This commit is contained in:
@@ -1,6 +1,7 @@
|
||||
#import "@preview/starter-journal-article:0.4.0": article, author-meta
|
||||
#import "@preview/tablem:0.2.0": tablem
|
||||
#import "@preview/physica:0.9.4": pdv
|
||||
#import "@preview/physica:0.9.4": pdv, super-T-as-transpose
|
||||
#show: super-T-as-transpose
|
||||
|
||||
#set par.line(numbering: "1")
|
||||
// TODO: fix indent of first line
|
||||
@@ -303,6 +304,31 @@ $
|
||||
mat(-a_4-epsilon_4,,;,a_4+epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)
|
||||
$
|
||||
|
||||
Before consider z-direction, it is important to note that, $a_1$ $a_2$ $a_3$ $a_4$ are not independent.
|
||||
Consider vibration along x+ direction (lets say the distance is $d$).
|
||||
System energy caused by external electric field and vibration is:
|
||||
$
|
||||
E^T (mat(,,2a_1;,,;2a_1,,) d) E
|
||||
$
|
||||
Apply 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(,,2a_1;,,;2a_1,,)(-1/2 d) + mat(,,;,,2a_3;,2a_3,)(sqrt(3)/2 d) )
|
||||
(mat(-1/2,-sqrt(3)/2,;sqrt(3)/2,-1/2,;,,1)E)
|
||||
$
|
||||
It 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
|
||||
$
|
||||
Thus:
|
||||
$
|
||||
1/2 a_1 + 3/2 a_3 = 2a_1 #linebreak()
|
||||
sqrt(3)/2 a_1 - sqrt(3)/2 a_3 = 0
|
||||
$
|
||||
Thus $a_1 = a_3$.
|
||||
Apply the same method, we get $abs(a_2) = abs(a_4)$.
|
||||
Since we have not define the sign of $a_4$, we could take $a_2 = a_4$.
|
||||
Same for $epsilon$.
|
||||
|
||||
Now consider what if we move the Si atom in A layer along z+ direction.
|
||||
If we move the Si atom in C layer along z+ direction, it is A1:
|
||||
$
|
||||
@@ -359,22 +385,22 @@ Frequency could be estimated by, how many atoms are moving towards its neighbor.
|
||||
table(columns: 4, align: center + horizon, inset: (x: 3pt, y: 5pt),
|
||||
[*Move Direction*], [x], [y], [z],
|
||||
[C A], [$mat(,a_2+eta_2,-a_1-eta_1;a_2+eta_2,,;-a_1-eta_1,,;)$],
|
||||
[$mat(a_4+eta_4,,;,-a_4-eta_4,-a_3-eta_3;,-a_3-eta_3,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
|
||||
[Si A], [$mat(,a_2,a_1;a_2,,;a_1,,;)$], [$mat(a_4,,;,-a_4,a_3;,a_3,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
|
||||
[$mat(a_2+eta_2,,;,-a_2-eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
|
||||
[Si 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, B1], [$mat(,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;-a_2-eta_2-zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
|
||||
[$mat(-a_4-eta_4-zeta_4,,;,a_4+eta_4+zeta_4,-a_3-eta_3-zeta_3;,-a_3-eta_3-zeta_3,;)$],
|
||||
[$mat(-a_2-eta_2-zeta_2,,;,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
|
||||
[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
|
||||
[Si B1], [$mat(,a_2+epsilon_2,a_1+epsilon_1;a_2+epsilon_2,,;a_1+epsilon_1,,;)$],
|
||||
[$mat(a_4+epsilon_4,,;,-a_4-epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)$],
|
||||
[$mat(a_2+epsilon_2,,;,-a_2-epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
|
||||
[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
|
||||
[C, C], [$mat(,-a_2-eta_2,-a_1-eta_1;-a_2-eta_2,,;-a_1-eta_1,,;)$],
|
||||
[$mat(-a_4-eta_4,,;,a_4+eta_4,-a_3-eta_3;,-a_3-eta_3,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
|
||||
[Si C], [$mat(,-a_2,a_1;-a_2,,;a_1,,;)$], [$mat(-a_4,,;,a_4,a_3;,a_3,;)$], [$mat(a_5,,;,a_5,;,,a_6;)$],
|
||||
[$mat(-a_2-eta_2,,;,a_2+eta_2,-a_1-eta_1;,-a_1-eta_1,;)$], [$mat(-a_5-eta_5,,;,-a_5-eta_5,;,,-a_6-eta_6;)$],
|
||||
[Si 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, B2], [$mat(,a_2+eta_2+zeta_2,-a_1-eta_1-zeta_1;a_2+eta_2+zeta_2,,;-a_1-eta_1-zeta_1,,;)$],
|
||||
[$mat(a_4+eta_4+zeta_4,,;,-a_4-eta_4-zeta_4,-a_3-eta_3-zeta_3;,-a_3-eta_3-zeta_3,;)$],
|
||||
[$mat(a_2+eta_2+zeta_2,,;,-a_2-eta_2-zeta_2,-a_1-eta_1-zeta_1;,-a_1-eta_1-zeta_1,;)$],
|
||||
[$mat(-a_5-eta_5-zeta_5,,;,-a_5-eta_5-zeta_5,;,,-a_6-eta_6-zeta_6;)$],
|
||||
[Si B2], [$mat(,-a_2-epsilon_2,a_1+epsilon_1;-a_2-epsilon_2,,;a_1+epsilon_1,,;)$],
|
||||
[$mat(-a_4-epsilon_4,,;,a_4+epsilon_4,a_3+epsilon_3;,a_3+epsilon_3,;)$],
|
||||
[$mat(-a_2-epsilon_2,,;,a_2+epsilon_2,a_1+epsilon_1;,a_1+epsilon_1,;)$],
|
||||
[$mat(a_5+epsilon_5,,;,a_5+epsilon_5,;,,a_6+epsilon_6;)$],
|
||||
)},
|
||||
caption: ["Raman tensor" caused by single atom],
|
||||
@@ -382,26 +408,37 @@ Frequency could be estimated by, how many atoms are moving towards its neighbor.
|
||||
)<table-singleatom>]
|
||||
|
||||
// Raman Tensor for A1: line1 xz/yz; 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
|
||||
// Relative Vibration Direction: col1 C ABCB col2 Si ABCB
|
||||
#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);
|
||||
table(columns: 4, align: center + horizon, inset: (x: 3pt, y: 5pt),
|
||||
[*Representation in C#sub[6v]*], m3[A1],
|
||||
[*Representation in C#sub[2v]*], m3[A1],
|
||||
[*x*], m2[0.5], [1],
|
||||
[*Relative Vibration Direction*], [$++--++--$], [$+--++--+$], [$+-+-+-+-$],
|
||||
[*Vibration Direction*], m3[z],
|
||||
[*Raman Tensor Predicted*], [$2(-epsilon_5+zeta_5)$ #linebreak() $2(-epsilon_6+zeta_6)$],
|
||||
let m4(content) = table.cell(colspan: 4, content);
|
||||
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]],
|
||||
[*x*], m2[0.5], [1], m2[0.5], [1], m2[0.25], m2[0.75],
|
||||
[*Relative Vibration Direction*],
|
||||
[$++\ --\ ++\ --$], [$+-\ -+\ +-\ -+$], [$+-\ +-\ +-\ +-$],
|
||||
[$++\ --\ ++\ --$], [$+-\ -+\ +-\ -+$], [$+-\ +-\ +-\ +-$],
|
||||
[$++\ +-\ --\ -+$], [$++\ --\ --\ ++$], [$++\ -+\ --\ +-$], [$+-\ ++\ -+\ --$],
|
||||
[*Vibration Direction*], m3[z], m3[x/y], m4[x/y],
|
||||
[*Raman Tensor Predicted*], [$2(zeta_5-epsilon_5)$ #linebreak() $2(zeta_6-epsilon_6)$],
|
||||
[$2(epsilon_5+zeta_5)$ #linebreak() $2(epsilon_6+zeta_6)$],
|
||||
[$-4(2a_5+eta_5+epsilon_5+zeta_5)$ #linebreak() $-4(2a_6+eta_6+epsilon_6+zeta_6)$],
|
||||
[*Raman Intensity Predicted*], m2[weak], [strong],
|
||||
[$2(zeta_1-epsilon_1)$], [$2(epsilon_1+zeta_1)$], [$-4(2a_1+eta_1+epsilon_1+zeta_1)$],
|
||||
[$-2(zeta_2+epsilon_2)$], [$2(2eta_2+zeta_2-epsilon_2)$], [$2(4a_2+2eta_2+zeta_2+epsilon_2)$],
|
||||
[$2(epsilon_2-zeta_2)$],
|
||||
[*Raman Intensity Predicted*], m2[weak], [strong], m2[weak], [strong], m2[weak], [strong], [weak],
|
||||
[*Raman Tensor Calculated*],
|
||||
[0.10 #linebreak() -1.33], [-1.68 #linebreak() 1.34], [7.68 #linebreak() -21.65],
|
||||
[*Atom-pair that Move Relatively In-plane*], [4], [0], [4],
|
||||
[*Atom-pair that Move Relatively Out-plane*], [0], [4], [4],
|
||||
[*Predicted Frequency*], [low], [medium], [high],
|
||||
[*Calculated Frequency*], [591.90], [812.87], [933.80],
|
||||
[-1.56], [-0.30], [7.32], [1.06], [0.41], [9.41], [0.17],
|
||||
[*Atom-pair that Move Relatively In-plane*], [4], [0], [4], [4], [0], [4], [0], [2], [4], [4],
|
||||
[*Atom-pair that Move Relatively Out-plane*], [0], [4], [4], [0], [4], [4], [2], [0], [2], [2],
|
||||
[*Predicted Frequency*], [low], [medium], [high], [medium], [low], [high], [low], [medium], m2[high],
|
||||
[*Calculated Frequency*],
|
||||
[591.90], [812.87], [933.80], [746.91], [257.35], [776.57], [197.84], [190.51], [756.25], [764.33]
|
||||
)},
|
||||
caption: [Predicted modes and their "Raman tensor"],
|
||||
placement: none,
|
||||
|
||||
Reference in New Issue
Block a user