In this blog post, we will explore the secrets of conductivity in solids in an easy-to-understand and in-depth manner, focusing on the wave nature and localization of electrons.
Solids are one of the most common states of matter around us. However, we do not often have the opportunity to think about how the atoms that make up solids are arranged and how this arrangement affects the movement of electrons. At the microscopic level, solids are three-dimensional structures in which atoms are arranged in fixed positions relative to each other. When this arrangement is regular, the solid is called a “crystalline solid,” and when the arrangement is irregular and disordered, the solid is called an “amorphous solid.” As such, the structure of solids varies depending on the arrangement of atoms, which in turn greatly affects the behavior of electrons.
Electrons are not viewed solely as classical particles. In modern physics, electrons are considered to have dual nature, existing as both particles and waves. The wave nature of electrons plays a key role in explaining how electrons propagate within solids. The waves of electrons are mathematically expressed as complex “displacements.” This displacement consists of two elements, the amplitude and phase of the wave, and squaring the amplitude gives the probability of finding an electron at a specific location. In other words, to predict where an electron will be, both the amplitude and phase of the wave must be considered. The phase is a function that changes with time and space and is very important for explaining the interference and superposition phenomena of waves.
This wave nature is essential to understanding how electrons move within solids or remain in specific areas. In particular, the phenomenon of localization, in which electron waves are trapped in certain areas within solids and cannot spread further, is of great importance in physics.
Localization originally means that a specific object remains within a certain range. In physics, localization refers to the phenomenon in which electron waves are confined to a specific space and cannot spread outside. This phenomenon is not a simple physical constraint, but a result of complex interactions between the phase of electron waves, interference of paths, and structural disorder within solids. This localization of electrons appears in various forms, among which Anderson localization is the most widely known. Anderson localization is a phenomenon in which electron waves are completely trapped in disordered solids, i.e., amorphous solids.
In amorphous solids, atoms are arranged irregularly, so electrons encounter numerous collisions when trying to move from one position to another. Electron waves can propagate through various paths, but each path has a unique phase, and these phases often change randomly. When electron waves overlap on multiple paths, their phase signs cancel each other out, and the total displacement becomes almost zero. This means that the probability of an electron reaching a position far from its original position is almost zero, and eventually the electron waves can no longer spread and become spatially confined.
An important concept in this phenomenon is “localization length.” This refers to the range of space in which electron waves are confined, and the shorter this length, the more strongly the electrons are confined. In other words, a short localization length means that electrons can only move within a very narrow space, which is directly related to the insulating properties of solids.
However, in order for electrons to exist as waves and become localized, one condition must be met. That is, the wave phase of the electrons must be consistently defined in terms of time and space, which is called coherence. A coherent state means that the wave phase of the electrons has a certain regularity, which allows meaningful interference between waves to occur. The degree of coherence is expressed by a physical quantity called the coherence length. It is important that the coherence length be longer than the localization length, i.e., that the range in which the waves are confined be smaller than the distance over which the wave phase is sufficiently maintained, for localization to occur.
However, this coherence state is easily affected by the external environment. For example, as the temperature rises, the interaction between electrons becomes more active, and the atoms fluctuate, causing confusion in the wave phase of the electrons. As a result, the coherence length becomes shorter and shorter, and eventually localization does not occur. This explains why the localization phenomenon is more clearly observed at low temperatures.
On the other hand, localization also varies depending on the spatial dimension of the solid. For example, in a one-dimensional structure, electrons have little room to avoid obstacles, so localization occurs easily and the material behaves like an insulator. Conversely, in a three-dimensional structure, electrons can bypass obstacles in various directions, so they may not localize and the material may retain its conductive properties. This dependence on dimension is one of the key factors determining the conductivity and insulating properties of solids.
Localization does not necessarily manifest itself as complete confinement. In some cases, the electron waves do not come to a complete stop, but their propagation is temporarily impeded. This phenomenon is called weak localization. It is particularly useful in explaining changes in electrical resistance, especially the difference between the presence and absence of a magnetic field.
When electron waves start from random points inside a solid and move along various paths, some paths form closed curves that return to their starting points. At this point, electrons can rotate in both clockwise and counterclockwise directions, and since the two paths have the same length, their phases become the same. As a result, the two waves interfere with each other, increasing the magnitude of the displacement, which means that the electrons are more likely to return to their starting point. This interference has the effect of hindering the movement of electrons, resulting in an increase in electrical resistance. However, when an external magnetic field is applied, a phase difference occurs between the two paths, eliminating the interference effect and reducing the weak localization phenomenon. This is why electrical resistance varies depending on the presence or absence of a magnetic field.
On the other hand, localization can occur not only in disordered solids but also under completely different physical conditions. This is called dynamical localization, which occurs in chaotic systems. A chaotic system is a system in which even very small differences in initial conditions are amplified into very large differences over time. In classical mechanics, particles in such a chaotic environment spread along complex and unpredictable paths. However, the situation is different in quantum mechanics.
Despite the complex motion of electrons within chaotic systems, their waves are observed to be confined to specific spaces and do not spread. This is very similar to the Anderson localization phenomenon, in which electrons collide with disordered atomic structures and become localized within amorphous solids. Dynamical localization occurs in areas where the opposing concepts of classical chaos and quantum stability act simultaneously, making it a very interesting topic in physics.
The wave nature and localization of electrons in solids are not merely theoretical concepts. They play a key role in explaining various physical properties of solids, such as their conductivity, insulating properties, and changes in electrical properties depending on magnetic fields. Furthermore, understanding localization is an important foundation for practical technologies such as semiconductor devices, nanotechnology, quantum computers, and the development of new materials.
Through the regularity of waves that emerge even in disorder and the subtle interference and stagnation of electrons, we can glimpse the deep essence of solid matter. This is also a gateway to understanding both the complexity and beauty of the material world.
