Envelope Function Approximation for Slowly Varying Wavefunctions

Hello! Welcome to your second lesson.

In our last session, we established a model for electrons and holes in a semiconductor crystal. By using k·p theory, we saw how the complex periodic potential of the lattice could be encapsulated in the concept of an effective mass, $m^{*}$ , and how this leads to the rich valence band structure of heavy, light, and split-off holes.

Today, we address the next logical question: what happens when we introduce an additional, weak, slowly-varying potential, $U (r)$ ? This is the central problem for understanding impurities, quantum wells, and, most importantly for us, the Coulomb interaction between an electron and a hole that forms an exciton. Our goal is to develop and apply the envelope function approximation (EFA), a powerful method that separates the electron's wavefunction into a rapidly oscillating part tied to the crystal lattice and a slowly varying "envelope" that describes its response to the new potential.

1. The Physical Basis: Wavepackets and Slow Potentials

The envelope function approximation is fundamentally based on the properties of wavepackets in Fourier space. A potential $U (r)$ that is "slowly varying" means its characteristic length scale is much larger than the lattice constant $a$ . In Fourier terms, its transform $U (q)$ is only significant for small wavenumbers $q$ .

When an electron in a Bloch state $ψ_{n, k}$ scatters off this potential, it transitions to a new state $ψ_{n, k^{'}}$ . Because the potential only provides small momentum kicks ħq, the final state k' will be very close to the initial state k. Therefore, a stationary state in the presence of U(r) will be a superposition of Bloch states from a very narrow region of the Brillouin zone.

A wavefunction constructed from a narrow range of k-vectors is, by the principles of Fourier analysis, necessarily spread out in real space. This is the "envelope."

Wavepackets and Fourier representation

To build intuition for this relationship, please watch the following video from MIT OpenCourseWare. It discusses how the spatial extent of a wavepacket is inversely related to its extent in k-space.

Please watch from the beginning to 11:15. Focus on these key ideas: A general wavefunction can be represented as a superposition (an integral) of plane waves, weighted by a function \phi(k) (0:00 - 3:54). The relationship between the width of the wavepacket in real space (\Delta x) and the width of the weighting function in k-space (\Delta k) (7:01 - 11:15). A narrow \Delta k implies a wide \Delta x.

In our case, the slowly varying potential U(r) ensures that the perturbed wavefunction is built from Bloch states with a narrow spread $Δ k$ around the band minimum. The resulting wavefunction will therefore have a slowly varying envelope modulating the underlying, fast-oscillating Bloch function of the band edge.

2. Deriving the Envelope Function Approximation

Let's now formalize this intuition. We will derive the structure of the wavefunction and the equation that governs its envelope.

Envelope functions and effective mass approximation

The following document provides a clear, step-by-step derivation of the envelope function approximation and the resulting effective mass equation. We will work through the key steps together.

Please read section 4.1, from 'The problem on the basis of Bloch-states' up to the end of the section, concluding with the derivation of the effective mass Schrödinger equation (Eq. 4.4). We will break this down below.

Let's trace the logic from the reading.

Step 1: Expand the Wavefunction
We start with the Schrödinger equation for an electron in the crystal potential H₀ plus the weak external potential U(r):
$[H_{0} + U (r)] Ψ (r) = E Ψ (r)$
Since the Bloch functions $ψ_{n, k} (r)$ form a complete basis, we can expand the unknown wavefunction $Ψ (r)$ in terms of them:
$Ψ (r) = \sum_{n, k} F_{n} (k) ψ_{n, k} (r) = \sum_{n, k} F_{n} (k) e^{i k \cdot r} u_{n, k} (r)$
The goal is to find the coefficients $F_{n} (k)$ .

Step 2: Key Approximations
As detailed in the reading under "Simplifying approximations," we make three crucial assumptions:

Slowly varying potential: U(r) varies over distances much larger than the lattice constant. This means its Fourier components U(q) are negligible for large q.
Weak potential: U(r) is small compared to the band gaps. This prevents significant mixing between different bands (n ≠ n'). The perturbation primarily shifts states within a single band.
States are near a band extremum: We are interested in energies close to a band edge (e.g., the conduction band minimum at k=0). Therefore, the expansion coefficients $F_{n} (k)$ are only significant for small k.

Step 3: Simplify the Wavefunction
Under these approximations, we are confined to a single band n, and only small k vectors contribute. This allows us to approximate the periodic part of the Bloch function, $u_{n, k} (r)$ , by its value at the band edge, $u_{n, 0} (r)$ , since it's a smooth function of k. The total wavefunction simplifies dramatically:

$Ψ (r) = \sum_{k} F_{n} (k) e^{i k \cdot r} u_{n, k} (r) \approx u_{n, 0} (r) (\sum_{k} F_{n} (k) e^{i k \cdot r})$

We now define the term in the parentheses as the envelope function, $F_{n} (r)$ :
$F_{n} (r) \equiv \sum_{k} F_{n} (k) e^{i k \cdot r}$
This is the inverse Fourier transform of our expansion coefficients. The total wavefunction takes its characteristic form:

$Ψ (r) \approx u_{n, 0} (r) F_{n} (r)$

This is the central result of the approximation: the full wavefunction is a product of the rapidly oscillating Bloch function at the band edge and a slowly varying envelope function that describes the larger-scale behavior of the particle.

Envelope Function Approximation (EFA) Mathematical Formulation and Illustration — This diagram summarizes the EFA formalism. The total wavefunction Ψ is a product of the slowly varying envelope function F(r) and the rapidly oscillating, lattice-periodic Bloch function u(r). The full Hamiltonian is simplified into an effective mass Hamiltonian that acts only on the envelope function.

3. The Effective Mass Equation

We've determined the form of the wavefunction, but what equation does the envelope F(r) obey?

As derived in the reading (section "Approximating the dispersion" and "Equation of motion in real space"), we take the equation for the coefficients $F_{n} (k)$ , insert the parabolic band approximation for the energy which we found in the last lesson, $E_{n} (k) \approx E_{n, 0} + \frac{ℏ ^{2} k ^{2}}{2 m ^{*}}$ , and perform an inverse Fourier transform.

The result is a Schrödinger-like equation for the envelope function, known as the effective mass equation:
$[- \frac{ℏ ^{2}}{2 m ^{*}} \nabla^{2} + U (r)] F_{n} (r) = (E - E_{n, 0}) F_{n} (r)$
This is a remarkable simplification. The entire complexity of the periodic crystal potential has been absorbed into the effective mass $m^{*}$ . We are left with a familiar single-particle Schrödinger equation for the envelope function, which "lives" in the external potential $U (r)$ and whose energy eigenvalues are measured relative to the band edge energy $E_{n, 0}$ .

Test your understanding!

The total wavefunction in the EFA is $Ψ (r) \approx u_{n, 0} (r) F_{n} (r)$ . If we want to calculate the probability of finding the electron at a position $r$ , we need $∣Ψ (r) ∣^{2}$ . Given that $u_{n, 0} (r)$ oscillates on the scale of the lattice constant a and $F_{n} (r)$ varies on a much larger scale $L ≫ a$ , what does the probability density look like, and what quantity determines its overall shape?

Show answer

The probability density is $∣Ψ (r) ∣^{2} = ∣ u_{n, 0} (r) ∣^{2} ∣ F_{n} (r) ∣^{2}$ . This is a rapidly oscillating function, $∣ u_{n, 0} (r) ∣^{2}$ , whose amplitude is modulated by the slowly varying function $∣ F_{n} (r) ∣^{2}$ . The overall spatial shape of the electron's probability distribution is determined by the envelope function's squared magnitude, $∣ F_{n} (r) ∣^{2}$ . The Bloch part, $∣ u_{n, 0} (r) ∣^{2}$ , describes how the probability is distributed within each unit cell.

4. Application: Hydrogenic Impurities

Let's apply this framework to a concrete and important example: a shallow donor impurity in a semiconductor (e.g., a Si atom replacing a Ga atom in a GaAs lattice). The Si atom provides an extra electron and a net positive charge. This positive core creates a Coulomb potential that can bind the electron.

Because the potential is embedded in a polarizable semiconductor medium, the Coulomb interaction is screened by the material's static dielectric constant, $ϵ$ . The potential is:
$U (r) = - \frac{e ^{2}}{4 π ϵ ϵ _{0} r}$
Plugging this into our effective mass equation for a conduction band electron gives:
$[- \frac{ℏ ^{2}}{2 m _{e}^{*}} \nabla^{2} - \frac{e ^{2}}{4 π ϵ ϵ _{0} r}] F_{c} (r) = (E - E_{c}) F_{c} (r)$
This is mathematically identical to the Schrödinger equation for the hydrogen atom. Your quantum mechanics background tells you the solutions immediately, but with two modifications: the electron mass $m_{0}$ is replaced by the effective mass $m_{e}^{*}$ , and the permittivity $ϵ_{0}$ is replaced by $ϵ ϵ_{0}$ .

Envelope functions and effective mass approximation

The document we used for the derivation also provides the solution for this hydrogen-like impurity problem.

Please read the subsection 'Hydrogen-like impurities'. Pay close attention to the expressions for the effective Bohr radius (a_B^*) and the effective Rydberg energy (E_{Ry}^*), and note the calculated values for GaAs.

The key results are:

Effective Bohr radius: $a_{B}^{*} = a_{B} \frac{m _{0}}{m _{e}^{*}} ϵ \approx 100$ Å in GaAs.
Effective Rydberg (binding) energy: $E_{R y}^{*} = E_{R y} \frac{m _{e}^{*}}{m _{0}} \frac{1}{ϵ ^{2}} \approx 5.7$ meV in GaAs.

The ground state energy level for this bound electron is $E = E_{c} - E_{R y}^{*}$ , lying just below the conduction band edge.

Crucially, the large effective Bohr radius ( $a_{B}^{*} ≫ a$ ) confirms the validity of our initial "slowly varying" assumption. The envelope function, which has the form $F_{c} (r) \propto e^{- r / a_{B}^{*}}$ , is indeed slowly varying on the scale of the lattice.

Lesson Summary

In this lesson, we have developed a crucial tool for semiconductor physics.

We established that a weak, slowly varying potential primarily mixes Bloch states from a narrow region of k-space, leading to a wavefunction with a slowly varying envelope.
We formally derived the envelope function approximation (EFA), which expresses the total wavefunction as $Ψ (r) \approx u_{n, 0} (r) F_{n} (r)$ .
We showed that the envelope function obeys the effective mass equation, a Schrödinger equation where the periodic potential's effects are captured entirely by the effective mass $m^{*}$ .
We applied this to a hydrogenic impurity, demonstrating how the EFA reduces a complex solid-state problem to a familiar quantum mechanics problem, and how it correctly predicts binding energies and the large spatial extent of impurity wavefunctions.

Preview of the Next Lesson

The hydrogenic impurity problem is the perfect warm-up for the main event. In the next lesson, we will finally construct the exciton. We will consider an electron and a hole interacting via their mutual (screened) Coulomb attraction. We will treat this as a two-body problem, separate the center-of-mass and relative motion coordinates, and apply the envelope function approximation to the relative motion part. This will yield a hydrogen-like Schrödinger equation for the exciton, allowing us to derive its binding energy and Bohr radius.

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