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2.6: The Particle in a Box

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    50021

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    Modeling our electronic material as a box allows us to ignore atoms and assume that the material is perfectly homogeneous. We will consider boxes in different dimensions: either three dimensions (typical bulk materials), 2-d (quantum wells), 1-d (quantum wires), or 0-dimensions (this is a quantum dot). The label "quantum‟ here refers to the confinement of electrons. When we say that an electron is "confined‟ in a low dimensional material we mean that critical dimensions of the material are on the order of the wavelength of an electron. We‟ve seen that when particles are confined, their energy levels become discrete.

    Screenshot 2021-04-15 at 21.26.54.png
    Figure \(\PageIndex{1}\): The "particle in a box‟ takes a complex structure like a molecule and approximates it by a homogeneous box. All details, such as atoms, are ignored.

    In quantum dots, electrons are confined in all three dimensions, in quantum wires, electrons are confined in only two dimensions and so on. So when we say that a given structure is 2-d, we mean that the electron is unconfined in 2 dimensions. In the unconfined directions, we will assume that the electron described by a plane wave.

    Screenshot 2021-04-15 at 21.28.05.png
    Figure \(\PageIndex{2}\): (a) In quantum wells, electrons are confined only in one dimension. Quantum wells are usually implemented by burying the confining material within a barrier material. (b) Quantum wires confine electrons in two dimensions. The electron is not confined along the wire. (c) In a quantum dot, an electron is confined in three dimensions.

    This page titled 2.6: The Particle in a Box is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marc Baldo (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.