Understanding k·p Theory: From Parabolic Bands to Multiband Models

Why Do We Need Another Band-Structure Model?

In our introduction to parabolic quantum-well bands, we represented an electron's energy using a simple parabola. Its curvature was determined by an effective mass, allowing us to turn each confined quantum-well state into a two-dimensional electronic subband. This is a very useful approximation, particularly close to the bottom of a simple conduction band.

However, a real semiconductor contains several electronic bands. They can lie close together, bend away from a parabolic shape, and interact with one another. This is especially important for the valence bands, where heavy-hole, light-hole and split-off states cannot always be treated as independent particles with a single mass.

The k·p method provides a way of describing these effects without recalculating the full atomic-scale electronic structure at every wavevector. We start with electronic states at one well-understood point in the crystal's band structure and calculate how they change as the wavevector moves away from that point. Most importantly, we allow different electronic bands to interact with each other.

The aim of this page is to explain where the k·p equations come from, why the effective mass appears naturally, and how a collection of coupled equations becomes the multiband Hamiltonian used in a quantum-well solver.

Starting with an Electron in a Crystal

For an electron moving in a semiconductor crystal, we begin with the familiar time-independent Schrödinger equation:

\[ \hat H_0\Psi(\mathbf r)=E\Psi(\mathbf r), \qquad \hat H_0=\frac{\hat{\mathbf p}^{\,2}}{2m_0}+V_{\mathrm{crystal}}(\mathbf r). \]

Here, \(m_0\) is the ordinary (free-electron) mass, \(\hat{\mathbf p}=-i\hbar\nabla\) is the momentum operator and \(V_{\mathrm{crystal}}\) is the potential experienced by the electron due to the periodic arrangement of atoms. We have deliberately used \(m_0\), not the semiconductor effective mass: at this stage the influence of the crystal has not yet been incorporated into an approximate mass.

A key property of an ideal crystal is that its potential repeats from one unit cell to the next. If \(\mathbf R\) describes a lattice translation, then:

\[ V_{\mathrm{crystal}}(\mathbf r+\mathbf R) =V_{\mathrm{crystal}}(\mathbf r). \]

We could try to solve Schrödinger's equation directly throughout this repeating potential, but Bloch's theorem gives us a much more useful way of expressing its solutions.

Bloch's Theorem: Separating the Wave from the Crystal

Bloch's theorem states that an electronic state in a periodic crystal can be written as the product of a plane wave and a function that repeats with the crystal lattice:

\[ \boxed{ \Psi_{n\mathbf k}(\mathbf r) =e^{i\mathbf k\cdot\mathbf r}\,u_{n\mathbf k}(\mathbf r) } \]

The two parts have different jobs. The plane-wave factor describes how the overall phase changes as we move through the crystal. The function \(u_{n\mathbf k}(\mathbf r)\) describes the detail of the wave within each unit cell and has the same periodicity as the crystal:

\[ u_{n\mathbf k}(\mathbf r+\mathbf R) =u_{n\mathbf k}(\mathbf r). \]

The label \(n\) tells us which electronic band we are considering, while \(\mathbf k=(k_x,k_y,k_z)\) is the crystal wavevector. The quantity \(\hbar\mathbf k\) is called crystal momentum. It is closely related to the momentum used in our earlier cricket-ball analogy, although the two should not be confused in a periodic crystal.

This is the first important simplification: instead of describing a new atomic-scale wavefunction throughout the entire crystal for every \(\mathbf k\), we can work with the cell-periodic functions \(u\). The k·p method investigates how those functions and their energies change when we move a small distance in wavevector space.

Where Does the k·p Term Come From?

Let us substitute Bloch's expression into Schrödinger's equation. When the momentum operator acts on the product of the plane wave and the cell-periodic function, the result is:

\[ \hat{\mathbf p}\left(e^{i\mathbf k\cdot\mathbf r}u\right) =e^{i\mathbf k\cdot\mathbf r} \left(\hat{\mathbf p}+\hbar\mathbf k\right)u. \]

In other words, acting on a Bloch wave is equivalent to replacing the momentum operator acting on \(u\) by \(\hat{\mathbf p}+\hbar\mathbf k\). Applying this to the kinetic-energy term and cancelling the common plane-wave factor gives:

\[ \left[ \frac{(\hat{\mathbf p}+\hbar\mathbf k)^2}{2m_0} +V_{\mathrm{crystal}}(\mathbf r) \right]u_{n\mathbf k} =E_n(\mathbf k)u_{n\mathbf k}. \]

Expanding the square is the crucial step. Because \(\mathbf k\) is a parameter, not a spatial derivative, we obtain:

\[ \boxed{ \left[ \hat H_0 +\frac{\hbar}{m_0}\mathbf k\cdot\hat{\mathbf p} +\frac{\hbar^2 k^2}{2m_0} \right]u_{n\mathbf k} =E_n(\mathbf k)u_{n\mathbf k} } \]

The first term is our original crystal Hamiltonian. The final term is a simple quadratic energy contribution. The middle term, \(\hbar\mathbf k\cdot\hat{\mathbf p}/m_0\), is where the method gets its name: k·p.

Physically, it tells us that moving away from our chosen reference wavevector can mix electronic states through the momentum operator. To see what that means, we need to choose the states between which this mixing can occur.

Choosing a Reference Point and a Basis

Semiconductor band structures are often most interesting close to a band minimum or maximum. For a direct-gap semiconductor such as GaAs, the conduction-band minimum and valence-band maximum occur at the zone centre, known as the Gamma point (\(\Gamma\)). For the following explanation we therefore expand about \(\mathbf k=0\). The same general idea can be applied about another reference point when the relevant band extremum is elsewhere in the Brillouin zone.

At \(\mathbf k=0\), the k·p term vanishes. Suppose we already know several cell-periodic crystal states there:

\[ \hat H_0|u_{n0}\rangle=E_n(0)|u_{n0}\rangle. \]

These known states form our basis. A state at a nearby wavevector can then be described as a combination of them:

\[ |u_{\mathbf k}\rangle =c_1|u_{10}\rangle+c_2|u_{20}\rangle+\cdots. \]

The coefficients \(c_1,c_2,\ldots\) describe how much of each reference state contributes to the new state. They are precisely what our calculation needs to determine. We do this by inserting the expansion into the k·p equation and projecting it onto each basis state.

The result is a matrix equation. Its elements are:

\[ \boxed{ H_{nm}(\mathbf k) =\left[E_n(0)+\frac{\hbar^2k^2}{2m_0}\right]\delta_{nm} +\frac{\hbar}{m_0}\mathbf k\cdot\mathbf p_{nm} } \]

\[ \mathbf p_{nm} =\langle u_{n0}|\hat{\mathbf p}|u_{m0}\rangle. \]

The symbol \(\delta_{nm}\) equals one when the two band labels are the same and zero otherwise. Consequently, the diagonal of this matrix contains the reference band energies and free-electron quadratic terms. Its off-diagonal elements contain the momentum couplings between different bands.

These momentum matrix elements are central to k·p theory. If a coupling is zero, the corresponding states do not mix through that term. If it is appreciable, their energies and wavefunctions can change significantly as we move away from the reference point.

A Worked Example: What Happens When Two Bands Interact?

Before considering eight or ten bands, it is helpful to see what a two-band model does. Imagine that we retain just one conduction-band state and one valence-band state, with reference energies \(E_c\) and \(E_v\). Their separation at the reference point is the band gap, \(E_g=E_c-E_v\).

For clarity, we consider motion along one crystal direction, with wavevector \(k\), and choose the phases of our basis states so that the coupling parameter \(P\) is real. The illustrative Hamiltonian is:

\[ \boxed{ H(k)= \begin{pmatrix} E_c+\dfrac{\hbar^2 k^2}{2m_0} & Pk\\[6pt] Pk & E_v+\dfrac{\hbar^2 k^2}{2m_0} \end{pmatrix} } \]

Here, \(P=\hbar\langle u_{c0}|\hat p|u_{v0}\rangle/m_0\) for the selected direction. Notice that the two off-diagonal terms vanish at \(k=0\), but increase as we move away from the reference point. The model is therefore not describing two independent parabolas: the conduction- and valence-band states can mix.

To obtain the permitted energies we solve an eigenvalue problem, just as we did when finding the confined states of a quantum well:

\[ H(k)\begin{pmatrix}c_c\\c_v\end{pmatrix} =E(k)\begin{pmatrix}c_c\\c_v\end{pmatrix}. \]

Instead of finding a single energy, the two-by-two matrix produces two energy branches. For this simplified example they are:

\[ E_{\pm}(k)= \frac{E_c+E_v}{2}+\frac{\hbar^2k^2}{2m_0} \pm\sqrt{\left(\frac{E_g}{2}\right)^2+P^2k^2}. \]

At \(k=0\), the two results reduce to \(E_c\) and \(E_v\), as expected. As \(k\) increases, the coupling changes both energies and mixes the underlying basis states. This square-root dependence also shows why a coupled band need not remain perfectly parabolic away from its extremum.

The physical lesson is more important than memorising this expression: the shape of a band depends not only on that band's own energy, but also on its coupling to neighbouring bands.

Where Does the Effective Mass Come From?

In the parabolic quantum-well tutorial, we used effective mass as an input parameter. The two-band example now allows us to see where it comes from. Very close to \(k=0\), the conduction-band result can be approximated by expanding the square root for small \(k\):

\[ E_+(k)\simeq E_c+ \left(\frac{\hbar^2}{2m_0}+\frac{P^2}{E_g}\right)k^2. \]

Compare this with our familiar parabolic expression:

\[ E(k)\simeq E_c+\frac{\hbar^2k^2}{2m_e^{*}}. \]

Both have the same mathematical shape. Matching their quadratic coefficients gives the effective mass of this simplified two-band model:

\[ \boxed{ \frac{1}{m_e^{*}}= \frac{1}{m_0}+\frac{2P^2}{\hbar^2E_g} } \]

This is a useful result. The effective mass is not an arbitrary replacement for the electron's mass. It arises because the electron's state is coupled to other electronic bands within the crystal. In this particular example, a smaller band gap or stronger coupling produces a greater conduction-band curvature and a lighter effective mass.

A real semiconductor has many other bands, not just the two in our example. If we treat their coupling perturbatively, the corresponding correction to an isolated band's energy is:

\[ E_n(\mathbf k)\simeq E_n(0) +\frac{\hbar^2k^2}{2m_0} +\frac{\hbar^2}{m_0^2} \sum_{m\ne n} \frac{|\mathbf k\cdot\mathbf p_{nm}|^2} {E_n(0)-E_m(0)}. \]

Each neighbouring band contributes according to its momentum coupling and its separation in energy. The resulting curvature is generally direction-dependent, so the most complete description is an effective-mass tensor, rather than one scalar mass. A scalar effective mass is appropriate when symmetry or the chosen approximation allows us to treat the curvature as the same in the directions of interest.

There is one important limitation: the perturbative expression becomes unreliable when the denominator is small, meaning that two bands lie close together in energy. In that situation, it is better to retain both bands explicitly in the Hamiltonian, rather than hide their interaction inside a single effective mass.

From Two Bands to a Multiband Hamiltonian

The next step is conceptually straightforward. Instead of starting with just two crystal states, we retain several relevant states in our basis:

\[ |u_{\mathbf k}\rangle =\sum_{n=1}^{N}c_n(\mathbf k)|u_{n0}\rangle. \]

This produces an \(N\)-by-\(N\) Hamiltonian. Each diagonal element describes a reference band and its quadratic correction; each permitted off-diagonal element describes a coupling between two basis states. Solving the matrix for a chosen wavevector gives all the resulting band energies and the amount of each basis state in the corresponding eigenvector.

This matters particularly in the valence band. Heavy-hole and light-hole states may be degenerate at the zone centre, and their mixing cannot be represented faithfully by assigning each one an independent constant mass. A multiband model allows those states to interact naturally as the wavevector changes.

We must still decide which bands to retain. The general approach is to keep the states that are most relevant to the energies and physics of interest. Coupling to more distant, excluded states can be included approximately through corrections to the parameters of the retained Hamiltonian. This is why different k·p models can have different numbers of bands and different material parameters.

For example, a common zincblende eight-band model retains an s-like conduction state and three p-like valence orbital states, each with two spin components. The detailed choice of basis, symmetry terms and matrix entries is explained in our basis and Hamiltonians chapter and the material-specific model pages.

Why Do We Need Spin and Spin-Orbit Coupling?

Up to this point, our simple two-band example has left out spin. Electrons, however, have two spin components. This means that each orbital state must be represented by spin-up and spin-down basis states (or a suitable equivalent spin basis).

If spin had no effect on the energies, this would mainly duplicate the problem. Real semiconductor crystals also exhibit spin-orbit coupling, which connects the electron's spin to its motion through the crystal potential. In a simplified Pauli description, the additional Hamiltonian is:

\[ \hat H_{\mathrm{so}}= \frac{\hbar}{4m_0^2c^2} \boldsymbol{\sigma}\cdot \left(\nabla V_{\mathrm{crystal}}\times\hat{\mathbf p}\right), \]

where \(c\) is the speed of light and \(\boldsymbol{\sigma}\) denotes the Pauli matrices. We do not need to derive this relativistic term here; the important point is its physical consequence. Spin and orbital character are no longer independent, and certain states which would otherwise have the same energy become separated.

In the familiar zincblende valence band, spin-orbit coupling gives rise to the heavy-hole, light-hole and split-off branches, with the split-off energy commonly denoted by \(\Delta_{\mathrm{so}}\). These are not simply three unrelated parabolas: their character and interaction depend on the wavevector and the underlying crystal symmetry.

This explains the name eight-band k·p: four orbital basis states (one conduction and three valence), each with two spin components, produce an eight-component basis. The actual Hamiltonian includes the permitted momentum and spin-orbit couplings between them. A complete quantum-well calculation using this model is demonstrated in the GaAs/AlGaAs eight-band zincblende k·p tutorial.

How Does This Become a Quantum-Well Calculation?

So far, we have been describing a bulk crystal, in which the electron energy changes with a three-dimensional wavevector. To describe a quantum well, we must now return to the geometry introduced on the previous tutorial page.

Suppose the semiconductor layers extend parallel to the \(x\)-\(y\) plane and are stacked along \(z\). The in-plane wavevector, \(k_{\parallel}\), remains a useful label because the structure is approximately uniform along the layers. However, the electron is confined in the \(z\) direction, so we must solve for its spatial variation across the well.

The standard envelope-function approach replaces the growth-direction wavevector in the bulk k·p Hamiltonian by a differential operator:

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

The band edges and other relevant material parameters can also vary with \(z\), reflecting the different semiconductor layers. The problem is no longer an ordinary finite matrix evaluated at one wavevector: it is a set of coupled differential equations for the envelope functions across the device. When parameters vary at interfaces, operator ordering and matching conditions must be chosen consistently so that the Hamiltonian remains physically meaningful and Hermitian.

The simplest parabolic model solved one scalar Schrödinger equation for an electron or a hole. A multiband k·p model instead solves for several coupled envelope components. The resulting eigenvalues give the confined energy levels, while their eigenvectors tell us how the retained band characters contribute to each state.

By repeating the calculation at different values of \(k_{\parallel}\), we build up the quantum-well subband structure. Unlike a collection of independent parabolas, these subbands can mix, curve non-parabolically and change character as the in-plane wavevector increases.

What Does the Extra Physics Allow Us to Calculate?

The benefit of a multiband calculation is not simply that it produces more complicated-looking band diagrams. It gives us a more realistic description of the electronic states that participate in transport and optical processes.

In our optical transitions tutorial, we introduced the probability of an electron making a transition between an occupied and an empty state. The transition strength depends on a matrix element between the initial and final wavefunctions. A multiband eigenvector contains information about the mixed orbital and spin character of each state, which is essential when calculating the direction and strength of optical transitions.

This is particularly useful when studying polarisation-dependent transitions, optical gain, spontaneous emission and systems in which valence-band mixing is substantial. The detailed optical matrix elements depend on the chosen Hamiltonian and basis, but their calculation follows naturally once the coupled eigenstates have been obtained.

k·p Models in OghmaNano

OghmaNano provides several models for different semiconductor structures. They share the general k·p philosophy developed here, but retain different electronic states and use the material parameters appropriate to each crystal system.

• Eight-band zincblende k·p includes the conduction and p-like valence states, together with spin and spin-orbit coupling.
• Ten-band zincblende k·p retains an extended electronic basis for problems requiring additional coupled states.
• Eight-band wurtzite k·p uses the symmetry and band parameters appropriate to wurtzite crystals.

The next chapter introduces the actual basis states and explains how they are arranged into the Hamiltonian matrices used by these solvers.