wannier functions for electrons in crystalline solids. Results are presented for the lowest few TE and TM bands in two-dimensional photonic crystals consisting of square and triangular lattices of holes and rods.

Possibility of electron-neutron emission spectroscopy.

The possibility of exciting electrons in crystalline solids by means of energetic neutrons is examined. As expected, the electron-neutron emission is not competitive with respect to familiar photoemission; however, it could be useful in studying special cases, like (for instance) the indirect gap of nonconducting systems.

Optical properties of amorphous semiconductors

Abstract The density of states and the dielectric constant of electrons in crystalline solids can be discussed easily by using the band structure scheme. This is due to the fact that a crystal is defined by both a short range order and a long range order of atoms. If the long range order is relaxed this procedure no longer works and one has to calculate the measurable quantities directly, which involves some configurational averaging process.

Theoretical Physics in Industry

IN considering tMe behaviour of electrons in crystalline solids, a irfrell-known theorem by F. Bloch is of great importance. According to this theorem, electrons in a perfectly periodic lattice move freely without b uig scattered. This does not mean, however hat all electrons contribute to the electric conductivity, because this would also require the possibility of accelerating electrons. To investigate this question we notice that the energy spectrum of electrons in a crystal consists of bands which are well separated in the low-energy region but which overlap at high energies. For simple structures each band contains N levels, where N is the number of unit cells. According to the Pauli exclusion principle, each level can be occupied by no more than two electrons. It thus follows that each energy band accommodates 2N electrons. The average velocity of all electrons in a completely filled band vanishes. Thus in the energy region in which bands do not overlap, a completely filled band does not contribute to the conductivity, because there are no empty levels into which an electron can be accelerated. This case is realized in insulators at low temperatures, where the highest occupied level coincides with the upper edge of an energy band in the region where bands do not overlap. In metals, on the other hand, there is at least one energy band which is not completely filled.