3.7: Calculation of Current

Let's model the net current at each contact/molecule interface as the sum of two components: the contact current, which is the current that flows into the molecule, and the molecule current, which is the current that flows out of the molecule.

Figure \(\PageIndex{1}\): The net current at a contact/molecule interface can be broken into a contact current – the current that flows out of the contact - and a molecule current – the current that flows out of the molecule. At equilibrium, these currents must balance.

The contact current

This current is the number of available states in the molecule filled per second. Electrons in the contact are filled to its chemical potential. They cannot jump into higher energy states in the molecule. The total number of electrons that can be transferred is simply equal to the number of states.

At the source contact, we get

\[ N_{S} = \int^{\infty}_{-\infty}g(E-U)f(E,\mu_{S})dE \nonumber \]

where \(g(E-U)\) is the molecular density of states shifted by the net potential change. Similarly, if at the drain contact then the number of electrons, \(N_{D}\), that could be transferred level is

\[ N_{D} = \int^{\infty}_{-\infty}g(E-U)f(E,\mu_{D})dE \nonumber \]

Let's define the transfer rate at the source and drain contacts as \(1/\tau_{S}\) and \(1/\tau_{D}\), respectively. Then the contact currents are

\[ I^{C}_{S} = q\frac{N_{S}}{\tau_{S}},\ \ \ I^{C}_{D} = -q\frac{N_{D}}{\tau_{D}} \nonumber \]

Note that we have defined electron flow out of the source and into the drain as positive.

Figure \(\PageIndex{2}\): The contact current is the rate of charge transfer from the contact to the molecule. Only states in the molecule with energies below the chemical potential of the contact may be filled. The transfer rate of a single electron from the contact is \(1/\tau_{S}\).

The molecule current

Now, if we add electrons to the molecule, these electrons can flow back into the contact, creating a current opposing the contact current. The molecule current is the number of electrons transferred from the molecule to the contact per second.

Thus, the molecule currents into the source and drain contacts are

\[ I^{M}_{S} = -q\frac{N}{\tau_{S}},\ \ \ I^{M}_{D} = q\frac{N}{\tau_{D}} \nonumber \]

where we have again defined electron flow out of the source and into the drain as positive.

Figure \(\PageIndex{3}\): The molecule current is the rate of charge transfer from the molecule to the contact. Note that the electrons on the molecule are not necessarily in equilibrium. The lifetime of a single electron on the molecule is \(\tau\).

From Equations (3.7.3) and (3.7.4) the net current at the source contact is

\[ I_{S} = \frac{q}{\tau_{S}}(N_{S}-N) \nonumber \]

and the net current at the drain contact is

\[ I_{D} = \frac{q}{\tau_{D}}(N-N_{D}) \nonumber \]

Note that we have assumed that the transfer rates in and out of each contact are identical. For example, let's define \(\tau^{M}_{S}\) as the lifetime of an electron in the molecule and \(1/\tau^{C}_{S}\) as the rate of electron transfer from the source contact. It is perhaps not obvious that \(\tau^{M}_{S}=\tau^{C}_{S}\), but examination of the inflow and outflow currents at equilibrium confirms that it must be so. When the source-molecule junction is at equilibrium, no current flows. From Equations (3.6.6), (3.6.7) and (3.7.1), we have \(N_{S} =N\). Thus, for \(I_{S} =0\) we must have \(\tau_{1}^{M} = \tau_{1}^{C}\).

Equating the currents in Equations. (3.7.5) and (3.7.6) gives\(^{†}\)

\[ I=q\int^{\infty}_{-\infty}g(E-U)\frac{1}{\tau_{S}+\tau_{D}} (f(E,\mu_{S})-f(E,\mu_{D}))dE \nonumber \]

and

\[ N = \int^{\infty}_{-\infty} g(E-U)\frac{\tau_{D}f(E,\mu_{S})+\tau_{S}f(E,\mu_{D})}{\tau_{S}+\tau_{D}}dE \nonumber \]

The difficulty in evaluating the current is that it depends on U and hence N. But Equation (3.7.8) is not a closed form solution for N, since the right hand side also contains a N dependence via U. Except in simple cases, this means we must iteratively solve for N, and then use the solution to get I. This will be discussed in greater detail in the problems accompanying this Part.

Figure \(\PageIndex{4}\): A flow diagram describing an iterative solution to the IV characteristics of two terminal molecular devices. Adapted from F. Zahid, M. Paulsson, and S. Datta, "Electrical conduction in molecules". In Advanced Semiconductors and Organic Nanotechniques, ed. H. Korkoc. Academic Press (2003).

\(^{†}\)F. Zahid, M. Paulsson, and S. Datta, "Electrical conduction in molecules‟. In Advanced Semiconductors and Organic Nanotechniques, ed. H. Korkoc. Academic Press (2003).