@@ -6,268 +6,251 @@
由于同种 原子的局部环境相似、只有次近邻才有不同，因此我们将同种原子导致 的拉曼效应 的差视为一个小量。
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 $ ） ，
剩余的部分视为一个大量($ a $ ).
$ epsilon $ , $ eta $ and $ zeta $ ） ，
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；
center principle of our approximation is to assign the Raman tensor
(i.e., change of polarizability caused by atomic displacement)
to each atom in the unit cell.
Because the local environment of the same type of atom is similar and only different for the next nearest neighbors ,
we consider the difference in Raman effect caused by the s ame type of atom as a small quantity.
For the atoms with similar but not exactly the same local environment ,
we consider the difference in their R aman tensors as quantities with small absolute value
($ epsilon $ , $ eta $ and $ zeta $ ),
while the remaining part is treated as quantities with large absolute value ($ a $ ).
Raman tensor of a phonon mode can be obtained
by summing the contributions of all atoms participating in the vibration.
the Raman tensor of a mode only contains quantities with small absolute value ($ epsilon $ , $ eta $ and $ zeta $ ),
it indicates that the Raman effect caused by atomic vibrations in this mode is largely canceled out,
and thus the mode has weak Raman activity.
if the Raman tensor contains quantities with large absolute value ($ a $ ),
it indicates that the mode has strong Raman activity.
推导 A/C 层 Si 原子沿 x 方向振动时的拉曼张量 。
使用 A/B1/C/B2 层的表述，而不是 ABCB， B 层 。
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。
this section, AB#sub [ 1] CB#sub [ 2] instead of ABCB was used to denote the four bilayers in 4H-SiC primative cell
to clearly distinguish the two B layers.
is because the local environment of the two B layers is mirror symmetric rather than translationally symmetric,
thus their Raman tensors are not equal.
=== Raman tensor of Si atoms in A and C layers
A/C 层 Si 原子沿 x 方向振动时的拉曼张量。
向和反 向振动时，它们分别 属于 E1(C6v) B2(C2v) 和 E2(C6v) A2(C2v) 表示，因此它们的 拉曼张量分别 为：
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：
first derive the Raman tensor of Si atoms in A/ C layer vibrating along x direction.
the two atoms vibrate in the same directio n and op posite direction,
they belong to the representation of E#sub [ 1] of C#sub [ 6v] or B#sub [ 2] of C#sub [ 2v]
and E#sub [ 2] of C#sub [ 6v] or A#sub [ 2] of C#sub [ 2v] , respectively.
their Raman tensors are in the form of:
first derive the Raman tensor of Si atoms in A and layer vibrating along x direction.
the two atoms vibrate synchronously i n the positive x direction,
they belong to the representation of E#sub [ 1] of C#sub [ 6v] or B#sub [ 2] of C#sub [ 2v] .
Thus, their Raman tensor can be written as:
$ mat ( , , 2 a _ 1 ; , , ; 2 a _ 1 , , ; ) , mat ( , 2 a _ 2 , ; 2 a _ 2 , , ; , , ; ) , $
$ mat ( , , 2 a _ 1 ; , , ; 2 a _ 1 , , ; ) $
$ a _ 1 $ 和 $ a _ 2 $ 是两个 未知的常数。
$ a _ 1 $ 是 未知的常数。
$ a _ 1 $ and $ a _ 2 $ are two constants with unknown values
$ a _ i ( i = 1 "to" 6 ) $ are unknown constants.
将上述结果相加或相减，得到 A 层和 C 层 Si 原子沿 x 方向振动时的拉曼张量分别为：
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，
它们属于 E#sub [ 2] of C#sub [ 6v] or A#sub [ 2] of C#sub [ 2v] 表示，
拉曼张量可以写为：
By adding or subtract ing the above results,
we get the Raman tensors of Si atoms in A and C layers vibrating along x direction:
When the Si atom in A layer vibrates in the positive x direction
while the Si atom in C layer vibrates in the negative x direction,
belong to the representation of E#sub [ 2] of C#sub [ 6v] or A#sub [ 2] of C#sub [ 2v] .
their Raman tensor can be written as:
$ mat ( , a _ 2 , a _ 1 ; a _ 2 , , ; a _ 1 , , ; ) , mat ( , - a _ 2 , a _ 1 ; - a _ 2 , , ; - a _ 1 , , ; ) , $
$ mat ( , 2 a _ 2 , ; 2 a _ 2 , , ; , , ; ) $
近似给出 B 层 Si 原子沿 x 方向振动时的拉曼张量。
因此， 层和 C 层 Si 原子沿 x 方向振动时的拉曼张量分别为：
注意到 B 层 Si 原子与 A 层 Si 原子
Thus, the Raman tensors of Si atoms in A and C layers vibrating in the positive x direction are:
$ mat ( , a _ 2 , a _ 1 ; a _ 2 , , ; a _ 1 , , ; ) , mat ( , - a _ 2 , a _ 1 ; - a _ 2 , , ; a _ 1 , , ; ) , $
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：
Raman tensors of Si atoms in A and C layers vibrating along positive y direction
can be obtained using the method, which gives:
$ mat ( a _ 4 , , ; , - a _ 4 , a _ 3 ; , a _ 3 , ; ) , mat ( - a _ 4 , , ; , a _ 4 , a _ 3 ; , a _ 3 , ; ) $
$ \ { 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，
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：
$ \ { a _ 3 , a _ 4 \ } $ are not independent of $ \ { a _ 1 , a _ 2 \ } $ .
determine the relationship between $ \ { a _ 1 , a _ 2 \ } $ and $ \ { a _ 3 , a _ 4 \ } $ ,
the system is rotated by $ 120 degree $ around the z axis.
point located at $ r $ before rotation should be at $ C _ 3 r $ after rotation,
where $ r $ is a column vector and $ C _ 3 $ is a rotation matrix:
$ C _ 3 = mat ( - 1 / 2 , - sqrt ( 3 ) / 2 , 0 ; sqrt ( 3 ) / 2 , - 1 / 2 , 0 ; 0 , 0 , 1 ; ) $
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 ' $ ，
沿 x 和 y 正方向振动的拉曼张量则为 $ alpha _ x $ 和 $ alpha _ y $ 。
$ alpha ' $ 可以从 $ alpha _ x $ 出发，将体系旋转 $ 120 degree $ 得到；
另一方面，$ alpha ' $ 也可以由 $ alpha _ x $ 和 $ alpha _ y $ 通过线性组合得到：
Raman tensor of Si atoms in A layer vibrating along the direction at an angle of $ 120 degree $ with the x axis
is denoted as $ alpha ' $ ,
while those vibrating in the positive x and y directions are denoted as $ alpha _ x $ and $ alpha _ y $ , respectively.
one hand, $ alpha ' $ can be obtained from $ alpha _ x $ by rotating the system by $ 120 degree $ ;
on the other hand, $ alpha ' $ can also be expressed as a linear combination of $ alpha _ x $ and $ alpha _ y $ :
$ alpha ' = C _ 3 alpha _ x C _ 3 ^ T = - 1 / 2 alpha _ x + sqrt ( 3 ) / 2 alpha _ y $
the above equations, we have:
$ a _ 3 = a _ 1 , a _ 4 = a _ 2 $
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。
Raman tensors of C atoms in A and C layers vibrating along z direction can be derived similarly.
the results are summarized as follows:
#figure (
table ( columns : 6 , align : center + horizon , inset : ( x : 3 pt , y : 5 pt ) ,
table . cell ( colspan : 3 ) [ *Vibration Direction* ] , [ x] , [ y] , [ z] ,
table . cell ( rowspan : 2 ) [ *Raman tensor #linebreak() of atoms* ] ,
[ A layer] , [ Si] ,
[ $ mat ( , a _ 2 , a _ 1 ; a _ 2 , , ; a _ 1 , , ; ) $ ] ,
[ $ mat ( a _ 2 , , ; , - a _ 2 , a _ 1 ; , a _ 1 , ; ) $ ] ,
[ $ mat ( a _ 5 , , ; , a _ 5 , ; , , a _ 6 ; ) $ ] ,
[ C layer] , [ Si] ,
[ $ mat ( , - a _ 2 , a _ 1 ; - a _ 2 , , ; a _ 1 , , ; ) $ ] ,
[ $ mat ( - a _ 2 , , ; , a _ 2 , a _ 1 ; , a _ 1 , ; ) $ ] ,
[ $ mat ( a _ 5 , , ; , a _ 5 , ; , , a _ 6 ; ) $ ] ,
) ,
placement : none ,
)
=== Raman tensor of Si atoms in B#sub[1] and B#sub[2] layers
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：
Raman tensor of Si atoms in B#sub [ 1] and B#sub [ 2] layer vibrating along positive x direction
can be written out similarily as that in A and C layer:
$ mat ( , a ' _ 2 , a ' _ 1 ; a ' _ 2 , , ; a ' _ 1 , , ; ) , mat ( , - a ' _ 2 , a ' _ 1 ; - a ' _ 2 , , ; a ' _ 1 , , ; ) $
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），
因此可以推测它们的拉曼张量只有较小的不同，即：
the local environment of Si atoms in B layer is very similar to that in A layer (as shown in @figure-same ),
we can assume that their Raman tensors differ only by small quantities, i.e.,
$
a ' _ 1 = a _ 1 + epsilon _ 1 , abs ( epsilon _ 1 ) << abs ( a _ 1 ) , \
a ' _ 2 = a _ 2 + epsilon _ 2 , abs ( epsilon _ 2 ) << abs ( a _ 2 ) ,
$
#include "fig-same.typ"
B 层 Si 原子沿 x 方向振动时的拉曼张量：
e center principle is t o assign the Raman tensor (i.e., change of polarizability caused by atomic displacement)
to each atom i n th e unit cell.
including the following steps:
- Write out the change of polarizability caused by displacement of Si atom in A and C layer,
Where unknown non-zero components are denoted by $ a _ 1 $ , $ a _ 2 $ , $ a _ 5 $ , $ a _ 6 $ .
For example, when we move the Si atom in A layer slightly towards the x+ direction in $ d $ distance,
the change of polarizability should be $ mat ( , a _ 2 , a _ 1 ; a _ 2 , , ; a _ 1 , , ) d $ .
This could be done by conclusion above.
- The Si atom in B layer have similar local environment as the A and C layer, with only a little difference.
We denote these difference by $ epsilon _ 1 $ , $ epsilon _ 2 $ , $ epsilon _ 5 $ , $ epsilon _ 6 $ ,
and the absolute value of $ epsilon _ i $ should be much smaller than $ a _ i $ .
For example, when we move the Si atom in B layer slightly towards the x+ direction in $ d $ distance,
the change of polarizability should be $ mat ( , a _ 2 + epsilon _ 2 , a _ 1 + epsilon _ 1 ; a _ 2 + epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ) d $ .
- The local environment of C atom in A layer is similar to the Si atom in A layer with charge reversed and
the system reversed along xy plane.
We denote these difference by $ eta _ 1 $ , $ eta _ 2 $ , $ eta _ 5 $ , $ eta _ 6 $ ,
and the absolute value of $ epsilon _ i $ should be much smaller than $ a _ i $ .
For example, when we move the C atom in A layer slightly towards the x+ direction in $ d $ distance,
the change of polarizability should be $ mat ( , a _ 2 + eta _ 2 , - a _ 1 - eta _ 1 ; a _ 2 + eta _ 2 , , ; - a _ 1 - eta _ 1 , , ) d $ .
- Similar to the case in Si atoms, we derive the change of polarizability
caused by moving C atom in B layer slightly towards the x+ direction in $ d $ distance,
which should be $ mat ( , - a _ 2 - eta _ 2 - zeta _ 2 , - a _ 1 - eta _ 1 - zeta _ 1 ; - a _ 2 - eta _ 2 - zeta _ 2 , , ; - a _ 1 - eta _ 1 - zeta _ 1 , , ) d $ .
us, the Raman tensor of Si atoms in B#sub [ 1] and B#sub [ 2] layers vibrating along x direction
ca n b e written as:
assign Raman tensor onto each atom.
is, Raman tensor is derivative of the polarizability with respect to the atomic displacement:
$
alpha = pdv ( chi , u )
$
$ u $ should be the displacement of the atom corresponding to a phonon mode.
even when $ u $ is *NOT* the displacement of a phonon
(for example, lets only slightly move Si atom in A layer, keeping other atoms fixed),
the (high-frequency) polarizability is still well-defined,
and the will still cause a change in the polarizability.
more, the group representation theory is still applicable in this condition:
the only thing that matters is, when applying $ g $ to the system,
the tensor transformed into $ g ^ ( - 1 ) alpha g $ or $ g alpha g ^ ( - 1 ) $ ,
no matter $ alpha $ is Raman tensor or something else, or it is related to a phonon or not.
we can, in principle, "assign" Raman tensor of a phonon, to each atom.
"assign" is unique since both the atom movement and all phonons have 24 dimensions.
we consider what these single-atom-caused "Raman tensors" looks like.
example, what happens if we move the Si atom in A layer slightly along the x+ direction?
also move the Si atom in C layer slightly, along x+ or x- direction.
about the Raman tensor caused by the both two atoms?
first case, this is B2 representation in E1 representation. Thus the Raman tensor should be something like:
$
mat ( , , 2 a _ 1 ; , , ; 2 a _ 1 , , ; )
$
the second case, it is A2 in E2. It turns out:
$
mat ( , 2 a _ 2 , ; 2 a _ 2 , , ; , , ; )
$
average of these two tensors should be the s"Raman tensor" cause by move only the Si atom in A layer,
slightly towards x+ direction.
$
mat ( , a _ 2 , a _ 1 ; a _ 2 , , ; a _ 1 , , ; )
$
difference should be the "Raman tensor" of the second atom.
$
mat ( , - a _ 2 , a _ 1 ; - a _ 2 , , ; a _ 1 , , ; )
$
// This approach applied relied on the fact that, all Si atom in 4H-SiC is "distinguishable" by the symmetry operations.
// I mean, what will happen if we have two Si atoms in A layer?
// Apparently, we could not extract the "Raman tensor" of only one of the two atoms.
// This is the case for the 6H-SiC.
// Hence, we will provide a more general approach to estimate the "Raman tensor" of a single atom.
the Si atom in the B1 layer.
lives in an environment quite similar to the A layer.
the "Raman tensor" caused by it should be similar to the one caused by the A layer:
$
mat ( , a _ 2 + epsilon _ 2 , a _ 1 + epsilon _ 1 ; a _ 2 + epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ; )
$
to the Si atom in B2 layer:
$
mat ( , a _ 2 + epsilon _ 2 , a _ 1 + epsilon _ 1 ; a _ 2 + epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ; ) ,
mat ( , - a _ 2 - epsilon _ 2 , a _ 1 + epsilon _ 1 ; - a _ 2 - epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ; )
$
Same approach applied for Si atom vibrate in y direction.
When we move both Si atoms in A and C layer in y+ direction,
it is B1 in E1, thus the "Raman tensor" should be:
$
mat ( , , ; , , 2 a _ 3 ; , 2 a _ 3 , ; )
$
if we move Si in A layer towards y+ but Si in C layer towards y-,
it is A2 in E2:
$
mat ( 2 a _ 4 , , ; , - 2 a _ 4 , ; , , ; )
$
we get the "Raman tensor" of Si atom in A layer sololy move towards y+ direction:
$
mat ( a _ 4 , , ; , - a _ 4 , a _ 3 ; , a _ 3 , ; )
$
the "Raman tensor" of Si atom in C layer towards y+ direction:
$
mat ( - a _ 4 , , ; , a _ 4 , a _ 3 ; , a _ 3 , ; )
$
applied for the Si atom in B layer:
$
mat ( a _ 4 + epsilon _ 4 , , ; , - a _ 4 - epsilon _ 4 , a _ 3 + epsilon _ 3 ; , a _ 3 + epsilon _ 3 , ; )
$
$
mat ( - a _ 4 - epsilon _ 4 , , ; , a _ 4 + epsilon _ 4 , a _ 3 + epsilon _ 3 ; , a _ 3 + epsilon _ 3 , ; )
$
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。
总结如下。
Before consider z-direction, it is important to note that, $ a _ 1 $ $ a _ 2 $ $ a _ 3 $ $ a _ 4 $ are n ot independent.
Consider vibration along x+ direction (lets say the distance is $ d $ ).
System energy caused by external electric field and vibration i s:
$
E ^ T ( mat ( , , 2 a _ 1 ; , , ; 2 a _ 1 , , ) d ) E
$
C#sub [ 3] to atom vibration and external field, energy should not change. We got:
$
( mat ( - 1 / 2 , - sqrt ( 3 ) / 2 , ; sqrt ( 3 ) / 2 , - 1 / 2 , ; , , 1 ) E ) ^ T ( mat ( , , 2 a _ 1 ; , , ; 2 a _ 1 , , ) ( - 1 / 2 d ) + mat ( , , ; , , 2 a _ 3 ; , 2 a _ 3 , ) ( sqrt ( 3 ) / 2 d ) )
( mat ( - 1 / 2 , - sqrt ( 3 ) / 2 , ; sqrt ( 3 ) / 2 , - 1 / 2 , ; , , 1 ) E )
$
is equal to:
$
E ^ T ( mat ( , , 1 / 2 a _ 1 + 3 / 2 a _ 3 ; , , sqrt ( 3 ) / 2 a _ 1 - sqrt ( 3 ) / 2 a _ 3 ; 1 / 2 a _ 1 + 3 / 2 a _ 3 , sqrt ( 3 ) / 2 a _ 1 - sqrt ( 3 ) / 2 a _ 3 , ) d ) E
$
$
1 / 2 a _ 1 + 3 / 2 a _ 3 = 2 a _ 1 #linebreak ( )
sqrt ( 3 ) / 2 a _ 1 - sqrt ( 3 ) / 2 a _ 3 = 0
$
$ a _ 1 = a _ 3 $ .
the same method, we get $ abs ( a _ 2 ) = abs ( a _ 4 ) $ .
we have not define the sign of $ a _ 4 $ , we could take $ a _ 2 = a _ 4 $ .
for $ epsilon $ .
The Raman tensors of S i atoms in B#sub [ 1] and B#sub [ 2] layers vibrating along other directions
can be obtained using similar method,
and the results are summarized as follow
Now consider what if we move the Si atom in A layer along z+ direction.
we move the Si atom in C layer along z+ direction, it is A1:
$
mat ( 2 a _ 5 , , ; , 2 a _ 5 , ; , , 2 a _ 6 ; )
$
we move the Si atom in C layer along z- direction, it is B1:
$
0
$
we get the "Raman tensor" of Si atom in A or C layer towards z+ direction:
$
mat ( a _ 5 , , ; , a _ 5 , ; , , a _ 6 ; )
$
consider the C atom in A layer.
should be somehow similar to the Si atom in A layer, but with a negative sign in some places,
and then add or subtract some little value.
the "transformation" of Si atom in A layer to C atom in A layer applied in the following steps:
- reverse charge.
- reverse system along xy plane.
we consider the first step.
the define of electricity tenser:
$
P = chi E
$
reverse charge of the system, say we now have electricity tensor $ chi ' $ . We get:
$
- P = chi ' ( - E )
$
we get $ chi ' = chi $ , the first step does not change the electricity tensor, nor the "Raman tensor".
we consider the second step.
electricity tensor, it will become:
$
mat ( 1 , , ; , 1 , ; , , - 1 ) chi mat ( 1 , , ; , 1 , ; , , - 1 )
$
$ u $ , when it is along x or y direction, it will not change. When it is along z direction, it will become $ - u $ .
in conclusion, Raman tensor of C atom in A layer could be estimated from the Raman tensor of Si atom in A layer, by:
- for movement alone x and y direction, xz yz should be applied a negative sign.
- for movement alone z direction, xx xy yy zz should be applied a negative sign.
"Raman tensor" of C atom in C layer from C atom in A layer, in the same way.
consider the C atom in B1 layer.
it similar to the C atom in A layer, just like that for Si atom?
It turns out to be similar to the C atom in C layer.
summarize these stuff into @table-singleatom .
now, we only consider the "Raman tensor" caused by single atom or atoms move in the same amplitudes.
that is not the case in real phonon.
- In some A1 modes, only Si or C atom moves. If we take the magnitude of eigenvector as 1,
then amplitude of each atom is $ 1 / ( 4 sqrt ( m _ #text [ Si] ) ) $ or $ 1 / ( 4 sqrt ( m _ #text [ C] ) ) $ .
- In other cases, the amplitude of Si and C are in the ration of $ m _ #text [ C] : m _ #text [ Si] $ .
thus the amplitude of Si atom is $ 1 / 2 sqrt ( 1 / ( m _ #text [ Si] + m _ #text [ Si] ^ 2 / m _ #text [ C] ) ) $ , so do the C atom.
we list predicted modes and their Raman tensors, in @table-predmode .
- $ a $ : Raman tensor of Si atom in A layer, large value.
- $ epsilon $ : Difference of Raman tensors of Si atom in A and B1 layer, small value.
- $ eta $ : Difference of Raman tensors of C and Si atom in A layer, small value.
- $ zeta $ : Difference of Raman tensors of C atoms in A and B layer, small value.
#page ( flipped : true ) [ #figure ( {
table ( columns : 4 , align : center + horizon , inset : ( x : 3 pt , y : 5 pt ) ,
[ *Move Direction* ] , [ x] , [ y] , [ z] ,
[ Si A] , [ $ mat ( , a _ 2 , a _ 1 ; a _ 2 , , ; a _ 1 , , ; ) $ ] , [ $ mat ( a _ 2 , , ; , - a _ 2 , a _ 1 ; , a _ 1 , ; ) $ ] , [ $ mat ( a _ 5 , , ; , a _ 5 , ; , , a _ 6 ; ) $ ] ,
[ C A] , [ $ mat ( , a _ 2 + eta _ 2 , - a _ 1 - eta _ 1 ; a _ 2 + eta _ 2 , , ; - a _ 1 - eta _ 1 , , ; ) $ ] ,
[ $ mat ( a _ 2 + eta _ 2 , , ; , - a _ 2 - eta _ 2 , - a _ 1 - eta _ 1 ; , - a _ 1 - eta _ 1 , ; ) $ ] , [ $ mat ( - a _ 5 - eta _ 5 , , ; , - a _ 5 - eta _ 5 , ; , , - a _ 6 - eta _ 6 ; ) $ ] ,
[ Si B1] , [ $ mat ( , a _ 2 + epsilon _ 2 , a _ 1 + epsilon _ 1 ; a _ 2 + epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ; ) $ ] ,
[ $ mat ( a _ 2 + epsilon _ 2 , , ; , - a _ 2 - epsilon _ 2 , a _ 1 + epsilon _ 1 ; , a _ 1 + epsilon _ 1 , ; ) $ ] ,
[ $ mat ( a _ 5 + epsilon _ 5 , , ; , a _ 5 + epsilon _ 5 , ; , , a _ 6 + epsilon _ 6 ; ) $ ] ,
[ C, B1] , [ $ 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 , , ; ) $ ] ,
[ $ mat ( - a _ 2 - eta _ 2 - zeta _ 2 , , ; , a _ 2 + eta _ 2 + zeta _ 2 , - a _ 1 - eta _ 1 - zeta _ 1 ; , - a _ 1 - eta _ 1 - zeta _ 1 , ; ) $ ] ,
[ $ mat ( - a _ 5 - eta _ 5 - zeta _ 5 , , ; , - a _ 5 - eta _ 5 - zeta _ 5 , ; , , - a _ 6 - eta _ 6 - zeta _ 6 ; ) $ ] ,
[ Si C] , [ $ mat ( , - a _ 2 , a _ 1 ; - a _ 2 , , ; a _ 1 , , ; ) $ ] , [ $ mat ( - a _ 2 , , ; , a _ 2 , a _ 1 ; , a _ 1 , ; ) $ ] , [ $ mat ( a _ 5 , , ; , a _ 5 , ; , , a _ 6 ; ) $ ] ,
[ C, C] , [ $ mat ( , - a _ 2 - eta _ 2 , - a _ 1 - eta _ 1 ; - a _ 2 - eta _ 2 , , ; - a _ 1 - eta _ 1 , , ; ) $ ] ,
[ $ mat ( - a _ 2 - eta _ 2 , , ; , a _ 2 + eta _ 2 , - a _ 1 - eta _ 1 ; , - a _ 1 - eta _ 1 , ; ) $ ] , [ $ mat ( - a _ 5 - eta _ 5 , , ; , - a _ 5 - eta _ 5 , ; , , - a _ 6 - eta _ 6 ; ) $ ] ,
[ Si B2] , [ $ mat ( , - a _ 2 - epsilon _ 2 , a _ 1 + epsilon _ 1 ; - a _ 2 - epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ; ) $ ] ,
[ $ mat ( - a _ 2 - epsilon _ 2 , , ; , a _ 2 + epsilon _ 2 , a _ 1 + epsilon _ 1 ; , a _ 1 + epsilon _ 1 , ; ) $ ] ,
[ $ mat ( a _ 5 + epsilon _ 5 , , ; , a _ 5 + epsilon _ 5 , ; , , a _ 6 + epsilon _ 6 ; ) $ ] ,
[ C, B2] , [ $ 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 , , ; ) $ ] ,
[ $ mat ( a _ 2 + eta _ 2 + zeta _ 2 , , ; , - a _ 2 - eta _ 2 - zeta _ 2 , - a _ 1 - eta _ 1 - zeta _ 1 ; , - a _ 1 - eta _ 1 - zeta _ 1 , ; ) $ ] ,
[ $ mat ( - a _ 5 - eta _ 5 - zeta _ 5 , , ; , - a _ 5 - eta _ 5 - zeta _ 5 , ; , , - a _ 6 - eta _ 6 - zeta _ 6 ; ) $ ] ,
) } ,
caption : [ "Raman tensor" caused by single atom] ,
#figure (
table ( columns : 6 , align : center + horizon , inset : ( x : 3 pt , y : 5 pt ) ,
table . cell ( colspan : 3 ) [ *Vibration Direction* ] , [ x] , [ y] , [ z] ,
table . cell ( rowspan : 2 ) [ *Raman tensor #linebreak() of atoms* ] ,
[ B#sub [ 1] layer] , [ Si] ,
[ $ mat ( , a _ 2 + epsilon _ 2 , a _ 1 + epsilon _ 1 ; a _ 2 + epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ; ) $ ] ,
[ $ mat ( a _ 2 + epsilon _ 2 , , ; , - a _ 2 - epsilon _ 2 , a _ 1 + epsilon _ 1 ; , a _ 1 + epsilon _ 1 , ; ) $ ] ,
[ $ mat ( a _ 5 + epsilon _ 5 , , ; , a _ 5 + epsilon _ 5 , ; , , a _ 6 + epsilon _ 6 ; ) $ ] ,
[ B#sub [ 2] layer] , [ Si] ,
[ $ mat ( , - a _ 2 - epsilon _ 2 , a _ 1 + epsilon _ 1 ; - a _ 2 - epsilon _ 2 , , ; a _ 1 + epsilon _ 1 , , ; ) $ ] ,
[ $ mat ( - a _ 2 - epsilon _ 2 , , ; , a _ 2 + epsilon _ 2 , a _ 1 + epsilon _ 1 ; , a _ 1 + epsilon _ 1 , ; ) $ ] ,
[ $ mat ( a _ 5 + epsilon _ 5 , , ; , a _ 5 + epsilon _ 5 , ; , , a _ 6 + epsilon _ 6 ; ) $ ] ,
) ,
placement : none ,
) <table-singleatom> ]
)
=== Raman tensor of C atoms
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：
Raman tensors of C atoms in A and C layers can be obtained using a similar method:
$
mat ( , b _ 2 , b _ 1 ; b _ 2 , , ; b _ 1 , , ; ) , mat ( , - b _ 2 , b _ 1 ; - b _ 2 , , ; b _ 1 , , ; )
$
$ { b _ 1 , b _ 2 } $ 与 $ { a _ 1 , a _ 2 } $ 之间的关系。
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：
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。
relationship between $ \ { b _ 1 , b _ 2 \ } $ and $ \ { a _ 1 , a _ 2 \ } $ needs to be estimated.
environment of C atoms in A layer can be obtained from that of Si atoms in A layer
by first taking a mirror image along the basal plane,
then reversing the charge and adjusting the atomic mass and other factors,
as shown in @figure-sitoc .
change of Raman tensor during this process could be derived following these two steps.
#include "fig-sitoc.typ"
Si 原子拉曼张量为 $ alpha $ 和 $ alpha ' $ 。
$ E $ 作用下， 原子沿 x 方向振动导致的系统能量变化为：
Raman tensor of the Si atom in A layer before and after taking the mirror image
was denoted as $ alpha $ and $ alpha ' $ , respectively.
the mirror image,
the system energy change caused by the vibration of this atom along x direction
under an external electric field $ E $ is:
$ Delta E = E ^ T alpha E , # [ where ] alpha = mat ( , a _ 2 , a _ 1 ; a _ 2 , , ; a _ 1 , , ; ) $
the electric field is also flipped during the mirror image,
the total energy does not change, i.e.,
$ E ^ T alpha E = Delta E = ( sigma E ) ^ T alpha ' ( sigma E ) , # [ where ] sigma = mat ( 1 , 0 , 0 ; 0 , 1 , 0 ; 0 , 0 , - 1 ; ) $
$ alpha ' = mat ( , a _ 2 , - a _ 1 ; a _ 2 , , ; - a _ 1 , , ; ) $
， 原子拉曼张量为 $ alpha ' ' $ 。
// TODO: 能量写明是 Delta E
Raman tensor of Si atom after charge reversal was denoted as $ alpha ' ' $ .
energy does not change when the direction of the external electric field is also reversed, i.e.,
$ E ^ T alpha ' E = ( sigma ' E ) ^ T alpha ' ' ( sigma ' E ) , # [ where ] sigma ' = - 1 $
$ alpha ' ' = alpha ' $
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。
we can write out the Raman tensors of Si atoms virbrating along other directions
and the Raman tensors of C atoms,
and thus the Raman tensors of all phonon modes.
results are summarized in @table-singleatom and @table-predmode .
#include "table-singleatom.typ"
#include "table-predmode.typ"
