227 lines
10 KiB
Typst
227 lines
10 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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The center principle is to assign the Raman tensor (i.e., change of polarizability caused by atomic displacement)
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to each atom in the unit cell.
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This including the following steps:
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- Write out the change of polarizability caused by displacement of Si atom in A and C layer,
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Where unknown non-zero components are denoted by $a_1$, $a_2$, $a_5$, $a_6$.
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For example, when we move the Si atom in A layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2,a_1;a_2,,;a_1,,)d$.
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This could be done by conclusion above.
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- The Si atom in B layer have similar local environment as the A and C layer, with only a little difference.
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We denote these difference by $epsilon_1$, $epsilon_2$, $epsilon_5$, $epsilon_6$,
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and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
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For example, when we move the Si atom in B layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,)d$.
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- The local environment of C atom in A layer is similar to the Si atom in A layer with charge reversed and
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the system reversed along xy plane.
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We denote these difference by $eta_1$, $eta_2$, $eta_5$, $eta_6$,
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and the absolute value of $epsilon_i$ should be much smaller than $a_i$.
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For example, when we move the C atom in A layer slightly towards the x+ direction in $d$ distance,
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the change of polarizability should be $mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,)d$.
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- Similar to the case in Si atoms, we derive the change of polarizability
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caused by moving C atom in B layer slightly towards the x+ direction in $d$ distance,
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which should be $mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,)d$.
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Lets assign Raman tensor onto each atom.
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That is, Raman tensor is derivative of the polarizability with respect to the atomic displacement:
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$
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alpha = pdv(chi, u)
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$
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where $u$ should be the displacement of the atom corresponding to a phonon mode.
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But, even when $u$ is *NOT* the displacement of a phonon
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(for example, lets only slightly move Si atom in A layer, keeping other atoms fixed),
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the (high-frequency) polarizability is still well-defined,
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and the will still cause a change in the polarizability.
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Even more, the group representation theory is still applicable in this condition:
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the only thing that matters is, when applying $g$ to the system,
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the tensor transformed into $g^(-1) alpha g$ or $g alpha g^(-1)$,
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no matter $alpha$ is Raman tensor or something else, or it is related to a phonon or not.
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Thus, we can, in principle, "assign" Raman tensor of a phonon, to each atom.
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This "assign" is unique since both the atom movement and all phonons have 24 dimensions.
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Next, we consider what these single-atom-caused "Raman tensors" looks like.
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For example, what happens if we move the Si atom in A layer slightly along the x+ direction?
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Consider also move the Si atom in C layer slightly, along x+ or x- direction.
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How about the Raman tensor caused by the both two atoms?
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In first case, this is B2 representation in E1 representation. Thus the Raman tensor should be something like:
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$
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mat(,,2a_1;,,;2a_1,,;)
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$
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In the second case, it is A2 in E2. It turns out:
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$
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mat(,2a_2,;2a_2,,;,,;)
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$
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The average of these two tensors should be the s"Raman tensor" cause by move only the Si atom in A layer,
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slightly towards x+ direction.
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$
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mat(,a_2,a_1;a_2,,;a_1,,;)
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$
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The difference should be the "Raman tensor" of the second atom.
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$
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mat(,-a_2,a_1;-a_2,,;a_1,,;)
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$
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// This approach applied relied on the fact that, all Si atom in 4H-SiC is "distinguishable" by the symmetry operations.
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// I mean, what will happen if we have two Si atoms in A layer?
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// Apparently, we could not extract the "Raman tensor" of only one of the two atoms.
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// This is the case for the 6H-SiC.
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// Hence, we will provide a more general approach to estimate the "Raman tensor" of a single atom.
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Consider the Si atom in the B1 layer.
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It lives in an environment quite similar to the A layer.
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Thus, the "Raman tensor" caused by it should be similar to the one caused by the A layer:
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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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$
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Similar to the Si atom in B2 layer:
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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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$
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Same approach applied for Si atom vibrate in y direction.
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When we move both Si atoms in A and C layer in y+ direction,
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it is B1 in E1, thus the "Raman tensor" should be:
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$
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mat(,,;,,2a_3;,2a_3,;)
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$
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And if we move Si in A layer towards y+ but Si in C layer towards y-,
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it is A2 in E2:
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$
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mat(2a_4,,;,-2a_4,;,,;)
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$
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Thus we get the "Raman tensor" of Si atom in A layer sololy move towards y+ direction:
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$
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mat(a_4,,;,-a_4,a_3;,a_3,;)
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$
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and the "Raman tensor" of Si atom in C layer towards y+ direction:
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$
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mat(-a_4,,;,a_4,a_3;,a_3,;)
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$
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Same applied for the Si atom in B layer:
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$
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mat(a_4+epsilon_4,,;,-a_4-epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
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$
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$
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mat(-a_4-epsilon_4,,;,a_4+epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
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$
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Before consider z-direction, it is important to note that, $a_1$ $a_2$ $a_3$ $a_4$ are not independent.
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Consider vibration along x+ direction (lets say the distance is $d$).
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System energy caused by external electric field and vibration is:
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$
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E^T (mat(,,2a_1;,,;2a_1,,) d) E
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$
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Apply C#sub[3] to atom vibration and external field, energy should not change. We got:
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$
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(mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)^T ( mat(,,2a_1;,,;2a_1,,)(-1/2 d) + mat(,,;,,2a_3;,2a_3,)(sqrt(3)/2 d) )
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(mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)
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$
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It is equal to:
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$
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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
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$
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Thus:
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$
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1/2 a_1 + 3/2 a_3 = 2a_1 #linebreak()
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sqrt(3)/2 a_1 - sqrt(3)/2 a_3 = 0
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$
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Thus $a_1 = a_3$.
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Apply the same method, we get $abs(a_2) = abs(a_4)$.
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Since we have not define the sign of $a_4$, we could take $a_2 = a_4$.
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Same for $epsilon$.
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Now consider what if we move the Si atom in A layer along z+ direction.
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If we move the Si atom in C layer along z+ direction, it is A1:
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$
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mat(2a_5,,;,2a_5,;,,2a_6;)
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$
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If we move the Si atom in C layer along z- direction, it is B1:
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$
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0
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$
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Thus we get the "Raman tensor" of Si atom in A or C layer towards z+ direction:
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$
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mat(a_5,,;,a_5,;,,a_6;)
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$
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Lets consider the C atom in A layer.
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It should be somehow similar to the Si atom in A layer, but with a negative sign in some places,
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and then add or subtract some little value.
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Actually, the "transformation" of Si atom in A layer to C atom in A layer applied in the following steps:
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- reverse charge.
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- reverse system along xy plane.
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First we consider the first step.
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Taking the define of electricity tenser:
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$
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P = chi E
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$
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Lets reverse charge of the system, say we now have electricity tensor $chi'$. We get:
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$
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-P = chi'(-E)
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$
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Thus we get $chi' = chi$, the first step does not change the electricity tensor, nor the "Raman tensor".
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Now we consider the second step.
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For electricity tensor, it will become:
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$
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mat(1,,;,1,;,,-1) chi mat(1,,;,1,;,,-1)
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$
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For $u$, when it is along x or y direction, it will not change. When it is along z direction, it will become $-u$.
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So in conclusion, Raman tensor of C atom in A layer could be estimated from the Raman tensor of Si atom in A layer, by:
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- for movement alone x and y direction, xz yz should be applied a negative sign.
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- for movement alone z direction, xx xy yy zz should be applied a negative sign.
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Export "Raman tensor" of C atom in C layer from C atom in A layer, in the same way.
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Now consider the C atom in B1 layer.
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Is it similar to the C atom in A layer, just like that for Si atom?
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No. It turns out to be similar to the C atom in C layer.
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We summarize these stuff into @table-singleatom.
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Until now, we only consider the "Raman tensor" caused by single atom or atoms move in the same amplitudes.
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However, that is not the case in real phonon.
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- In some A1 modes, only Si or C atom moves. If we take the magnitude of eigenvector as 1,
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then amplitude of each atom is $1/(4sqrt(m_#text[Si]))$ or $1/(4sqrt(m_#text[C]))$.
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- In other cases, the amplitude of Si and C are in the ration of $m_#text[C] : m_#text[Si]$.
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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.
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Furthermore, we list predicted modes and their Raman tensors, in @table-predmode.
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- $a$: Raman tensor of Si atom in A layer, large value.
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- $epsilon$: Difference of Raman tensors of Si atom in A and B1 layer, small value.
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- $eta$: Difference of Raman tensors of C and Si atom in A layer, small value.
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- $zeta$: Difference of Raman tensors of C atoms in A and B layer, small value.
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#page(flipped: true)[#figure({
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table(columns: 4, align: center + horizon, inset: (x: 3pt, y: 5pt),
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[*Move Direction*], [x], [y], [z],
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[Si A], [$mat(,a_2,a_1;a_2,,;a_1,,;)$], [$mat(a_2,,;,-a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
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[C A], [$mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,;)$],
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[$mat(a_2+eta_2,,;,-a_2-eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
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[Si B1], [$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$],
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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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[C, B1], [$mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
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[$mat(-a_2-eta_2-zeta_2,,;,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
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[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
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[Si C], [$mat(,-a_2,a_1;-a_2,,;a_1,,;)$], [$mat(-a_2,,;,a_2,a_1;,a_1,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
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[C, C], [$mat(,-a_2-eta_2,-a_1-eta_1;-a_2-eta_2,,;-a_1-eta_1,,;)$],
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[$mat(-a_2-eta_2,,;,a_2+eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
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[Si B2], [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$],
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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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[C, B2], [$mat(,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;a_2+eta_2+zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
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[$mat(a_2+eta_2+zeta_2,,;,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
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[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
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)},
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caption: ["Raman tensor" caused by single atom],
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placement: none,
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)<table-singleatom>]
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