Solid State Computing Peter Ballo
Jan 04, 2016
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 ?
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 rRrRVm
,.,2
2
Hamiltonian operator
Kinetic EnergyPotential Energy
Coulombic 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
rrrr
2
1
Exchange correlation funtionalContains:ExchangeCorrelationInteracting part of K.E.
Electrons are fermions (antisymmetric wave function)
Exchange correlation functional
At 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:
,XCEGGA
LDA
Bulk properties •zero temperature equations of state (bulk modulus, lattice constant, cohesive energy)•structural energy difference (FCC,HCP,BCC)
distance
en
erg
y
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 Space
biBrillouin Zone
(Inverting effect)
k-vector (or k-point)
sin(k.r)
See X-Ray diffraction for instance
Also, Fourier transform and Bloch theorem
Band structure
Molecule
E
Crystal
Energy levels (eigenvalues of SE)
The k-point mesh
Brillouin 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 2.8665
ECOH (eV/atom) -4.2993 - - c-4.28 -4.316
Bulk Modulus (GPa)
179 175.65 *180 c168.3 1.89
C` 53.14 57.73 - c59.40 -
C44 83.56 - a142 d112 116
C11 250.59 252.62 a250 d242 243.4
C12 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 - - - 3.68
μ (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