From c53569a26a8f247a34a5c5ee332146044c3d307d Mon Sep 17 00:00:00 2001 From: chn Date: Mon, 21 Jul 2025 16:14:52 +0800 Subject: [PATCH] --- paper/appendix/predict/default.typ | 172 +++++++++++--------- paper/appendix/predict/fig-sitoc.typ | 2 +- paper/appendix/predict/table-singleatom.typ | 24 +-- 3 files changed, 109 insertions(+), 89 deletions(-) diff --git a/paper/appendix/predict/default.typ b/paper/appendix/predict/default.typ index 9a56801..4316b68 100644 --- a/paper/appendix/predict/default.typ +++ b/paper/appendix/predict/default.typ @@ -11,7 +11,7 @@ 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，说明该模式的拉曼效应较强。 +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) @@ -33,7 +33,7 @@ Otherwise, if the Raman tensor contains quantities with large absolute value ($a 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 rather than translationally symmetric, +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 @@ -42,8 +42,8 @@ This is because the local environment of the two B layers is mirror symmetric ra 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. -When the two atoms vibrate synchronously in the positive x direction, - they belong to the representation of E#sub[1] of C#sub[6v] or B#sub[2] of C#sub[2v]. +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,,;) $ @@ -56,9 +56,9 @@ where $a_i (i = 1 "to" 6)$ are unknown constants. 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： -When the Si atom in A layer vibrates in the positive x direction - while the Si atom in C layer vibrates in the negative x direction, -they belong to the representation of E#sub[2] of C#sub[6v] or A#sub[2] of C#sub[2v]. +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,,;,,;) $ @@ -72,33 +72,31 @@ $ 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 method, which gives: + 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，我们考虑将体系绕 z 轴旋转 120 度， - 同一个点的坐标旋转前后分别为 $r$ 和 $C_3 r$，其中 $r$ 是一个列向量，$C_3$ 是一个旋转矩阵： +$\{a_1, a_2, a_3, a_4\}$ 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 system is rotated by $120 degree$ around the z axis. -The point located at $r$ before rotation should be at $C_3 r$ after rotation, - where $r$ is a column vector and $C_3$ is a rotation matrix: + 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: -$ C_3 = mat(-1/2,-sqrt(3)/2,0;sqrt(3)/2,-1/2,0;0,0,1;) $ - -记 A 层中 Si 原子沿与 x 轴夹角 $120 degree$ 的方向振动所导致的拉曼张量为 $alpha'$， - 沿 x 和 y 正方向振动的拉曼张量则为 $alpha_x$ 和 $alpha_y$。 -一方面，$alpha'$ 可以从 $alpha_x$ 出发，将体系旋转 $120 degree$ 得到； - 另一方面，$alpha'$ 也可以由 $alpha_x$ 和 $alpha_y$ 通过线性组合得到： - -The Raman tensor of Si atoms in A layer vibrating along the direction at an angle of $120 degree$ with the x axis - is denoted as $alpha'$, - while those vibrating in the positive x and y directions are denoted as $alpha_x$ and $alpha_y$, respectively. -On one hand, $alpha'$ can be obtained from $alpha_x$ by rotating the system by $120 degree$; - on the other hand, $alpha'$ can also be expressed as a linear combination of $alpha_x$ and $alpha_y$: - -$ alpha' = C_3 alpha_x C_3^T = -1/2 alpha_x + sqrt(3)/2 alpha_y $ +$ + 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： @@ -109,8 +107,7 @@ $ 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. -Thus the results are summarized as follows: - +The results are summarized as follows: #figure( table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt), @@ -141,7 +138,7 @@ $ mat(,a'_2,a'_1;a'_2,,;a'_1,,;), mat(,-a'_2,a'_1;-a'_2,,;a'_1,,;) $ 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 small quantities, i.e., + 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), \ @@ -152,7 +149,7 @@ $ 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 x direction +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: $ @@ -185,72 +182,95 @@ The Raman tensors of Si atoms in B#sub[1] and B#sub[2] layers vibrating along ot === Raman tensor of C atoms -考虑 A/C 层 C 原子的拉曼张量，使用与 Si 原子类似的方法，可以得到： +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 in A and C layers can be obtained using a similar method: +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: -$ - mat(,b_2,b_1;b_2,,;b_1,,;), mat(,-b_2,b_1;-b_2,,;b_1,,;) -$ +#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 原子通过以下操作得到： +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。 -我们分两步来推导这个过程中拉曼张量的变化。 +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 - by first taking a mirror image along the basal plane, - then reversing the charge and adjusting the atomic mass and other factors, - as shown in @figure-sitoc. -The change of Raman tensor during this process could be derived following these two steps. +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" -记翻转前后的 Si 原子拉曼张量为 $alpha$ 和 $alpha'$。 -考虑在外场 $E$ 作用下，Si 原子沿 x 方向振动导致的系统能量变化为： +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。 -The Raman tensor of the Si atom in A layer before and after taking the mirror image - was denoted as $alpha$ and $alpha'$, respectively. -Before the mirror image, - the system energy change caused by the vibration of this atom along x direction - under an external electric field $E$ is: +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. -$ Delta E = E^T alpha E, #[where] alpha = mat(,a_2,a_1;a_2,,;a_1,,;) $ +$ + 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;), +$ -若在翻转的过程中，将电场同样翻转，则总能量不变。因此： +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 $alpha = alpha'$。 -When the electric field is also flipped during the mirror image, - the total energy does not change, i.e., +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 $alpha = alpha'$. -$ E^T alpha E = Delta E = (sigma E)^T alpha' (sigma E), #[where] sigma = mat(1,0,0;0,1,0;0,0,-1;) $ - -整理可得： +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: -$ alpha' = mat(,a_2,-a_1;a_2,,;-a_1,,;) $ +$ + 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), +$ -记电荷反转后，Si 原子拉曼张量为 $alpha''$。 -若将外加电场方向同时反转，则能量不变。即： +=== Summary -// TODO: 能量写明是 Delta E +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。 -The Raman tensor of Si atom after charge reversal was denoted as $alpha''$. -The energy does not change when the direction of the external electric field is also reversed, i.e., - -$ E^T alpha' E = (sigma' E)^T alpha'' (sigma' E), #[where] sigma' = -1 $ - -Thus: - -$ alpha'' = alpha' $ - -使用类似的方法，得到 Si 原子沿其它方向和 C 原子沿各个方向的拉曼张量，以及各个模式的拉曼张量。 - -Similarily, we can write out the Raman tensors of Si atoms virbrating along other directions - and the Raman tensors of C atoms, - and thus the Raman tensors of all phonon modes. -The results are summarized in @table-singleatom and @table-predmode. +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" \ No newline at end of file diff --git a/paper/appendix/predict/fig-sitoc.typ b/paper/appendix/predict/fig-sitoc.typ index 6a8aa92..fc95c86 100644 --- a/paper/appendix/predict/fig-sitoc.typ +++ b/paper/appendix/predict/fig-sitoc.typ @@ -4,4 +4,4 @@ where the nearest neighbors are exactly the same, and only half of the next nearest neighbors are different. ], placement: none, -) +) diff --git a/paper/appendix/predict/table-singleatom.typ b/paper/appendix/predict/table-singleatom.typ index 57aa19a..c044e01 100644 --- a/paper/appendix/predict/table-singleatom.typ +++ b/paper/appendix/predict/table-singleatom.typ @@ -5,36 +5,36 @@ table.cell(rowspan: 8)[*Raman tensor #linebreak() of atoms*], r2[A layer], [C], - [$mat(,b_2,b_1;b_2,,;b_1,,;)$], - [$mat(b_2,,;,-b_2,b_1;,b_1,;)$], - [$mat(b_5,,;,b_5,;,,b_6;)$], + [$mat(,a_2+zeta_2,-a_1-zeta_1;a_2+zeta_2,,;-a_1-zeta_1,,;)$], + [$mat(a_2+zeta_2,,;,-a_2-zeta_2,-a_1-zeta_1;,-a_1-zeta_1,;)$], + [$mat(-a_5-zeta_5,,;,-a_5-zeta_5,;,,-a_6-zeta_6;)$], [Si], [$mat(,a_2,a_1;a_2,,;a_1,,;)$], [$mat(a_2,,;,-a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$], r2[B#sub[1] layer], [C], - [$mat(,-b_2-eta_2,b_1+eta_1;-b_2-eta_2,,;b_1+eta_1,,;)$], - [$mat(-b_2-eta_2,,;,b_2+eta_2,b_1+eta_1;,b_1+eta_1,;)$], - [$mat(b_5+eta_5,,;,b_5+eta_5,;,,b_6+eta_6;)$], + [$mat(,-a_2-zeta_2-eta_2,-a_1-zeta_1+eta_1;-a_2-zeta_2-eta_2,,;-a_1-zeta_1+eta_1,,;)$], + [$mat(-a_2-zeta_2-eta_2,,;,a_2+zeta_2+eta_2,-a_1-zeta_1+eta_1;,-a_1-zeta_1+eta_1,;)$], + [$mat(-a_5-zeta_5+eta_5,,;,-a_5-zeta_5+eta_5,;,,-a_6-zeta_6+eta_6;)$], [Si], [$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$], [$mat(a_2+epsilon_2,,;,-a_2-epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$], [$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$], r2[C layer], [C], - [$mat(,-b_2,b_1;-b_2,,;b_1,,;)$], - [$mat(-b_2,,;,b_2,b_1;,b_1,;)$], - [$mat(b_5,,;,b_5,;,,b_6;)$], + [$mat(,-a_2-zeta_2,-a_1-zeta_1;-a_2-zeta_2,,;-a_1-zeta_1,,;)$], + [$mat(-a_2-zeta_2,,;,a_2+zeta_2,-a_1-zeta_1;,-a_1-zeta_1,;)$], + [$mat(-a_5-zeta_5,,;,-a_5-zeta_5,;,,-a_6-zeta_6;)$], [Si], [$mat(,-a_2,a_1;-a_2,,;a_1,,;)$], [$mat(-a_2,,;,a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$], r2[B#sub[2] layer], [C], - [$mat(,b_2+eta_2,b_1+eta_1;b_2+eta_2,,;b_1+eta_1,,;)$], - [$mat(b_2+eta_2,,;,-b_2-eta_2,b_1+eta_1;,b_1+eta_1,;)$], - [$mat(b_5+eta_5,,;,b_5+eta_5,;,,b_6+eta_6;)$], + [$mat(,a_2+zeta_2+eta_2,-a_1-zeta_1+eta_1;a_2+zeta_2+eta_2,,;-a_1-zeta_1+eta_1,,;)$], + [$mat(a_2+zeta_2+eta_2,,;,-a_2-zeta_2-eta_2,-a_1-zeta_1+eta_1;,-a_1-zeta_1+eta_1,;)$], + [$mat(-a_5-zeta_5+eta_5,,;,-a_5-zeta_5+eta_5,;,,-a_6-zeta_6+eta_6;)$], [Si], [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$], [$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],