From e01e9999a5fcfd098acf9bae615b1f7634c6bb83 Mon Sep 17 00:00:00 2001 From: chn Date: Mon, 26 May 2025 10:46:53 +0800 Subject: [PATCH] =?UTF-8?q?=E5=88=86=E5=89=B2=E6=96=87=E4=BB=B6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- test-typst/main.typ | 356 +----------------- test-typst/section/introduction.typ | 48 +++ test-typst/section/perfect/bec.typ | 15 + test-typst/section/perfect/default.typ | 28 ++ .../section/perfect/non-polar/default.typ | 94 +++++ .../section/perfect/non-polar/discont.typ | 15 + .../section/perfect/non-polar/nopol.typ | 73 ++++ .../section/perfect/non-polar/predmode.typ | 47 +++ .../section/perfect/non-polar/raman.typ | 11 + test-typst/section/perfect/non-polar/rep.typ | 18 + 10 files changed, 352 insertions(+), 353 deletions(-) create mode 100644 test-typst/section/introduction.typ create mode 100644 test-typst/section/perfect/bec.typ create mode 100644 test-typst/section/perfect/default.typ create mode 100644 test-typst/section/perfect/non-polar/default.typ create mode 100644 test-typst/section/perfect/non-polar/discont.typ create mode 100644 test-typst/section/perfect/non-polar/nopol.typ create mode 100644 test-typst/section/perfect/non-polar/predmode.typ create mode 100644 test-typst/section/perfect/non-polar/raman.typ create mode 100644 test-typst/section/perfect/non-polar/rep.typ diff --git a/test-typst/main.typ b/test-typst/main.typ index acb62f3..d133e9e 100644 --- a/test-typst/main.typ +++ b/test-typst/main.typ @@ -55,54 +55,7 @@ = Introduction -// SiC 是很好的材料。 -// 其中，4H-SiC 是SiC的一种多型，它的性质更好，近年来随着外延工艺的成熟而获得了更多的关注。 -SiC is a promising wide-bandgap semiconductor material - with high critical electric field strength and high thermal conductivity. - It has been widely used in power electronic devices and has long attracted a lot of research - @casady_status_1996 @okumura_present_2006. -The 4H-SiC has a wider bandgap, higher critical electric field strength, - higher thermal conductivity, and higher electron mobility along the c-axis than other polytypes. -Currently, the 4H-SiC has gradually received more attention than other polytypes, - thanks to the development of epitaxy technology and the increasing application in the new energy industry - @tsuchida_recent_2018 @harada_suppression_2022 @sun_selection_2022. // TODO: 多引用一些近年来的文献，有很多 - -// 声子（量子化的原子振动）在理解晶体的原子结构以及热电性质方面起着重要作用。 -// 声子可以通过多种实验技术来探测，包括 EELS、IR 吸收谱等。 -// 拉曼光谱是最常用的方法，它提供了一种无损、非接触、快速和局部的声子测量方法，已被广泛用于确定晶体的原子结构（包括区分 SiC 的多型）。 -Phonons (quantized atomic vibrations) play a fundamental role - in understanding the atomic structure - as well as the thermal and electrical properties - of crystals (including 4H-SiC). -They could be probed by various experimental techniques, - such as electron energy loss spectroscopy and infrared absorption spectroscopy. -Among these techniques, - Raman spectroscopy is the most commonly used method, - as it provides non-destructive, non-contact, rapid and spatially localized measurement of phonons - that near the #sym.Gamma point in reciprocal space. -Studies in Raman scattering of 4H-SiC have been conducted since as early as 1983 - and have been widely employed to identification of different SiC polytypes. -// TODO: 增加引用文献 - -// 近年来，更多信息被从拉曼光谱中挖掘出来。 -// LOPC 已经被用于快速估计 n 型 SiC 的掺杂浓度。 -// 层错的拉曼光谱也已经被研究，可以被用于检测特定结构层错的存在和位置。 -// 掺杂对拉曼光谱的潜在影响也已经被研究。 -// 然而，拉曼光谱上仍有一些不知来源的峰；同时，一些也缺少一些理论上预测应该存在的峰。 -// 此外，预测掺杂导致的新峰也没有说明原因。 -Increasingly rich information has been extracted from Raman spectra of 4H-SiC. -Longitudinal optical phonon–plasmon coupling (LOPC) peek - has been utilized to rapidly estimate the doping concentration in n-type SiC. -Peeks associated with some stacking faults have also been investigated - and used to detect the presence and location of specific structural faults. -Moreover, the potential effects of doping on Raman spectra have been explored. -However, some unidentified peaks still appear in the Raman spectra, - while certain phonon modes predicted by theory remain unobserved. -In addition, the origins of newly emerged peaks induced by doping are often unclear or unexplained. -// TODO: 多举例，增加引用文献 - -In this paper, we do some things. Especially we do something for the first time. -// TODO: 完善 +#include "section/introduction.typ" = Method @@ -137,314 +90,11 @@ experiment == Phonons in Perfect 4H-SiC -#par()[#text()[#h(0.0em)]] - -(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。 -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. -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. -The 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 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, - and the same holds for the C atoms, - leading to cancellations of macroscopic polarity. -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, - resulting in a strong dipole moment. - -#figure({ - set text(size: 9pt); - table(columns: 4, align: center + horizon, - table.cell(colspan: 2)[], table.cell(colspan: 2)[*BEC* (unit: |e|)], - 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)[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, calculated using first principle method. - ], - placement: none, -) +#include "section/perfect/default.typ" === Phonons with Negligible Polarities -#par()[#text()[#h(0.0em)]] - -(We investigate phonons at Gamma instead of the exact location near Gamma.) - -Phonons at the #sym.Gamma point were used - 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, // TODO: cite - 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 - (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, - see orange dotted line in @figure-discont), - and their dispersion at #sym.Gamma point is continuous with vanishing derivatives. -Therefore, negligible-polar phonons involved in Raman processes - 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. - -#figure( - image("/画图/声子不连续/embed.svg"), - caption: [ - (a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path. - Gray lines represent negligible-polar phonon modes, - while colored lines indicate strong-polar phonon modes. - The green, red and blue lines indicate the mode along the z-direction, y-direction and x-direction, respectively. - Along A-#sym.Gamma path, strong-polar modes along x- and y-directions are degenerated, - showing as a single purple line. - (b) Magnified view of the boxed region in (a). - The orange dashed lines mark the phonon wavevectors involved in Raman scattering - with incident light along the z- and y-directions. - ], - placement: none, -) - -#par()[#text()[#h(0.0em)]] - -(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, - and the 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group: - 2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]. -Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate, - while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate. -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 are potentially be visible in Raman experiment under appropriate polarization configurations. -However, whether 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. - -#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, -) - -#par()[#text()[#h(0.0em)]] - -(We propose a method to estimate the magnitudes of the Raman tensors of these phonons, - without first-principle calculations. -Here we only write out results, details are in appendix.) - -// 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 of these phonons by symmetry analysis. -The method only takes the vibration directions of each atom in each phonon mode, - leaving the amplitudes unconsidered (see appendix for details), - and the result was summarized in @table-predmode. -In the Raman tensors in @table-predmode, - $a_i$ corresponding to the change of polarizability caused by movement of the Si atoms in A and C layers, - $epsilon_i$, $eta_i$ and $eta_i$ corresponding to the difference between different bilayers and different atoms. -Due to the similarity of environment in different bilayers and around different atoms, - the absolute values of $epsilon_i$, $eta_i$ and $zeta_i$ are expected to be much smaller than that of $a_i$, - thus the Raman tensors containing $a_i$ are expected to be much larger than those not containing $a_i$. - -// 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: $-A_#text[C]zeta_6$], - [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*], - [-1.68 #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, -)] - -The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods, - and the results are compared with experiment and theory (@table-nopol). -Calculated frequencies of these phonons are consistent with the experimental results - with a low-estimated error of about 2% to 5%, which might be due to the PBE functional used in the calculation (cite). -The Raman tensors of these phonons are also consistent with the experimental and theoretical results, - where E#sub[2] mode experimentally at 776 is the most intense phonon mode, - followed by four modes with lesser intensities - (E#sub[2] modes at 195.5 and 203.3, E#sub[1] mode at 269.7, A#sub[1] mode at 609.5). -The Raman scatter of the E#sub[1] mode calculately at 746.91 and E#sub[2] mode calculately at 756.25 - are much weaker than the E#sub[2] mode calculated at 756.25 but located near it, according to our calculation, - thus it could not be distinguished from E#sub[2] mode calculated at 756.25, - which explains why they are not observed in experiments. -Moveever, the A#sub[1] mode calculated at 812.87 - have a very weak Raman intensity in the basal plane (xx and yy, only 0.01) - but an observable intensity in the zz configuration (1.78). -Thus, this mode could not be observed in most Raman experiments (cite), - but could be observable when incident light propagate not along the z-direction (our experiment), - or the incident light wavelength is near the resonance condition (cite). - -Besides, there are other peeks in the experiment. -The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later. -Besides, there are small peeks at xxx, - which could not be explained in perfect 4H-SiC and will be discussed in the next section. - -// TODO: 将一部分 phonons 改为 phonon modes -// 在论文中我们这样来称呼：phonon 对应某一个特征向量，而 modes 对应于一个子空间。 -// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode - -#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 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]]; - 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)], - // TODO: 整理表格，使用 m2 m3 来代替 - [*Number of Phonon*], - // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1 - [1], m2[2], [3], m2[4], [5], [6], [7], [8], m3[9], [10], [11], [12], m2[13], [14], m2[15], m3[16], [17], [18], - [*Vibration Direction*], - // E2 E2 E1 2B1 A1 - [x], m2[y], [x], m(2)[y], [x], [y], m(2)[z], m(3)[z], - // E1 E2 E2 A1 2B1 - [x], [y], [x], m(2)[y], [x], m(2)[y], m(3)[z], m(2)[z], - [*Representation #linebreak() in Group C#sub[6v]*], - m(3, E2), m(3, E2), m(2, E1), B1, B1, m(3, A1), m(2, E1), m(3, E2), m(3, E2), m(3, A1), B1, B1, - [*Raman-active or Not*], - m(8)[Raman-active], m(2)[Raman-inactive], m(14)[Raman-active], m(2)[Raman-inactive], - // [*Representation in Group C#sub[2v]*], - // // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1 - // A2, m(2, A1), A2, m(2, A1), B2, B1, B1, B1, m(3, A1), B2, B1, A2, m(2, A1), A2, m(2, A1), m(3, A1), B1, B1, - [*Scattering in Polarization #linebreak() (non-zero Raman #linebreak() tenser components)*], - // E2 E2 E1 2B1 A1 - [xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], m(2)[-], [xx], [yy], [zz], - // E1 E2 E2 A1 2B1 - [xz], [yz], [xy], [xx], [yy], [xy], [xx], [yy], [xx], [yy], [zz], m(2)[-], - [*Raman Intensity (a.u.)*], - // E2 E2 E1 2B1 A1 - m(3)[0.17], m(3)[1.13], m(2)[2.43], m(2)[0], m(2)[2.83], [1.79], - // E1 E2 E2 A1 2B1 - m(2)[0.09], m(3)[88.54], m(3)[0.50], m(2)[0.01], [1.78], m(2)[0], - [*Visible in Common #linebreak() Raman Experiment or Not*], - // E2 E2 E1 2B1 A1 - m(8)[Visible], m(2)[-], m(3)[Visible], - // E1 E2 E2 A1 2B1 - m(2)[Invisible], m(3)[Visible], m(5)[Invisible], [Visible], m(2)[-], - [*Wavenumber #linebreak() (Simulation) (cm#super[-1])*], - // E2 E2 E1 2B1 A1 - m(3)[190.51], m(3)[197.84], m(2)[257.35], [389.96], [397.49], m(3)[591.90], - // E1 E2 E2 A1 2B1 - m(2)[746.91], m(3)[756.25], m(3)[764.33], m(3)[812.87], [885.68], [894.13], - [*Wavenumber #linebreak() (Experiment) (cm#super[-1])*], - // E2 E2 E1 2B1 A1 - m(3)[195.5], m(3)[203.3], m(2)[269.7], m(2)[-], m(3)[609.5], - // E1 E2 E2 A1 2B1 - m(2)[-], m(3)[776], m(5)[-], [839], m(2)[-], - [*FWHM #linebreak() (Simulation) (cm#super[-1])*], - // E2 E2 E1 2B1 A1 - 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)[-], - // 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)[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)[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], - )}, - caption: [Negaligible-polarized Phonons at $Gamma$ Point], -)] - -#figure( - image("/画图/拉曼整体图/main.svg"), - caption: [ - (a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path. - Gray lines represent negligible-polar phonon modes, - while colored lines indicate strong-polar phonon modes. - (b) Magnified view of the boxed region in (a). - The orange dashed lines mark the phonon wavevectors involved in Raman scattering - with incident light along the z- and y-directions. - ] -) - -// TODO: 画一个模拟的图，与实验图对比。 - -// 实验与计算基本相符。对于声子频率，计算总是低估大约 3%。 -// 此外，一些较强的模式在预测无法看到的偏振中也可以看到。例如，一些在 xy 偏振中不应该看到的模式可以被看到了。 -// 这个现象可以由 4度的斜切所解释：我们将材料略微踮起一些角度，就可以使得该模式减小。 -// 这个现象也可以由材料或偏振片的微小角度来解释。 -// 例如，我们将偏振方向转动 5 度，就可以得到这个模拟结果。 -// 此外，由于使用的材料是沿着 c 轴切片的，所以我们在测量 y 入射时不得不将片子以略小于 90 度（约 75 度）的角度放置。这也导致实验与计算的偏差。 -// TODO: 翻译成英文 +#include "section/perfect/non-polar/default.typ" === Strong-polar Phonons diff --git a/test-typst/section/introduction.typ b/test-typst/section/introduction.typ new file mode 100644 index 0000000..72d46a4 --- /dev/null +++ b/test-typst/section/introduction.typ @@ -0,0 +1,48 @@ +// SiC 是很好的材料。 +// 其中，4H-SiC 是SiC的一种多型，它的性质更好，近年来随着外延工艺的成熟而获得了更多的关注。 +SiC is a promising wide-bandgap semiconductor material + with high critical electric field strength and high thermal conductivity. + It has been widely used in power electronic devices and has long attracted a lot of research + @casady_status_1996 @okumura_present_2006. +The 4H-SiC has a wider bandgap, higher critical electric field strength, + higher thermal conductivity, and higher electron mobility along the c-axis than other polytypes. +Currently, the 4H-SiC has gradually received more attention than other polytypes, + thanks to the development of epitaxy technology and the increasing application in the new energy industry + @tsuchida_recent_2018 @harada_suppression_2022 @sun_selection_2022. // TODO: 多引用一些近年来的文献，有很多 + +// 声子（量子化的原子振动）在理解晶体的原子结构以及热电性质方面起着重要作用。 +// 声子可以通过多种实验技术来探测，包括 EELS、IR 吸收谱等。 +// 拉曼光谱是最常用的方法，它提供了一种无损、非接触、快速和局部的声子测量方法，已被广泛用于确定晶体的原子结构（包括区分 SiC 的多型）。 +Phonons (quantized atomic vibrations) play a fundamental role + in understanding the atomic structure + as well as the thermal and electrical properties + of crystals (including 4H-SiC). +They could be probed by various experimental techniques, + such as electron energy loss spectroscopy and infrared absorption spectroscopy. +Among these techniques, + Raman spectroscopy is the most commonly used method, + as it provides non-destructive, non-contact, rapid and spatially localized measurement of phonons + that near the #sym.Gamma point in reciprocal space. +Studies in Raman scattering of 4H-SiC have been conducted since as early as 1983 + and have been widely employed to identification of different SiC polytypes. +// TODO: 增加引用文献 + +// 近年来，更多信息被从拉曼光谱中挖掘出来。 +// LOPC 已经被用于快速估计 n 型 SiC 的掺杂浓度。 +// 层错的拉曼光谱也已经被研究，可以被用于检测特定结构层错的存在和位置。 +// 掺杂对拉曼光谱的潜在影响也已经被研究。 +// 然而，拉曼光谱上仍有一些不知来源的峰；同时，一些也缺少一些理论上预测应该存在的峰。 +// 此外，预测掺杂导致的新峰也没有说明原因。 +Increasingly rich information has been extracted from Raman spectra of 4H-SiC. +Longitudinal optical phonon–plasmon coupling (LOPC) peek + has been utilized to rapidly estimate the doping concentration in n-type SiC. +Peeks associated with some stacking faults have also been investigated + and used to detect the presence and location of specific structural faults. +Moreover, the potential effects of doping on Raman spectra have been explored. +However, some unidentified peaks still appear in the Raman spectra, + while certain phonon modes predicted by theory remain unobserved. +In addition, the origins of newly emerged peaks induced by doping are often unclear or unexplained. +// TODO: 多举例，增加引用文献 + +In this paper, we do some things. Especially we do something for the first time. +// TODO: 完善 diff --git a/test-typst/section/perfect/bec.typ b/test-typst/section/perfect/bec.typ new file mode 100644 index 0000000..2186132 --- /dev/null +++ b/test-typst/section/perfect/bec.typ @@ -0,0 +1,15 @@ +#figure({ + set text(size: 9pt); + table(columns: 4, align: center + horizon, + table.cell(colspan: 2)[], table.cell(colspan: 2)[*BEC* (unit: |e|)], + 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)[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, calculated using first principle method. + ], + placement: none, +) diff --git a/test-typst/section/perfect/default.typ b/test-typst/section/perfect/default.typ new file mode 100644 index 0000000..23762c8 --- /dev/null +++ b/test-typst/section/perfect/default.typ @@ -0,0 +1,28 @@ +(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。 +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. +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. +The 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 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, + and the same holds for the C atoms, + leading to cancellations of macroscopic polarity. +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, + resulting in a strong dipole moment. + +#include "bec.typ" diff --git a/test-typst/section/perfect/non-polar/default.typ b/test-typst/section/perfect/non-polar/default.typ new file mode 100644 index 0000000..c9b80a4 --- /dev/null +++ b/test-typst/section/perfect/non-polar/default.typ @@ -0,0 +1,94 @@ +// We investigate phonons at Gamma instead of the exact location near Gamma. +Phonons at the #sym.Gamma point were used + 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, // TODO: cite + 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 + (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, + see orange dotted line in @figure-discont), + and their dispersion at #sym.Gamma point is continuous with vanishing derivatives. +Therefore, negligible-polar phonons involved in Raman processes + 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" + +// 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, + and the 18 negligible-polar phonons correspond to 12 irreducible representations of the C#sub[6v] point group: + 2A#sub[1] + 4B#sub[1] + 2E#sub[1] + 4E#sub[2]. +Phonons belonging to the A#sub[1] and B#sub[1] representations vibrate along the z-axis and are non-degenerate, + while those belonging to the E#sub[1] and E#sub[2] representations vibrate in-plane and are doubly degenerate. +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 are potentially be visible in Raman experiment under appropriate polarization configurations. +However, whether 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. + +#include "rep.typ" + +// We propose a method to estimate the magnitudes of the Raman tensors of these phonons, +// without first-principle calculations. +// Here we only write out results, details are in appendix. + +// 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 of these phonons by symmetry analysis. +The method only takes the vibration directions of each atom in each phonon mode, + leaving the amplitudes unconsidered (see appendix for details), + and the result was summarized in @table-predmode. +In the Raman tensors in @table-predmode, + $a_i$ corresponding to the change of polarizability caused by movement of the Si atoms in A and C layers, + $epsilon_i$, $eta_i$ and $eta_i$ corresponding to the difference between different bilayers and different atoms. +Due to the similarity of environment in different bilayers and around different atoms, + the absolute values of $epsilon_i$, $eta_i$ and $zeta_i$ are expected to be much smaller than that of $a_i$, + thus the Raman tensors containing $a_i$ are expected to be much larger than those not containing $a_i$. + +#include "predmode.typ" + +The Raman tensors and frequencies of the negligible-polar phonons were calculated using first-principles methods, + and the results are compared with experiment and theory (@table-nopol). +Calculated frequencies of these phonons are consistent with the experimental results + with a low-estimated error of about 2% to 5%, which might be due to the PBE functional used in the calculation (cite). +The Raman tensors of these phonons are also consistent with the experimental and theoretical results, + where E#sub[2] mode experimentally at 776 is the most intense phonon mode, + followed by four modes with lesser intensities + (E#sub[2] modes at 195.5 and 203.3, E#sub[1] mode at 269.7, A#sub[1] mode at 609.5). +The Raman scatter of the E#sub[1] mode calculately at 746.91 and E#sub[2] mode calculately at 756.25 + are much weaker than the E#sub[2] mode calculated at 756.25 but located near it, according to our calculation, + thus it could not be distinguished from E#sub[2] mode calculated at 756.25, + which explains why they are not observed in experiments. +Moreover, the A#sub[1] mode calculated at 812.87 + have a very weak Raman intensity in the basal plane (xx and yy, only 0.01) + but an observable intensity in the zz configuration (1.78). +Thus, this mode could not be observed in most Raman experiments (cite), + but could be observable when incident light propagate not along the z-direction (our experiment), + or the incident light wavelength is near the resonance condition (cite). + +Besides, there are other peeks in the experiment. +The peek at 796 and 980 are caused by strong-polar phonons which will be discussed later. +Besides, there are small peeks at xxx, + which could not be explained in perfect 4H-SiC and will be discussed in the next section. + +// TODO: 将一部分 phonons 改为 phonon modes +// 在论文中我们这样来称呼：phonon 对应某一个特征向量，而 modes 对应于一个子空间。 +// 也就是说，简并的里面有两个或者无数个 phonon，但只有一个 mode + +#include "nopol.typ" + +#include "raman.typ" + +// TODO: 画一个模拟的图，与实验图对比。 + +// 实验与计算基本相符。对于声子频率，计算总是低估大约 3%。 +// 此外，一些较强的模式在预测无法看到的偏振中也可以看到。例如，一些在 xy 偏振中不应该看到的模式可以被看到了。 +// 这个现象可以由 4度的斜切所解释：我们将材料略微踮起一些角度，就可以使得该模式减小。 +// 这个现象也可以由材料或偏振片的微小角度来解释。 +// 例如，我们将偏振方向转动 5 度，就可以得到这个模拟结果。 +// 此外，由于使用的材料是沿着 c 轴切片的，所以我们在测量 y 入射时不得不将片子以略小于 90 度（约 75 度）的角度放置。这也导致实验与计算的偏差。 +// TODO: 翻译成英文 \ No newline at end of file diff --git a/test-typst/section/perfect/non-polar/discont.typ b/test-typst/section/perfect/non-polar/discont.typ new file mode 100644 index 0000000..3c40388 --- /dev/null +++ b/test-typst/section/perfect/non-polar/discont.typ @@ -0,0 +1,15 @@ +#figure( + image("/画图/声子不连续/embed.svg"), + caption: [ + (a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path. + Gray lines represent negligible-polar phonon modes, + while colored lines indicate strong-polar phonon modes. + The green, red and blue lines indicate the mode along the z-direction, y-direction and x-direction, respectively. + Along A-#sym.Gamma path, strong-polar modes along x- and y-directions are degenerated, + showing as a single purple line. + (b) Magnified view of the boxed region in (a). + The orange dashed lines mark the phonon wavevectors involved in Raman scattering + with incident light along the z- and y-directions. + ], + placement: none, +) diff --git a/test-typst/section/perfect/non-polar/nopol.typ b/test-typst/section/perfect/non-polar/nopol.typ new file mode 100644 index 0000000..4751c21 --- /dev/null +++ b/test-typst/section/perfect/non-polar/nopol.typ @@ -0,0 +1,73 @@ +#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 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]]; + 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)], + // TODO: 整理表格，使用 m2 m3 来代替 + [*Number of Phonon*], + // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1 + [1], m2[2], [3], m2[4], [5], [6], [7], [8], m3[9], [10], [11], [12], m2[13], [14], m2[15], m3[16], [17], [18], + [*Vibration Direction*], + // E2 E2 E1 2B1 A1 + [x], m2[y], [x], m(2)[y], [x], [y], m(2)[z], m(3)[z], + // E1 E2 E2 A1 2B1 + [x], [y], [x], m(2)[y], [x], m(2)[y], m(3)[z], m(2)[z], + [*Representation #linebreak() in Group C#sub[6v]*], + m(3, E2), m(3, E2), m(2, E1), B1, B1, m(3, A1), m(2, E1), m(3, E2), m(3, E2), m(3, A1), B1, B1, + [*Raman-active or Not*], + m(8)[Raman-active], m(2)[Raman-inactive], m(14)[Raman-active], m(2)[Raman-inactive], + // [*Representation in Group C#sub[2v]*], + // // E2 E2 E1 2B1 A1 E1 E2 E2 A1 2B1 + // A2, m(2, A1), A2, m(2, A1), B2, B1, B1, B1, m(3, A1), B2, B1, A2, m(2, A1), A2, m(2, A1), m(3, A1), B1, B1, + [*Scattering in Polarization #linebreak() (non-zero Raman #linebreak() tenser components)*], + // E2 E2 E1 2B1 A1 + [xy], [xx], [yy], [xy], [xx], [yy], [xz], [yz], m(2)[-], [xx], [yy], [zz], + // E1 E2 E2 A1 2B1 + [xz], [yz], [xy], [xx], [yy], [xy], [xx], [yy], [xx], [yy], [zz], m(2)[-], + [*Raman Intensity (a.u.)*], + // E2 E2 E1 2B1 A1 + m(3)[0.17], m(3)[1.13], m(2)[2.43], m(2)[0], m(2)[2.83], [1.79], + // E1 E2 E2 A1 2B1 + m(2)[0.09], m(3)[88.54], m(3)[0.50], m(2)[0.01], [1.78], m(2)[0], + [*Visible in Common #linebreak() Raman Experiment or Not*], + // E2 E2 E1 2B1 A1 + m(8)[Visible], m(2)[-], m(3)[Visible], + // E1 E2 E2 A1 2B1 + m(2)[Invisible], m(3)[Visible], m(5)[Invisible], [Visible], m(2)[-], + [*Wavenumber #linebreak() (Simulation) (cm#super[-1])*], + // E2 E2 E1 2B1 A1 + m(3)[190.51], m(3)[197.84], m(2)[257.35], [389.96], [397.49], m(3)[591.90], + // E1 E2 E2 A1 2B1 + m(2)[746.91], m(3)[756.25], m(3)[764.33], m(3)[812.87], [885.68], [894.13], + [*Wavenumber #linebreak() (Experiment) (cm#super[-1])*], + // E2 E2 E1 2B1 A1 + m(3)[195.5], m(3)[203.3], m(2)[269.7], m(2)[-], m(3)[609.5], + // E1 E2 E2 A1 2B1 + m(2)[-], m(3)[776], m(5)[-], [839], m(2)[-], + [*FWHM #linebreak() (Simulation) (cm#super[-1])*], + // E2 E2 E1 2B1 A1 + 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)[-], + // 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)[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)[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], + )}, + caption: [Negaligible-polarized Phonons at $Gamma$ Point], +)] diff --git a/test-typst/section/perfect/non-polar/predmode.typ b/test-typst/section/perfect/non-polar/predmode.typ new file mode 100644 index 0000000..7a82aec --- /dev/null +++ b/test-typst/section/perfect/non-polar/predmode.typ @@ -0,0 +1,47 @@ +// 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: $-A_#text[C]zeta_6$], + [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*], + [-1.68 #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/test-typst/section/perfect/non-polar/raman.typ b/test-typst/section/perfect/non-polar/raman.typ new file mode 100644 index 0000000..272765f --- /dev/null +++ b/test-typst/section/perfect/non-polar/raman.typ @@ -0,0 +1,11 @@ +#figure( + image("/画图/拉曼整体图/main.svg"), + caption: [ + (a) Phonon dispersion of 4H-SiC along the A–#sym.Gamma–K high-symmetry path. + Gray lines represent negligible-polar phonon modes, + while colored lines indicate strong-polar phonon modes. + (b) Magnified view of the boxed region in (a). + The orange dashed lines mark the phonon wavevectors involved in Raman scattering + with incident light along the z- and y-directions. + ] +) diff --git a/test-typst/section/perfect/non-polar/rep.typ b/test-typst/section/perfect/non-polar/rep.typ new file mode 100644 index 0000000..884177f --- /dev/null +++ b/test-typst/section/perfect/non-polar/rep.typ @@ -0,0 +1,18 @@ +#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, +)