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k·p Band-Structure and Quantum-Well Solver

Calculated conduction and valence subbands from an eight-band zincblende k p quantum-well simulation — Conduction- and valence-subband dispersion calculated for a GaAs/AlGaAs quantum well using the eight-band zincblende k·p model.

Quantum well electron and hole states, subbands and optical transitions — Confined electron and hole states form two-dimensional subbands from which optical transitions can be calculated.

S-like conduction state coupled to X Y and Z valence states in the eight-band k p model — Multiband k·p models explicitly couple conduction and valence basis states rather than treating every band as an independent parabola.

Optical gain spectrum calculated from k p quantum-well states — Electronic states can be carried forward into calculations of absorption, stimulated emission, optical gain and spontaneous emission.

1. Introduction

OghmaNano contains a family of semiconductor band-structure, quantum-well and k·p solvers for calculating electronic states in both two-dimensional quantum wells and bulk semiconductors. The models calculate confined electron and hole states, electronic band dispersion, wavefunctions, band mixing and optical matrix elements, and can be coupled to calculations of optical absorption, spontaneous emission, stimulated emission, gain, carrier scattering and mobility.

A range of band-structure models is available for different semiconductor materials and crystal structures:

Parabolic effective-mass quantum-well solver — a simple Schrödinger-equation model for electron and hole confinement in semiconductor quantum wells. It is useful for straightforward direct-gap heterostructures such as GaAs/AlGaAs quantum wells. A complete worked example is available in the GaAs/AlGaAs parabolic quantum-well tutorial .
8-band zincblende k·p solver — a multiband model for zincblende semiconductors such as GaAs, InAs and InP and related III–V alloys. The Hamiltonian includes the conduction band together with the heavy-hole, light-hole and split-off valence bands, allowing non-parabolicity, spin-orbit coupling and valence-band mixing to be calculated explicitly. See the 8-band GaAs/AlGaAs k·p tutorial .
10-band zincblende k·p solver — an extended zincblende model for dilute-nitride semiconductors such as GaAsN and GaInNAs. Additional nitrogen-derived resonant states allow the model to reproduce conduction-band anticrossing and the strong changes in band gap and electron dispersion that occur in dilute nitrides. See the GaInNAs 10-band k·p tutorial .
8-band wurtzite k·p solver — a multiband model for hexagonal semiconductors including GaN, AlN and InN and their alloys. The model includes crystal-field splitting, spin-orbit coupling, anisotropic band structure and the electronic-state mixing required to calculate TE and TM optical transitions. See the GaN/AlGaN 8-band wurtzite tutorial .
Silicon six-Δ-valley quantum-well solver — a model for electron states in silicon and Si/SiGe quantum wells. It treats the six anisotropic silicon conduction-band valleys, including their longitudinal and transverse effective masses and the effects of confinement and strain. The resulting states can also be used to calculate scattering rates and two-dimensional electron mobility. See the Si/SiGe electron quantum-well tutorial .
6-band Luttinger–Kohn valence-band solver — a multiband hole model for Si, Ge and SiGe heterostructures. It treats the coupled heavy-hole, light-hole and spin-orbit split-off bands and can be used to calculate confined valence subbands, band mixing, scattering and two-dimensional hole mobility. See the Si/SiGe Luttinger–Kohn tutorial .
8-band bulk perovskite k·p solver — a bulk band-structure model for metal-halide perovskites, including CsPbCl₃, CsPbBr₃, CsPbI₃, MAPbCl₃, MAPbBr₃ and MAPbI₃, together with mixed-halide compositions. The Hamiltonian is evaluated around the perovskite R point and includes the strong spin-orbit coupling characteristic of lead-halide perovskites. The calculated bulk electronic states can be used to obtain effective masses, carrier statistics, Fröhlich electron–phonon scattering, relaxation times and carrier mobility. See the perovskite band-structure and mobility tutorial .

Together, these models allow OghmaNano to calculate both two-dimensional quantum-well subbands and three-dimensional bulk semiconductor band structures. For quantum wells, carriers are confined along the growth direction while remaining free to move parallel to the layers. In a multiband k·p calculation the growth-direction wavevector is replaced by the envelope-function operator

\[ k_z \longrightarrow -i\frac{d}{dz}, \]

and the resulting coupled equations are solved across the semiconductor heterostructure. Repeating the calculation as a function of in-plane wavevector produces the two-dimensional subband dispersion. Bulk models, such as the perovskite solver, instead solve the Hamiltonian directly as a function of the three-dimensional wavevector \(\mathbf{k}=(k_x,k_y,k_z)\).

The calculated eigenvalues and eigenvectors provide more than a band diagram. They form the starting point for simulations of optical transitions , absorption, spontaneous emission, stimulated emission and optical gain, as well as carrier scattering, relaxation times and mobility where supported. These capabilities make the band-structure solvers applicable to the modelling of semiconductor lasers, LEDs, photodetectors, quantum-well heterostructures, high-mobility semiconductor channels, silicon/SiGe devices and metal-halide perovskite solar-cell and light-emitting materials.

For a general introduction to the underlying physics, see quantum wells and carrier confinement and the introduction to k·p theory .

2. Applications of band theory: optical transitions, gain, scattering and mobility

Calculating the electronic band structure is only the first stage of the simulation. Once the energies and wavefunctions of the electronic states are known, OghmaNano can use them to calculate a range of optical and charge-transport properties. These include optical absorption, spontaneous and stimulated emission, optical gain, phonon scattering rates, carrier relaxation times and charge-carrier mobility.

The same basic idea underlies many of these calculations: a carrier makes a transition from an initial electronic state to a final state, with the transition probability determined by the coupling between the two states. In the weak-interaction limit this is described by Fermi's golden rule,

\[ W_{i\rightarrow f} = \frac{2\pi}{\hbar} \left|\langle f|\hat H_{\mathrm{int}}|i\rangle\right|^2 \delta(E_f-E_i). \]

The interaction Hamiltonian \(\hat H_{\mathrm{int}}\) depends on the physical process being considered. For an optical transition it describes the interaction between the electronic states and the electromagnetic field. For carrier scattering it can instead describe interactions with lattice vibrations, including acoustic phonons or longitudinal optical (LO) phonons. The calculated band structure therefore provides a common electronic basis from which both optical and transport properties can be derived.

2.1 Optical transitions, absorption and gain

For optical calculations, OghmaNano combines the calculated conduction- and valence-band states with their optical matrix elements and carrier occupations. Transitions between occupied and unoccupied states can then be summed across the calculated band structure to obtain absorption, stimulated emission, net optical gain and spontaneous emission.

Because multiband k·p eigenvectors retain information about the orbital and spin character of each state, the optical calculation can go beyond simply matching energy levels. Band mixing changes the transition matrix elements and therefore the strength of the optical response. In anisotropic systems such as wurtzite semiconductors, this also allows TE- and TM-polarised optical transitions to be treated separately.

Optical gain calculated from an eight-band k p quantum-well band structure — Stimulated emission and optical gain calculated from the electronic states.

Optical absorption calculated from an eight-band k p quantum-well band structure — Calculated optical absorption spectrum.

Spontaneous emission calculated from an eight-band k p quantum-well band structure — Spontaneous emission calculated from occupied electron and hole states.

2.2 Carrier scattering from the calculated band structure

The electronic states can also be used to calculate the probability that a carrier is scattered from one part of the band structure to another. Depending on the material and model, OghmaNano can include mechanisms such as acoustic-phonon scattering and Fröhlich polar-optical-phonon scattering . These processes determine carrier lifetimes, momentum relaxation and ultimately charge-carrier mobility.

Fröhlich scattering is particularly important in polar semiconductors such as GaAs, InP, GaN and metal-halide perovskites. A longitudinal optical phonon creates an oscillating polarisation field that interacts with an electron or hole. The strength of this interaction is described by the Fröhlich matrix element,

\[ M(q) = -\frac{i}{q} \left[ \frac{e^2\hbar\omega_{LO}}{2V\varepsilon_0} \left( \frac{1}{\varepsilon_\infty} - \frac{1}{\varepsilon_s} \right) \right]^{1/2}. \]

Here \(q\) is the phonon wavevector, \(\omega_{LO}\) is the LO-phonon angular frequency, and \(\varepsilon_\infty\) and \(\varepsilon_s\) describe the electronic and static dielectric screening of the material. The \(1/q\) dependence reflects the long-range nature of the polar interaction.

Applying Fermi's golden rule and integrating over the available final states gives the carrier scattering rate. For LO-phonon absorption, for example,

\[ \begin{aligned} \frac{1}{\tau_{\mathrm{abs}}(\mathbf{k})} ={}& \frac{2\pi}{\hbar} \frac{V}{(2\pi)^3} \int d^3q\; \frac{e^2\hbar\omega_{LO}}{2V\varepsilon_0} \left( \frac{1}{\varepsilon_\infty} - \frac{1}{\varepsilon_s} \right) \frac{N_{LO}}{q^2} \\[4pt] &\times \delta\left( E(\mathbf{k}+\mathbf{q})- E(\mathbf{k})- \hbar\omega_{LO} \right). \end{aligned} \]

The Dirac delta function enforces conservation of energy, while the integration over \(\mathbf{q}\) sums over the phonon wavevectors capable of connecting the initial and final electronic states. LO-phonon emission is calculated in the same way, except that the carrier loses an energy \(\hbar\omega_{LO}\). In contrast, acoustic-phonon scattering describes interactions with the lower-energy lattice vibrations associated with elastic deformation of the crystal.

Fröhlich interaction between an electron and a longitudinal optical phonon in a polar semiconductor — A longitudinal optical phonon generates an electric polarisation field which interacts with charge carriers in a polar semiconductor.

LO phonon absorption and emission transitions within a semiconductor band — LO-phonon absorption and emission move carriers between states separated by the phonon energy \(\hbar\omega_{LO}\).

Calculated Fröhlich LO phonon scattering rates as a function of electron energy — Calculated LO-phonon absorption, emission and total scattering rates.

2.3 From scattering rates to carrier mobility

A scattering event does not necessarily destroy all of a carrier's directed momentum. Forward scattering, for example, changes the carrier direction only slightly, while backscattering strongly opposes the electrical current. For transport calculations OghmaNano can therefore use a momentum-relaxation rate rather than simply taking the inverse of the total scattering rate.

Once the transport relaxation time has been calculated across the electronic band structure, the mobility can be obtained by averaging the velocity and scattering properties of the electronic states that participate in transport. Within the relaxation-time approximation,

\[ \mu_{ij} = \frac{e}{n} \sum_b \int \frac{d^3k}{(2\pi)^3} v_{b,i}(\mathbf{k}) v_{b,j}(\mathbf{k}) \tau_{\mathrm{tr},b}(\mathbf{k}) \left( -\frac{\partial f}{\partial E} \right)_{E_b(\mathbf{k})}. \]

The carrier velocity is obtained directly from the calculated electronic dispersion,

\[ \mathbf{v}_b(\mathbf{k}) = \frac{1}{\hbar} \nabla_{\mathbf{k}}E_b(\mathbf{k}). \]

This connects the calculated band structure directly to an experimentally measurable transport quantity. Rather than assuming a single constant effective mass and scattering time, the calculation can account for the way carrier velocity and scattering vary across the electronic bands.

For Si/SiGe quantum wells, OghmaNano can calculate acoustic-phonon scattering and two-dimensional electron or hole mobility using the silicon Δ-valley and Luttinger–Kohn models. For bulk metal-halide perovskites, the calculated eight-band electronic structure can be combined with Fröhlich electron–phonon scattering to obtain carrier relaxation times and phonon-limited mobility.

Worked examples are available in the Si/SiGe electron mobility tutorial , Si/SiGe hole mobility tutorial and perovskite band-structure and mobility tutorial .

3. Getting started

The examples supplied with OghmaNano provide ready-made starting points for each band-structure model. For quantum-well optical calculations, the GaAs/AlGaAs eight-band k·p tutorial demonstrates the calculation of electronic subbands, absorption, stimulated emission, optical gain and spontaneous emission.

For bulk calculations, the perovskite band-structure and mobility tutorial shows how to calculate the electronic bands of CsPbX₃ and MAPbX₃ materials and use the resulting states to calculate Fröhlich scattering and charge-carrier mobility.

Further theoretical background is available in the k·p theory guide , optical-transition theory and Fröhlich scattering theory .

5. Getting started

For an introduction to quantum confinement, begin with quantum wells and carrier confinement . The parabolic-band tutorial introduces the simplest quantum-well model, while the k·p theory guide explains how several electronic bands are coupled in a multiband Hamiltonian.

For a practical starting point, try the GaAs/AlGaAs eight-band k·p tutorial or the bulk perovskite band-structure and mobility tutorial .

Explore the quantum and band-structure modelling tools.

Start with the quantum-well introduction, read the k·p theory guide, or run the 8-band GaAs/AlGaAs tutorial.