Link between tunneling rate and conductivity

Question

Generally speaking, in the theory of single-electron tunneling events (one electron tunneling through a quantum well), the rate under which this event occurs is calculated using Fermi's Golden Rule, and the result based on the different of energy between initial and final state of the electron $\Delta F=F_f-F_i,$ is given by: $$ \Gamma(\Delta F) \propto \frac{-\Delta F}{1-e^{\Delta F/k_B T}} \tag{1} $$

Moreover, in (particle, or pn-junction) systems where conductivity stems from single electron tunneling events, the conductance between a pair of particles is often taken to decay exponentially as a function of distance between the two $g_{ij}\propto e^{-d_{ij}}$.

My question is, how do get to the latter given that we know the rate given by $(1)$? In other words, starting from the functional form of Eq. $(1)$ could we have predicted that conductivity via single e-tunneling must decay exponentially as function of distance?

Answer 1

To be frank, I don't think you are proposing a good method for understanding tunneling. If you want to understand the exponential dependence, then just treat transmission thru a square barrier the same way you would for a step potential, and the exponential dependence becomes quite clear. Moreover, I disagree with your assertion that a pn-junction (diode?) works by tunneling. A standard pn-junction diode does not work by tunneling (altho a Zener diode does).

However, I'll try to approach tunneling using Fermi's golden rule.

First, I must disagree with your characterization of Fermi's golden rule. I'm not sure where you got your first Formula, but that looks like some temperature-dependent equation that was derived from Fermi's golden rule in some particular situation. Tunneling thru a barrier isn't really a temperature-dependent phenomena (unless you add in phonon-assisted tunneling and the likes, which is a much higher level of complicated), so I don't think your first equation is relevant to the current discussion.

Fermi's golden rule is often stated as it is on wikipedia

$$\Gamma_{i \to f} = \left|\left<i\right|H^\prime\left|f\right>\right|^2 \rho$$

This says, roughly, that the scattering rate due to a perturbation $H^\prime$ from an initial state $\left|i\right>$ to a final state $\left|f\right>$ is proportional to $\left|\left<i\right|H^\prime\left|f\right>\right|^2$.

This is not normally where you would start to talk about tunneling in the context of conductivity and transport because Fermi's golden rule is derived using first order perturbation theory, and for transport problems, your barrier is often a large feature, not a perturbation.

Never the less, I think that you can start with Fermi's golden rule if you have a weak potential to tunnel thru, altho it's a little awkward. Suppose your potential is zero with the exception of a single, small square barrier, and you treat that single square barrier as the perturbation $H^\prime$. Without the barrier, you would have modes that look like $e^{\pm ikx}$, and we can label those states by their wavenumber. Those are the unperturbed states. The probability of reflection from the barrier would be something like $\left|\left<k\right|H^\prime\left|-k\right>\right|^2$ --- i.e. the initial state is right traveling with some $k$, and the final state is left traveling (i.e. back scattered) with the same $k$. The larger $H^\prime$ (i.e. a taller or wider barrier), the larger the reflection probability. You might be able show an exponential dependence (altho I'm not certain). The rate of transmission would be something like $1 - \left|\left<k\right|H^\prime\left|-k\right>\right|^2$.

Note that I don't think that you can write the transmission as $\left|\left<k\right|H^\prime\left|k\right>\right|^2$ because I don't think that Fermi's golden rule really applies there. Fermi's golden rule is for calculating the rate out of scattering out of some state into another, not for the rate of one state into itself.