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陈浩南
2025-05-21 12:02:21 +08:00
parent 780bc471e7
commit 7915f9e9dd
1 changed files with 63 additions and 41 deletions
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test-typst/main.typ
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@@ -117,10 +117,9 @@ experiment
- 无缺陷：
我们将声子分为两类，一类是极性比较弱的（18个），一类是比较强的（3个）。
- 弱极性的声子：
- 使用 Gamma 点的声子模式来近似。根据对称性可以明确预测它们的拉曼张量。
- 使用 Gamma 点的声子模式来近似。根据对称性可以预测它们的拉曼张量。
- 我们提出了一个方法，直接根据对称性来估计声子模式的拉曼张量，或者反过来，估计拉曼光谱中峰对应的原子振动模式。估计的结果大多数是正确的。、
- TODO: 可能换成使用原子对（键）来估计，要比使用原子来估计要更合理。
- TODO: 描述估计频率的方法，以及确认 x 的定义的正确性。
- 我们使用第一性原理计算了各种性质，它与实验、预测相符。
- TODO: 将峰宽列出来，将模拟图画出来。
- 某某峰 is reported 在某人的实验中可以看到而在某人的实验中看不到。我们 propose 它的确存在，但只能通过共振拉曼或者zz偏振才能看到。
@@ -277,41 +276,49 @@ The absolute values of $a_i$ is expected to be much larger than that of $epsilon
It could be seen that,
our prediction is mostly consistent with the first principle calculation and experiment.
// Raman Tensor for A1: line1 xz/yz; line2 zz
// Raman Tensor for A1: line1 xx/yy; line2 zz
// Raman Tensor for E1: x-dirc xz or y-dirc yx
// Raman Tensor for E2: x-dirc xy or y-dirc xx or y-dirc -yy
// Relative Vibration Direction: col1 C ABCB col2 Si ABCB
// TODO: remove LO TO or not?
#page(flipped: true)[#figure({
let m(n, content) = table.cell(colspan: n, content);
let m2(content) = table.cell(colspan: 2, content);
let m3(content) = table.cell(colspan: 3, content);
let m4(content) = table.cell(colspan: 4, content);
let A1 = [A#sub[1]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
set text(size: 9pt);
set par(justify: false)
table(columns: 9, align: center + horizon, inset: (x: 2pt, y: 5pt),
// TODO: explain where x comes from
// TODO: 校验这个表的数据（确认没有标错列、没有标错正负号）
[*$x$*], m2[0.25], m4[0.5], m2[0.75],
[*Representation in C#sub[6v]*], m2(E2), E1, A1, E1, A1, m2(E2),
table.cell(rowspan: 2, [*Vibration Direction* (ABCB layer)]), m2[x/y], [x/y], [z], [x/y], [z], m2[x/y],
[Si: $+--+$ #linebreak() C: $++--$], [Si: $+--+$ #linebreak() C: $+--+$],
m2[Si: $+-+-$ #linebreak() C: $+-+-$], m2[Si: $-+-+$ #linebreak() C: $+-+-$],
[Si: $++--$ #linebreak() C: $+--+$], [Si: $-++-$ #linebreak() C: $++--$],
[*Raman Tensor Predicted*],
[$-2(zeta_2+epsilon_2)$], [$2(2eta_2+zeta_2-epsilon_2)$],
[$2(zeta_1-epsilon_1)$], [$2(zeta_5-epsilon_5)$ #linebreak() $2(zeta_6-epsilon_6)$],
[$2(epsilon_1+zeta_1)$], [$2(epsilon_5+zeta_5)$ #linebreak() $2(epsilon_6+zeta_6)$],
[$2(4a_2+2eta_2+zeta_2+epsilon_2)$], [$2(epsilon_2-zeta_2)$],
[*Raman Intensity Predicted*], m2[weak], m4[weak], [strong], [weak],
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: $-A_#text[C]zeta_6$],
[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*],
[1.06], [0.41], [-1.56], [0.10 #linebreak() -1.33], [-0.30], [-1.68 #linebreak() 1.34], [9.41], [0.17],
[*Move-towards Atom-pairs* (In-plane/Out-plane)], [0/2], [2/0], m2[4/0], m2[0/4], m2[4/2],
[*Predicted Frequency*], [low], [medium], [medium], [low], [low], [medium], m2[high],
[-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*],
[197.84], [190.51], [746.91], [591.90], [257.35], [812.87], [756.25], [764.33]
[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,
@@ -353,6 +360,8 @@ The atomic vibration amplitudes are listed separately in the Appendix.
// let B2 = [B#sub[2]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
set text(size: 9pt);
set par(justify: false);
table(columns: 27, align: center + horizon, inset: (x: 3pt, y: 5pt),
// [*Direction of Incident & Scattered Light*],
// m(26)[Any direction (not depend on direction of incident & scattered light)],
@@ -402,11 +411,12 @@ The atomic vibration amplitudes are listed separately in the Appendix.
m(3)[0.08], m(3)[0.09], m(2)[0.08], m(2)[-], m(3)[0.61],
// E1 E2 E2 A1 2B1
m(2)[3.97], m(3)[4.62], m(3)[4.01], m(3)[0.89], m(2)[-],
[*FWHM #linebreak() (Experiment) (cm#super[-1])*],
// TODO: 怎么选取用于比较的合适的实验？
[*FWHM #linebreak() (Experiment, zxxz) (cm#super[-1])*],
// E2 E2 E1 2B1 A1
m(3)[1.11], m(3)[1.11], m(2)[1.11], m(2)[-], m(3)[591.90],
m(3)[2.61], m(3)[2.09], m(2)[1.98], m(2)[-], m(3)[2.64],
// E1 E2 E2 A1 2B1
m(2)[-], m(3)[1.11], m(3)[-], m(3)[1.11], m(2)[-],
m(2)[-], m(3)[3.27], m(3)[-], m(3)[-], m(2)[-],
[*Electrical Polarity*],
// E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1
m(6)[None], m(2)[Weak], m(2)[None], m(5)[Weak], m(6)[None], m(3)[Weak], m(2)[None],
@@ -463,8 +473,10 @@ The atomic vibration amplitudes are listed separately in the Appendix.
let yzmix = [y-z mixed#linebreak() (LO-TO mixed)];
let lopc = [Yes#linebreak() (LOPC)];
let overf = [Yes#linebreak() (overfocused)];
set text(size: 9pt);
set par(justify: false);
table(columns: 20, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Direction of Incident & Scattered Light*], m(5)[z], m(5)[y], m(9)[between z and y, 10#sym.degree to z],
[*Direction of Incident & Scattered Light*], m(5)[z], m(5)[y], m(9)[between z and y, 45#sym.degree to z],
// z y 45 y&z
[*Number of Phonon*], [1], [2], m(3)[3], m(3)[1], [2], [3], m(4)[1], [2], m(4)[3],
[*Vibration Direction*],
@@ -473,7 +485,7 @@ The atomic vibration amplitudes are listed separately in the Appendix.
m(4, yzmix), [x#linebreak() (TO)], m(4, yzmix), // 45 y&z
[*Representation in Group C#sub[6v]*], m(2, E1), m(3, A1), m(14, NA),
// z y 45 y&z
[*Representation in Group C#sub[2v]*], B2, B1, m(3, A1), m(3, A1), B2, B1, m(4, NA), B2, m(4, NA),
[*Representation in Group C#sub[2v]*], B2, B1, m(3, A1), m(3, A1), B2, B1, m(9, NA),
[*Scattering in Polarization*],
[xz], [yz], [xx], [yy], [zz], // z
[xx], [yy], [zz], [xz], [yz], // y
@@ -485,7 +497,7 @@ The atomic vibration amplitudes are listed separately in the Appendix.
[*Visible in Common Raman Experiment*],
m(2)[Yes], m(2, lopc), [No], // z
overf, [No], overf, [Yes], lopc, // y
m(4)[???], [???], m(4)[???], // 45 y&z
m(4)[], [], m(4)[], // 45 y&z
[*Wavenumber (Simulation) (cm#super[-1])*],
// z y 45 y&z
m(2)[776.57], m(3)[933.80], m(3)[761.80], [776.57], [941.33], m(4)[762.76], [776.57], m(4)[940.86],
@@ -668,6 +680,7 @@ For $u$, when it is along x or y direction, it will not change. When it is along
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.
@@ -675,6 +688,15 @@ 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.
@@ -686,24 +708,24 @@ Furthermore, we list predicted modes and their Raman tensors, in @table-predmode
#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 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, 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 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 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, B2], [$mat(,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;a_2+eta_2+zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
[$mat(a_2+eta_2+zeta_2,,;,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
[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,