Electronic band structure - Wikipedia, the free encyclopedia. In solid- state physics, the electronic band structure (or simply band structure) of a solid describes the range of energies that an electron within the solid may have (called energy bands, allowed bands, or simply bands) and ranges of energy that it may not have (called band gaps or forbidden bands). Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules. Band theory has been successfully used to explain many physical properties of solids, such as electrical resistivity and optical absorption, and forms the foundation of the understanding of all solid- state devices (transistors, solar cells, etc.). Why bands and band gaps occur. Each orbital forms at a discrete energy level. When multiple atoms join together to form into a molecule, their atomic orbitals combine to form molecular orbitals, each of which forms at a discrete energy level. As more atoms are brought together, the molecular orbitals extend larger and larger, and the energy levels of the molecule will become increasingly dense. Eventually, the collection of atoms form a giant molecule, or in other words, a solid. For this giant molecule, the energy levels are so close that they can be considered to form a continuum. Band gaps are essentially leftover ranges of energy not covered by any band, a result of the finite widths of the energy bands. The bands have different widths, with the widths depending upon the degree of overlap in the atomic orbitals from which they arise. Band structure of solids. 3.4 Examples of band structures band structure E gap point. IMPRS band structure pdf.ppt. Solid is one of the four fundamental. Band Structures of Crystalline Solids. An Introduction to Band Theory, A Molecular Orbital Approach. An introduction to the concept of band structure. Band structure is one of the most important concepts in solid state. Modern theory of solids: Band structure. Concepts in Materials Science I VBS/MRC Band Theory . Handbook Of The Band Structure Of Elemental Solids Pdf Read/Download. Papaconstantopoulos, in Handbook of the Band Structure of Elemental. Two adjacent bands may simply not be wide enough to fully cover the range of energy. For example, the bands associated with core orbitals (such as 1s electrons) are extremely narrow due to the small overlap between adjacent atoms. As a result, there tend to be large band gaps between the core bands. Higher bands involve larger and larger orbitals with more overlap, becoming progressively wider and wider at high energy so that there are no band gaps at high energy. Basic concepts. The concept of band structure can be extended to systems which are only . Practically, this means that band structure describes the bulk inside a uniform piece of material. Non- interactivity: The band structure describes . The existence of these states assumes that the electrons travel in a static potential without dynamically interacting with lattice vibrations, other electrons, photons, etc. The above assumptions are broken in a number of important practical situations, and the use of band structure requires one to keep a close check on the limitations of band theory: Inhomogeneities and interfaces: Near surfaces, junctions, and other inhomogeneities, the bulk band structure is disrupted. Not only are there local small- scale disruptions (e. These charge imbalances have electrostatic effects that extend deeply into semiconductors, insulators, and the vacuum (see doping, band bending). Along the same lines, most electronic effects (capacitance, electrical conductance, electric- field screening) involve the physics of electrons passing through surfaces and/or near interfaces. The full description of these effects, in a band structure picture, requires at least a rudimentary model of electron- electron interactions (see space charge, band bending). Small systems: For systems which are small along every dimension (e. The crossover between small and large dimensions is the realm of mesoscopic physics. Strongly correlated materials (for example, Mott insulators) simply cannot be understood in terms of single- electron states. The electronic band structures of these materials are poorly defined (or at least, not uniquely defined) and may not provide useful information about their physical state. Crystalline symmetry and wavevectors. Note that Si and Ge are indirect band gap materials, while Ga. As and In. As are direct. Band structure calculations take advantage of the periodic nature of a crystal lattice, exploiting its symmetry. The single- electron Schr. For each value of k, there are multiple solutions to the Schr. Each of these energy levels evolves smoothly with changes in k, forming a smooth band of states. For each band we can define a function En(k), which is the dispersion relation for electrons in that band. The wavevector takes on any value inside the Brillouin zone, which is a polyhedron in wavevector space that is related to the crystal's lattice. Wavevectors outside the Brillouin zone simply correspond to states that are physically identical to those states within the Brillouin zone. Special high symmetry points in the Brillouin zone are assigned labels like . In scientific literature it is common to see band structure plots which show the values of En(k) for values of k along straight lines connecting symmetry points. Another method for visualizing band structure is to plot a constant- energy isosurface in wavevector space, showing all of the states with energy equal to a particular value. The isosurface of states with energy equal to the Fermi level is known as the Fermi surface. Energy band gaps can be classified using the wavevectors of the states surrounding the band gap: Direct band gap: the lowest- energy state above the band gap has the same k as the highest- energy state beneath the band gap. Indirect band gap: the closest states above and beneath the band gap do not have the same k value. Asymmetry: Band structures in non- crystalline solids. As a result, virtually all of the existing theoretical work on the electronic band structure of solids has focused on crystalline materials. Density of states. It appears in calculations for optical absorption where it provides both the number of excitable electrons and the number of final states for an electron. The Fermi level of a solid is directly related to the voltage on that solid, as measured with a voltmeter. Conventionally, in band structure plots the Fermi level is taken to be the zero of energy (an arbitrary choice). The density of electrons in the material is simply the integral of the Fermi. The preferred value for the number of electrons is a consequence of electrostatics: even though the surface of a material can be charged, the internal bulk of a material prefers to be charge neutral. The condition of charge neutrality means that N/V must match the density of protons in the material. For this to occur, the material electrostatically adjusts itself, shifting its band structure up or down in energy (thereby shifting g(E)), until it is at the correct equilibrium with respect to the Fermi level. Names of bands near the Fermi level (conduction band, valence band). However, most of the bands simply have too high energy, and are usually disregarded under ordinary circumstances. These low- energy core bands are also usually disregarded since they remain filled with electrons at all times, and are therefore inert. The bands and band gaps near the Fermi level are given special names, depending on the material: In a semiconductor or band insulator, the Fermi level is surrounded by a band gap, referred to as the band gap (to distinguish it from the other band gaps in the band structure). The closest band above the band gap is called the conduction band, and the closest band beneath the band gap is called the valence band. In semimetals the bands are usually referred to as . In many metals, however, the bands are neither electron- like nor hole- like, and often just called . Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b. Now, any periodic potential V(r) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non- vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as: V(r)=. This approximation allows use of Bloch's Theorem which states that electrons in a periodic potential have wavefunctions and energies which are periodic in wavevector up to a constant phase shift between neighboring reciprocal lattice vectors. The consequences of periodicity are described mathematically by the Bloch wavefunction. In such materials the overlap of atomic orbitals and potentials on neighbouring atoms is relatively large. In that case the wave function of the electron can be approximated by a (modified) plane wave. The band structure of a metal like Aluminium even gets close to the empty lattice approximation. Tight binding model. This tight binding model assumes the solution to the time- independent single electron Schr. Index n refers to an atomic energy level and R refers to an atomic site. A more accurate approach using this idea employs Wannier functions, defined by. Here index n refers to the n- th energy band in the crystal. The Wannier functions are localized near atomic sites, like atomic orbitals, but being defined in terms of Bloch functions they are accurately related to solutions based upon the crystal potential. Wannier functions on different atomic sites R are orthogonal. The Wannier functions can be used to form the Schr. Band structures of materials like Si, Ga. As, Si. O2 and diamond for instance are well described by TB- Hamiltonians on the basis of atomic sp. In transition metals a mixed TB- NFE model is used to describe the broad NFE conduction band and the narrow embedded TB d- bands. The radial functions of the atomic orbital part of the Wannier functions are most easily calculated by the use of pseudopotential methods. NFE, TB or combined NFE- TB band structure calculations. Within these regions, the potential experienced by an electron is approximated to be spherically symmetric about the given nucleus. In the remaining interstitial region, the screened potential is approximated as a constant.
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