Theory of Band Structures in Crystals
The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch waves as treated generally in the dynamical theory of diffraction. 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 (b1,b2,b3). 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:
where K = m1b1 + m2b2 + m3b3 for any set of integers (m1,m2,m3).
From this theory, an attempt can be made to predict the band structure of a particular material, however most ab initio methods for electronic structure calculations fail to predict the observed band gap.
Read more about this topic: Electronic Band Structure
Famous quotes containing the words theory of, theory, band, structures and/or crystals:
“The theory of truth is a series of truisms.”
—J.L. (John Langshaw)
“The theory of the Communists may be summed up in the single sentence: Abolition of private property.”
—Karl Marx (1818–1883)
“And the heavy night hung dark
The hills and waters o’er,
When a band of exiles moored their bark
On the wild New England shore.”
—Felicia Dorothea Hemans (1783–1835)
“The philosopher believes that the value of his philosophy lies in its totality, in its structure: posterity discovers it in the stones with which he built and with which other structures are subsequently built that are frequently better—and so, in the fact that that structure can be demolished and yet still possess value as material.”
—Friedrich Nietzsche (1844–1900)
“It is clear that everybody interested in science must be interested in world 3 objects. A physical scientist, to start with, may be interested mainly in world 1 objects—say crystals and X-rays. But very soon he must realize how much depends on our interpretation of the facts, that is, on our theories, and so on world 3 objects. Similarly, a historian of science, or a philosopher interested in science must be largely a student of world 3 objects.”
—Karl Popper (1902–1994)