276 lines
13 KiB
Typst
276 lines
13 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 <appd-predict>
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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（$a$），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 with each other,
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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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The vibration where the two atoms vibrate synchronously along the positive x direction,
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belongs to the representation of E#sub[1] of C#sub[6v] and 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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For the vibration where the Si atom in A layer vibrates towards the positive x direction
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and the Si atom in C layer vibrates towards the negative x direction,
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it belongs to the representation of E#sub[2] of C#sub[6v] and 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 same 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 A 层中 Si 原子沿 b 轴正方向振动所导致的拉曼张量（记为 $alpha'$）。
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一方面，它可以看作由向 x 和 y 正方向振动的拉曼张量（记为 $alpha_x$ 和 $alpha_y$）通过线性组合得到；
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另一方面，它也可以看作由 $alpha_x$ 通过将体系绕 z 轴旋转 $120 degree$ 得到。
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因此：
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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 Raman tensor of Si atoms in A layer vibrating along the positive b-axis direction
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(denoted as $alpha'$) was considered.
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On one hand,
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it can be expressed as a linear combination of the Raman tensors vibrating in the positive x and y directions
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(denoted as $alpha_x$ and $alpha_y$, respectively);
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on the other hand,
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it can also be obtained from $alpha_x$ by rotating the system by $120 degree$.
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Thus:
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$
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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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"where" C_3 = mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1;),
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alpha_x = mat(,a_2,a_1;a_2,,;a_1,,;), alpha_y = mat(a_4,,;,-a_4,a_3;,a_3,;)
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$
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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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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 quantities with small absolute values, 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 positive 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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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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例如 A 层 C 原子的拉曼张量可以写为：
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The Raman tensors of C atoms can be obtained using a similar method as that of Si atoms.
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For example, the Raman tensor of C atom in A layer can be written as:
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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*], table.cell(rowspan: 2)[A layer],
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[C],
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[$alpha_"Cx" = mat(,b_2,b_1;b_2,,;b_1,,;)$],
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[$alpha_"Cy" = mat(b_2,,;,-b_2,b_1;,b_1,;)$],
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[$alpha_"Cz" = mat(b_5,,;,b_5,;,,b_6;)$],
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[Si],
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[$alpha_"Six" = mat(,a_2,a_1;a_2,,;a_1,,;)$],
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[$alpha_"Siy" = mat(a_2,,;,-a_2,a_1;,a_1,;)$],
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[$alpha_"Siz" = mat(a_5,,;,a_5,;,,a_6;)$],
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),
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placement: none,
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)
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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 原子通过以下三步操作得到：
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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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我们分别考虑这些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 with the following three steps:
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first taking a mirror image along the basal plane, then reversing the charge,
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and finally adjusting the atomic mass and other factors, as shown in @fig-sitoc.
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The change of Raman tensor during these processes would be discussed separately.
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#include "fig-sitoc.typ"
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第一步中，记翻转后的拉曼张量为 $alpha'_"Six"$、$alpha'_"Siy"$ 和 $alpha'_"Siz"$。
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对于 x 和 y 方向振动的拉曼张量，只需要将群元素 $sigma_"h" = op("diag") (1, 1, -1)$ 作用上去即可；
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对于 z 方向振动的拉曼张量，还需要乘以 $-1$，因为在这个过程中振动的方向发生了改变。
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In the first step,
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the Raman tensors after taking the mirror image were denoted as $alpha'_"Six"$, $alpha'_"Siy"$ and $alpha'_"Siz"$.
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For $alpha'_"Six"$ and $alpha'_"Siy"$, they are connected with $alpha_"Six"$ and $alpha_"Siy"$
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by the group element $sigma_"h" = op("diag") (1, 1, -1)$.
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For $alpha'_"Siz"$, an additional factor of $-1$ is needed,
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because the direction of vibration has changed during this process.
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$
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alpha'_"Six" = sigma_"h" alpha_"Six" sigma_"h"^T = mat(,a_2,-a_1;a_2,,;-a_1,,;), \
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alpha'_"Siy" = sigma_"h" alpha_"Siy" sigma_"h"^T = mat(a_2,,;,-a_2,-a_1;,-a_1,;), \
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alpha'_"Siz" = -sigma_"h" alpha_"Siz" sigma_"h"^T = mat(-a_5,,;,-a_5,;,,-a_6;),
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$
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翻转电荷的过程不会导致拉曼张量的变化。
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这可以通过考虑在外场 $E$ 作用下的能量变化来得知。
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记电荷翻转前后的拉曼张量分别为 $alpha$ 和 $alpha'$。
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若在翻转电荷的过程中，外加电场同样翻转，则总能量不变。
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因此 $E^T alpha E = (-E)^T alpha' (-E)$，因此 $alpha = alpha'$。
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In the second step (reversing the charge), the Raman tensor does not change.
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This can be derived by considering the energy caused by an external electric field $E$.
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The Raman tensors before and after charge reversal were denoted as $alpha$ and $alpha'$.
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When the direction of the external electric field is also reversed during the charge reversal,
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the total energy does not change, i.e., $E^T alpha E = (-E)^T alpha' (-E)$.
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Thus, we have $alpha = alpha'$.
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第三步中，我们假定原子质量和其它因素的变化对拉曼张量的影响较小，
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即 $alpha'_"Six"$、$alpha'_"Siy"$ 和 $alpha'_"Siz"$ 与 $alpha_"Six"$、$alpha_"Siy"$ 和 $alpha_"Siz"$ 之间仅有较小的差异。
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因此：
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In the third step, we assume that the change in atomic mass and other factors has a small effect on the Raman tensor,
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i.e., $alpha'_"Six"$, $alpha'_"Siy"$ and $alpha'_"Siz"$ differ from $alpha_"Six"$, $alpha_"Siy"$ and $alpha_"Siz"$
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only by small quantities, respectively.
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Thus:
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$
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b_1 = -a_1 - zeta_1, abs(zeta_1) << abs(a_1), \
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b_2 = a_2 + zeta_2, abs(zeta_2) << abs(a_2), \
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b_5 = -a_5 - zeta_5, abs(zeta_5) << abs(a_5), \
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b_6 = -a_6 - zeta_6, abs(zeta_6) << abs(a_6),
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$
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=== Summary
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我们将各个原子的拉曼张量总结于 @table-singleatom，用它推测了各个模式的拉曼张量并与第一性原理计算对比，
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结果如 @table-predmode 所示。
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We summarized the Raman tensors of each atom in @table-singleatom.
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The result was used to predict the Raman tensors of each mode and comparing with first-principles calculations,
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which is shown in @table-predmode.
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#include "table-singleatom.typ"
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#include "table-predmode.typ" |