The six-band Luttinger–Kohn valence model

The six-band Luttinger–Kohn (LK) model describes the valence bands of diamond and zincblende semiconductors — heavy hole, light hole and split-off — without treating the conduction band explicitly. It is the natural valence model for silicon, germanium and SiGe, where the conduction physics is handled separately by the Δ-valley solver. OghmaNano implements the LK Hamiltonian in the total-angular-momentum basis with the classic \(P, Q, R, S\) structure.

1. Basis

The model uses the \(J=3/2\) and \(J=1/2\) hole states, ordered as

\[ \{\,\text{HH}\!\uparrow,\ \text{LH}\!\uparrow,\ \text{LH}\!\downarrow,\ \text{HH}\!\downarrow,\ \text{SO}\!\uparrow,\ \text{SO}\!\downarrow\,\}, \]

so on an \(N\)-node grid the Hamiltonian is a \(6N\) complex Hermitian matrix. In this basis spin-orbit coupling is diagonal — the split-off states simply sit \(\Delta_{\text{so}}\) below the \(J=3/2\) states — at the price of the kinetic terms acquiring the \(P, Q, R, S\) structure below.

2. Kinetic terms

With \(A_0=\hbar^2/2m_0\) and the Luttinger parameters \(\gamma_1,\gamma_2,\gamma_3\), OghmaNano computes four kinetic quantities:

\[ \begin{aligned} P_k &= -A_0\,\gamma_1\,(k_x^2+k_y^2+k_z^2), \\ Q_k &= -A_0\,\gamma_2\,(k_x^2+k_y^2-2k_z^2), \\ R_k &= A_0\,\sqrt{3}\,\big[\gamma_2(k_x^2-k_y^2) - 2i\,\gamma_3\,k_x k_y\big], \\ S_k &= 2\sqrt{3}\,A_0\,\gamma_3\,(k_x - i k_y)\,k_z. \end{aligned} \]

\(P_k\) is the average (isotropic) kinetic energy, \(Q_k\) the tetragonal splitting between heavy and light holes, \(R_k\) an in-plane warping term and \(S_k\) a term that couples the bands at finite \(k_z\). In the quantum well \(k_z\to -i\,d/dz\): the \(k_z^2\) parts of \(P_k, Q_k\) become symmetric second derivatives, and the linear \(k_z\) in \(S_k\) becomes a symmetrised first derivative, exactly as on the discretisation page.

3. Adding strain

The Bir–Pikus strain terms have the same \(P, Q, R, S\) labels and are simply added to the kinetic ones,

\[ P = P_k + P_\varepsilon,\quad Q = Q_k + Q_\varepsilon,\quad R = R_k + R_\varepsilon,\quad S = S_k + S_\varepsilon, \]

with \(P_\varepsilon=a_v\mathrm{Tr}\varepsilon\), \(Q_\varepsilon=\tfrac12 b(\varepsilon_{xx}+ \varepsilon_{yy}-2\varepsilon_{zz})\), and the shear \(R_\varepsilon, S_\varepsilon\) as given on the Bir–Pikus page. This is the elegance of the LK form: kinetic and strain physics share the same four building blocks.

4. The 6×6 Hamiltonian

The building blocks assemble into the Hamiltonian (measuring from the valence edge \(E_v\), with \(\Delta\equiv\Delta_{\text{so}}\)):

\[ H = E_v\,\mathbb{1} + \begin{pmatrix} P{+}Q & S & R & 0 & -\tfrac{S}{\sqrt2} & -\sqrt2\,R \\ S^\dagger & P{-}Q & 0 & R & \sqrt2\,Q & \sqrt{\tfrac32}\,S \\ R^\dagger & 0 & P{-}Q & -S & \sqrt{\tfrac32}\,S^\dagger & -\sqrt2\,Q \\ 0 & R^\dagger & -S^\dagger & P{+}Q & \sqrt2\,R^\dagger & -\tfrac{S^\dagger}{\sqrt2} \\ -\tfrac{S^\dagger}{\sqrt2} & \sqrt2\,Q & \sqrt{\tfrac32}\,S & \sqrt2\,R & P{-}\Delta & 0 \\ -\sqrt2\,R^\dagger & \sqrt{\tfrac32}\,S^\dagger & -\sqrt2\,Q & -\tfrac{S}{\sqrt2} & 0 & P{-}\Delta \end{pmatrix}. \]

The upper-left \(4\times4\) block describes the coupled heavy and light holes; the lower-right \(2\times2\) block is the split-off doublet lying \(\Delta\) lower; and the off-diagonal blocks couple them through \(Q, R, S\). At \(k_\parallel=0\) and zero strain, \(R=S=0\) and \(Q\) contains only \(k_z^2\), so the heavy and light holes decouple — the familiar quantum-well limit. At finite \(k_\parallel\) the \(R\) and \(S\) terms mix them, producing the warped, non-parabolic valence subbands that the parabolic model cannot represent.