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Solid State Computing

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Solid State Computing

Peter Ballo

- Classical:
- Quantum mechanical:
H = E

- Semi-empirical methods
- Ab-initio methods

- 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

Dangerously

classical

representation

Cores

Electrons

Wave function

Potential Energy

Kinetic Energy

Coulombic interaction

External Fields

Energy levels

Hamiltonian operator

Very Complex many body Problem !!

(Because everything interacts)

- 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

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

- Exchange correlation funtional
- Contains:
- Exchange
- Correlation
- Interacting part of K.E.

Electrons are fermions (antisymmetric wave function)

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

- 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)

Crystal structure

Periodicity

(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

E

Energy levels (eigenvalues of SE)

Crystal

Molecule

Corresponds to a supercell 36 time bigger than the primitive cell

Brillouin Zone

Question:

Which require a finer mesh, Metals or Insulators ??

(6x6) mesh

- 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

- 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