Question

(a) What is quantum tunneling?

(b) Based on the concept of quantum tunneling, describe the working principle of a scanning tunneling microscope.

(c) Explain what is quantum confinement?

(d) You have a metallic nanowire. The length of the nanowire is 1 mm (Comparing with the wave length of electrons, you can think the length is infinitely long.) The width of the nanowire is 50 nm. The height of the nanowire is 20 nm. Based on the simple model we discussed in our lecture, write out the formula of the energy of electrons in the nanowire (formula is enough, no calculation is required).

Answer 1

Solution :

(a)

QUANTUM TUNNELING :

If a particle has insufficient energy, it is quite impossible to move out of the potential barrier in a classical system. Owing to the dual nature, in quantum world, particles are sometimes observed to behave like waves. These quantum waves when encountered with a potential barrier decrease their amplitude exponentially. When amplitude decreases, the probability of finding the particle in the potential barrier decreases significantly. (If the barrier is supposedly thin, then the amplitude of the quantum wave will be non-zero.) This phenomenon of a particle passing through a potential barrier is QUANTUM TUNNELING.

(b)

WORKING PRINCIPLE OF SCANNING TUNNELING MICROSCOPE (STM) :

Scanning Tunneling Microscope is a non-contact electron microscope based on the principle of quantum tunneling used to resolve single atoms. The conducting tip in STM is moved close to the sample and it is noted that the spacing between the tip and the surface is reduced to a value comparable to the spacing between neighboring atoms in the lattice. In the above conditions, the tunneling electron can move either to the adjacent atoms in the lattice or to the atom on the tip of the probe.The tunneling current to the tip measures the density of electrons at the surface of the sample, and this information is displayed in the image.

When the specimen and the tip are brought close to each other, there is only a narrow space of empty region between them. On either side electrons are present up to Fermi Energy. It is essential to overcome a barrier to move from the tip to the specimen or the other way round. If the distance between the specimen and the tip is small enough, the electrons can tunnel through the vacuum barrier. When a voltage V is applied between specimen and tip, the tunneling effect results in a net electron current.This is the tunneling current.

(c)

QUANTUM CONFINEMENT :

QUANTUM CONFINEMENT refers to a change in optical and electronic properties of when the electrons and holes in the specimen (specimen is considerably of small size, say 10 nm or less) are compressed into a dimension that approaches exciton Bohr radius.

(d)

FORMULA FOR THE ENERGY OF ELECTRONS BASED ON EFFECTIVE- MASS APPROXIMATION MODEL :