Band structure of optical semiconductors

October 18th, 2016

The training of specialists in the field fiber-optic communication lines involves the study of such difficult topics as “Light-emitting diode (LED)” and “Semiconductor Lasers” (Zvelto, 1990). Clarification of the physical foundations of these devices requires the use of educational information which is studied in the course of quantum chemistry.

Physical basis of the work of LEDs.

LEDs are the semiconductor devices which emit light at passing through it of an electric current. The first LED which radiated light in the optical area of the spectrum was created in 1962. LED has one transition, but the difference between an ordinary semiconductor diodes and LEDs consists in that the LEDs are manufactured from optical band semiconductors. Only in the optical band semiconductors recombination of majority carriers is accompanied by the process of an emission of light. The main difficulty in understanding of the physical basis of the work of LEDs is the concept of an “optical band semiconductor”. The formation of this concept demands to involve such a category of quantum chemistry as a “dispersion law”. The term of the “dispersion law” in turn, follows from the theory of the formation of zones of “Bloch’s chain” (Levin, 1974). The initial idea of the appearance of bonding and anti-bonding energy levels from which are formed the energy bands in crystals has to be formed in the example of a hydrogen molecule. For these reasons, a method of formation of the concept of “optical band semiconductor” should contain the following items:

1. Fundamentals of molecular orbitals as combinations of atomic orbitals (MO LKAO). Two-centers task in the method of MO LKAO.

2. Bloch’s functions for the one-dimensional chain.

3.Energy bands in crystals, the band structure of optical semiconductors.

The basic concept for the realization of such methods is the concept formed at the study of topic “Schrodinger equation” in course of general physics. Let us consider the educational content of these items.

Fundamentals of MO LKAO. Two-centers task in the method of MO LKAO.

The mathematical basis of the method MO LKAO is the presentation of a wave function of the physical system (molecule, molecular ion cluster, crystal) as a linear combination of atomic functions, which satisfy the normalization conditions. The simplest view of the molecular orbital for a hydrogen molecule (1):


Let’s substitute the schedule for \psi in the Schrodinger equation for hydrogen molecule:

opt_semicond_2 Using the linear properties of the Hamiltonian operator, we receive: opt_semicond_3   Multiplying the last equality at first by atomic functions and integrating across whole space, we obtain a system of two equations:








The system of linear homogeneous equations has a unique solution when the main

Determinant of system (age Determinant) is equal to zero:


from where we obtain two solutions for the energy:


Because the hybrid integral  (β) has a negative value, the Coulomb integral (α) has a positive value, the level E(+) is lower than E(−) (Fig. 1):


Fig.1. The scheme of energy levels of homonuclear diatomic molecule

Thus, the formation of a molecule leads to splitting of the atomic levels for two energy levels, one of which lies below the atomic and is called a bonding, and the other is above the atomic and is called anti-bonding.

Bloch’s functions for the one-dimensional chain.

The simplest model of solids, which include semiconductors, is the one-dimensional chain, in which the atoms are placed at equal distance from each other and with one valence atomic orbital. For the equivalence of the atoms the chain is locked in the ring (Fig. 2).


Fig. 2. Cyclic chain of N atoms

The wave functions of atoms in the chain have the properties of translational symmetry, which mathematically expressed in the fact that the wave function of each next atom multiplied by the multiplier exp(-2πin/N).

By entering the radius-vector concept \vec R=m \vec a, where  - number of the atom, and of vector \mid \vec k \mid = \frac {2 \pi n}{Na},

basic Bloch’s \psi_j can be written as:


Energy bands in crystals, the band structure of optical semiconductors.

Hamiltonian eigenfunctions of the chain will look, according to the method of MO

LKAO, as a linear combination:


Then the age determinant will have the order equal to m:


and provides m solutions, or m branches of the dispersion law:


Let us consider the relationship between the branches of the dispersion law and the concept of energy bands in a crystal. Let us construct the inverse lattice, a period of which is equal to  \frac{2\pi}{a}.Let’s divide each of the elementary cells of the inverse lattice to N parts. Then the obtained set of vectors \vec k, (Fig. 3) which is called the \vec k– space. A set of projection of points of the dispersion law for the axis of energy, creates the energy zone. Optical band semiconductor is such semiconductor, in which the minimum of the conduction band and maximum of the valence band projected at the same point in \vec k– space (Fig. 3), non-optical band is such semiconductor, in which the minimum of the conduction band and maximum of the valence band projected at the different points in – \vec k-space (Fig. 4).


Fig. 3. The inverse lattice \vec k – space, branches of the law of dispersion, band structure


Fig. 4. The band structure of the optical semiconductor (A) and non-optical semiconductor (B)

The band structure with the optical band have semiconductors of AIIIBV (GaAs,GaP,GaN, InP ) and AIIBIV (ZnSe,CdTe) type. Diodes made from non-optical band semiconductors, almost don’t emit a light.Diodes made from non-optical semiconductors, don’t emit a light.

(The information has been taken from the article: Shvets V. D. Interdisciplinary Connections as a Tool of Learning Process Management// Socialinis ugdymas (Social Education). – 2014. – Vol. 1. - № 37. – P. 155-161. More information can be found at )

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