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陈浩南
2025-05-26 11:28:09 +08:00
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test-typst/main.typ
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@@ -1,28 +1,6 @@
#import "@preview/starter-journal-article:0.4.0": article, author-meta #import "@preview/starter-journal-article:0.4.0": article, author-meta
#import "@preview/tablem:0.2.0": tablem #import "@preview/tablem:0.2.0": tablem
#import "@preview/physica:0.9.5": pdv, super-T-as-transpose
#show: super-T-as-transpose
#set par.line(numbering: "1")
#set par(justify: true)
// 思源宋体，也算是宋体吧
#set text(font: ("Times New Roman", "Source Han Serif SC"))
// TODO: fix indent of first line
#show figure.caption: it => {
set text(10pt)
// TODO: how to align correctly?
align(center, box(align(left, it), width: 80%))
}
#set page(
// paper: "us-letter",
// header: align(right)[
// A fluid dynamic model for
// glacier flow
// ],
numbering: "1/1",
)
// TODO: why globally set placement not work?
// #set figure(placement: none)
#show: article.with( #show: article.with(
title: "Article Title", title: "Article Title",
authors: ( authors: (
@@ -40,17 +18,32 @@
), ),
abstract: [#lorem(100)], abstract: [#lorem(100)],
keywords: ("Typst", "Template", "Journal Article"), keywords: ("Typst", "Template", "Journal Article"),
// template: (body: (body) => {
// show heading.where(level: 1): it => block(above: 1.5em, below: 1.5em)[
// #set pad(bottom: 2em, top: 1em)
// #it.body
// ]
// set par(first-line-indent: (amount: 2em, all: true))
// set footnote(numbering: "1")
// body
// })
) )
// 行号
#set par.line(numbering: "1")
// 两端对齐
#set par(justify: true)
// 中文使用思源宋体，英文使用 Times New Roman
#set text(font: ("Times New Roman", "Source Han Serif SC"))
// 图表标题
#show figure.caption: it => {
set text(10pt)
align(center, box(align(left, it), width: 80%))
}
// 页码
#set page(
numbering: "1/1",
)
// TODO: why globally set placement not work?
// #set figure(placement: none)
// 标题序号
#set heading(numbering: "1.") #set heading(numbering: "1.")
= Introduction = Introduction
@@ -59,34 +52,23 @@
= Method = Method
// TODO #include "section/method.typ"
calc
experiment
= Results and Discussion = Results and Discussion
- 无缺陷： // - 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）。 // 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： // - 弱极性的声子：
- 使用 Gamma 点的声子模式来近似。根据对称性可以预测它们的拉曼张量。 // - TODO: 确认一下最后一次实验中，峰偏移等是否与掺杂有明显关系，以及这个关系与之前是否相同。
- 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。、 // - 强极性的声子：
- 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。 // - 强极性声子在 Gamma 附近散射谱不连续，它的声子模式由入射光的方向决定。在入射光不沿 z 轴的情况下，使用 C6v 群不再适用。
- 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: 写文字
- TODO: 将峰宽列出来，将模拟图画出来。 // - 在接近 y 轴入射时，可以看到分裂。这个模式可能对表面敏感。
- 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。 // - TODO: 佐证它对表面敏感
- TODO: 引用文献。 // - 对于 LO，可能形成 LOPC
- TODO: 确认一下最后一次实验中，峰偏移等是否与掺杂有明显关系，以及这个关系与之前是否相同。 // - 有缺陷的情况：
- 强极性的声子： // - TODO: 描述缺陷原子的振动
- 强极性声子在 Gamma 附近散射谱不连续，它的声子模式由入射光的方向决定。在入射光不沿 z 轴的情况下，使用 C6v 群不再适用。 // - TODO: 计算拉曼张量，描述光谱的可能变化
- TODO: 写文字
- 在接近 y 轴入射时，可以看到分裂。这个模式可能对表面敏感。
- TODO: 佐证它对表面敏感
- 对于 LO，可能形成 LOPC
- 有缺陷的情况：
- TODO: 描述缺陷原子的振动
- TODO: 计算拉曼张量，描述光谱的可能变化
== Phonons in Perfect 4H-SiC == Phonons in Perfect 4H-SiC
@@ -157,229 +139,7 @@ experiment
) )
] ]
The center principle is to assign the Raman tensor (i.e., change of polarizability caused by atomic displacement) #include "section/appendix/default.typ"
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>]
#include "掺杂晶格变化.typ" #include "掺杂晶格变化.typ"
#include "晶格变化导致的频率变化.typ" #include "晶格变化导致的频率变化.typ"

226
test-typst/section/appendix/default.typ Normal file
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@@ -0,0 +1,226 @@
#import "@preview/physica:0.9.5": pdv, super-T-as-transpose
#show: super-T-as-transpose
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>]

1
test-typst/section/method.typ Normal file
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@@ -0,0 +1 @@
Only back-scattering configurations were considered in this study.

14
test-typst/section/perfect/default.typ
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@@ -1,10 +1,7 @@
(There are 21 phonons in total. // There are 21 phonons in total.
We classified them into two categories: 18 negligible-polar phonons and 3 strong-polar phonons.) // We classified them into two categories: 18 negligible-polar phonons and 3 strong-polar phonons.
// 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
// 根据这些声子模式的极性，我们将这些声子分成两类。
The phonons involved in Raman scattering are located in reciprocal space around the #sym.Gamma point, The phonons involved in Raman scattering are located in reciprocal space around the #sym.Gamma point,
at the exact positions are determined by the wavevectors of the incident and scattered light. and the exact positions are determined by the wavevectors of the incident and scattered light.
At each such position, there are 21 phonon modes (degenerate modes are counted as their multiplicity). At each such position, there are 21 phonon modes (degenerate modes are counted as their multiplicity).
We classify these 21 phonons into two categories based on their polarities. We classify these 21 phonons into two categories based on their polarities.
The 18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity), The 18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
@@ -12,8 +9,7 @@ The 18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons
and the other three phonons are strong-polar phonons, and the other three phonons are strong-polar phonons,
where the polarity gives rise to observable effects in the Raman spectra. where the polarity gives rise to observable effects in the Raman spectra.
(This classification make sense.) // This classification make sense.
This classification is based on the fact that This classification is based on the fact that
the four Si atoms in the primitive cell of 4H-SiC carry similar positive Born effective charges (BECs), the four Si atoms in the primitive cell of 4H-SiC carry similar positive Born effective charges (BECs),
and the four C atoms carry similar negative BECs (see @table-bec). and the four C atoms carry similar negative BECs (see @table-bec).
@@ -25,4 +21,4 @@ In contrast, in the three strong-polar phonons,
all Si atoms vibrate in the same direction, and all the C atoms vibrate in the opposite direction, all Si atoms vibrate in the same direction, and all the C atoms vibrate in the opposite direction,
resulting in a strong dipole moment. resulting in a strong dipole moment.
#include "bec.typ" #include "table-bec.typ"

13
test-typst/section/perfect/non-polar/default.typ
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@@ -1,7 +1,8 @@
// We investigate phonons at Gamma instead of the exact location near Gamma. // We investigate phonons at Gamma instead of the exact location near Gamma.
Phonons at the #sym.Gamma point were used Phonons at the #sym.Gamma point were used
to approximate negligible-polar phonons that participating in Raman processes of any incident/scattered light. to approximate negligible-polar phonons that participating in Raman processes
This approximation is widely adopted and justified by the fact that, // TODO: cite regardless of the wavevector of the incident and scattered light.
This approximation is widely adopted (cite) and justified by the fact that,
although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point, although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
they are very close to the #sym.Gamma point in reciprocal space they are very close to the #sym.Gamma point in reciprocal space
(about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light, (about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
@@ -9,11 +10,9 @@ This approximation is widely adopted and justified by the fact that, // TODO: ci
see orange dotted line in @figure-discont), see orange dotted line in @figure-discont),
and their dispersion at #sym.Gamma point is continuous with vanishing derivatives. and their dispersion at #sym.Gamma point is continuous with vanishing derivatives.
Therefore, negligible-polar phonons involved in Raman processes Therefore, negligible-polar phonons involved in Raman processes
have nearly indistinguishable properties from those at the #sym.Gamma point, have nearly indistinguishable properties from those at the #sym.Gamma point.
and the phonon participating in Raman processes of different incident/scattered light directions
are all nearly identical to the phonons at the #sym.Gamma point.
#include "discont.typ" #include "figure-discont.typ"
// Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.) // Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
Phonons at the #sym.Gamma point satisfy the C#sub[6v] point group symmetry, Phonons at the #sym.Gamma point satisfy the C#sub[6v] point group symmetry,
@@ -30,7 +29,7 @@ However, whether a mode is sufficiently strong to be experimentally visible
depends on the magnitudes of its Raman tensor components, depends on the magnitudes of its Raman tensor components,
which cannot be determined solely from symmetry analysis. which cannot be determined solely from symmetry analysis.
#include "rep.typ" #include "table-rep.typ"
// We propose a method to estimate the magnitudes of the Raman tensors of these phonons, // We propose a method to estimate the magnitudes of the Raman tensors of these phonons,
// without first-principle calculations. // without first-principle calculations.

0
test-typst/section/perfect/non-polar/discont.typ → test-typst/section/perfect/non-polar/figure-discont.typ
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18
test-typst/section/perfect/non-polar/rep.typ
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@@ -1,18 +0,0 @@
#figure({
let m2(content) = table.cell(colspan: 2, content);
set text(size: 9pt);
table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
[*Representations in C#sub[6v]*], [A#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
[*Representations in C#sub[2v]*], [A#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
[*Vibration Direction*], [z], [x], [y], [x], [y],
[*Raman Tensor of #linebreak() Individual Phonons*],
[$mat(a,,;,a,;,,b)$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
[*Raman Intensity with Different #linebreak() Polarization Configurations*],
[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0],
m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
)},
caption: [
Raman-active representations of C#sub[6v] and C#sub[2v] point groups.
],
placement: none,
)<table-rep>

25
test-typst/section/perfect/non-polar/table-rep.typ Normal file
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@@ -0,0 +1,25 @@
#figure({
set text(size: 9pt);
set par(justify: false);
let m2(content) = table.cell(colspan: 2, content);
let A1 = [A#sub[1]];
let A2 = [A#sub[2]];
let B1 = [B#sub[1]];
let B2 = [B#sub[2]];
let E1 = [E#sub[1]];
let E2 = [E#sub[2]];
table(columns: 7, align: center + horizon,
[*Representations in C#sub[6v]*], A1, B1, m2(E1), m2(E2),
[*Representations in C#sub[2v]*], A1, B1, B2, B1, A2, A1,
[*Vibration Direction*], [z], [z], [x], [y], [x], [y],
[*Raman Tensor*],
[$mat(a,,;,a,;,,b)$], [$0$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
[*Raman scatter Intensity* #linebreak() (polarization of incident and scattered light)],
[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0], [0],
m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
)},
caption: [
Irreducible representations and raman tensors of phonons in 4H-SiC.
],
placement: none,
)<table-rep>

0
test-typst/section/perfect/polar/default.typ Normal file
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10
test-typst/section/perfect/bec.typ → test-typst/section/perfect/table-bec.typ
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@@ -1,12 +1,10 @@
#figure({ #figure({
set text(size: 9pt); set text(size: 9pt);
set par(justify: false);
table(columns: 4, align: center + horizon, table(columns: 4, align: center + horizon,
table.cell(colspan: 2)[], table.cell(colspan: 2)[*BEC* (unit: |e|)], table.cell(colspan: 2, rowspan: 2)[], table.cell(colspan: 2)[BEC (unit: |e|)], [x / y direction], [z direction],
table.cell(colspan: 2)[], [x / y direction], [z direction], table.cell(rowspan: 2)[Si atom], [A/C layer], [2.667], [2.626], [B layer], [2.674], [2.903],
table.cell(rowspan: 2)[Si atom], [A/C layer], [2.667], [2.626], table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730], [B layer], [-2.648], [-2.800],
[B layer], [2.674], [2.903],
table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730],
[B layer], [-2.648], [-2.800],
)}, )},
caption: [ caption: [
Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method. Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method.