Effective Mass and Valence Bands from k·p Theory

1. From Free Electrons to Electrons in a Crystal

An electron in the periodic potential of a crystal lattice is not free. Its behavior is governed by the Schrödinger equation with a periodic potential, V(r) = V(r + R), where R is a lattice vector. As you'll know from your solid-state physics courses, the solutions to this equation are not simple plane waves but Bloch waves:

${\psi }_{n,\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{n,\mathbf{k}}(\mathbf{r})$

Here, $u_{n,\mathbf{k}}(\mathbf{r})$ is a function with the same periodicity as the crystal lattice, n is the band index, and $\mathbf{k}$ is the crystal momentum. This leads to the formation of energy bands, $E_n(\mathbf{k})$.

For a good visual and conceptual recap of these ideas, please watch the following video.

As the video explains, for many applications, we are only interested in electrons near the bottom of the conduction band or holes near the top of thevalence band. In these regions, the E(k) dispersion relation can often be approximated by a parabola:

$E(\mathbf{k})\approx E_0+\frac{{\hslash }^2|\mathbf{k}{|}^2}{2m^{\ast }}$

This is the parabolic band approximation. The parameter $m^{\ast }$, the effective mass, encapsulates the influence of the crystal potential. It's defined by the curvature of the band:

$\frac{1}{m^{\ast }}=\frac{1}{{\hslash }^2}\frac{{\partial }^{2}E}{\partial k^2}$

A small effective mass (high curvature) means the particle responds readily to external forces, while a large effective mass (low curvature) means it is "heavier" and less mobile. This approximation allows us to treat electrons and holes as quasi-classical particles, but with a mass modified by the crystal environment.

2. Deriving Band Structure with k·p Theory

How do we systematically calculate the band structure E(k) and the resulting effective masses? The most common method is k·p perturbation theory. The name comes from a term in the Hamiltonian proportional to $\mathbf{k}\cdot \mathbf{p}$.

The goal is to find the energy E(k) for small k around a high-symmetry point (like the center of the Brillouin zone, k=0, also called the Γ-point), where we already know the energies E_n(0) and wavefunctions u_n,0(r).

To see how this works, let's derive the central equation of k·p theory.

As shown in the reading, substituting the Bloch wave into the Schrödinger equation gives an equation for the periodic part $u_{n,\mathbf{k}}$:

$\left(\frac{p^2}{2m_0}+V(\mathbf{r})+\frac{\hslash }{m_0}\mathbf{k}\cdot \mathbf{p}+\frac{{\hslash }^2{k}^2}{2m_0}\right)u_{n,\mathbf{k}}(\mathbf{r})=E_{n,\mathbf{k}}{u}_{n,\mathbf{k}}(\mathbf{r})$

Here, $m_0$ is the free electron mass. We can group the terms:

• $H_0=\frac{p^2}{2m_0}+V(\mathbf{r})$ is the Hamiltonian at k=0. Its eigenvalues are the known band-edge energies $E_{n,0}$.
• $H^{\prime }=\frac{\hslash }{m_0}\mathbf{k}\cdot \mathbf{p}+\frac{{\hslash }^2{k}^2}{2m_0}$ is the k-dependent perturbation.

We can now use perturbation theory to find E_n(k).

The 1-Band Model: The Effective Mass Tensor

Let's start with the simplest case: a non-degenerate band (like the conduction band in many direct-gap semiconductors). We can use second-order non-degenerate perturbation theory to find the energy.

The result from the reading is crucial. The energy up to second order in k is:
$E_{n,\mathbf{k}}\approx E_{n,0}+\frac{{\hslash }^2{k}^2}{2m_0}+\frac{{\hslash }^2}{m_0^2}{\sum }_{m\ne n}\frac{|⟨u_{n,0}|\mathbf{k}\cdot \mathbf{p}|u_{m,0}⟩{|}^2}{E_{n,0}-E_{m,0}}$
This can be written compactly as:
$E_{n,\mathbf{k}}\approx E_{n,0}+\frac{{\hslash }^2}{2}{\sum }_{\alpha ,\beta }{k}_{\alpha }{\left(\frac{1}{m^{\ast }}\right)}_{\alpha \beta }{k}_{\beta }$
where the inverse effective mass tensor is:
${\left(\frac{1}{m^{\ast }}\right)}_{\alpha \beta }=\frac{{\delta }_{\alpha \beta }}{m_0}+\frac{2}{m_0^2}{\sum }_{m\ne n}\frac{⟨u_{n,0}|p_{\alpha }|u_{m,0}⟩⟨u_{m,0}|p_{\beta }|u_{n,0}⟩}{E_{n,0}-E_{m,0}}$

This equation beautifully illustrates the physical origin of effective mass: it arises from the k·p coupling between different bands. The strength of this coupling is determined by the momentum matrix elements $⟨u_n|p|u_m⟩$ and the energy separation between the bands. A nearby band with strong coupling can significantly alter the effective mass from the free electron mass $m_0$.

3. Valence Band Structure: Heavy, Light, and Split-Off Holes

The situation is more complex for the valence bands of most common semiconductors (like GaAs, Si, Ge). At the Γ-point (k=0), the valence band is formed from atomic p-orbitals (p_x, p_y, p_z). Including electron spin, this results in a six-fold degeneracy. This degeneracy is lifted by two effects: k·p interactions and spin-orbit coupling.

Degenerate Perturbation Theory and Anisotropy

First, let's ignore spin. The three p-like valence bands (|X>, |Y>, |Z>) are degenerate at k=0. We must use degenerate perturbation theory. The k·p interaction couples these states to each other (via other bands) and lifts the degeneracy for k ≠ 0.

The Role of Spin-Orbit Coupling

The final piece of the puzzle is spin-orbit coupling, a relativistic effect where the electron's spin interacts with the magnetic field it experiences due to its motion around the atomic nucleus. Your background in advanced quantum mechanics will be useful here. The spin-orbit interaction Hamiltonian is $H_{so}\propto (\nabla V\times \mathbf{p})\cdot \mathbf{\sigma }$.

This interaction splits the six degenerate valence-band states (3 p-orbitals × 2 spin states) at k=0 into two groups according to their total angular momentum, J = L + S.

1. A quadruply-degenerate set of states with J=3/2.
2. A doubly-degenerate set of states with J=1/2, which is pushed down in energy.

When we now "turn on" the k·p interaction for k ≠ 0, the J=3/2 states further split into two bands:

• Heavy-Hole (hh) band: Characterized by a smaller curvature and thus a larger effective mass.
• Light-Hole (lh) band: Characterized by a larger curvature and a smaller effective mass.

The J=1/2 states form the Split-Off (so) band, separated from the hh/lh bands at k=0 by an energy Δ_so, the spin-orbit splitting energy.

The full picture is obtained by diagonalizing a 6x6 Hamiltonian matrix that includes both the k·p terms and the spin-orbit interaction.

The resulting band structure is a fundamental feature of most semiconductors you will encounter.

This complex valence band structure is essential for understanding optical properties. Transitions can occur from any of these valence bands to the conduction band, and the properties of the resulting excitons will depend on which type of hole (heavy, light, or split-off) is involved.

Lesson Summary

In this lesson, we have laid the theoretical groundwork for describing electrons and holes in semiconductors.

• We reviewed the effective mass approximation, which allows us to treat electrons and holes as quasi-particles whose dynamics are dictated by the curvature of the energy bands.
• We used k·p perturbation theory to derive a formal expression for the effective mass, showing that it originates from the quantum mechanical coupling between different energy bands.
• We saw how the combination of orbital degeneracy at the valence band maximum and spin-orbit coupling leads to the splitting of the valence band into the characteristic heavy-hole, light-hole, and split-off bands.

Preview of the Next Lesson

Now that we have a model for the constituent particles (electrons and holes with their respective effective masses), we are ready to start building the exciton. In the next lesson, we will introduce the envelope function approximation. This will allow us to write down a simplified two-particle wavefunction for the electron-hole pair, leading us directly to the Schrödinger equation for the exciton.