Lorenz number

Why are materials that conduct electricity well, like copper, also excellent conductors of heat? This seemingly simple observation points to a deep and fundamental connection in the physics of matter, quantified by the Wiedemann-Franz law and its universal constant, the Lorenz number. For over a century, understanding this relationship has driven physicists from classical intuition to the strange realities of the quantum world. This article addresses the core question of why this law exists and what its implications are, revealing the Lorenz number as a powerful tool for exploring the electronic properties of materials.

The journey will unfold across two main chapters. In "Principles and Mechanisms," we will trace the historical and theoretical development of the law, contrasting the partially successful classical Drude model with the triumph of the quantum Sommerfeld model, which reveals the true nature of electrons in a metal. Subsequently, "Applications and Interdisciplinary Connections" will demonstrate how this principle is applied in diverse fields, from cryogenic engineering to the study of exotic states of matter. We will see how adherence to the law confirms our models and, more excitingly, how its violation provides crucial clues into new and unexplained physical phenomena.

Imagine you have two metal rods, one made of copper and the other of glass. You stick one end of each rod into a campfire. Which one would you rather be holding at the other end? The copper, of course, will get hot very quickly, while the glass will remain cool for much longer. Now, imagine you use these same two rods to complete a circuit with a light bulb and a battery. The copper rod will make the bulb shine brightly, while the glass rod will leave it dark.

This isn't a coincidence. It seems nature has a rule: materials that are good at conducting electricity are also good at conducting heat. This simple observation was quantified in the 19th century by Gustav Wiedemann and Rudolph Franz. They discovered that for most metals, the ratio of thermal conductivity, $\kappa$ (kappa), to electrical conductivity, $\sigma$ (sigma), is not just random but is directly proportional to the absolute temperature $T$.

$\frac{\kappa}{\sigma} = L T$

The constant of proportionality, $L$, was named the ​​Lorenz number​​. The truly startling discovery, made by Ludvig Lorenz, was that this number $L$ is remarkably similar for a vast range of different metals. From copper to aluminum to lead, the value hovered around a constant. This simple relationship, connecting two seemingly distinct properties, hinted at a deep, underlying unity in the physics of metals. It posed a challenge to the physicists of the time: why should this be so? What are the principles and mechanisms that tie heat and electricity together so intimately?

The first serious attempt to explain this law came from Paul Drude in 1900. His model was beautifully simple. He pictured a metal not as a solid, rigid lattice, but as a container filled with a gas of electrons, flitting about like tiny billiard balls and bouncing off the stationary metal ions.

In this picture, electrical conduction is easy to visualize. An electric field creates a gentle "wind," causing the electron gas to drift in one direction, producing a current. Thermal conduction is also intuitive: if you heat one end of the metal, the electrons there move faster. They then zip to the colder end, collide with slower electrons, and transfer their energy. Heat flows because the energetic electrons themselves flow.

Using the well-established kinetic theory of gases, Drude derived expressions for both $\sigma$ and $\kappa$. When he took their ratio, he found that many of the unknown parameters—like the density of electrons and the average time between their collisions—miraculously canceled out. His model successfully predicted the Wiedemann-Franz law! It even gave a value for the Lorenz number based only on fundamental constants: $L_D = \frac{3}{2} (\frac{k_B}{e})^2$, where $k_B$ is the Boltzmann constant and $e$ is the electron's charge.

This was a triumph, but a short-lived one. When the numbers were plugged in, the theoretical value was about $1.11 \times 10^{-8} \, \text{W}\Omega\text{K}^{-2}$. Experimental values at room temperature were more than twice as large, clustering around $2.3 \times 10^{-8} \, \text{W}\Omega\text{K}^{-2}$ for many metals like copper. The theory was in the right ballpark, but it was clearly missing something.

The fascinating part of the story, a detail that would make any physicist smile, is that Drude's model wasn't just a little bit wrong; it was spectacularly wrong in two different ways that just happened to cancel each other out. This "cancellation of errors" is a wonderful lesson in how a flawed model can sometimes give a deceptively good answer.

The first flaw was in the ​​heat capacity​​. The Drude model, treating electrons as a classical gas, predicted a large contribution to the metal's heat capacity. But experiments showed this was completely false; the electrons' contribution was over 100 times smaller than predicted. The second flaw was in the ​​electron's speed​​. The model assumed the electrons' kinetic energy was set by the temperature, $\frac{1}{2} m v^2 = \frac{3}{2} k_B T$. This wildly underestimated how fast the electrons responsible for conduction were actually moving.

In the calculation of thermal conductivity, the model used a heat capacity that was far too large and a velocity-squared that was far too small. The two errors, one in the numerator and one in the denominator, largely canceled, leading to a Lorenz number that was only off by a factor of about two. It was a fortunate accident, a hint that the underlying physics was far stranger than a simple gas of billiard balls.

The real answer, as is so often the case in modern physics, came from a quantum leap in thinking. The Sommerfeld model, developed in the late 1920s, kept Drude's idea of a "free electron gas" but treated the electrons correctly, as quantum particles.

Electrons are ​​fermions​​, and they obey the ​​Pauli exclusion principle​​: no two electrons can occupy the same quantum state. Imagine filling a giant bucket with water; you can't put all the water at the bottom. It fills up level by level. In a metal, the available energy states are like the space in the bucket, and the electrons are the water. At absolute zero temperature, the electrons fill every available energy state up to a sharp cutoff energy called the ​​Fermi energy​​, $E_F$. This vast collection of electrons is called the ​​Fermi sea​​.

This picture immediately solves the puzzles of the Drude model.

First, the heat capacity: The Fermi energy in a typical metal is huge, corresponding to a temperature of tens of thousands of Kelvin. This means at room temperature, the thermal energy $k_B T$ is just a tiny ripple on the surface of this deep sea. The vast majority of electrons are deep within the sea and cannot be excited; there are no empty states nearby for them to jump to. Only the electrons right at the top, on the ​​Fermi surface​​, have a place to go. This is why the electronic heat capacity is so small.

Second, the electron's speed: The electrons that do all the work—conducting heat and electricity—are the ones at the Fermi surface. And their energy isn't determined by the room's temperature, but by the enormous Fermi energy. They are moving at tremendous speeds, the ​​Fermi velocity​​, typically a hundred times faster than the classical model predicted.

When we re-derive the Lorenz number using these quantum mechanical ingredients, something beautiful happens. We once again find that parameters like the electron density and mass cancel out. The final expression for the Lorenz number, $L_0$, depends only on the most fundamental constants of nature:

$L_0 = \frac{\pi^2}{3} \left(\frac{k_B}{e}\right)^2 \approx 2.44 \times 10^{-8} \, \text{W}\Omega\text{K}^{-2}$

This is the celebrated ​​Sommerfeld value​​. When we compare it to the experimental value for copper, which is around $2.3 \times 10^{-8} \, \text{W}\Omega\text{K}^{-2}$, the agreement is stunning. The mystery is solved. The Wiedemann-Franz law is not an accident; it's a direct and profound consequence of the quantum nature of electrons in metals. The ratio of thermal to electrical conductivity is universal because both phenomena are governed by the same agents—the high-speed electrons at the Fermi surface—and the result is etched into the very fabric of physics by the constants $\pi$, $k_B$, and $e$.

The success of the Sommerfeld model is remarkable, but nature is always more subtle. The model's prediction holds true under one critical assumption: that electron collisions are ​​elastic​​. An elastic collision is like two billiard balls colliding; they change direction, but the total kinetic energy is conserved. For electrons scattering off static impurities in a crystal lattice at low temperatures, this is a very good approximation.

But what happens when collisions are ​​inelastic​​? This occurs, for example, when an electron collides with a vibration of the crystal lattice—a quantum of sound called a ​​phonon​​—and loses a significant chunk of its energy.

Think about the difference between electricity and heat flow. An electric current is a net flow of charge. A thermal current is a net flow of energy.

• To disrupt an ​​electric current​​, any collision that randomizes an electron's direction is effective. A small-angle elastic scattering event can do the job.
• To disrupt a ​​heat current​​, you need to stop the transport of energy. An inelastic collision that takes a high-energy electron and leaves it with low energy is devastatingly effective.

This means that inelastic scattering processes hinder thermal conductivity much more than they hinder electrical conductivity. When these processes become dominant (for instance, at intermediate temperatures where electron-phonon scattering is strong), the thermal conductivity $\kappa$ drops more than $\sigma$. As a result, the measured Lorenz number $L = \kappa/(\sigma T)$ falls below the ideal Sommerfeld value $L_0$. This deviation from the law is not a failure of physics; it's a new clue, telling us about the types of collisions the electrons are experiencing.

The Wiedemann-Franz law is a hallmark of being a metal. What happens if we look at materials that are not metals?

Consider an ​​insulator​​, like glass or a diamond. Here, there is no Fermi sea of free electrons. The electrons are all tightly bound to their atoms. Heat is transported not by electrons, but by the propagation of lattice vibrations—phonons. Charge, on the other hand, is hardly transported at all. The electrical conductivity is practically zero. If we were to naively form the Lorenz ratio, we would be dividing a finite thermal conductivity by a near-zero electrical conductivity. The result would be a gigantic number that diverges as the temperature approaches zero, bearing no resemblance to the universal value $L_0$. The law fails because its basic premise is violated: the carriers of heat (phonons) and the carriers of charge (electrons) are completely different entities.

An even more exotic and illustrative case is the ​​superconductor​​. Below a critical temperature, a material like lead loses all electrical resistance. Its conductivity, $\sigma$, becomes effectively infinite. What does this mean for the Lorenz number? If we plug $\sigma = \infty$ into the formula, we'd get $L=0$. But this is a red herring.

In a superconductor, electrons form ​​Cooper pairs​​, which can move through the lattice as a quantum superfluid, carrying charge with zero resistance. Here's the catch: this superfluid carries no entropy. It is in a perfect, ordered ground state and cannot be used to transport heat. Heat is still carried by the few remaining "normal" electrons (quasiparticles) and by phonons.

Once again, the carriers of charge (Cooper pairs) and heat (quasiparticles, phonons) have been decoupled. As in the insulator, the fundamental premise of the Wiedemann-Franz law is broken. Therefore, the very concept of a Lorenz number, which hinges on this unified transport mechanism, becomes physically inappropriate and meaningless for a superconductor.

The journey of the Lorenz number, from a curious empirical rule to a cornerstone of quantum solid-state physics, reveals the power and beauty of a physical law. It shows us how a simple ratio can be a deep probe, telling us not only about the quantum world of electrons in a metal, but also what it fundamentally means for a material to be an insulator or a superconductor.

After a journey through the fundamental principles of the Wiedemann-Franz law, one might be tempted to think of it as a neat, but perhaps niche, piece of physics. A curiosity of the antechamber of quantum mechanics. But nothing could be further from the truth. The story of what happens when we apply this law—and more importantly, what happens when it seems to fail—is a passport to some of the most fascinating and active frontiers of science. The Lorenz number, $L = \kappa/(\sigma T)$, is not just a constant; it is a character witness for the behavior of electrons in matter. It tells us whether they are behaving as the simple, independent particles of our textbooks, or whether they are embroiled in some deeper, more complex collective drama.

In its simplest guise, the Wiedemann-Franz law is an incredibly practical tool. Imagine you are an engineer designing a cryogenic system, a machine that operates at temperatures just a few degrees above absolute zero. You need to connect two components, but you want to do so with a material that provides a strong thermal link, allowing heat to flow easily. On the other hand, for a different part of your apparatus, you need a support structure that is a terrible conductor of heat, a thermal insulator, to keep the coldest parts from warming up. How do you choose your materials? You could embark on a series of difficult and expensive experiments to measure the thermal conductivity, $\kappa$, of various alloys at cryogenic temperatures. Or, you could remember the Wiedemann-Franz law.

At these frigid temperatures, electron scattering is dominated by static impurities and defects—a process that is almost perfectly elastic. This is the ideal stage for the Wiedemann-Franz law to perform. It tells us that $\kappa$ is directly proportional to $\sigma T$. Electrical conductivity, $\sigma$ (or its inverse, resistivity, $\rho$), is vastly easier to measure than thermal conductivity. So, our engineering problem is solved in a stroke: to find the best thermal conductor, we just need to find the best electrical conductor. The law provides a beautifully simple and effective design principle, born from the deep quantum nature of electrons.

But the law's success is not limited to such simple cases. It is a defining feature of a state of matter known as a ​​Fermi liquid​​. This is a concept of profound power and subtlety. In many materials, especially those with strong electronic interactions, electrons don't move as lone wolves. Their mutual pushing and shoving create a complex, correlated dance. Yet, remarkably, Landau's Fermi liquid theory tells us that the collective behavior of this electronic "liquid" can still be described by "quasiparticles"—excitations that act very much like individual particles, albeit with modified properties, such as a different mass.

Consider ​​heavy-fermion systems​​. In these exotic materials, interactions are so strong that the quasiparticles behave as if they are hundreds or even thousands of times heavier than a free electron. It's a "syrupy" sea of electrons. One might intuitively expect our simple law, born from a picture of a "gas" of electrons, to fail spectacularly. But it doesn't. At very low temperatures, these heavy quasiparticles form a perfectly respectable Fermi liquid, and the Wiedemann-Franz law holds with astonishing accuracy. The enormous effective mass and other effects of the strong interactions cancel out perfectly in the ratio of $\kappa$ to $\sigma T$. This triumph of the law is a powerful confirmation of the Fermi liquid concept itself, showing that the underlying principles are robust even when the characters in the play are heavily disguised.

The law has thus evolved into a crucial diagnostic tool. Physicists probing the mysteries of ​​high-temperature superconductors​​, for example, can suppress the superconductivity with an immense magnetic field. They can then measure $\kappa$ and $\sigma$ in the underlying "normal" state. If they find that the Lorenz number approaches the universal value $L_0$, it's a strong piece of evidence that, in this regime, the material is behaving as a Fermi liquid, despite all its other complexities. The law acts as a standard-bearer, telling us when we are on familiar ground.

If the success of the Wiedemann-Franz law is beautiful, its failure is, in many ways, even more exciting. A deviation from the universal Lorenz number $L_0$ is a red flag, a signpost telling us that the electrons are up to something unusual. These violations are not "errors"; they are clues pointing toward new physics.

​​The In-Between World of Semiconductors​​

Our first stop on this tour of the exotic is the familiar semiconductor. Unlike a metal, where the Fermi sea is vast and deep, a semiconductor has a limited number of charge carriers. This leads to two key ways the Wiedemann-Franz law can be violated. First, the electrons are often "non-degenerate," and their scattering rates can depend strongly on their energy, which is enough to shift the Lorenz Number away from $L_0$.

But a far more dramatic effect can occur at high temperatures: the ​​bipolar effect​​. As the semiconductor gets hot, thermal energy can create pairs of charge carriers: a negative electron and a positive "hole". Now, imagine a temperature gradient across the material. On the hot side, pairs are constantly being created. They can then diffuse to the cold side, where they recombine and release their formation energy—the band-gap energy, $E_g$—as heat.

This process constitutes a new, highly effective channel for heat transport. It's like a secret courier service for heat, carried by pairs of oppositely charged particles that create no net electric current. This "bipolar thermal conductivity" adds to the regular electronic thermal conductivity. If an unsuspecting engineer were to estimate the thermal conductivity of a thermoelectric device using the simple Wiedemann-Franz law, they would completely miss this enormous bipolar contribution and dramatically miscalculate the device's performance. Understanding this "failure" of the law is absolutely critical for the technology of converting heat directly into electricity.

​​The Symphony of Inelasticity​​

Even in metals, the law can break down if electron scattering is ​​inelastic​​—that is, if the electron loses a significant chunk of its energy in a collision. Elastic scattering is like a billiard ball hitting a stationary steel post; inelastic scattering is like it hitting a drum, setting it vibrating and losing energy in the process.

Heat current is disproportionately carried by the most energetic electrons. Inelastic scattering processes are often more effective at scattering these high-energy electrons than their lower-energy brethren. Consequently, inelastic scattering acts as a "tax" that falls more heavily on the heat current than on the charge current. This typically leads to a Lorenz number smaller than the universal value $L_0$.

This is precisely what is believed to happen in the mysterious "strange metal" phase of cuprate superconductors. The electrons are thought to scatter inelastically from a sea of magnetic fluctuations (ripples in the magnetic order of the material). By measuring the deviation from the Wiedemann-Franz law, physicists gain invaluable insight into the nature of this bizarre magnetic "soup" in which the electrons swim.

​​Life in One Dimension​​

What happens when we confine electrons to a one-dimensional wire? They can't go around each other; they are stuck in a line, like cars in a single-lane tunnel. The consequences are profound. Electrons lose their individual identity and are replaced by collective, wave-like excitations. In this strange new world, called a ​​Luttinger liquid​​, it's possible for an excitation of charge to propagate at a different speed from an excitation of heat!

This "spin-charge separation" is one of the most radical ideas in modern physics, and it rips the heart out of the Wiedemann-Franz law. Since charge and heat are no longer carried by the same "packets," there is no reason for their conductivities to be related in the same way. The Lorenz number is no longer the universal constant $L_0$. Instead, it becomes dependent on the strength of the interactions between the electrons. The 1D world reveals that the very concept of an electron as a discrete carrier of both charge and heat is an illusion, an emergent property of higher dimensions.

​​On the Edge of Chaos: Quantum Criticality​​

Perhaps the most dramatic violations of the Wiedemann-Franz law occur at a ​​quantum critical point​​ (QCP). This is a tipping point at absolute zero temperature between two distinct quantum phases of matter, such as a magnet and a non-magnet. At this critical point, the system is a seething, scale-invariant fractal of fluctuations. The notion of a quasiparticle breaks down completely.

Out of this chaos, however, a new kind of universality emerges. Transport coefficients, including the Lorenz number, take on new universal values that are the fingerprints of the specific critical point. These are not the familiar $L_0$ of the Fermi liquid but are new fundamental constants of nature. Theorists have predicted such universal, non-$L_0$ values for the Lorenz number at the metal-insulator transition, at deconfined quantum critical points described by exotic gauge theories, and in models of "strange metals" derived from string theory and holography. Measuring the Lorenz number in a real material and finding one of these new universal values would be a stunning discovery, a glimpse into a world of physics far beyond our conventional understanding of metals.

The profound connection between heat and charge transport is not just a feature of condensed matter. Let us look to a completely different realm: a ​​plasma​​, the hot, ionized gas of which stars are made. Here too, free electrons zip around, carrying charge when an electric field is applied and carrying thermal energy when there's a temperature gradient. Using the kinetic theory of classical gases, we can derive a Wiedemann-Franz-like law for a plasma. The structure of the law is identical, but the numerical prefactor changes. For a common model of a high-Z plasma, one finds $L \approx 3.2 (k_B/e)^2$, a value surprisingly close to the solid-state result of $L_0 = (\pi^2/3) (k_B/e)^2 \approx 3.29 (k_B/e)^2$. The difference arises from the different statistical rules governing the electrons (classical Maxwell-Boltzmann in the plasma versus quantum Fermi-Dirac in the metal) and the details of their collisions. Yet, the deep link between thermal and electrical conduction remains.

The Lorenz number is a simple ratio, born from a simple observation. Yet, chasing it through the myriad forms of matter takes us on a grand tour of modern physics. Where it holds, it solidifies our understanding of metals and gives us a powerful engineering tool. Where it bends, it reveals the subtle dances of bipolar diffusion and inelastic scattering. And where it breaks, it heralds the presence of entirely new states of matter—Luttinger liquids, quantum critical points, strange metals—where our most basic intuitions about electrons fall apart. The simple rule is beautiful. But the rich and complex tapestry of reasons for its exceptions is, perhaps, where the truest beauty lies.