Phy5670/Phonon in Graphene

Structure of Graphene

Carbon atoms in graphene are constructed on a honeycomb lattice, which is shown in Fig. 1. The circle and solid point together combine into a two-point basis in the Bravais lattice. The carbon-carbon distance is 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 a\simeq 1.42{\AA}} . The primitive vectors are

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}_1=\frac{\sqrt{3}}{2}a\hat{x}+\frac{3}{2}a\hat{y}}

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}_2=-\frac{\sqrt{3}}{2}a\hat{x}+\frac{3}{2}a\hat{y}}

Then the reciprocal lattice parameters can by generated by using 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{b_i}\cdot \vec{a_j}=2\pi\delta_{ij}, (i,j=1,2)}

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{b}_1=\frac{2\pi}{a}(\frac{\sqrt{3}}{3}\hat{x}+\frac{1}{3}\hat{y})}

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{b}_2=\frac{2\pi}{a}(-\frac{\sqrt{3}}{2}a\hat{x}+\frac{3}{2}a\hat{y})}

General phonon dispersion relations

Before we move on to solving phonon in graphene, we discuss how to solve phonon problem for general case. Suppose we have particle system, which is constructed by unit cells. For each unit cell, there are N atoms. Let's use 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=(x_i, y_i, z_i)} to represent the displacement 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 i^{th}} particle coordinate. Thus, the equations of motion for these N atoms in each unit are:

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 M_i\ddot\vec{u}_i=\sum_{j}K^{(ij)}(\vec{u}_j-\vec{u}_i)} 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 \qquad (i=1,2,...N)}

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 M_i} is the mass of each particle, 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 K^{(ij)}} is 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 3*3} force constant tensor between 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 i^{th}} and 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 j^{th}} particles.

By Fourier transform, we have

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=\frac{1}{\sqrt{N_\Omega}}\sum_{\vec{k^'}}e^{-i(\vec{k^'}\cdot\vec{R}_i-\omega t)}\vec{u}_\vec{k^'}^{(i)}}

and, in turn,

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{k}^{(i)}=\frac{1}{\sqrt{N_\Omega}}\sum_{\vec{R}_i}e^{i(\vec{k}\cdot\vec{R_i}-\omega t)}\vec{u_i}}

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 N_\Omega} is the number of the wave vectors 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_i}} in the first Brillouin zone, 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{R_i}} is the original position 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 i^{th}} particle.

We assume that all 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}} have the same eigenfreguency 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 means:

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 \ddot{\vec{u}_i}=-\omega^2\vec{u}_i}

So,

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 (\sum_{j} K^{(ij)}-M_i\omega^2)\sum_{\vec{k^'}}e^{-i\vec{k^'}\cdot\vec{R}_i}\vec{u}_{\vec{k^'}}^{(i)}=\sum_{j}K^{(ij)}\sum_{\vec{k^'}}e^{-i\vec{k^'}\cdot\vec{R}_j}\vec{u}_\vec{k^'}^{(j)}}

Multiplying 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 e^{i\vec{k}\cdot\vec{R}_i}} and taking the summation 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 \vec{R}_i} , we obtain:

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 \sum_{\vec{R_i}}\exp{i(\vec{k}-\vec{k^'})\cdot\vec{R_i}}=N_\Omega\delta_{\vec{k},\vec{k^'}}}

Thus,

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 (\sum_{j} K^{(ij)}-M_i\omega^2(\vec{k})I)\vec{u}_\vec{k}^{(i)}-\sum_{j}K^{(ij)}e^{i\vec{k}\cdot\Delta\vec{R}_{ij}}\vec{u}_\vec{k}^{(j)}=0}

where, I is a 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 3\times 3} unit matrix, 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 \Delta\vec{R_{ij}}=\vec{R_i}-\vec{R_j}} is the relative 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 i^{th}} particle and 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 j^{th}} particle.

By using boundary conditions, all j can be shifted to a site in the original unit cell. And, when j and j' sites are equivalent to each other (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}_j} 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{R}_{j^'}} ) differ by a lattice vector), we have 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{k}^{(j)}=\vec{u}_\vec{k}^{(j^')}} . Thus, 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{k}^{(j)}} is contained within the original unit cell.

For 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')}} , there will be a phase factor 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 e^{i\vec{k}\cdot\Delta\vec{R}_{ij'}}}

Thus, for a 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}} , eigenfuctions can be written as:

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 D(\vec{k})\vec{u}_\vec{k}=0}

in which, 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 D(\vec{k})} is 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 3N\times3N} dynamical matrix, and is constructed by small 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 3\times 3} matrices 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 D^{(ij)}(\vec{k}) \quad (i,j=1,2,...,N)} .

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 D^{(ij)}(\vec{k})=(\sum_{j''} K^{(ij'')}-M_i\omega^2(\vec{k})I)\delta_{ij}-\sum_{j'}K^{(ij)}e^{i\vec{k}\cdot\Delta\vec{R}_{ij'}}}

Phonon dispersion relations for graphene

Now, under the master of calculation method for phonon dispersion in general case, we now can study phonon dispersion relations for graphene.