The 8-band wurtzite k·p model

Wurtzite nitrides (GaN, AlN, InN and their alloys) are the basis of blue and ultraviolet light-emitting devices. Their hexagonal symmetry singles out the c-axis, so the band structure is anisotropic and the valence bands are split by a crystal field even without spin- orbit coupling. The OghmaNano 8-band wurtzite model captures this with anisotropic conduction masses, two Kane parameters, six valence parameters and three splitting energies.

1. Basis and energy reference

The basis is the same Cartesian set as the zincblende model, \(\{|S\!\uparrow\rangle,|S\!\downarrow\rangle,|X\!\uparrow\rangle,\dots,|Z\!\downarrow\rangle\}\), giving an \(8N\) matrix, but here the c-axis is taken along \(Z\) (the growth direction). The valence-band edge input \(E_v\) is placed at the highest valence eigenvalue at \(\Gamma\). Because the crystal-field and spin-orbit terms shift the diagonal, OghmaNano subtracts the computed \(\Gamma\)-point valence maximum so that the topmost state sits exactly at \(E_v\); the average valence energy used on the diagonal is \(E_{v,\text{av}} = E_v - \Delta_{v}^{\Gamma}\), where \(\Delta_v^{\Gamma}\) is the larger of \(\delta_1+\delta_2\) and \(\tfrac{1}{2}(\delta_1-\delta_2)+\sqrt{\tfrac{1}{4}(\delta_1-\delta_2)^2+2\delta_3^2}\).

2. Anisotropic conduction band and Kane coupling

Hexagonal symmetry gives the conduction band different curvatures along and across the c-axis, described by two parameters \(S_1\) (c-axis) and \(S_2\) (basal plane):

\[ H_{cc} = E_c + A_0\left(S_2\,k_\parallel^2 + S_1\,k_z^2\right), \qquad A_0=\frac{\hbar^2}{2m_0}. \]

Similarly there are two Kane parameters. \(P_2\) couples the conduction state to the basal orbitals \(X, Y\), and \(P_1\) couples it to the c-axis orbital \(Z\):

\[ P_1 = \sqrt{A_0\,E_{p1}}, \qquad P_2 = \sqrt{A_0\,E_{p2}}, \]

appearing as \(iP_2 k_x\) (S–X), \(iP_2 k_y\) (S–Y) and \(iP_1 k_z\) (S–Z), with the \(k_z\) part symmetrised as usual. This split into c-axis and basal coupling is the origin of the strong TE/TM polarisation anisotropy discussed below.

3. Valence parameters from A1–A6

The wurtzite valence kinetic terms follow the Rashba–Sheka–Pikus form and are built from the six parameters \(A_1\dots A_6\). OghmaNano forms the combinations

\[ \begin{aligned} L_1 &= A_0(A_5+A_4+A_2-1), & L_2 &= A_0(A_1-1), \\ M_1 &= A_0(A_4+A_2-A_5-1), & M_2 &= A_0(A_1+A_3-1), \\ M_3 &= A_0(A_2-1), & N_1 &= 2A_0 A_5, \quad N_2 = \sqrt{2}\,A_0 A_6, \end{aligned} \]

and assembles the valence diagonal (with the free-electron term \(A_0 k^2\) included) as

\[ \begin{aligned} H_{XX} &= E_{v,\text{av}} + A_0 k^2 + L_1 k_x^2 + M_1 k_y^2 + M_2 k_z^2, \\ H_{YY} &= E_{v,\text{av}} + A_0 k^2 + M_1 k_x^2 + L_1 k_y^2 + M_2 k_z^2, \\ H_{ZZ} &= E_{v,\text{av}} + A_0 k^2 + M_3 k_\parallel^2 + L_2 k_z^2, \end{aligned} \]

with off-diagonal couplings \(H_{XY}=N_1 k_x k_y\), \(H_{XZ}=N_2 k_x k_z\) and \(H_{YZ}=N_2 k_y k_z\). The distinct \(A_1/A_2\) (c-axis versus basal) reflect the hexagonal anisotropy of the valence bands.

4. Crystal-field and spin-orbit splitting

Two effects that are absent or degenerate in zincblende are essential here. The crystal field \(\delta_1\) raises the basal \(X, Y\) orbitals relative to the c-axis \(Z\) orbital even without spin-orbit coupling. Spin-orbit coupling then enters through two further parameters: \(\delta_2\), which couples \(X\) and \(Y\) within the same spin, and \(\delta_3\), which couples the orbitals across spin. Together \(\delta_1, \delta_2, \delta_3\) split the top of the valence band into three closely spaced doublets, conventionally labelled heavy hole, light hole and crystal-field/split-off hole. The near-degeneracy of these bands is why nitride valence physics is genuinely multiband.

5. Strain

Strain is built from the hydrostatic deformation potentials \(a_1, a_2\) for the conduction band and the shear deformation potentials \(D_1\dots D_6\) for the valence band. OghmaNano forms the conduction-band shift \(a_z\varepsilon_{zz} + a_{xy}(\varepsilon_{xx}+\varepsilon_{yy})\) with \(a_z = a_1 + D_1\) and \(a_{xy} = a_2 + D_2\), and valence-band shifts using the combinations \(l_1=D_5+D_4+D_2\), \(l_2=D_1\), \(m_1=D_4+D_2-D_5\), \(m_2=D_1+D_3\) and \(m_3=D_2\), exactly mirroring the kinetic combinations. For pseudomorphic c-plane growth the shear strains vanish, so \(D_6\) (the shear deformation potential) does not contribute; it is retained for completeness.

6. TE and TM optical polarisation

Because the c-axis and basal Kane couplings differ, the optical matrix elements are strongly polarisation dependent. OghmaNano computes two separate quantities from the conduction– valence envelope overlaps,

\[ M^2_{\text{TE}} = \frac{|M_x|^2 + |M_y|^2}{2\,P_2^2}, \qquad M^2_{\text{TM}} = \frac{|M_z|^2}{P_1^2}, \]

where \(M_x, M_y\) use \(P_2\) and the \(X, Y\) valence character, and \(M_z\) uses \(P_1\) and the \(Z\) valence character. TE light (electric field in the plane) therefore couples to the basal \(X, Y\) character, while TM light (field along the c-axis) couples to the \(Z\) character. Whichever valence doublet lies uppermost sets the polarisation of the band-edge emission; this is why strain and confinement, which reorder the valence doublets, change the TE/TM ratio of nitride lasers. The gain page uses these two matrix-element sets to produce separate TE and TM spectra.