Understanding the Six-Band Luttinger–Kohn Valence Model

Why Do We Need a Separate Model for Holes?

In the parabolic-band tutorial, we treated electrons and holes using separate parabolas. That is a useful starting point, but the upper valence bands of many semiconductors are considerably more complicated. Rather than a single hole band, we encounter several closely spaced bands that can interact with one another.

These are conventionally called the heavy-hole (HH), light-hole (LH) and split-off (SO) bands, illustrated in ??. The names heavy and light refer to their different band curvatures and therefore different effective masses; they do not mean that there are two different physical kinds of hole.

The six-band Luttinger–Kohn (LK) model describes these valence states together. It is particularly useful for silicon, germanium and SiGe quantum wells, where the conduction states can be treated using a separate silicon valley model. Unlike an eight-band model, the six-band LK model does not explicitly include the conduction band: it concentrates on the physics of holes.

Where Do the Six States Come From?

To understand the number six, it helps to start with the valence-band orbitals. Near the zone centre, the relevant states in diamond and zincblende semiconductors have three p-like orbital components, which we can label X, Y and Z. An electron also has two possible spin components. Three orbital components multiplied by two spin components therefore give six states.

The crystal's spin–orbit interaction couples orbital and spin angular momentum. It is then convenient to replace the separate orbital and spin labels with states of total angular momentum, written as J and its projection m_J. The six states separate into a group of four with J = 3/2 and a group of two with J = 1/2:

\[ \underbrace{4\text{ states}}_{J=3/2} \quad+\quad \underbrace{2\text{ states}}_{J=1/2} \quad=\quad 6\text{ states}. \]

The four J = 3/2 states form the heavy- and light-hole manifold at the zone centre. The remaining pair forms the split-off band, displaced downwards by the spin–orbit energy, \(\Delta_{\mathrm{so}}\). These are not six unrelated parabolas: as a carrier moves away from the zone centre, the different components can mix, which is why they need to be solved together.

Choosing the Six-Band Basis

OghmaNano uses the total-angular-momentum basis, with the following ordering of states:

\[ \begin{aligned} &\left|\tfrac32,+\tfrac32\right\rangle,\quad \left|\tfrac32,+\tfrac12\right\rangle,\quad \left|\tfrac32,-\tfrac12\right\rangle,\\[4pt] &\left|\tfrac32,-\tfrac32\right\rangle,\quad \left|\tfrac12,+\tfrac12\right\rangle,\quad \left|\tfrac12,-\tfrac12\right\rangle. \end{aligned} \]

The first and fourth states have heavy-hole character, the middle two states of the J = 3/2 group have light-hole character, and the last two states are split-off states. This ordering is useful because the spin–orbit splitting is already included in the basis: at the zone centre, the split-off pair sits \(\Delta_{\mathrm{so}}\) below the upper valence states.

We can therefore construct a 6 × 6 Hamiltonian, a matrix which describes the individual states and the interactions between them. The allowed energies are obtained by finding its eigenvalues, while the eigenvectors tell us the heavy-hole, light-hole and split-off character of each resulting state.

How Do We Describe the Motion of a Hole?

In a single parabolic band, one effective mass determines how quickly the energy changes with wavevector. Here we have several coupled valence states, so one mass is no longer enough. Instead, the Luttinger–Kohn model uses three material parameters, \(\gamma_1\), \(\gamma_2\) and \(\gamma_3\), to describe the curvature and mixing of the bands.

These parameters are collected into four convenient mathematical quantities, traditionally called \(P\), \(Q\), \(R\) and \(S\). Their physical roles are easier to understand before seeing their complete expressions:

• \(P\): the common contribution to the curvature of the valence states.
• \(Q\): distinguishes different orientations of the heavy- and light-hole states, and can separate their energies.
• \(R\): couples valence components through motion in the plane of the semiconductor layers.
• \(S\): couples states through a combination of in-plane motion and motion across the layers.

The underlying wavevector has three components: \(k_x\) and \(k_y\) describe motion parallel to the quantum-well layers, while \(k_z\) refers to the growth direction, perpendicular to them. We also define \(A_0=\hbar^2/(2m_0)\), where \(m_0\) is the free-electron mass.

In the sign convention used by the supplied OghmaNano formulation, the kinetic contributions are:

\[ \begin{aligned} P_k &= -A_0\gamma_1(k_x^2+k_y^2+k_z^2),\\[5pt] Q_k &= -A_0\gamma_2(k_x^2+k_y^2-2k_z^2),\\[5pt] R_k &= \sqrt{3}A_0\left[\gamma_2(k_x^2-k_y^2) -2i\gamma_3k_xk_y\right],\\[5pt] S_k &= 2\sqrt{3}A_0\gamma_3(k_x-ik_y)k_z. \end{aligned} \]

We do not need to memorise these expressions to understand the model. Notice instead that some terms depend on \(k_x\) and \(k_y\) differently from \(k_z\). This is the mathematics that allows the model to describe different heavy- and light-hole behaviour along different crystal directions. The imaginary factors in \(R\) and \(S\) are part of the chosen quantum basis; the complete Hamiltonian remains Hermitian and has real energies.

A Useful Example: A Hole Moving Along the Growth Direction

Before looking at the full matrix, consider the simple case where the in-plane wavevector is zero, so \(k_x=k_y=0\). The kinetic terms then reduce to:

\[ \begin{aligned} P_k &=-A_0\gamma_1k_z^2,\\[3pt] Q_k &=+2A_0\gamma_2k_z^2,\\[3pt] R_k &=0,\qquad S_k=0. \end{aligned} \]

Consequently, the heavy-hole diagonal contains \(P+Q\), while the light-hole diagonal contains \(P-Q\). Even without calculating the complete eigenvalues, we can see how different curvatures emerge:

\[ \begin{aligned} E_{\mathrm{HH}}(k_z)&=E_v-A_0(\gamma_1-2\gamma_2)k_z^2,\\[4pt] E_{\mathrm{LH,diag}}(k_z)&=E_v-A_0(\gamma_1+2\gamma_2)k_z^2. \end{aligned} \]

These expressions show the different curvature scales along the growth direction. There is an important qualification: the light-hole part of the full six-band Hamiltonian can still couple to the split-off states through \(Q\). Thus the second expression is a diagonal contribution, not in general the exact light-hole eigenvalue of the six-band model. This illustrates precisely why we need to solve a matrix rather than assuming each valence band is independently parabolic.

Putting the Six States Together

Now that the individual terms have a physical meaning, we can assemble the full Hamiltonian. Its rows and columns correspond, in order, to the six states introduced earlier. We measure the energies relative to the valence-band reference \(E_v\) and write \(\Delta=\Delta_{\mathrm{so}}\):

\[ \small H=E_v\mathbb{1}+ \begin{pmatrix} P+Q & S & R & 0 & -S/\sqrt2 & -\sqrt2 R\\ S^\dagger & P-Q & 0 & R & \sqrt2 Q & \sqrt{3/2}S\\ R^\dagger & 0 & P-Q & -S & \sqrt{3/2}S^\dagger & -\sqrt2 Q\\ 0 & R^\dagger & -S^\dagger & P+Q & \sqrt2 R^\dagger & -S^\dagger/\sqrt2\\ -S^\dagger/\sqrt2 & \sqrt2 Q & \sqrt{3/2}S & \sqrt2 R & P-\Delta & 0\\ -\sqrt2 R^\dagger & \sqrt{3/2}S^\dagger & -\sqrt2 Q & -S/\sqrt2 & 0 & P-\Delta \end{pmatrix}. \]

This looks much more complicated than a single Schrödinger equation, but its structure is straightforward. The first four rows and columns describe the heavy- and light-hole manifold. The last two describe the split-off pair. Entries on the diagonal describe the energies of individual components, while the off-diagonal terms mix them. The dagger symbol denotes the Hermitian conjugate of a term; for a scalar coefficient this means its complex conjugate.

At the zone centre and without strain, the kinetic terms vanish: the four J = 3/2 states share the valence-band edge and the split-off pair lies \(\Delta\) below it. Away from the zone centre, the mixing changes the energies and the character of the resulting bands.

What Happens When the Semiconductor Is Strained?

Silicon and germanium quantum wells are often grown on a substrate with a slightly different lattice constant. This introduces strain, which can change the valence energies and lift the degeneracy of the heavy- and light-hole states.

A useful feature of the Luttinger–Kohn formulation is that strain can be introduced using another set of \(P,Q,R,S\) terms. We simply combine the kinetic and strain contributions:

\[ \begin{aligned} P&=P_k+P_{\varepsilon}, &Q&=Q_k+Q_{\varepsilon},\\ R&=R_k+R_{\varepsilon}, &S&=S_k+S_{\varepsilon}. \end{aligned} \]

For example, the hydrostatic strain contribution responds to the overall change in volume, while the biaxial component distinguishes deformation along the growth direction from deformation in the plane:

\[ \begin{aligned} P_{\varepsilon}&=a_v(\varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz}),\\[4pt] Q_{\varepsilon}&=\frac{b}{2} (\varepsilon_{xx}+\varepsilon_{yy}-2\varepsilon_{zz}). \end{aligned} \]

The remaining strain components contribute to \(R_{\varepsilon}\) and \(S_{\varepsilon}\), allowing shear strain to mix the valence states. Their full expressions and deformation-potential conventions are covered on the Bir–Pikus strain page. We can then solve the same six-band matrix with the strain terms included.

Applying the Model to a Quantum Well

Until now, we have described the band structure using three wavevector components. In a quantum well, the carriers are confined along the growth direction \(z\), while remaining free to move along \(x\) and \(y\). The in-plane wavevectors are therefore retained as numbers, but motion along \(z\) must be represented by an operator:

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

All terms containing \(k_z\) become derivatives of the envelope wavefunctions. The six-band problem is then solved across the semiconductor layers, accounting for the potential profile, material parameters and strain. The result is a set of confined valence subbands, each with a six-component envelope wavefunction. These components tell us how much heavy-hole, light-hole and split-off character is present in each state.

If the growth-direction grid contains \(N_z\) spatial points, the discrete Hamiltonian has dimensions \(6N_z\) by \(6N_z\). OghmaNano uses this multiband calculation for the valence states while the silicon Δ-valley solver handles the conduction states separately. Unlike a simple parabolic treatment, this approach can describe valence-band mixing that changes with in-plane wavevector. A complete Si/SiGe calculation using this model is demonstrated in the six-band Luttinger–Kohn quantum-well tutorial.

How Does This Relate to the Eight-Band Model?

The six-band LK model and the eight-band zincblende k·p model describe related physics, but they make different choices about which states are included explicitly. Both retain the interacting valence states. The eight-band model also adds the two spin-resolved, s-like conduction states and their Kane coupling to the valence band.

In a semiconductor such as silicon, where the lowest conduction minima occur away from the zone centre, a zone-centred eight-band Hamiltonian is not a direct description of those six valleys. It is therefore useful to treat the valence band with the six-band LK model and the conduction band with the dedicated valley model.