basics for ab initio calculations - universitetet i...
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations
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Basics for ab initio Calculations
22 February 2011; V172
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Unit Cell: the smallest group of atoms possessing thesystem of the crystal which, when repeated in alldirections, will develop the crystal lattice.Primitive Cell: is a type of minimum-volume cell thatwill fill all space by the repetition of suitable translationoperation.
Primitive Cell
Unit Cell
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations
6Bravais Lattice and Crystal structures
The Fourteen Space (Bravais) Lattice : well knownP - primitive cell;I - body-centered cell;F - face-centered cell,C - Centered: additional point in the center of each endR - Rhombohedral: Hexagonal class only
Monoclinic andTriclinic Cells
Orthorhombic Cells
p
FC
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Hexagonal andTriagonal Cells
Tetragonal Cells
Cubic Cells
Rhombohedral
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RotationReflectionInversion
32 point groups
Translation
14 Space Lattices
ScrewGlide
230 Space Groups
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▲Miller Indices:Indices for Direction: [x,y,z] represents a specific
crystallographic direction; <x,y,z> represents allof the directions of the same form
Indices for Planes: (x,y,z) represents a specific plane;{x,y,z} represents all of the planes of the sameform [x,y,z] is the directional normal of plane(x,y,z)
Miller Indices for Hexagonal Crystals: 4-digit system(hklm); h+k=-l => reduced to 3-digit system
a1
a2
a3
c hkl: reciprocals of the intercepts witha1 , a2 , and a3 , m reciprocals of theintercepts with c.
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Miller Indices (Example)
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12The Brillouin zone (BZ)
• Irreducible BZ (IBZ)– The irreducible wedge – Region, from which the
whole BZ can be obtained by applying all symmetry operations
• Bilbao Crystallographic Server:– www.cryst.ehu.es/cryst/– The IBZ of all space groups
can be obtained from this server
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Wigner-Seitz CellForm connection to all neighbors and span a plane normal
to the connecting line at half distance
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Wigner-Seitz Cell & Brillouin Zone1
st Brillouin Zone is the Wigner-Seitz Cell of the reciprocal lattice
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BCC WS cell BCC BZ
FCC WS cell FCC BZ
Real Space Reciprocal Space
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Muffin-Tin Potential
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Sampling of the k-points
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K-point sampling in the BZ
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Linear Interpolation of eigen values to improve the BZ integration
Brillouin Zone Integration – Tetrahedrons method
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D(ε)
D
Density of States (DOS)
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42Bandstructure and DOS
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43LDA vs. LSDA
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations
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What we can calculate
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Murnaghan Equation of State is usually used to extract Bulk modulus from Total energy curve
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations
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What About the Kohn-Sham Eigenvalues?
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Attempts on Improving LDA
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Slab structure of Al(110)
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Surface Relaxation and Reconstruction
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Surface Relaxation
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Surface Relaxation
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Relaxation of Pt (210)
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Surface Reconstruction: Missing Row Reconstruction
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations
62Semiconductor Surfaces: Si(100)