Exciton Drift-Diffusion and Mobility

Hello! Welcome to our next lesson.

In our previous session, we examined how ortho- and para-excitons produce distinct signatures in absorption and photoluminescence spectra, highlighting their different optical activities and, crucially, their vastly different lifetimes. We concluded by noting that excitons are dynamic: they relax, convert between states, and move through the crystal.

Today, we delve into the spatial aspect of these dynamics. Our goal is to model exciton transport using a drift-diffusion equation and relate mobility to momentum relaxation time. This is a foundational topic in understanding how excitons propagate from their point of creation, which is essential for both interpreting experiments and designing excitonic devices. We'll build this important equation from the ground up by considering the two primary modes of transport: random thermal motion (diffusion) and directed motion in response to a force (drift).

1. Diffusion: The Random Walk of Excitons

Imagine creating a dense cloud of excitons at one spot in a crystal using a focused laser. Even at low temperatures, these excitons possess kinetic energy and are constantly in motion. As they travel, they scatter off crystal imperfections and, more importantly, phonons (lattice vibrations). The result of these countless random scattering events is a net motion from regions of high concentration to regions of low concentration. This process is called diffusion.

The most intuitive way to understand how this collective spreading arises from individual random walks is through a discrete model that is then taken to a continuous limit.

The diffusion equation | Week 12 | MIT 18.S191 Fall 2020 | Grant Sanderson

To build a strong intuition for diffusion, I recommend this excellent video by Grant Sanderson (3Blue1Brown). It masterfully derives the diffusion equation from a simple 1D random walk, providing a very clear visual and mathematical foundation.

Please watch from the beginning until 15:49. Focus on these key steps: The setup of the discrete random walk model. The derivation of the change in particle number in a cell as a 'difference of differences' (the discrete version of a second derivative). The transition to the continuous limit, which results in the diffusion equation. Pay close attention to the physical meaning of the second spatial derivative (curvature) as the driver of diffusion.

As the video shows, this process is mathematically described by the diffusion equation:

\frac{\partial n ( r , t )}{\partial t} = D \nabla^{2} n (r, t)

Here:

$n (r, t)$ is the exciton density at position $r$ and time $t$ .
$D$ is the diffusion coefficient, a constant that quantifies how quickly the excitons spread out. It has units of area/time (e.g., cm²/s).
$\nabla^{2}$ is the Laplacian operator ( $\frac{\partial ^{2}}{\partial x ^{2}} + \frac{\partial ^{2}}{\partial y ^{2}} + \frac{\partial ^{2}}{\partial z ^{2}}$ ). The term $\nabla^{2} n$ represents the curvature of the density profile. Diffusion is fastest where the density profile is most "curved" (i.e., at sharp peaks) and stops when the profile is flat (uniform density).

The flux of particles due to diffusion is given by Fick's first law:

J_{diff} = - D \nabla n (r, t)

This states that the net flow of particles is proportional to the negative of the density gradient.

2. Drift: Directed Motion under a Force

Diffusion describes random motion. However, excitons can also be pushed in a specific direction by a force, $F$ . This directed motion is called drift. Although excitons are neutral, they can be subject to forces from:

An inhomogeneous electric field (acting on their small induced dipole moment).
A gradient in the crystal lattice's strain, which modifies the bandgap.
A temperature gradient (thermophoresis).
Repulsive interactions with other excitons, which creates a force pushing them away from high-density regions.

When a constant force is applied, an exciton doesn't accelerate indefinitely. Its motion is impeded by the same scattering processes that cause diffusion. The exciton accelerates, scatters, loses its directed momentum, and then accelerates again. The net result is a constant average drift velocity, $v_{d}$ .

This is analogous to the Drude model for electrical conductivity. A simple classical derivation provides a clear link between drift velocity, the applied force, and the microscopic scattering time.

Carrier Drift Current: Mobility

This video provides a clear, semi-classical derivation for the drift of charge carriers in an electric field. The core physics is identical for an exciton experiencing a general force \mathbf{F}.

Please watch from 03:16 to 07:09. The key idea is using Newton's law between collisions to find the average velocity. Pay attention to how the average time between collisions, \tau, determines the final drift velocity.

As the video explains for an electric force, the key result is that the drift velocity is proportional to the applied force. We can generalize this:

v_{d} = μ F

The constant of proportionality, $μ$ , is called the mobility. It measures how readily an exciton moves in response to a force.

From the derivation in the video, we can see that the mobility is directly related to the average time between momentum-randomizing collisions, $τ$ , and the exciton's effective mass, $M$ :

μ = \frac{τ}{M}

This $τ$ is formally known as the momentum relaxation time. It's the characteristic timescale over which an exciton "forgets" its direction of motion due to scattering. This directly answers the second part of our learning outcome: mobility is simply the momentum relaxation time divided by the effective mass.

Fundamentally, the momentum relaxation time arises from the scattering term in the Boltzmann transport equation, which describes how a particle distribution f returns to its equilibrium state f₀ after being perturbed.

Boltzmann Transport Equation and Relaxation Time Approximation — The relaxation time approximation simplifies the complex scattering integral in the Boltzmann transport equation. It assumes that any deviation of the particle distribution function `f` from its equilibrium value `f₀` will decay exponentially with a characteristic time constant, the relaxation time `τ`.

3. The Full Picture: The Drift-Diffusion Equation

In any real system, both drift and diffusion happen simultaneously. The total flux of excitons, $J_{total}$ , is simply the sum of the drift and diffusion fluxes:

J_{total} = J_{drift} + J_{diff} = n v_{d} - D \nabla n = n μ F - D \nabla n

Given your physics background, you'll recognize the next step. The rate of change of the exciton density is governed by the continuity equation, which expresses particle conservation: $\frac{\partial n}{\partial t} = - \nabla \cdot J_{total}$ . Substituting our expression for the total flux gives:

\frac{\partial n}{\partial t} = - \nabla \cdot (n μ F - D \nabla n) = \nabla \cdot (D \nabla n - n μ F)

Finally, we must account for the finite lifetime of excitons. Excitons can recombine and disappear. We can add a simple phenomenological decay term, $- n / τ_{life}$ , where $τ_{life}$ is the exciton population lifetime (which we discussed in the last lesson).

This gives us the complete drift-diffusion equation:

\frac{\partial n ( r , t )}{\partial t} = \nabla \cdot (D \nabla n - n μ F) - \frac{n}{τ _{life}}

This equation is a cornerstone for modeling exciton transport. Let's see how it appears in modern research literature.

arXiv:2207.03942v1 [cond-mat.mes-hall] 8 Jul 2022

Now that we have constructed the drift-diffusion equation from physical principles, let's look at its formal statement and application in a recent research paper on exciton transport.

First, please read the section 'COULOMB-DRIVEN EXCITON PROPAGATION' on page 3. Focus on Equation (7), which is the drift-diffusion equation. Notice how the authors define the drift term in the context of a force arising from exciton-exciton interactions (\Delta E(n)). Second, given your preference for mathematical rigor, I suggest skimming Section III: 'DRIFT-DIFFUSION EQUATION' in the supplementary material (page 6). This shows how the equation is derived more formally from the Wigner function formalism, starting from the Heisenberg equation of motion. This connects the semi-classical picture we've built to a more complete quantum-mechanical framework.

From the paper, we see the equation written in a slightly different form, but the three physical components are clear:

Diffusion: ∇ · (D(n)∇n) — The spreading due to random motion. The paper notes that D itself can depend on density n due to quantum statistics (boson bunching).
Drift: μ_m∇ · (∆E(n) ∇n(r, t)) (this is a simplified form used in some contexts, the more general form is ∇·(nμF)) — The directed motion caused by a force. In this paper's specific example, the force arises from the gradient of the interaction energy, $F = - \nabla (Δ E (n))$ .
Recombination: -n(r,t)/τ — The decay of the exciton population over time.

A crucial final link is the Einstein relation, which connects the diffusion coefficient and mobility. For a classical gas, this relation is:

D = μ k_{B} T

This shows that diffusion and the response to a force (drift) are not independent; they are two manifestations of the same underlying random thermal motion.

Test your understanding!

An experiment on excitons in a GaAs quantum well at T = 10 K measures a low-density diffusion coefficient $D_{0} = 10$ cm²/s and an exciton effective mass $M = 0.22 m_{e}$ , where $m_{e}$ is the free electron mass.

Using the Einstein relation, calculate the exciton mobility $μ$ .
From the mobility, calculate the momentum relaxation time $τ$ . How does this compare to a typical exciton lifetime (which can be ~1 ns)?

(Constants: $k_{B} \approx 1.38 \times 1 0^{- 23}$ J/K, $m_{e} \approx 9.11 \times 1 0^{- 31}$ kg)

Show answer

Mobility Calculation:
First, let's find the mobility using the Einstein relation $D = μ k_{B} T$ .
$μ = \frac{D _{0}}{k _{B} T} = \frac{10 \times 1 0 ^{- 4} m ^{2} / s}{( 1.38 \times 1 0 ^{- 23} J/K ) \times ( 10 K )} \approx 7.25 \times 1 0^{18} \frac{m ^{2}}{V \cdot s \cdot J}$
The units seem strange. A more standard unit for mobility is m²/(V·s) for charged particles, but here the force is general. Let's use the unit s/kg, which is more fundamental.
Mobility units from $μ = D / (k_{B} T)$ are (m²/s) / J = (m²/s) / (kg·m²/s²) = s/kg.
$μ = \frac{10 \times 1 0 ^{- 4} m ^{2} / s}{( 1.38 \times 1 0 ^{- 23} J/K ) \times ( 10 K )} \approx 7.25 \times 1 0^{18} s/kg$
Relaxation Time Calculation:
Now use the relation $μ = τ / M$ .
The exciton mass is $M = 0.22 \times (9.11 \times 1 0^{- 31} kg) \approx 2.0 \times 1 0^{- 31}$ kg.
$τ = μ \times M = (7.25 \times 1 0^{18} s/kg) \times (2.0 \times 1 0^{- 31} kg) \approx 1.45 \times 1 0^{- 12} s$
So, the momentum relaxation time is approximately 1.45 picoseconds (ps).

Comparison: This momentum relaxation time (1.45 ps) is about three orders of magnitude shorter than a typical radiative lifetime (~1 ns = 1000 ps). This tells us that an exciton undergoes thousands of scattering events that randomize its momentum during its lifetime, justifying the random walk picture of diffusion.

Conclusion

Today, we've developed a comprehensive model for exciton transport. By combining the concepts of diffusion and drift, we arrived at the drift-diffusion equation, a powerful tool for describing the spatiotemporal evolution of an exciton population.

Key Takeaways:

Exciton transport is governed by two main processes: diffusion (random motion down a concentration gradient) and drift (directed motion in response to a force).
The drift-diffusion equation, $\frac{\partial n}{\partial t} = \nabla \cdot (D \nabla n - n μ F) - \frac{n}{τ _{life}}$ , combines these effects with population decay.
Mobility ( $μ$ ) connects the drift velocity to the applied force ( $v_{d} = μ F$ ).
Mobility is fundamentally determined by the momentum relaxation time ( $τ$ ) and the exciton effective mass ( $M$ ): $μ = τ / M$ . This relaxation time is typically on the picosecond scale, far shorter than the exciton lifetime.
The Einstein relation ( $D = μ k_{B} T$ ) provides a deep connection between the seemingly separate phenomena of diffusion and drift.

Preview of the Next Lesson

The drift-diffusion equation includes a phenomenological decay term, $- n / τ_{life}$ . But what microscopic processes contribute to this lifetime? In our next lesson, we will dissect this term and describe radiative and non-radiative recombination mechanisms (e.g., Auger processes) and their impact on exciton lifetime. We will explore the different pathways by which an exciton population can disappear, moving from modeling where they go to understanding how long they last.

Recommended resources

1. Diffusion: The Random Walk of Excitons

2. Drift: Directed Motion under a Force

3. The Full Picture: The Drift-Diffusion Equation

Conclusion

Preview of the Next Lesson