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Background theory: Introduction to phonons Original notebook: theory_phonon_2d.ipynb
Introduction
Please find a description of the notation used in the phonon 1D theory notebook .
2D monoatomic lattice
In the 2D case, displacements are now 2D vectors :
u n ⃗ = [ u n m , x ( t ) u n m , y ( t ) ] = [ A x e i ( k ⃗ ⋅ R n m ⃗ − ω t ) A y e i ( k ⃗ ⋅ R n m ⃗ − ω t ) ] \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} u n = [ u nm , x ( t ) u nm , y ( t ) ] = [ A x e i ( k ⋅ R nm − ω t ) A y e i ( k ⋅ R nm − ω t ) ] with k = [ k x k y ] k=\begin{bmatrix} k_x\\k_y\end{bmatrix} k = [ k x k y ] R n m = [ n ⋅ a x m ⋅ a y ] R_{nm}=\begin{bmatrix}n\cdot a_x\\m\cdot a_y\end{bmatrix} R nm = [ n ⋅ a x m ⋅ a y ]
Let h ⃗ = u ⃗ n ′ , m ′ − u ⃗ n m \vec{h}=\vec{u}_{n',m'}-\vec{u}_{nm} h = u n ′ , m ′ − u nm h ^ = h ⃗ ∣ ∣ h ⃗ ∣ ∣ \hat{h}=\frac{\vec{h}}{||\vec{h}||} h ^ = ∣∣ h ∣∣ h
Then the vector dynamical equation is given by:
M ∂ u ⃗ n m ∂ t = C [ ( u ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ M\frac{\partial \vec{u}_{nm}}{\partial t}=C\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} M ∂ t ∂ u nm = C [ ( u n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ and the component dynamical equations are :
M ∂ u ⃗ n m , x ∂ t = C [ ( u ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] ( h ^ ⋅ x ⃗ ) M ∂ u ⃗ n m , y ∂ t = C [ ( u ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] ( h ^ ⋅ y ⃗ ) . \begin{aligned}
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{aligned} M ∂ t ∂ u nm , x M ∂ t ∂ u nm , y = C [ ( u n ′ , m ′ − u n , m ) ⋅ h ^ ] ( h ^ ⋅ x ) = C [ ( u n ′ , m ′ − u n , m ) ⋅ h ^ ] ( h ^ ⋅ y ) . If we now consider all neighbouring atoms, we get:
M ∂ u ⃗ n m ∂ t = C 1 ∑ 1 s t neigh. [ ( u ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ + C 2 ∑ 2 n d neigh. [ ( u ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ + C 3 ∑ 3 r d neigh. [ ( u ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ + ⋯ \begin{aligned}
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} \\
& + C_3\sum_{3^{rd} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \\
& + \cdots
\end{aligned} M ∂ t ∂ u nm = C 1 1 s t neigh. ∑ [ ( u n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ + C 2 2 n d neigh. ∑ [ ( u n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ + C 3 3 r d neigh. ∑ [ ( u n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ + ⋯ which, when projected onto x x x y y y A ω ⃗ = [ A x A y ] \vec{A_{\omega}}=\begin{bmatrix}
A_x \\
A_y
\end{bmatrix} A ω = [ A x A y ] ω \omega ω
2D diatomic lattice
Tackling on now the 2D diatomic lattice, the displacement vectors are :
u n ⃗ = [ u n m , x ( t ) u n m , y ( t ) ] = [ A 1 , x e i ( k ⃗ ⋅ R n m ( 1 ) ⃗ − ω t ) A 1 , y e i ( k ⃗ ⋅ R n m ( 1 ) ⃗ − ω t ) ] , v n ⃗ = [ v n m , x ( t ) v n m , y ( t ) ] = [ A 2 , x e i ( k ⃗ ⋅ R n m ( 2 ) ⃗ − ω t ) A 2 , y e i ( k ⃗ ⋅ R n m ( 2 ) ⃗ − ω t ) ] \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} u n = [ u nm , x ( t ) u nm , y ( t ) ] = [ A 1 , x e i ( k ⋅ R nm ( 1 ) − ω t ) A 1 , y e i ( k ⋅ R nm ( 1 ) − ω t ) ] , v n = [ v nm , x ( t ) v nm , y ( t ) ] = [ A 2 , x e i ( k ⋅ R nm ( 2 ) − ω t ) A 2 , y e i ( k ⋅ R nm ( 2 ) − ω t ) ] with R n m ( 1 ) = [ n ⋅ a x m ⋅ a y ] R_{nm}^{(1)}=\begin{bmatrix}n\cdot a_x\\m\cdot a_y\end{bmatrix} R nm ( 1 ) = [ n ⋅ a x m ⋅ a y ] R n m ( 2 ) = [ n ⋅ a x + c ⋅ a x m ⋅ a y + d ⋅ a y ] R_{nm}^{(2)}=\begin{bmatrix}n\cdot a_x+c\cdot a_x\\m\cdot a_y +d\cdot a_y\end{bmatrix} R nm ( 2 ) = [ n ⋅ a x + c ⋅ a x m ⋅ a y + d ⋅ a y ] c = 1 c=1 c = 1 d = 0 d=0 d = 0
The vector dynamical equation is given by:
M ∂ u ⃗ n m ∂ t = C [ ( v ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ M\frac{\partial \vec{u}_{nm}}{\partial t}=C\left[(\vec{v}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} M ∂ t ∂ u nm = C [ ( v n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ but now, v ⃗ n ′ , m ′ \vec{v}_{n',m'} v n ′ , m ′ u ⃗ n ′ , m ′ \vec{u}_{n',m'} u n ′ , m ′
Considering all neighboring atoms, we would now get:
M ∂ u ⃗ n m ∂ t = C 1 ∑ 1 s t neigh. [ ( v ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ + C 2 ∑ 2 n d neigh. [ ( v ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ + C 3 ∑ 3 r d neigh. [ ( v ⃗ n ′ , m ′ − u ⃗ n , m ) ⋅ h ^ ] h ^ + ⋯ \begin{aligned}
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} \\
& + C_3\sum_{3^{rd} \text{neigh.}}\left[(\vec{v}_{n',m'}-\vec{u}_{n,m})\cdot \hat{h}\right]\hat{h} \\
& + \cdots
\end{aligned} M ∂ t ∂ u nm = C 1 1 s t neigh. ∑ [ ( v n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ + C 2 2 n d neigh. ∑ [ ( v n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ + C 3 3 r d neigh. ∑ [ ( v n ′ , m ′ − u n , m ) ⋅ h ^ ] h ^ + ⋯ and
M ∂ v ⃗ n m ∂ t = C 1 ∑ 1 s t neigh. [ ( u ⃗ n ′ , m ′ − v ⃗ n , m ) ⋅ h ^ ] h ^ + C 2 ∑ 2 n d neigh. [ ( u ⃗ n ′ , m ′ − v ⃗ n , m ) ⋅ h ^ ] h ^ + C 3 ∑ 3 r d neigh. [ ( u ⃗ n ′ , m ′ − v ⃗ n , m ) ⋅ h ^ ] h ^ + ⋯ \begin{aligned}
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} \\
& + C_3\sum_{3^{rd} \text{neigh.}}\left[(\vec{u}_{n',m'}-\vec{v}_{n,m})\cdot \hat{h}\right]\hat{h} \\
& + \cdots
\end{aligned} M ∂ t ∂ v nm = C 1 1 s t neigh. ∑ [ ( u n ′ , m ′ − v n , m ) ⋅ h ^ ] h ^ + C 2 2 n d neigh. ∑ [ ( u n ′ , m ′ − v n , m ) ⋅ h ^ ] h ^ + C 3 3 r d neigh. ∑ [ ( u n ′ , m ′ − v n , m ) ⋅ h ^ ] h ^ + ⋯ from which both expression can be projected on x x x y y y A w = ( A 1 , x A 1 , y A 2 , x A 2 , y ) A_w=\begin{pmatrix}A_{1,x}\\A_{1,y}\\A_{2,x}\\A_{2,y}\end{pmatrix} A w = A 1 , x A 1 , y A 2 , x A 2 , y
Further references
G. Grosso and G. Pastori Parravicini, Solid state physics, Second edition. Amsterdam: Academic Press, an imprint of Elsevier, 2014.