Understanding the 8-Band Zincblende k·p Model

Why Do We Need Eight Bands?

In our parabolic-band tutorial, we treated the conduction and valence bands as independent parabolas. This is a useful starting point: solving Schrödinger's equation across the well gives the confined states, and allowing carriers to move parallel to the layers turns those states into electronic subbands. But it assumes that one effective mass can describe each band.

Real semiconductor bands are not independent. As the electron's wavevector changes, states derived from different atomic orbitals can interact, altering their energies and the shapes of their wavefunctions. The effect is especially important in the valence band, where the heavy-hole, light-hole and split-off bands lie close together. An eight-band k·p model describes these interactions directly rather than trying to absorb all of them into one effective mass.

Here we consider zincblende semiconductors, including materials such as GaAs, InAs and InP. The aim is to build up the eight-band model in stages: first identify the states, then see how they interact, and finally understand what solving the resulting matrix tells us about a quantum well.

What Are the Eight States?

Near the centre of the Brillouin zone (the Gamma point), a useful starting point is to describe the conduction-band state as having predominantly S-like symmetry. The valence states have predominantly p-like symmetry, which we label X, Y and Z to indicate their Cartesian orientations. These are crystal states, not individual atomic orbitals, but the orbital labels help us understand the interactions between them.

Without considering spin, we therefore have four states:

\[ |S\rangle,\qquad |X\rangle,\qquad |Y\rangle,\qquad |Z\rangle. \]

Each state can occur with spin up or spin down. The actual basis used in this Cartesian representation is:

\[ \begin{gathered} |S\uparrow\rangle,\quad |S\downarrow\rangle,\\[3pt] |X\uparrow\rangle,\quad |Y\uparrow\rangle,\quad |Z\uparrow\rangle,\\[3pt] |X\downarrow\rangle,\quad |Y\downarrow\rangle,\quad |Z\downarrow\rangle. \end{gathered} \]

This is where the name eight-band comes from: four orbital functions multiplied by two spin components. We will initially ignore the spin interaction so that we can understand the simpler four-state problem first. Spin-orbit coupling will be introduced after the orbital interactions are clear.

What Does It Mean for Two Bands to Couple?

Imagine that we know the energies of the S and X states at the Gamma point. If the two states never interacted, changing the wavevector would simply change their individual energies. In a real crystal, however, motion through the periodic potential can couple them. The electron's wavefunction can then contain a mixture of S-like and X-like character, rather than belonging exclusively to one state.

We represent this by writing the wavefunction as a sum of basis states. For a four-orbital example:

\[ |\Psi\rangle = c_S|S\rangle+c_X|X\rangle+c_Y|Y\rangle+c_Z|Z\rangle. \]

The coefficients tell us how much of each basis state contributes to a particular solution. Instead of solving a separate equation for every orbital, we solve a matrix Schrödinger equation that determines all of the coefficients together:

\[ H\,\mathbf{c}=E\,\mathbf{c},\qquad \mathbf{c}=\begin{pmatrix}c_S\\c_X\\c_Y\\c_Z\end{pmatrix}. \]

The entries along the diagonal describe the energy of each state. Entries away from the diagonal describe interactions between different states. If those off-diagonal entries are zero, the states remain independent. If they are non-zero, solving the matrix changes both the permitted energies and the composition of the resulting wavefunctions.

The Kane Coupling: Connecting Conduction and Valence Bands

In k·p theory, the interaction between the S-like conduction state and the X, Y and Z valence states depends on the electron's wavevector. This interaction is described by the Kane coupling. For example, the S and X states are coupled by a term proportional to the wavevector component along x; the corresponding S-to-Y and S-to-Z couplings depend on the y and z components.

Using the energy-times-length convention for the Kane parameter P, we can write the couplings schematically as:

\[ H_{SX}\propto iPk_x,\qquad H_{SY}\propto iPk_y,\qquad H_{SZ}\propto iPk_z. \]

Here, P controls the strength of the interaction. Its magnitude is often expressed through the Kane energy, \(E_p\):

\[ P=\sqrt{A_0E_p},\qquad A_0=\frac{\hbar^2}{2m_0}, \]

where \(m_0\) is the free-electron mass. A larger Kane parameter produces stronger mixing of the conduction and valence states as the wavevector moves away from Gamma. The imaginary factors and signs in the full Hamiltonian depend on the chosen basis convention; the important physical idea is that the off-diagonal terms connect the bands.

A Two-State Example

Before considering all eight states, it is useful to look at the simplest possible example. Suppose we retain only one conduction state and one valence state. If their uncoupled energies are \(E_c\) and \(E_v\), a convenient Hermitian model is:

\[ H=\begin{pmatrix} E_c & Pk\\ Pk & E_v \end{pmatrix}. \]

In this illustrative two-state model, we have chosen the phases of the basis functions so that the coupling is real. Solving the matrix gives two energies:

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

At zero wavevector, the coupling vanishes and we recover the original band edges. Away from zero, the upper and lower solutions move apart. The separation depends on the coupling strength P, and the corresponding wavefunctions become mixtures of the two original basis states. This simple example illustrates the central principle of the full eight-band calculation: coupling changes the band dispersion, rather than merely shifting an existing parabola.

Why Does Coupling Change the Effective Mass?

Recall the parabolic-band equation from the previous tutorial. Close to a conduction-band minimum, the energy depends approximately on the square of the wavevector, and the effective mass determines the curvature. The two-state model above explains why other bands can influence that mass.

If the coupling \(Pk\) is small compared with the band gap \(E_g=E_c-E_v\), the upper solution can be expanded near \(k=0\):

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

Notice that the extra term is proportional to \(k^2\): even though we introduced a coupling that is linear in k, its effect on the energy close to the band edge is quadratic. It therefore changes the curvature of the conduction band and, consequently, the electron's effective mass.

In the actual model there is also a direct kinetic-energy contribution and coupling to several valence states, including the spin-orbit split-off states. These contributions must be combined to reproduce the observed band-edge mass. This is why the effective mass is not an arbitrary replacement for the free-electron mass: it summarises the influence of the surrounding crystal bands on the motion of an electron near a particular band edge.

Why Are There Three Valence Orbitals?

The conduction state alone is comparatively simple. The valence band requires more care because it contains X-, Y- and Z-like states that can interact with one another. Their behaviour depends on the direction of the wavevector, not just its magnitude. A single downward-facing parabola therefore misses important features of the valence-band structure.

In the Cartesian representation, we construct a valence-band block with a diagonal entry for each orbital and off-diagonal entries describing their coupling. Using the coefficients L, M and N, a typical orbital block has the form:

\[ H_{vv}^{\mathrm{orb}}= \begin{pmatrix} D_X & Nk_xk_y & Nk_xk_z\\ Nk_xk_y & D_Y & Nk_yk_z\\ Nk_xk_z & Nk_yk_z & D_Z \end{pmatrix}, \]

where the diagonal terms take the form:

\[ \begin{aligned} D_X&=E_{v,\mathrm{av}}+A_0k^2+Lk_x^2+M(k_y^2+k_z^2),\\[3pt] D_Y&=E_{v,\mathrm{av}}+A_0k^2+Lk_y^2+M(k_x^2+k_z^2),\\[3pt] D_Z&=E_{v,\mathrm{av}}+A_0k^2+Lk_z^2+M(k_x^2+k_y^2). \end{aligned} \]

We do not need to memorise this matrix to understand its purpose. L and M determine how the energies of the individual orbital components vary with direction; N controls the mixing between different orbital components. The full eight-band calculation includes this valence block for both spin components and couples it to the conduction states.

Where Do Heavy Holes, Light Holes and the Split-Off Band Come From?

We have so far described S, X, Y and Z as orbital states. The electron also carries spin, giving an up-spin and a down-spin component for each orbital. Simply doubling the basis would be straightforward if spin never interacted with the orbital motion. However, a semiconductor contains an additional interaction called spin-orbit coupling.

In the valence band this coupling mixes the orbital and spin components. At the Gamma point of an unstrained zincblende crystal, the six p-like spin-orbital states separate into a fourfold group, containing the heavy-hole and light-hole states, and a twofold group called the split-off band. The energy separation is denoted by \(\Delta_{\mathrm{so}}\).

The names heavy hole and light hole refer to the different curvatures of these bands near Gamma, and hence their different effective masses. The bands need not remain independent once the wavevector changes. In a quantum well, confinement and strain can mix them further, which is one of the main reasons to use the full multiband model.

With the valence-band maximum chosen as the reference energy, the average unperturbed valence energy in the Cartesian representation is often written:

\[ E_{v,\mathrm{av}}=E_v-\frac{\Delta_{\mathrm{so}}}{3}. \]

The spin-orbit matrix then separates the upper valence states from the split-off states, reproducing their required energies at Gamma. OghmaNano retains this coupling explicitly rather than attempting to model all of the valence bands with one hole effective mass.

Putting the Eight-Band Hamiltonian Together

We can now see how the different pieces fit together. There are two S-like conduction states (one for each spin) and six p-like valence states. The Hamiltonian is divided into four blocks:

\[ \boxed{ H=\begin{pmatrix} H_{cc} & H_{cv}\\[3pt] H_{cv}^{\dagger} & H_{vv} \end{pmatrix} } \]

The conduction block \(H_{cc}\) contains the conduction-band edge and its kinetic energy. The valence block \(H_{vv}\) contains the X/Y/Z orbital terms and spin-orbit interaction. The coupling block \(H_{cv}\) contains the Kane interactions connecting the conduction and valence states. The symbol \(\dagger\) denotes the Hermitian conjugate, ensuring that the Hamiltonian has the required Hermitian structure.

Although the full matrix has eight rows and eight columns for a uniform bulk material at a specified wavevector, its individual entries are now less mysterious: each entry either assigns an energy to a basis state or describes an interaction between two states. We solve the matrix eigenvalue problem to obtain the permitted band energies and the composition of their wavefunctions.

Where Do the Model Parameters Come From?

Experimental and theoretical semiconductor data are usually expressed in terms of the band gap \(E_g\), the conduction effective mass, the Kane energy \(E_p\), the spin-orbit splitting \(\Delta_{\mathrm{so}}\), and the Luttinger parameters \(\gamma_1\), \(\gamma_2\) and \(\gamma_3\). These quantities determine the model's diagonal dispersions and the strengths of the interactions between states.

There is an important bookkeeping issue. In the parabolic model, the measured effective mass already includes the influence of neighbouring bands. In an eight-band model, some of those neighbouring bands are now included explicitly through Kane coupling. We must therefore adjust the remaining kinetic-energy coefficients so that the same physical interaction is not counted twice.

In OghmaNano's parameterisation, the conduction remote-band coefficient is denoted \(A_c\), and the valence orbital coefficients are written L, M and N. The conversion from measured material parameters is part of the model definition; the resulting coefficients should not be confused with the bare effective masses of independent parabolic bands.

The Conduction-Band Correction

One expression used for the conduction remote-band contribution is:

\[ A_c=A_0\left( \frac{m_0}{m_e^{*}} -\frac{2E_p}{3E_g} -\frac{E_p}{3(E_g+\Delta_{\mathrm{so}})} \right), \]

with \(A_0=\hbar^2/(2m_0)\). The first term sets the measured band-edge curvature, while the two terms involving \(E_p\) remove the part produced by the valence states that are now coupled explicitly. Here \(m_e^{*}\) is a mass with SI dimensions; if a material database stores the relative mass \(m_e^{*}/m_0\), its reciprocal is used instead.

Similar corrections enter L, M and N through the Luttinger parameters. Exact parameter conventions must be kept consistent with the chosen basis and Hamiltonian; mixing coefficients taken from different conventions can give incorrect band curvature even when the matrix dimensions appear correct.

What Changes When We Put the Model Inside a Quantum Well?

Everything so far has concerned a semiconductor at a specified three-dimensional crystal wavevector. For a quantum well, we again choose \(z\) perpendicular to the layers and \(x\) and \(y\) parallel to them. Carriers can move freely along the layers, so \(k_x\) and \(k_y\) remain ordinary in-plane wavevector components.

Across the layers, however, we must solve for the confined wavefunctions. The z component is consequently represented by a differential operator:

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

This is the multiband extension of the Schrödinger-equation approach introduced on the previous pages. Instead of solving for just one scalar envelope function, we solve for eight coupled envelope components:

\[ \boldsymbol{\Psi}(z)= \begin{pmatrix} \psi_{S\uparrow}(z)\\ \psi_{S\downarrow}(z)\\ \psi_{X\uparrow}(z)\\ \psi_{Y\uparrow}(z)\\ \psi_{Z\uparrow}(z)\\ \psi_{X\downarrow}(z)\\ \psi_{Y\downarrow}(z)\\ \psi_{Z\downarrow}(z) \end{pmatrix}. \]

Each numerical grid point across the well carries eight coupled values. With N spatial grid points, the discretised Hamiltonian therefore has dimension \(8N\). Solving it gives the confined energies and the orbital and spin composition of every calculated subband. A complete example of this calculation is given in the GaAs/AlGaAs eight-band zincblende k·p quantum-well tutorial.

Material parameters can also change across an interface. When coefficients multiply differential operators, their ordering must be chosen carefully to preserve Hermiticity and the correct interface behaviour. For example, a spatially varying Kane coefficient is commonly introduced through a symmetrised operator rather than treating it as a constant outside the derivative.

Where Does Strain Enter?

When one semiconductor layer is grown on another with a different lattice constant, the layer may be compressed or stretched. This produces strain, which changes the relative energies of the conduction and valence states.

The conduction band experiences a hydrostatic energy shift, while the valence bands can move relative to one another and become mixed. In zincblende models these effects are commonly represented using the Bir–Pikus strain Hamiltonian. Adding this term to the eight-band Hamiltonian lets us calculate how confinement and strain act together, rather than correcting the bands independently after solving the quantum well.

Why Does This Matter for Optical Calculations?

The optical-transitions tutorial explained how the electron and hole subbands determine possible photon energies. In the parabolic approximation, transition strengths can be estimated using the overlap of two scalar envelope functions and a bulk optical matrix element.

In the eight-band model, each state is instead a mixture of S, X, Y and Z components. Its optical response depends on these components and on the polarisation of the light. This is why a multiband calculation can predict changes in transition strength and TE/TM response that are absent from a single-mass description.

The central improvement is not simply that we calculate more curves. We calculate what the states are made of, including how their orbital character changes with wavevector, confinement and strain.