Background theory: Introduction to phonons¶
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Source code: https://github.com/osscar-org/quantum-mechanics/tree/master/notebook/lattice-vibration/theory/theory_phonon_2d.ipynb
Introduction¶
Please find a description of the notation used in the phonon 1D theory notebook.
2D monoatomic lattice¶
$$\require{cancel}$$
In the 2D case, displacements are now 2D vectors :
\begin{equation}
\vec{u_n}=
\begin{bmatrix}
u_{nm,x}(t) \\
u_{nm,y}(t)
\end{bmatrix}
=
\begin{bmatrix}
A_{x}e^{i(\vec{k}\cdot\vec{R_{nm}}-\omega t)} \\
A_{y}e^{i(\vec{k}\cdot\vec{R_{nm}}-\omega t)}\
\end{bmatrix}
\end{equation}
with $k=\begin{bmatrix} k_x\\k_y\end{bmatrix}$ and $R_{nm}=\begin{bmatrix}n\cdot a_x\\m\cdot a_y\end{bmatrix}$.
Let $\vec{h}=\vec{u}_{n',m'}-\vec{u}_{nm}$ the vector between two atoms and $\hat{h}=\frac{\vec{h}}{||\vec{h}||}$.
Then the vector dynamical equation is given by:
\begin{equation}
M\frac{\partial \vec{u}_{nm}}{\partial t}=C\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h}
\end{equation}
and the component dynamical equations are :
\begin{align}
M\frac{\partial \vec{u}_{nm,x}}{\partial t} & =C\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right](\hat{h}\cdot\vec{x}) \\
M\frac{\partial \vec{u}_{nm,y}}{\partial t} & =C\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right](\hat{h}\cdot\vec{y}).
\end{align}
If we now consider all neighbouring atoms, we get:
\begin{align}
M\frac{\partial \vec{u}_{nm}}{\partial t} & =C_1\sum_{1^{st} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \\
& + C_2\sum_{2^{nd} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \nonumber \\
& + C_3\sum_{3^{rd} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \nonumber \\
& + \cdots\nonumber
\end{align}
which, when projected onto $x$ and $y$, would give an eigenvalue problem with $\vec{A_{\omega}}=\begin{bmatrix}
A_x \\
A_y
\end{bmatrix}$ the eigenvector of the $\omega$ eigenvalue.
2D diatomic lattice¶
Tackling on now the 2D diatomic lattice, the displacement vectors are :
\begin{equation}
\vec{u_n}=
\begin{bmatrix}
u_{nm,x}(t) \\
u_{nm,y}(t)
\end{bmatrix}
=
\begin{bmatrix}
A_{1,x}e^{i(\vec{k}\cdot\vec{R_{nm}^{(1)}}-\omega t)} \\
A_{1,y}e^{i(\vec{k}\cdot\vec{R_{nm}^{(1)}}-\omega t)}\
\end{bmatrix}, \quad
\vec{v_n}=
\begin{bmatrix}
v_{nm,x}(t) \\
v_{nm,y}(t)
\end{bmatrix}
=
\begin{bmatrix}
A_{2,x}e^{i(\vec{k}\cdot\vec{R_{nm}^{(2)}}-\omega t)} \\
A_{2,y}e^{i(\vec{k}\cdot\vec{R_{nm}^{(2)}}-\omega t)}\
\end{bmatrix}
\end{equation}
with $R_{nm}^{(1)}=\begin{bmatrix}n\cdot a_x\\m\cdot a_y\end{bmatrix}$ and $R_{nm}^{(2)}=\begin{bmatrix}n\cdot a_x+c\cdot a_x\\m\cdot a_y +d\cdot a_y\end{bmatrix}$ where $c=1$, $d=0$ for the honeycomb lattice.
The vector dynamical equation is given by:
\begin{equation}
M\frac{\partial \vec{u}_{nm}}{\partial t}=C\left[(\vec{v}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h}
\end{equation}
but now, $\vec{v}_{n',m'}$ could as well be $\vec{u}_{n',m'}$ depending on whether the neighbouring atom is of the same kind or not. For notational simplicity, we will consider that the neighboring atom is of different kind.
Considering all neighboring atoms, we would now get:
\begin{align}
M\frac{\partial \vec{u}_{nm}}{\partial t} & =C_1\sum_{1^{st} \text{neigh.}}\left[(\vec{v}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \\
& + C_2\sum_{2^{nd} \text{neigh.}}\left[(\vec{v}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \nonumber \\
& + C_3\sum_{3^{rd} \text{neigh.}}\left[(\vec{v}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \nonumber \\
& + \cdots\nonumber
\end{align}
and
\begin{align}
M\frac{\partial \vec{v}_{nm}}{\partial t} & =C_1\sum_{1^{st} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{v}_{n,m})\cdot \hat{h}\right]\hat{h} \\
& + C_2\sum_{2^{nd} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{v}_{n,m})\cdot \hat{h}\right]\hat{h} \nonumber \\
& + C_3\sum_{3^{rd} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{v}_{n,m})\cdot \hat{h}\right]\hat{h} \nonumber \\
& + \cdots\nonumber
\end{align}
from which both expression can be projected on $x$ and $y$ to get a 4-dimensional eigenvalue problem with $A_w=\begin{pmatrix}A_{1,x}\\A_{1,y}\\A_{2,x}\\A_{2,y}\end{pmatrix}$ the eigenvector.
Further references¶
G. Grosso and G. Pastori Parravicini, Solid state physics, Second edition. Amsterdam: Academic Press, an imprint of Elsevier, 2014.