Bloch's theorem and wavevectors

According to quantum mechanics (in the single-electron approximation), the quasi-free electrons in any solid are characterized by wavefunctions which are eigenstates of the following stationary Schrödinger equation:

${\displaystyle \left({\frac {p^{2}}{2m}}+V\right)\psi =E\psi }$

where p is the quantum-mechanical momentum operator, V is the potential, and m is the vacuum mass of the 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:

${\displaystyle \psi _{n,\mathbf {k} }(\mathbf {x} )=e^{i\mathbf {k} \cdot \mathbf {x} }u_{n,\mathbf {k} }(\mathbf {x} )}$

where k is a vector (called the wavevector), n is a discrete index (called the band index), and 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. 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):^[1]

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

where the Hamiltonian is

${\displaystyle H_{\mathbf {k} }={\frac {p^{2}}{2m}}+{\frac {\hbar \mathbf {k} \cdot \mathbf {p} }{m}}+{\frac {\hbar ^{2}k^{2}}{2m}}+V}$

Note that k is a vector consisting of three real numbers with dimensions of inverse length, while p is a vector of operators; to be explicit,

${\displaystyle \mathbf {k} \cdot \mathbf {p} =k_{x}(-i\hbar {\frac {\partial }{\partial x}})+k_{y}(-i\hbar {\frac {\partial }{\partial y}})+k_{z}(-i\hbar {\frac {\partial }{\partial z}})}$

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

${\displaystyle H_{\mathbf {k} }=H_{0}+H_{\mathbf {k} }',\;\;H_{0}={\frac {p^{2}}{2m}}+V,\;\;H_{\mathbf {k} }'={\frac {\hbar ^{2}k^{2}}{2m}}+{\frac {\hbar \mathbf {k} \cdot \mathbf {p} }{m}}}$

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 ${\displaystyle H_{\mathbf {k} }'}$. 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 E_n,k and u_n,k in terms of the energies and wavefunctions at k = 0.

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

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]

${\displaystyle u_{n,\mathbf {k} }=u_{n,0}+{\frac {\hbar }{m}}\sum _{n'\neq n}{\frac {\langle u_{n',0}|\mathbf {k} \cdot \mathbf {p} |u_{n,0}\rangle }{E_{n,0}-E_{n',0}}}u_{n',0}}$

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

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:

${\displaystyle \langle u_{n,0}|\mathbf {k} \cdot \mathbf {p} |u_{n',0}\rangle =\mathbf {k} \cdot \langle u_{n,0}|\mathbf {p} |u_{n',0}\rangle }$

Therefore, one can calculate the energy at any k using only a few unknown parameters, namely E_n,0 and ${\displaystyle \langle u_{n,0}|\mathbf {p} |u_{n',0}\rangle }$. 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.

k·p model with spin–orbit interaction

Including the spin–orbit interaction, the Schrödinger equation for u is:^[2]

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

where^[7]

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

where ${\displaystyle {\vec {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})}$ 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 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"),^[8] and the "Kane model".^[7]

Generally, an effective Hamiltonian ${\displaystyle H^{\rm {eff}}}$ is introduced, and to the first order, its matrix elements can be expressed as

${\displaystyle H_{\mathbf {k} ,mn}^{\rm {eff}}=\langle u_{m,0}|H_{0}|u_{n,0}\rangle +\mathbf {k} \cdot \langle u_{m,0}|\nabla _{\mathbf {k} }H_{\mathbf {k} }'|u_{n,0}\rangle }$

After solving it, the wave functions and energy bands are obtained.