Understanding the 8-Band Wurtzite k·p Model

Why Do We Need a Different Model for Wurtzite?

In the eight-band zincblende tutorial, we allowed the conduction and valence states to interact rather than treating each one as an independent parabola. We now want to apply the same approach to materials such as GaN, AlN, InN and their alloys, which are widely used in LEDs and lasers.

These materials commonly have the wurtzite crystal structure. Unlike cubic zincblende, its hexagonal symmetry distinguishes one direction: the crystal's c-axis. Throughout this page we take this to be the \(z\) direction, with \(x\) and \(y\) lying in the perpendicular basal plane. This distinction matters because the electronic energy, the mixing of orbitals and the interaction with light can depend on direction.

We will introduce the physical reason for each new term first, then show how the eight-band Hamiltonian describes it. The result will connect directly to quantum-well subbands and to the polarisation of emitted light.

The c-Axis and Crystal-Field Splitting

Consider the three p-like orbital states used to represent the upper valence bands. We label them \(X\), \(Y\) and \(Z\) according to their orbital character. They are electronic basis functions, not three different locations in a device. In wurtzite the \(X\) and \(Y\) states are symmetry-related, whereas \(Z\) is distinguished by the c-axis.

This allows the crystal potential to shift the energy of the \(Z\)-like state relative to the \(X,Y\)-like states, even before spin is included. The effect is called crystal-field splitting, illustrated in ??. Spin-orbit coupling subsequently mixes orbital and spin character and produces three valence doublets, often called the A, B and C bands.

Their orbital character matters. A band containing mostly in-plane \(X,Y\) character generally couples differently to polarised light from one containing substantial c-axis \(Z\) character. This is why crystal-field splitting is not merely a small correction to the band gap; it can affect the optical properties of nitride quantum wells.

Where Do the Eight Bands Come From?

As in zincblende, we retain one predominantly s-like conduction orbital, \(S\), and three predominantly p-like valence orbitals, \(X\), \(Y\) and \(Z\). Each orbital has two spin components, giving eight basis states:

\[ \begin{aligned} \{&|S\uparrow\rangle,\ |S\downarrow\rangle,\\ &|X\uparrow\rangle,\ |Y\uparrow\rangle,\ |Z\uparrow\rangle,\\ &|X\downarrow\rangle,\ |Y\downarrow\rangle,\ |Z\downarrow\rangle\}. \end{aligned} \]

Instead of solving eight independent equations, we put these states into a single Hamiltonian matrix. Its diagonal entries describe the energies associated with the individual basis states; off-diagonal entries describe how they interact. It is useful to divide the matrix into conduction, valence and coupling blocks:

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

The conduction block contains two spin states, the valence block contains six, and \(H_{cv}\) couples them. The symbol \(\dagger\) denotes the Hermitian conjugate. Once we diagonalise this matrix, we obtain energies and eigenvectors whose conduction, valence, orbital and spin components are allowed to mix.

Why Are There Two Conduction-Band Masses?

In the parabolic-band tutorial, the curvature of a band was controlled by an effective mass. Wurtzite makes us distinguish motion along the c-axis from motion in the basal plane. Using \(k_{\parallel}^2=k_x^2+k_y^2\), a simple directional dispersion can be written as:

\[ E_c(\mathbf{k})=E_c+ \frac{\hbar^2 k_{\parallel}^2}{2m_{e,\parallel}^{*}} +\frac{\hbar^2 k_z^2}{2m_{e,z}^{*}}. \]

The electron can therefore gain energy at different rates depending on whether its motion is parallel or perpendicular to the layers. This is especially relevant for a c-plane quantum well, where confinement occurs along \(z\) but the subbands are plotted against \(k_{\parallel}\).

Within the multiband Hamiltonian OghmaNano uses two conduction parameters, \(S_1\) for the c-axis and \(S_2\) for the basal plane:

\[ H_{cc}=E_c+A_0(S_2k_{\parallel}^2+S_1k_z^2), \qquad A_0=\frac{\hbar^2}{2m_0}. \]

These parameters describe the conduction block's residual curvature. The final band-edge masses also depend on coupling to the valence states, which we include explicitly below. Consequently the parameters must be chosen consistently so that the same interband contribution is not counted twice.

Why Does Wurtzite Need Two Kane Parameters?

In the introduction to k·p theory, we encountered momentum matrix elements that connect different orbital states. For example, an s-like conduction state couples to the three p-like valence states. The strengths of these interactions are known as Kane parameters.

Wurtzite requires separate coupling strengths for the c-axis and basal plane:

• \(P_2\): coupling between \(S\) and the in-plane \(X,Y\) orbitals through \(k_x,k_y\).
• \(P_1\): coupling between \(S\) and the c-axis \(Z\) orbital through \(k_z\).

In the convention used here the representative Hamiltonian elements are:

\[ \begin{aligned} H_{SX}&=iP_2k_x,\\ H_{SY}&=iP_2k_y,\\ H_{SZ}&=iP_1k_z. \end{aligned} \]

Their Hermitian conjugates appear on the other side of the matrix. The parameters are often specified through the Kane energies:

\[ P_1=\sqrt{A_0E_{p1}},\qquad P_2=\sqrt{A_0E_{p2}}. \]

Having two couplings means that electronic states respond differently to interactions along and perpendicular to the c-axis. It will also give us a direct explanation for TE and TM optical selection rules later.

Why Do We Need Six Valence Parameters?

The conduction band began with a single s-like orbital for each spin. The valence sector contains three p-like orbitals per spin, closely spaced in energy. As wavevector changes, these states can mix instead of following three independent parabolas.

The wurtzite model describes the directional curvature and mixing of these states using six commonly tabulated parameters, \(A_1\) to \(A_6\), known as the Rashba–Sheka–Pikus parameters. Some determine how the band energies change with motion along or across the c-axis; others control coupling between the orbital components.

Suppressing spin and other terms temporarily, the wavevector-dependent Cartesian valence block has the schematic form:

\[ H_v^{\mathrm{kin}}= \begin{pmatrix} H_{XX}&N_1k_xk_y&N_2k_xk_z\\ N_1k_xk_y&H_{YY}&N_2k_yk_z\\ N_2k_xk_z&N_2k_yk_z&H_{ZZ} \end{pmatrix}. \]

The diagonal entries describe predominantly \(X\), \(Y\) and \(Z\) character. The off-diagonal entries mix them when the relevant wavevector components are present. This mixing is one reason why the calculated valence subbands cannot generally be replaced by three simple parabolas.

How OghmaNano Uses the A1–A6 Parameters

The following definitions document the Cartesian coefficient convention given in the existing OghmaNano wurtzite model. We first form:

\[ \begin{aligned} L_1&=A_0(A_5+A_4+A_2-1),\\ L_2&=A_0(A_1-1),\\ M_1&=A_0(A_4+A_2-A_5-1),\\ M_2&=A_0(A_1+A_3-1),\\ M_3&=A_0(A_2-1),\\ N_1&=2A_0A_5,\\ N_2&=\sqrt{2}\,A_0A_6. \end{aligned} \]

With the free-electron contribution \(A_0k^2\) written separately, the diagonal valence entries are:

\[ \begin{aligned} H_{XX}&=E_{v,\mathrm{av}}+A_0k^2+L_1k_x^2+M_1k_y^2+M_2k_z^2,\\[3pt] H_{YY}&=E_{v,\mathrm{av}}+A_0k^2+M_1k_x^2+L_1k_y^2+M_2k_z^2,\\[3pt] H_{ZZ}&=E_{v,\mathrm{av}}+A_0k^2+M_3k_{\parallel}^2+L_2k_z^2. \end{aligned} \]

The off-diagonal elements are \(H_{XY}=N_1k_xk_y\), \(H_{XZ}=N_2k_xk_z\) and \(H_{YZ}=N_2k_yk_z\), with the Hermitian counterparts understood. The reference \(E_{v,\mathrm{av}}\) is chosen consistently with the crystal-field and spin-orbit terms. Parameter conventions vary across publications, so tabulated numbers should not be transferred between Hamiltonian conventions without checking where free-electron and interband contributions have been included.

Adding Spin: The Three Valence Doublets

We can now return to the crystal-field diagram. In the OghmaNano convention the relevant splitting parameters are denoted \(\delta_1\), \(\delta_2\) and \(\delta_3\). The crystal-field term \(\delta_1\) distinguishes the basal orbitals from the c-axis orbital. The remaining two parameters describe spin-orbit coupling, which mixes orbital and spin character.

Diagonalising the valence block at the zone centre produces three closely spaced, two-fold-degenerate valence levels (A, B and C). Their ordering is determined by the material's parameters rather than by an assumption that one particular orbital must always be the highest. OghmaNano sets its valence energy reference so that the highest resulting zone-centre valence level agrees with the chosen valence-band edge \(E_v\).

This matters optically: the character of the uppermost valence level determines which polarisation can couple most strongly to the lowest-energy interband transitions.

What Does Strain Do to the Wurtzite Bands?

A quantum-well layer is normally grown on another semiconductor. If the materials have different natural lattice dimensions, a thin layer may be forced to match the in-plane spacing of its neighbours. This produces strain, which shifts and mixes the electronic states.

In a c-plane structure the in-plane strain components can be similar to one another, while the strain along the c-axis is different. Wurtzite therefore needs directional deformation potentials. OghmaNano uses \(a_1,a_2\) for the conduction-band contribution and \(D_1\)–\(D_6\) for the valence-band strain matrix.

In the convention of the existing model documentation, the conduction shift is written:

\[ \Delta E_c=a_z\varepsilon_{zz} +a_{xy}(\varepsilon_{xx}+\varepsilon_{yy}), \quad a_z=a_1+D_1,\quad a_{xy}=a_2+D_2. \]

The valence strain coefficients are formed from combinations of the \(D_i\), analogous to the directional kinetic coefficients:

\[ \begin{aligned} l_1&=D_5+D_4+D_2,&l_2&=D_1,\\ m_1&=D_4+D_2-D_5,&m_2&=D_1+D_3,\\ m_3&=D_2. \end{aligned} \]

For ideal pseudomorphic c-plane growth without shear strain, the terms proportional to shear components vanish. They remain necessary for more general growth orientations and strain fields. The important outcome is that strain can change the band gap and reorder the upper valence states, with consequences for optical polarisation.

From Orbital Character to TE and TM Optical Transitions

The optical-transitions tutorial explained how the energy difference between a conduction and valence state sets the photon energy. But energy conservation alone does not tell us how strongly a transition couples to light. The orientation of the light's electric field matters too.

For the usual c-plane quantum-well convention, we distinguish:

• TE polarisation: electric field within the basal plane, perpendicular to the c-axis. It primarily probes the \(X,Y\) valence character through the \(P_2\) Kane coupling.
• TM polarisation: electric field along the c-axis. It probes the \(Z\) valence character through \(P_1\).

For a light polarisation direction \(\mathbf e\), the optical transition amplitude involves the momentum matrix element:

\[ \mathcal{M}_{cv}(\mathbf e)= \langle\Psi_c|\mathbf e\cdot\mathbf p|\Psi_v\rangle, \qquad \text{transition strength}\propto |\mathcal{M}_{cv}(\mathbf e)|^2. \]

The matrix element includes both the appropriate directional Kane coupling and the overlap of the electron and hole envelope components. This explains why the same quantum well can produce different TE and TM optical strengths, even for transitions between the same energy levels.

We can now see why crystal-field splitting, spin-orbit coupling and strain were worth introducing first: they determine the orbital mixture of the valence states, which determines how those states couple to differently polarised light. This is important when modelling nitride LEDs and lasers. The resulting TE- and TM-polarised transitions can be explored in the GaN/AlGaN eight-band wurtzite k·p quantum-well tutorial.

How Is the Model Applied to a Quantum Well?

So far we have used the wavevector components \(k_x,k_y,k_z\) to describe bulk bands. In a quantum well grown along the c-axis, electrons and holes are confined along \(z\), but can still move within the \(x\)-\(y\) plane. As in the zincblende model, the growth- direction wavevector becomes a differential operator:

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

The eight-band Hamiltonian is therefore solved for eight coupled envelope components across the layer stack, at selected in-plane wavevectors. The potential, effective-mass parameters, Kane coupling and strain can all change between the well and barrier materials. Position-dependent coefficients and derivative terms need a consistent Hermitian interface discretisation.

On a grid with \(N_z\) points, there are eight components at each point, giving a coupled numerical problem with \(8N_z\) unknowns. Its eigenvalues provide the confined states; the eigenvectors describe the spatial, orbital and spin content of those states. Repeating the solution across in-plane wavevectors produces the electronic subbands. These are then used to calculate polarisation-dependent optical transitions.

What Has the Eight-Band Model Added?

The parabolic model is a useful starting point for quantum confinement, but it cannot describe coupling between closely spaced valence states or the full directional optical response of wurtzite. The eight-band model retains the same overall picture of confined states and subbands, while adding the physics needed for the hexagonal crystal structure.