|
|
(One intermediate revision by one other user not shown) |
Line 121: |
Line 121: |
| e^\frac{-\mathbf{G} \cdot \epsilon^\infty \cdot \mathbf{G}}{4\lambda^2} | | e^\frac{-\mathbf{G} \cdot \epsilon^\infty \cdot \mathbf{G}}{4\lambda^2} |
| </math> | | </math> |
| with <math display="inline">\epsilon^\infty</math> the clamped ion dielectric tensor, <math display="inline">\mathbf{Z}^*_{I\alpha}</math> the Born effective charges, <math display="inline">\alpha</math> the Ewald parameter which is chosen such that the contributions from <math display="inline">e^\frac{-\mathbf{G} \cdot \epsilon^\infty \cdot \mathbf{G}}{4\lambda^2}</math> are negligible within a certain <math display="inline">\mathbf{G}</math> vector cutoff sphere {{TAG|PHON_G_CUTOFF}}. | | with <math display="inline">\epsilon^\infty</math> the ion-clamped dielectric tensor, <math display="inline">\mathbf{Z}^*_{I\alpha}</math> the Born effective charges, <math display="inline">\alpha</math> the Ewald parameter which is chosen such that the contributions from <math display="inline">e^\frac{-\mathbf{G} \cdot \epsilon^\infty \cdot \mathbf{G}}{4\lambda^2}</math> are negligible within a certain <math display="inline">\mathbf{G}</math> vector cutoff sphere {{TAG|PHON_G_CUTOFF}}. |
|
| |
|
| This also allows us to separate the dynamical matrix into short and long-range parts | | This also allows us to separate the dynamical matrix into short and long-range parts |
Line 281: |
Line 281: |
| The results should be equivalent to the ones obtained using {{TAG|LCALCEPS}} and {{TAG|LEPSILON}}. | | The results should be equivalent to the ones obtained using {{TAG|LCALCEPS}} and {{TAG|LEPSILON}}. |
|
| |
|
| == Related Tags and Sections == | | == Related tags and sections == |
| {{TAG|IBRION}}, | | {{TAG|IBRION}}, |
| [[Phonons from finite differences]], | | [[Phonons from finite differences]], |
Phonons are the collective excitation of nuclei in an extended periodic system.
Taylor expansion in ionic displacements
To compute the phonon modes and frequencies we start by Taylor expanding the total energy () around the set of equilibrium positions of the nuclei ()
where the positions of the nuclei.
The first derivative of the total energy with respect to the nuclei corresponds to the forces
- ,
and the second derivative to the second-order force-constants
Changing variables in the Taylor expansion of the total energy with that corresponds to the displacement of the atoms with respect to their equilibrium position leads to
Dynamical matrix and phonon modes
If we take the system to be in equilibrium, the forces on the atoms are zero and then the Hamiltonian of the system is
with the mass of the -th nucleus.
The equation of motion is then given by
We then look for solutions in the form of plane waves traveling parallel to the wave vector , i.e.
where are the phonon mode eigenvectors and
the amplitudes.
Replacing it in the equation of motion we obtain the following eigenvalue problem
with
the dynamical matrix in the harmonic approximation.
Now by solving the eigenvalue problem above we can obtain the phonon modes
and frequencies
at any arbitrary q point.
We can write the positions of the atoms in the supercell in terms of integer multiples of the lattice vectors of the unit cell such that
with being the position of the ion in the unit cell.
The force constants then become .
The dynamical matrix is then given by
with chosen using the minimal image convention.
This allows us to compute the phonons in the unit cell using the following equation
with the dynamical matrix having dimension with the number of atoms in the unit cell.
Long-range interatomic force constants (LO-TO splitting)
For semiconductors or insulators, the electronic screening of the ions is incomplete which leads to long-range (LR) interatomic force constants. To compute them explicitly would require infinitely large supercell calculations. For practical calculations, a finite size truncation is needed which leads to Gibbs oscillations in the phonon dispersion. Fortunately, this long-range behavior can be modeled by looking at the analytic form of the ion-ion contribution to the total energy.
For that, follow the approach outlined in Ref. [1] and start by splitting the second-order force constants into short-range and long-range parts,
with the long-range part being obtained from the analytic derivative of the long-range part of the ion-ion contribution to the total energy . This contribution is typically evaluated using an Ewald sum technique in which we separate the ion-ion contribution to the total energy into two part, one is evaluated in real space and captures the short-range part and the other one in reciprocal space which captures the long-range part . The separation is governed by an Ewald parameter which represents a truncation length.
This leads to the following analytical expression for the long-range interatomic force constants,
with the ion-clamped dielectric tensor, the Born effective charges, the Ewald parameter which is chosen such that the contributions from are negligible within a certain vector cutoff sphere PHON_G_CUTOFF.
This also allows us to separate the dynamical matrix into short and long-range parts
with the long-range part of the dynamical matrix
The equations above give us the practical method for computing the phonon dynamical matrices including the long-range force constants using a moderately sized supercell calculation with the steps:
- Compute using a finite size supercell
- Compute
- Compute using
- Compute in the unit cell and add to
The treatment is done automatically inside VASP using the LPHON_POLAR=.TRUE. tag and specifying the dielectric tensor with PHON_DIELECTRIC and the Born effective charges with PHON_BORN_CHARGES.
Finite differences
The second-order force constants are computed using finite differences of the forces when each ion is displaced in each independent direction.
This is done by
creating systems with finite ionic displacement of atom in direction with magnitude ,
computing the orbitals and the forces for these systems.
The second-order force constants are then computed using
where corresponds to the displacement of atom in the cartesian direction and retrieves the set of forces acting on all the ions given the KS orbitals.
Similarly, the internal strain tensor is
where computes the strain tensor given the KS orbitals.
Density functional perturbation theory
Density-functional-perturbation theory provides a way to compute the second-order linear response to ionic displacement, strain, and electric fields. The equations are derived as follows.
In density-functional theory, we solve the Kohn-Sham (KS) equations
where
is the DFT Hamiltonian,
is the overlap operator and,
and
are the KS eigenstates.
Taking the derivative with respect to the ionic displacements , we obtain the Sternheimer equations
Once the derivative of the KS orbitals is computed, we can write
where is a small numeric value to use in the finite differences formulas below.
The second-order response to ionic displacement, i.e., the force constants or Hessian matrix
can be computed using the same equation used in the case of the finite differences approach
where again yields the forces for a given set of KS orbitals.
Similarly, the internal strain tensor is computed using
where computes the strain tensor given the KS orbitals.
The Born effective charges are then computed using Eq. (42) of Ref. [1].
where is the atom index, the direction of the displacement of the atom, the polarization direction,
and is the polarization vector defined in Eq. (30) in Ref. [2].
The results should be equivalent to the ones obtained using LCALCEPS and LEPSILON.
Related tags and sections
IBRION,
Phonons from finite differences,
Phonons from density-functional-perturbation theory
References
- ↑ a b X. Gonze and C. Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory, Phys. Rev. B 55, 10355 (1997).
- ↑ M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 73, 045112 (2006).