Kronig–Penney Model – Theory, Derivation & Band Structure

Bloch's Theorem

Bloch’s theorem describes the nature of wave functions for quantum mechanical particle ( like electron) in a periodic potential.

\[ \psi(r) = e^{ik.r}U(r)\] where r is position, \(\psi\) is the wave function, \(U\) is a periodic function with the same periodicity as the crystal, the wave vector \(k\) is the crystal momentum vector, \(e\) is Euler's number, and \(i\) is the imaginary unit.

Kronig-Penny Model

The Kronig–Penney model is a simple way to understand how electrons move inside a crystal (solid). In a crystal, the atoms (ions) are arranged in a regular repeating pattern, and this creates a periodic potential for the electrons. According to this model:

1. Near the positive ions, the potential energy of the electron is almost zero (very small).
2. Between two ions (midway), the potential energy becomes the highest.

This model helps us explain how the repeating structure of a crystal affects the motion and energy of electrons in solids.

Figure 1: Kronig–Penney model diagram illustrating periodic potential wells and barriers in a crystal lattice.

The potential energy for region-I and region-II can be written as, \[ V(x) = \begin{cases} 0, & 0 \leq x \leq a \\ V_0, & -b< x < a \end{cases} \quad \tag{1} \] The time independent Schrodinger equation for a particle in one dimensional box is given by \[ \frac{d^2 \psi}{dx^2} + \frac{2m(E - V)}{\hbar^2} \psi = 0 \] Region-\(\mathrm{I}\): \[ \frac{d^2 \psi}{dx^2} + \alpha^2 \psi = 0 \quad ; \quad \alpha^2 = \frac{2mE}{\hbar^2} \tag{2}\] Region-\(\mathrm{II}\): \[ \frac{d^2 \psi}{dx^2} + \beta^2 \psi = 0; \quad \beta^2 = \frac{2m(E-V_0)}{\hbar^2} \tag{3}\] The solution of this equation have the form (According to Bloch's theorem) \[ \psi(x) = U_k(x) e^{ikx} \] Where \(U_k(x)\) is periodic with periodicity of the lattice, That is \[U_k(x+a) = U_k(a) \] The form \(U_k\) depends on the exact nature of potential field. The solutions for the Eqn(2) are possible only for energies given by the relation: \[ \boxed{\frac{Psin\alpha a}{\alpha a}+cos \alpha a = cos ka }\tag{4} \] Where P is scattering power of the potantial barrier, \[ P = \frac{4\pi^2ma}{h^2}V_0 w \] The equation (4) tells us that the right-hand side of the function can only take values between +1 and −1. This means that not all values of α are possible—only certain values are allowed. As a result, the energy E of an electron cannot take any random value; instead, it is restricted to specific ranges. These ranges of allowed energy values are what we call energy bands (or zones).

Figure 2: Equation and potential well representation used in the derivation of the Kronig–Penney model.

Observations

When \( P \rightarrow 0 \)

\[ cos \alpha a = cos ka\] Thus \(k = \alpha \), \[ k^2 = \alpha^2 = \frac{2mE}{\hbar^2} \] \[ E = \frac{\hbar^2k^2}{2m} \] \[ E = \frac{4h^2\pi^2}{8m\lambda^2\pi^2} ( \because k = \frac{2\pi}{\lambda} \& \hbar = \frac{h}{2\pi}) \] \[ E = \frac{h^2}{2m\lambda^2} \] \[ E = \frac{h^2p^2}{2mh^2} ( \because \lambda = \frac{h}{p})\] \[ E = \frac{p^2}{2m} = \frac{\hbar^2 k^2}{2m} = \frac{1}{2}mv^2 \] This indicates that the total energy of the electron comes only from kinetic energy and the electron can move freely inside the crystal without being restricted by potential energy. In such a case, the material behaves like a conductor, because free electrons are available to carry electric current.

When \( P \rightarrow \infty \);

Similarly, we get \[E = \frac{n^2h^2}{8ma^2} \] This indicates that the energy values are discrete (fixed, not continuous). This means the electron is trapped inside the potential well and cannot move freely. Therefore, this situation represents an insulator, where no free electrons are available for conduction.

The Kronig–Penney model explains the motion of electrons in a periodic potential of a crystal lattice. It shows that allowed and forbidden energy bands are formed. This model helps in classifying solids as conductors, semiconductors, or insulators based on their band structure.

Applications of Kronig Penny Model

1. It explains the origin of allowed and forbidden energy bands in solids.
2. It helps in classifying materials into conductors, semiconductors, and insulators.
3. It improves the free electron theory by including the effect of periodic potential of ions.
4. It provides the foundation for band theory, which is essential to study solid-state devices.
5. It helps in understanding the electrical, optical, and thermal properties of solids.

Interactive Visualisation of Allowed and Forbidden bands

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Use the interactive visualization below to explore how different parameters affect the band structure in the Kronig-Penney model. Adjust the potential strength and barrier width to see how energy bands and band gaps change.

Band Structure Visualization

Potential Strength (P):

P = 3.0

Barrier Width Ratio (b/a):

b/a = 0.3