Developing phonon dispersion codefor simulated Raman spectrum byusing ﬁnite diﬀerence method
2015/12/21
Yung-Ting Lee
Korea Advanced InsFtute of Science and TechnologyandInsFtute for Solid State Physics
1Outline• IntroducFon• Formula of phonon dispersion and Ramanspectrum• Flowchart of phonon dispersion• Phonon dispersion Code• Example

2

PBE

PBE+U

ScienFﬁc Reports 3 (2013): 2999.

3Formula of phonon dispersion
Interatomic Force constants (IFCs)!!"#$ ! − !!

! ! !!"!
!!!"==−!!!" (!)!!!" (!′)!!!" (!′)(1)

The displacement of I-th atom along the direcFon α in Bravais laZce R

Dynamical matrix!!"#$ (!) =1!! !!

!!"#$!! − !

!
!!∙(!
!!)!

(2)

!,! !

Mass of J-th atom!! (!)!!" (!) = !!"#$ (!)!!" (!)(3)
4

Formula of Raman spectrum
The Raman intensity for a
Stokes process is describedby the Placzek’s expression:!∝!

The Bose-Einstein distribuFon(!! + 1)!
!
!! ∙ ! ∙ !!
!!

(4)!! = incident polarized vector of light!! = scaaered polarized vector of light

The frequency m of opFcal phonon

Raman tensor!
!!"= Ω!
!!"#! !!,!
!,!

the eigenvector m at atom Ialong k direcFon(5)

The electronic polarizaFon
The third ordersuscepFbility tensor!
!!!"! ! !!!"
1 ! ! !!,!
!!"#$ ===−!!!,! !!!,! !ℰ!
Ω !ℇ! !ℰ!

(6)
5

The third ordersuscepFbility tensor!
!!!"! ! !!!"
1 ! ! !!,!
!!"#$ ===−!!!,! !!!,! !ℰ!
Ω !ℇ! !ℰ!

(6)

The displacement of the I-th atom in the k direcFon

The electronic polarizaFon along the direcFon of electric ﬁeld!
!!"!
! = ! ! ! − ℰ ∙ (!!"# + !!" [!])

(7)

The ionic polarizaFon! [!]! ! !!"!
!!!!" [!]!!" ==!ℇ! !ℰ!
!ℰ!

(8)
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y =the direcFon of the propagaFon of
the incident light (ki)x =the direcFon of the polarizaFon of
the incident light (ei)z =the direcFon of the polarizaFon of
the scaaed light (es)-y =the direcFon of the propagaFon of
the scaaed light (ks)
ScienFﬁc Reports 3 (2013): 2999.

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Born eﬀecFve charge and dielectric
constant
Born eﬀecFve charge tensor:!
!"!!! !
!!
Ω!!
[!]
!
!,!
!"!
!
∗!!,!"===−! !!!,!
!ℰ! !!!,!
!ℰ!

(9)

The high-frequency dielectric constant:!
!!"= !!,! + 4!!!"(10)

The staFc dielectric constant:!!!
!!! (!")! = !
!!
!! (!")

Lyddane-Sachs-Teller relaFon(11)

s = electric ﬁeld polarizaFons parallel and perpendicular to the polar axis.

8Physical quanFFes and properFes• Force constants, phonon dispersion relaFon,phonon modes, PD-DOS.
• Raman tensor, suscepFbility tensor• Electronic polarizaFon (Pel)• Raman spectrum• Born eﬀecFve charge and dielectric constant9

The classical equaFon of moFon-1! ! !!! (!)!!" = !! !!" = !!
!" !

!!" = −!!!"!
!!!" (!)!!"#$ !! − ! !!" (!! )=−! ! !!" (!)
!!
=−
!" !

(12)

(13)

!!

!!"#$ !! − ! !!" (!! )

(14)

!!

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The classical equaFon of moFon-2!!" ! = !!" (!)! !!∙! ! !!"#

(15)!!!" (!)= −!"!!" (!)! !!∙! ! !!"#!"(16)! ! !!" (!)! ! (!)! !!∙! ! !!"#=−!
!"!" !

(17)! ! !!" (!)!!
=−!" !

!!"#$ !! − ! !!" (!! )

(14)

!!

Combining Eq. 14-17,!

!!"#$ !! − ! !!" (!)! !!∙! ! !!"#−!! !! !!" (!)! !!∙! ! !!"# = −!!

(18)
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The classical equaFon of moFon-3−!! !! !!" (!)! !!∙! ! !!"# = −!!"#$!! − !

!!"! !!"#
!!∙!
(!)!
!

(18)

!

(19)

!!

Removing signs and exp(-iwt),!! !! !!" (!)! !!∙! =Moving exp(iqR) to the right hand side,!!"#$ !! − ! !!" (!)! !!∙!
!!

(20)

!

!! !! !!" (!) =!!"#$ !! − ! !!" (!)! !!∙(! !!)!!

Reorganizing Eq. 19,!! !! !!" (!) =!!"#$!! − !

!
!!∙(!
!!) ! (!)
!
!"

(21)

!!

SubsFtuFng dynamical matrix for force constant matrix,!! !! !!" (!) = !!"#$ (!)!!" (!)

(22)

12

The classical equaFon of moFon-4
Assume!!" (!)! !!∙! ! !!"#!!" ! =!!

(23)! ! !!" (!) −!! !!" (!)! !!∙! ! !!"#=!" !
!!

!!∙! ! !!"#!
(!)!
!"−!! !!
=−!!

(24)

!

!!"#$ !! − !

!!! (!)! !!∙! ! !!"#(25)!!

!!

Removing signs and exp(-iwt),!!" (!)! !!∙!
!
!! !
=!!

!!"#$ !! − !
!!

!!" (!)! !!∙!
!!

!

(26)
13

The classical equaFon of moFon-5!! ! !

!!"(!)! !!∙!
!!

!! !! !!" (!)! !!∙! =!! !!" (!) =Assume!!"#$ (!) =LAPACK à zheev() subrou2ne!!"#$ !! − !

=!!

1
!!

!!" (!)! !!∙!

!

(27)

!!

!!"#$ !! − ! !!" (!)! !!∙!

!

(28)

!,! !

1!! !!
1!! !!

!

!!"#$ !! − ! ! !!∙(! !!) !!" (!)

(29)

!,! !
!

!!"#$ !! − ! ! !!∙(! !!)

(30)

!,! !

!! !!" = !!"#$ (!)!!"(31)
14Dynamical matrix
LaZce b from 1 to N

Original laZce constant(30)

Dynamical matrix[3xI+α][3xJ+β]

force constant matrix[a][b][3xI+α][3xJ+β]

phase factor

Atomic weight15

Force constant(1)

Center diﬀerence approxima2on(31)
Displacement
Atomic coordinate x at equilibrium state16

Bravais la?ce vectors
Reciprocal la?ce vectors(Eq. 31)(Eq. 33)(Eq. 32)

The primi=ve vectors of the reciprocal and Bravais vectors have the property(Eq. 34)

Dot product between reciprocal la?ce vectors and Bravais la?ce vectors(Eq. 35)(Eq. 36)17

Flowchart of phonon dispersion
Construct a supercell ( 3 X 3 X 1 ) of graphene
Displace atoms in the unit cell along 3 direcFons (x/y/z).
Calculate all of atomic forces at each iteraFon!!"#$ ! − !

!

! ! !!"!
!!!"==−!!!" (!)!!!" (!′)!!!" (!′)

Construct force constant matrix(s)Applying acousFc sum rule

Calculate real part and imagery part of dynamical matrix
with wave vectors q.

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Construct a supercell (3x3x1) ofGraphene
18 atoms in the supercell

054318762Index of originalprimiFve cell

19

Displace atomic coordinates with hx, -x, y, -y, z, -z
Total iteraFons = # of atoms X 6876hNumber of atoms
543

Index of iteraFon

forces[iter][noa][3]012x, y, z 20

Construct 9 force constant matrixs

Each force constant matrix is 6 X 6.

[K6X6]

21

Force constant matrix K6X6(a,b)
a=4

K6X6(4,6) K6X6(4,7) K6X6(4,8)

K6X6(4,3)
K6X6(4,4)

K6X6(4,5)

K6X6(4,0) K6X6(4,1) K6X6(4,2)
22ConstrucFng force constant matrix
FC [ # of atoms X 3 X size of supercell ] [ # of atoms X 3 X size of supercell ]For example:Supercell (3x3x1) à 3 X 3 X 1 = 9Dimension of FC matrix = [ 2 X 3 X 9 ] [ 2 X 3 X 9 ] = [ 54 ] [ 54 ]First derivaFve of force by center diﬀerence approximaFon
noa3 = # of atom X 3, numberofatom = # of atom, disp2 = displacement23

Sum rules

Rule 1

Rule 2Rule 324

AcousFc sum rule (ASR)

Both oﬀ-diagonal terms are averaged.

k=αo=β25

Results with ASR

Unit = cm-1

For example 1: Graphene (1x1x1), cutoﬀ 150 Ry, Gamma point
No rules applied
Rule 1

Rule 1 and rule 2

For example 2: Graphene (3x3x1), cutoﬀ 150 Ry, Gamma point
Rule 1 and rule 2

For example 3: Graphene (5x5x1), cutoﬀ 150 Ry, Gamma point
Rule 1 and rule 2
26

Calculate dynamical matrix
i=Ix3+αj=Jx3+β!!"#$ (!) =1!

!! !!

!!"#$ !! − ! ! !!∙(! !!)!,! !

// (2)

Sqrt ( I-th mass X J-th mass)

27OpFmizaFon of graphene
2 carbon atoms
a1= a2= 2.46 angstrom
a3= 15.00 angstrom

Energy Cutoﬀ = 150 Ry
K-points = 20 X 20 X 1
SCF Criterion = 10-9 hartree
MD type = BFGS
MD.Opt.Criterion = 10-7 hartree/Bohra2

a1= 2.46 angstrom
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Input ﬁle29

Supercell (3x3x1)

Supercell (5x5x1)

Supercell (7x7x1)

Displacement =0.05 Ang.

30

References• Scien6ﬁc Reports 3 (2013): 2999.
• Phys. Rev. B 55 (1997) 10355.
• J. Phys: Condens. MaCer 9 (1997) 7861.
• J. Phys: Condens. MaCer 25 (2013) 144201.
• N. W. Ashroo, N. D. Mermin, Solid statephysics
Thanks for your aaenFon.
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