Advanced Solid State Physics

Electrostatics of a charging network

Electrostatics of a system of N conductors
Consider a system consisting of N conductors. A capacitance can be defined between each conductor and every other conductor as well as a capacitance from each of the N conductors to ground. This results in a total of N(N + 1)/2 capacitors. The capacitor between node j and node k has a capacitance c_jk and stores a charge q_jk. The total charge on node j is the sum of the charges on all of the capacitors connected to node j,

[1]

Here V_j is the electrostatic potential of node j and ground is defined to be at zero potential, V₀ = 0. The charges on the nodes are linear functions of the potentials of the nodes so this can be expressed more compactly in matrix form,

[2]

where C is called the capacitance matrix. A diagonal element of the capacitance matrix, C_jj, is the total capacitance of node j,

[3]

An off-diagonal element of the capacitance matrix is minus the capacitance between node j and node k, C_jk = C_kj = - c_jk.

The electrostatic energy of this system of conductors is the sum of the electrostatic energy stored on the N(N + 1)/2 capacitors,

[4]

The electrostatic energy of the system can be conveniently expressed using the capacitance matrix,

[5]

By differentiating the above expression for the energy it is possible to show that the charges and the voltages can be expressed as,

and

[6]

Tunneling between charge nodes
If tunnel junctions are included in the network, then charge can tunnel between the charge nodes. When an electron tunnels from one charge node to another, there is a change in the electrostatic energy. The system tunnels from a state of higher electrostatic energy to a state of lower electrostatic energy and the difference in energy is dissipated as heat.

A differential change in electrostatic energy is,

[7]

This can be written in terms of the voltages,

If a charge dQ_j is added to node j, then the differential change in energy is,

[9]

Integrating the change in energy from 0 to |e| gives the energy needed to add an electron to node j,

[10]

Often one is interested in the difference in the electrostatic energy when an electron tunnels from one node to another. Consider the case where an electron is transported from node i to node f where the initial voltage on node i is V_i and the initial voltage on node f is V_f. If the electron is first removed from node i, the change in electrostatic energy will be,

[11]

Because the charge on node i has changed, the voltages on the nodes throughout the network have changed. Let V_i' and V_f' be the voltages on nodes i and f after the electron has been removed,

[12]

The electron is then placed on node f. The change in energy for this second step is

[13]

The total change in energy when an electron is transported from node i to node f is

[14]

Note that the change in electrostatic energy depends only on the difference in voltage between the initial and final nodes plus a term that is independent of the charge state of the system.

Voltage Sources
Voltage sources can be included in the network by treating them as nodes with large capacitances to ground and large charges on them such that V = Q/C. In this case, it is numerically difficult to compute the inverse of the capacitance matrix since it contains large elements. However, it is not necessary to invert the entire capacitance matrix since the voltages on the voltage sources are already known. Only the voltages on the other nodes need to be determined. These voltages can be determined by writing the relation between the charges and the voltages as,

[15]

Here and are the charges and the voltages on the charge nodes, and are the charges and the voltages on the voltage sources, and the capacitance matrix has been expressed in terms of four submatrices.

The voltages on the charge nodes are then,

[16]

Once all of the voltages have been determined, the change in electrostatic energy can be calculated using Eq. 8. The change in electrostatic energy when an electron tunnels from charge node i to charge node f is,

[17]

The change in electrostatic energy when an electron tunnels from charge node i to a voltage source f is,

[18]

Offset charges
Offset charges or background charges are charges on extra charge nodes that appear unintentionally in circuits. They are usually due to charged defects in the oxide surrounding the intentionally fabricated charge nodes. These nodes can also be included in the relation between charge and voltage,

[19]

Here and are the charges and the voltages on the offset nodes and the capacitance matrix has been divided into nine submatrices.

Since the offset charge nodes are unintentional, the capacitance between an offset charge node and the other charge nodes are not known. Usually these capacitances are assumed to be small and are ignored in the diagonal elements of the capacitance matrix. The voltage on the charge nodes can then be calculated with,

[20]

where the unknown offset charge capacitances and voltages are lumped into an offset charge vector, . The change of electrostatic energy can then be obtained using Eq. 8.

Thermodynamic formulation
The tunneling of electrons in networks of capacitors and tunnel junctions is often discussed in terms of the chemical potentials of the nodes and the free energy of the circuit. This description is equivalent to the description in terms of electrostatic energy given above.

Consider a closed system consisting of capacitors, tunnel junctions, and voltage sources. The total internal energy of such a system is conserved. When an electron tunnels, the decrease in electrostatic energy is dissipated as heat. The internal energy of the system is,

Here Q_therm is the thermal energy. The internal energy of a closed system is expressed in terms of all of the extensive variables in the problem. Extensive variables are variables that scale with the size of the system. In this case, the extensive variables are the charges on the nodes and the entropy.

A differential change in internal energy is,

[22]

where S is the entropy of the system. The voltages on the nodes and the temperature are partial derivatives of the internal energy,

and

[23]

The differential of the internal energy can then be written,

When an electron tunnels, the internal energy is conserved, dU = 0,

The left-hand side is a differential change in the electrostatic energy of the system and the right-hand side is the thermal energy dissipated, i.e. -ΔE = ΔQ_therm.

Since the experiments are performed at fixed temperature instead of fixed entropy, it is useful to construct the Helmholtz free energy of this system,

F = U - TS,

[26]

As long as the temperature is kept constant, changes in the Helmholtz free energy are equal to changes in the electrostatic energy of the network,

Since an electron can tunnel if it decreases the electrostatic energy of the system, one can equivalently say that an electron can tunnel if it decreases the Helmholtz free energy of the system.

The electrochemical potential of node j is the energy needed to add a charge |e| to node j. If only the charge on node j is changed, then the differential change in energy is,

[28]

Integrating the change in energy from 0 to |e| gives the energy needed to add an electron to node j, which is the electrochemical potential for node j,

[29]

Note that if a charge |e| is removed from node j the change in energy is,

[30]

The energy of adding an electron to node j is not simply minus the energy of taking an electron away from node j.

The change in the Helmholtz free energy when an electron tunnels can be calculated by adding the chemical potential for removing an electron from node i to the chemical potential of adding an electron to node f. This calculation is the same as the calculation leading to Eq. 14. The result is:

[31]

Like the change in electrostatic energy, the change in Helmholtz free energy depends only on the voltage difference between the initial and final nodes plus a term that is independent of the charge state of the system.

When voltage sources are included in the network, it is convenient to construct the generalized Gibbs free energy,

[32]

A differential change in the Gibbs free energy is,

[33]

At constant temperature and constant voltage on the voltage sources, the change in the Gibbs free is the change in electrostatic energy of the charge nodes,

[34]

Here E' is the electrostatic energy of the just the charge nodes, excluding the electrostatic energy of the voltage sources. The tunnel rate depends on the change in the electrostatic energy of the whole network when an electron tunnels. This can be expressed as the change in the electrostatic energy of just the charge nodes plus the work done by the voltage sources,

[35]

This seems to imply that the change in the free energy depends on the state of the state of the entire network before and after the tunneling has taken place. This is not really true and Eq. 35 is a numerically inefficient way to calculate the change in free energy. The change in free energy only depends on the voltage difference between the node that the electron tunnels from and the node that the electron tunnels to. (See Eq. 31.)

When an electron is added to one of the charge nodes, the change in the Gibbs free energy is equal to the change in the Helmholtz free energy which is equal to the change in electrostatic energy of the whole network which is equal to the change in electrostatic energy of just the charge nodes. Thus the electrochemical potential is,

Although all of these expressions for the electrochemical potential are equivalent, none of them are numerically efficient. The electrochemical potential of node j depends only on the voltage of node j. The numerically efficient way to calculate the electrochemical potential is given in Eq. 29.