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

Bloch's theorem

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 $\lambda =e^{-2i\pi k_{n}a}$ with $k_{n}=n/a$ . On another hand, eigenfunction can always be written:

$\psi _{k}(r)=e^{ikr}u_{k}(r).$

Since $u_{k}$ is periodical theorem is proved. }}

Free electron model

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

$u_{k}({\bar {r}})=e^{iK_{n}r}$

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

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:

$V(x)=V_{box}+\epsilon e^{iK_{1}r}$

In the free electron model functions

${\begin{matrix}\psi _{1k}&=&e^{ikr}\\\psi _{2k}&=&e^{ikr}e^{iK_{1}r}\end{matrix}}$

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

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.