next up previous contents
Next: Treatment of the Coulomb Up: OpenPhonon: an open source Previous: Introduction   Contents


Basic concept of lattice dynamics

Lattice Hamiltonian is developed to second order around the nominal equilibrium position, given by the experimental crystal structure. This corresponds to a simple implementation of the quasi-harmonic approximation (the positions of the atom at rest are choosen at the minimum of the free energy, in thermodynamic equilibrium) instead of the harmonic approximation (the position of the atom at rest are choose at the minimum of the potential energy, in mechanical equilibrium)

The Hamiltonian is given by the sum of the kinetic part plus the sum of interatomic two-body central potentials. The Hamiltonian degrees of freedom are the atomic cores and atomic shells coordinates. The atomic shells have zero mass and does not contribute to the kinetic part.

The potentials implemented up to now in OpenPhonon are:

1)
The Born-von-Karman potential: $V_{BK}$, which describes an empirical central potential by its second derivative (longitudinal and transverse components) for each interacting couple of atoms. One describes a Born von Karman potential by the force constants $F_L$ (longitudinal) and $F_T$ (transverse) which are given by the formula:


\begin{displaymath}
F_L=\frac{d^2V_{BK}(\kappa,\ell,\kappa ' \ell ')}{dr^2}
\vert _{r=(r_{\kappa' \ell'}^\circ - r_{\kappa \ell}^\circ)}
\end{displaymath} (1)

and


\begin{displaymath}
F_T=\frac{1}{r} \frac{dV_{BK}(\kappa,\ell,\kappa ' \ell ')}{dr}
\vert _{r=(r_{\kappa' \ell'}^\circ - r_{\kappa \ell}^\circ)}
\end{displaymath} (2)

2)
The Coulomb potential:


\begin{displaymath}
U_c(\vert\vec{R}_{\kappa,\ell}-\vec{R}_{\kappa ',\ell '}\vert) Z_{\kappa} Z_{\kappa '}
\end{displaymath} (3)

where $ U_c=\frac{1}{r} $ and $Z_{\kappa }$ is the charge of the $\kappa$ ion.

3)
The Born-Mayer potential:


\begin{displaymath}
V_{BM}=
V_{\circ} \exp\biggl(-\frac{r}{R^\circ_{\kappa}+R^\circ_{\kappa'}}\biggr)
\end{displaymath} (4)

4)
The Van der Waals potential:


\begin{displaymath}
V_{VW}=-V_{\circ} \biggl(\frac{R^\circ_{\kappa}+R^\circ_{\kappa'}}{r}\biggr)^6
\end{displaymath} (5)

5)
The Lennard-Jones potential:


\begin{displaymath}
V_{VW}=V_{\circ} \biggl[
\biggl(\frac{R^\circ_{\kappa}+R^\ci...
...(\frac{R^\circ_{\kappa}+R^\circ_{\kappa'}}{r}\biggr)^6
\biggr]
\end{displaymath} (6)


Finally, in condensed matter (in particular for metals) the Coulomb potential is renormalized by the screening effect of the surrounding charges. This effect can be calculated in many ways, we have implemented the simplest approach: the Thomas-Fermi one, i.e. replacing the Coulomb potential by the Yukawa potential:


\begin{displaymath}
U_{TF}=\frac{\exp\bigl({\frac{-r}{R_{TF}}}\bigr)}{r}
\end{displaymath} (7)

.

The motion of the atom labelled $\kappa$ in the $\ell$th cell is described by the following equations:

$\displaystyle m_{\kappa} \frac{d^2 u_{\kappa\ell}}{dt^2} =$ $\textstyle -X_{\kappa} \sum_{(\kappa,\ell)\neq(\kappa',\ell')}
V_{(\kappa \ell ...
... X_{\kappa ' } u_{\kappa ' \ell ' } +
Y_{\kappa ' } w_{\kappa ' \ell ' } \bigl)$   (8)
  $\textstyle + X_{\kappa} \sum_{(\kappa,\ell)\neq(\kappa',\ell')}
V_{(\kappa \ell  , \kappa ' \ell ' )}^{C}
Z_{\kappa ' } u_{\kappa \ell }$    
  $\textstyle - \sum_{(\kappa,\ell)\neq(\kappa',\ell')}\Bigl[
V_{(\kappa \ell  , \...
... \kappa ' \ell ' )}^{T} w_{\kappa ' \ell ' }\Bigl] +
K_{\kappa} w_{\kappa \ell}$    
  $\textstyle + \Bigl[ \sum_{(\kappa,\ell)\neq(\kappa',\ell')}
\bigl( V_{(\kappa \...
...kappa \ell  , \kappa ' \ell ' )}^{T}
\bigl) - K_{\kappa} \Bigl] u_{\kappa \ell}$    


$\displaystyle \mu_{\kappa} \frac{d^2 w_{\kappa\ell}}{dt^2} =$ $\textstyle -Y_{\kappa} \sum_{(\kappa,\ell)\neq(\kappa',\ell')}{
V_{(\kappa \ell...
... X_{\kappa ' } u_{\kappa ' \ell ' } +
Y_{\kappa ' } w_{\kappa ' \ell ' } \bigl)$   (9)
  $\textstyle + Y_{\kappa} \sum_{(\kappa,\ell)\neq(\kappa',\ell')}
V_{(\kappa \ell  , \kappa ' \ell ' )}^{C}
Z_{\kappa ' } w_{\kappa \ell }$    
  $\textstyle - \sum_{(\kappa,\ell)\neq(\kappa',\ell')} \Bigl[
V_{(\kappa \ell  , ...
...\kappa ' \ell ' )}^{T'}\Bigl] u_{\kappa ' \ell ' } +
K_{\kappa} u_{\kappa \ell}$    
  $\textstyle + \Bigl[ \sum_{(\kappa,\ell)\neq(\kappa',\ell')}
\bigl( V_{(\kappa \...
...appa \ell  , \kappa ' \ell ' )}^{T'}
\bigl) - K_{\kappa} \Bigl] w_{\kappa \ell}$    

where equation (8) describes the motion of the centre of mass of each atom while equation (9) refers to the motion of the shell associated to each atom, and its mass $\mu_{\kappa} = 0$. The parameter $V^C$ is given by


\begin{displaymath}
V^C=-\frac{\partial^2 U_c(\vert\vec{r}_1-\vec{r}_2\vert)}{\partial\vec{r}_1\partial\vec{r}_2}
\end{displaymath} (10)

$K$ are the spring constants which tie the shells to the cores. Symbols $V^{SR}$,$V^{T}$,$V^{T'}$,$V^{S}$ denote non-coulombian interaction. The $V^{SR}$ represent interactions between the cores. Although coulombian interactions act between the cores too, they need a special treatment because of their long range, and have thus their own writing $V^C$. $V^{T}$ are the elastic interactions between a shell and a core on another site, $V^{T'}$ between a core and another site shell and $V^{S}$ are the interactions between two different shells. These terms are the sum of analytical potentials ( Born-Mayer or Van de Waals ) and Born-von-Karman potentials. In the case of analytical potentials the $V$'s are derived from the analogous of equation (10) while in the Born-von-Karman case they are derived from the longitudinal and transverse second derivatives $F_L$ and $F_T$, namely :

\begin{displaymath}
V_{(\kappa \ell  , \kappa ' \ell ' )}= - \hat{r} \otimes \hat{r}
(F_L-F_T) - F_T
\end{displaymath} (11)

$\hat{r}$ being the unit vector going from $(  \kappa ' \ell ' )$ to $(\kappa \ell   )$

One can cast together $X_{\kappa \ell}$ (core charge) and $Y_{\kappa \ell}$ (shell charge) in equations (8) and (9) to get $Z_{\kappa \ell}$ (ion charge=core+shell) and $Y_{\kappa \ell}$ by using a different set of displacement variables ( $u_{\kappa \ell}$ and $\bar w_{\kappa \ell}$ ) where $\bar w_{\kappa \ell} = w_{\kappa \ell} - u_{\kappa \ell}$. By performing such substitution, by summing together equations (8) and (9) and considering the limit $\mu_{\kappa} \rightarrow 0$, one obtains :


$\displaystyle m_{\kappa} \frac{d^2 u_{\kappa\ell}}{dt^2} =
-Z_{\kappa} \sum_{(\...
...kappa ' } u_{\kappa ' \ell ' } + Y_{\kappa ' } \bar w_{\kappa ' \ell ' } \bigl)$      
$\displaystyle + Z_{\kappa} \sum_{(\kappa,\ell)\neq(\kappa',\ell')}
V_{(\kappa \ell  , \kappa ' \ell ' )}^{C}
Z_{\kappa ' } u_{\kappa \ell }$      
$\displaystyle + Y_{\kappa} \sum_{(\kappa,\ell)\neq(\kappa',\ell')}
V_{(\kappa \ell  , \kappa ' \ell ' )}^{C}
Z_{\kappa ' } \bar w_{\kappa \ell }$     (12)
$\displaystyle - \sum_{(\kappa,\ell)\neq(\kappa',\ell')}\Bigl[
V_{(\kappa \ell  ...
...ll  , \kappa ' \ell ' )}^{T'}
\Bigl] (u_{\kappa ' \ell ' } - u_{\kappa \ell } )$      
$\displaystyle - \sum_{(\kappa,\ell)\neq(\kappa',\ell')}\Bigl[
V_{(\kappa \ell  ...
...S} +
V_{(\kappa \ell  , \kappa ' \ell ' )}^{T}
\Bigl] \bar w_{\kappa ' \ell ' }$      
$\displaystyle + \sum_{(\kappa,\ell)\neq(\kappa',\ell')}\Bigl[
V_{(\kappa \ell  ...
...}^{S} +
V_{(\kappa \ell  , \kappa ' \ell ' )}^{T'}
\Bigl] \bar w_{\kappa \ell }$      

and for the zero-mass shells

$\displaystyle 0= -Y_\kappa \sum_{(\kappa,\ell)\neq(\kappa',\ell')} V_{(\kappa \...
... \kappa ' \ell ' )}^{C} Z_{\kappa ' } ( u_{\kappa ' \ell ' } - u_{\kappa \ell})$      
$\displaystyle -Y_\kappa \sum_{(\kappa,\ell)\neq(\kappa',\ell')} V_{(\kappa \ell...
...} V_{(\kappa \ell  , \kappa ' \ell ' )}^{C} Z_{\kappa ' } \bar w_{\kappa \ell }$      
$\displaystyle -K_x \bar w_{\kappa \ell }
- \sum_{(\kappa,\ell)\neq(\kappa',\ell...
...)}^{S} +
V_{(\kappa \ell  , \kappa ' \ell ' )}^{T'}
\Bigl] u_{\kappa ' \ell ' }$     (13)
$\displaystyle - \sum_{(\kappa,\ell)\neq(\kappa',\ell')} V_{(\kappa \ell  , \kappa ' \ell ' )}^{S} \bar w_{\kappa ' \ell ' }$      
$\displaystyle \sum_{(\kappa,\ell)\neq(\kappa',\ell')}\Bigl[
V_{(\kappa \ell  , ...
...l  , \kappa ' \ell ' )}^{T'}
\Bigl] ( u_{\kappa \ell } + \bar w_{\kappa \ell })$      

Phonon eigenvectors can be written as Bloch waves:

$\displaystyle u_{\kappa \ell } =u_{\kappa} e^{i \bf {Q} . {\bf {R}}_{\kappa \ell} }$     (14)
$\displaystyle \bar w_{\kappa \ell } =\bar w_{\kappa} e^{i \bf {Q} . {\bf {R}}_{\kappa \ell} }$      

where $\vec R_{\kappa \ell}$ is the equilibrium position of the atom $(\kappa \ell )$. The eigenproblem can then be written as :
$\displaystyle -\omega^2 u$ $\textstyle = - Z   C   Z   u +Z   C_0   u - V^{SM}   u +V_0^{SM}   u$   (15)
  $\textstyle +(Y   C_0 - Z   C   Y - V^S - V^T + V_0^S+V_0^{T'})   \bar w$    
$\displaystyle \bar w$ $\textstyle =( V_0^S+V_0^T-V^S-K+Y   C_0 - Y   C   Y )^{-1} .$    
  $\textstyle (Y   C   Z -Y   C_0 + V^S + V^{T'} - V_0^S - V_0^{T'} ) . u$    

where $u$ and $\bar w$ are now $3N$ dimensional vectors ( N being the number of atoms in a cell), Y,Z,K are $3N$ dimensional diagonal matrices whose diagonals are equal to the shell charges, ionic charge and spring constants respectively. The matrices C (coulombian), and the V's are obtained substituting equation 14 into 15 and summing over all the cells up to convergence. The subscript ``$_o$'' in $C_0$ and in $V_0$'s denotes the so said ``self-forces''. The dynamical matrix is found from equation (15) by elimination of $\bar w$. The summation of $C$ and $C_0$ over the cells needs a special treatment because of the Coulomb long range.


next up previous contents
Next: Treatment of the Coulomb Up: OpenPhonon: an open source Previous: Introduction   Contents
Alessandro Mirone 2003-11-17