Phonon dispersion relation

Phonon Dispersion Relation

Yanming Wang, Saad Bhamla and Wei Cai

May, 2012

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METHODS

Small displacement method for calculating the phonon dispersion relation

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

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(\{ \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, ${\vec {r}}_{i}$ is the coordinate of the $ith$ atom, ${\vec {R}}_{i}$ is the position of the $ith$ atom at emuilibrium and ${\vec {u}}$ is a small displacement from the equilibrium position: ${\vec {u}}_{i}={\vec {r}}_{i}-{\vec {R}}_{i}$.

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.

$K_{ij}=[{\frac {d^{2}V}{d{\vec {r}}_{i}{\vec {r}}_{j}}}]=\left[{\begin{array}{ccc}{\frac {\partial ^{2}V}{\partial x_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial x_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial y_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial y_{i}\partial z_{j}}}\\{\frac {\partial ^{2}V}{\partial z_{i}\partial x_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial y_{j}}}&{\frac {\partial ^{2}V}{\partial z_{i}\partial z_{j}}}\\\end{array}}\right]\qquad (2)$

From Equation 2, we can calculate the force on the atoms, as shown in Equation 3 as follows, ${\vec {f}}_{i}=-{\frac {dV}{d{\vec {r}}_{i}}}{\bigg |}_{{\vec {R}}_{i}}=-\sum _{j}K_{ij}{\vec {u}}_{j}\qquad (3)$

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

${\vec {f}}_{i}=m{\ddot {u}}=-\sum _{j}K_{ij}{\vec {u}}_{j}\qquad (4)$

Obviously, equation 4 is a second order differential equation about ${\vec {u}}$,and ${\vec {u}}$ can be derived from a general solution of a plane wave shown in Equation 5,

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}(\vec{R},t) = \vec{A}\cdot exp[i(\vec{k} \cdot \vec{R} - \omega t)] \qquad(5) }

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 \vec{A}} determines the amplitude and direction of vibration, 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 \omega} represents the vibration modes of the system, 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{k}} represents the wave vector. Substituting Equation 5 in Equation 4, we obtain the following eigenfunction expression:

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{A} \cdot \xi = Q(\vec{k)} \cdot \vec{A} \qquad(6) }

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 \xi = m \omega^{2}} 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 Q(\vec{k)}} is the dynamical matrix defined by:

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 Q(\vec{k)} = \sum_{j} K_{ij} \cdot exp[i(\vec{k} \cdot \vec{R} - \omega t)] \qquad(7) }

Equation 6 is an eigenfunction 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 \xi} represents the eigenvalues, 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{A}} represents the eigenvectors of the dynamical matrix 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 Q(\vec{k})} . We may notice each given 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{k}} coressponds to a unique 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 Q(\vec{k)}} and by solving the eigenvalues of 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 Q(\vec{k)}} , finally we can obtain the relation between 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{k}} 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 \omega } , which is the phonon dispersion relation.