# Everything’s a spring

Solids are made of atoms which are bonded together chemically. Many solids have some periodic order, or structure, while others are amorphous. Either way, they have atoms that are bonded together and those bonds could be thought of as springs. The truth is that atoms do not stop moving but instead are constantly **vibrating**. Their back-and-forth, or oscillatory, motion can be represented by a bunch of balls attached to each other with springs.

When one atom moves, the ones next to it move as well. The atoms bonded to those atoms start moving, and so on throughout the block of material. The array of atoms in a block of material is called its **lattice**. The motion of a spring can be represented like a wave, with its oscillatory motion representing the amplitude. Imagine how complicated it would be to model the motion of all of the atoms in a lattice. To give some perspective, consider a grain of sand. In just one grain of sand, there are in the realm of 1-100 quintillion atoms (on the order of 10^{18}-10^{20}).

Why do we care about this in the first place? It turns out that understanding these oscillations is highly important in understanding the electrical and thermal properties of materials. This includes electronics, superconductivity, heat transfer, and material failure. These vibrations have discrete values which are proportional to their frequency (oscillations per second). In other words, atomic vibrations are quantized and their quanta are called **phonons**.

## Phonons are freaky

Phonons are the quanta of lattice vibration. They put numbers to the vibrations in solids that we cannot see. The lowest energy vibration occurs at 0 K, i.e., *atoms never stop moving*. Phonons are quantum mechanical objects that obey different laws than other particles. Phonons are Bosons, with an integer spin of 0. This means that they don’t obey the Pauli Exclusion Principle, so more than one can exist in the same state at the same spot. There are 3N modes* *of phonons in a bulk material, for N is the number of atoms. Every mode is an independent vibration state for which a phonon can occupy. Since phonons are Bosons, any number of phonons in the same state can exist on top of each other. Up to 3N modes of phonons can exist in a lattice at once while there could be any number of phonons in each occupied state.

Phonons can interact with each other, electrons, atoms, and defects in materials. The electrical properties of materials must be described with regards to phonons because they can reduce conductivity. When phonons hit electrons, they knock them off of their course and cause them to take longer to move around the circuit. This reduces conductivity, whereas in semiconductors, it is the phonon’s energy that brings about conduction in the first place. Their energy breaks apart chemical bonds and frees up electrons. Electrical conduction in semiconductors is a thermally activated process.

Phonons are a quanta of lattice vibration *and* of heat. Remember how there can be up to 3N modes of phonons in a lattice? Thermal energy produces more and more modes of phonons up until a certain point (Debye temperature). In each material, there is a certain cutoff frequency for phonons. This means that phonons can exist in any frequency up until this cutoff (Debye) frequency. Once the material is at the Debye temperature, all frequencies up to the Debye frequency are excited. No more frequencies can be excited and thus every *type* of phonon has been created (they are a quasiparticle, so they can be created or destroyed). As a solid heats further, more phonons are created. The *number of* phonons increases with temperature linearly while the *number of modes* of phonons increases cubically.