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2.3: Current

  • Page ID
    50018
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    The electron distribution within a material determines its conductivity. As an example, let's consider some moving electrons in a Gaussian wavepacket. The wavepacket in turn can be described by the weighted superposition of plane waves. Now, we know from the previous section that, if the wavepacket is not centered on k = 0 in k-space, then it will move and current will flow.

    There is another way to look at this. Note that there are plane wave components with both \(+k_{z}\) and \(-k_{z}\) wavenumbers. Thus, even when the electron is stationary, components of the wavepacket are traveling in both directions. But if each component moving in the \(+k_{z}\) direction is balanced by an component moving in the \(-k_{z}\) direction, there is no net current.

    Screenshot 2021-04-15 at 21.08.12.png
    Figure \(\PageIndex{1}\): A wavepacket with no net velocity. Note that each plane wave component with a \(+k_{z}\) wavenumber is compensated by a plane wave component with a \(-k_{z}\) wavenumber. (b) A wavepacket with a net velocity in the positive z direction is asymmetric about \(k_{z} = 0\).

    We‟ll show in this section that we can apply a similar analysis to electrons within a conductor. For example, electrons in a wire occupy states with different wavenumbers, known as k-states. Each of these states can be modeled by a plane wave and there are states that propagate in both directions.

    Recall that for a plane wave

    \[ E = \frac{\hbar^{2}k_{z}^{2}}{2m} \nonumber \]

    This relation between energy and wavenumber is known as a dispersion relation. For plane waves it is a parabolic curve. Below the curve there are no electron states. Thus, the electrons reside within a certain band of energies.\(^{†}\)

    Screenshot 2021-04-15 at 21.11.38.png
    Figure \(\PageIndex{2}\): Electrons in a wire occupy states with different energies and wavenumbers.

    Under equilibrium conditions, the wire is filled with electrons up to the Fermi energy, \(E_{F}\). The electrons fill both \(+k_{z}\) and \(-k_{z}\) states, and propagate equally in both directions. No current flows. We say that these electrons are compensated.

    Screenshot 2021-04-15 at 21.12.58.png
    Figure \(\PageIndex{3}\): (a) Under equilibrium conditions, electrons fill up the lowest energy k-states first. Since equal numbers of \(+k_{z}\) and \(-k_{z}\) states are filled there is no net current. (b) When \(+k_{z}\) and \(-k_{z}\) states are filled to different levels, there is a net current.

    For a net current to flow there must be difference in the number of electrons moving in each direction. Thus, electrons traveling in one direction cannot be in equilibrium with electrons traveling in the other. We define two quasi Fermi levels: \(F^{+}\) is the energy level when the states with \(k_{z}>0\) are half full, \(F^{-}\) is the corresponding energy level for states with \(k_{z}<0\). We can see that current flow is associated with a difference in the quasi Fermi levels, and the presence of electron states between the quasi Fermi levels. If there are no electrons between \(F^{+}\) and \(F^{-}\), then the material is an insulator and cannot conduct charge.

    Screenshot 2021-04-15 at 21.15.55.png
    Figure \(\PageIndex{4}\): Examples of a metal (a), and an insulator (b).

    To summarize: current is carried by uncompensated electrons.

    \(^{†}\)Strictly a band needs an upper as well as a lower limit to the allowed energy, whereas the simple plane wave model yields only a lower limit. Later we'll also find upper limits in more accurate models of materials. Note also that 0-d materials such as molecules or quantum dots do not have bands because the electrons are confined in all directions and cannot be modeled by a plane wave in any direction.


    This page titled 2.3: Current 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.