Phonon dispersion relation

Phonon Dispersion Relation of graphene

Yanming Wang, Saad Bhamla and Wei Cai

June, 2012

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Background knowledge

The following links may be helpful to read at first.

What is phonon?

What is phonon dispersion relation?

What is plane wave?

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

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

where $V$ is the potential function, ${\vec {r}}_{i}$ 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) }

Obviously, equation 4 is a second order differential equation about 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}} ,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}} 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.

TEST CASE

The tcl script Graphene_md.tcl can provide the atoms' configurations and the Hessian Matrix for graphene dispersion curve calculation. First, you need to compile the REBO potential in MD++ using the command like,

make rebo build=R SYS=mac

Please refer the details for compiling to Introduction to MD++.

Then the following command can be run to generate the data for the calculation.

bin/rebo_mac scripts/ME346/graphene_md.tcl 0 0

After the simulation, you can find two files named as perf.cn and hessian.out. The matlab script Graphene_phonon.m can extract data from the files and draw the phonon dispersion curve for graphene.

In the end, you will have the same figure shown below,

You may want to compare this result with DFT calculation. Here is the plot. The red curves are from this small displacement method using REBO potential and the black dots are the data from DFT calculation.

The three phonon dispersion branches, which originate from 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 \Gamma} -point of the Brilliouin zone correspond to acoustic modes: an out-of plane mode (ZA), an in-plane transverse mode (TA), and in-plane longitudinal (LA), listed in order of increasing energy. The remaining three branches correspond to optical modes: one out-of plane mode (ZO), and two in-plane modes (TO) and (LO).

We observe that the REBO potential captures the acoustic modes quite well, but there is a slight deviation in the optical modes. This has also been reported in other literatures. Since it is known that the thermal conductivity is mainly dependent on the acoustic modes, this strategy is satisfactory.