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4.7: The Landauer Formula†

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    51311
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    We are now going to generalize the result of Equation (4.6.9) by considering conduction at higher temperatures and in the presence of a scattering site.

    Electrons flowing through the wire may be reflected by the scatterer. We define the transmission probability \(\Im\), of the scatterer, and assume that it acts equally on electrons flowing in either direction in the wire.

    Screenshot 2021-05-12 at 19.43.25.png
    Figure \(\PageIndex{1}\): A quantum wire containing a scatterer with transmission probability \(\Im\).

    Let‟s define \(i_{S}^{+}\) as the current carried by all electrons (compensated and uncompensated) in the \(+k_{z}\) states in the wire adjacent to the source. Let \(i_{S}^{-}\) be the current carried by all electrons in the \(-k_{z}\) states in the wire adjacent to the source. Similarly, we define \(i_{D}^{+}\) and \(i_{D}^{-}\) as the currents entering and leaving the drain, respectively.

    Generalizing Equation (4.6.5) for wires with multiple modes and arbitrary temperatures, we calculate the number of electrons traveling in the \(+k_{z}\) states adjacent to the source:

    \[ N_{S}^{+} = 2 \int^{\infty}_{0} \frac{dk}{2\pi /L}\ M(E(k))f(E(k), \mu_{S}) \nonumber \]

    where the number of modes at energy E is M(E), and as before \(f(E,\mu)\) is the probability that a state of energy E is filled given the chemical potential \(\mu\). It follows that

    \[ i_{S}^{+} = \frac{2q}{h} \int^{\infty}_{0} M(E)f(E, \mu_{S}) dE \nonumber \]

    \[ i_{D}^{-} = \frac{2q}{h} \int^{\infty}_{0} M(E)f(E, \mu_{D}) dE \nonumber \]

    and

    \[ i_{D}^{+} = \frac{2q}{h} \int^{\infty}_{0} \Im M(E)f(E,\mu_{S}) + (1-\Im) M(E) f(E,\mu_{D})dE \nonumber \]

    \[ i_{S}^{+} = \frac{2q}{h} \int^{\infty}_{0} \Im M(E)f(E,\mu_{D}) + (1-\Im) M(E) f(E,\mu_{S})dE \nonumber \]

    The total current is \(I = i_{S}^{+}-i_{S}^{-} = i_{D}^{+}-i_{D}\), this gives us the Landauer Formula

    \[ I = \frac{2q}{h} \int^{\infty}_{0} \Im M(E)(f(E,\mu_{S})-f(E, \mu_{D}))dE \nonumber \]

    \(^{†}\)This section is adapted from S. Datta, "Electronic Transport in Mesoscopic Systems" Cambridge (1995).


    This page titled 4.7: The Landauer Formula† 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.