- Bloch wave
A Bloch wave or Bloch state, named after
Felix Bloch , is thewavefunction of a particle (usually, anelectron ) placed in a periodic potential. It consists of the product of aplane wave envelope function and aperiodic function ("periodic Bloch function") which has the same periodicity as the potential::
The result that the
eigenfunction s can be written in this form for a periodic system is called Bloch's theorem. The corresponding energy eigenvalue is Єn(k)= Єn(k + K), periodic with periodicity K of areciprocal lattice vector. Because the energies associated with the index "n" vary continuously with wavevector k we speak of an "energy band" with "band index" "n". Because the eigenvalues for given "n" are periodic in k, all distinct values of Єn(k) occur for k-values within the firstBrillouin zone of the reciprocal lattice.More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic
dielectric inelectromagnetism leads tophotonic crystal s, and a periodic acoustic medium leads tophononic crystal s. It is generally treated in the different forms of thedynamical theory of diffraction .The plane wave wavevector (or "Bloch wavevector") k (multiplied by the reduced
Planck's constant , this is the particle's "crystal momentum") is unique only up to a reciprocal lattice vector, so one only needs to consider the wavevectors inside the first Brillouin zone. For a given wavevector and potential, there are a number of solutions, indexed by "n", toSchrödinger's equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete)band gap . The band structure is the collection of energy eigenstates within thefirst Brillouin zone . All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within theindependent electron approximation .A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the
group velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and thatelectrical resistance in a crystallineconductor only results from imperfections and finite size which break the periodicity and interaction withphonon s.It can be shown that the eigenfunctions of a particle in a periodic potential can always be chosen in this form by proving that translation operators (by lattice vectors) commute with the Hamiltonian. More generally, the consequences of symmetry on the eigenfunctions are described by representation theory.
The concept of the Bloch state was developed by
Felix Bloch in 1928, to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: byGeorge William Hill (1877),Gaston Floquet (1883), andAlexander Lyapunov (1892). As a result, a variety of nomenclatures are common: applied toordinary differential equations , it is calledFloquet theory (or occasionally the "Lyapunov-Floquet theorem"). Various one-dimensional periodic potential equations have special names, for example,Hill's equation :cite book
author=W Magnus and S Winkler
title=Hill's Equation
year= 2004
page=p. 11
publisher=Courier Dover
isbn=0-0486495655
url=http://books.google.com/books?id=ML5wm-T4RVQC&dq=%22hill's+equation%22&printsec=frontcover&source=web&ots=kWOt7bG8n0&sig=qB8ilFCc4_g6eCNoLnVDyY3Wo-U#PPA11,M1]::,
where the θ's are constants. Hill's equation is very general, as the θ-related terms may viewed as a
Fourier series expansion of a periodic potential. Other much studied periodic one-dimensional equations are theKronig-Penney model and Mathieu's equation.References
ee also
*
Electronic band structure
*Tight-binding model
*Nearly-free electron model
*Wannier function Further reading
* Charles Kittel, "Introduction to Solid State Physics" (Wiley: New York, 1996).
* Neil W. Ashcroft and N. David Mermin, "Solid State Physics" (Harcourt: Orlando, 1976).
* Felix Bloch, "Über die Quantenmechanik der Elektronen in Kristallgittern," "Z. Physik" 52, 555-600 (1928).
* George William Hill, "On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon," "Acta. Math." 8, 1-36 (1886). (This work was initially published and distributed privately in 1877.)
* Gaston Floquet, "Sur les équations différentielles linéaires à coefficients périodiques," "Ann. École Norm. Sup." 12, 47-88 (1883).
* Alexander Mihailovich Lyapunov, "The General Problem of the Stability of Motion" (London: Taylor and Francis, 1992). Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).
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