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@@ -4,6 +4,7 @@
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#show: super-T-as-transpose
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#set par.line(numbering: "1")
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#set par(justify: true)
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// TODO: fix indent of first line
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#show figure.caption: it => {
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set text(10pt)
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@@ -48,6 +49,8 @@
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// })
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)
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#set heading(numbering: "1.")
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= Introduction
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// SiC 是很好的材料。
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@@ -111,19 +114,24 @@ experiment
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== Phonons in Perfect 4H-SiC
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(There are 21 phonons in total.
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We classified them into two categories: 18 negligible-polar phonons and 3 strong-polar phonons.)
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// 拉曼活性的声子模式对应于 Gamma 点附近的声子模式。
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// 根据这些声子模式的极性，我们将这些声子分成两类。
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Raman scattering peeks correspond to atom vibrations (phonons) located near the #sym.Gamma point in reciprocal space,
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and the exact location of these phonons is determined by the wavevectors of incident and scattered light.
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On each site of the Brillouin zone near the #sym.Gamma point,
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there are 21 phonon modes in 4H-SiC.
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We classified these phonons into two categories based on their polarities.
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18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
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The phonons involved in Raman scattering are located in reciprocal space
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at positions determined by the difference between the wavevectors of the incident and scattered light.
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At each such position, there are 21 phonon modes (excluding translational modes).
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We classify these 21 phonons into two categories based on their polarities.
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The 18 of 21 phonons are classified into negligible-polar phonons (i.e., phonons with zero or very weak polarity),
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for which the effect of polarity can be ignored in the Raman scattering process;
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and the other three phonons are strong-polar phonons,
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where the polarity gives rise to observable effects in the Raman spectra.
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(This classification make sense.)
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This classification is based on the fact that
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the four Si atoms in the primitive cell carry similar positive Born effective charges (BECs),
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the four Si atoms in the primitive cell of 4H-SiC carry similar positive Born effective charges (BECs),
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and the four C atoms carry similar negative BECs (see @table-bec).
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In the 18 negligible-polar phonons,
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the vibrations of two Si atoms are approximately opposite to those of the other two Si atoms,
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@@ -142,22 +150,21 @@ While in the three strong-polar phonons,
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table.cell(rowspan: 2)[C atom], [A/C layer], [-2.693], [-2.730],
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[B layer], [-2.648], [-2.800],
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),
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caption: [Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC.],
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caption: [
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Born effective charges of Si and C atoms in A/B/C/B layers of 4H-SiC, calculated using first principle method.
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],
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placement: none,
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)<table-bec>
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=== Phonons with Negligible Polarities
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// 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。
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// 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：
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// 尽管拉曼过程中起作用的声子并不是那些严格在 Gamma 点的，
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// 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%）。
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// 因此，它们的性质与 Gamma 点的声子模式区别不大。
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(We investigate phonons at Gamma instead of the exact location near Gamma.)
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Phonons at the #sym.Gamma point were used
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to approximate negligible-polar phonons that participating in Raman processes.
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This approximation is widely adopted and justified by the fact that,
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to approximate negligible-polar phonons that participating in Raman processes of any incident/scattered light.
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This approximation is widely adopted and justified by the fact that, // TODO: cite
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although the phonons participating in Raman processes are not these strictly located at the #sym.Gamma point,
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their dispersion near the #sym.Gamma point is continuous with vanishing derivatives,
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dispersion of negligible-polar phonons near the #sym.Gamma point is continuous with vanishing derivatives,
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and their wavevector is very small (about 0.01 nm#super[-1] in back-scattering configurations with 532 nm laser light,
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which corresponds to only 1% of the smallest reciprocal lattice vector of 4H-SiC),
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as shown by the orange dotted line in @figure-discont.
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@@ -182,43 +189,77 @@ Therefore, negligible-polar phonons involved in Raman processes
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placement: none,
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)<figure-discont>
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// 这18个声子对应于 $\mathrm{C_{6v}}$ 点群的 14 个表示：2A1 + 4B1 + 2E_1 + 4E2
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// 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
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// 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。
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The 18 negligible-polar phonons correspond to 14 irreducible representations of the C#sub[6v] point group:
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2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
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Phonons belonging to A#sub[1] and B#sub[1] representations vibration along z-axis and are non-degenerate,
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while phonons belonging to E#sub[1] and E#sub[2] representations vibrate in plane and are doubly degenerate.
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Phonons belonging to B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
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// TODO: 调整英语
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In contrast, phonons belonging to other representations are Raman-active,
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and the non-zero components of Raman tensor of each phonon
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could be determined by considering further in the C#sub[2v] point group @table-rep.
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These Raman-active phonons might be visible in Raman experiment under appropriate polarization configurations.
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However, the actual visibility of each phonon depends on the magnitudes of its Raman tensor components,
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which cannot be computed solely from symmetry analysis.
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(Representation of these 18 phonons, and the shape of their Raman tensors could be determined in advance.)
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Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
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In contrast, phonons of the other representations are Raman-active,
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and the non-zero components of their Raman tensors
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can be determined by further considering the C#sub[2v] point group (see @table-rep).
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These Raman-active phonons may appear in Raman spectra under appropriate polarization configurations.
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However, the actual visibility of each mode depends on the magnitude of its Raman tensor components,
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which cannot be determined solely from symmetry analysis.
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The 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group:
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2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2].
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Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate,
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while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate.
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Phonons of the B#sub[1] representation are Raman-inactive, as their Raman tensors vanish.
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In contrast, phonons of the other representations are Raman-active,
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and the non-zero components of their Raman tensor
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can be determined by further considering their representation in the C#sub[2v] point group (see @table-rep).
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These Raman-active phonons might be visible in Raman experiment under appropriate polarization configurations.
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However, whethear a mode is sufficiently strong to be experimentally visible
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depends on the magnitudes of its Raman tensor components,
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which cannot be determined solely from symmetry analysis.
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// TODO: 完善表格
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#figure({
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let m2(content) = table.cell(colspan: 2, content);
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table(columns: 7, align: center + horizon, inset: (x: 3pt, y: 5pt),
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[*Rep in C6v*], [A#sub[1]], [B#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
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[*Rep in C2v*], [A#sub[1]], [B#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
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[*Vib Direction*], [z], [z], [x], [y], [x], [y],
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[*Raman Tensor*],
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[$mat(a, , ; , a, ; , , b)$], [$0$], [$mat( , , a; , , ; a, , ;)$], [$mat( , , ; , , a; , a, ;)$],
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[$mat( , a, ; a, , ; , , ;)$], [$mat( a, , ; , -a, ; , , ;)$],
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[*Raman Intensity*],
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[$mat(a^2, , ; , a^2, ; , , b^2)$], [$0$], m2[$mat( , , a^2; , , a^2; a^2, a^2, ;)$],
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m2[$mat( a^2, a^2, ; a^2, a^2, ; , , ;)$]
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table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
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[*Representations in C6v*], [A#sub[1]], m2[E#sub[1]], m2[E#sub[2]],
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[*Representations in C2v*], [A#sub[1]], [B#sub[2]], [B#sub[1]], [A#sub[2]], [A#sub[1]],
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[*Vibration Direction*], [z], [x], [y], [x], [y],
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[*Raman Tensor of #linebreak() Individual Phonons*],
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[$mat(a,,;,a,;,,b)$], [$mat(,,a;,,;a,,;)$], [$mat(,,;,,a;,a,;)$], [$mat(,a,;a,,;,,;)$], [$mat(a,,;,-a,;,,;)$],
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[*Raman Intensity with Different #linebreak() Polarization Configurations*],
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[xx/yy: $a^2$ #linebreak() zz: $b^2$ #linebreak() others: 0],
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m2[xz/yz: $a^2$ #linebreak() others: 0], m2[xx/xy/yy: $a^2$ #linebreak() others: 0],
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)},
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caption: [Rep],
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caption: [
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Raman-active representations of C#sub[6v] and C#sub[2v] point groups.
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],
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placement: none,
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)<table-rep>
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Here we propose a method to estimate the magnitudes of the Raman tensors of these phonons.
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(We propose a method to estimate the magnitudes of the Raman tensors of these phonons.
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Here we write out its main steps, details are in appendix.)
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// TODO: 写出来这个方法，并验证。
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// TODO: maybe it is better to assign Raman tensor to each bond, instead of atom
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We propose a method to estimate the magnitudes of the Raman tensors by symmetry analysis (see appendix for details).
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The center principle is to assign the Raman tensor (i.e., change of polarizability caused by atomic displacement)
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to each atom in the unit cell.
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This including the following steps:
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- Write out the change of polarizability caused by displacement of Si atom in A and C layer,
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Where unknown non-zero components are denoted by $a_1$, $a_2$, $a_5$, $a_6$.
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For example, when we move the Si atom in A layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2,a_1;a_2,,;a_1,,)d$.
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This could be done by conclusion above.
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- The Si atom in B layer have similar local environment as the A and C layer, with only a little difference.
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We denote these difference by $epsilon_1$, $epsilon_2$, $epsilon_5$, $epsilon_6$,
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and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
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For example, when we move the Si atom in B layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,)d$.
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- The local environment of C atom in A layer is similar to the Si atom in A layer with charge reversed and
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the system reversed along xy plane.
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We denote these difference by $eta_1$, $eta_2$, $eta_5$, $eta_6$,
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and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
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For example, when we move the C atom in A layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,)d$.
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- Similar to the case in Si atoms, we derive the change of polarizability
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caused by moving C atom in B layer slightly towards the x+ direction in $d$ distance,
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which should be $mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,)d$.
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Lets assign Raman tensor onto each atom.
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That is, Raman tensor is derivative of the polarizability with respect to the atomic displacement:
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@@ -379,6 +420,11 @@ No. It turns out to be similar to the C atom in C layer.
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We summarize these stuff into @table-singleatom.
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Furthermore, we list predicted modes and their Raman tensors, in @table-predmode.
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- $a$: Raman tensor of Si atom in A layer, large value.
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- $epsilon$: Difference of Raman tensors of Si atom in A and B1 layer, small value.
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- $eta$: Difference of Raman tensors of C and Si atom in A layer, small value.
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- $zeta$: Difference of Raman tensors of C atoms in A and B layer, small value.
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Frequency could be estimated by, how many atoms are moving towards its neighbor.
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#page(flipped: true)[#figure({
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@@ -444,39 +490,6 @@ Frequency could be estimated by, how many atoms are moving towards its neighbor.
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placement: none,
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)<table-predmode>]
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/*
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这里应该有办法来估计。下面是我总结的规律：
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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：
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* 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）
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* 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
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* 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
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* 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
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我们或许可以通过这个路径来探索：
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* 首先，根据 C3v 点群的表示，写出每个键的拉曼张量。这包括：
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* 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。
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* 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。
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* 对于 A 到 B1 的横着的键，它是 C3v 。写下这个结果。
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* 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。
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* 对于其它键，根据对称性由上面的结果直接写出。
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* 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。
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*/
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//
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// 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。
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// 在第一个空间中，它的形式为：
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//
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// $
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// mat(a, , ; , -a, ; , , ;)
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// mat(, a, ; a, , ; , , ;)
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// $
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//
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// 在第二个空间中，它的形式为：
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//
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// $
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// mat(, , a; , , ; a, , ;)
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// mat(, , ; , , a; , a, ;)
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// $
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// 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。
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// 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。
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// 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。
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