Introduction to Solid State Physics - SFU to Solid State Physics - SFU
Solid State Computing
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Transcript of Solid State Computing
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Solid State Computing
Peter Ballo
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Models Classical:
Quantum mechanical:H = E
Semi-empirical methods Ab-initio methods
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Molecular Mechanics atoms = spheres bonds = springs math of spring
deformation describes bond stretching, bending, twisting
Energy = E(str) + E(bend) + E(tor) + E(NBI)
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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 ?
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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
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Using DFT Practical Issues
Input File(s) Output files Configuration K-points mesh Pseudopotentials Control Parameters
LDA/GGA ‘Diagonalisation’
Applications Isolated molecule Bulk Surface
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The Basic ProblemDangerously classical representation
Cores
Electrons
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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
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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 .,
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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
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DFT energy functional XCNI EdddvTE
rrrr
rrrr21
Exchange correlation funtionalContains:ExchangeCorrelationInteracting part of K.E.
Electrons are fermions (antisymmetric wave function)
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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
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Bulk properties •zero temperature equations of state (bulk modulus, lattice constant, cohesive energy)•structural energy difference (FCC,HCP,BCC)
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distance
ener
gy
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M. I. Baskes, Phys. Rev. B 46, 2727 (1992)M. I. Baskes, Matter. Chem. Phys. 50, 152 (1997)
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And now, for something completely different: A little bit of Solid State Physics
Crystal structure
Periodicity
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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
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Band structure
Molecule
E
Crystal
Energy levels (eigenvalues of SE)
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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 ??
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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
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Solid State: Summary Quantities can be
calculated in the direct or reciprocal space
k-point Mesh Plane wave basis
set, Ecut
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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
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# 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)
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#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)
+ + =
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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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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