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
2

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.
7/16/2015
9

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!
7/16/2015
10

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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