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

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Bloch's theorem[edit]

sectheobloch

Consider following spectral problem:

Problem:

Find \psi(r) and \epsilon such that


[-\frac{\hbar^2}{2m}\nabla^2+V(r)]\psi(r)=\epsilon\psi(r)

where V(r) 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 V(r) is periodic then wave function \psi solution of the spectral problem can bne written:

\psi(r)=\psi_k(r)=e^{ikr}u_k(r)

with u_k(r)=U_k(r+R) (function u_k has the lattice's periodicity).

Proof:

Operator -\frac{\hbar^2}{2m}\nabla^2+V(r) commutes with translations \tau_j defined by \tau_a\psi(r)=\psi(r+a). Eigenfunctions of \tau_a are such that: {IMP/label

\tau_a\psi=\psi

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

\delta(x-a)*\psi(r)=\psi(r)

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

e^{-2i\pi ka}\hat{\psi}=\hat{\psi}

That is the eigenvalue is \lambda=e^{-2i\pi k_na} with k_n=n/a [1]. On another hand, eigenfunction can always be written:

\psi_k(r)=e^{ikr}u_k(r).

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

Free electron model[edit]

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


H=-\frac{\hbar^2}{2m}\nabla^2+V(r)

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

Potential in the free electron approximation.}
figpotperioboit

Eigenfunctions of H are eigenfunctions of \nabla^2 (translation invariance) that verify boundary conditions. Bloch's theorem implies that \phi can be written:


\psi_k=e^{ikr}u_k(\bar{r})

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


u_k(\bar{r}+\bar{R}_i)=u_k(\bar{r})

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

u_k(\bar{r})=e^{iK_nr}

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

E_k=\frac{\hbar^2}{2m}|k+K_n|^2

where K_n can take values \frac{2n\pi}{a}, where a is lattice's period and n is an integer. Plot of E as a function of k is represented in figure figeneeleclib.

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

Quasi-free electron model[edit]

Let us show that if the potential is no more the potential of a periodic box, degeneracy at k=\frac{K_1}{2} is erased. Consider for instance a potential V(x) defined by the sum of the box periodic potential plus a periodic perturbation:


V(x)=V_{box}+\epsilon e^{iK_1r}

In the free electron model functions

\begin{matrix}
\psi_{1k}&=&e^{ikr}\\
\psi_{2k}&=&e^{ikr}e^{iK_1r}
\end{matrix}

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 model[edit]

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:


\psi(r)=\sum_j c_j\phi_{at}(r-R_j)

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


\psi_k(r)=e^{ikr}u_k(r)

Identifying u_k(r) and u_k(r+R_i), it can be shown that c_l=e^{ikK_l}. 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 k. This representation is labelled \Gamma_k.
  2. Indeed, let us write in two ways the action of \tau_a on \phi_k:

    
\tau_a\psi_k=e^{ika}\psi_k

    and

    
\tau_a\psi_k=e^{ik(r+a)}u_k(r+a)