From 40018dc3534b8337d00a78b1fc4579c6e506131d Mon Sep 17 00:00:00 2001 From: chn Date: Sun, 20 Jul 2025 18:28:12 +0800 Subject: [PATCH] --- paper/appendix/default.typ | 1 - paper/appendix/predict/default.typ | 451 ++++++++++---------- paper/appendix/predict/fig-same.typ | 4 +- paper/appendix/predict/fig-sitoc.typ | 7 + paper/appendix/predict/table-predmode.typ | 31 ++ paper/appendix/predict/table-singleatom.typ | 45 ++ paper/appendix/predmode.typ | 47 -- 画图/SiC相似/atom.csv | 12 + 8 files changed, 315 insertions(+), 283 deletions(-) create mode 100644 paper/appendix/predict/fig-sitoc.typ create mode 100644 paper/appendix/predict/table-predmode.typ create mode 100644 paper/appendix/predict/table-singleatom.typ delete mode 100644 paper/appendix/predmode.typ create mode 100644 画图/SiC相似/atom.csv diff --git a/paper/appendix/default.typ b/paper/appendix/default.typ index 26c2b91..162a893 100644 --- a/paper/appendix/default.typ +++ b/paper/appendix/default.typ @@ -1,4 +1,3 @@ = Appendix #include "predict/default.typ" -#include "predmode.typ" diff --git a/paper/appendix/predict/default.typ b/paper/appendix/predict/default.typ index 926078e..9a56801 100644 --- a/paper/appendix/predict/default.typ +++ b/paper/appendix/predict/default.typ @@ -6,268 +6,251 @@ 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。 -由于同种原子的局部环境相似、只有次近邻才有不同，因此我们将同种原子导致的拉曼效应的差视为一个小量。 +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，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. -Because the local environment of the same type of atom is similar and only different for the next nearest neighbors, - we consider the difference in Raman effect caused by the same type of atom as a small quantity. +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. -推导 A/C 层 Si 原子沿 x 方向振动时的拉曼张量。 +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 rather than translationally symmetric, + 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，当这两个原子同向和反向振动时，它们分别属于 E1(C6v) B2(C2v) 和 E2(C6v) A2(C2v) 表示，因此它们的拉曼张量分别为： +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/C layer vibrating along x direction. -When the two atoms vibrate in the same direction and opposite direction, - they belong to the representation of E#sub[1] of C#sub[6v] or B#sub[2] of C#sub[2v] - and E#sub[2] of C#sub[6v] or A#sub[2] of C#sub[2v], respectively. -Thus, their Raman tensors are in the form of: +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]. +Thus, their Raman tensor can be written as: -$ mat(,,2a_1;,,;2a_1,,;), mat(,2a_2,;2a_2,,;,,;), $ +$ mat(,,2a_1;,,;2a_1,,;) $ -其中 $a_1$ 和 $a_2$ 是两个未知的常数。 +其中 $a_1$ 是未知的常数。 -where $a_1$ and $a_2$ are two constants with unknown values. +where $a_i (i = 1 "to" 6)$ are unknown constants. -将上述结果相加或相减，得到 A 层和 C 层 Si 原子沿 x 方向振动时的拉曼张量分别为： +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： -By adding or subtracting the above results, - we get the Raman tensors of Si atoms in A and C layers vibrating along x direction: +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]. +Thus, their Raman tensor can be written as: -$ mat(,a_2,a_1;a_2,,;a_1,,;), mat(,-a_2,a_1;-a_2,,;-a_1,,;), $ +$ mat(,2a_2,;2a_2,,;,,;) $ -近似给出 B 层 Si 原子沿 x 方向振动时的拉曼张量。 +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： -注意到 B 层 Si 原子与 A 层 Si 原子 +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 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： + +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: + +$ 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 $ + +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. +Thus 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 small quantities, 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： -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 $mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,)d$. +Thus, the Raman tensor of Si atoms in B#sub[1] and B#sub[2] layers vibrating along x direction + can be written as: -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,,;), 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,;) -$ +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。 -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 $epsilon$. +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: -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], +#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 + +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： + +The Raman tensors of C atoms in A and C layers can be obtained using a similar method: + +$ + mat(,b_2,b_1;b_2,,;b_1,,;), mat(,-b_2,b_1;-b_2,,;b_1,,;) +$ + +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： + 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。 + +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. + +#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： + +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: + +$ Delta E = E^T alpha E, #[where] alpha = mat(,a_2,a_1;a_2,,;a_1,,;) $ + +若在翻转的过程中，将电场同样翻转，则总能量不变。因此： + +When the electric field is also flipped during the mirror image, + the total energy does not change, i.e., + +$ 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： + +Thus: + +$ alpha' = mat(,a_2,-a_1;a_2,,;-a_1,,;) $ + +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： + +// TODO: 能量写明是 Delta E + +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. + +#include "table-singleatom.typ" +#include "table-predmode.typ" \ No newline at end of file diff --git a/paper/appendix/predict/fig-same.typ b/paper/appendix/predict/fig-same.typ index df1fbd4..e1415fa 100644 --- a/paper/appendix/predict/fig-same.typ +++ b/paper/appendix/predict/fig-same.typ @@ -1,5 +1,7 @@ #figure( image("/画图/AB相似/embed.svg"), - caption: [Light incidence configurations in our Raman experiments.], + caption: [Local environment of the Si atoms in A and B#sub[1] layers, + 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/fig-sitoc.typ b/paper/appendix/predict/fig-sitoc.typ new file mode 100644 index 0000000..6a8aa92 --- /dev/null +++ b/paper/appendix/predict/fig-sitoc.typ @@ -0,0 +1,7 @@ +#figure( + image("/画图/AB相似/embed.svg"), + caption: [Local environment of the Si atoms in A and B#sub[1] layers, + 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-predmode.typ b/paper/appendix/predict/table-predmode.typ new file mode 100644 index 0000000..7d44b65 --- /dev/null +++ b/paper/appendix/predict/table-predmode.typ @@ -0,0 +1,31 @@ +#figure({ + set par(justify: false); + let c(n, content) = table.cell(colspan: n, content); + let r(n, content) = table.cell(rowspan: n, content); + let r2(content) = r(2, content); + table(columns: 6, align: center + horizon, inset: (x: 1pt, y: 5pt), + r2[*Representation #linebreak() in C#sub[6v]*], r2[*Calculated Frequency* (THz)], + r2[*Relative Vibration Direction*], c(3)[*Raman Tensor*], [Component], [Predicted], [Calculated (a.u.)], + r(6)[A#sub[1]], + r2[591.90], r2[Si: $+-+-$ #linebreak() C: none], + [xx, yy], [$-2epsilon_5$], [-1.68], + [zz], [$-2epsilon_6$], [1.34], + r2[812.87], r2[Si: none #linebreak() C: $+-+-$], + [xx, yy], [$-2eta_5$], [0.10], + [zz], [$-2eta_6$], [-1.33], + r2[933.80], r2[Si: $++++$ #linebreak() C: $----$], + [xx, yy], [$4a_5+4b_5+2epsilon_5+2eta_5$], [-7.68], + [zz], [$4a_6-4b_6+2epsilon_6-2eta_6$], [21.65], + r(3)[E#sub[1]], + [257.35], [Si: $+-+-$ #linebreak() C: $-+-+$], [xz, yz], [$-2epsilon_1+2eta_1$], [-1.56], + [746.91], [Si: $+-+-$ #linebreak() C: $+-+-$], [xz, yz], [$-2epsilon_1-2eta_1$], [-0.30], + [776.57], [Si: $++++$ #linebreak() C: $----$], [xz, yz], [$4a_1-4b_1+2epsilon_1-2eta_1$], [7.32], + r(4)[E#sub[2]], + [190.51], [Si: $++--$ #linebreak() C: $-++-$], [xx, -yy], [$-2epsilon_1+2eta_1$], [-1.56], + [197.84], [Si: $+--+$ #linebreak() C: $++--$], [xx, -yy], [$-2epsilon_1-2eta_1$], [-0.30], + [756.25], [Si: $++--$ #linebreak() C: $+--+$], [xx, -yy], [$4a_1-4b_1+2epsilon_1-2eta_1$], [7.32], + [764.33], [Si: $+--+$ #linebreak() C: $--++$], [xx, -yy], [$4a_1-4b_1+2epsilon_1-2eta_1$], [7.32], + )}, + caption: [Predicted modes and their Raman tensor.], + placement: none, +) diff --git a/paper/appendix/predict/table-singleatom.typ b/paper/appendix/predict/table-singleatom.typ new file mode 100644 index 0000000..57aa19a --- /dev/null +++ b/paper/appendix/predict/table-singleatom.typ @@ -0,0 +1,45 @@ +#figure({ + let r2(content) = table.cell(rowspan: 2, content); + table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt), + table.cell(colspan: 3)[*Vibration Direction*], [x], [y], [z], + table.cell(rowspan: 8)[*Raman tensor #linebreak() of atoms*], + r2[A layer], + [C], + [$mat(,b_2,b_1;b_2,,;b_1,,;)$], + [$mat(b_2,,;,-b_2,b_1;,b_1,;)$], + [$mat(b_5,,;,b_5,;,,b_6;)$], + [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], + [$mat(,-b_2-eta_2,b_1+eta_1;-b_2-eta_2,,;b_1+eta_1,,;)$], + [$mat(-b_2-eta_2,,;,b_2+eta_2,b_1+eta_1;,b_1+eta_1,;)$], + [$mat(b_5+eta_5,,;,b_5+eta_5,;,,b_6+eta_6;)$], + [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], + [$mat(,-b_2,b_1;-b_2,,;b_1,,;)$], + [$mat(-b_2,,;,b_2,b_1;,b_1,;)$], + [$mat(b_5,,;,b_5,;,,b_6;)$], + [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], + [$mat(,b_2+eta_2,b_1+eta_1;b_2+eta_2,,;b_1+eta_1,,;)$], + [$mat(b_2+eta_2,,;,-b_2-eta_2,b_1+eta_1;,b_1+eta_1,;)$], + [$mat(b_5+eta_5,,;,b_5+eta_5,;,,b_6+eta_6;)$], + [Si], + [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$], + [$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$], + [$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$], + )}, + caption: [Raman tensor of each atoms.], + placement: none, +) diff --git a/paper/appendix/predmode.typ b/paper/appendix/predmode.typ deleted file mode 100644 index 38ba565..0000000 --- a/paper/appendix/predmode.typ +++ /dev/null @@ -1,47 +0,0 @@ -// Raman Tensor for A1: line1 xx/yy; line2 zz -// Raman Tensor for E1: x-dirc xz or y-dirc yx -// Raman Tensor for E2: x-dirc xy or y-dirc xx or y-dirc -yy -// TODO: remove LO TO or not? -#page(flipped: true)[#figure({ - let m(n, content) = table.cell(colspan: n, content); - let m2(content) = table.cell(colspan: 2, content); - let m3(content) = table.cell(colspan: 3, content); - let m4(content) = table.cell(colspan: 4, content); - set text(size: 9pt); - set par(justify: false); - table(columns: 11, align: center + horizon, inset: (x: 3pt, y: 5pt), - [*Representation in C#sub[6v]*], m3[A#sub[1]], m3[E#sub[1]], m4[E#sub[2]], - [*Relative Vibration Direction*], - [Si: $+-+-$ #linebreak() C: $0000$], [Si: $0000$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$], - [Si: $+-+-$ #linebreak() C: $-+-+$], [Si: $+-+-$ #linebreak() C: $+-+-$], [Si: $++++$ #linebreak() C: $----$], - [Si: $++--$ #linebreak() C: $-++-$], [Si: $+--+$ #linebreak() C: $++--$], - [Si: $++--$ #linebreak() C: $+--+$], [Si: $+--+$ #linebreak() C: $--++$], - [*Vibration Direction*], m3[z], m3[x/y], m4[x/y], - [*Raman Tensor Predicted*], [xx/yy: $-2A_#text[Si] epsilon_5$ #linebreak() zz: $-2A_#text[Si]epsilon_6$], - [xx/yy: $-2A_#text[C]zeta_5$ #linebreak() zz: $-2A_#text[C]zeta_6$], - [xx/yy: $2A_#text[Si] (2a_5+epsilon_5) + 2A_#text[C] (2a_5+eta_5+zeta_5)$ #linebreak() zz: $2A_#text[Si] (2a_6+epsilon_6) + 2A_#text[C] (2a_6+eta_6+zeta_6)$], - [xz/yz: $-2A_#text[Si]epsilon_1-2A_#text[C]zeta_1$], - [xz/yz: $-2A_#text[Si]epsilon_1+2A_#text[C]zeta_1$], - [xz/yz: $2A_#text[Si] (2a_1+epsilon_1) +2A_#text[C] (2a_1+2eta_1+zeta_1))$], - [xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) -2A_#text[C] (2a_2+2eta_2+zeta_2))$], - [xx/-yy/xy: $-2A_#text[Si]epsilon_2-2A_#text[C]zeta_2$], - [xx/-yy/xy: $2A_#text[Si] (2a_2+epsilon_2) +2A_#text[C] (2a_2+2eta_2+zeta_2))$], - [xx/-yy/xy: $-2A_#text[Si]epsilon_2+2A_#text[C]zeta_2$], - [*Raman Intensity Predicted*], m2[weak], [strong], m2[weak], [strong], m2[weak], [strong], [weak], - [*Raman Tensor Calculated*], - [-1.68 #linebreak() 1.34], [0.10 #linebreak() -1.33], [-7.68 #linebreak() 21.65], - [-1.56], [-0.30], [7.32], [-0.41], [1.06], [9.41], [-0.71], - // [*x*], [1 axial acoustic], [0 axial optical], [1 axial optical], - // [0 axial acoustic], [1 axial optical], [1 axial optical], - // m2[0.5 acoustic], m2[0.5 optical], - [*Type*], [axial acoustic], [axial optical], [longitudinal optical], - [planer acoustic], [planer optical], [transverse optical], - m2[planer acoustic], m2[planer optical], - [*Move-towards Atom-pairs* (In-plane/Out-plane)], [4/0], [0/4], [4/4], [0/4], [4/0], [4/4], [0/2], [2/0], m2[4/2], - // [*Predicted Frequency*], [low], [medium], [high], [medium], [low], [high], [low], [medium], m2[high], - [*Calculated Frequency*], - [591.90], [812.87], [933.80], [257.35], [746.91], [776.57], [190.51], [197.84], [756.25], [764.33] - )}, - caption: [Predicted modes and their "Raman tensor"], - placement: none, -)] diff --git a/画图/SiC相似/atom.csv b/画图/SiC相似/atom.csv new file mode 100644 index 0000000..b02e540 --- /dev/null +++ b/画图/SiC相似/atom.csv @@ -0,0 +1,12 @@ +type,x,y,z,radius +0,3.08813,0.00000,10.10781,1.18 +0,0.00000,0.00000,10.10781,1.18 +0,1.54407,2.67440,10.10781,1.18 +1,1.54407,0.89147, 9.48201,0.77 +0,1.54407,0.89147, 7.58104,1.18 +1,1.54407,2.67440, 6.94819,0.77 +1,3.08813,0.00000, 6.94819,0.77 +1,0.00000,0.00000, 6.94819,0.77 +0,3.08813,0.00000, 5.05312,1.18 +0,0.00000,0.00000, 5.05312,1.18 +0,1.54407,2.67440, 5.05312,1.18