How are band structures calculated in perovskites? Eight-band k·p theory

Why Calculate the Electronic Band Structure?

Calculating the electronic band structure of a perovskite means determining which electron energies are allowed at different wavevectors within its crystal. The resulting energy bands provide information about the band gap, effective masses, carrier velocities and the electronic states available to electrons and holes. An example calculated for MAPbI₃ is shown in ??.

Why is this useful? Once the electronic band structure and corresponding wavefunctions have been calculated, they provide the starting point for predicting a wide range of semiconductor properties. These include acoustic phonon scattering, polar optical phonon (Fröhlich) scattering, carrier relaxation times, electron and hole mobilities, carrier concentrations, density of states, optical absorption, spontaneous emission, stimulated emission and optical gain. The electronic structure can also be used to investigate how these properties change with temperature, carrier density and material composition. In this way, calculating the band structure is often the first step towards understanding both the electrical and optical behaviour of a semiconductor.

Here we explain how OghmaNano calculates the band structures of three-dimensional CsPbX₃ and MAPbX₃ perovskites (X = Cl, Br or I), starting with Schrödinger's equation and progressing to the eight-band k·p model. The mathematical formulation and material parameterisation follow Gawarecki et al., “Eight-band k·p description and material gain for selected cubic and pseudo-cubic perovskites”, Physical Review Applied 22, 014058 (2024): published paper (DOI) | freely accessible arXiv preprint. The explanations and illustrations on this page describe the numerical workflow independently; the original publication contains the complete matrix formulation and derivation.

For instructions on running a simulation, choosing a material or exploring mixed-halide compositions, see the practical perovskite band structure and mobility tutorial. A more detailed explanation of how electronic states interact with polar optical phonons is available on our Fröhlich scattering theory page.

Starting with Schrödinger's Equation

To understand how electronic bands form in a semiconductor, consider an electron moving through a crystal. Rather than experiencing the potential of a single isolated atom, the electron encounters a repeating arrangement of atoms, each contributing to the overall crystal potential. This is illustrated in ??, where the potential of an isolated atom is compared with the periodic potential of a crystal.

The important feature is that the crystal potential repeats at regular intervals, determined by the lattice spacing \(a\). Mathematically, this means that \(V(x+a)=V(x)\). An electron moving through such a periodic potential can occupy a range of allowed energy states, which collectively form the electronic bands of the material.

To calculate these states, we start with the time-independent Schrödinger equation:

\[ \hat H\psi(\mathbf r)=E\psi(\mathbf r). \]

Here, \(\psi\) is the electron wavefunction, \(E\) is its energy, and \(\hat H\) is the Hamiltonian: the mathematical operator describing the energy of the electron. In a simple one-electron description of the crystal, the Hamiltonian contains two contributions: the kinetic energy of the electron and the potential energy associated with the atomic lattice:

\[ \hat H_0 = -\frac{\hbar^2}{2m_0}\nabla^2 + V_{\mathrm{crystal}}(\mathbf r). \]

The first term describes the electron's kinetic energy, while \(V_{\mathrm{crystal}}(\mathbf r)\) represents the periodic potential shown in ??. Solving Schrödinger's equation for this potential gives the allowed electron energies and corresponding wavefunctions, from which the electronic band structure can be constructed.

In principle, the electronic structure can be calculated by resolving the full atomic potential, for example using density functional theory (DFT). However, if we are primarily interested in the electronic states close to the band gap, it is often more efficient to describe a smaller number of important states using an effective Hamiltonian.

This is the idea behind the k·p method. We retain Schrödinger's eigenvalue equation but replace the simple one-electron Hamiltonian with a matrix describing several interacting electronic bands. For metal halide perovskites, OghmaNano uses the eight-band formulation developed by Gawarecki et al., which includes the strong spin–orbit coupling characteristic of these materials. The numerical parameters of this Hamiltonian were obtained by fitting the model to first-principles DFT band structures (Gawarecki et al., Sections III–IV).

Where in k-Space Should We Calculate the Band Structure?

When calculating the electronic band structure of a semiconductor, we do not necessarily need to calculate every possible electronic state throughout the crystal's momentum space (k-space). For many applications, particularly charge carrier scattering and mobility, we are most interested in the states close to the conduction-band minimum and the valence-band maximum. This is because electrons tend to occupy states near the bottom of the conduction band, while holes occupy states near the top of the valence band.

The position of these band extrema depends on the material. For example, in the familiar semiconductor gallium arsenide (GaAs), both occur at the centre of the Brillouin zone, known as the Γ (Gamma) point. This can be seen in ??. Consequently, when using the k·p method to describe the electronic states close to the band gap of GaAs, we normally construct the Hamiltonian around the Γ point.

Lead halide perovskites are different. In the cubic perovskites considered here, the conduction-band minimum and valence-band maximum occur at a corner of the Brillouin zone, known as the R point. This is illustrated for MAPbI₃ in ??. We therefore construct the perovskite Hamiltonian around R, rather than Γ, since this is the region containing the electronic states most relevant to near-band-edge transport (Gawarecki et al., Fig. 1).

To calculate the electronic bands around R, we start at that position and gradually move away from it in k-space. Mathematically, we describe this movement using a small wavevector \(\mathbf{k}\):

\[ \mathbf{K}=\mathbf{K}_R+\mathbf{k}, \qquad \mathbf{k}=(k_x,k_y,k_z). \]

Here, \(\mathbf{K}_R\) is the position of the R point and \(\mathbf{k}\) describes our displacement from it. Thus, \(\mathbf{k}=0\) corresponds to the R point itself. By changing the components of \(\mathbf{k}\) and solving the Hamiltonian at each position, we obtain the electronic band structure around the band gap.

There is one further difference between GaAs and these perovskites. In GaAs, the valence-band edge is predominantly p-like and the conduction-band edge is s-like. In lead halide perovskites, this ordering is reversed: the valence band is predominantly s-like, while the nearby conduction bands are p-like. This distinction determines which electronic states we need to include when constructing the eight-band Hamiltonian (Gawarecki et al., Fig. 1).

Constructing the Perovskite Hamiltonian

Having established that we want to calculate the electronic states around the R point, we can now construct the Hamiltonian used to describe them. Recall that we started with the time-independent Schrödinger equation:

\[ \hat H\psi = E\psi. \]

Rather than explicitly solving for the full atomic potential, we replace the original Hamiltonian with an eight-band effective Hamiltonian, following the formulation developed by Gawarecki et al. (Section IV and Appendix A). This Hamiltonian is an \(8\times8\) matrix describing the energies and interactions of eight electronic states.

These states correspond to the three groups of bands shown in ??:

• Valence band (2 states): The predominantly s-like states at the top of the valence band.
• Lower conduction band (2 states): The lowest predominantly p-like conduction band, with total angular momentum \(j=1/2\).
• Upper conduction bands (4 states): The higher-energy p-like states, with total angular momentum \(j=3/2\), separated from the lowest conduction band by spin–orbit coupling.

Writing Schrödinger's equation in matrix form

In the eight-band model, the wavefunction is represented by eight coefficients, one for each basis state. Schrödinger's equation therefore becomes a matrix eigenvalue problem:

\[ \boxed{ H(\mathbf{k})\,\mathbf{u}_n(\mathbf{k}) = E_n(\mathbf{k})\,\mathbf{u}_n(\mathbf{k}) } \]

Here, \(H(\mathbf{k})\) is our \(8\times8\) Hamiltonian, \(E_n(\mathbf{k})\) is one of its allowed energies, and \(\mathbf{u}_n(\mathbf{k})\) is the corresponding eight-component eigenvector.

The explicit matrix in Appendix A of Gawarecki et al. uses a total-angular-momentum basis. The states are ordered as follows, with the colours matching the bands in our earlier MAPbI₃ diagram:

The labels indicate the total angular momentum and its projection. Notice that the two valence states come first in this particular ordering, followed by the four upper conduction states and finally the two lower conduction states.

The complete eight-band Hamiltonian

We can now substitute the full Hamiltonian into Schrödinger's equation. Following the matrix formulation given in Appendix A, Eq. (A1), of Gawarecki et al., we obtain:

Although the matrix initially looks complicated, its structure is relatively straightforward. The coloured diagonal entries describe the three groups of electronic bands:

• Orange — \(E_v\): The two valence-band states.
• Purple — \(E_{hc}\) and \(E_{lc}\): The four upper conduction-band states, often described as the heavy- and light-electron branches.
• Blue — \(E_{sc}\): The two lowest conduction-band states, split from the upper conduction bands by spin–orbit coupling.

The remaining entries describe the interactions between these states. These terms cause the bands to mix and influence their energies and curvature as we move through k-space. The asterisk denotes complex conjugation. The Hamiltonian is Hermitian, meaning that entries on opposite sides of the diagonal are complex conjugates of one another.

What do the individual matrix terms mean?

The symbols in the Hamiltonian are calculated from the material parameters and the wavevector \(\mathbf{k}=(k_x,k_y,k_z)\). First, the diagonal energies are given by:

Here, \(m_0\) is the free-electron mass and \(k^2=k_x^2+k_y^2+k_z^2\). The band gap \(E_g\) determines the energy of the lowest conduction band, while \(\Delta_c\) describes the separation produced by spin–orbit coupling. The parameter \(\delta\) introduces an additional splitting of the upper conduction states when the crystal symmetry is reduced.

The remaining terms describe the coupling between different electronic states:

The parameters \(P_{\parallel}\) and \(P_z\) describe the coupling between the valence and conduction states in different crystal directions. The parameters \(\gamma'_1\), \(\gamma'_2\) and \(\gamma'_3\) influence the curvature, mixing and directional dependence of the electronic bands. Finally, \(\zeta\) introduces an additional coupling associated with reduced symmetry.

These expressions, together with the complete Hamiltonian above, follow Appendix A of Gawarecki et al.. The numerical values for all six materials are provided in the parameter table later on this page.

Solving the Hamiltonian to Calculate the Band Structure

Having constructed the eight-band Hamiltonian, we can now solve it to obtain the electronic band structure. At the R point, where \(k_x=k_y=k_z=0\), the Hamiltonian becomes considerably simpler. Taking the valence-band maximum as our energy reference, the principal band-edge energies can be identified from:

\[ \begin{aligned} \color{#db790d}{E_v(0)} &= 0, \\[6pt] \color{#3e72d5}{E_{sc}(0)} &= E_g, \\[6pt] \color{#7948bd}{E_{hc}(0)} &= E_g+\Delta_c+\frac{\delta}{3}, \\[6pt] \color{#7948bd}{E_{lc}(0)} &= E_g+\Delta_c-\frac{\delta}{3}. \end{aligned} \]

These expressions identify the main energy scales of the valence band, the lowest conduction band and the two upper conduction branches. Their separation is illustrated in ??. In the cubic limit, \(\delta=\zeta=0\) and \(P_{\parallel}=P_z\). For pseudo-cubic materials, additional symmetry-dependent coupling can modify the exact eigenvalues, so we obtain the final energies by diagonalising the complete Hamiltonian.

As we move away from R, the wavevector terms become non-zero. The matrix elements change, the electronic states interact, and the resulting eigenvalues begin to disperse. This produces the curved electronic bands seen in our band-structure diagrams.

In practice, OghmaNano performs a repeated eigenvalue calculation. For each selected wavevector, it substitutes \(k_x\), \(k_y\), \(k_z\) and the material parameters into the expressions introduced above, assembles the numerical \(8\times8\) Hamiltonian, and solves Schrödinger's equation:

\[ \boxed{ H(\mathbf{k})\mathbf{u}_n(\mathbf{k}) = E_n(\mathbf{k})\mathbf{u}_n(\mathbf{k}), \qquad n=1,\ldots,8. } \]

This gives eight eigenvalues (the allowed electron energies) and eight corresponding eigenvectors (the electronic states). The calculation is then repeated at successive positions in k-space. For example, to calculate the band structure along the x direction, we evaluate the Hamiltonian at:

\[ \begin{aligned} \mathbf{k}_0 &= (0,0,0),\\ \mathbf{k}_1 &= (\Delta k,0,0),\\ \mathbf{k}_2 &= (2\Delta k,0,0),\\ &\ \vdots \end{aligned} \]

At every point, the eight calculated eigenvalues give the electronic energies at that wavevector:

\[ H(\mathbf{k}_i) \quad\longrightarrow\quad \left\{ E_1(\mathbf{k}_i),\ldots,E_8(\mathbf{k}_i) \right\}. \]

Plotting these energies as a function of wavevector gives the familiar E–k band-structure diagram. Other crystal directions can be explored by changing the k-space path, while a full three-dimensional calculation samples \(k_x\), \(k_y\) and \(k_z\) together. The complete numerical workflow is summarised in ??.

The calculated band structure also provides information about the effective masses of electrons and holes. Close to a simple band extremum, the effective mass in a particular direction can be obtained from the curvature of the energy band:

\[ \frac{1}{m_i^*} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i^2}, \]

with the corresponding positive-mass convention used for holes. However, as we shall see below, the full electronic bands and wavefunctions provide considerably more information than a single effective-mass value.

Material Parameters and Where They Enter the Model

To construct the eight-band Hamiltonian, we need a set of numerical parameters describing the electronic properties of the material. Gawarecki et al. obtained these parameters by fitting the eight-band k·p model to electronic band structures calculated using hybrid-functional density functional theory (DFT), including spin–orbit coupling (see Sections III–IV of their paper).

The values below are taken from Table I of Gawarecki et al. and cover the six pure metal halide perovskites included in the OghmaNano material library.

The different parameters determine how the Hamiltonian is constructed and how the resulting electronic bands behave:

• \(E_g\): The fundamental band gap. This establishes the energy of the lowest conduction band relative to the valence-band maximum.
• \(\Delta_c\): The main spin–orbit splitting between the lower and upper conduction-band groups. It shifts the upper conduction states relative to the lowest conduction band.
• \(\delta\) and \(\zeta\): Additional splitting and coupling parameters associated with reduced crystal symmetry. Both are zero for the cubic CsPbX₃ parameter sets, but can be non-zero for pseudo-cubic MAPbX₃.
• \(P_{\parallel}\) and \(P_z\): Describe the coupling between the predominantly s-like valence states and p-like conduction states. Separate values are used for the in-plane and vertical crystal directions. Their units are eV·Å.
• \(\gamma'_1\), \(\gamma'_2\) and \(\gamma'_3\): Coefficients of the wavevector-dependent terms in the Hamiltonian. They influence the average band curvature, anisotropy and mixing between the conduction states, including contributions from bands outside the eight-band model.
• \(a_{\mathrm{exp}}\) and \(a_{\mathrm{DFT}}\): The experimental and DFT-calculated lattice constants, respectively. These are provided for structural reference and are not additional fitting coefficients in the displayed Hamiltonian.
• \(m_v\) and the unprimed parameters \(\gamma_1,\gamma_2,\gamma_3\): Derived quantities included in the original publication. The Hamiltonian uses the primed parameters because the coupling between the explicitly included bands is already accounted for in the matrix. See Eqs. (2)–(4) of Gawarecki et al..

In OghmaNano, the pure-material scripts contain the tabulated parameter values. For mixed-halide perovskites, the material scripts interpolate between the corresponding endpoint parameters before constructing the Hamiltonian. This allows the same eight-band calculation to be performed for a user-selected iodine, bromine and chlorine composition. The procedure is explained in the mixed-halide perovskite tutorial.

What Can We Calculate from the Band Structure?

Once the electronic band structure and corresponding wavefunctions have been calculated, we can use them as the starting point for predicting a wide range of semiconductor properties. These include effective masses, carrier concentrations, acoustic and optical phonon scattering, carrier relaxation times, electron and hole mobilities, optical absorption, spontaneous emission and optical gain.

For example, a major intrinsic scattering mechanism in polar metal halide perovskites is Fröhlich scattering, in which electrons or holes interact with longitudinal optical (LO) phonons. Using Fermi's golden rule, the scattering rate can be calculated by integrating over the available final electronic states:

\[ \frac{1}{\tau(\mathbf{k})} = \frac{2\pi}{\hbar} \sum_{\pm} \int\frac{d^3k'}{(2\pi)^3} \,|M_{\pm}(\mathbf{k},\mathbf{k}')|^2 \delta\!\left( E(\mathbf{k}')-E(\mathbf{k}) \mp\hbar\omega_{\mathrm{LO}} \right). \]

Here, \(M_{\pm}\) represents the phonon absorption or emission matrix element, including the appropriate phonon occupation factor. The Dirac delta function ensures that energy is conserved during the transition. The scattering calculation can then be used to obtain momentum relaxation times and, together with the calculated band velocities and carrier occupations, determine the Fröhlich-limited mobility.

Similarly, the electronic bands and optical transition matrix elements can be used to calculate optical gain. Within a Fermi's golden rule treatment, its spectral dependence can be expressed schematically as:

\[ g(\omega)\propto \frac{1}{\omega} \sum_{c,v} \int\frac{d^3k}{(2\pi)^3} \left| \mathbf{e}\cdot\mathbf{p}_{cv}(\mathbf{k}) \right|^2 \left[f_c(\mathbf{k})-f_v(\mathbf{k})\right] \delta\!\left( E_c(\mathbf{k})-E_v(\mathbf{k}) -\hbar\omega \right). \]

This expression describes optical transitions between the calculated valence and conduction bands. Their energies, wavefunctions and carrier occupations determine the spectral position and strength of absorption or stimulated emission. Additional scattering and optical parameters are required for quantitative predictions.

A full explanation of LO phonon absorption, emission and the calculation of carrier mobility is available on our Fröhlich scattering theory page. To see how the perovskite band-structure and mobility calculations are performed in practice, including pure and mixed-halide materials, see the step-by-step OghmaNano tutorial.

References and further reading

Primary model: K. Gawarecki, M. Wiśniewski, M. Polak, R. Kudrawiec and M. Gładysiewicz, “Eight-band k·p description and material gain for selected cubic and pseudo-cubic perovskites”, Physical Review Applied 22, 014058 (2024). DOI: 10.1103/PhysRevApplied.22.014058; arXiv:2306.08643 (free preprint). See especially Section IV and Appendix A for the full Hamiltonian, and Table I for the parameters.