Everything about Band Structure totally explained
In
solid-state physics, the
electronic band structure (or simply
band structure) of a
solid describes ranges of
energy that an
electron is "forbidden" or "allowed" to have. It is due to the
diffraction of the quantum mechanical electron waves in the periodic
crystal lattice. The band structure of a material determines several characteristics, in particular its electronic and optical properties.
Why bands occur in materials
The electrons of a single free-standing atom occupy
atomic orbitals, which form a discrete set of
energy levels. If several atoms are brought together into a molecule, their atomic orbitals split, as in a
coupled oscillation. This produces a number of
molecular orbitals proportional to the number of atoms. When a large number of atoms (of order
.
Ab initio Density-functional theory
In present days physics literature, the large majority of the electronic structures and band plots is calculated using the
density-functional theory (DFT) which isn't a model but rather an
ab initio theory, for example a microscopical first-principle theory of
condensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of an
exchange-correlation term in the functional of the
electronic density. DFT calculated bands are found in many cases in agreement with experimental measured bands, for example by
angle-resolved photoemission spectroscopy (ARPES). In particular, the band shape seems well reproduced by DFT. But also there are systematic errors of DFT bands with respect to the experiment. In particular, DFT seems to underestimate systematically by a 30-40% the band gap in insulators and semiconductors.
It must be said that DFT is in principle an exact theory to reproduce and predict
ground state properties (for example the
total energy, the
atomic structure, etc.). However DFT isn't a theory to address
excited state properties, such as the band plot of a solid that represents the excitation energies of electrons injected or removed from the system. What in literature is quoted as a DFT band plot is a representation of the DFT
Kohn-Sham energies, that's the energies of a fictive non-interacting system, the
Kohn-Sham system, which has no physical interpretation at all. The Kohn-Sham electronic structure must not be confused with the real,
quasiparticle electronic structure of a system, and there's no
Koopman's theorem holding for Kohn-Sham energies, like on the other hand for Hartree-Fock energies that can be truly considered as an approximation for
quasiparticle energies. Hence in principle DFT isn't a band theory, not a theory suitable to calculate bands and band-plots.
Green's function methods and the ab initio GW approximation
To calculate the bands including electron-electron interaction many-body effects, one can resort to so called
Green's function methods. Indeed, the knowledge of the Green's function of a system provides both ground (the total energy) and also excited state observables of the system. The
poles of the Green's function are the quasiparticle energies, the bands of a solid. The Green's function can be calculated by solving the
Dyson equation once the
self-energy of the system is known. For real systems like solids, the self-energy is a very complex quantity and usually approximations are needed to solve the problem. One of such approximations is the
GW approximation, so called from the mathematical form the self-energy takes as product
of the Green's function
and the
dynamically screened interaction . This approach is more pertinent to address the calculation of band plots (and also quantities beyond, such as the
spectral function) and can be also formulated in a completely
ab initio way. The GW approximation seems to provide band gaps of insulators and semiconductors in agreement with the experiment and hence to correct the systematic DFT underestimation.
Mott insulators
Although the nearly-free electron approximation is able to describe many properties of electron band structures, one consequence of this theory is that it predicts the same number of electrons in each unit cell. If the number of electrons is odd, we'd then expect that there's an unpaired electron in each unit cell, and thus that the valence band isn't fully occupied, making the material a conductor. However, materials such as
CoO that have an odd number of electrons per unit cell are insulators, in direct conflict with this result. This kind of material is known as a
Mott insulator, and requires inclusion of detailed electron-electron interactions (treated only as an averaged effect on the crystal potential in band theory) to explain the discrepancy. The
Hubbard model is an approximate theory that can include these interactions.
Other
Calculating band structures is an important topic in theoretical solid state physics. In addition to the models mentioned above, other models include the following:
The Kronig-Penney Model, a one-dimensional rectangular well model useful for illustration of band formation. While simple, it predicts many important phenomena, but isn't quantitative.
Bands may also be viewed as the large-scale limit of molecular orbital theory. A solid creates a large number of closely spaced molecular orbitals, which appear as a band.
Hubbard model
The band structure has been generalised to wavevectors that are complex numbers, resulting in what is called a complex band structure, which is of interest at surfaces and interfaces.
Each model describes some types of solids very well, and others poorly. The nearly-free electron model works well for metals, but poorly for non-metals. The tight binding model is extremely accurate for ionic insulators, such as metal halide salts (for example NaCl).
Further Information
Get more info on 'Band Structure'.
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