K·p Perturbation Theory
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In
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state phy ...
, the k·p
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
is an approximated semi-empirical approach for calculating the
band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
(particularly effective 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 (after
Joaquin Mazdak Luttinger Joaquin (Quin) Mazdak Luttinger (December 2, 1923 – April 6, 1997) was an American physicist well known for his contributions to the theory of interacting electrons in one-dimensional metals (the electrons in these metals are said to be in ...
and
Walter Kohn Walter Kohn (; March 9, 1923 – April 19, 2016) was an Austrian-American theoretical physicist and theoretical chemist. He was awarded, with John Pople, the Nobel Prize in Chemistry in 1998. The award recognized their contributions to the un ...
), and of the Kane model (after Evan O. Kane).


Background and derivation


Bloch's theorem and wavevectors

According to
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
(in the single-electron approximation), the quasi-free
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s in any solid are characterized by
wavefunction In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
s which are eigenstates of the following stationary
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
: :\left(\frac+V\right)\psi = E\psi where p is the quantum-mechanical momentum operator, ''V'' is the
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
, and ''m'' is the vacuum mass of the electron. (This equation neglects the spin–orbit effect; see below.) In a
crystalline solid A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
, ''V'' is a
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
, with the same periodicity as the
crystal lattice In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystal, crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that ...
.
Bloch's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, ...
proves that the solutions to this differential equation can be written as follows: :\psi_(\mathbf) = e^ u_(\mathbf) where k is a vector (called the ''wavevector''), ''n'' is a discrete index (called the ''
band Band or BAND may refer to: Places *Bánd, a village in Hungary * Band, Iran, a village in Urmia County, West Azerbaijan Province, Iran * Band, Mureș, a commune in Romania * Band-e Majid Khan, a village in Bukan County, West Azerbaijan Province, ...
index''), and ''u''''n'',k is a function with the same periodicity as the crystal lattice. For any given ''n'', the associated states are called a
band Band or BAND may refer to: Places *Bánd, a village in Hungary * Band, Iran, a village in Urmia County, West Azerbaijan Province, Iran * Band, Mureș, a commune in Romania * Band-e Majid Khan, a village in Bukan County, West Azerbaijan Province, ...
. 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 (simply, a direct expansion of the Schrödinger equation with a Bloch-type wave function): :H_ u_=E_u_ where the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
is :H_ = \frac + \frac + \frac + V Note that k is a vector consisting of three real numbers with dimensions of
inverse length Reciprocal length or inverse length is a quantity or measurement used in several branches of science and mathematics, defined as the reciprocal of length. Common units used for this measurement include the reciprocal metre or inverse metre (symbol ...
, while p is a vector of operators; to be explicit, :\mathbf\cdot\mathbf = k_x (-i\hbar \frac) + k_y (-i\hbar \frac) + k_z (-i\hbar \frac) In any case, we write this Hamiltonian as the sum of two terms: :H_=H_0+H_', \;\; H_0 = \frac+V, \;\; H_' = \frac + \frac This expression is the basis for
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. The "unperturbed Hamiltonian" is ''H''0, which in fact equals the exact Hamiltonian at k = 0 (i.e., at the gamma point). The "perturbation" is the term H_'. 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_' 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 entire
Brillouin zone In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry whic ...
.


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 an
extremum In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' ...
at k = 0, and with no spin–orbit coupling, the result of k''·''p perturbation theory is (to lowest nontrivial order): :u_ = u_+\frac\sum_\frac u_ :E_ = E_+\frac + \frac \sum_ \frac Since k is a vector of real numbers (rather than a vector of more complicated linear operators), the matrix element in these expressions can be rewritten as: :\langle u_ , \mathbf\cdot\mathbf , u_ \rangle = \mathbf \cdot \langle u_ , \mathbf , u_ \rangle Therefore, one can calculate the energy at ''any'' k using only a ''few'' unknown parameters, namely ''E''''n'',0 and \langle u_ , \mathbf , u_ \rangle. The latter are called "optical matrix elements", closely related to transition dipole moments. These parameters are typically inferred from experimental data. 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.


Effective mass

Using the expression above for the energy dispersion relation, a simplified expression for the effective mass in the conduction band of a semiconductor can be found. To approximate the dispersion relation in the case of the conduction band, take the energy ''En0'' as the minimum conduction band energy ''Ec0'' and include in the summation only terms with energies near the valence band maximum, where the energy difference in the denominator is smallest. (These terms are the largest contributions to the summation.) This denominator is then approximated as the band gap ''Eg'', leading to an energy expression: :E_c(\boldsymbol k ) \approx E_ +\frac +\frac\sum_n The effective mass in direction ℓ is then: : \frac = \sum_ \cdot \approx \frac+\frac\sum_ Ignoring the details of the matrix elements, the key consequences are that the effective mass varies with the smallest bandgap and goes to zero as the gap goes to zero. A useful approximation for the matrix elements in direct gap semiconductors is:A ''direct gap'' semiconductor is one where the valence band maximum and conduction band minimum occur at the same position in k-space, usually the so-called Γ-point where k = 0. :\frac\sum_ \approx 20\mathrm \frac \ , which applies within about 15% or better to most group-IV, III-V and II-VI semiconductors.Se
Table 2.22
in Yu & Cardona, ''op. cit.''
In contrast to this simple approximation, in the case of valence band energy the ''spin–orbit'' interaction must be introduced (see below) and many more bands must be individually considered. The calculation is provided in Yu and Cardona.See Yu & Cardona, ''op. cit.'' pp. 75–82 In the valence band the mobile carriers are ''
holes A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
''. One finds there are two types of hole, named ''heavy'' and ''light'', with anisotropic masses.


k·p model with spin–orbit interaction

Including the
spin–orbit interaction In quantum mechanics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin– ...
, the Schrödinger equation for ''u'' is: :H_ u_=E_u_ where :H_ = \frac + \frac\mathbf\cdot\mathbf + \frac + V + \frac (\nabla V \times (\mathbf+\hbar\mathbf))\cdot \vec \sigma where \vec \sigma=(\sigma_x,\sigma_y,\sigma_z) is a vector consisting of the three
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
. 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 In solid-state physics, the valence band and conduction band are the bands closest to the Fermi level, and thus determine the electrical conductivity of the solid. In nonmetals, the valence band is the highest range of electron energies in ...
s in certain materials such as
gallium arsenide Gallium arsenide (GaAs) is a III-V direct band gap semiconductor with a Zincblende (crystal structure), zinc blende crystal structure. Gallium arsenide is used in the manufacture of devices such as microwave frequency integrated circuits, monoli ...
, 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". Generally, an effective Hamiltonian H^ is introduced, and to the first order, its matrix elements can be expressed as :H^_=\langle u_, H_0, u_\rangle + \mathbf\cdot \langle u_, \nabla _\mathbf H_\mathbf', u_\rangle After solving it, the wave functions and energy bands are obtained.


See also

Electronic band structure *
Electronic band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
*
Nearly free electron model In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model ...
*
Kronig–Penney model In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic Crystal structure, crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromag ...
Band properties *
Band gap In solid-state physics and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to t ...
* Effective mass *
Density of states In condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or quantum state, states per unit energy range. The density of states is defined as where N(E)\delta E is the number of states in the syste ...
*
Fermi surface In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied electron states from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and sym ...
Wavefunctions *
Wannier functions The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems. The Wannier functions f ...
*
Bloch's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, ...
Fundamental theory *
Kohn–Sham equations The Kohn-Sham equations are a set of mathematical equations used in quantum mechanics to simplify the complex problem of understanding how electrons behave in atoms and molecules. They introduce fictitious non-interacting electrons and use them to ...
*
Local-density approximation Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space (and ...


Notes and references

{{DEFAULTSORT:K P Perturbation Theory Electronic structure methods