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

== Approximation of Raman tensor of 4H-SiC

近似的核心思路。

我们的近似方法基于这样一个原则：将拉曼张量（即原子位移引起的极化率变化）分配给单位晶胞中的每个原子。
一些原子的局部环境相似但不完全相同，我们将它们的拉曼张量的差视为一个小量（$epsilon$, $eta$ and $zeta$），
  剩余的部分视为一个大量($a$).
将一个模式中所有参与振动的原子的贡献相加，就可以得到该模式的拉曼张量。
若模式的该拉曼张量只包含小量（$epsilon$, $eta$ and $zeta$），
  说明该模式中，原子振动导致的拉曼效应互相抵消了大部分，该模式的拉曼活性较弱；
而若该拉曼张量包含较大的常数项（$a$），说明该模式的拉曼效应较强。

The center principle of our approximation is to assign the Raman tensor
    (i.e., change of polarizability caused by atomic displacement)
  to each atom in the unit cell.
For the atoms with similar but not exactly the same local environment,
  we consider the difference in their Raman tensors as quantities with small absolute value
    ($epsilon$, $eta$ and $zeta$),
  while the remaining part is treated as quantities with large absolute value ($a$).
The Raman tensor of a phonon mode can be obtained
  by summing the contributions of all atoms participating in the vibration.
If the Raman tensor of a mode only contains quantities with small absolute value ($epsilon$, $eta$ and $zeta$),
  it indicates that the Raman effect caused by atomic vibrations in this mode is largely canceled out,
  and thus the mode has weak Raman activity.
Otherwise, if the Raman tensor contains quantities with large absolute value ($a$),
  it indicates that the mode has strong Raman activity.

使用 A/B1/C/B2 层的表述，而不是 ABCB，来区分两个 B 层。
这是因为两个 B 层的原子局部环境互相镜面对称而不是平移对称，导致它们的拉曼张量不相等。

In this section, AB#sub[1]CB#sub[2] instead of ABCB was used to denote the four bilayers in 4H-SiC primative cell
  to clearly distinguish the two B layers.
This is because the local environment of the two B layers is mirror symmetric with each other,
  thus their Raman tensors are not equal.

=== Raman tensor of Si atoms in A and C layers

我们首先推导 A/C 层 Si 原子沿 x 方向振动时的拉曼张量。
根据前文，我们知道，当这两个原子同步地沿 x 正方向振动时，它们属于 E1(C6v) B2(C2v) 表示，拉曼张量可以写为：

We first derive the Raman tensor of Si atoms in A and C layer vibrating along x direction.
The vibration where the two atoms vibrate synchronously along the positive x direction,
  belongs to the representation of E#sub[1] of C#sub[6v] and B#sub[2] of C#sub[2v].
Thus, their Raman tensor can be written as:

$ mat(,,2a_1;,,;2a_1,,;) $

其中 $a_1$ 是未知的常数。

where $a_i (i = 1 "to" 6)$ are unknown constants.

当 A 层 Si 原子沿 x 正方向振动而 C 层 Si 原子沿 x 负方向振动时，
  它们属于 E#sub[2] of C#sub[6v] or A#sub[2] of C#sub[2v] 表示，
  拉曼张量可以写为：

For the vibration where the Si atom in A layer vibrates towards the positive x direction
    and the Si atom in C layer vibrates towards the negative x direction,
  it belongs to the representation of E#sub[2] of C#sub[6v] and A#sub[2] of C#sub[2v].
Thus, their Raman tensor can be written as:

$ mat(,2a_2,;2a_2,,;,,;) $

因此，A 层和 C 层 Si 原子沿 x 方向振动时的拉曼张量分别为：

Thus, the Raman tensors of Si atoms in A and C layers vibrating in the positive x direction are:

$ mat(,a_2,a_1;a_2,,;a_1,,;), mat(,-a_2,a_1;-a_2,,;a_1,,;), $

接下来讨论 A/C 层 Si 原子沿 y 方向振动时的拉曼张量，使用相似的方法可以得到：

The Raman tensors of Si atoms in A and C layers vibrating along positive y direction
  can be obtained using the same method, which gives:

$ mat(a_4,,;,-a_4,a_3;,a_3,;), mat(-a_4,,;,a_4,a_3;,a_3,;) $

$\{a_1, a_2, a_3, a_4\}$ 之间并不独立。为了确定它们之间的关系，我们考虑 A 层中 Si 原子沿 b 轴正方向振动所导致的拉曼张量（记为 $alpha'$）。
一方面，它可以看作由向 x 和 y 正方向振动的拉曼张量（记为 $alpha_x$ 和 $alpha_y$）通过线性组合得到；
  另一方面，它也可以看作由 $alpha_x$ 通过将体系绕 z 轴旋转 $120 degree$ 得到。
因此：

where $\{a_3, a_4\}$ are not independent of $\{a_1, a_2\}$.
To determine the relationship between $\{a_1, a_2\}$ and $\{a_3, a_4\}$,
  the Raman tensor of Si atoms in A layer vibrating along the positive b-axis direction
    (denoted as $alpha'$) was considered.
On one hand,
  it can be expressed as a linear combination of the Raman tensors vibrating in the positive x and y directions
    (denoted as $alpha_x$ and $alpha_y$, respectively);
  on the other hand,
    it can also be obtained from $alpha_x$ by rotating the system by $120 degree$.
Thus:

$
  alpha' = C_3 alpha_x C_3^T = -1/2 alpha_x + sqrt(3)/2 alpha_y, \
  "where" C_3 = mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1;),
    alpha_x = mat(,a_2,a_1;a_2,,;a_1,,;), alpha_y = mat(a_4,,;,-a_4,a_3;,a_3,;)
$

化简可得：

Simplifying the above equations, we have:

$ a_3 = a_1, a_4 = a_2 $

与 x 和 y 方向的情况类似，可以推导出 A/C 层 C 原子沿 z 方向振动时的拉曼张量。总结如下。

The Raman tensors of C atoms in A and C layers vibrating along z direction can be derived similarly.
The results are summarized as follows:

#figure(
  table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
    table.cell(colspan: 3)[*Vibration Direction*], [x], [y], [z],
    table.cell(rowspan: 2)[*Raman tensor #linebreak() of atoms*],
      [A layer], [Si],
        [$mat(,a_2,a_1;a_2,,;a_1,,;)$],
        [$mat(a_2,,;,-a_2,a_1;,a_1,;)$],
        [$mat(a_5,,;,a_5,;,,a_6;)$],
      [C layer], [Si],
        [$mat(,-a_2,a_1;-a_2,,;a_1,,;)$],
        [$mat(-a_2,,;,a_2,a_1;,a_1,;)$],
        [$mat(a_5,,;,a_5,;,,a_6;)$],
  ),
  placement: none,
)

=== Raman tensor of Si atoms in B#sub[1] and B#sub[2] layers

与 A/C 层原子类似，同理可以给出 B#sub[1]B#sub[2] 层原子沿 x 方向振动的拉曼张量：

The Raman tensor of Si atoms in B#sub[1] and B#sub[2] layer vibrating along positive x direction
  can be written out similarily as that in A and C layer:

$ mat(,a'_2,a'_1;a'_2,,;a'_1,,;), mat(,-a'_2,a'_1;-a'_2,,;a'_1,,;) $

注意到 B 层 Si 原子与 A 层 Si 原子的局部环境非常相似（最近邻完全相同，次近邻也只有一半不同，如图所示），
  因此可以推测它们的拉曼张量只有较小的不同，即：

Because the local environment of Si atoms in B layer is very similar to that in A layer (as shown in @figure-same),
  we can assume that their Raman tensors differ only by quantities with small absolute values, i.e.,

$
 a'_1 = a_1 + epsilon_1, abs(epsilon_1) << abs(a_1), \
 a'_2 = a_2 + epsilon_2, abs(epsilon_2) << abs(a_2),
$

#include "fig-same.typ"

由此可以写出 B 层 Si 原子沿 x 方向振动时的拉曼张量：

Thus, the Raman tensor of Si atoms in B#sub[1] and B#sub[2] layers vibrating along positive x direction
  can be written as:

$
  mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;),
  mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)
$

同理，可以得到 B#sub[1] 和 B#sub[2] 层 Si 原子沿其它方向振动的拉曼张量。
总结如下。

The Raman tensors of Si atoms in B#sub[1] and B#sub[2] layers vibrating along other directions
    can be obtained using similar method,
  and the results are summarized as follows:

#figure(
  table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
    table.cell(colspan: 3)[*Vibration Direction*], [x], [y], [z],
    table.cell(rowspan: 2)[*Raman tensor #linebreak() of atoms*],
      [B#sub[1] layer], [Si],
        [$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$],
        [$mat(a_2+epsilon_2,,;,-a_2-epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
        [$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
      [B#sub[2] layer], [Si],
        [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$],
        [$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
        [$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
  ),
  placement: none,
)

=== Raman tensor of C atoms

C 原子的拉曼张量，使用与 Si 原子类似的方法，可以得到。
例如 A 层 C 原子的拉曼张量可以写为：

The Raman tensors of C atoms can be obtained using a similar method as that of Si atoms.
For example, the Raman tensor of C atom in A layer can be written as:

#figure(
  table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
    table.cell(colspan: 3)[*Vibration Direction*], [x], [y], [z],
    table.cell(rowspan: 2)[*Raman tensor #linebreak() of atoms*], table.cell(rowspan: 2)[A layer],
      [C],
        [$alpha_"Cx" = mat(,b_2,b_1;b_2,,;b_1,,;)$],
        [$alpha_"Cy" = mat(b_2,,;,-b_2,b_1;,b_1,;)$],
        [$alpha_"Cz" = mat(b_5,,;,b_5,;,,b_6;)$],
      [Si],
        [$alpha_"Six" = mat(,a_2,a_1;a_2,,;a_1,,;)$],
        [$alpha_"Siy" = mat(a_2,,;,-a_2,a_1;,a_1,;)$],
        [$alpha_"Siz" = mat(a_5,,;,a_5,;,,a_6;)$],
  ),
  placement: none,
)

我们需要估计 ${b_1, b_2}$ 与 ${a_1, a_2}$ 之间的关系。
考虑 A 层 C 原子的环境，它可以由 A 层 Si 原子通过以下三步操作得到：
  先沿基平面取镜像，然后反转电荷，再调整原子质量等其它因素，如图所示。
我们分别考虑这些过程中拉曼张量的变化。

The relationship between $\{b_1, b_2\}$ and $\{a_1, a_2\}$ needs to be estimated.
The environment of C atoms in A layer can be obtained from that of Si atoms in A layer with the following three steps:
  first taking a mirror image along the basal plane, then reversing the charge,
  and finally adjusting the atomic mass and other factors, as shown in @fig-sitoc.
The change of Raman tensor during these processes would be discussed separately.

#include "fig-sitoc.typ"

第一步中，记翻转后的拉曼张量为 $alpha'_"Six"$、$alpha'_"Siy"$ 和 $alpha'_"Siz"$。
对于 x 和 y 方向振动的拉曼张量，只需要将群元素 $sigma_"h" = op("diag") (1, 1, -1)$ 作用上去即可；
  对于 z 方向振动的拉曼张量，还需要乘以 $-1$，因为在这个过程中振动的方向发生了改变。

In the first step,
  the Raman tensors after taking the mirror image were denoted as $alpha'_"Six"$, $alpha'_"Siy"$ and $alpha'_"Siz"$.
For $alpha'_"Six"$ and $alpha'_"Siy"$, they are connected with $alpha_"Six"$ and $alpha_"Siy"$
  by the group element $sigma_"h" = op("diag") (1, 1, -1)$.
For $alpha'_"Siz"$, an additional factor of $-1$ is needed,
  because the direction of vibration has changed during this process.

$
  alpha'_"Six" = sigma_"h" alpha_"Six" sigma_"h"^T = mat(,a_2,-a_1;a_2,,;-a_1,,;), \
  alpha'_"Siy" = sigma_"h" alpha_"Siy" sigma_"h"^T = mat(a_2,,;,-a_2,-a_1;,-a_1,;), \
  alpha'_"Siz" = -sigma_"h" alpha_"Siz" sigma_"h"^T = mat(-a_5,,;,-a_5,;,,-a_6;),
$

翻转电荷的过程不会导致拉曼张量的变化。
这可以通过考虑在外场 $E$ 作用下的能量变化来得知。
记电荷翻转前后的拉曼张量分别为 $alpha$ 和 $alpha'$。
若在翻转电荷的过程中，外加电场同样翻转，则总能量不变。
因此 $E^T alpha E = (-E)^T alpha' (-E)$，因此 $alpha = alpha'$。

In the second step (reversing the charge), the Raman tensor does not change.
This can be derived by considering the energy caused by an external electric field $E$.
The Raman tensors before and after charge reversal were denoted as $alpha$ and $alpha'$.
When the direction of the external electric field is also reversed during the charge reversal,
  the total energy does not change, i.e., $E^T alpha E = (-E)^T alpha' (-E)$.
Thus, we have $alpha = alpha'$.

第三步中，我们假定原子质量和其它因素的变化对拉曼张量的影响较小，
  即 $alpha'_"Six"$、$alpha'_"Siy"$ 和 $alpha'_"Siz"$ 与 $alpha_"Six"$、$alpha_"Siy"$ 和 $alpha_"Siz"$ 之间仅有较小的差异。
因此：

In the third step, we assume that the change in atomic mass and other factors has a small effect on the Raman tensor,
  i.e., $alpha'_"Six"$, $alpha'_"Siy"$ and $alpha'_"Siz"$ differ from $alpha_"Six"$, $alpha_"Siy"$ and $alpha_"Siz"$
  only by small quantities, respectively.
Thus:

$ 
  b_1 = -a_1 - zeta_1, abs(zeta_1) << abs(a_1), \
  b_2 = a_2 + zeta_2, abs(zeta_2) << abs(a_2), \
  b_5 = -a_5 - zeta_5, abs(zeta_5) << abs(a_5), \
  b_6 = -a_6 - zeta_6, abs(zeta_6) << abs(a_6),
$

=== Summary

我们将各个原子的拉曼张量总结于 @table-singleatom，用它推测了各个模式的拉曼张量并与第一性原理计算对比，
  结果如 @table-predmode 所示。

We summarized the Raman tensors of each atom in @table-singleatom.
The result was used to predict the Raman tensors of each mode and comparing with first-principles calculations,
  which is shown in @table-predmode.

#include "table-singleatom.typ"
#include "table-predmode.typ"