k·p perturbation theory

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In solid-state physics, k·p perturbation theory is an approximation scheme for calculating the band structure and optical properties of crystalline solids.[1][2] 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 and Walter Kohn), and of the Kane model (after Evan O. Kane).

Contents

[edit] Background and derivation

[edit] Bloch's theorem and wavevectors

See also: Bloch wave

According to quantum mechanics (in the single-electron approximation), the electrons in any material have wavefunctions which can be described by the following Schrodinger equation:

(\frac{p^2}{2m}+V)\psi = E\psi

where p is the quantum-mechanical momentum operator, V is the potential, and m is the mass of an electron. (This equation neglects the spin-orbit effect; see below.)

In a crystalline solid, V is a periodic function, with the same periodicity as the crystal lattice. Bloch's theorem proves that the solutions to this differential equation can be written as follows:

\psi_{n,\mathbf{k}}(\mathbf{x}) = e^{i\mathbf{k}\cdot\mathbf{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 un,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 En,k, called the band dispersion. Calculating this dispersion is one of the primary applications of k·p perturbation theory.

[edit] Perturbation theory

See also: Perturbation theory (quantum mechanics)

The periodic function un,k satisfies the following Schrodinger-type equation:[1]

H_{\mathbf{k}} u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}

where the Hamiltonian is

H_{\mathbf{k}} = \frac{p^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{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}\cdot\mathbf{p} = k_x (-i\hbar \frac{\partial}{\partial x}) + k_y (-i\hbar \frac{\partial}{\partial y}) + k_z (-i\hbar \frac{\partial}{\partial z})

In any case, we write this Hamiltonian as the sum of two terms:

H=H_0+H_{\mathbf{k}}', \;\; H_0 = \frac{p^2}{2m}+V, \;\; H_{\mathbf{k}}' = \frac{\hbar^2 k^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m}

This expression is the basis for perturbation theory. The "unperturbed Hamiltonian" is H0, which in fact equals the exact Hamiltonian at k=0 (i.e., at the Gamma point). The "perturbation" is the term H_{\mathbf{k}}'. 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 En,k and un,k in terms of the energies and wavefunctions at k=0.

Note that the "perturbation" term H_{\mathbf{k}}' 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.

[edit] 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 at k=0, and with no spin-orbit coupling, the result of k·p perturbation theory is (to lowest nontrivial order):[1]

u_{n,\mathbf{k}} = u_{n,0}+\frac{\hbar}{m}\sum_{n' \neq n}\frac{\langle u_{n,0} | \mathbf{k}\cdot\mathbf{p} | u_{n',0} \rangle}{E_{n,0}-E_{n',0}} u_{n',0}
E_{n,\mathbf{k}} = E_{n,0}+\frac{\hbar^2 k^2}{2m} + \frac{\hbar^2}{m^2} \sum_{n'\neq n} \frac{|\langle u_{n,0} | \mathbf{k}\cdot\mathbf{p} | u_{n',0} \rangle |^2}{E_{n,0}-E_{n',0}}

The parameters that are required to do these calculations, namely Em,0 and \langle u_{n,0} | \mathbf{p} | u_{n',0} \rangle, 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.

[edit] k·p model with spin-orbit interaction

Including the spin-orbit interaction, the Schrodinger equation for u is:[2]

H_{\mathbf{k}} u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}

where

H_{\mathbf{k}} = \frac{p^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m} + \frac{\hbar^2 k^2}{2m} + V + \frac{1}{4 m^2 c^2} (\vec \sigma \times \nabla 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.

[edit] Calculation in degenerate case

For degenerate or nearly-degenerate bands, in particular the valence bands in certain materials such as gallium arsenide, the equations can be analyzed by the methods of degenerate perturbation theory.[1][2] Models of this type include the "Luttinger-Kohn model" (a.k.a. "Kohn-Luttinger model")[3], and the "Kane model".[4][5]

[edit] References

  1. ^ a b c d P. Yu, M. Cardona, Fundamentals of Semiconductors, 3rd edition, ISBN 3-540-41323-5, 2005. Section 2.6.
  2. ^ a b c C. Kittel, Quantum Theory of Solids, Second revised printing, 1987, ISBN 0-471-62412-8. Pages 186-190.
  3. ^ Luttinger and Kohn, Phys. Rev. 97 (1955), p869
  4. ^ Powerpoint presentation by Eran Rabani
  5. ^ Evan O. Kane, "Band Structure of Indium Antimonide", J. Phys. Chem. Solids 1, p249 (1957).
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