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

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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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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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

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

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

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Hexagonal andTriagonal Cells

Tetragonal Cells

Cubic Cells

Rhombohedral

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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RotationReflectionInversion

32 point groups

Translation

14 Space Lattices

ScrewGlide

230 Space Groups

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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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.

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Miller Indices (Example)

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

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

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Wigner-Seitz CellForm connection to all neighbors and span a plane normal

to the connecting line at half distance

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Wigner-Seitz Cell & Brillouin Zone1

st Brillouin Zone is the Wigner-Seitz Cell of the reciprocal lattice

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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BCC WS cell BCC BZ

FCC WS cell FCC BZ

Real Space Reciprocal Space

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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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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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Muffin-Tin Potential

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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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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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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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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Sampling of the k-points

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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K-point sampling in the BZ

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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Linear Interpolation of eigen values to improve the BZ integration

Brillouin Zone Integration – Tetrahedrons method

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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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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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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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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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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D(ε)

D

Density of States (DOS)

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

42Bandstructure and DOS

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

43LDA vs. LSDA

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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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

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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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Murnaghan Equation of State is usually used to extract Bulk modulus from Total energy curve

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 About the Kohn-Sham Eigenvalues?

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Attempts on Improving LDA

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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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Slab structure of Al(110)

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Surface Relaxation and Reconstruction

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Surface Relaxation

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Surface Relaxation

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Relaxation of Pt (210)

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

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Surface Reconstruction: Missing Row Reconstruction

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 22 February 2011 Basics For Ab initio Calculations

62Semiconductor Surfaces: Si(100)