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DFT Density Functional theory and the description of Nature Edison Z. da Silva Instituto de Física "Gleb Wataghin", UNICAMP, Campinas - SP, Brazil

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Outline DFT, What it is, how it came about, what it is used for..... Solids and DFT Probing the Earth´s Inner Core Gold Nanowires

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We have a well defined system Nuclei + Electrons Atoms Molecules Solids Liquids

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Description of known materials, prediction of new materials, explanation of many questions Description of known materials, prediction of new materials, explanation of many questions DFT, theoretical framework Electrons are the glue which holds matter together

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Fundamental laws of electronic and nuclear motions well known Electromagnetic Forces + Quantum Mechanics Electromagnetic Forces + Quantum Mechanics H ( {r i, r } ) = E TOT ( {r i, r } ) ^ H = ( -½ 2 i ) + ( -½ 2 ) + ( - ) + + Z r i ___ i

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There is an oral tradition that, shortly after Schrödingers equation for the electronic wavefunction had been put forward and spectacularly validated for small systems like He and H 2, P. A. M. Dirac declared that chemistry had come to an end - its content was entirely contained in that powerful equation. Too bad, he is said to have added, that in almost all cases, this equation was far too complex to allow solution. W. Kohn, Reviews of Modern Physics 71, 1253 (1999).

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Therefore, we have a well posed problem, but too complex and difficult to solve Impossible to have exact solution except in very few cases Need of approximations

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Born-Oppenheimer (1927) Approximation (Qualitatively) Based on the fact that nuclear masses are much larger than the electronic ones Therefore, electrons move on a different time-scale than the nuclei (nuclei move much more slowly) Thus, electrons follow the nuclei instantaneously This allows us to consider the nuclei fixed when solving for the electronic motion

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Born-Oppenheimer Approximation H elec a elec ({r i };{r }) = E a elec ({r }) a elec ({r i };{r }) ^ a elec ({r i };{r }) depends explicitly on the electronic coordinates {r i } but only parametrically on the nuclear coordinates {r }, as does E a elec ({r }) H elec = ( -½ 2 i ) + ( - ) + + Z r i ___ i

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Born-Oppenheimer Approximation As the { a elec (r;Q)}, for a fixed Q, form a complete orthonormalized set, one can write: (r, Q) = a (Q) a elec (r;Q) a [T n + E b elec (Q)] b (Q) + ba (Q) a (Q) = E tot b (Q) ^ a Roughly speaking, if the kinetic energy of the nuclei is small when compared to |E b elec (Q) - E a elec (Q)|, the non-diagonal terms ba can be neglected

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Born-Oppenheimer Approximation [T n + E b elec (Q)] b (Q) = E tot b (Q) ^ The total electronic energy E b elec (Q) acts as a potential energy for the nuclear motion, which occurs in a single electronic state b elec. The nuclear motion simply deforms the electronic distribution, and does not cause transitions between different electronic states. (r, Q) = b (Q) b elec (r;Q)

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Born-Oppenheimer Approximation Important concepts: –Molecular geometry

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Born-Oppenheimer Approximation Important concepts: –Molecular geometry –Crystal structure

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Born-Oppenheimer Approximation Important concepts: –Molecular geometry –Crystal structure Geometrical coordinate PES Ex: Carbon Amorphous Diamond Graphite –Potential Energy Surface (PES)

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Two issues How to calculate (as accurately as possible) the PES –electronic degrees of freedom How to sample the PES (or how to move the atoms) –nuclear degrees of freedom

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Traditional way of thinking would be to describe the wavefunction as well as possible However, WF for a many electron system is a very complicated object! How to solve H elec elec = E elec elec

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Electronic density e (r) as the fundamental object (and not wavefunction elec ) Electronic density e (r) as the fundamental object (and not wavefunction elec ) YES Is there another way to solve the problem (ab initio)?

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Density Functional Theory (DFT) Formally well founded in the works of W. Kohn –P. Hohenberg and W. Kohn, Phys. Rev 136, 864B (1964) –W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) Chemistry Nobel prize Roots in the works of Thomas (1926), Fermi (1928), Dirac (1930), Slater, etc. Proc. Cambridge Philos. Soc. 26, 376 (1930)

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DFT - Two basic theorems P. Hohenberg and W. Kohn, Phys. Rev 136, 864B (1964) Lets consider a system of N interacting electrons in the ground-state associated with an external potential v(r) 1) The ground state density 0 e (r) uniquely determines the potential v(r) –Since with v(r) the full electronic Hamiltonian is known, 0 e (r) completely determines all properties of the system 2) There exists a functional E[ e (r)] which has its minimum for the ground-state density 0 e (r) –Can use this variational principle to calculate 0 e (r)

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DFT - How to calculate 0 e W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) E[ e ] = v(r) e (r) d 3 r + T[ e ] + U[ e ] I dont know how to solve this problem, but I know how to solve the following problem: –If I had a system of non-interacting electrons, in an external potential v eff (r), then: E s [ e ] = v eff (r) e (r) d 3 r + T s [ e ] T s [ e ] = kinetic energy of a non-interacting system with density e

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DFT - How to calculate 0 e W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) The ground-state wavefunction in this case is a Slater determinant of N orbitals i (i=1,N), which satisfy the equations: [-½ 2 + v eff (r)] i (r) = i i (r) And the ground-state density can be written as: 0 e (r) = | i (r)| 2 lowest N i

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DFT - How to calculate 0 e W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) However, from the Hohenberg-Kohn theorem, a completely equivalent way of calculating 0 e (r) would be through the minimization of E[ e (r)]: e (r) E s [ e ] - e (r) d 3 r = 0 + v eff (r) - = 0 T s [ e ] e (r) Gives same density as 0 e (r) = | i (r)| 2 i

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DFT - How to calculate 0 e W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) Application of the Hohenberg-Kohn theorem to E[ e ] gives e (r) E[ e ] - e (r) d 3 r = 0 T s [ e ] + v(r) e (r) d 3 r + U H [ e ] + E XC [ e ] - e (r) d 3 r e (r) = 0 + v(r) + d 3 r e (r)/|r-r| + T s [ e ] e (r) E XC [ e ] e (r) - = 0

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DFT - How to calculate 0 e W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) Comparison with similar equation for non-interacting electrons gives: + v(r) + d 3 r e (r)/|r-r| + T s [ e ] e (r) E XC [ e ] e (r) - = 0 Interacting electrons + v eff (r) - = 0 T s [ e ] e (r) Non-interacting electrons v(r) + d 3 r e (r)/|r-r| + E XC [ e ] e (r) v eff (r) =

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DFT - How to calculate 0 e W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) We can, therefore, calculate exactly (in principle) the problem of the interacting electrons through the solution of the following equations: v(r) + d 3 r e (r)/|r-r| + E XC [ e ] e (r) v eff (r) = [-½ 2 + v eff (r)] i (r) = i i (r) 0 e (r) = | i (r)| 2 lowest N i

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DFT - How to calculate 0 e W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A (1965) Problem of many-electrons mapped exactly (in principle) into the problem of non-interacting electrons subjected to an effective potential! v(r) + d 3 r e (r)/|r-r| + E XC [ e ] e (r) v eff (r) = However, we dont know how to calculate this effective potential: We need approximations for E XC [ e ] and v XC [ e ] = E XC [ e ] e (r)

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DFT - Local Density Approximation (LDA) Lets consider the exchange-correlation energy, per particle, for a homogeneous electron gas, e h XC This quantity can be calculated very accurately (Quantum Monte Carlo) Lets imagine the real system divided in small cells: i N i V i i = N i / V i

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DFT - Local Density Approximation (LDA) If we consider that the density is approximately homogeneous inside cell i, then we can write for the XC energy in cell i i N i V i i = N i / V i E i XC e h XC ( i ) N i = e h XC ( i ) N i (V i / V i ) = = e h XC ( i ) (N i / V i ) V i = e h XC ( i ) i V i

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DFT - Local Density Approximation (LDA) Summing over all the cells i will give for the total XC energy i N i V i i = N i / V i E XC e h XC ( i ) i V i i E XC e h XC ( (r)) (r) d 3 r = E LDA XC

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DFT - Local Density Approximation (LDA) And from this we can get the XC potential v LDA XC i N i V i i = N i / V i E LDA XC e v LDA XC = = e h XC ( (r)) (r) d 3 r e = e h XC ( ) d d e

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DFT - Self-consistency cycle Still a problem: v eff depends on e and e depends on v eff Solution: self-consistency cycle 1. Choose an initial density 0 e and determine the effective potential v eff ( 0 e ) 2. Solve the Kohn-Sham equations [-½ 2 + v eff (r)] i (r) = i i (r) 1 e (r) = | i (r)| 2 lowest N i 3. Calculate the density 4. Check if 1 e = 0 e. If yes, end of cycle. If not, go back to 2. (with 0 e = 1 e )

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Crystal - Bloch functions v eff has the periodicity of the crystal According to Blochs theorem, the i will have the following form: (n,k) (r) = exp(i kr) u (n,k) (r) The function u (n,k) also has the periodicity of the crystal Therefore, it can be expanded in a discrete set of plane waves whose wave vectors are reciprocal lattice vectors of the crystal u(r) = c G exp(i Gr) G

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Crystal - Bloch functions Therefore For a crystal, the density will be given by (n,k) (r) = c n k+G exp[i (k+G)r] G e (r) = | i (r)| 2 = 2 | (n,k) (r)| 2 lowest N i occ. bands n k BZ

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Crystal - Bloch functions Two approximations: –1) Maximum value of G=G C number of basis function (plane wave) e (r) = 2 | (n,k) (r)| 2 occ. bands n k SP (n,k) (r) = c n k+G exp[i (k+G)r] G Gc –To determine G C one gives the kinetic energy associated with the G C –Cutoff energy E C = G C 2 /2m –2) Number of Brillouin-zone points - set of special points

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All electron valence WF Crystal - Bloch functions Strong oscillations in the core region

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These strong oscillations are bad for plane-wave expansion (large value of G C to describe small features in real space) Either use a different basis function in the core region, or change the function in the core region

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Pseudopotentials rcrc r V, Valence electrons are the chemically relevant ones Removes the core electrons and replaces them and the strong potential by an effective, weaker potential that acts on the valence electrons

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All electron Pseudo

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Si, The credibility of DFT

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How to solve the KS equations After all these approximations, I have to find the c n k+G through the self-consistent solution of the KS equations (n,k) (r) = c n k+G exp[i (k+G)r] G Gc H KS i (r) = i i (r)

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Traditional method - diagonalization Write the matrix for the KS Hamiltonian in the plane wave basis set and diagonalize it H KS k+G,k+G c n k+G = n k+G c n k+G G H KS c = c

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Perfect crystal

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Crystal with a defect

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Crystal with a defect

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Crystal with a defect

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Surfaces

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Surfaces

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Surfaces

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Applications Understanding the Earth´s inner core Breaking of Gold nanowires

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Elasticity of Iron at the temperature of the Earth´s inner core G. Stainle-Neumann, L. Stixrude, R. Cohen & O. Gulseren Nature, 413, 57 (2001)

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General picture

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Seismological Studies: Compressional waves travel faster along polar paths than in equatorial plane. Earth´s center, 2,440 km across is nearly pure Iron At high temperatures and pressures iron goes to an hcp phase called -phase

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Iron at Temperatures and pressures of the Earth´s core. Crystal structure distorts unexpectedly Inner core believed to be made up of Iron crystals in hcp structure Observed seismic anisotropy must arise from differences in elastic response of hcp Iron Densities for iron 7.87 Mg m -3 atmospheric pressure Mg m -3 at Earth´s core pressures

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Iron at Temperatures and pressures of the Earth´s core. DFT calculations of elastic constants Thermodynamics given by the Helmholtz free energy F(V,T) = E 0 (V) + E el (V,T) -T S el (V,T) + F vib (V,T) E 0 (V) static total energy E el (V,T) due to thermal excitations S el (V,T) electronic entropy F vib (V,T) phonon contribution (particle in a cell )

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Equilibrium structure of hcp iron over a range of densities and temperatures of the Earth´s core. Computing F for different c/a and finding the minima (Mg m -3 )

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Elastic constants Elasticity of hcp iron for a density Mgm -3

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Velocity Acoustic properties of iron for the Earth´s core. Adiabatic bulk modulus (K S ) and shear modulus ( ) as function of temperature compared with inner core at the same density Mgm -3 Travel time anomaly Solid line, model with 1/3 of crystals with basal planes aligned with rotation axis

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Conclusions Seismological Studies: Compressional waves travel faster along polar paths than in equatorial plane. At high temperatures and pressures iron goes to an hcp phase called -phase. With c/a = 1.7 seismic waves travel 12 % faster in the ab plane. This anisotropy is too strong, if crystals are aligned it accounts for the experimental findings.

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Breaking of Gold Nanowires: A Computer Simulation study E. Z. da Silva Instituto de Física "Gleb Wataghin", UNICAMP, Campinas - SP, Brazil

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Participants, motivation... Theory Antonio José R. da Silva(IF-USP) Adalberto Fazzio (IF-USP) Frederico D. Novaes (PhD-IF-USP) Experimental motivation Daniel Ugarte (LNLS) Varlei Rodrigues

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When we get to the very, very small world - say circuits of seven atoms - we have a lot of new things that would happen that represent completelly new opportunities for design. Atoms on a small scale behave like nothing on the large scale, for they satisfy the laws of quantum mechanics. Richard Feynman Plenty of Room at the Bottom(1959)

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Program Why study Au nanowires? Experiments Theory TBMD simulations Impurities Conclusions

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Why study Au nanowires? Gold nanowires are produced in the lab! Understanding of its properties is fundamental. Gold is the electrical contact in nanotechnology. When Gold nanowires become very thin, interesting surprises are found: Conductivity is quantized. Possibility of helical structures. One-atom thick wires are produced.

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Gold in devices J Reichert, et al., Phys. Rev. Lett., 88, (2002)

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U.Landman, W.D.Luedtke, B.E.Salisbury, and R.L.Whetten, Phys. Rev. Lett. 77, 1362, (1996) Experiments, How it all started! Pin-plate experiment: nanowire formed conductance during elongation and contraction

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O. Gulseren, F Ercolessi, and E. Tosatti, Phys. Rev. Lett. 80, 3775, (1998) Theoretical Predictions Computer simulations with a glue type empirical many-body potential (for Al and Pb) predicted: As the diameter decrease, thin metal wires develop exotic stable noncrystalline structures: Existence of critical diameter Formation of helical, spiral-structured wires

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O. Gulseren, F Ercolessi, and E. Tosatti, Phys. Rev. Lett. 80, 3775, (1998) Theory Weird Nanowires

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How to make a wire

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Theorist´s view: production and imaging process!

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Stable Au Nanowires from surface structures Production Au thin films (3nm) in UHVTEM electron beam irradiation (100 A/cm 2 ) produced many holes nanowires Imaging Straight nanowires images produced with electron beam irradiation (5 A/cm 2 ) Y. Kondo and K. Takayanagi, Phys. Rev. Lett. 79, 3455, (1997)

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Interpretation of the Experiments

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Cylindrical folding of a triangular lattice for an (m, n) tube, with views of several coaxial tube nanowires. Each atom is pictured as a sphere of atomic radius. The (7, 3) gold nanowire (note its chirality) was reported to be magic in (3). Folding of a (111) plane

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V. Rodrigues and D. Ugarte, Phys. Rev. B 63, , (2001) Real time imaging of one-atom-thick Au nanowires The film irradiation technique from Kondo and Takayanagi produced stable one-atom-thick Au- nanowires. Time sequence of formation, elongation and fracture of a chain of gold atoms.

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Is it possible to simulate the evolution of a gold nanowire under stress with a reliable electronic structure method? Dynamical Simulations of Gold nanowires Wire under tension. It forms, starts to thin ( atoms going to the wire edges) neck is produced (one atom constriction). Atoms from the wire,( near the tips) are draw back and are incorporated into the one -atom thick neck. This thin neck grows to 4-5 atoms The wire finally breaks

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ACCURACY Number of atoms MD with Classical Potentials Embedded-atom, Effective medium, Finnis-Sinclair, Second Moment(TB) (No electrons!) Ab Initio DFT Methods (costly in terms of computer time!) NRL-TBMD keeps the electrons (three orders of magnitude faster)

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The Tight -Binding Method Slater and Koster proposed in 1954 a modified combination of atomic orbitals (LCAO) as a interpolation scheme to determine energy bands. In 1986 Papaconstantopoulos published a handbook of SK parameters for most elements in the periodic table. As DFT ab initio methods improve, so does the Papaconstantopoulos TB fits with updates the of SK parameters for most elements in the periodic table.

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M.J.Mehl and D. A.Papacontantopoulos, Phys. Rev. B 54, 4519 (1996) The NRL tight-binding method Procedure, The method fits TB parameters to reproduce ab initio total energies (DFT) calculated with full-potential linear aumented plane wave method. Slater-Koster parameters are fitted to LAPW bands and total energies as function of volume for 10 fcc structures, 6 bcc structures and 5 sc structures. Perdue-Wang parametrization to LDA

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Tight-binding and Molecular Dynamics TB basis set The Slater-Koster (SK) parameters Tight-binding Fit of the ab initio Potential energy surface Ab initio molecular dynamics

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TB basis set Minimum basis set of atomic like orbitals, for transition metals; Number of basis functions per atom N 0. Number of atoms N a, dimension space spanned by this basis set N b = N 0 N a. States are expressed as linear combinations of these orbitals:

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The Slater-Koster parameters Overlap terms Hopping terms Matrix elements of the Hamiltonian and overlap gives: SK-parameters

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Tight-Binding Fit of the ab initio Potential energy surface Formalism Parametrization

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Complicated Transition Metal-Mn Manganese crystallizes in the -Mn structure with 29 atoms Above C, Mn transforms to the -Mn with 20 atoms Calculated TB total energy

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F.Kirchhhoff et al. Phys. Rev. B, (2001) Fitting Procedure for Au, using the static TB code Equation of state Phonon dispersion, T = 0 How good is the TB parametrization for fcc bulk Au? Data basis: 10 fcc structures, 6 bcc structures, 5 sc structures. Fcc structure, lowest energy

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Ab initio molecular dynamics steps of the simulation Input, R i, positions of the atoms H the Hamiltonian The electronic ground state of the system is calculated. Where states are expanded using the TB orbitals Eigenvectors Forces on the atoms. Forces (EOM) integrated and new positions found.

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F.Kirchhhoff et al. Phys. Rev. B, (2001) Electronic DOS for Au, using TBMD code Solid, T dependence Liquid Au, T = 1773 K MD simulations: Microcanonical ensemble, EM with Verlet algo, t = 2 fs, point sampling.

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Au-nanowires, simulations E. Z. da Silva. A J. R.da Silva and A. Fazzio, Phys. Rev. Lett., 87, (2001) Starting structures: nanowire along the along the(111) direction - 10 atom planes - 7atoms in each plane. Top view Front view

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Simulation Protocol EM integrated using Verlet algo, t =1 fs The system is heated to T = 600 K, 7000 MD steps, 7ps Elongation of the wire by L = 0.5 Å The system is heated to T = 400 K, 4000 MD steps, 4ps Previous two steps are repeated Procedure done for 24.0 Å < L W < 44.0 Å

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Au nanowire, 6 (111) planes of 7 Au atoms, relaxed structure unrelaxed structure

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Similarities for the breaking the one-atom necklace Experiment Simulation

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Evolution and breaking Atomic configurations (in Å) (a) 25.5, (b) 33.0, (c) 37.0 (a) 38.0, (b) 40.5, (d) 41.0

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Defect structure, one atom neck (i) Defect structure(iii) 4-5 atoms planes (iv) atoms planes (v) one atom neck

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Forces Saw-tooth behavior

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Before breaking the one-atom necklace The nanowire develops a one-atom thick necklace with five atoms in its final structure. Just before breaking, the apex-apex distance gets to d a-a ~ Å.

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Animation of the simulation See this animation at

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Pending problems Are TBMD results reproducible using ab initio DFT. Large Au-Au distances seen in the experiments

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HRTEM image of atom chain with three gold atoms. Atomic positions are the dark spots Large Au-Au distances!!!! Schematic representation Real time imaging of one-atom-thick Au nanowires V. Rodrigues and D. Ugarte, Phys. Rev. B 63, , (2001)

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Before breaking: one-atom necklace Simulations give shorter Au-Au distances

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Ab initio DFT (Siesta) Fully self-consistent DFT LCAO –Extremely fast simulation using minimal basis set Kohn-Sham equations –Exchange-correlation (GGA) –Norm conserving pseudopotentials (Troullier-Martins) Total energy Forces

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Ab initio evolution Starting structure Before breaking After breaking Au-Au distance 3.0 Å

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Large distances, Impurities? The Images Au-Au distances 3.6±0.2 The statistics Legoas,Galvão, Rodrigues, Ugarte, PRL, 88, , (2002) (Au-X-Au) Max 3.6 Å X = ? (Au-Au) Max 3.1 Å

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Reasoning for impurities TEM simulation Structure simulation

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Carbon in Au nanowire Carbon intermediate structure, 3.73 Carbon, initial structure Au-nanowire Carbon, structure right before breaking Quebra

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Impurities and the Au-Au Bond Length Frederico D. Novaes, A.J.R. da Silva, E.Z. da Silva and A.Fazzio, Phys. Rev. Lett. 90, (2003)22 january Use ab initio DFT to study contamination of the pure Au nanowire by impurities. Considered the effect of the impurities, H, B, C, N, O and S Experiment gives Au-?-Au = 3.6 Å Au-??-Au = 4.8 Å

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Contamination by H and S

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H Impurities? Experiments: Legoas,Galvão, Rodrigues, Ugarte, PRL, 88, , (2002) Our simulation

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General Conclusions Importance of Gold nanowires. Simulations can help understand the dynamical evolution and finally the breaking of metal nanowires. TBMD is reliable and fast. Can be used to study dynamical evolution of metal nanowires. Future: Study of other directions for wire formation. Investigation of nanowires of other metals (Ag, Pt etc..) Use nanowires and their tips to design devices.

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Conclusions DFT, A effective way of studying and projecting interesting materials. Solids and DFT Probing the Earth´s Inner Core Gold Nanowires

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