The 8-band zincblende k·p model

The 8-band zincblende model is the workhorse for arsenide, phosphide and antimonide quantum wells such as GaAs/AlGaAs, InGaAs/GaAs and InAs/GaSb. It keeps the s-like conduction band and the three p-like valence bands explicitly, coupled by the Kane momentum matrix element, and therefore reproduces conduction-band non-parabolicity, heavy-/light-hole mixing, the split-off band and strain effects within a single Hermitian Hamiltonian. This page documents the form that OghmaNano actually assembles.

1. Basis and matrix size

OghmaNano works in the Cartesian orbital basis

\[ \{\,|S\!\uparrow\rangle,\ |S\!\downarrow\rangle,\ |X\!\uparrow\rangle,\ |Y\!\uparrow\rangle,\ |Z\!\uparrow\rangle,\ |X\!\downarrow\rangle,\ |Y\!\downarrow\rangle,\ |Z\!\downarrow\rangle\,\}, \]

that is, one s-like conduction orbital and three p-like valence orbitals, each with two spins: eight states per spatial node. After discretising the growth direction on \(N\) nodes, the Hamiltonian is a complex Hermitian sparse matrix of dimension \(8N\). Choosing the Cartesian \(|X\rangle,|Y\rangle,|Z\rangle\) orbitals rather than the \(|J,m_j\rangle\) combinations keeps the Kane and strain couplings simple and real; spin-orbit coupling then appears as an explicit off-diagonal term.

2. Block structure

The Hamiltonian divides into a conduction block, a valence block and their Kane coupling,

\[ H = \begin{pmatrix} H_{cc} & H_{cv} \\[2pt] H_{cv}^\dagger & H_{vv} \end{pmatrix}. \]

Each block is developed below in the terms OghmaNano uses. Throughout, \(\hbar^2/2m_0\) is written \(A_0\); \(E_g\) is the gap, \(E_v\) the valence-band edge (the energy zero is the structure valence-band maximum), \(\Delta_{\text{so}}\) the spin-orbit splitting, \(m_e\) the band-edge electron mass, \(E_p\) the Kane energy, and \(\gamma_1,\gamma_2,\gamma_3\) the Luttinger parameters.

3. Conduction block

The conduction edge is placed at \(E_c = E_v + E_g\). Its dispersion carries a remote-band effective-mass parameter \(A_c\) that has had the explicit coupling to the valence band removed, so that the total curvature (explicit Kane coupling plus \(A_c\)) reproduces the measured mass:

\[ A_c = A_0\left(\frac{1}{m_e} - \frac{2 E_p}{3 E_g} - \frac{E_p}{3(E_g+\Delta_{\text{so}})}\right), \qquad A_0 = \frac{\hbar^2}{2 m_0}. \]

The conduction diagonal is then \(H_{cc} = E_c + a_c\,\mathrm{Tr}(\varepsilon) + A_c\,k_\parallel^2 + A_c\,k_z^2\), where the last term is discretised as the symmetric second derivative and \(a_c\,\mathrm{Tr}(\varepsilon)\) is the hydrostatic strain shift.

4. Kane conduction–valence coupling

The Kane matrix element couples the s-like conduction state to the p-like valence states. In the units used internally,

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

The coupling is \(iP\,k_j\) between \(|S\rangle\) and \(|j\rangle\) for \(j = X,Y,Z\) and the same spin. The in-plane parts (\(k_x, k_y\)) are local, while the \(k_z\) part becomes the symmetrised derivative \(\tfrac{1}{2}(P\,d/dz + d/dz\,P)\), so that the operator remains anti-Hermitian and the assembled matrix stays Hermitian across interfaces where \(P\) changes. This explicit coupling is what ties the conduction non-parabolicity to the valence structure.

5. Valence block: the L, M, N parameters

The valence kinetic terms use the Dresselhaus \(L, M, N\) parameters, written so that the free- electron term \(A_0 k^2\) is added back separately and the coupling already included through the Kane term is subtracted out:

\[ \begin{aligned} M &= A_0\,(-\gamma_1 + 2\gamma_2 - 1), \\ L &= A_0\,(-\gamma_1 - 4\gamma_2 - 1) + A_0\,\frac{E_p}{E_g}, \\ N &= A_0\,(-6\gamma_3) + A_0\,\frac{E_p}{E_g}. \end{aligned} \]

The \(+A_0 E_p/E_g\) terms in \(L\) and \(N\) remove the valence-band coupling to the conduction band that is now treated explicitly by the Kane term; \(M\) is not renormalised because the \(|X\rangle\) orbital couples to \(|S\rangle\) only through \(p_x\), which contributes to \(L\) and \(N\) but not \(M\). Including the free-electron term, the diagonal valence entries are

\[ \begin{aligned} H_{XX} &= E_{v,\text{av}} + A_0 k^2 + L k_x^2 + M k_y^2 + M k_z^2, \\ H_{YY} &= E_{v,\text{av}} + A_0 k^2 + M k_x^2 + L k_y^2 + M k_z^2, \\ H_{ZZ} &= E_{v,\text{av}} + A_0 k^2 + M k_x^2 + M k_y^2 + L k_z^2, \end{aligned} \]

with the off-diagonal valence couplings \(H_{XY} = N k_x k_y\), \(H_{XZ} = N k_x k_z\) and \(H_{YZ} = N k_y k_z\). The average valence energy is \(E_{v,\text{av}} = E_v - \Delta_{\text{so}}/3\); the spin-orbit interaction below raises the highest valence states back to \(E_v\) at \(\Gamma\).

6. Spin-orbit coupling and strain

Spin-orbit coupling enters as an off-diagonal term of magnitude \(\Delta_{\text{so}}/3\) that couples the \(X, Y, Z\) orbitals with imaginary factors and spin flips. It splits the six-fold p-manifold into the four-fold \(J=3/2\) heavy- and light-hole states at \(E_v\) and the two-fold \(J=1/2\) split-off states \(\Delta_{\text{so}}\) below. Strain adds the hydrostatic shift \(a_c\,\mathrm{Tr}(\varepsilon)\) to the conduction block and the full Bir–Pikus matrix, expressed in the same Cartesian \(X/Y/Z\) basis, to the valence block; the strain potential is spin-independent, so the same \(3\times3\) block is applied to both spins.

7. Why the multiband model matters

Compared with the parabolic model, the 8-band Hamiltonian adds four things that are essential for realistic devices. Conduction bands become non-parabolic, so higher subbands are correctly spaced. The valence subbands mix at finite \(k_\parallel\), so their dispersions are warped and non-parabolic rather than simple parabolas. Strain is included consistently in both bands. And the optical matrix elements acquire the correct polarisation dependence through the S–X, S–Y and S–Z envelope overlaps rather than a single scalar overlap. OghmaNano solves the conduction and valence states with the shared shift-invert eigensolver, targeting the conduction edge and the valence-band maximum separately.