Boltzmann Transport for Conductivity

Hello! Welcome to our next lesson.

In our previous session, we analyzed the p-n junction in thermal equilibrium. We saw how diffusion and drift lead to a static situation with a depletion region and a built-in potential. That was a problem of electrostatics. However, the real power of semiconductors and metals comes from their ability to conduct charge and heat when we apply external stimuli like voltages or temperature gradients. To describe these dynamic, non-equilibrium processes, we need a more powerful framework.

Today, we will introduce that framework: the Boltzmann Transport Equation (BTE). It is the central tool in the semi-classical theory of transport, providing a bridge between the microscopic quantum states of electrons and the macroscopic transport properties we can measure, like conductivity.

Our learning outcome is to apply the Boltzmann transport equation in the relaxation time approximation to derive electrical and thermal conductivity. We'll start by building an intuition for the BTE, introduce a crucial simplification known as the relaxation time approximation (RTA), and then apply it to derive these two fundamental transport coefficients.

1. The Semi-classical Picture and the Boltzmann Equation

The core idea of semi-classical transport is to treat electrons as wave packets that have both a well-defined position $\mathbf{r}$ and a well-defined crystal momentum $\hslash \mathbf{k}$ (or Bloch wavevector $\mathbf{k}$). This picture is valid as long as the applied fields are slowly varying on the scale of the lattice constant.

We describe the state of the electron gas using a distribution function, $f(\mathbf{r},\mathbf{k},t)$, which gives the probability that an electronic state with wavevector $\mathbf{k}$ near position $\mathbf{r}$ is occupied at time $t$. In equilibrium, with no external fields, this is simply the Fermi-Dirac distribution, $f_0({ϵ}_{\mathbf{k}})$.

When we apply external fields, the distribution function changes. The BTE is essentially a statement of continuity in the six-dimensional $(\mathbf{r},\mathbf{k})$ phase space. It states that the total change in the distribution function at a point in phase space is the sum of changes due to electrons drifting in and out, plus the changes due to collisions.

The full BTE is written as:

$$\frac{\partial f}{\partial t}+\dot{\mathbf{r}}\cdot {\nabla }_{\mathbf{r}}f+\dot{\mathbf{k}}\cdot {\nabla }_{\mathbf{k}}f={\left(\frac{\partial f}{\partial t}\right)}_{\text{coll}}$$

Let's break this down:

• The terms on the left are the "drift" or "streaming" terms. They describe how $f$ changes smoothly as electrons move according to the semi-classical equations of motion:
  • $\dot{\mathbf{r}}={\mathbf{v}}_{\mathbf{k}}=\frac{1}{\hslash }{\nabla }_{\mathbf{k}}{ϵ}_{\mathbf{k}}$ (group velocity)
  • $\hslash \dot{\mathbf{k}}={\mathbf{F}}_{\text{ext}}=q(\mathbf{E}+{\mathbf{v}}_{\mathbf{k}}\times \mathbf{B})$ (force from electric and magnetic fields)
• The term on the right, ${\left(\frac{\partial f}{\partial t}\right)}_{\text{coll}}$, is the collision integral. It accounts for the abrupt changes in an electron's momentum due to scattering from impurities, defects, or phonons. In the absence of this term, the equation describes a collisionless flow in phase space (Liouville's theorem). The collision term is what drives the system towards equilibrium.

The following reading provides a concise derivation of the BTE and introduces the key concepts.

2. The Relaxation Time Approximation (RTA)

The collision integral is notoriously difficult to handle; it's an integro-differential term that couples the state $\mathbf{k}$ to all other states ${\mathbf{k}}^{\prime }$. The most common and effective simplification is the Relaxation Time Approximation (RTA).

The RTA makes a simple, physically motivated assumption: if the system is perturbed from its local equilibrium distribution $f^0$, it will relax back to equilibrium exponentially with a characteristic time ${\tau }_{\mathbf{k}}$, called the relaxation time or scattering time.

Mathematically, we write the collision integral as:

$${\left(\frac{\partial f}{\partial t}\right)}_{\text{coll}}=-\frac{f-f^0}{{\tau }_{\mathbf{k}}}=-\frac{\delta f}{{\tau }_{\mathbf{k}}}$$

where $\delta f=f-f^0$ is the deviation from equilibrium. This approximation assumes that the scattering processes at $\mathbf{k}$ only depend on the occupation at $\mathbf{k}$ and that they always tend to restore the equilibrium population.

3. Deriving Electrical Conductivity

Let's use the BTE with the RTA to derive the electrical conductivity, $\sigma$. Consider a uniform, static electric field $\mathbf{E}$ applied to a solid, with no magnetic field or temperature gradient.

The BTE simplifies to:

• $\frac{\partial f}{\partial t}=0$ (steady state)
• ${\nabla }_{\mathbf{r}}f=0$ (spatial uniformity)
• $\mathbf{B}=0$

This leaves us with:

$$\frac{q}{\hslash }\mathbf{E}\cdot {\nabla }_{\mathbf{k}}f={\left(\frac{\partial f}{\partial t}\right)}_{\text{coll}}=-\frac{f-f^0}{{\tau }_{\mathbf{k}}}$$

We are in the linear response regime, meaning the applied field $\mathbf{E}$ is small. This implies that the deviation $\delta f$ is also small, so we can approximate $f\approx f^0$ on the left-hand side:

$$\frac{q}{\hslash }\mathbf{E}\cdot {\nabla }_{\mathbf{k}}{f}^0\approx -\frac{\delta f}{{\tau }_{\mathbf{k}}}$$

Using the chain rule, ${\nabla }_{\mathbf{k}}{f}^0=\frac{\partial f^0}{\partial ϵ}{\nabla }_{\mathbf{k}}ϵ=\hslash {\mathbf{v}}_{\mathbf{k}}\frac{\partial f^0}{\partial ϵ}$. Substituting this in, we can solve for the deviation $\delta f$:

$$\delta f=-q{\tau }_{\mathbf{k}}(\mathbf{E}\cdot {\mathbf{v}}_{\mathbf{k}})\frac{\partial f^0}{\partial ϵ}$$

This is a key result. It tells us that the deviation from equilibrium is proportional to the applied field, the scattering time, and the energy derivative of the Fermi function. The term $(-\partial f^0/\partial ϵ)$ is a bell-shaped curve peaked at the Fermi energy, ${ϵ}_F$, which means that only electrons near the Fermi surface contribute to transport.

Now, we can calculate the electric current density $\mathbf{j}$. The contribution from the equilibrium distribution $f^0$ is zero due to symmetry ($f^0(\mathbf{k})=f^0(-\mathbf{k})$ and ${\mathbf{v}}_{-\mathbf{k}}=-{\mathbf{v}}_{\mathbf{k}}$). So the current comes entirely from $\delta f$:

$$\mathbf{j}=\int \frac{2d^{3}k}{(2\pi {)}^3}q{\mathbf{v}}_{\mathbf{k}}\delta f=\left[2q^2\int \frac{d^{3}k}{(2\pi {)}^3}{\tau }_{\mathbf{k}}{\mathbf{v}}_{\mathbf{k}}\otimes {\mathbf{v}}_{\mathbf{k}}\left(-\frac{\partial f^0}{\partial ϵ}\right)\right]\mathbf{E}$$

The term in the brackets is the conductivity tensor, $\sigma$. For a system with cubic symmetry, the tensor is diagonal with ${\sigma }_{\alpha \beta }=\sigma {\delta }_{\alpha \beta }$, and we get the scalar conductivity:

$$\sigma =\frac{2q^2}{3}\int \frac{d^{3}k}{(2\pi {)}^3}{\tau }_{\mathbf{k}}{v}_{\mathbf{k}}^2\left(-\frac{\partial f^0}{\partial ϵ}\right)$$

For a metal at low temperatures, where $(-\partial f^0/\partial ϵ)\approx \delta (ϵ-{ϵ}_F)$, this expression can be shown to reduce to the familiar Drude formula, $\sigma =\frac{n{q}^2\tau }{m^{\ast }}$, putting it on a much firmer theoretical ground.

The following resource walks through this derivation in detail.

4. Deriving Thermal Conductivity and the Wiedemann-Franz Law

The BTE framework is powerful because it can describe multiple coupled transport phenomena. Let's now consider the case where we have both an electric field $\mathbf{E}$ and a temperature gradient $\nabla T$.

The deviation from equilibrium, $\delta f$, now has two driving forces:

$$\delta f=-{\tau }_{\mathbf{k}}{\mathbf{v}}_{\mathbf{k}}\cdot \left[q\mathbf{E}+\frac{{ϵ}_{\mathbf{k}}-\mu }{T}\nabla T\right]\left(-\frac{\partial f^0}{\partial ϵ}\right)$$

We need to calculate both the electric current $\mathbf{j}$ and the heat current ${\mathbf{j}}_Q$. The heat current is the flow of energy relative to the chemical potential:

$${\mathbf{j}}_Q=\int \frac{2d^{3}k}{(2\pi {)}^3}({ϵ}_{\mathbf{k}}-\mu ){\mathbf{v}}_{\mathbf{k}}\delta f$$

Substituting $\delta f$ into the expressions for $\mathbf{j}$ and ${\mathbf{j}}_Q$ gives us a set of coupled linear equations:
\begin{align*}
\mathbf{j} &= L_{11} (\mathbf{E}') - L_{12} \frac{\nabla T}{T} \
\mathbf{j}Q &= L{21} (\mathbf{E}') - L_{22} \frac{\nabla T}{T}
\end{align*}
where ${\mathbf{E}}^{\prime }$ is the electrochemical field and the $L$ coefficients are matrices involving integrals over the band structure (see eq. 1.181-1.183 in the upcoming reading).

The thermal conductivity $\kappa$ is defined under the condition of zero electric current ($\mathbf{j}=0$). This is the typical experimental setup for measuring $\kappa$: you heat one end of a sample and measure the temperature difference and heat flow, while ensuring no net electrical current flows.
Setting $\mathbf{j}=0$ allows us to solve for the electric field that must build up to counteract the diffusion of hot electrons (this is the Seebeck effect, the topic of our next lesson!). Substituting this field back into the equation for ${\mathbf{j}}_Q$ yields the relation ${\mathbf{j}}_Q=-\kappa \nabla T$.

For metals at low temperatures, a remarkable result emerges from this formalism: the Wiedemann-Franz Law. It states that the ratio of thermal to electrical conductivity is a universal constant, proportional to temperature:

$$\frac{\kappa }{\sigma T}=\frac{{\pi }^2}{3}{\left(\frac{k_B}{q}\right)}^2\equiv L_0$$

The constant $L_0$ is known as the Lorentz number. This law holds surprisingly well for many simple metals and is a triumph of the semi-classical transport theory. It arises because in a metal, the same electrons are responsible for both electrical and thermal conduction.

Test your understanding!

Why does the Wiedemann-Franz law often fail in semiconductors? (Hint: Think about what other quasi-particles can carry heat).

Show answer

In semiconductors, heat is carried not only by electrons and holes but also significantly by phonons (lattice vibrations). Our derivation for $\kappa$ only included the electronic contribution. Since phonons contribute to $\kappa$ but not to $\sigma$, the ratio $\kappa /(\sigma T)$ will be much larger than the Lorentz number, especially in lightly doped semiconductors where the phonon contribution to thermal conductivity dominates.

The following reading guides you through the derivation of the coupled transport equations and the thermal conductivity.

Conclusion

Today, we've developed a powerful and general tool for understanding how electrons move in solids. By starting from a phase-space distribution function and accounting for both drift in external fields and scattering processes, the Boltzmann Transport Equation provides a rigorous foundation for transport phenomena.

Key Takeaways:

• The Boltzmann Transport Equation describes the evolution of the electron distribution function $f(\mathbf{r},\mathbf{k},t)$ under the influence of external fields and internal scattering.
• The Relaxation Time Approximation (RTA) simplifies the collision term, assuming that deviations from equilibrium, $\delta f$, decay exponentially with a characteristic time $\tau$.
• By solving the linearized BTE in the RTA, we found that the deviation from equilibrium is $\delta f\propto (\mathbf{E}\cdot {\mathbf{v}}_{\mathbf{k}})(-\partial f^0/\partial ϵ)$, highlighting that only electrons near the Fermi surface contribute to transport in metals.
• We derived general expressions for the electrical conductivity tensor $\sigma$ and the thermal conductivity $\kappa$ based on integrals over the electronic band structure.
• For metals, this framework successfully recovers the Drude model for electrical conductivity and predicts the Wiedemann-Franz law, a universal relationship between electrical and thermal conductivity.

Preview of Next Lesson:
The coupled transport equations we derived today contain more information than just $\sigma$ and $\kappa$. They also describe thermoelectric phenomena, where temperature gradients create voltages and vice-versa. In our next lesson, we will use the exact same formalism to calculate the Seebeck coefficient, which quantifies the voltage produced per unit temperature gradient and is the basis for thermocouples and thermoelectric generators.