Purpose: easy drawing of accurate and beautiful phonon dispersion in first-principles calculations See also: Draw phonon dispersion of Si with Quantum Espresso https://gist.github.com/t-nissie/32c10a148a7fc054b836

1. Takeshi Nishimatsu

3. Dispersion and DOS of phonon for 7LiF
and 6LiF Beautiful!!!

4. } Calculation of inter-atomic force constant (IFC)
matrix is time consuming. ---> [1] interpolation
} Band connectivity requires group theory
(compatibility relation) such as [Yépez, Calles and
Castro: A simple algorithm for the group
theoretical classification of quantum states,
Applied Mathematics and Computation 133,
119-130 (2002)]. ---> [2] band connectivity
without group theory
} Easy programming with Fortran 90 and GNUPLOT
Purpose: easy drawing of accurate and
beautiful phonon dispersion

5. } With DFPT, calculate inter-atomic force constant (IFC)
matirx in sparse q-point mesh (4x4x4 for example),
effective charges tensor Z*Iαβ for each ion, optical
dielectric constant tensor ε∞
αβ, then interpolation [X. Gonze
and C. Lee: Dynamical matrices, Born effective charges,
dielectric permittivity tensors and interatomic force
constants, Phys. Rev. B 55, 10355 (1997)]
} Efficient especially for ionic crystals
} Implemented in http://www.ABINIT.org/ by Xavier Gonze
} Implemented in http://loto.sourceforge.net/loto/ by
Takeshi Nishimatsu
Efficient algorithm 1: interpolation in
reciprocal space

6. Phonon Hamiltonian
Inter-atomic interaction
α, β = x, y, z

7. Fourier transform (FT) of phonon
Hamiltonian
q-local
harmonic-oscillator-like
Hamiltonian
inter-atomic force
constant (IFC) matrix
Dynamical matrix

8. Interpolation in reciprocal space for normal
modes of phonon of ionic crystals
X. Gonze and C. Lee,
Phys. Rev. B 55,
10355 (1997).
ABINIT

9. LO-TO splitting as a non-analytic term
9
I, J
3x3 submatrices in 3Nx3N matrix
dyadic product
depends on from which direction q → 0

10. Example of LO-TO splitting as a non-
analytic term, two ±Z* ions in cubic cell
Direction of q → ●→○→ acoustic phonon (parallel translation)
●→←○ optical phonon

11. } [Oleg V. Yazyev, Konstantin N. Kudin, and Gustavo
E. Scuseria: Efficient algorithm for band
connectivity resolution, Phys. Rev. B 65, 205117
(2002)] for electronic band structure.
} For phonon, implemented in
} ABINT http://www.ABINIT.org/ by Takeshi Nishimatsu
} Quantum Espresso http://www.quantum-espresso.org/
by Takeshi Nishimatsu
Efficient algorithm 2: band connectivity with
similarity of eigenvectors

12. q
Energy
1
2
3
2
1
3
?
S =|<i,q | j,q >|
absolute values of
overlap of eigenvectors:
0.21 0.70 0.01
0.61 0.10 0.10
0.07 0.01 0.78
j i
q
Energy
1
2
3
2
1
3
j i
q qlast new
new lastij
q qlast new
Algorithm for band connectivity
Similarity between j and i is the largest → Connect them

13. Implementation of band connectivity
} Use maxloc(ary,mask) of Fortran 90 with mask
|<i,qnew|j,qlast>|

14. Dispersion
Gallery
14

15. PbTiO3: No Good
q

16. PbTiO3: Good
q

17. } Rock salt structure (fcc)
} Lattice constant a = 4.026 Å
} Calculated a = 4.076 Å
LiF crystal

18. Dispersion and DOS of phonon for 7LiF
and 6LiF
q

19. Put LO-TO splitting by hands
q

20. z uz
Pz
Ed=-4!Pz
誘電体薄膜の縦波振動．分極の薄膜に垂直な成分Pzは薄膜に±Pzの表面電荷を生じさせ
る．表面電荷は薄膜内にEd=-4πPzの反分極場を作る．反分極場によりPzには復元力が働く
ので「振動」する.

21. Simple cubic dipole array, which has only
optical phonon
q

22. non-analytic term at q=0 for
Z*-dipole simple cubic crystal
Only LO has this element

23. Introduce LO-TO splitting by hand
q

24. } space group:
P42/mnm (No.136)
} tetragonal
a = b = 4.59373 Å
c = 2.95812 Å
TiO2 in rutile structure
Ti
O

25. 0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
A M Γ X M Γ Z R A Z
phononenergy[Hartree]
q
rutile.20.ecut60.k8x8x8.TeterLDA.phonon-SR11000
V Σ ∆ Y Σ Λ U T S
Phonon Dispersion for rutile TiO2
discontinuous at the Γ point

26. 2
2.2
2.4
2.6
2.8
3
3.2
3.4
1 1.5 2 2.5 3 3.5 4 4.5 5
’foo.dat’
} Data file for GNUPLOT.
If there is a space line,
GNUPLOT does not connect
data with a line.
Tips: Using GNUPLOT, You can easily plot
discontinuous functions

27. } Two efficient algorithms for drawing accurate and
beautiful phonon dispersion
◦ Interpolation in reciprocal space
◦ Band connectivity from similarity of eigenvectors
} They are implemented in http://www.ABINIT.org/
and http://www.quantum-espresso.org/
} PbTiO3, LiF
} Dipolar crystal
} Ionic crystals have LO-TO splitting
} Easy programming with Fortran 90 and GNUPLOT
Summary