Takeshi Nishimatsu

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

}  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

}  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

Phonon Hamiltonian

Inter-atomic interaction

α, β = x, y, z

Fourier transform (FT) of phonon Hamiltonian

q-local harmonic-oscillator-like Hamiltonian

inter-atomic force constant (IFC) matrix

Dynamical matrix

Interpolation in reciprocal space for normal modes of phonon of ionic crystals

X. Gonze and C. Lee, Phys. Rev. B 55,

10355 (1997). ABINIT

LO-TO splitting as a non-analytic term

9

I, J

3x3 submatrices in 3Nx3N matrix

dyadic productdepends on from which direction q → 0

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

}  [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

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

Implementation of band connectivity }  Use maxloc(ary,mask) of Fortran 90 with mask

|<i,qnew|j,qlast>|

Dispersion Gallery

14

PbTiO3: No Good

q

PbTiO3: Good

q

}  Rock salt structure (fcc)}  Lattice constant a = 4.026 Å }  Calculated a = 4.076 Å

LiF crystal

Dispersion and DOS of phonon for 7LiF and 6LiF

q

Put LO-TO splitting by hands

q

zuz

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.

Simple cubic dipole array, which has only optical phonon

q

non-analytic term at q=0 forZ*-dipole simple cubic crystal

Only LO has this element

Introduce LO-TO splitting by hand

q

}  space group: P42/mnm (No.136)

}  tetragonal a = b = 4.59373 Å c = 2.95812 Å

TiO2 in rutile structure

Ti

O

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

phonon e

ner

gy [

Har

tree

]

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

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 connectdata with a line.

Tips: Using GNUPLOT, You can easily plot discontinuous functions

}  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