3.4: Equilibrium between contacts and the conductor

In this section we will consider the combination of a molecule and a single contact.

In the absence of a voltage source, the isolated contact and molecule are at the same potential. Thus, their vacuum energies (the potential energy of a free electron) are identical in isolation. When the contact is connected with the molecule, equilibrium must be established in the combined system. To prevent current flow, there must be a uniform Fermi energy in both the contact and the molecule. But if the Fermi energies are different in the isolated contact and molecules, how is equilibrium obtained?

Figure \(\PageIndex{1}\): The energy lineup of an isolated contact and an isolated molecule. If there is no voltage source in the system, the energy of a free electron is identical at the contact and molecule locations. Thus, the vacuum energies align. The Fermi energies may not, however. But at equilibrium, the Fermi energies are forced into alignment by charge transfer.

Since Fermi levels change with the addition or subtraction of charge, equilibrium is obtained by charge transfer between the contact and the molecule. Charge transfer changes the potential of the contact relative to the molecule, shifting the relative vacuum energies. This is known as "charging". Charge transfer also affects the Fermi levels as electrons fill some states and empty out of others. Both charging and state filling effects can be modeled by capacitors. We‟ll consider electron state filling first.

The Quantum Capacitance

Figure \(\PageIndex{2}\): An analogy for electron transfer at the interface between a metal and a molecule. The size of the water tank is equivalent to the density of states. The Fermi level is equivalent to the water level. If the „metal‟ has a sufficiently large density of states, then the change in its water level is imperceptible.

But a molecule will not necessarily have a uniform density of states as shown in Figure \(\PageIndex{2}\). It is also possible that only a fractional amount of charge will be transferred. For example, imagine that some fractional quantity \(\delta n\) electrons are transferred from the contact to the molecule. It is possible for the wavefunction of the transferred electron to include both the contact and the molecule. Since part of the shared wavefunction resides on the molecule, this is equivalent to a fractional charge transfer.

But if \(\delta n\) were equal to +1, the LUMO would be half full and hence the Fermi energy would lie on the LUMO, while if \(\delta n\) were -1, the HOMO would be half full and hence the Fermi energy would lie on the HOMO. In general, the number of charges on the molecule is given by

\[ n=\int^{+\infty}_{-\infty} g(E)f(E,E_{F})dE \nonumber \]

where g(E) is the density of molecular states per unit energy. For small shifts in the Fermi energy, we can linearize Equation (3.4.1) to determine the effect of charge transfer on \(E_{F}\). We are interested in the quantity \(dE_{F}/dn\). For degenerate systems we can simplify Equation (3.4.1):

\[ n = \int^{E_{F}}_{-\infty}g(E)dE \nonumber \]

taking the derivative with respect to the Fermi energy gives:

\[ \frac{dn}{dE_{F}} =g(E_{F}) \nonumber \]

We can re-arrange this to get:

\[ \delta E_{F} = \frac{\delta n}{g(E_{F})} \nonumber \]

Thus after charge transfer the Fermi energy within the molecule changes by \(\delta n/g\), where g is the density of states per unit energy.

Sometimes it is convenient to model the effect of filling the density of states by the "quantum capacitance" which we will define as:

\[ C_{Q}=q^{2}g(E_{F}) \nonumber \]

i.e.

\[ \delta E_{F} = \frac{q^{2}}{C_{Q}}\delta n \nonumber \]

If the molecule has a large density of states at the Fermi level, its quantum capacitance is large, and more charge must be transferred to shift the Fermi level.

Figure \(\PageIndex{3}\): Transferring charge changes the Fermi level in a conductor. The magnitude of the change is determined by the density of states at the Fermi level, and often expressed in terms of a "quantum capacitance".

We can also calculate the quantum capacitance of the contact. Metallic contacts contain a large density of states at the Fermi level, meaning that a very large number of electrons must be transferred to shift its Fermi level. Thus, we say that the Fermi energy of the contact is "pinned" by the density of states. Another way to express this is that the quantum capacitance of the contact is approximately infinite.

The quantum capacitance can be employed in an equivalent circuit for the metalmolecule junction. But we have generalized the circuit such that each node potential is the Fermi level, not just the electrostatic potential as in a conventional electrical circuit.

In the circuit below, the metal is modeled by a voltage source equal to the chemical potential \(\mu_{1}\) of the metal. Prior to contact, the Fermi level of the molecule is \(E_{F}^{0}\). The contact itself is modeled by a resistor that allows current to flow when the Fermi levels on either side of the contact are misaligned. Charge flowing from the metal to the molecule develops a potential across the quantum capacitance. But note that this is a change in the Fermi level, not an electrostatic potential. It is also important to note that the quantum capacitance usually depends on the Fermi level in the molecule. The only exception is if the density of states is constant as a function of energy. Thus, a constant value of \(C_{Q}\) can only be employed for small deviations between \(\mu_{1}\) and \(E_{F}^{0}\).

Figure \(\PageIndex{4}\): A small signal model for the metal-molecule junction. The effects of charging are not included. The resistor will be characterized further in later sections.

Electrostatic Capacitance

Unfortunately, the establishment of equilibrium between a contact and the molecule is not as simply as water flow between two tanks. Electrons, unlike water, are charged. Thus, the transfer of electrons from the contact to a molecule leaves a net positive charge on the contact and a net negative charge on the molecule.

Charging at the interface changes the potential of the molecule relative to the metal and is equivalent to shifting the entire water tanks up and down. Charging assists the establishment of equilibrium and it reduces the number of electrons that are transferred after contact is made.

Figure \(\PageIndex{5}\): Electrons carry charge and shift the potential when they are transferred between a metal and a molecule. The resulting change in potential is equivalent to lifting up the "molecule" column of water. The water levels must ultimately match, but now less water is required to be transferred.

The contact and the molecule can be considered as the two plates of a capacitor. In Figure 3.4.6 we label this capacitor, \(C_{ES}\) - the electrostatic capacitance, to distinguish it from the quantum capacitance discussed in the previous section.

When charge is transferred at the interface, the capacitor is charged, a voltage is established and the molecule changes potential. The change in the molecule‟s potential per electron transferred is known as the charging energy and is reflected in a shift in the vacuum energy. From the fundamental relation for a capacitor:

\[ C_{ES}=\frac{Q}{V} \nonumber \]

where V is the voltage across the capacitor. We can calculate the change in potential due to charging:

\[ U_{C} = qV = \frac{q^{2}}{C_{ES}}\delta n \nonumber \]

Figure \(\PageIndex{6}\): A contact and a molecule can be modeled as two plates of a parallel capacitor. When charge is transferred, this electrostatic capacitance determines the change in electrostatic potential, and hence the shift in the vacuum energy.

We will find that \(\delta n\) is a dynamic quantity – it changes with current flow. It can be very important in nanodevices because the electrostatic capacitance is so small. For the small spacings between contact and conductor typical of nanoelectronics (e.g. 1 nm), the charging energy can be on the order of 1V per electron.

Summarizing these effects, we find that the Fermi energy of the neutral molecule, \(E_{F}^{0}\), is related to the Fermi energy of the metal-molecule combination, \(E_{F}\), by

\[ E_{F} = \delta n/g +\frac{q^{2}}{C_{ES}}\delta n +E_{F}^{0} \nonumber \]

Figure \(\PageIndex{7}\): A small signal model for the metal-molecule junction, including the effects of charging. The resistor will be characterized further in later sections.

Or, in terms of the quantum capacitance:

\[ E_{F} = \frac{q^{2}}{C_{Q}} \delta n + \frac{q^{2}}{C_{ES}} \delta n + E_{F}^{0} \nonumber \]

Figure \(\PageIndex{8}\): Changes in energy level alignment when charge is transferred from the metal to a molecule. Charging of the molecule corresponds to applying a voltage across an interfacial capacitor, thereby changing the potential of the molecule. Consequently, the vacuum level shifts at the molecule‟s location, shifting all the molecular states along with it. In addition, the transferred charge fills some previous empty states in the molecule. Both effects change the Fermi energy in the molecule.