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- potentia

l ..

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 ?

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. …

The Framework of DFT

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 ProblemDangerously classical representation

Cores

Electrons

Schroedinger’s Equation iiii rRrRV

m,.,

22

Hamiltonian operator

Kinetic EnergyPotential EnergyCoulombic interactionExternal Fields

Very Complex many body Problem !!(Because everything interacts)

Wave function

Energy levels

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

iiii rRrR .,

Self consistent loop

Solve the independents K.S. =>wave functions

From density, work out Effective potential

New density ‘=‘ input density ??

Deduce new density from w.f.

Initial density

Finita la musica

YES

NO

DFT energy functional XCNI EdddvTE

rrrr

rrrr21

Exchange correlation funtionalContains:ExchangeCorrelationInteracting part of K.E.

Electrons are fermions (antisymmetric wave function)

Exchange correlation functionalAt this stage, the only thing we need is: XCE

Still a functional (way too many variables)#1 approximation, Local Density Approximation:Homogeneous electron gasFunctional becomes function !! (see KS3)Very good parameterisation for XCE

Generalised Gradient Approximation: ,XCE

GGA

LDA

Bulk properties •zero temperature equations of state (bulk modulus, lattice constant, cohesive energy)•structural energy difference (FCC,HCP,BCC)

distance

ener

gy

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

Real Space

ai

ijji ba .2

Reciprocal SpacebiBrillouin

Zone

(Inverting effect)

k-vector (or k-point)

sin(k.r)

See X-Ray diffraction for instanceAlso, Fourier transform and Bloch theorem

Band structure

Molecule

E

Crystal

Energy levels (eigenvalues of SE)

The k-point meshBrillouin Zone

(6x6) mesh

Corresponds to a supercell 36 time bigger than the primitive cell

Question:Which require a finer mesh, Metals or Insulators ??

Plane wavesProject the wave functions on a basis setTricky integrals become linear algebraPlane Wave for Solid StateCould 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 cellacell 3*10.18 # This is equivalent to 10.18 10.18 10.18rprim 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 typesntypat 1 # There is only one type of atomznucl 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 atomsnatom 2 # There are two atomstypat 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 setecut 8.0 # Maximal kinetic energy cut-off, in Hartree

#Definition of the k-point gridkptopt 1 # Option for the automatic generation of k points, taking # into account the symmetryngkpt 2 2 2 # This is a 2x2x2 grid based on the primitive vectorsnshiftk 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 procedurenstep 10 # Maximal number of SCF cyclestoldfe 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

DEPARTMENT OF PHYSICS AND DEPARTMENT OF NUCLEAR PHYSICS AND TECHNOLOGY, FACULTY OF ELECRICAL ENGINEERING AND INFORMATION

TECHNOLOGY, SLOVAK UNIVERSITY OF TECHNOLOGY

“Fe” RESULTS

This workab-initio Experiment fAckland

et al. potential

EAM (nonmag

.)

ab-initio (mag.)

aBCC (Å) 2.866 2.831 *2.88 c2.87 2.8665ECOH (eV/atom) -4.2993 - - c-4.28 -4.316Bulk Modulus

(GPa)179 175.65 *180 c168.3 1.89

C` 53.14 57.73 - c59.40 -C44 83.56 - a142 d112 116C11 250.59 252.62 a250 d242 243.4C12 144.3 137.16 a145 d145.6 145

EVFA (eV) 1.9112 - b1.93-2.02, *2.07

e2.02±0.2 1.89

aFCC (Å) 3.630 - - - 3.68μ (μB) - 2.19 *2.31 *2.22 -

EBCC – EFCC (eV) -0.0495 - - - -

* Fu CC, Williame F., Phys.Rev.Lett. 2004, 94, 175503

(a) Mehl MJ, Papaconstantopoulos DA, Yip S., editor. Handbook of materials modeling

(b) Domain C., Becquart C., Phys.Rev. B 2002, 65, 024103

(c) Kittel C., Introduction to solid state physics, NY,Wiley, 1986

(d) Hirth JP, Lothe J., Theory of dislocation, 2.edition, NY, Wiley,1982

(e) Schepper LD et al., Phys.Rev. B , 1983, 27, 5257