- K·p perturbation theory
In

solid-state physics ,**k·p perturbation theory**is an approximation scheme for calculating theband structure (particularlyeffective mass ) and optical properties of crystalline solids. It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the**Luttinger-Kohn**model (afterJoaquin Mazdak Luttinger andWalter Kohn ), and of the**Kane model**(afterEvan O. Kane ).**Background and derivation****Bloch's theorem and wavevectors**According to

quantum mechanics (in the single-electron approximation), theelectron s in any material havewavefunction s which can be described by the followingSchrodinger equation ::$(frac\{p^2\}\{2m\}+V)psi\; =\; Epsi$where**p**is the quantum-mechanical momentum operator, "V" is the potential, and "m" is the mass of an electron. (This equation neglects thespin-orbit effect ; see below.)In a

crystalline solid , "V" is aperiodic function , with the same periodicity as thecrystal lattice . Bloch's theorem proves that the solutions to this differential equation can be written as follows::$psi\_\{n,mathbf\{k(mathbf\{x\})\; =\; e^\{imathbf\{k\}cdotmathbf\{x\; u\_\{n,mathbf\{k(mathbf\{x\})$where**k**is a vector (called the "wavevector"), "n" is a discrete index (called the "band index"), and "u"_{"n",k}is a function with the same peridicity as the crystal lattice.For any given "n", the associated states are called a band. In each band, there will be a relation between the wavevector

**k**and the energy of the state "E"_{"n",k}, called the band dispersion. Calculating this dispersion is one of the primary applications of "k"·"p" perturbation theory.**Perturbation theory**The periodic function "u"

_{"n",k}satisfies the following Schrödinger-type equation:P. Yu, M. Cardona, "Fundamentals of Semiconductors", 3rd edition, ISBN 3-540-41323-5, 2005. Section 2.6.] :$H\_\{mathbf\{k\; u\_\{n,mathbf\{k=E\_\{n,mathbf\{ku\_\{n,mathbf\{k$where the Hamiltonian is :$H\_\{mathbf\{k\; =\; frac\{p^2\}\{2m\}\; +\; frac\{hbar\; mathbf\{k\}cdotmathbf\{p\{m\}\; +\; frac\{hbar^2\; k^2\}\{2m\}\; +\; V$Note that**k**is a vector consisting of three real numbers with units of length^{-1}, while**p**is a vector of operators; to be explicit,:$mathbf\{k\}cdotmathbf\{p\}\; =\; k\_x\; (-ihbar\; frac\{partial\}\{partial\; x\})\; +\; k\_y\; (-ihbar\; frac\{partial\}\{partial\; y\})\; +\; k\_z\; (-ihbar\; frac\{partial\}\{partial\; z\})$In any case, we write this Hamiltonian as the sum of two terms::$H=H\_0+H\_\{mathbf\{k\text{'},\; ;;\; H\_0\; =\; frac\{p^2\}\{2m\}+V,\; ;;\; H\_\{mathbf\{k\text{'}\; =\; frac\{hbar^2\; k^2\}\{2m\}\; +\; frac\{hbar\; mathbf\{k\}cdotmathbf\{p\{m\}$This expression is the basis for perturbation theory. The "unperturbed Hamiltonian" is "H"_{0}, which in fact equals the exact Hamiltonian at**k**=0 (i.e., at theGamma point ). The "perturbation" is the term $H\_\{mathbf\{k\text{'}$. The analysis that results is called "k·p perturbation theory", due to the term proportional to "k"·"p". The result of this analysis is an expression for "E"_{"n",k}and "u"_{"n",k}in terms of the energies and wavefunctions at**k**=0.Note that the "perturbation" term $H\_\{mathbf\{k\text{'}$ gets progressively smaller as

**k**approaches zero. Therefore, k·p perturbation theory is most accurate for small values of**k**. However, if enough terms are included in the perturbative expansion, then the theory can in fact be reasonably accurate for any value of**k**in the entireBrillouin zone .**Expression for a nondegenerate band**For a nondegenerate band (i.e., a band which has a different energy at

**k**=0 from any other band), with anextremum at**k**=0, and with nospin-orbit coupling , the result of "k"·"p" perturbation theory is (to lowest nontrivial order)::$u\_\{n,mathbf\{k\; =\; u\_\{n,0\}+frac\{hbar\}\{m\}sum\_\{n\text{'}\; eq\; n\}frac\{langle\; u\_\{n,0\}\; |\; mathbf\{k\}cdotmathbf\{p\}\; |\; u\_\{n\text{'},0\}\; angle\}\{E\_\{n,0\}-E\_\{n\text{'},0\; u\_\{n\text{'},0\}$:$E\_\{n,mathbf\{k\; =\; E\_\{n,0\}+frac\{hbar^2\; k^2\}\{2m\}\; +\; frac\{hbar^2\}\{m^2\}\; sum\_\{n\text{'}\; eq\; n\}\; frac\{|langle\; u\_\{n,0\}\; |\; mathbf\{k\}cdotmathbf\{p\}\; |\; u\_\{n\text{'},0\}\; angle\; |^2\}\{E\_\{n,0\}-E\_\{n\text{'},0$The parameters that are required to do these calculations, namely "E"

_{"m",0}and $langle\; u\_\{n,0\}\; |\; mathbf\{p\}\; |\; u\_\{n\text{'},0\}\; angle$, are typically inferred from experimental data. (The latter are called "optical matrix elements".)In practice, the sum over "n

**often includes only the nearest one or two bands, since these tend to be the most important (due to the denominator). However, for improved accuracy, especially at larger**k"', more bands must be included, as well as more terms in the perturbative expansion than the ones written above.**"k"·"p" model with spin-orbit interaction**Including the

spin-orbit interaction , the Schrödinger equation for "u" is:C. Kittel, "Quantum Theory of Solids", Second revised printing, 1987, ISBN 0-471-62412-8. Pages 186-190.] :$H\_\{mathbf\{k\; u\_\{n,mathbf\{k=E\_\{n,mathbf\{ku\_\{n,mathbf\{k$where:$H\_\{mathbf\{k\; =\; frac\{p^2\}\{2m\}\; +\; frac\{hbar\; mathbf\{k\}cdotmathbf\{p\{m\}\; +\; frac\{hbar^2\; k^2\}\{2m\}\; +\; V\; +\; frac\{1\}\{4\; m^2\; c^2\}\; (vec\; sigma\; imes\; abla\; V)cdot\; (mathbf\{k\}+mathbf\{p\})$where $vec\; sigma=(sigma\_x,sigma\_y,sigma\_z)$ is a vector consisting of the three Pauli matrices. This Hamiltonian can be subjected to the same sort of perturbation-theory analysis as above.**Calculation in degenerate case**For degenerate or nearly-degenerate bands, in particular the

valence band s in certain materials such asgallium arsenide , the equations can be analyzed by the methods of degenerate perturbation theory. Models of this type include the "Luttinger-Kohn model" (a.k.a. "Kohn-Luttinger model") [*[*] , and the "Kane model". [*http://dx.doi.org/10.1103/PhysRev.97.869 Luttinger and Kohn, Phys. Rev.*]**97**(1955), p869*[*] [*http://people.bu.edu/theochem/history/pdfs/20022003/rabani.pdf Powerpoint presentation by Eran Rabani*]*Evan O. Kane, "Band Structure of Indium Antimonide", J. Phys. Chem. Solids*]**1**, p249 (1957).**References**

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