#import "@preview/physica:0.9.5": pdv, super-T-as-transpose
#show: super-T-as-transpose

= Appendix

#include "predmode.typ"

The center principle is to assign the Raman tensor (i.e., change of polarizability caused by atomic displacement)
  to each atom in the unit cell.
This including the following steps:
  - Write out the change of polarizability caused by displacement of Si atom in A and C layer,
      Where unknown non-zero components are denoted by $a_1$, $a_2$, $a_5$, $a_6$.
    For example, when we move the Si atom in A layer slightly towards the x+ direction in $d$ distance,
      the change of polarizability should be $mat(,a_2,a_1;a_2,,;a_1,,)d$.
    This could be done by conclusion above.
  - The Si atom in B layer have similar local environment as the A and C layer, with only a little difference.
    We denote these difference by $epsilon_1$, $epsilon_2$, $epsilon_5$, $epsilon_6$,
      and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
    For example, when we move the Si atom in B layer slightly towards the x+ direction in $d$ distance,
      the change of polarizability should be $mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,)d$.
  - The local environment of C atom in A layer is similar to the Si atom in A layer with charge reversed and 
      the system reversed along xy plane.
    We denote these difference by $eta_1$, $eta_2$, $eta_5$, $eta_6$,
      and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
    For example, when we move the C atom in A layer slightly towards the x+ direction in $d$ distance,
      the change of polarizability should be $mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,)d$.
  - Similar to the case in Si atoms, we derive the change of polarizability
      caused by moving C atom in B layer slightly towards the x+ direction in $d$ distance,
      which should be $mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,)d$.

Lets assign Raman tensor onto each atom.
That is, Raman tensor is derivative of the polarizability with respect to the atomic displacement:
$
  alpha = pdv(chi, u)
$
where $u$ should be the displacement of the atom corresponding to a phonon mode.
But, even when $u$ is *NOT* the displacement of a phonon
  (for example, lets only slightly move Si atom in A layer, keeping other atoms fixed),
  the (high-frequency) polarizability is still well-defined,
  and the will still cause a change in the polarizability.
Even more, the group representation theory is still applicable in this condition:
  the only thing that matters is, when applying $g$ to the system,
  the tensor transformed into $g^(-1) alpha g$ or $g alpha g^(-1)$,
  no matter $alpha$ is Raman tensor or something else, or it is related to a phonon or not.

Thus, we can, in principle, "assign" Raman tensor of a phonon, to each atom.
This "assign" is unique since both the atom movement and all phonons have 24 dimensions.

Next, we consider what these single-atom-caused "Raman tensors" looks like.
For example, what happens if we move the Si atom in A layer slightly along the x+ direction?
Consider also move the Si atom in C layer slightly, along x+ or x- direction.
How about the Raman tensor caused by the both two atoms?
In first case, this is B2 representation in E1 representation. Thus the Raman tensor should be something like:
$
  mat(,,2a_1;,,;2a_1,,;)
$
In the second case, it is A2 in E2. It turns out:
$
  mat(,2a_2,;2a_2,,;,,;)
$
The average of these two tensors should be the s"Raman tensor" cause by move only the Si atom in A layer,
  slightly towards x+ direction.
$
  mat(,a_2,a_1;a_2,,;a_1,,;)
$
The difference should be the "Raman tensor" of the second atom.
$
  mat(,-a_2,a_1;-a_2,,;a_1,,;)
$

// This approach applied relied on the fact that, all Si atom in 4H-SiC is "distinguishable" by the symmetry operations.
// I mean, what will happen if we have two Si atoms in A layer?
// Apparently, we could not extract the "Raman tensor" of only one of the two atoms.
// This is the case for the 6H-SiC.
// Hence, we will provide a more general approach to estimate the "Raman tensor" of a single atom.

Consider the Si atom in the B1 layer.
It lives in an environment quite similar to the A layer.
Thus, the "Raman tensor" caused by it should be similar to the one caused by the A layer:
$
  mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)
$
Similar to the Si atom in B2 layer:
$
  mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)
$

Same approach applied for Si atom vibrate in y direction.
When we move both Si atoms in A and C layer in y+ direction, 
  it is B1 in E1, thus the "Raman tensor" should be:
$
  mat(,,;,,2a_3;,2a_3,;)
$
And if we move Si in A layer towards y+ but Si in C layer towards y-,
  it is A2 in E2:
$
  mat(2a_4,,;,-2a_4,;,,;)
$
Thus we get the "Raman tensor" of Si atom in A layer sololy move towards y+ direction:
$
  mat(a_4,,;,-a_4,a_3;,a_3,;)
$
and the "Raman tensor" of Si atom in C layer towards y+ direction:
$
  mat(-a_4,,;,a_4,a_3;,a_3,;)
$
Same applied for the Si atom in B layer:
$
  mat(a_4+epsilon_4,,;,-a_4-epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
$
$
  mat(-a_4-epsilon_4,,;,a_4+epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
$

Before consider z-direction, it is important to note that, $a_1$ $a_2$ $a_3$ $a_4$ are not independent.
Consider vibration along x+ direction (lets say the distance is $d$).
System energy caused by external electric field and vibration is:
$
  E^T (mat(,,2a_1;,,;2a_1,,) d) E
$
Apply C#sub[3] to atom vibration and external field, energy should not change. We got:
$
  (mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)^T ( mat(,,2a_1;,,;2a_1,,)(-1/2 d) + mat(,,;,,2a_3;,2a_3,)(sqrt(3)/2 d) )
  (mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)
$
It is equal to:
$
  E^T (mat(,,1/2 a_1 + 3/2 a_3;,,sqrt(3)/2 a_1 - sqrt(3)/2 a_3;1/2 a_1 + 3/2 a_3,sqrt(3)/2 a_1 - sqrt(3)/2 a_3,) d) E
$
Thus:
$
  1/2 a_1 + 3/2 a_3 = 2a_1 #linebreak()
  sqrt(3)/2 a_1 - sqrt(3)/2 a_3 = 0
$
Thus $a_1 = a_3$.
Apply the same method, we get $abs(a_2) = abs(a_4)$.
Since we have not define the sign of $a_4$, we could take $a_2 = a_4$.
Same for $epsilon$.

Now consider what if we move the Si atom in A layer along z+ direction.
If we move the Si atom in C layer along z+ direction, it is A1:
$
  mat(2a_5,,;,2a_5,;,,2a_6;)
$
If we move the Si atom in C layer along z- direction, it is B1:
$
  0
$
Thus we get the "Raman tensor" of Si atom in A or C layer towards z+ direction:
$
  mat(a_5,,;,a_5,;,,a_6;)
$

Lets consider the C atom in A layer.
It should be somehow similar to the Si atom in A layer, but with a negative sign in some places,
  and then add or subtract some little value.
Actually, the "transformation" of Si atom in A layer to C atom in A layer applied in the following steps:
  - reverse charge.
  - reverse system along xy plane.
First we consider the first step.
Taking the define of electricity tenser:
$
  P = chi E
$
Lets reverse charge of the system, say we now have electricity tensor $chi'$. We get:
$
  -P = chi'(-E)
$
Thus we get $chi' = chi$, the first step does not change the electricity tensor, nor the "Raman tensor".

Now we consider the second step.
For electricity tensor, it will become:
$
  mat(1,,;,1,;,,-1) chi mat(1,,;,1,;,,-1)
$
For $u$, when it is along x or y direction, it will not change. When it is along z direction, it will become $-u$.

So in conclusion, Raman tensor of C atom in A layer could be estimated from the Raman tensor of Si atom in A layer, by:
  - for movement alone x and y direction, xz yz should be applied a negative sign.
  - for movement alone z direction, xx xy yy zz should be applied a negative sign.

Export "Raman tensor" of C atom in C layer from C atom in A layer, in the same way.

Now consider the C atom in B1 layer.
Is it similar to the C atom in A layer, just like that for Si atom?
No. It turns out to be similar to the C atom in C layer.

We summarize these stuff into @table-singleatom.

Until now, we only consider the "Raman tensor" caused by single atom or atoms move in the same amplitudes.
However, that is not the case in real phonon.
- In some A1 modes, only Si or C atom moves. If we take the magnitude of eigenvector as 1,
  then amplitude of each atom is $1/(4sqrt(m_#text[Si]))$ or $1/(4sqrt(m_#text[C]))$.
- In other cases, the amplitude of Si and C are in the ration of $m_#text[C] : m_#text[Si]$.
  thus the amplitude of Si atom is $1/2 sqrt(1/(m_#text[Si]+m_#text[Si]^2/m_#text[C]))$, so do the C atom.


Furthermore, we list predicted modes and their Raman tensors, in @table-predmode.

- $a$: Raman tensor of Si atom in A layer, large value.
- $epsilon$: Difference of Raman tensors of Si atom in A and B1 layer, small value.
- $eta$: Difference of Raman tensors of C and Si atom in A layer, small value.
- $zeta$: Difference of Raman tensors of C atoms in A and B layer, small value.


#page(flipped: true)[#figure({
  table(columns: 4, align: center + horizon, inset: (x: 3pt, y: 5pt),
    [*Move Direction*], [x], [y], [z],
    [Si A], [$mat(,a_2,a_1;a_2,,;a_1,,;)$], [$mat(a_2,,;,-a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
    [C A], [$mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,;)$],
      [$mat(a_2+eta_2,,;,-a_2-eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
    [Si B1], [$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$],
      [$mat(a_2+epsilon_2,,;,-a_2-epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
      [$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
    [C, B1], [$mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
      [$mat(-a_2-eta_2-zeta_2,,;,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
      [$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
    [Si C], [$mat(,-a_2,a_1;-a_2,,;a_1,,;)$], [$mat(-a_2,,;,a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
    [C, C], [$mat(,-a_2-eta_2,-a_1-eta_1;-a_2-eta_2,,;-a_1-eta_1,,;)$],
      [$mat(-a_2-eta_2,,;,a_2+eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
    [Si B2], [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$],
      [$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
      [$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
    [C, B2], [$mat(,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;a_2+eta_2+zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
      [$mat(a_2+eta_2+zeta_2,,;,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
      [$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
  )},
  caption: ["Raman tensor" caused by single atom],
  placement: none,
)<table-singleatom>]
