Phonon dispersion relation

Phonon Dispersion Relation

Yanming Wang, Saad Bhamla and Wei Cai

May, 2012

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METHODS

For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1

$V(\{{\vec {r}}_{i}\})=V(\{{\vec {r}}_{i}\})+{\frac {1}{2}}\sum _{ij}{\vec {u}}_{i}^{T}\cdot [{\frac {d^{2}V}{d{\vec {r}}_{i}{\vec {r}}_{j}}}]\cdot {\vec {u}}_{j}\qquad (1)$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is the potential function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{ r}_{i}} is the coordinate of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ith} atom, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{ R}_{i}} is the position of the $ith$ atom at emuilibrium and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}} is a small displacement from the equilibrium position: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}_{i} = \vec{r}_{i} - \vec{R}_{i} } .

In Equation 1, the linear terms in the Taylor expansion are at 0K with the minimum energy state and the higher order terms are neglected using the Harmonic approximation. The second derivative of the potential energy evaluated at the equilibrium position and is called the force constant matrix or the Hessian matrix and is obtained from MD++ using the calHessian function.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_{ij} =[\frac{d^{2}V}{d \vec{r}_i \vec{r}_j}] = \left[ {\begin{array}{ccc} \frac{\partial^2V}{\partial x_i\partial x_j} & \frac{\partial^2V}{\partial x_i\partial y_j} & \frac{\partial^2V}{\partial x_i\partial z_j} \\ \frac{\partial^2V}{\partial y_i\partial x_j} & \frac{\partial^2V}{\partial y_i\partial y_j} & \frac{\partial^2V}{\partial y_i\partial z_j} \\ \frac{\partial^2V}{\partial z_i\partial x_j} & \frac{\partial^2V}{\partial z_i\partial y_j} & \frac{\partial^2V}{\partial z_i\partial z_j} \\ \end{array}} \right] \qquad(2) }

From Equation 2, we can calculate the force on the atoms, as shown in Equation 3 as follows, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{f}_{i} = -\frac{d V}{d \vec{r}_{i}}\bigg|_{\vec{R}_{i}} = - \sum_{j} K_{ij} \vec{u}_{j} \qquad(3) }

This force acting on the atoms leads to vibrations in the lattice. Applying the Newtonian equation of motion, we obtain Equation 4,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{f}_{i} = m\ddot{u} = - \sum_{j} K_{ij} \vec{u}_{j} \qquad(4) }