Introduction to Mathematical Physics/N body problem in quantum mechanics/Crystals

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Bloch's theoremEdit

sectheobloch

Consider following spectral problem:

Problem:

Find   and   such that

 

where   is a periodical function.

Bloch's theorem [ma:equad:Dautray5], [ph:solid:Kittel67], [ph:physt:Diu89] allows\index{Bloch theorem} to look for eigenfunctions under a form that takes into account symmetries of considered problem.

Theorem:

theobloch

Bloch's theorem. If   is periodic then wave function   solution of the spectral problem can bne written:

 

with   (function   has the lattice's periodicity).

Proof:

Operator   commutes with translations   defined by  . Eigenfunctions of   are such that: {IMP/label

 

Properties of Fourier transform\index{Fourier transform} allow to evaluate the eigenvalues of  . Indeed, equation tra can be written:

 

where   is the space convolution. Applying a Fourier transform to previous equation yields to:

 

That is the eigenvalue is   with   [1]. On another hand, eigenfunction can always be written:

 

Since   is periodical[2] theorem is proved. }}

Free electron modelEdit

Hamiltonian can be written ([ph:solid:Kittel67],[ph:solid:Callaway64]) here:

 

where   is the potential of a periodical box of period   (see figure figpotperioboit) figeneeleclib.

 
Potential in the free electron approximation.}
figpotperioboit

Eigenfunctions of   are eigenfunctions of   (translation invariance) that verify boundary conditions. Bloch's theorem implies that   can be written:

 

where   is a function that has crystal's symmetry\index{crystal}, that means it is translation invariant:

 

Here (see [ph:solid:Callaway64]), any function   that can be written

 

is valid. Injecting this last equation into Schr\"odinger equation yields to following energy expression:

 

where   can take values  , where   is lattice's period and   is an integer. Plot of   as a function of   is represented in figure figeneeleclib.

 
Energy of mode   in the free electron approximation (electron in a box).}
figeneeleclib

Quasi-free electron modelEdit

Let us show that if the potential is no more the potential of a periodic box, degeneracy at   is erased. Consider for instance a potential   defined by the sum of the box periodic potential plus a periodic perturbation:

 

In the free electron model functions

 

are degenerated. Diagonalization of Hamiltonian in this basis (perturbation method for solving spectral problems, see section chapresospec) shows that degeneracy is erased by the perturbation.

Thigh binding modelEdit

Tight binding approximation [ph:solid:Ashcroft76] consists in approximating the state space by the space spanned by atomic orbitals centred at each node of the lattice. That is, each eigenfunction is assumed to be of the form:

 

Application of Bloch's theorem yields to look for   such that it can be written:

 

Identifying   and  , it can be shown that  . Once more, symmetry considerations fully determine the eigenvectors. Energies are evaluated from the expression of the Hamiltonian. Please refer to [ph:solid:Ashcroft76] for more details.

  1. So, each irreducible representation\index{irreducible representation} of the translation group is characterized by a vector  . This representation is labelled  .
  2. Indeed, let us write in two ways the action of   on  :

     

    and