Skip to main content
Engineering LibreTexts

3.2: Quantum Dot / Single Molecule Conductors

  • Page ID
    50024
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    As we saw in Chapter 2, a quantum dot is a 0-d conductor; its electrons are confined in all dimensions. A good example of a quantum dot is a single molecule that is isolated in space. We can approximate our quantum dot or molecule by a square well that confines electrons in all dimensions. One consequence of this confinement is that the energy levels in the isolated quantum dot or molecule are discrete. Typically, however, the simple particle-in-a-box model does not generate sufficiently accurate estimates of the discrete energy levels in the dot. Rather, the material in the quantum dot or the structure of the molecule defines the actual energy levels.

    Figure 3.2.1 shows a typical square well with its energy levels. We will assume that these energy levels have already been accurately determined. Each energy level corresponds to a different molecular orbital. Energy levels of bound states within the well are measured with respect to the Vacuum Energy, typically defined as the potential energy of a free electron in a vacuum. Note that if an electric field is present the vacuum energy will vary with position.

    Next we add electrons to the molecule. Each energy level takes two electrons, one of each spin. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are particularly important. In most chemically stable materials, the HOMO is completely filled; partly filled HOMOs usually enhance the reactivity since they tend to readily accept or donate electrons.

    Earlier we stated that charge transport occurs only in partly filled states. This is best achieved by adding electrons to the LUMO, or subtracting electrons from the HOMO. Modifying the electron population in all other states requires much more energy. Hence we will ignore all molecular orbitals except for the HOMO and LUMO.

    Figure 3.2.1 also defines the Ionization Potential (IP) of a molecule as the binding energy of an electron in the HOMO. The binding energy of electrons in the LUMO is defined as the Electron Affinity (EA) of the molecule.

    Screenshot 2021-04-24 at 15.43.14.png
    Figure \(\PageIndex{1}\): A square well approximation of a molecule. Energy levels within the molecule are defined relative to the vacuum energy – the energy of a free electron at rest in a vacuum.

    This page titled 3.2: Quantum Dot / Single Molecule Conductors 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.