Seebeck Coefficient: Calculation and Physical Origin

Hello! Let's dive into our next lesson.

In our last session, we developed the Boltzmann Transport Equation (BTE) framework and used it to derive the electrical conductivity ($\sigma$) and the electronic part of the thermal conductivity ($\kappa$). We saw that applying an electric field or a temperature gradient drives the system out of equilibrium, resulting in electric and heat currents. We summarized this with 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 $\mathbf{j}$ is the charge current, ${\mathbf{j}}_Q$ is the heat current, and ${\mathbf{E}}^{\prime }$ is the electrochemical field. So far, we've focused on the "diagonal" responses: $\mathbf{j}$ due to ${\mathbf{E}}^{\prime }$ ($\sigma$) and ${\mathbf{j}}_Q$ due to $\nabla T$ ($\kappa$).

Today, we'll explore the fascinating "off-diagonal" or cross-coupled phenomena. Specifically, we'll investigate how a temperature gradient can generate an electric field. This is known as the Seebeck effect, and it forms the basis of thermoelectric devices. Our goal is to calculate the Seebeck coefficient and explain its physical origin in the context of thermoelectric devices.

1. The Seebeck Effect: Definition and Calculation

The Seebeck effect describes the emergence of a voltage across a conductor when a temperature difference is maintained across it. To measure this voltage, we must ensure no net charge flows. This "open-circuit" condition, $\mathbf{j}=0$, is the key to defining the Seebeck coefficient.

Let's start with our equation for the charge current, assuming for now isotropic materials where the $L$ coefficients are scalars:

$$\mathbf{j}=L_{11}{\mathbf{E}}^{\prime }-L_{12}\frac{\nabla T}{T}$$

Setting $\mathbf{j}=0$, we find that a steady-state electrochemical field ${\mathbf{E}}^{\prime }$ must be established to perfectly counteract the current driven by the temperature gradient:

$${\mathbf{E}}^{\prime }=\frac{L_{12}}{L_{11}T}\nabla T$$

Since the electrochemical field for charge carriers $q$ is ${\mathbf{E}}^{\prime }=-\nabla {\mu }^{\prime }/q$, where ${\mu }^{\prime }$ is the electrochemical potential, this generated field creates a measurable voltage. The Seebeck coefficient (also called thermopower), denoted by $S$ (or $Q$ in some texts), is defined as the proportionality constant between the induced electric field ($\mathbf{E}=-\nabla \varphi$, where ${\mu }^{\prime }=\mu +q\varphi$) and the temperature gradient under this zero-current condition. The exact definition can vary, but a common one is:

$$S=\frac{E_{induced}}{\nabla T}{|}_{\mathbf{j}=0}$$

From our equations, this gives:

$$S=\frac{1}{q}\frac{L_{12}}{L_{11}T}$$

The following reading, from the same lecture notes as last time, formally defines the full matrix of transport coefficients and derives the expression for the thermopower.

As you saw in the reading, the Seebeck coefficient $Q$ (which we'll call $S$ from now on) can be expressed using the integrals $J_n$ that we encountered in the previous lesson:

$$S=-\frac{1}{eT}(J_0{)}^{-1}{J}_1\text{where}{J}_n\propto \int dϵ\tau (ϵ)(ϵ-\mu {)}^n\left(-\frac{\partial f^0}{\partial ϵ}\right)(\dots )$$

This expression is the starting point for both calculating $S$ and understanding its physical meaning.

2. The Physical Origin of Thermopower

The formula for $S$ might seem abstract, but it has a clear physical interpretation. Let's watch a clip that explains the microscopic origin.

To summarize the key idea:

1. A temperature gradient ($\nabla T$) is applied. The "hot" end has more high-energy electrons (and/or holes) than the "cold" end.
2. These energetic carriers diffuse from the hot end to the cold end, creating a net charge current.
3. This charge builds up at the cold end, establishing an internal electric field that opposes further diffusion.
4. A steady state is reached when the drift current caused by the internal E-field exactly cancels the diffusion current from the $\nabla T$, resulting in zero net current ($\mathbf{j}=0$). The voltage measured across the material in this state is the Seebeck voltage.

The sign and magnitude of $S$ depend on the asymmetry of carrier transport around the Fermi level. The crucial term in the integral for $J_1$ is $(ϵ-\mu )$. States above the chemical potential ($ϵ>\mu$) and below it ($ϵ<\mu$) contribute with opposite signs. If transport were perfectly symmetric around $\mu$, the Seebeck coefficient would be zero. However, because the density of states, group velocity, and scattering time are all energy-dependent, the contributions don't cancel.

• In a simple metal, the density of states typically increases with energy. This means the "hot" electrons ($ϵ>\mu$) are more numerous than the "cold holes" ($ϵ<\mu$) participating in transport, leading to a net flow of negative charge to the cold end. This results in a negative Seebeck coefficient.
• In a p-type semiconductor, the dominant carriers are holes, which have positive charge. They also diffuse from hot to cold, but since their charge is positive, this builds up a positive potential at the cold end. This results in a positive Seebeck coefficient.

Therefore, the sign of the Seebeck coefficient is a reliable experimental tool for determining the dominant carrier type (n-type vs. p-type) in a material.

Calculation for a Degenerate Electron Gas

For a metal at low temperatures ($T\ll T_F$), we can evaluate the integrals for $J_0$ and $J_1$ using the same Sommerfeld expansion technique from our last lesson. This gives the well-known Mott formula, which for a parabolic band simplifies to:

$$S\approx -\frac{{\pi }^2}{2}\frac{k_B^{2}T}{e{ϵ}_F}$$

This result shows that thermopower in metals is typically small and linear in temperature. You can find a parallel derivation for this in the LibreTexts resource below.

3. Thermoelectric Devices and the Figure of Merit

The Seebeck effect is not just a scientific curiosity; it's the foundation of practical thermoelectric devices for power generation and temperature measurement (thermocouples). The inverse phenomenon, the Peltier effect, where an electrical current drives a heat flow, is used for solid-state cooling.

A key relationship, rooted in the Onsager reciprocity relations of non-equilibrium thermodynamics, connects the Peltier coefficient ($\Pi$) and the Seebeck coefficient ($S$):

$$\Pi =S\cdot T$$

This is known as the Kelvin relation. It means a material with a large Seebeck effect will also exhibit a strong Peltier effect.

The efficiency of a thermoelectric material for either power generation or cooling is captured by a single dimensionless quantity: the thermoelectric figure of merit, ZT.

The figure of merit is defined as:

$$ZT=\frac{S^2\sigma T}{\kappa }$$

where $\kappa ={\kappa }_e+{\kappa }_{ph}$ is the total thermal conductivity (from both electrons and phonons). To build a good thermoelectric device, we need to maximize ZT. This requires a material that simultaneously has:

1. A large Seebeck coefficient ($S$) to generate a large voltage per unit temperature difference.
2. High electrical conductivity ($\sigma$) to minimize resistive (Joule) heating losses.
3. Low thermal conductivity ($\kappa$) to maintain a large temperature gradient.

This presents a major challenge in materials science. From the Wiedemann-Franz law we saw last lesson, ${\kappa }_e$ and $\sigma$ are proportional in metals. Improving $\sigma$ usually increases ${\kappa }_e$, which works against maximizing ZT. Furthermore, both $S$ and $\sigma$ are sensitive functions of carrier concentration (doping), typically peaking at different values.

The key to high ZT materials is often to find ways to reduce the lattice thermal conductivity, ${\kappa }_{ph}$, without harming the electronic properties ($S^2\sigma$, the "power factor"). This has led to research into complex crystal structures and nanostructured materials that scatter phonons effectively without scattering electrons as much—an approach often described as "electron crystal, phonon glass."

Your experience with computational tools like Python and Matlab is directly applicable here. Researchers use codes that solve the BTE to calculate transport coefficients and guide the search for new materials.

Test your understanding!

A material has a very high ZT value at 800 K, but a very low one at 300 K. Based on the formula for ZT and the typical temperature dependence of $S$, $\sigma$, and $\kappa$, can you suggest a reason why this might be the case?

Show answer

There are several contributing factors. A key one is often the behavior of semiconductors. In many thermoelectric semiconductors, the Seebeck coefficient $S$ increases with temperature until it peaks and then drops due to the bipolar effect (thermal excitation of both electrons and holes, whose contributions to S can cancel). Electrical conductivity $\sigma$ in a semiconductor also increases with temperature as more carriers are activated. Thermal conductivity $\kappa$ typically decreases with temperature (e.g., as $1/T$) due to increased phonon-phonon scattering. The product of these dependencies, $S(T{)}^2\sigma (T)T/\kappa (T)$, can easily lead to a ZT value that is strongly peaked at a specific high temperature, making the material suitable for high-temperature waste heat recovery but not for room-temperature applications.

Conclusion

Today, we've explored the Seebeck effect, one of the cornerstones of thermoelectricity. By applying the BTE framework under the zero-current condition, we derived a formal expression for the Seebeck coefficient and understood its physical origin.

Key Takeaways:

• The Seebeck effect is the generation of a voltage in a material subjected to a temperature gradient under open-circuit ($\mathbf{j}=0$) conditions.
• The Seebeck coefficient ($S$) quantifies this effect. Its sign indicates the dominant charge carrier type (negative for electrons, positive for holes).
• Its physical origin lies in the asymmetry of charge transport around the Fermi energy. Hotter carriers diffuse, and the resulting charge buildup creates a counteracting field.
• The Peltier effect is the inverse phenomenon, and the two are linked by the Kelvin relation, $\Pi =ST$.
• The efficiency of thermoelectric devices is determined by the figure of merit, $ZT=S^2\sigma T/\kappa$. Designing high-ZT materials involves a challenging optimization of conflicting electronic and thermal properties.

Preview of Next Lesson:
We've now analyzed transport driven by $\mathbf{E}$ fields and $\nabla T$. The final piece of the puzzle in our semi-classical picture is to see what happens in the presence of a magnetic field, $\mathbf{B}$. In the next lesson, we will return to the full BTE, including the Lorentz force term, to calculate the Hall coefficient and magnetoresistance.