Exact calculation of tunneling current?

[QMrocks]

Supposed we have a metal sandwitched between two thin dielectrics. If i contact an electrical voltage on the dielectric, there will be tunneling current due to electron tunneling from the metal across the dielectric. i could attempt to calculate this current from a semiclassical approach using *Current=N x Freq x Prob* (where N is number density of electrons. Freq is the impact frequency of an electron on the dielectric wall. Prob is the probability it will penetrate the dielectric, taken to be the probability it can be found in the dielectric.)

Now, the Prob can be calculate very accurately using numerical approach for any arbitrary dielectric potential barrier. But how does one account for Freq "quantum mechanically" ?

 

[ZapperZ]

Supposed we have a metal sandwitched between two thin dielectrics. If i contact an electrical voltage on the dielectric, there will be tunneling current due to electron tunneling from the metal across the dielectric. i could attempt to calculate this current from a semiclassical approach using *Current=N x Freq x Prob* (where N is number density of electrons. Freq is the impact frequency of an electron on the dielectric wall. Prob is the probability it will penetrate the dielectric, taken to be the probability it can be found in the dielectric.)
Now, the Prob can be calculate very accurately using numerical approach for any arbitrary dielectric potential barrier. But how does one account for Freq "quantum mechanically" ?

Your problem here is a bit puzzling. You have a metal in between 2 dielectrics? Are you sure it's not the other way around? Usually you have tunneling across the dielectric, so you sandwich the dielectric in between 2 metals.

In any case, there are problems with the scenario you are describing because you neglected to describe how "sophisticated" you want this to be. The rate of tunneling, or the tunneling current, can be described via the Fermi Golden Rule. This depends not only on the tunneling probability as characterized by the tunneling matrix element, but also the density of states of the material. What this means is that not only is the tunneling current depends on how probable it is for a charge carrier to get across, but once it gets across, what is the probability that there is an available state for that carrier to occupy.

Contrary to what you just said, the calculation of the tunneling probability is usually not trivial and usually is the one that requires several approximations. The square barrier model usually is not sufficient for most practical application. The usual approach is to use the WKB approximations. In most cases, there are NO analytical solutions to solve such a thing.

The density of states, especially for a metal, is known. Now this would have a loose connection with your "freq" since it gives the number of possible carriers that can participate in tunneling process. However, there is no one-to-one transformation from this to your freq.

Zz.

 

[QMrocks]

Your problem here is a bit puzzling. You have a metal in between 2 dielectrics? Are you sure it's not the other way around? Usually you have tunneling across the dielectric, so you sandwich the dielectric in between 2 metals.

i am also interested in the case you mentioned i.e. Metal-Insulator-Metal. The case i described should be Metal-Insulator(thin)-Metal(thin)-Insulator(thin)-Metal. In other words, the electrons in the center metal are also confined.

In any case, there are problems with the scenario you are describing because you neglected to describe how "sophisticated" you want this to be. The rate of tunneling, or the tunneling current, can be described via the Fermi Golden Rule. This depends not only on the tunneling probability as characterized by the tunneling matrix element, but also the density of states of the material. What this means is that not only is the tunneling current depends on how probable it is for a charge carrier to get across, but once it gets across, what is the probability that there is an available state for that carrier to occupy.
Contrary to what you just said, the calculation of the tunneling probability is usually not trivial and usually is the one that requires several approximations. The square barrier model usually is not sufficient for most practical application. The usual approach is to use the WKB approximations. In most cases, there are NO analytical solutions to solve such a thing.
The density of states, especially for a metal, is known. Now this would have a loose connection with your "freq" since it gives the number of possible carriers that can participate in tunneling process. However, there is no one-to-one transformation from this to your freq.
Zz.

i see. Does it mean that tunneling can only be modeled using a perturbative approach like Fermi-golden rule? Suppose i can calculate the wavefunction 'exactly' using numerical approach to solve Schrodinger equation, is there a way of finding the tunneling current without using perturbation approach?

Yes, my definition of 'freq' kinda loose. My apology. I'm trying to say that the confined electron if regarded classically is like a particle bombarding between the two dielectric wall with a certain frequency to be calculated classically. Which is a totally classical treatment.

 

[da_willem]

You can obtain, using some reasonable assumptions, and elementary physics an expression for the tunneling current that contains the essential feautures of tunneling. I'm not sure this is what you're looking for, but I'll post it anyway. I derived this once for some report on an STM.
Consider two conductors separated by a vacuum (e.g. an STM tip and some conducting surface), this can be modeled as a finite potential (V) well. Treating the situation as effectively one-dimensional the time-independent Schrödinger equation is as follows

$$\hat{H}\psi = E \psi$$ with $$\hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V$$

which, for the region between the two conductors, can be rewritten as

$$\frac{d^2 \psi}{dx^2}=\kappa ^2 \psi$$ with $$\kappa = \frac{\sqrt{2m(V-E)}}{\hbar}$$

where kappa is real as V>E, it is tunneling after all. The solution to this equation is ofcourse an exponential (in the metal kappa is imaginary so that the wavefunction is oscillatory instead)(the positive sign solution is not physical)

$$\psi \propto e^{-\kappa x}$$

The essential point in this 'derivation' is that the probability for an electron on one of the conductors to be found on the other conductor is a measure of the conductance. With s the distance between the two conductors the tunneling-conductance is

$$e^{-2\kappa s}$$ so $$I \propto Ue^{-frac{2 \sqrt{2m(V-E)}}{\hbar}}$$

with U the voltage between the two conductors...

 

[QMrocks]

The essential point in this 'derivation' is that the probability for an electron on one of the conductors to be found on the other conductor is a measure of the conductance. With s the distance between the two conductors the tunneling-conductance is
$$e^{-2\kappa s}$$ so $$I \propto Ue^{-frac{2 \sqrt{2m(V-E)}}{\hbar}}$$
with U the voltage between the two conductors...

May i know based on what argument you can associate the tunneling conductance with that exponential?

 

[QMrocks]

For the Metal-Insulator-Metal case:
How about considering an incident normalised Gaussian wave packet and solve its time dependent Schrodinger solution (assume i can solve it numerically)? The solution will consist of a reflected and transmitted packet. The total prob of transmitted gives probabilty of an electron penetrating the barrier. Using the Maxwellian distribution for velocity in the Fermion gas (metal) and its Fermi distribution, we can know the rate of electron (Gaussian packet) hitting insulator wall per unit time (in a 1D case of course). Together wif the tunneling prob, this will gives us tunneling current.

Is this an accurate approach?

 

[da_willem]

May i know based on what argument you can associate the tunneling conductance with that exponential?

Formally the answer can be found within the Landauer formalism, where conductance is proportional to transmission probablility. Intuitively I at least feel that the probablility of an electron to be found at the other side is proportional to frequency with which electrons make this 'jump'. And is thus proportional to the conductivity.

 

[QMrocks]

Formally the answer can be found within the Landauer formalism, where conductance is proportional to transmission probablility. Intuitively I at least feel that the probablility of an electron to be found at the other side is proportional to frequency with which electrons make this 'jump'. And is thus proportional to the conductivity.

OK. So you are referring to the transmission prob factor in Landaur formalism in conductance calculation. From this point of view, i guess what you posted is reasonable and apparently is the conventional approach to ballistic calculation (except the transmission prob will be described by Airy function when we acount for the potential bending in the dielectric, and also have to consider density of states, and maybe mass difference at metal-dielectric interface).

i supposed i can also calculate the more complicated structure metal-insulator-metal-insulator-metal structure using this Landauer approach if i were to attack it numerically. i think my previous confusion is because i tried to view the tunneling from the center metal peice.. i should instead view the source from the left or right metal which provides the thermal equilibrium source of electrons. Guess the Landauer approach will be the best for this kind of coherent transport problems.

 

[QMrocks]

Another curious point. If i want to solve the problem using path integral approach, is it possible? Is tunneling supposed to be described by some instanton equations?