The Kronig-Penney model

When we talk about the conductivity of a material, we generally divide it into three parts: Conductors, Insulators, and Semiconductors. But do you know, which model proves why some materials behave like conductors, insulators, and semiconductors, and what is the reason behind it? This property of a material is verified by Bloch, and two German scientists named Kronig-Penny gave his model to prove it, which is called ” The Kronig-Penney Model”. This model helps in determining the electrical, thermal, and magnetic properties of a solid. In this article, we will discuss this model in detail and understand its advantages, properties, and specific features.

What is Kronig-Penney Model?

Kronig-Penney’s model states that “electrons inside the metal crystal feel the periodic potential”, and the periodicity depends on the lattice of the crystal. The potential near the +ve immobile ions is almost equal to zero, and it reaches its maximum in between the two +ve ions. This model tells us about the different properties associated with a solid, like its magnetic, electrical, and thermal properties. This model came into existence after Bloch made some changes in the assumptions of the free electron model. Bloch changed the assumption, which states that “electrons inside the metal crystal feel zero constant potential.

According to Bloch’s theorem:

The periodic potential function will be Ѱ = e^ikxu_k(x) where u_k(x) is the periodic function of x and e^ikx is representing the plane wave and stated that electrons experience the periodic potential inside the metal crystal.

History

The first theory which describes the different properties of metal was discovered by Drude and Lorentz in the 1900s, which was called “The free electron theory” that was based on the following assumptions:

• Electrons inside the metal crystal move freely as the molecules of a perfect gas in the container.
• During their motion, they collide with the immobile positive ions present in the lattice and other electrons.
• All the collisions are elastic in nature, so the kinetic energy before and after the collision is the same, and hence there is no energy loss in the collision.
• Electrons inside the metal crystal, experience the zero constant potential and hence move freely.
• The mutual repulsion between the electrons is zero.
• All the electrons move in a random direction, in the absence of an external electric field.
• When an external electric field is applied, all the electrons accelerated in the opposite direction of an applied electric field.

Metal crystal: According to the free electron theory

This theory explains successfully the phenomenon of specific heat, electrical conductivity, thermionic emission, thermal conductivity, and para-magnetism. But the free electron model fails to explain why some solids are good conductors of electricity, some are semiconductors and others are insulators.

But we know that inside a real crystal, there is an infinite array of lattice points and there is a periodic arrangement of positively charged ions and all the electrons move through them. The potential of electrons near or at the +ve ion site is zero and is the maximum exactly in between the 2 +ve ions.

According to the Kronig-Penny model, an electron move in a periodic potential produced by +ve ions.

The Kronig-Penny Model

The potential varies periodically with the same period as the lattice.

V(x) = V(x+a)

Potential Variation inside the real crystal

Formula

After solving the Schrodinger time-independent wave equation for regions 1 and 2 :

d²Ѱ/dx² + 8П²m/h²[E – V(x) ]Ѱ = 0 ———-(1)

Region 1: ( V(x) = 0 ) ( 0 < x < a )

Region 2: ( V(x) = V_o ) ( -b < x < 0 )

and applying Bloch theorem and boundary conditions: Ѱ = e^ikxu_k(x) ————- (2)

We get [ P*Sinαa/αa + Cosαa = Coska ] ———-(Final equation)

where P represents the strength with which ions attract the electrons toward each other.

Case 1: When P → 0

We get Cosαa = Coska

αa = ka

α = k

α² = k²

8π²mE/h² = (2П/λ)²

E = p²/2m ———-(3)

From equation (3) we can say that the total energy of an electron is purely kinetic energy, and the electron is free to move inside the solid crystal. Hence we can conclude that it is the case for Conductors.

Case 2: When P →∞

Similarly, we get

E = n²h²/8ma² ———–(4)

From equation (4) we can say that the energy values are discrete and the electron is bound in the potential well and hence it does not move. Hence we can conclude that it is the case for Insulators.

Conclusion

We can conclude that the Kronig-Penney model was successful in proving the different properties of solids by assuming the real crystal behavior and believing that electrons present inside the crystal experience the periodic potential and the periodicity depends on the lattice and it also corrects some of the assumptions in the free electron theory. With the help of the Kronig-Penney equation, we can easily determine the nature of a solid. It also explains the band theory in the solid crystal with the help of their potential curve for the electron.

FAQs: Kronig-Penney Model

1. What are the drawbacks of free electron theory?

The wrong assumptions made by the Drude and Lorentz are the major drawbacks of quantum free electron theory like:

• Electrons inside the crystal, experiences zero constant potential.
• The mutual repulsion between electrons is considered to be zero.

Due to this, the free electron theory does not explains the phenomenon of photoelectric effect, the superconductivity in the metals etc.

2. What is the energy associated with the electron at temperature T?

The energy associated with the electron at temperature T was = 3/2kT and it is related to the kinetic energy with the equation: 3/2kT =1/2mv²_th. Where k = Boltzmann’s Constant

T = Temperature in kelvin

V_th = Thermal velocity of electron

3. How does Kronig-Penney model helps in determining of energy bands?

With the help of their potential well curve, Kronig-Penney describes the concept of allowed and forbidden energy bands. The portion in which Potential V(x) = 0, called forbidden energy gap in which no electrons is allowed to be present inside it. The the portion of curve where potential V(x) ≠ 0 are knows as Valence and Conduction bands.

Energy bands and forbidden energy gap

4. What are the application of kronig-Penney Model?

The real life application of Kronig-Penney model are:

• In the development of Semiconductor chips.
• Used to select the correct material according to the need in the manufacturing of different electronic devices.
• Understanding the behavior of material.
• Identification of nature of material.