256 lines
11 KiB
Typst
256 lines
11 KiB
Typst
#import "@preview/physica:0.9.5": pdv, super-T-as-transpose
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#show: super-T-as-transpose
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== Approximation of Raman tensor of 4H-SiC
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近似的核心思路。
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我们的近似方法基于这样一个原则：将拉曼张量（即原子位移引起的极化率变化）分配给单位晶胞中的每个原子。
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一些原子的局部环境相似但不完全相同，我们将它们的拉曼张量的差视为一个小量（$epsilon$, $eta$ and $zeta$），
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剩余的部分视为一个大量($a$).
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将一个模式中所有参与振动的原子的贡献相加，就可以得到该模式的拉曼张量。
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若模式的该拉曼张量只包含小量（$epsilon$, $eta$ and $zeta$），
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说明该模式中，原子振动导致的拉曼效应互相抵消了大部分，该模式的拉曼活性较弱；
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而若该拉曼张量包含较大的常数项，说明该模式的拉曼效应较强。
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The center principle of our approximation is to assign the Raman tensor
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(i.e., change of polarizability caused by atomic displacement)
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to each atom in the unit cell.
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For the atoms with similar but not exactly the same local environment,
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we consider the difference in their Raman tensors as quantities with small absolute value
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($epsilon$, $eta$ and $zeta$),
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while the remaining part is treated as quantities with large absolute value ($a$).
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The Raman tensor of a phonon mode can be obtained
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by summing the contributions of all atoms participating in the vibration.
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If the Raman tensor of a mode only contains quantities with small absolute value ($epsilon$, $eta$ and $zeta$),
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it indicates that the Raman effect caused by atomic vibrations in this mode is largely canceled out,
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and thus the mode has weak Raman activity.
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Otherwise, if the Raman tensor contains quantities with large absolute value ($a$),
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it indicates that the mode has strong Raman activity.
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使用 A/B1/C/B2 层的表述，而不是 ABCB，来区分两个 B 层。
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这是因为两个 B 层的原子局部环境互相镜面对称而不是平移对称，导致它们的拉曼张量不相等。
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In this section, AB#sub[1]CB#sub[2] instead of ABCB was used to denote the four bilayers in 4H-SiC primative cell
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to clearly distinguish the two B layers.
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This is because the local environment of the two B layers is mirror symmetric rather than translationally symmetric,
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thus their Raman tensors are not equal.
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=== Raman tensor of Si atoms in A and C layers
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我们首先推导 A/C 层 Si 原子沿 x 方向振动时的拉曼张量。
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根据前文，我们知道，当这两个原子同步地沿 x 正方向振动时，它们属于 E1(C6v) B2(C2v) 表示，拉曼张量可以写为：
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We first derive the Raman tensor of Si atoms in A and C layer vibrating along x direction.
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When the two atoms vibrate synchronously in the positive x direction,
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they belong to the representation of E#sub[1] of C#sub[6v] or B#sub[2] of C#sub[2v].
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Thus, their Raman tensor can be written as:
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$ mat(,,2a_1;,,;2a_1,,;) $
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其中 $a_1$ 是未知的常数。
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where $a_i (i = 1 "to" 6)$ are unknown constants.
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当 A 层 Si 原子沿 x 正方向振动而 C 层 Si 原子沿 x 负方向振动时，
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它们属于 E#sub[2] of C#sub[6v] or A#sub[2] of C#sub[2v] 表示，
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拉曼张量可以写为：
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When the Si atom in A layer vibrates in the positive x direction
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while the Si atom in C layer vibrates in the negative x direction,
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they belong to the representation of E#sub[2] of C#sub[6v] or A#sub[2] of C#sub[2v].
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Thus, their Raman tensor can be written as:
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$ mat(,2a_2,;2a_2,,;,,;) $
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因此，A 层和 C 层 Si 原子沿 x 方向振动时的拉曼张量分别为：
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Thus, the Raman tensors of Si atoms in A and C layers vibrating in the positive x direction are:
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$ mat(,a_2,a_1;a_2,,;a_1,,;), mat(,-a_2,a_1;-a_2,,;a_1,,;), $
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接下来讨论 A/C 层 Si 原子沿 y 方向振动时的拉曼张量，使用相似的方法可以得到：
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The Raman tensors of Si atoms in A and C layers vibrating along positive y direction
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can be obtained using the method, which gives:
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$ mat(a_4,,;,-a_4,a_3;,a_3,;), mat(-a_4,,;,a_4,a_3;,a_3,;) $
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$\{a_1, a_2, a_3, a_4\}$ 之间并不独立。为了确定它们之间的关系，我们考虑将体系绕 z 轴旋转 120 度，
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同一个点的坐标旋转前后分别为 $r$ 和 $C_3 r$，其中 $r$ 是一个列向量，$C_3$ 是一个旋转矩阵：
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where $\{a_3, a_4\}$ are not independent of $\{a_1, a_2\}$.
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To determine the relationship between $\{a_1, a_2\}$ and $\{a_3, a_4\}$,
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the system is rotated by $120 degree$ around the z axis.
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The point located at $r$ before rotation should be at $C_3 r$ after rotation,
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where $r$ is a column vector and $C_3$ is a rotation matrix:
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$ C_3 = mat(-1/2,-sqrt(3)/2,0;sqrt(3)/2,-1/2,0;0,0,1;) $
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记 A 层中 Si 原子沿与 x 轴夹角 $120 degree$ 的方向振动所导致的拉曼张量为 $alpha'$，
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沿 x 和 y 正方向振动的拉曼张量则为 $alpha_x$ 和 $alpha_y$。
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一方面，$alpha'$ 可以从 $alpha_x$ 出发，将体系旋转 $120 degree$ 得到；
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另一方面，$alpha'$ 也可以由 $alpha_x$ 和 $alpha_y$ 通过线性组合得到：
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The Raman tensor of Si atoms in A layer vibrating along the direction at an angle of $120 degree$ with the x axis
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is denoted as $alpha'$,
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while those vibrating in the positive x and y directions are denoted as $alpha_x$ and $alpha_y$, respectively.
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On one hand, $alpha'$ can be obtained from $alpha_x$ by rotating the system by $120 degree$;
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on the other hand, $alpha'$ can also be expressed as a linear combination of $alpha_x$ and $alpha_y$:
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$ alpha' = C_3 alpha_x C_3^T = -1/2 alpha_x + sqrt(3)/2 alpha_y $
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化简可得：
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Simplifying the above equations, we have:
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$ a_3 = a_1, a_4 = a_2 $
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与 x 和 y 方向的情况类似，可以推导出 A/C 层 C 原子沿 z 方向振动时的拉曼张量。总结如下。
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The Raman tensors of C atoms in A and C layers vibrating along z direction can be derived similarly.
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Thus the results are summarized as follows:
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#figure(
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table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
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table.cell(colspan: 3)[*Vibration Direction*], [x], [y], [z],
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table.cell(rowspan: 2)[*Raman tensor #linebreak() of atoms*],
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[A layer], [Si],
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[$mat(,a_2,a_1;a_2,,;a_1,,;)$],
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[$mat(a_2,,;,-a_2,a_1;,a_1,;)$],
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[$mat(a_5,,;,a_5,;,,a_6;)$],
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[C layer], [Si],
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[$mat(,-a_2,a_1;-a_2,,;a_1,,;)$],
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[$mat(-a_2,,;,a_2,a_1;,a_1,;)$],
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[$mat(a_5,,;,a_5,;,,a_6;)$],
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),
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placement: none,
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)
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=== Raman tensor of Si atoms in B#sub[1] and B#sub[2] layers
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与 A/C 层原子类似，同理可以给出 B#sub[1]B#sub[2] 层原子沿 x 方向振动的拉曼张量：
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The Raman tensor of Si atoms in B#sub[1] and B#sub[2] layer vibrating along positive x direction
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can be written out similarily as that in A and C layer:
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$ mat(,a'_2,a'_1;a'_2,,;a'_1,,;), mat(,-a'_2,a'_1;-a'_2,,;a'_1,,;) $
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注意到 B 层 Si 原子与 A 层 Si 原子的局部环境非常相似（最近邻完全相同，次近邻也只有一半不同，如图所示），
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因此可以推测它们的拉曼张量只有较小的不同，即：
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Because the local environment of Si atoms in B layer is very similar to that in A layer (as shown in @figure-same),
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we can assume that their Raman tensors differ only by small quantities, i.e.,
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$
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a'_1 = a_1 + epsilon_1, abs(epsilon_1) << abs(a_1), \
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a'_2 = a_2 + epsilon_2, abs(epsilon_2) << abs(a_2),
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$
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#include "fig-same.typ"
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由此可以写出 B 层 Si 原子沿 x 方向振动时的拉曼张量：
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Thus, the Raman tensor of Si atoms in B#sub[1] and B#sub[2] layers vibrating along x direction
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can be written as:
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$
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mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;),
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mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)
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$
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同理，可以得到 B#sub[1] 和 B#sub[2] 层 Si 原子沿其它方向振动的拉曼张量。
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总结如下。
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The Raman tensors of Si atoms in B#sub[1] and B#sub[2] layers vibrating along other directions
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can be obtained using similar method,
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and the results are summarized as follows:
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#figure(
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table(columns: 6, align: center + horizon, inset: (x: 3pt, y: 5pt),
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table.cell(colspan: 3)[*Vibration Direction*], [x], [y], [z],
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table.cell(rowspan: 2)[*Raman tensor #linebreak() of atoms*],
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[B#sub[1] layer], [Si],
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[$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$],
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[$mat(a_2+epsilon_2,,;,-a_2-epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
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[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
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[B#sub[2] layer], [Si],
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[$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$],
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[$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
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[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
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),
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placement: none,
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)
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=== Raman tensor of C atoms
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考虑 A/C 层 C 原子的拉曼张量，使用与 Si 原子类似的方法，可以得到：
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The Raman tensors of C atoms in A and C layers can be obtained using a similar method:
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$
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mat(,b_2,b_1;b_2,,;b_1,,;), mat(,-b_2,b_1;-b_2,,;b_1,,;)
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$
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我们需要估计 ${b_1, b_2}$ 与 ${a_1, a_2}$ 之间的关系。
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考虑 A 层 C 原子的环境，它可以由 A 层 Si 原子通过以下操作得到：
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先沿基平面取镜像，然后反转电荷，再调整原子质量等其它因素，如图所示。
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我们分两步来推导这个过程中拉曼张量的变化。
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The relationship between $\{b_1, b_2\}$ and $\{a_1, a_2\}$ needs to be estimated.
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The environment of C atoms in A layer can be obtained from that of Si atoms in A layer
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by first taking a mirror image along the basal plane,
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then reversing the charge and adjusting the atomic mass and other factors,
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as shown in @figure-sitoc.
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The change of Raman tensor during this process could be derived following these two steps.
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#include "fig-sitoc.typ"
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记翻转前后的 Si 原子拉曼张量为 $alpha$ 和 $alpha'$。
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考虑在外场 $E$ 作用下，Si 原子沿 x 方向振动导致的系统能量变化为：
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The Raman tensor of the Si atom in A layer before and after taking the mirror image
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was denoted as $alpha$ and $alpha'$, respectively.
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Before the mirror image,
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the system energy change caused by the vibration of this atom along x direction
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under an external electric field $E$ is:
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$ Delta E = E^T alpha E, #[where] alpha = mat(,a_2,a_1;a_2,,;a_1,,;) $
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若在翻转的过程中，将电场同样翻转，则总能量不变。因此：
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When the electric field is also flipped during the mirror image,
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the total energy does not change, i.e.,
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$ E^T alpha E = Delta E = (sigma E)^T alpha' (sigma E), #[where] sigma = mat(1,0,0;0,1,0;0,0,-1;) $
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整理可得：
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Thus:
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$ alpha' = mat(,a_2,-a_1;a_2,,;-a_1,,;) $
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记电荷反转后，Si 原子拉曼张量为 $alpha''$。
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若将外加电场方向同时反转，则能量不变。即：
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// TODO: 能量写明是 Delta E
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The Raman tensor of Si atom after charge reversal was denoted as $alpha''$.
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The energy does not change when the direction of the external electric field is also reversed, i.e.,
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$ E^T alpha' E = (sigma' E)^T alpha'' (sigma' E), #[where] sigma' = -1 $
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Thus:
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$ alpha'' = alpha' $
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使用类似的方法，得到 Si 原子沿其它方向和 C 原子沿各个方向的拉曼张量，以及各个模式的拉曼张量。
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Similarily, we can write out the Raman tensors of Si atoms virbrating along other directions
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and the Raman tensors of C atoms,
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and thus the Raman tensors of all phonon modes.
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The results are summarized in @table-singleatom and @table-predmode.
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#include "table-singleatom.typ"
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#include "table-predmode.typ" |