When electrons tunnel through an insulating barrier, the tunnel rate can be calculated using Fermi's golden rule. This situation is typically described quantum mechanically by a Hamilitonian containing three terms.

H_total = H_left + H_right + H_t

Here H_left is the Hamiltonian electrons on the left side of the barrier, H_right is the Hamiltonian for electrons on the right side of the barrier, and H_t is the tunnel Hamiltonian that describes how the left side and right side are coupled. If H_t is small enough that it can be treated as a perturbation, then Fermi's golden rule can be applied. The tunnel rate is given by,

Here Ψ_i is the initial state and Ψ_f is the final state. If the electron tunnels from left to right, then Ψ_i would be an eigenstate of H_left and Ψ_f would be an eigenstate of H_right. The Dirac delta function, δ(E_i - E_f), ensures that tunneling only takes place if the initial and final states have the same energy.

If both the left and right electrodes are metals, then there will be many initial states and many final states. The total rate will be given by a double sum over all possible initial states and all possible final states. There are so many states in a metal that the double sum can be converted into a double integral over energies. One of the integrals is readily performed using the properties of the delta function. If we assume that the tunneling matrix element that couples every initial state to every final state is the same, |<Ψ_i|H_t|Ψ_f>|² = |t|², then the total rate for electron tunneling is,

Here ρ_L(E)f(E-E_L) is the number of occupied states on the left at energy E, ρ_R(E)f(E-E_R) is the number of empty states on the right at energy E, ρ_L(E) and ρ_R(E) are the densities of states on the left and the right, E_L and E_R are the Fermi energies on the left and right, and f(E-E_L,R) = 1/(1+exp((E-E_L,R)/k_BT)) is the Fermi function. Typically, the densities of states are nearly constant near the Fermi energy and can be taken out of the integral. The integral over the product of Fermi functions can then be evaluated.

The contants can be collected together and the difference in Fermi energies can then be written in terms of the voltage drop across the junction, E_R - E_L = eV. The tunnel rate can then be writen as,

Here R = h/(4π²e²|t|²ρ_L(E_L)ρ_R(E_R)) is the tunnel resistance. For |eV| >> k_BT, the tunnel rate can be approximated by,

In the limit |eV| << k_BT, the tunnel rate is proportional to the temperature.

While the tunnel rate is temperature dependent, the tunnel current is not. The current flows in the opposite direction to the electron tunneling. The current is,

I_L→R = -|e|(Γ_L→R - Γ_R→L).

Using the expression for the tunnel rates, this can be written as,

A tunnel junction behaves like an Ohmic resistor.

When a quantum dot, small particle, or molecule is placed between electrodes, tunneling can take place from a filled state in the metal electrode to a single quantum state on the particle. Assuming that all of the states in the metal couple equally to the single quantum state on the particle, the tunnel rate would be,

Here E_P is the energy of the state on the particle and E_M is the Fermi energy of the metal. If an electron tunnels from the particle to an empty state in the metal, the rate will be,