Tunneling

Quantum mechanics let us understand many interesting aspects of microscopic scales. With the help of quantum mechanics, we have understood concepts like particles confined in a box of infinite potential that gave us important formulae such as wave function of the particle, expression of energy that elaborated the fact of discreteness. Similar to this quantum physics introduced one more concept that is known as tunneling or quantum tunnelling. 

Quantum tunneling is a phenomenon that says if the energy of the particle more than the energy of the potential barrier then the particle can tunnel through the barrier. This is a major discovery of quantum physics that contradicts the classical mechanical aspects of physics. According to classical physics, the particle can pass through the barrier, but it can not penetrate the barrier. But quantum mechanics explains how an electron can penetrate the barrier and enter the region which is classically forbidden and this process is called tunneling.

Barrier Penetration

The quantum mechanical tunneling or the barrier penetration allows a confined particle within a finite potential well to penetrate through the classically impenetrable potential wall. Quantum mechanical tunneling is just a special case of a particle confined in a finite potential well, such that when the energy of the particle is greater than the energy (or the barrier height) of the potential barrier. 

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According to classical mechanics, when the particles are allowed to strike against a wall of finite potential will get reflected back completely, i.e., 100% reflectance if the energy of the particle is less than the barrier height the same has been accepted quantum mechanically too. When the energy of the particle is considered to be more than the height of the barrier then according to classical theory all the particles will transmit i.e., 100% transmittance. But this is not completely true, there is an equal probability of particles getting tunnel through the barrier and reach the classically forbidden region. This phenomenon is known the barrier penetration or quantum mechanical tunneling.

Quantum Mechanical Tunneling:

According to classical mechanics, if the energy of the particle is less than the energy or height of the barrier then the particle will reflect back because if the particle surpasses the height of the barrier the kinetic energy attained by the particle will be negative which classically impossible. Because we know that kinetic energy of the particle is given by \[\frac {1}{2}\]mv² and mass of the particle is always positive and square of the velocity will also be positive and hence negative kinetic energy is not possible. 

Thus, according to classical mechanics if the energy of the particle is less than the height of the potential barrier, then there will be always a total reflection of particles, quantum-mechanically this is true if and only if the height of the barrier is almost equal to infinity or of infinite height.

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At the same time if the particle strikes the potential barrier of finite height, then there is no complete reflection. There is a fair possibility that after collision with the wall the particle passing through the barrier by barrier penetration. This can be achieved if the energy of the particle is more than the barrier height. The barrier penetration or quantum mechanical tunneling depends on the width of the potential well, the height of the potential barrier, and the energy of the particle.

Now, consider three regions of the potential barrier such that in first region x>0,in second region 0<x<aand finally in third region x>a.Where x is the width of the barrier.

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For the first region, Time independent Schrodinger equation is given by:

\[\Rightarrow \frac{d^{2}\Psi _{1}}{dx^{2}}+\frac{2m}{h^2}E\Psi _{1}=0\]

\[\Rightarrow \frac{d^{2}\Psi _{1}}{dx^{2}}+k_{1}^{2}\Psi _{1}=0\]……….(1) 

Where k1is the angular wavenumber and is given by,

\[K_{1}=\sqrt{\frac{2mE}{h^{2}}}\]

On solving equation (1) we find the solution of the second-order differential equation which in turn gives rise to the wave equation of the particle in the first region. Therefore, we write:

\[\Rightarrow \Psi _{1}=Ae^{-ik_{1}x}+Be^{ik_{1}x}\]……. (2)

Similarly, the time-independent Schrodinger equation for the second region (0<x<a) is given by:

\[\Rightarrow \frac{d^{2}\Psi _{2}}{dx^{2}}+\frac{2m}{h^2}(E-v_{0})\Psi _{2}=0\]

\[\Rightarrow \frac{d^{2}\Psi _{2}}{dx^{2}}+k_{'2}^{2}\Psi _{2}=0\]…….(3)

Where \[K_{2}^{'}\] an imaginary constant and is given by,

\[K_{2}^{'}=\sqrt{\frac{2m(E-v_{0})}{h^{2}}}\]

On solving equation (3) we find the solution of the second-order differential equation which in turn gives rise to the wave equation of the particle in the second region. Therefore, we write:

\[\Rightarrow \Psi _{1}=Ce^{-k_{2}x}+De^{k_{2}x}\]………. (4)

Where k₂ is the angular wavenumber and is given by,

\[K_{2}=-iK_{2}^{'}\]

The solution of TISE in the second region consists of the real exponentials in contrary to the first and third regions.

For the third region, the Time-independent Schrodinger equation is given by:

\[\Rightarrow \frac{d^{2}\Psi _{3}}{dx^{2}}+\frac{2m}{h^2}E\Psi _{3}=0\]

\[\Rightarrow \frac{d^{2}\Psi _{1}}{dx^{2}}+k_{1}^{2}\Psi _{1}=0\]……….(5) 

Since region one and region three are the same, then the value of wave number will also be the same. Thus, k1is the angular wavenumber and is given by,

\[K_{1}=\sqrt{\frac{2mE}{h^{2}}}\]

On solving equation (5) we find the solution of the second-order differential equation which in turn gives rise to the wave equation of the particle in the third region. Therefore, we write:

\[\Rightarrow \Psi _{3}=Fe^{-ik_{1}x}+Ge^{ik_{1}x}\]………. (6)

Here, G=0, thus the solution of the equation (5) is given by:

\[\Rightarrow \Psi _{3}=Fe^{-ik_{1}x}\]…….(7)

The coefficient A is an arbitrary coefficient whereas B, C, D, F, and G are depending on E. solving for these we get the equation for transmission and reflection coefficients respectively. 

Thus, the transmission coefficient is given by:

\[\Rightarrow T=\frac{1}{cosh^{2}(\beta x)+(\frac{y}{2})^{2}sinh^{2}(\beta x)}\]

This explains the tunneling of electrons through the potential barrier and it is known as quantum mechanical tunneling.

Tunneling Microscope:

Scanning tunneling microscope (also known as STM), is a type of microscope whose fundamental principle of working is based on the quantum mechanical phenomenon known as tunneling, in which the wavelike properties of electrons permit them to the tunnel (barrier penetration) beyond the surface of a solid into regions of space that are forbidden to them under the influence of rules of classical physics. The probability of locating such tunneling electrons will be decreasing exponentially as the distance from the surface increases. 

The STM was invented in the year 1981 when two Swiss physicists Gerd Binnig and Heinrich Rohrer set out to build a tool for studying the local conductivity of surfaces. Binnig and Rohrer chose the surface of gold for their first image. When the image was displayed on the screen of a television monitor, they saw rows of precisely (equally) spaced atoms and observed broad terraces separated by steps one atom in height. Binnig and Rohrer had discovered in the STM a simple method for creating a direct image of the atomic structure of surfaces. Their discovery opened a new era for surface physics, and their impressive achievement was recognized with the honourable award of the Nobel Prize for Physics in the year 1986.

Did You Know?

Quantum mechanical tunneling shows how profoundly particles such as electrons differ from bigger objects. Throw a ball at the wall and it bounces backward, let it roll to the bottom of a valley and it stays there. But when a particle will occasionally hop through the wall. It has the possibility of slipping through the mountain and escaping from the valley, as two physicists wrote in Nature in 1928, in one of the earliest descriptions of tunneling.

Physicists noticed an amazing fact that particles’ ability to tunnel through barriers solved many mysteries. It explained various chemical bonds and radioactive decays and how hydrogen nuclei in the sun are capable to overcome their mutual repulsion and fuse, producing sunlight.