Quantum confinement effect: controlled dance of electrons at microscopic level

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Every solid material is made up of atoms and atoms have electrons. The arrangement of electrons in a material determines the properties of the material. In bulk materials (when the size of the material is big enough, let’s say more than 100 nm), the electrons are free to move in all the three dimensions (similar to us, as we also live in 3D) and at this level classical physics can explain all the behaviour of the materials. However, if we reduce the size of the material to microscopic scale, where electron movement is restricted at least in once dimension, the properties of the material change dramatically. At this scale, the properties of the materials can not be explained by classical physics and quantum physics is required.  

What Happens During Quantum Confinement?

Imagine a dance floor where movement is progressively restricted:

(1) Suppose you are at the dance floor and allowed to move only in two direction (x-y plane) but can not jump (restricted in the z-direction), this is called 1D confinement.

(2) Imagine you can dance but you can only move in a line (bongo line dance), now you are limited by 2 dimensions and this is 2D confinement.

(3) Suppose you are put in a dancing cage where you can not move in any direction and it is 3D confinement.

The same behaviour occurs with electrons;

(1) electrons can move only in two directions in 1D confinement,

(2) electrons can move only in one direction in 2D confinement, and

(3) electrons are not free to move in any direction in 3D confinement.

Confinement limit: When do we say that the electron is confined?

An interesting question arises when we talk about the electron confinement: in reality, we can not completely eliminate a dimension. So, how small a dimension should be for it to be called “confined”? For instance, imagine a cube which has length (l), width (b), and height (h). If we gradually reduce the height to confine the electrons in 2D plane, at what point can we consider the height negligible? You will find the answer in next two paragraphs.

Well, for a moment, let us discuss about electrons in a solid material. Electrons exhibit wave-like nature and can be described by wave function. Square of the wave function gives the probability of finding the electron in the material at a given time. Since electrons behave as waves, they are associated with a wavelength, which is called de Broglie wavelength. De Broglie wavelength of an electron depends inversely on the momentum of the electron (λ= h/p, where h is Planck’s constant, p is momentum). The momentum of an electron in a material depends on the effective mass of the electron in that material and its velocity, which is different for every material.  Hence, for every material the de-Broglie wavelength of an electron changes.

Coming back to quantum confinement effect, the quantum confinement effect starts to play the role when a materials dimension (such as height h in a cube) is reduced to a scale comparable to the de-Broglie wavelength of the electron. In other words, when the size of a dimension approaches the electron’s de Broglie wavelength, the system enters the quantum confinement regime. The critical size for quantum confinement typically ranges between 1 and 10 nm, though it varies depending on the material. This defines the threshold dimension at which confinement effects become significant.

When one dimension of a material is reduced to a size comparable to the de Broglie wavelength of an electron, the system exhibits 1D quantum confinement. A well known example of this is a quantum well, in which electrons are restricted in one dimension but can move freely in the other two.

When two of the three spatial dimensions of a material are reduced to the order of the de Broglie wavelength of the electrons, the system exhibits quantum confinement in two dimensions, commonly referred to as 2D confinement. A prominent example of this phenomenon is a nanowire. Nanowires with specific diameters can be fabricated from various materials, such as TiO₂, ZnO, and InSb, enabling unique electronic and optical properties tailored for advanced applications.

When all three spatial dimensions of a material are confined to the scale of the de Broglie wavelength of the electron, the material exhibits quantum confinement in all three dimensions, known as 3D confinement. A prominent example of this phenomenon is the quantum dot, which exhibits size dependent electronic and optical properties due to quantum confinement effects. The significance of quantum dots was recognized with the awarding of the 2023 Nobel Prize in Chemistry to Moungi G. Bawendi, Louis E. Brus, and Alexey I. Ekimov for their discovery and synthesis of these nanocrystals. Quantum dots have been synthesized from a variety of materials, including carbon (carbon quantum dots), cadmium selenide (CdSe), and cadmium telluride (CdTe).

To summarize it in a tabular form:

Quantum Structure	Confinement Type	Electron Freedom	Real-World Example

Quantum Well

Confinement in 1 dimension

Free to move in 2D (x, y), restricted in 1D (z)

Semiconductor lasers, LEDs

Quantum Wire

Confinement in 2 dimensions

Free in 1D (x), restricted in 2D (y, z)

Carbon nanotubes, nanowire transistors

Quantum Dot

Confinement in all 3 dimensions

Completely trapped, behaves like an atom

QLED displays, quantum computing

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