First Principal Calculations of Oxide Perovskite
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Transcript First Principal Calculations of Oxide Perovskite
Pádraig Ó Conbhuí & David-Alexander Robinson
Prof. Stefano Sanvito
Computational Spintronics Group
Physics Department, The University of Dublin, Trinity College
Layered crystals- Large volume of research
Recent literature (J. Coleman) suggests that when trying to peal off
flakes of TiS2 that an undesired reaction occurs with the water
TiS2 + xH2O → TiOxS2−x + xH2S = “smell of eggs”
Want to see if this reaction is occurring. With computers!
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Layered crystals- Large volume of research
Recent literature (J. Coleman) suggests that when trying to peal off
flakes of TiS2 that an undesired reaction occurs with the water
TiS2 + xH2O → TiOxS2−x + xH2S = “smell of eggs”
Want to see if this reaction is occurring. With computers!
Use an approximation method called Density Functional
Theory to calculate the overall energy.
• Replaces the many-electron problem with a single-particle in an effective
potential.
• Provides an excellent description of the ground-state properties, and overall
crystal energies
7/16/2015
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Layered crystals- Large volume of research
Recent literature (J. Coleman) suggests that when trying to peal off
flakes of TiS2 that an undesired reaction occurs with the water
TiS2 + xH2O → TiOxS2−x + xH2S = “smell of eggs”
Want to see if this reaction is occurring. With computers!
Use an approximation method called Density Functional
Theory to calculate the overall energy.
• Replaces the many-electron problem with a single-particle in an effective
potential.
• Provides an excellent description of the ground-state properties, and overall
crystal energies
•
Use a spring model to get a phonon spectrum.
This reduces to a simple eigenvalue and eigenvector problem.
The crystal structure can then be inferred from the phonon spectrum.
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Build a unit cell based on the geometry of the crystal
Allow the z-primitive vector to be very large, to avoid interplane interactions. Can model Graphene, say.
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Build a unit cell based on the geometry of the crystal
Allow the z-primitive vector to be very large, to avoid interplane interactions. Can model Graphene, say.
Unit cell for an 𝐴𝐵2 layered crystal
S
𝑎𝑧
Mo
𝑎𝑥
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Can run structure relaxation calculations using Quantum
Espresso. This will check our lattice structure’s
plausibility.
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Can run structure relaxation calculations using Quantum
Espresso. This will check our lattice structure’s
plausibility.
Like VASP, this is a computational method of implementing the DFT.
An open source code, with free pseudo-potentials available online.
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Can run structure relaxation calculations using Quantum
Espresso. This will check our lattice structure’s
plausibility.
Like VASP, this is a computational method of implementing the DFT.
An open source code, with free pseudo-potentials available online.
Should get a Lennard-Jones potential plot, with a potential well
at the stable lattice vector.
E
n
e
r
g
y
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Want to model a large unit cell, so that point defects can be
included in calculations.
This would be the way to build our proposed layered structure of the
TiOx𝑆2−x flakes.
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Want to model a large unit cell, so that point defects can be
included in calculations.
This would be the way to build our proposed layered structure of the
TiOx𝑆2−x flakes.
However there is realistically a doping concentration of only
1 × 10−6 , or maybe even less.
This would require a unit cell with one million or more atoms!
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Want to model a large unit cell, so that point defects can be
included in calculations.
This would be the way to build our proposed layered structure of the
TiOx𝑆2−x flakes.
However there is realistically a doping concentration of only
1 × 10−6 , or maybe even less.
This would require a unit cell with one million or more atoms!
This is very huge, especially when we note that our computational
time goes as 𝑁 3 , where N is the number of atoms in the unit cell.
Thus we will only ever be able to get an approximate answer, with
1,000 atoms in our unit cell say. (And even this is pushing it!)
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We
look first at the 1D diatomic case with internal
spring coefficient K and external coefficient G.
d
G
K
G
A
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We
look first at the 1D diatomic case with internal
spring coefficient K and external coefficient G.
d
G
K
G
A
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Potential:
Equations
of Motion:
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Ansatz: Time derivative + Bloch Theorem:
All
this gives us 2 linear equations and 2 unknowns:
This
is now an eigenvalue problem with eigenvalue
𝜔 and normal modes (𝜖1 , 𝜖2 )
We
can find the band structure by solving this matrix
𝜋 𝜋
for each k in the interval [− , ].
𝑎 𝑎
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Now
generalize to 3D!
Many
things look familiar.
Notation:
Write
like
Define:
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This
gives us the most general form of our potential:
This
gives the equations of motion similarly:
Using
Get
a similar ansatz, changed slightly for 3D:
the matrix equation
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𝐷(𝑘)
here is known as the Dynamical matrix and is
defined:
Solving
this matrix equation for a given k vector gives
us a value for 𝜔. A band can be plotted by defining a
mesh of k vectors and solving for each value.
The
generalization to a multi-atomic basis is similar to
the 1D case, but just a little more awkward.
Eigenvalues will be solved for 𝜔 = 𝜔𝑠 and
eigenvectors 𝜖𝜇𝑠 .
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Hopefully the program for calculating these will be
finished soon.
Most
awkward part is building these matrices. They
can be solved easily using the LAPACK function
DGEEV.
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Questions?
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