Vibrational Normal Modes or “Phonon” Dispersion Relations in Crystalline Materials
Two efficient algorithms for drawing accurate and beautiful phonon dispersion
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Transcript of Two efficient algorithms for drawing accurate and beautiful phonon dispersion
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Takeshi Nishimatsu
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Dispersion and DOS of phonon for 7LiF and 6LiF Beautiful!!!
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} 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
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} 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
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Phonon Hamiltonian
Inter-atomic interaction
α, β = x, y, z
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Fourier transform (FT) of phonon Hamiltonian
q-local harmonic-oscillator-like Hamiltonian
inter-atomic force constant (IFC) matrix
Dynamical matrix
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Interpolation in reciprocal space for normal modes of phonon of ionic crystals
X. Gonze and C. Lee, Phys. Rev. B 55,
10355 (1997). ABINIT
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LO-TO splitting as a non-analytic term
9
I, J
3x3 submatrices in 3Nx3N matrix
dyadic productdepends on from which direction q → 0
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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
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} [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
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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
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Implementation of band connectivity } Use maxloc(ary,mask) of Fortran 90 with mask
|<i,qnew|j,qlast>|
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Dispersion Gallery
14
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PbTiO3: No Good
q
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PbTiO3: Good
q
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} Rock salt structure (fcc)} Lattice constant a = 4.026 Å } Calculated a = 4.076 Å
LiF crystal
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Dispersion and DOS of phonon for 7LiF and 6LiF
q
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Put LO-TO splitting by hands
q
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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.
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Simple cubic dipole array, which has only optical phonon
q
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non-analytic term at q=0 forZ*-dipole simple cubic crystal
Only LO has this element
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Introduce LO-TO splitting by hand
q
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} space group: P42/mnm (No.136)
} tetragonal a = b = 4.59373 Å c = 2.95812 Å
TiO2 in rutile structure
Ti
O
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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
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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
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} 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