Understanding Magnetic Dichroism and Hybridization Gaps: Band Symmetry Question

[fk08]

hello,

i am trying to get familiar with magnetic dichroism, first question on that is, why bands of same symmetry are not allowd to cross each other and why as a consequence hybridization gaps are formed.

thanks for help

 

[kanato]

It's easier to answer the question of why bands with different symmetry can cross. If two basis states |a> and |b> are of different symmetry, then H|b> will modify |b> somehow but it does not change the symmetry. Then the product <a|H|b> will be zero, because states of different symmetry will integrate to zero.

If two bands have the same symmetry, the Hamiltonian can and will mix them. That is, the product <a|H|b> will be non-zero if a and b have the same symmetry, so there will be an off diagonal element in the Hamiltonian. At the value of k where <a|H|a> = <b|H|b> there will be off diagonal elements, so when diagonalized the energies will be different, and this will result in the hybridization gap.

 

[charng]

Magnetic dichroism is a phenomenon that occurs when a material exhibits different absorption or emission of light depending on the orientation of its magnetic field. This can be observed in materials with anisotropic properties, such as crystals, where the symmetry of the material plays a crucial role.

To understand why bands of the same symmetry are not allowed to cross each other, we need to look at the underlying principles of band structure. In a solid material, the energy levels of electrons are grouped into bands, and the electrons are distributed among these bands according to the Pauli exclusion principle, which states that no two electrons can occupy the same quantum state.

In a crystal, the electrons can only occupy certain allowed energy states, known as energy bands. These bands are determined by the symmetry of the crystal lattice and the interaction between the electrons. When two bands of the same symmetry approach each other, they cannot cross because this would result in two electrons occupying the same state, violating the Pauli exclusion principle.

As a consequence, hybridization gaps are formed, where the energy levels of the two bands are split apart due to the repulsive forces between the electrons. This creates an energy barrier that prevents the bands from crossing and leads to the formation of hybridized states.

In summary, the reason why bands of the same symmetry are not allowed to cross each other is due to the Pauli exclusion principle, which governs the behavior of electrons in materials. Understanding this principle is crucial in studying magnetic dichroism and the formation of hybridization gaps.