3.10: The Ideal Contact Limit†

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Marc Baldo
Massachusetts Institute of Technology via MIT OpenCourseWare

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Interfaces between molecules and contacts vary widely in quality. Much depends on how close we can bring the molecule to the contact surface. Here, we have modeled the source and drain interfaces with the parameters \(\tau_{S}\) and \(\tau_{D}\). If electron injection is unencumbered by barriers or defects then these lifetimes will be very short. We might expect that the current should increase indefinitely as the injection rates decrease. But in fact we find a limit – known as the quantum limit of conductance. We will examine this limit rigorously in the next section but for the moment, we will demonstrate that it also holds in this system.

We model ideal contacts by considering the current under the limit that \(\tau_{S}=\tau_{D} \rightarrow 0\). Note that the uncertainty principle requires that the uncertainty in energy must increase if the lifetime of an electron on the molecule decreases. Thus, the density of states must change as the lifetime of an electron on a molecule changes.

Let‟s assume that the energy level in the isolated molecule is discrete. In Chapter 2, we found a Lorentzian density of states for a single molecular orbital with net decay rate \(\tau_{S}^{-1}+\tau_{D}^{-1}\):

\[ g(E-U)dE = \frac{2}{\pi} \frac{(\hbar/2\tau_{S} + \hbar/2\tau_{D})}{\pi(E-U-E_{0})^{2}+(\hbar/2\tau_{S} + \hbar/2\tau_{D})^{2}}dE \nonumber \]

If we take the limit, we find that the molecular density of states is uniform in energy:

\[ \text{lim }\tau_{S} = \tau_{D}\rightarrow0 \ \ \ g(E-U)dE = \frac{8}{h}\frac{1}{1/\tau_{S}+1/\tau_{D}}dE \nonumber \]

Substituting into Equation (3.7.7) for \(\tau_{S}=\tau_{D}\) gives

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

At T = 0K,

\[ f(E,\mu) = u(\mu - E) \nonumber \]

where u is the unit step function, and the integral in Equation (3.10.3) gives

\[ I=\frac{2q}{h}(\mu_{S}-\mu_{D}) \nonumber \]

Note \(-qV_{DS} = (\mu_{D}-\mu_{S})\), thus the conductance through a single molecular orbital is

\[ G=\frac{2q^{2}}{h} \nonumber \]

The equivalent resistance is \(G^{-1}=12.9\ k\Omega\). Thus, even for ideal contacts, this structure is resistive. We will see this expression again in the next section. It is the famous quantum limited conductance.

\(^{†}\)This derivation of the quantum limit of conductance is due to S. Datta, "Quantum transport: atom to transistor" Cambridge University Press (2005).