Charge distribution of nanoscale particles

The chemical activity of small metal particles can depend on the number of electrons on the particle. At room temperature, the energy needed to charge a nanometer diameter particle with one electron is typically lower than the thermal energy k_BT. This means that a collection of small metal particles will have a distribution of charge states. This charge distribution can be shifted by applying an electric field.

It is assumed that charge can tunnel onto the particles from a conducting substrate. This would be the case if the particles were lying on a metal electrode covered with a thin insulating layer. The equivalent electrical circuit in this case is shown in the following figure.

Here the resistor models the tunneling of electrons on an off the particle. A positive bias voltage will pull electrons onto the particle and a negative will push electrons off the particle. The charge on the particle is related to the voltage on the particle by the equation,

Here V is the voltage of the particle and q is the charge on the particle. The voltage of the particle is,

The energy needed to add an infinitesimal charge dq to a particle is Vdq. As soon as the charge is added, the voltage of the particle changes. To calculate the energy needed to add an electron to the particle, Vdq must be integrated from 0 to -e. The energy needed to add an electron to the particle is,

Here the charge has been written as the number of electrons on the island plus an offset charge, q = -ne + q₀ where e is the positive elementary charge. ΔE₊(n) is the energy needed to add an electron to a particle with n electrons already on it and V(n) is the voltage of the particle when there are n electrons on it. The second term is called the charging energy, E_c.

Similarly, the energy needed to remove an electron from the particle is,

The tunnel rate for electrons tunneling to tunnel onto the island Γ₊ and the tunnel rate for electrons to tunnel off the island Γ_- are,

In a stationary state, the current of electrons tunneling on the particle plus the current of electrons tunneling off of the particle adds to zero. This can be expressed as,

Here P(n) is the probability that there are n electrons on the particle. The following form uses the above equations to calclate the charge distribution P(n).