Tunneling

TUNNELING

Definition

The phenomena in which a particle, like an electron, encounters an energy barrier in an electronic structure and suddenly pentrates is known as Tunneling.

With quantum dots, there are no physical electrical connections at all i.e., no wires leading in or out. Quantum dots are islands.

As practical example, we can see that Andaman & Nichobar is also an island, surrounded by an ocean, where in people try to disconnect from the hustle and bustle of the mainland. And yet there are thousands of cars on Andaman & Nichobar. There is even traffic (making it more difficult to feel very disconnected). Cars usually come and go from islands by boat.

Electrons Tunnel

The insulating material that surrounds a quantum dot is an electrical barrier. If electrons were conventional particles, they would hit such a barrier and bounce off it, as if it were a wall.

But electrons wave nature makes it possible for them to hit the barrier and all of a sudden reappear on the opposite side. This phenomenon, called electron tunneling, happens all the time. And it is due to the fact that electrons occupy space in rippling patterns of probability.

When a particle such as an electron is confined inside a potential energy well with walls of a given height, the electron's wave function, Ψ, extends outside the walls. Hence, the probability density, |Ψ|², is used to find an electron at a given location. This means there is a slim chance‒but a chance nonetheless‒that the electron will be found outside the well.

In a potential well which was infinitely high, and the wave function was constrained, then both Ψ and |Ψ|² values will be equal to zero.

The outside of the well was mathematically off‒limits to the electron. This situation is nearly achievable in real devices, and does well to model the energy quantization of atoms and quantum devices.

However, there is actually no way of making a potential barrier that is infinitely high. Very, very, high‒yes. Infinitely high‒no.

The wave functions and probability densities of an electron inside a potential well of finite height are shown in Fig. 3.10.

Fig 5.10 A particle in a potential energy well of finite depth. Graphed here are the wave function Ψ and probability distribution |Ψ|² of the particle. The lowest two energy states are shown. Both Ψ and are nonzero at the well's boundaries and decay exponentially outside the well. The well's width is L.

The width of the well is L wide, with x being the electron's location; the wave functions shown are those of the two lowest energy states.

The potential energy at the walls, while greater than the energy of the electron, is not infinity. For this reason, the prior boundary condition is no longer in effect: the wave function does not have to equal zero at the walls.

As we can see, this changes both the wave function and the probability density, enabling both of these functions to spill out the sides of the well.

There are three regions of interest here namely, the two regions outside the well boundaries (regions I and III) and the well itself (region II). The general forms of the wave functions for these three regions are tabulated in Table 3.2.

Table 3.2: Wave functions in all three regions

Here, A, B, C, F, and G are constants. The variable k determines the wavelength λ of the function in region II, such that k = 2π/λ.

The wave function is sinusoidal within the well, as expected, and decays exponentially beyond the walls. The probability density |Ψ|² is also nonzero outside the well's boundaries.

This does not make sense when viewed through the lens of classical physics, but it does make sense according to quantum mechanics. Most importantly, it is what actually happens with electrons.

When an electron meets with an energy barrier of finite height and width, it can some‒times penetrate it and appear on the opposite side. This is tunneling and it is shown in Fig 3.11.

Here we see the wave function of a particle that has an energy, E_p. The potential energy barrier has energy, E_B, which is greater than E_p. (If by chance the particle's energy were instead greater than that of the barrier, the wave could pass right over it.)

As it is, the wave function is incident on the barrier, then decays exponentially while propagating through it. What we do know is that if the barrier is narrow and low enough, the wave reappears on the other side.

Fig 3.11 Tunneling. The wave function Ψ of a particle with energy, E_p arrives at the left side o a barrier with energy, E_B. The function decays exponentially while passing through the barrier by reemerging on the opposite side. In this way, electrons "tunnel" across energy barriers that would otherwise seem to prohibit passage.

Note: In the case of a quantum dot, the barrier is the energy needed to pass across the electrically Insulating material that surrounds long the dot. The physical distance separating the bulk electrons from the isolated electrons in a dot is often about a nanometer. A finite (and occasionally achievable) amount of energy keeps the electrons from moving back and forth from the dot.

A particularly innovative and useful application of electron tunneling is the scanning tunneling microscope (STM).