Solid State Computing
Peter Ballo
Models
Classical:
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Quantum mechanical:
H = E
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Semi-empirical methods
Ab-initio methods
Molecular Mechanics
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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
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Quantum calculation
First principles
Reliability proven within
the approximations
Basis sets,
functional,
all-electron or pseudopotential ..
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Computationally expensive
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Based on fitting parameters
Two body , three body…,
multi-body potential
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No theoretical background
empirical
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Applicability to large system
no self consistency loop
and no eigenvalue
computation
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Reliability ?
The Framework of DFT
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DFT: the theory
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Elements of Solid State Physics
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Schroedinger’s equation
Hohenberg-Kohn Theorem
Kohn-Sham Theorem
Simplifying Schroedinger’s
LDA, GGA
Reciprocal space
Band structure
Plane waves
And then ?
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Forces (Hellmann-Feynman theorem)
E.O., M.D., M.C. …
Using DFT
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Practical Issues
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Input File(s)
Output files
Configuration
K-points mesh
Pseudopotentials
Control Parameters
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LDA/GGA
‘Diagonalisation’
Applications
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Isolated molecule
Bulk
Surface
The Basic Problem
Dangerously
classical
representation
Cores
Electrons
Schroedinger’s Equation


2
 V

 2m
Kinetic Energy

   R i , ri    .  R i , ri 

Wave function
Potential Energy
Coulombic interaction
External Fields
Energy levels
Hamiltonian operator
Very Complex many body Problem !!
(Because everything interacts)
First approximations
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Adiabatic (or Born-Openheimer)
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Electrons are much lighter, and faster
Decoupling in the wave function
  R i , ri     R i .  ri 
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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.
NO
New density ‘=‘
input density ??
YES
Finita la musica
DFT energy functional
E     T NI    
1
  r v r d r  2  
Exchange correlation funtional
Contains:
Exchange
Correlation
Interacting part of K.E.
 r  
r  r
d r d r   E XC   
Electrons are fermions
(antisymmetric wave function)
Exchange correlation functional
At this stage, the only thing we need is: E XC   
Still a functional (way too many variables)
#1 approximation, Local Density Approximation:
Homogeneous electron gas
Functional becomes function !! (see KS3)
Very good parameterisation for E XC   
Generalised Gradient Approximation:
E XC   ,   
LDA
GGA
Bulk properties
•zero temperature equations of state (bulk modulus, lattice
constant, cohesive energy)
•structural energy difference (FCC,HCP,BCC)
e n e rg y
d is ta n c e
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
a i  b j  2  . ij
(Inverting effect)
sin(k.r)
Real Space
ai
Reciprocal Space
Brillouin Zone
bi
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)
Molecule
Crystal
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 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
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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
2
0.00000000000000
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
*
(a)
(b)
This work
EAM
(nonmag.
)
ab-initio
ab-initio
Experiment
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,
e2.02±0.2
1.89
aFCC (Å)
3.630
-
-
-
3.68
μ (μB)
-
2.19
*2.31
*2.22
-
-
-
-
EBCC
– EFCC (eV)
Fu CC, Williame
F., Phys.Rev.Lett.
2004, 94,-0.0495
175503
Mehl MJ, Papaconstantopoulos DA, Yip S., editor. Handbook
of materials modeling
Domain C., Becquart C., Phys.Rev. B 2002, 65, 024103
(mag.)
-
*2.07
fAckland
et
al. potential
(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