Solid State Computing
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Solid State Computing. Peter Ballo. Models. Classical: Quantum mechanical: H  = E  Semi-empirical methods Ab-initio methods. Molecular Mechanics. atoms = spheres bonds = springs

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Solid State Computing

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Solid State Computing

Peter Ballo


Models

  • Classical:

  • Quantum mechanical:

    H = E

    • Semi-empirical methods

    • Ab-initio methods


Molecular Mechanics

  • atoms = spheres

  • bonds = springs

  • math of spring deformation describes bond stretching, bending, twisting

Energy = E(str) + E(bend) + E(tor) + E(NBI)


From ab initio to (semi) empirical

  • Quantum calculation

  • First principles

  • Reliability proven within the approximations

    Basis sets,

    functional,

    all-electron or pseudo- potential ..

  • Computationally expensive

  • Based on fitting parameters

    Two body , three body…, multi-body potential

  • No theoretical background empirical

  • Applicability to large system

    no self consistency loop and no eigenvalue computation

  • Reliability ?


Climbing Mt. Psi


The Framework of DFT

  • DFT: the theory

    • Schroedinger’s equation

    • Hohenberg-Kohn Theorem

    • Kohn-Sham Theorem

    • Simplifying Schroedinger’s

    • LDA, GGA

  • Elements of Solid State Physics

    • Reciprocal space

    • Band structure

    • Plane waves

  • And then ?

    • Forces (Hellmann-Feynman theorem)

    • E.O., M.D., M.C. …


Using DFT

  • Practical Issues

    • Input File(s)

    • Output files

    • Configuration

    • K-points mesh

    • Pseudopotentials

    • Control Parameters

      • LDA/GGA

      • ‘Diagonalisation’

  • Applications

    • Isolated molecule

    • Bulk

    • Surface


The Basic Problem

Dangerously

classical

representation

Cores

Electrons


Schroedinger’s Equation

Wave function

Potential Energy

Kinetic Energy

Coulombic interaction

External Fields

Energy levels

Hamiltonian operator

Very Complex many body Problem !!

(Because everything interacts)


First approximations

  • Adiabatic (or Born-Openheimer)

    • Electrons are much lighter, and faster

    • Decoupling in the wave function

  • Nuclei are treated classically

    • They go in the external potential


Self consistent loop

Initial density

From density, work out Effective potential

Solve the independents K.S. =>wave functions

Deduce new density from w.f.

New density ‘=‘ input density ??

NO

YES

Finita la musica


DFT energy functional

  • Exchange correlation funtional

  • Contains:

  • Exchange

  • Correlation

  • Interacting part of K.E.

Electrons are fermions (antisymmetric wave function)


Exchange correlation functional

At this stage, the only thing we need is:

Still a functional (way too many variables)

  • #1 approximation, Local Density Approximation:

  • Homogeneous electron gas

  • Functional becomes function !! (see KS3)

  • Very good parameterisation for

LDA

Generalised Gradient Approximation:

GGA


DFT: Summary

  • The ground state energy depends only on the electronic density (H.K.)

  • One can formally replace the SE for the system by a set of SE for non-interacting electrons (K.S.)

  • Everything hard is dumped into Exc

  • Simplistic approximations of Exc work !

    LDA or GGA


  • Bulk properties

  • zero temperature equations of state (bulk modulus, lattice constant, cohesive energy)

  • structural energy difference (FCC,HCP,BCC)

  • two shear elastic constants in FCC structure


M. I. Baskes, Phys. Rev. B 46, 2727 (1992)

M. I. Baskes, Matter. Chem. Phys. 50, 152 (1997)


And now, for something completely different:A little bit of Solid State Physics

Crystal structure

Periodicity


Reciprocal space

(Inverting effect)

sin(k.r)

Reciprocal Space

bi

Real Space

ai

Brillouin Zone

k-vector (or k-point)

See X-Ray diffraction for instance

Also, Fourier transform and Bloch theorem


Band structure

E

Energy levels (eigenvalues of SE)

Crystal

Molecule


The k-point mesh

Corresponds to a supercell 36 time bigger than the primitive cell

Brillouin Zone

Question:

Which require a finer mesh, Metals or Insulators ??

(6x6) mesh


Plane waves

  • Project the wave functions on a basis set

  • Tricky integrals become linear algebra

  • Plane Wave for Solid State

  • Could be localised (ex: Gaussians)

+

+

=

Sum of plane waves of increasing frequency (or energy)

One has to stop: Ecut


Solid State: Summary

  • Quantities can be calculated in the direct or reciprocal space

  • k-point Mesh

  • Plane wave basis set, Ecut


if (i.LE.n) then

kx=kx-step ! Move to the Gamma point (0,0,0)

ky=ky-step

kz=kz-step

xk=xk+step

else if ((i.GT.n).AND.(i.LT.2*n)) then

kx=kx+2.0*step ! Now go to the X point (1,0,0)

ky=0.0

kz=0.0

xk=xk+step

else if (i.EQ.2*n) then

kx=1.0 ! Jump to the U,K point

ky=1.0

kz=0.0

xk=xk+step

else if (i.GT.2*n) then

kx=kx-2.0*step ! Now go back to Gamma

ky=ky-2.0*step

kz=0.0

xk=xk+step

end if


# Crystalline silicon : computation of the total energy

#

#Definition of the unit cell

acell 3*10.18 # This is equivalent to 10.18 10.18 10.18

rprim 0.0 0.5 0.5 # In lessons 1 and 2, these primitive vectors

0.5 0.0 0.5 # (to be scaled by acell) were 1 0 0 0 1 0 0 0 1

0.5 0.5 0.0 # that is, the default.

#Definition of the atom types

ntypat 1 # There is only one type of atom

znucl 14 # The keyword "znucl" refers to the atomic number of the

# possible type(s) of atom. The pseudopotential(s)

# mentioned in the "files" file must correspond

# to the type(s) of atom. Here, the only type is Silicon.

#Definition of the atoms

natom 2 # There are two atoms

typat 1 1 # They both are of type 1, that is, Silicon.

xred # This keyword indicate that the location of the atoms

# will follow, one triplet of number for each atom

0.0 0.0 0.0 # Triplet giving the REDUCED coordinate of atom 1.

1/4 1/4 1/4 # Triplet giving the REDUCED coordinate of atom 2.

# Note the use of fractions (remember the limited

# interpreter capabilities of ABINIT)


+

+

=

#Definition of the planewave basis set

ecut 8.0 # Maximal kinetic energy cut-off, in Hartree

#Definition of the k-point grid

kptopt 1 # Option for the automatic generation of k points, taking

# into account the symmetry

ngkpt 2 2 2 # This is a 2x2x2 grid based on the primitive vectors

nshiftk 4 # of the reciprocal space (that form a BCC lattice !),

# repeated four times, with different shifts :

shiftk 0.5 0.5 0.5

0.5 0.0 0.0

0.0 0.5 0.0

0.0 0.0 0.5

# In cartesian coordinates, this grid is simple cubic, and

# actually corresponds to the

# so-called 4x4x4 Monkhorst-Pack grid

#Definition of the SCF procedure

nstep 10 # Maximal number of SCF cycles

toldfe 1.0d-6 # Will stop when, twice in a row, the difference

# between two consecutive evaluations of total energy

# differ by less than toldfe (in Hartree)


iter Etot(hartree) deltaE(h) residm vres2 diffor maxfor

ETOT 1 -8.8611673348431 -8.861E+00 1.404E-03 6.305E+00 0.000E+00 0.000E+00

ETOT 2 -8.8661434670768 -4.976E-03 8.033E-07 1.677E-01 1.240E-30 1.240E-30

ETOT 3 -8.8662089742580 -6.551E-05 9.733E-07 4.402E-02 5.373E-30 4.959E-30

ETOT 4 -8.8662223695368 -1.340E-05 2.122E-08 4.605E-03 5.476E-30 5.166E-31

ETOT 5 -8.8662237078866 -1.338E-06 1.671E-08 4.634E-04 1.137E-30 6.199E-31

ETOT 6 -8.8662238907703 -1.829E-07 1.067E-09 1.326E-05 5.166E-31 5.166E-31

ETOT 7 -8.8662238959860 -5.216E-09 1.249E-10 3.283E-08 5.166E-31 0.000E+00

At SCF step 7, etot is converged :

for the second time, diff in etot= 5.216E-09 < toldfe= 1.000E-06

cartesian forces (eV/Angstrom) at end:

1 0.00000000000000 0.00000000000000 0.00000000000000

2 0.00000000000000 0.00000000000000 0.00000000000000

Metals (T=0.25eV)

ik=1

| eig [eV]: -5.8984 1.7993 1.9147 1.9147 2.8058 2.8058 141.3489 313.9870 313.9870

| focc: 2.0000 1.9999 1.9998 1.9998 1.9979 1.9979 0.0000 0.0000 0.0000


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