test-typst/main.typ

@@ -4,6 +4,7 @@
 #show: super-T-as-transpose
 
 #set par.line(numbering: "1")
+#set par(justify: true)
 // TODO: fix indent of first line
 #show figure.caption: it => {
   set text(10pt)
@@ -48,6 +49,8 @@
   // })
 )
 
+#set heading(numbering: "1.")
+
 = Introduction
 
 // SiC 是很好的材料。
@@ -111,19 +114,24 @@ experiment
 
 == Phonons in Perfect 4H-SiC
 
+(There are 21 phonons in total.
+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。
-Raman scattering peeks correspond to atom vibrations (phonons) located near the #sym.Gamma point in reciprocal space,
+The phonons involved in Raman scattering are located in reciprocal space
-  and the exact location of these phonons is determined by the wavevectors of incident and scattered light.
+  at positions determined by the difference between the wavevectors of the incident and scattered light.
-On each site of the Brillouin zone near the #sym.Gamma point,
+At each such position, there are 21 phonon modes (excluding translational modes).
-  there are 21 phonon modes in 4H-SiC.
+We classify these 21 phonons into two categories based on their polarities.
-We classified these 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),
-18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
     for which the effect of polarity can be ignored in the Raman scattering process;
   and the other three phonons are strong-polar phonons,
     where the polarity gives rise to observable effects in the Raman spectra.
+
+(This classification make sense.)
+
 This classification is based on the fact that
-  the four Si atoms in the primitive cell 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).
 In the 18 negligible-polar phonons,
   the vibrations of two Si atoms are approximately opposite to those of the other two Si atoms,
@@ -142,22 +150,21 @@ While in the three strong-polar phonons,
     table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730],
     [B layer], [-2.648], [-2.800],
   ),
-  caption: [Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC.],
+  caption: [
+    Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method.
+  ],
   placement: none,
 )<table-bec>
 
 === Phonons with Negligible Polarities
 
-// 我们使用 Gamma 点的声子模式来近似拉曼过程中的非极性声子。
+(We investigate phonons at Gamma instead of the exact location near Gamma.)
-// 这个近似被广泛使用，并且由于这个原因而被认为是可行的：
+
-// 尽管拉曼过程中起作用的声子并不是那些严格在 Gamma 点的，
-//  但这些声子模式的散射谱在 Gamma 附近连续且导数为零，且波矢很小（在本文中大约 0.01 A，只有c轴的大约2%）。
-// 因此，它们的性质与 Gamma 点的声子模式区别不大。
 Phonons at the #sym.Gamma point were used
-  to approximate negligible-polar phonons that participating in Raman processes.
+  to approximate negligible-polar phonons that participating in Raman processes of any incident/scattered light.
-This approximation is widely adopted and justified by the fact that,
+This approximation is widely adopted and justified by the fact that, // TODO: cite
   although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
-  their dispersion near the #sym.Gamma point is continuous with vanishing derivatives,
+  dispersion of negligible-polar phonons near the #sym.Gamma point is continuous with vanishing derivatives,
   and their wavevector is very small (about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
     which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC),
   as shown by the orange dotted line in @figure-discont.
@@ -182,43 +189,77 @@ Therefore, negligible-polar phonons involved in Raman processes
   placement: none,
 )<figure-discont>
 
-// 这18个声子对应于 $\mathrm{C_{6v}}$ 点群的 14 个表示：2A1 + 4B1 + 2E_1 + 4E2
+(Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
-// 其中，B1 表示没有拉曼活性，它的拉曼张量为零；其它表示的拉曼张量不为零
+
-// 但张量的大小是否足够大到可以在实验上看到，则还需要第一性原理计算，不能直接通过表示来判断。
+Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
-The 18 negligible-polar phonons correspond to 14 irreducible representations of the C#sub[6v] point group:
+In contrast, phonons of the other representations are Raman-active,
-  2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
+and the non-zero components of their Raman tensors
-Phonons belonging to A#sub[1] and B#sub[1] representations vibration along z-axis and are non-degenerate,
+can be determined by further considering the C#sub[2v] point group (see @table-rep).
-  while phonons belonging to E#sub[1] and E#sub[2] representations vibrate in plane and are doubly degenerate.
+These Raman-active phonons may appear in Raman spectra under appropriate polarization configurations.
-Phonons belonging to B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
+However, the actual visibility of each mode depends on the magnitude of its Raman tensor components,
-// TODO: 调整英语
+which cannot be determined solely from symmetry analysis.
-In contrast, phonons belonging to other representations are Raman-active,
+
-  and the non-zero components of Raman tensor of each phonon
+The 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group:
-  could be determined by considering further in the C#sub[2v] point group @table-rep.
+  2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
-These Raman-active phonons might be visible in Raman experiment under appropriate polarization configurations.
+Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate,
-However, the actual visibility of each phonon depends on the magnitudes of its Raman tensor components,
+  while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate.
-  which cannot be computed solely from symmetry analysis.
+Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
+In contrast, phonons of the other representations are Raman-active,
+  and the non-zero components of their Raman tensor
+  can be determined by further considering their representation in the C#sub[2v] point group (see @table-rep).
+These Raman-active phonons might be visible in Raman experiment under appropriate polarization configurations.
+However, whethear a mode is sufficiently strong to be experimentally visible
+    depends on the magnitudes of its Raman tensor components,
+  which cannot be determined solely from symmetry analysis.
 
-// TODO: 完善表格
 #figure({
   let m2(content) = table.cell(colspan: 2, content);
-  table(columns: 7, align: center + horizon, inset: (x: 3pt, y: 5pt),
+  table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
-    [*Rep in C6v*], [A#sub[1]], [B#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
+    [*Representations in C6v*], [A#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
-    [*Rep in C2v*], [A#sub[1]], [B#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
+    [*Representations in C2v*], [A#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
-    [*Vib Direction*], [z], [z], [x], [y], [x], [y],
+    [*Vibration Direction*], [z], [x], [y], [x], [y],
-    [*Raman Tensor*],
+    [*Raman Tensor of #linebreak() Individual Phonons*],
-      [$mat(a, , ; , a, ; , , b)$], [$0$], [$mat( , , a; , , ; a, , ;)$], [$mat( , , ; , , a; , a, ;)$],
+      [$mat(a,,;,a,;,,b)$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
-      [$mat( , a, ; a, , ; , , ;)$], [$mat( a, , ; , -a, ; , , ;)$],
+    [*Raman Intensity with Different #linebreak() Polarization Configurations*],
-    [*Raman Intensity*],
+      [xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0],
-      [$mat(a^2, , ; , a^2, ; , , b^2)$], [$0$], m2[$mat( , , a^2; , , a^2; a^2, a^2, ;)$],
+      m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
-      m2[$mat( a^2, a^2, ; a^2, a^2, ; , , ;)$]
   )},
-  caption: [Rep],
+  caption: [
+    Raman-active representations of C#sub[6v] and C#sub[2v] point groups.
+  ],
   placement: none,
 )<table-rep>
 
-Here 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.
+Here we write out its main steps, details are in appendix.)
 
-// TODO: 写出来这个方法，并验证。
+// TODO: maybe it is better to assign Raman tensor to each bond, instead of atom
+
+We propose a method to estimate the magnitudes of the Raman tensors by symmetry analysis (see appendix for details).
+
+
+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:
@@ -379,6 +420,11 @@ No. It turns out to be similar to the C atom in C layer.
 We summarize these stuff into @table-singleatom.
 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.
+
 Frequency could be estimated by, how many atoms are moving towards its neighbor.
 
 #page(flipped: true)[#figure({
@@ -444,39 +490,6 @@ Frequency could be estimated by, how many atoms are moving towards its neighbor.
   placement: none,
 )<table-predmode>]
 
-
-/*  
-这里应该有办法来估计。下面是我总结的规律：
-按照我们规定的 ABCB 层序，并将拉曼张量的大小归结为键长的变化的话：
-* 对于 E2 表示（AC层运动方向必须相反，B1/B2层运动方向必须相反，因此只讨论A和B1层）
-  * A 层内部的那个竖的键，同向运动会导致比较大的拉曼张量
-  * B1 层内部的那个竖的键，反向运动会导致比较大的拉曼张量
-  * A 层和 B1 层之间的那个横的键，反向运动会导致比较大的拉曼张量
-我们或许可以通过这个路径来探索：
-* 首先，根据 C3v 点群的表示，写出每个键的拉曼张量。这包括：
-  * 对于 A 内竖着的键，考虑连着的两个原子和第一近邻原子，对称性为 C3v。写出此时的拉曼张量。
-  * 对于 B1 内竖着的键，它也是 C3v，它此时的拉曼张量是 h 下稍微变动的结果。写下这个结果。
-  * 对于 A 到 B1 的横着的键，它是 C3v 。写下这个结果。
-  * 对于 B1 到 C 的横着的键，它是 C3v 。写下这个结果为之前的结果的微微变动。
-  * 对于其它键，根据对称性由上面的结果直接写出。
-* 写出各个模式的拉曼张量（上面的线性组合）。即可以直接看到结果。
-*/
-// 
-// 考虑 A 层的竖着的键。面内的振动模式对应的拉曼张量落在两个空间中。
-// 在第一个空间中，它的形式为：
-// 
-// $
-//   mat(a, , ; , -a, ; , , ;)
-//   mat(, a, ; a, , ; , , ;)
-// $
-// 
-// 在第二个空间中，它的形式为：
-// 
-// $
-//   mat(, , a; , , ; a, , ;)
-//   mat(, , ; , , a; , a, ;)
-// $
-
 // 我们计算了拉曼活性声子的频率及拉曼张量，并与实验对比，如表如图所示。
 // 其中有几个声子的拉曼活性较弱，有几个比较强。强的都可以在实验上看到；但弱的能否看到则取决于它是否恰好位于强模式的附近。
 // 其中，xxx 和xxx 位于强模式的附近，它们在实验上无法看到；xxx 只在 z 方向入射/散射时可以看到；xxx 则在任意方向都能看到。