Electron-phonon coupling in metals: Force constants from tight-binding models

Abstract

Electron-lattice interactions play a large role in determining physical properties of crystalline solids, such as electrical resistance and superconductivity. Properties like these are not only of theoretical interest, but also of practical importance when designing new materials. A key quantity describing these interactions is the electron-phonon coupling, which links atomic displacement to changes in the electronic free energy and the resulting lattice forces. This thesis investigates the interatomic force constants emerging from the electron-phonon coupling in a two-dimensional square lattice, within a tight-binding framework. By introducing a distance-dependent hopping parameter, mobile electrons couple to the lattice and the resulting forces can be derived from the electronic structure. The force constants are defined as the second derivative of the Helmholtz free energy of the electronic Hamiltonian. An analysis is given of how variables such as temperature, chemical potential, hopping strength and atomic displacement affects the force constants. Finally, the strength and limitations of this approach to electron-lattice coupling are discussed.

Popular Abstract

At first glance, a solid crystal might look perfectly still to the human eye. Take a small piece of metal for example, holding it in your hand, how could it be anything other than perfectly rigid? However, if we zoom in to the atomic scale, nothing is ever truly at rest. Even at the coldest possible temperature, the atoms are actually constantly moving, vibrating around their equilibrium positions in the lattice (the repeating arrangement of atoms). As the atoms vibrate, they influence each other through forces that behave very similarly to springs between them. This means that when one atom moves, it creates a ripple effect by dragging or pushing the other atoms around it, which in turn do the same to other atoms. This traveling motion of all atoms is known as phonons, the collective vibrational motion of atoms. Phonons are not only fun physical quirks, they are responsible for many of the crystals properties, such as how well it transfers heat, how sound propagates through it, and even how well it conducts electricity.

In metals, these effects get even more amplified due to presence of very mobile electrons. The electrons can move around through the lattice, exchanging energy with the phonons. At the same time, phonons in the crystal cause neighboring atoms to have varying distances, which affect the strength of which an electron will move from one atom to another. This complex interaction is known as electron-phonon coupling.

Studying real materials can be both hard and expensive, and instead we focus on building a simple computational model to look into these effects. Our starting point is the tight-binding model, which is just a set of simple, well tested rules that we can follow. Basically, this only means that we assume that the electrons are bound to their respective atom, but that they can hop to the closest neighboring atoms. (Even though it might sound a bit silly, “hop” is actually the most common physical term for this!). Despite being a very simple model, it has been shown that the tight-binding model is good enough to capture features of real materials. The key idea of the model is just like we mentioned, that hopping depends on how far apart the atoms are. When the atoms move closer together or farther apart, the hopping strength of the electrons changes. This in turn creates a feedback loop of atomic motion affecting electron hopping, and electron hopping in turn affecting the atomic motion.

Our main interests in this project is to understand how the electron-phonon coupling affect the forces between two given atoms, also known as the force constant (how strongly each atom resists being moved relative to the other). The force constant might be different for each atomic pair, but patterns will emerge based on their relative positions, which means that just looking at a small part of the lattice is enough. By finding all of these patterns for a given lattice structure, we have basically constructed a detailed map of the spring network in the crystal. These force constants are not just abstract numbers, they play a huge part in determining how vibrations spread through the material, which as we mentioned earlier, leads to many of its properties. What makes this especially interesting is that using relatively simple computer code, we can still look into these material defining effects. In the end, research like this might help us understand how behavior at the quantum level affects material properties.

• author: Möller, Jesper
• supervisor: Erik van Loon; Ali Sufyan
• organization: Department of Physics; Mathematical Physics
• course: FYSK04 20261
• year: 2026
• type: M2 - Bachelor Degree
• subject: Physics and Astronomy
• language: English
• id: 9239447
• date added to LUP: 2026-06-16 15:14:46
• date last changed: 2026-06-16 15:14:46