First-principles calculations: Exploration and understanding
Alberto García Institut de Ciencia de Materials de Barcelona (ICMAB-CSIC)
Alberto García, Matthieu Verstraete, Javier Junquera
Institut de Ciencia de Materials de Barcelona, (ICMAB-CSIC), Spain Department of Physics, Université de Liège, Belgium Dept. de Ciencias de la Tierra y Física de la Materia Condensada, Univ. de Cantabria, Santander, Spain
The problems Our goals
Practical differences in the handling of pseudopotential data Design a format that faithfully maps the basic ontology of the among codes hinder their interoperability norm-conserving pseudopotential domain
The lack of proper provenance information and other metadata Provide the means to generate and process the data in the new format hinders reproducibility and resource discovery Full details: http://esl.cecam.org/mediawiki/index.php/PSML
Typical legacy pseudopotential files do not contain enough The processing of the file by SIESTA involves a number of information for general use, and lack proper metadata: extra assumptions particular to the code.
Si ca nrl nc ATM3 20-FEB-98 Troullier-Martins 3s 2.00 r= 1.89/3p 2.00 r= 1.89/3d 0.00 r= 1.89/4f 0.00 r= 1.89/ To make sure that the same pseudopotential operator is used 4 0 1074 0.177053726905E-03 0.125000000000E-01 4.00000000000 Radial grid follows when interoperating with other code (say, Abinit), one would 0.222706172396E-05 0.448213643589E-05 0.676557649581E-05 0.907773869587E-05 0.114189843161E-04 0.137896791809E-04 0.161901937162E-04 0.186209030073E-04 have to re-implement in Abinit part of the SIESTA internals. ... Down Pseudopotential follows (l on next line) 0 -0.115795944561E-05 -0.233048422795E-05 -0.351775755637E-05 -0.471996494469E-05 To ease interoperability, we need to get as close as possible -0.593729424022E-05 -0.716993565309E-05 -0.841808178597E-05 -0.968192766418E-05 ... to the physical level which all codes share, and map the basic Valence charge follows domain ontology of radial functions and quantum numbers. 0.703913565639E-12 0.285118220480E-11 0.649627565371E-11 0.116952681014E-10 0.185058816439E-10 0.269875308733E-10 0.372013686330E-10 0.492103329670E-10 ... In addition, to support resilience and reproducibility (always desirable, but required in particular by high-throughput frameworks, The “psf” format of this example is part of the Froyen lineage of the data format should include proper metadata and provenance generation codes, and it is now used almost exclusively by SIESTA. information.
PSML format highlights
Provenance Support for different kinds of sets Radial function abstraction. Explicit grid(s).
The PSML library
Data extraction from PSML files with the PSML files can be generated with the libPSML Fortran library (C interface in development): xmlf90-wxml Fortran library (easily wrapped in C):
use xmlf90_wxml Basics Set selection Evaluation of functions type(xmlf_t) :: xf ... program test_psml … call xml_OpenFile("Si.psml",xf, indent=.false.) do ir = 1, npts setidx = ps_Potential_Indexes(ps,SET_LJ) ... r(ir) = (ir-1)*delta use m_psml do i = 1, size(setidx) call xml_NewElement(xf,"header") enddo type(ps_t) :: ps Abstract handle l = ps_Potential_L(ps,setidx(i)) call my_add_attribute(xf,"atomic-label","Si") n = ps_Potential_N(ps,setidx(i)) ... do ir = 1, npts call psml_reader(“Si.psml”,ps) Transparent parsing rc = ps_Potential_Rc(ps,setidx(i)) call xml_NewElement(xf,"exchange-correlation") val = ps_Potential_Value(ps,setidx(i),r(ir)) j = ps_Potential_J(ps,setidx(i)) call xml_NewElement(xf,"annotation") ... if (ps_HasPsOperator(ps)) ... enddo ... Simple query functions enddo call xml_EndElement(xf,"annotation") if (ps_HasCoreCorrections(ps)) ...... call xml_NewElement(xf,"libxc-info") The library (and PSML) is grid-agnostic. ... call ps_destroy(ps) Lower layer of routines provides full control. Evaluation is done by interpolation onto call xml_EndElement(xf,"libxc-info") end program test_psml A higher-level custom layer is easily built. call xml_EndElement(xf,"exchange-correlation") arbitrary points in a client’s grid. ... call xml_EndElement(xf,"header") ... Parsing is done by the lightweight and fast xmlf90-sax Fortran library call xml_Close(xf)
Milestones and Future plans
The pseudopotential generators ATOM (included in the SIESTA distribution) and ONCVPSP (by D.R. Hamann) can now produce PSML files.
Recent versions of SIESTA and Abinit use the PSML library to process PSML files, thus ensuring interoperability at the level of the pseudopotential operator. This allows convergence tests for localized basis sets, and also the leveraging of the capabilities of both codes: efficiency for large systems and richness of features.
The PSML project is part of CECAM’s Electronic Structure Library initiative. This will surely help in the further dissemination of the PSML format and associated tools. • Scientific method: experiment, modelization, prediction, experiment, model refinement… • We have the “ultimate model” for materials, and it involves the use of computers. • What do the calculations teach us? How can we use them well? Spring constant
Basic idea: Vibrations around an equilibrium point
!(k)
Parameters can be fitted to experiment Refinement of the model: polarizable electrons (shell model)
Better fit Internal structure to experiment of the atom acknowledged New phenomena ....
Electrons are the glue holding solids together
We know the basic equations: Quantum Mechanics and Electromagnetism The “ultimate model” for electrons in a material
e2 Z I 4⇡✏0 ri RI XI | |
Hˆ Ψ = EΨ Ψ(r1, r2, . . . , rn) We could compute “everything” Simulation of reality
Meteorology: We know the basic equations
Astrophysics: We know the basic equations. Little data Density-functional theory E = E[n] n(r)
2 + V [n](r) ψ = ε ψ One electron eqs. {−∇ eff } i i i Veff [n](r) = Vext(r) + VH[n](r) + Vxc[n](r) Internal electrons do not participate in the chemical bond
Effective potential for valence electrons Pseudopotential
V(r)
Veff r -Ze2 r
?
r (a.u.) Density-functional theory is a practical implementation of the “ultimate model” for atomic aggregates
Reasonably accurate Versatile
Dozens of codes available Siesta, Espresso, Abinit, Fleur, Vasp, BigDFT, FHI-Aims, Wien2k, CP2K, Dmol, ADF, Castep, OpenMX, …. Output of the program • Energy, forces, and stress for a given geometry • Charge density, wave functions, band energies, and other low-level technical information
As As
Ga Ga Ga Ga
* Calculation without Classic Standards is Dangerous. A Computer is Incapable of Setting its own Standards.
* By its Emphasis on Application of the Already Known, Computing can Delay Basic Discovery and thus Reduce the Field of Applications in the Future.
* Classic Theories used Inductive and Deductive Models. Computing Encourages Floating Models.
(Headings from the essay: "The Computer: Ruin of Science and Threat to Mankind", by Clifford Truesdell, in “An Idiot's Fugitive Guide to Science”, Springer, 1984)
A simple model can shed more light on Nature’s workings than a series of “ab-initio” calculations of individual cases, which, even if correct, are so detailed that they hide reality instead of revealing it. ... A perfect computation simply reproduces Nature, it does not explain it.
(P.W. Anderson) Uses in materials science
• Exploration and prediction, simulating experiments difficult or impossible in the laboratory. • Clarification/complement of experimental information by means of the precise control of simulation conditions. (The computer is a perfect control machine) • Design of materials with desired properties. Reduction of the “trial and error” loop. • Parametrization of simpler models Calculation of electronic charge density (Simulation of an X-ray experiment)
Synthetic diffraction diagram High-pressure Diamond-anvil experiment cell Sample
872 Mujica et al.: High-pressure phases of group-IV, III–V, and II–VI compounds energy or enthalpy states using the calculated values of the forces on the atomsTheoreticaland the components treatmentof the stress tensor (see, for example, Pfrommer et al., 1997b). dE Whether one minimizesE = E(V )the, energyp = at constant, p =vol-p(V ) ume or the enthalpy at constant pressure,−dVone faces the problem of the likely existence of many local minima. As one can never investigateEquationsall possible ofstarting State points for the minimization, one canPhasenever transitionsbe sure that the global minimum has been obtained. This is a difficult problem common to many branches of science, and one to which there is no wholly satisfactory solution. It should be remembered, of course, that Nature itself sometimes finds this problem difficult—hence the occur- rence of metastable phases—but Nature generally ex- plores far more possible structures than can be investi- gated in calculations. This fact has limited the predictive power of the calculations, although we will describe in- stances where successful predictions of new phases have been made, and where predictions are still waiting to be tested. Typically the search of configuration space is re- stricted to certain classes of configurations defined by FIG. 3. Typical output data from a series of total-energy cal- the number of atoms in the unit cell and particular culations: (a) Schematic diagram of the energy-volume curves space-group symmetries. In this case one cannot imme- for four phases of a material. For phases I and II we have diately decide whether the structures obtained would indicated by filled circles the discrete set of calculated E(V) even be local minima if the imposed restrictions were to points. The curves correspond to fitting of the calculated points be relaxed. There is, however, a solution to this problem, [in the present case, using a Murnaghan-type expression (Mur- which is becoming very important in first-principles naghan, 1944)]. (b) Schematic diagram of the enthalpy- studies of high-pressure phases. As described in Sec. pressure curves for the same four phases. (c) The difference in III.B, a crystal is locally stable if all of its phonon fre- enthalpy, ⌬H, from phase I. quencies are real and its elastic constants satisfy certain inequalities. These quantities can, however, be obtained directly from first-principles calculations using density- mon tangent gives the coexistence or equilibrium pres- functional perturbation theory (DFPT) (see Baroni sure, pe(I/II)ϭϪ(EIIϪEI)/(VIIϪVI). In Fig. 3(b) the et al., 2001). These developments have not so far been corresponding enthalpy-pressure relations are illus- used very widely in theoretical studies of phase stability, trated. The two H(p) curves for phases I and II cross at but it is clear that they offer considerable advantages the pressure pe(I/II). The large variation of the enthalpy and will prove very important in the future. It is unlikely with pressure, which obscures the relative stability of the that one can predict many phase transitions by analyzing phases (for which only the difference in enthalpies is local stability because the vast majority of transitions are relevant) has been subtracted in Fig. 3(c) by plotting the first order, i.e., the transition is from one locally stable enthalpy with respect to that of phase I, which is taken phase to another one. However, starting from an equi- as a reference. Phases iii and iv are not stable in any librium structure one can use the stability analysis to pressure range. However, the enthalpy of phase iii is determine whether the structure is stable or unstable rather close to those of phases I and II near pe(I/II) and with respect to infinitesimal changes. If it is unstable the it is conceivable that the effects of temperature or an analysis also reveals which small changes in the structure improvement in the calculations could result in the lower the enthalpy, which can be used to predict more emergence of a narrow field of stability for phase iii. stable configurations. Such calculations have been per- Even if phase iii is unstable it is possible that the I formed for several group-IVA, IIIA–VA, and IIB–VIA iii transition could be observed at a slightly larger materials by Ozolin¸ sˆ and Zunger (1999) and Kim et al. → (1999). pressure if the I II transition were suppressed by a large activation barrier→ . Conversely, a iii II transition In Fig. 3 we show schematic energy-volume ͓E(V)͔ ← and enthalpy-pressure ͓H(p)͔ curves for four phases of could be favorable on decrease of pressure from phase II if the I II transition were impeded. In both cases, some material. This figure illustrates typical output data ← from a series of total-energy calculations with fixed sym- phase iii would exist only as a metastable phase. Of metry and how it can be analyzed to obtain the transi- course, a proper theoretical study of these possibilities tion pressures. Phases I and II have equal enthalpies at would require the investigation of the actual mecha- the two points EI(VI) and EII(VII), respectively, where nisms of the transitions. We shall have an opportunity to the common tangent touches the energy-volume curves mention several instances of metastable phases obtained shown in Fig. 3(a). The negative of the slope of the com- upon decrease of pressure from high-pressure phases.
Rev. Mod. Phys., Vol. 75, No. 3, July 2003 Post-perovskite phase of MgSiO3 Oganov et al, Nature (2004)
Prediction of BN nanotubes Rubio, Corkill, Cohen, PRB (1994)
Proposal for a super-hard material Liu, Cohen, Science (1989) C3N4 REPORTS layers. On the basis of the bulk materials, we would therefore expect that thin films involve Structure of the Ultrathin hexagonal O planes and octahedrally and tetrahedrally coordinated metal atoms with an
Aluminum Oxide Film overall Al2O3 stoichiometry (17, 18). However, this simple assumption is in- on NiAl(110) validated by the room-temperature STM measurements (Fig. 1C) (19). We observed week ending 1 2 2 P H Y S I C A L R E V I E W L E T T E R S Georg Kresse, * Michael Schmid,PRL Evelyn94, 056103Napetschnig,(2005) square features on the surface (marked by 11 FEBRUARY 2005 Maxim Shishkin,1 Lukas Ko¨hler,1 Peter Varga2 green rectangles and squares). We explain unrealistically close separations, thusthemallobywinga squareonly arrangementfor functionof oxygenS r log r = 0, so that The well-ordered aluminum oxide film formed by oxidation of the NiAl(110) atoms, as shown in the final model (Fig. 1A 0 surface is the most intensely studied metalqualitativsurfacee oxide,comparisons.but its structureTo solve theseandtechnicaltable S1).difficul-The stacking sequence and stoichiometry of the film is 4(Al O Al 2O ) 3 r was previously unknown. We determinedties,the structurewe firstbynoteextensivethat abtheinitioeffective electron potential is 4f 6r d6 r^7 f r c r d r; with c r S r r : modeling and scanning tunneling microscopynearlyexperiments.flat in theBecausevacuumthe top-region, atandthethustypicalalso deviatestunnelfrom theZcommonly Z r assumed Al O stoichiometry (10, 16, 17). most aluminum atoms are pyramidally adistancesnd tetrahedraoflly c5o–or10dinaAte,d,whichthe sur- correspond to currents2 3 of face is different from all Al O bulk phases. The film is a wide-gap insulator, The oxide unit cell covers 16 NiAl surface unit (4) 2 3 7 12 10 –10 A (Fig. 1). Then, we usecethells anobservd is comationmensuofrate to the substrate in although the overall stoichiometry of the filmÿ is notÿ Al2O3 but Al10O13. We the direction of the yellow diagonal of the unit propose that the same building blocks canTHbethatfoundMtsoncanthebesurfaceseasilyofobtainedbulk if ’t is replaced by the After smoothing the delta function (i.e., substituting it by a oxides, such as the reduced corundum (0001)vacuumsurface.Green function G, obeyingcell (Bro2wGmatrchinRg[) (10–12).Gaussian-likeIn the second function of finite width) the 3D integrals can 2 directiorn, th e filÿm isinÿcommensurate, which The detailed study of oxide surfaces is often fromGarll prevRious models brased Ron A, lwhereO bulkR denotesnecessitatethes somtipe apposi-proximationbein performedthe model- conveniently in a regular grid. Furthermore, ÿ 2ÿm ÿ 2 3 hampered because many surface-sensitive tech- structures. Al 2O commonly crystallizes in the ing. We chose a parallelogramthe-shapecond suvolutionper- theorem can be applied in Eq. (3) to tion and 2 3 h2 " , where is the work function. niques cannot be applied to bulk oxides, which corundum structure (s appÿhire), adopted by cell with two oxide unit cells placed onto 33 Substitution into Eq. (2) gives express ’s r0 as an inverse Fourier transform: are almost perfect insulators and are therefore many other trivalent metals as well. In this NiAl unit cells to model the interface (19). The difficult to examine with methods involving crystal structure, O atoms are arranged in hex- positions of the 28 O atoms in the surface layer 3 charged particles (electrons and ions). Ultrathin agonal close-packed planes, and the Al atoms (O ) are determined by2 the room’-tesmrp0erature G r r0 As r Bs r G r r0 d r Mts R G r R ’s r ’s r s G r R d r^ Z ÿ ÿ r ÿ oxide films, which are thin enough to avoid occu py tw/o oZutof thr ee oÿctahedrral int erstiÿtial STMr ima gesÿ. Almost coplanar with these O charge accumulation, represent an elegant sites between the oxygen planes, forming a atoms, 24 Al surface atoms (Al ) are ar- s 1 ik r 3 solution to this problem and are now widely buckled honey’cosmRb la:ttice. In the other bulk ranged in a nearly h(3)exagonal pattern that g~ k A~ k ik B~ k e 0 d k; 2 3=2 Z s s used as microscopic model systems. Further- alumina structures, such as g and k alumina, fits the 24 bright protrusions observed in more, ultrathin oxides are technologically metal atoms are found in octahedral and low-temperature STM images under certain In the TH approach, Eq. (3) is used to replace Mts by ’s in (5) important; their uses range from protective Eq.tetrah(1),edraleliminatingsites between tthehe cluncertaintiesose-packed O fromtunneltheing ctipondicompo-tions (Fig. 1E) (14, 20). The films against corrosion and mechanical wear where g~ k 1= k2 2 , A~ k and B~ k are the to insulating films in semiconductor devices. It sition and structure and greatly simplifying the calculation. s s Fourier transforms ofG r , A r c r ’ r , and is thus not unexpected that ultrathin oxides are However, this simplification has its own drawbacks be- s s r s fascinating researchers worldwide (1–9). cause (i) the tip structure does matter in many cases and Bs r cs r ’s r . Then, to obtain the tunneling matrix The ultrathin alumina film formed by Surfaces (ii), as explained previously, ’ is difficult to obtain accu- elements, Eq. (5) is substituted into Eq. (2) (shifting tip oxidation of NiAl(110) is widely used as a s variables up to R), which leads to model system for technologically important rately far from the surface. Therefore, we use Eq. (3) in the oxide-supported catalysts. Despite enormous opposite direction, i.e., to find ’s accurately in the vacuum 2 effort, its structure has remained unresolved h region. Thus, as illustrated in Fig. 1, the sample states are M R B r0 R ’ r0 (10–15), impeding progress in detailed under- ts ÿ 2m Z t ÿ r s standing of the influence of the oxide support propagated from a surface s close to the sample, through ’ r A r R d3r on the catalytic reactivity. A surface x-ray dif- vacuum, up to another surface t (in the vicinity of the tip), ÿ s 0 t 0 ÿ 0 fraction (XRD) study recently presented a where both wave functions can be substituted accurately in h2 structural model for this surface (16). How- ~ ~ ~ Eq. (2). Notice that the method is entirely symmetric and At k ik Bt k g~ k As k ever, this structure has two bonds with un- 2m Z ÿ that it can be seen alternatively as the propagation of the tip physically short lengths (Al-Al, 2.08 ); Al-O, ik R 3 ik B~ k e d k: (6) 1.51 )) and was revealed to be unstable in our wave functions from t to s [9]. A convenient definition s ab initio calculations. Here, we combine of t and s is as isosurfaces of constant electron density, atomistic information gained by scanning Finally, in Eq. (1) we broaden the delta function with an r . The value must be chosen so that the tunneling microscopy (STM) with ab initio r 0 0 empirically fitted self-energy to account for the coupling of isosurfaces 2 are close enough to the physical surfaces to density functional theory (DFT) to produce a the finite calculated systems with their respective bulks. structural model, which differs considerably ensure an accurate description of the first-principles wave Thus, using fast Fourier transforms for the convolutions, functions,Fig. 1. (A) Topbuandt far(B) sideenoughview oftothemakeDFT- andadequateSTM-basedthemodelapproxi-for the ultrathin aluminum oxide film on NiAl(110). (C to E) Experimental STM images of the film at [(C) Mandts(D)]canroombe evaluated, for all tip positions and bias voltages, First work1Institut fu¨r Materialphysikon Si(100)and Centre for Computa- mationtemperatureofanda (E)constantlow temperature.efOxidationfectiveSamplepotentialbias voltage ofin andtheNiAltunnelingvacuumcurrent values are (C) tional Materials Science, Universita¨t Wien, A-1090 j2.5 mV/1.4 nA, (D) j0.2 V/0.9 nA, and (E) j0.5 V/0.3 nA. Two oxide unit cells inareamarkedsinglebycomputation. The starting point is the values of 2 region. In practice, surface integrals can be more effi- Wien, Austria. Institut fu¨r Allgemeine Physik, Tech- white rectangles, the diagonal along which the oxide is commensurate is yellow,c r , and’ rthe, and ’ r stored in the points of a uniform grid Yin, Cohen,nische Universita PRB¨t Wien, A-1040 (1981)Wien, Austria. ciently implementedKresseif they areet transformedal, Scienceinto volume(2005) s t parallelogram enclosed by the yellow and white lines delimits the simulation cell. Greenwithin rectanglesthe regions of the broadened surfaces and . *To whom correspondence should be addressed. integrals.and squares Thighlighto this end,oxygenweatomsrepresentedin a squarearrangement.by the constraintCircles indicate the Al and O s t E-mail: [email protected] positions from (A) and (B) in the corresponding colors. (F) Closeup of the structure. In the following we show simulations for the P H Y S I C A L R E V I E W L E T T E R S week ending PRL 94, 056103 (2005) Si 111 - 7 7 surface and compare them with experi- 11 FEBRUARY 2005 1440 3 JUNE 2005 VOL 308 SCIENCE www.sciencemag.org mental data. This surface presents a good experimental true r G constant(a) potential+1.5 V (b) reproducibility+1.4 V and a rich varietywith theof topographicfeedback loopand ofspec-f) in every point of the surface. r' potential troscopic characteristics [10],Thus,makingmapsitofanI x;idealy; Vbench-are obtained at many different bias Method for the simulation of mark for STM-STS simulations. The experiments were sample tip voltages V, with the tip-sample distance z x; y (the topog- carried out in ultrahigh vacuumraphy)(basedeterminedpressure beloby wa 5fixed control current and bias STM images 10 11 Torr) with a homebuilt variable-temperature STM Σt ÿ voltage. The CITS maps of @I x; y; V =@V are obtained vacuum described elsewhere [11]. Clean reconstructed surfaces region from I x; y; V by direct numerical differentiation [12]. were prepared by flashing the samples at 1150 C, after Paz et al, PRL (2005) Ab initio calculations of the Si 111 - 7 7 surface Σs sample Σs Σt tip carefully degassing at 600 C for several hours. The (c) -1.5 V (d) samples-1.5were V then slowly cooledwere predownviouslyto roomreportedtempera-[13,14]. In the simulations of Exp Theoryture (RT) and transferred tothisthe STM.work,Allthethesurfaceexperimentswas mimicked using a repeated FIG. 1. A schematic view of the propagation of the sample slab geometry with four layers of silicon (the lowest of wave function through the vacuum region up to the proximities were carried out at RT. For the present measurements we of the tip by means of the vacuum Green function (left panel). used W tips electrochemicallythemetchedsaturatedfrom polycrystallinewith hydrogen atoms). Two tips were con- The approximation of a flat effective potential in the vacuum wires. They were cleaned insidered:situ, intheultrahighfirst wasvacuum,a tungstenby bcc pyramid pointing in the region between the electrodes (right panel). heating at temperatures close(111)to 800direction,C and withby field20 emis-atoms; the second tip was made of 10 silicon atoms [15], in which all dangling bonds were 056103-2 (e) 0.6 Experiment saturated with hydrogen atoms except that of the apex. The Si tip wave functions of the surface and tips were calculated W tip 0.4 within DFT in its local density approximation [16], using ] the SIESTA code [17,18]. Core electrons were replaced by m n
[ norm-conserving pseudopotentials [19], whereas valence
Z 0.2 electrons were described using a double- plus polariza- tion (DZP) basis set. A real-space grid with a plane-wave 0.0 cutoff of 100 Ry was used to perform some integrals and to 0 1 2 3 4 project the final wave functions ’s and ’t. Only the ÿ point Distance [nm] was used in reciprocal space. The geometries of the surface FIG. 2 (color online). (a),(b) Empty state and (c),(d) occupied and tips were relaxed independently until the maximum re- state images of the Si 111 - 7 7 surface at 0.2 nA. The left sidual force was below 0:04 eV=A . The values of the tun- panels show experimental data, while the right panels represent nel current I x; y; z; V , calculated for each tip as explained simulations with a Si tip, using the same gray scale. (e) Profile previously, were dumped on files which were then read by comparison between experiment and theory for positive sample the experimental data-acquisition program [12] and pro- voltages, along the solid lines in (a) and (b). The three curves cessed in exactly the same way as the experimental data. have been shifted to make their minima coincide. Figure 2 shows experimental and theoretical topo- graphic images and profiles. The experimental images sion against a previously degassed Ta foil (typical voltages show typical features of the empty and occupied states, and currents were 1 kV and 10 A). Atomic resolution in like the different apparent heights of the faulted and un- STM images, however, was usually obtained after inten- faulted half cells, and the observation of rest atoms in the tional slight tip-sample contacts, which may lead to Si occupied states only, which are well reproduced in the termination of the original W tips. Constant-current STM simulation with a Si tip. The more quantitative profile topographies were measured at tunnel currents between 0.1 comparison with both tips, in the lower panel, shows that and 2 nA, with sample bias voltages between 2 and the agreement is excellent for the Si tip and considerably 2 V. Spectroscopic data were taken in the current-imag-ÿ worse for the W tip. In the former, the corrugation is ing-tunneling spectroscopy (CITS) mode [10], which in- appreciably higher, in agreement with the experimental volves acquiring, simultaneously with the constant-current observations that an enhanced resolution is usually ob- topography, a current vs voltage (I-V) curve, (measured tained after intentional tip-surface contacts.
FIG. 3 (color online). CITS comparison between experiments (top pictures) and simulations using a silicon tip (bottom pictures). The first column shows topographic images at the set point of 2 nA and 1:75 V. The remaining images represent @I=@V as a function of V and at the constant tip-sample distance determined by the set point. 056103-3 Clarification of the structure and prediction of a new surface phase of ZnSe
Ω
(a) (b)
µ
Garcia, Northrup, APL (1994) (c) (d)
(e) (f) Zn Se Precise control of simulation conditions
Point defects: Great experimental complexity
In a calculation they can be “prepared” (isolated or in complexes) and their energies of formation and bonding computed
Help in the analysis of experiments, and direct testing of hypothesis Mechanism for p-doping saturationZn in ZnSe Se
Zn Se +2 Vse
+2 Vse p n log p
Experimental Ec saturation level
Ev +2 Compensation of two holes by VSe
Garcia, Northrup, PRL (1995) log NSe letters to nature
and unusual monoclinic and orthorhombic phases for some values octahedra, neglected in our effective model of PSN. of the n parameter. The analysis of our results points to the different Here, we use the total energy of our alloy effective hamiltonian in valence of the B atoms (Sc and Nb) as the main reason for the Monte Carlo simulations to calculate ®nite-temperature properties
existence of these anomalous properties. of some selected Pb Sc12xNbxO3 structures. We consider structures Here, we use the numerical scheme proposed in refs 7 and 8, with the following sequence of four B-planes along the [001] which consists of constructing an effective hamiltonian for the alloy direction (Fig. 1): a niobium-poor plane for which x 0:5 2 n, a from ®rst-principles calculations, to predict the properties of second plane made of 50% scandium and 50% niobium (x 0:5), a 7,8 Pb Sc12xNbxO3 (PSN) structures. This effective hamiltonian niobium-rich plane with x 0:5 n, and a fourth plane similar to contains a local-mode self-energy (expanded in even-order terms the second one. The B atoms are randomly distributed within each up to fourth order of the phonon local soft modes), a long-range of these four planes. The studied structures only differ in the value of dipole±dipole interaction, a short-range interaction between soft the parameter n, which is allowed to vary from zero (in the case of modes (quadratic order of the soft modes), an elastic energy the disordered PSN alloy) to 0.5 (in the case in which the ®rst plane (expanded to second order of the strain variables), and an inter- is entirely made of Sc atoms and the third plane is fully occupied by action between the local modes and local strain (second order of the Nb atoms). We use 12 3 12 3 12 supercells, implying that the soft modes and ®rst order in strain). It also contains an energy term sequence of the four different B(001) planes is repeated three describing the effects of the atomic con®guration on the phonon times, to get well converged results7,8. local soft modes (which are directly related to the electrical Figure 2a shows the cartesian coordinates (ux, uy and uz)Ðalong polarization) which can be written as
¢E 2 Z*u 2 j S eà 1 0.08 ^ i× ^ j ji ji a i " j # 0.07 where i runs over all the cells and j runs over the three nearest- letters to nature 0.06 neighbour shells of cell i. Here ui is the local soft mode in cell i, Z* is the Born effective charge associated with the local soft modes, and eÃji 0.05
is the unit vector joining the B site j to the B site i. The variables {jj} u = u and unusual monoclinic and orthorhombic phases for some values octahedra, neglected in our effective model of PSN. 0.04 x y characterize the atomic con®guration: jj 1 21 indicates that of the n parameter. The analysis of our results points to the different Here, we use the total energy of our alloy effective hamiltonian in uz there is an Nb (Sc) ion in cell j. Sji is an alloy-related parameter that 0.03 valence of the B atoms (Sc and Nb) as the main reason for the Monte Carlo simulations to calculate ®nite-temperature properties Local modes (a.u.) only depends on the distance between the B sites i and j, and ux(LDA) = uy(LDA) 9±12 0.02 existence of these anomalous properties. of some selectedis derivedPbfr Scom12x®rst-principlesNbxO3 structurcalculationses. We consideron smallstructurescells. uz(LDA) Here, we use the numerical scheme proposed in refs 7 and 8, with theEquationfollowing(1) indicatessequencethatof¢fourE can B-planesbe viewed asalongthe interactionthe [001] 0.01 which consists of constructing an effective hamiltonian for the alloy directionenerg(Figy. 1):betweena niobium-poorthe dipole momentplaneZ*forui associatedwhich xwith0the:5 2siten,i a and an internal electric ®eld S 2 j S eà induced by the B ions of 0.00 from ®rst-principles calculations, to predict the properties of second plane made of 50% scandiumj andj ji50%ji niobium (x 0:5), a sites j on the site i. Moreover, we numerically ®nd that the S b 7,8 : n ji Pb Sc12xNbx O3 (PSN) structures. This effective hamiltonian niobium-richparameplaneters allwithhavexnegative0 5 signs., andAs a resufourlt,ththeplaneradialsimilarelectricto 5,000 3+ 5+ contains a local-mode self-energy (expanded in even-order terms the second®eldone.2jThejSjieÃjiBactingatomsonarsitee randomlyi and induceddistributeby a Scd w(Nbithin) ioneach ) up to fourth order of the phonon local soft modes), a long-range of these foursittingplanes.on siteThej is directestudiedd fstructurrom the sitees onlyi to thediffersite inj (respectivthe valueely,of –1 4,000 from site j to site i). This is consistent with an electrostatic picture of dipole±dipole interaction, a short-range interaction between soft the parameter n, which is allowed to vary from zero (in the case of (pC N 34 PSN since Sc (Nb) ions are negatively (positively) charged with d modes (quadratic order of the soft modes), an elastic energy the disordered PSN alloy) to 0.5 (in the case in which the ®rst plane 3,000 respect to the average B-ion valence of 4+. The effective hamiltonian ficient (expanded to second order of the strain variables), and an inter- is entirelyapproachmade ofyieldsSc atomsgood agreandementthe thirdwith directplane®rst-principlesis fully occupiedresultsby action between the local modes and local strain (second order of the Nb atoms).in PZTWande usePSN12allo3ys127,8. P3rev12ioussupercworksells,(see refsimply7±8)inghavethatfoundthe 2,000 soft modes and ®rst order in strain). It also contains an energy term sequencethatof athelinearfourrescaldiffering ofentthe simulationB(001) planestemperatureis repeaoftentedleadsthreeto 7,8 1,000 describing the effects of the atomic con®guration on the phonon times, togoodget wellagreementconvergedwith experiment.results . Adopting this approach again Piezoelectric coef local soft modes (which are directly related to the electrical Figurehere,2a showsour temperaturesthe cartesianare rescacoordinatesled down (byu a, factoru andofu2.5)Ðsoalongthat the theoretical Curie temperature in disorderx edyPSN is zforced to 0 polarization) which can be written as 13 match the experimental value . The need for such rescaling may be c 10,000 due to higher perturbative terms or the rotation of the oxygen
¢E 2 Z*u 2 j S eà 1 0.08 8,000 ^ i× ^ j ji ji a
i " j # 33 χ 0.07 6,000 where i runs over all the cells and j runs over the three nearest- 0.06 plane composition Pb(ScxNb1 x)O3 neighbour shells of cell i. Here ui is the local soft mode in cell i, Z* is 4,000 − the Born effective charge associated with the local soft modes, and eÃji 0.05 4 50% ection is the unit vector joining the B site j to the B site i. The variables {jj} u = u Dielectric susceptibility x y 2,000 0.04 3 Nb-rich characterize the atomic con®guration: jj 1 21 indicates that x = 0.5 ν uz there is an Nb (Sc) ion in cell j. Sji is an alloy-related parameter that [001] dir 0.03 Local modes (a.u.) 2 50% −0 only depends on the distance between the B sites i and j, and ux(LDA) = uy(LDA) 0.0 0.1 0.2 0.3 0.4 0.5 is derived from ®rst-principles calculations9±12 on small cells. 0.02 Parameter ν uz(LDA) 1 Sc-rich Equation (1) indicates that ¢E can be viewed as the interaction 0.01 x =Figure02 .Properties5 +of theνstudied structures as a function of the n parameter at 20 K. a, Average cartesian coordinates u , u and u of the local mode; b, the d piezoelectric energy between the dipole moment Z*ui associated with the site i x y z 34 Figure 1 0.00Schematic representation of the ferroelectric structures considered here. The coef®cient; c, the x33 dielectric susceptibility. The ®lled symbols in a represent the and an internal electric ®eld Sj 2 jjSjieÃji induced by the B ions of 3 5 3 5 3 5 materials are Pb Sc0:5nNb0:5 2 nO3=Pb Sc0:5 Nb0:5 O3=Pb Sc0:5 2 nNb0:5nO3= cartesian coordinates ux LDA, uy LDA and uz LDA of the local modes predicted by sites j on the site i. Moreover, we numerically ®nd that the Sji Pb bSc3Nb5O structures. The internal electric ®elds acting on the four B planes are direct ®rst-principles calculations made using the local density approximation9±12. a.u., 0:5 0:5 3 Design of materials parameters all have negative signs. As a result, the radial electric indicated5,000by arrows. arbitrary units. 3+ 5+ ®eld 2jjSjieÃji acting on site i and induced by a Sc (Nb ) ion with optimized )
NATURE–1 | VOL 413 | 6 SEPTEMBER 2001 | www.nature.com 55 sitting on site j is directed from the site i to the site j (respectively, 4,000 © 2001 Macmillan Magazines Ltd from site j to site i). This is consistent with an electrostatic picture of piezoelectric response (pC N 34 d PSN since Sc (Nb) ions are negatively (positively) charged with 3,000 respect to the average B-ion valence of 4+. The effective hamiltonian ficient George, Iñiguez, Bellaiche approach yields good agreement with direct ®rst-principles results in PZT and PSN alloys7,8. Previous works (see refs 7±8) have found 2,000 Nature 413, 54 (2001) that a linear rescaling of the simulation temperature often leads to 1,000 good agreement with experiment. Adopting this approach again Piezoelectric coef here, our temperatures are rescaled down by a factor of 2.5 so that the theoretical Curie temperature in disordered PSN is forced to 0 match the experimental value13. The need for such rescaling may be c 10,000 0 ν 0.5 due to higher perturbative terms or the rotation of the oxygen
8,000 33 χ 6,000 plane composition
4,000 4 50% ection Dielectric susceptibility 2,000 3 Nb-rich [001] dir 2 50% 0 0.0 0.1 0.2 0.3 0.4 0.5 Parameter ν 1 Sc-rich Figure 2 Properties of the studied structures as a function of the n parameter at 20 K. a, Average cartesian coordinates ux , uy and uz of the local mode; b, the d34 piezoelectric Figure 1 Schematic representation of the ferroelectric structures considered here. The coef®cient; c, the x33 dielectric susceptibility. The ®lled symbols in a represent the 3 5 3 5 3 5 materials are Pb Sc0:5nNb0:5 2 nO3=Pb Sc0:5 Nb0:5 O3=Pb Sc0:5 2 nNb0:5nO3= cartesian coordinates ux LDA, uy LDA and uz LDA of the local modes predicted by 3 5 9±12 Pb Sc0:5 Nb0:5 O3 structures. The internal electric ®elds acting on the four B planes are direct ®rst-principles calculations made using the local density approximation . a.u., indicated by arrows. arbitrary units.
NATURE | VOL 413 | 6 SEPTEMBER 2001 | www.nature.com © 2001 Macmillan Magazines Ltd 55 What of Anderson’s claim? Do we understand more? anion ion bond cation
Ionic Covalent valence electrons cation Electronegativity difference is enough!
Metallic Molecular Classification involving ionic radii Simulation as a route for comprehension (1)
It provides more “experimental data” to construct theoretical models Exploration
Can serve to test hypotheses in optimal conditions. Simulation as a route for comprehension (1I)
Low-level High-level theoretical ingredients physical concepts
Charge density Electronegativity Wave functions Bonding Energy Parametrization of simple models
One can use first-principles methods to compute parameters for simple but relevant and realistic models Parametrization of a Heisenberg model from the electronic structure.
Relevant for magnetic properties Ferroelectricity
Cubic Tetragonal
BaTiO3
Orthorhombic Rhombohedral + Ti Basic distortion Ba involved in ferroelectricity _ O (soft mode)
p Relevant degree of freedom
Model system
Local mode u
Lattice Strain Zhong, Vanderbilt, Rabe, PRL 73, 1861 (1994)
1 E ( u ) = J u u short { i} 2 ij,αβ iα jβ i=j αβ !̸ ! Effective-Hamiltonian parametrized ab-initio Phase transition sequence obtained from Monte Carlo simulations
T(K) P H Y S I C A L R E V I E W L E T T E R S week ending VOLUME 93, NUMBER 11 10 SEPTEMBER 2004
10 3 from 10ÿ to 10 atmospheres. Several calculations were CO. This result is at variance with the ‘‘constrained done on a system with 30 30 surface sites (450 bridge thermodynamics’’ study in which the CO2 formation plus 450 cus sites), but the vast amount of calculations was forbidden. Now we find that catalysis is practically was done for a 400 sites system. The results for both poisoned by adsorbed CO. The encountered situation is in system sizes were identical. The kMC simulations were fact close to that where the RuO2 catalyst will be reduced run until steady state is reached [cf. Fig. 1(b)]P H. TYhenS I tCheA L byRCOE VtoI EthWe purLe ERuT metT EaRl. S week ending VOLUME 93, NUMBER 11 10 SEPTEMBER 2004 movies [19] were recorded [cf. Fig. 1(c)] and the statistical Figure 2 summarizes the results by showing the various analysescorofrelatthione frefunctquenciesional of(e.gth.,ethevarGGA)ious elementare thenarynot stcready-stu- atmente suirmplfaceicitstrlyuctincludesures as twelhe el fafescat omapf O suofrfacethe vacancies. processes were performed. T66he latter also gives the toD.E.talJiang,TE.A.OFsCarterin (/pSurface, pScience) space583 (2005). The60–68highest activity is found to cial, as long as the trends of the energetics of the various ACOtotalOof2 26 different elementary processes are possible TOFs ofinttehreplayCOi2ngforpromatcessesion. Tablear3e described correctly. In bethiisn verony gothoisdlatagtreementice, and walilthwerethe caearefrlyulexplyeranimentalyzeadl by DFT to Adsorption energy [E = E E E ] of S atom on Fe(110), the smallest Fe–S distance (d ), the height of S atom over The results show that catalytic reaction condad S/Fe-slabitionsÀ aFe-slabre À resultS s by Peden and Goodman [14], aFenAdS occurs whenever sense, the frequentlyFe(1re1quest0) (h), andedthecheS vibrationalmical accufrequenciesracy, includingfor theobtadsorbateain –substratetheir patstretchhways(xFeASa)nandd energthe two yfrustratedbarrierstranslations[3,18]. Table I only established when both CO and O2 partial pressures the environmental conditions lead to the dynamic coex- the description of in(dxitv) idual processes appears to be a summarizes the adsorption energies and the diffusion and exceed certain values below which the surfacea is in a a a b c misleading concept. SiteAt least for theOTpresent systemist,SBencea phareactse iondescrLBenergibed aboy baverrunderiersLBuse(ii)d. RforecentExperimenthlye, WkaMCtng study, as thermodynamic equilibrium phase of RuO 110 , charac- Ead (eV) 2 4.408 (hos) et a5.271l. [15](ts) have also5.788mea(min)sured the TOFs– for the RuO– 2 110 careful combination of DF˚ T and stat istÀ ica l mechanicsÀis obtainedÀ from DFT calculations with a 1 2 surface teristic for low oxygen pressudFeresAS (Aa)nd routinely2.038observed su2.144rface at T 3502.180K for various pressu2.19 res, and2.17in Fig .3 more important. h (A˚ ) 1.917 1.655 unitcell.1.526In a systematic1.567study of CO1.43and O (co)adsorp- under ultrahigh vacuum (UHV) cond1 itions: all bridge we compare their results with ours. The agreement is xFeAS (cmÀ ) 396 353 325 – – The methodology will b1e used to study CO oxidation tion at various coverages, we found adsorbate-adsorbate sites are occupied by oxygenxtat(cmomÀ )s and all cus117i,sit130ies are aga125i,in 231far better 111,tha257n what one would– have –expected: over RuO 110 [the surface is sketched in Fig. 1(a)], as interaction to be always smaller than 150 meV. Thus, empty. For higher2 pressu res, Thethreenaturesuofrfacethe criticalcondpointitionis givens arein parenthethesestheoret(min = minimumical and, ts =extransitionperimentstate,aandl TOhosF=vhigheraluesorderat saddlethe optpoint).i- this system has recentalyThisrework.ceived considerable attention lateral interactions in this system are small (compared worth mentioning, which arbe, e.g., obtained for T mum pressures are practically identical, and also the as a highly active modelRef.cat[25]a.2lyst (see Ref3 . [15] and ref- to the other energies), and will therefore be neglected. 600 K, p 1 atm, and p c Ref.10[9]ÿ. ; 20; and 10 atm. optimum pressure conditions (when the ‘‘dynamic phase’’ O2 CO CO adsorption into vacant cus or bridge sites is non- i Aterencesthe lothereiw p n),. Ianllfactbridge, thisawndasaprell cusviouslysitescaalrleed thise Rreau lized) agree very well. Because the errors due to the CO dissociative, while oxygen adsorption is dissociative and cov ere catd waitlysth oxygen, but recentatomexps [cfer. iFigment. 1a(a)]l a.nTdhtheoretis is essenical-workGGAhas are in the range of 0.1–0.3 eV, this good agreement requires two vacant neighboring sites, i.e., a br-br, cus- tially tshohe twhern mothatdyatnareamiclisthigichOp2 pressuphasere[2, ,3]the. TRu(0here0is01a) surbetfaceween theory and experiment, seen in Fig. 3, may seem O2 cus, or br-cus pair. The adsorption rate per free site is is transformed into an epitaxial RuO2 110 film for(seetuitous. However, it is worth noting that the position of noticeable desorption/adsorption dynamics, i.e. , about ~ every 40Ref. s[,16two] andof referencesthe 900 O adatthereiomn)s.oIft tihes by30now30also estthab-e optimguivmencatbyaltyheticloefficiencycal stickiinngcTo;efficientp space , Sa,nadndtothe kinetic ~ simulatDisociationlisheion celd tlhatdesorthisb of(maRuO Hin2lyS110 finrom Fe(110)sucusrfacesites)actauats Oes.tThheecatalsomeytic extentimpialsongementthe va:lueÿadof thSp=e TOFp2aremknotBTdet. Hereerminemdis the mass 2 2 resultinreactg vacaionncies, and,araeltthoughen hrapiddomaly ifilnleboundad againries(waitnhdinstepsbyartehe energetof thicse gofas-phaa sinsegulamoler proculecess, aalonend k, BbutisbythteheBoltzmann 1 ns), presentmostJiang,of, thei tCarter,her tiinfluenceme Surf.with isScioxygennot (2005)sig. nOificanlyntra[1rely7]. TCOhe suractfaceion ofconmanstyaplayersnt. The. Apparate orentf thely, tihemeDFreTverse-GGAd caprolcu-cess (desorp- adsorbs,unaitndcelevenl is riefcittadonges,ulaitr aratndhercontdesorainsbstwagao adsorin thaptnion latsitiones s descrtion)ibe tthenhe diffolferenceslows betfromweentthhee vrelatariousionsu: r-ÿdes=ÿad initiates[3]:a reactbridgeion(.br)Theanodveracolordl COinat2 forivelymatunionsatraturate undered (cus) facesitesprocessesexp Fbetb tertha=nkBtheT i,ndwhereividuaFl babsolutis theefvreealuesenerg. y of the 12 2 1 ÿ these cond[cf. Figition. 1s(a)]is w. Eitithher0:9of t10hemcmcan bes emptveryyolorwo.ccupieIdnbtyhis respadsorect,beitdisspimeciesport(approxant to reaimatlizeedthatby Ea bcombi), andna- T; p is the ÿ ÿ ii OFoorrpCO, an20d adsoratm, battheesitdiuatffusionion corcarespn goondsbr totobr, brtionto of dchemifferenticalcaplculatotentionial so(employf the gains-phag disefferentmoleapcule- [2]. The CO what wacus,s precusviouslyto cus,suggestor ecusd totbeo brthat. Sofincehigthhchemis comprical isesproxtheimatSion~ Ts) aorreofthustheorobtyaianendd expfromerimentthe cacouldlculatehavd teotal energy activity [3], and this suggestionbr is now confirmed. The spoiled tsuhe rfacesdescr iptofiondesor. Tptheioncontogetsistenthertwreatithmentdetaiofledabllalance. The possibility of missing O atoms, we notFig.e t5.hatDFT-GGAour treatminimum- energy path for H2S dissociation on Fe(110). time tKinetic-Monteo reach the steady Carlostate imethods remarkably for long,catalysisin pa-- rparametrizationticipatoninlyg prouncercessestainimplty hereies thatarisesthe optfromimuthme mvibratix ofionaO l proper- particular when compared toH tSheonpicosethe LBcondsite. tTheimestructuresscale foranthed CinitialO attiestheofsuthertfacehpresencee trais ndescrsitofiontheibestnedatwelarbye l:wHhthichatom,e abutrcondaanmparedslatnceesoftointo uncer- 2 Reuter et al, PRL (2004) of the underlying atomisticandprofinacessesl states,[cftogether. Fig.wi1th(b)]severa. l intermediatetaintiesisoliatedn theHSdesor(a). Despiptionte this,rateH–Sby bonda factisorstillofsig-10 (at most Finally, an interesting mix ofstructuresCO mole, areculesalsoanshownd O atinomFig.s 5. We predict100). Fnificanor thtlye dlengtiffusionhened atpro1.49cessesA˚ . Wwee fouusendathatprefactor of at both bridge and cus sites isthatestablH2Sishebreakd [sa onetypicaH–Sl snapbond- by rotating one1012 HzHS, willwhichsponhataneouslys an uncerdissoctaiateinttoy Hofanda factS overor of 10. We H atom towards a nearby surface Fe atom and the opposite Fe atom if the HS group rotates away shot is shown in Fig. 1(c)], with average occupation carefully checked that these uncertainties do not affect br custhe H–S bond dissocbr iates over that Fe atom. The from H. numbers NCO 0:11, NCOdissoc0iation:70, barriNO er is0:0.1089, eV,anindicad ting the easeour belowFourrepFeoratomsted resultcoordinats ande arounconclusiond S whens. DetH Sails of the cus 2 N 0:29. The dynamics ofof breakthis suingrfacethe isfirstextH–Sremelybond of H S on resides at the LB site, with two of them farther O 2 employed new methodology and of the various test cal- fast. It is mainly due to CO desorFe(1pt10).ionAtanthed adsortransptitionionstat. Thee, the length of theculationaways wfromill bethe publS atomishethand elsethe otherwheretwo.[18]Fig. . A5 detailed breaking H–S bond is 1.45 A˚ , only 0.07 A˚ longer shows that H S breaks its first H–S bond on one rate of CO2 formation is a factor of 0.0004 lower, but truly balance of the scena2 rio was carefully checked by con- significant, namely 4:6 1018thancmthe2 inits 1ial. Itstatis e.domAfterinattheed transition state, of the two more distant Fe atoms, with a barrier one H ÿatomÿbreaks away and moves to the TF sitefirmingof t0.10hat eV.theWeearalsolier examinresultsedobtdissocaineiationd by of‘‘atomistic by the reaction COcus Ocus CO . The CObr with a 1.40 eV2 energy drop. Then the system des-thermoHdy2S naon moneicsof’’the[3]twoacloserre eFexactatomsly .reproThe baducerrierd by the Ocus CO and COcus Obr ! CO reactions also ! 2 cends !through2 a flat step to reach the final statepresentis st0.11atisteV,icaalmosl mtechathe nsameics tareats thementformer., if Thissurface reac- contribFuItGe., 1thoug(colorh tonheilirne)rat. eswith(a)arePerspanbyenaergyecfacttivedroorvpiofeowf 0.200of.30teV,haenRuOwhichd corres110 pondstions aindicare nottesathatlloweH2dS tcano occubreakr. its first H–S bond br br 2 FIG. 2 (color online). (a) Steady-state surface structures ob- 0.01 lower than the first oneto. benThedingCOof the HSO group towCOards2 H. In the final on any of the four Fe atoms coordinated around surface, illustrating thestate,two thepromHSinentgrou adsorp is stabipt!ionlizedsitesby inta0.30tinhedeVbybyab KiniitnetioS withkicMCMontacveryalculate lowCionarlobarrier.s atrTunsIn600wereourKearlierapnedrforvstudyarmeiousdof for about reactionrectisanginulasigrnsificaurfacent. unTithiscelresultl. Theseis sitdiesfferentare labelefromd as the br CO and O12 0pa0r0tiadlifpressuferentresT. I;npaCOll nonwh; pO2 itconde areaits,ionthse caoveragevering the tem- what one(bridge)wouldanedxtpheectcusfrom(coordtheinatreactivelyionunenergsaturaty baedr)rsitierse, and both br cus site occupatperation isurdome raingenated300(>90%K <) bTy
QM-MM: Precise treatment (QM) of a special part of the system. Rest treated at a lower level of quality.
Matching of atomistic methods with the continuum approximation Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) Escaping free-energy minima
Phase transition mechanisms, new crystal structures
Diapositiva 11
F(s)
s
Metadynamics
(Laio & Parrinello, 2002; Martonak et al., 2003) The Torii Analogy (Prof. H. Nakamura) First-principles calculations
Theory Experiment Thanks!
418 418
During fitting, the parameter K4 was fixed at the value towards room temperature by fitting data from room- determiDurinednginfitttheing,prevtheiousparametstudy (Fislerer Ket4 al.was2000)fix,edtemperaat theturevaluexperie ments.towards room temperature by fitting data from room- 418 whiledetermiK2 wasnedallowedin tothevary.prevTheiousrelaxedstudyfitting(Fisleralgo- etTheal. model2000),describetemperad in theturepaperexperiby ments.Fisler et al. During fitting, the parametrithmer alloK wswasthefixedsheatll theto movaluvee relativetowardsto roomits coretemperato (2000ture by) reprodfittingucesdataexperimefrom room-ntal structural data with a createwhilea dipole4 K2 .wasThis allowedreproducestothevary.polarizatiTheon relaxedof the highfittingdegreealgo-of accuracyThe, but doesmodelso at thedescribeexpensedofithen the paper by Fisler et al. determined in the previousrithmstudy allo(Fislerws ettheal. she2000)ll, totemperamoveturerelativeexperiments.to its core to (2000) reproduces experimental structural data with a while K was allowed tooxygevary.n Theatom.relaxedThe fittingCoulombalgo-interactionThe modelis excludeddescribestabilityd in theofpaperthe strubycture.FislerWeethaveal. reduced the accuracy 2 within each CO group. of the fit to produce a stable model, as given in Table 1. rithm allows the shell to createmove relativea dipole3 to its. Tcorehis toreprod(2000uces) reprodtheucespolarizatiexperimeonntalofstructurathe lhighdatadegreewith a of accuracy, but does so at the expense of the Potentials that act on the CO group from outside Table 2 shows how well the potentials created reproduce create a dipole. This reprodoxygeuces then polarizatiatom. onTheof theCoulo3 highmbdegreeinteofractionaccuracy, isbutexcludeddoes so at the expenstabilityse of theof the structure. We have reduced the accuracy interact with the oxygen shell; these include a Bucking- experimental data. The deviation of the elastic constants oxygen atom. The Coulomb interaction is excluded stability of the structure. We have reduced the accuracy hamwithO–Oinaneacd ahcationCO–O3 group.interaction. The A, C and from the experimentalof theresultsfit tois producenot large enougha stabltoe model, as given in Table 1. within each CO group. of the fit to produceq a stable model, as given in Table 1. 3 are parameters of the Buckingham function are opti- cause the crystal to distort unrealistically when pressure Potentials that act on the PotenCO grouptialsfromthatoutsidact eonTablethe 2COshows3 grouphow welfroml the potoutsidentialsecreatedTablereprod2 showsuce how well the potentials created reproduce mized during3 fitting. r is the interatomic distance. is applied. The main mechanism of structural change interact with the oxygen shell;interathesect includwithethea Bucoxygenking- experimshell; theseental datainclud. Theedeviaa Buctionking-of the elasticexperimconstantsental data. The deviation of the elastic constants comes from the pressure induced soft mode which is best ham O–O and a cation–O interaction.r=q The6A, C and from the experimental results is not large enough to InteratomicU potentialsrhamAeO–OÀ CranÀd :a cationq –O interaction. 2TheobserveA, Cdanthroughd q thefromphononthe distribuexperimetion ntalcurves.resultsThe is not large enough to are parameters of the Buckið Þ ¼ngham functionÀ are opti- cause the crystal toð Þdistort unrealistically when pressure are parameters of the Buckingham functionphononaredispersionopti- causcurvese werethe crystcalculataledtobetweendistorttheunrealistically when pressure mized during fitting. r isPotenthe tialsinteratomicassociateddistawince.th interactioisnsapplied.between atomsThe mainof mechanism of structural change gamma point and the wave vector [½ 0 2], as given in the mizedsame COduring3 group fitting.act on ther oxygeis comethen intecsofromre.ratomicThtheeseprearessuredistainducence.d soft modeiswhichapplied.is best The main mechanism of structural change r=q 6 Fig. 1, and the gamma point and [0 0 1.5], shown in U r AeÀ CrÀ :an O–O Buckingham interaction2 andobservea C–Od throughMorse in-the phonon distributioncomecurves.s fromThe the pressure induced soft mode which is best ð Þ ¼ À teraction. For the rMorse=q intera6ð ctionÞ the A and D values Fig. 2. Although the dispersion curves were not used Potentials associated with interactioU r nsAebetwÀ een atomsCrÀ of: phonon dispersion curves were calculat2 ed between the are optimið Þ ¼zed by fittinÀg, r is the predetermined bond during the fitting,ð Þ theyobservestill showd throughthe basic featuresthe phononof the distribution curves. The the same CO group act on the oxygen core. These0 are gamma point and the wave vector [½ 0 2], as given in 3 lengthPotenandtialsr is theassodistanceciatedbetwewithen interactioatoms. ns betwexperimeen atomsental ofcurvesphononwith a smalldispersionsystematic curverror.es were calculated between the an O–O Buckingham interaction and a C–O Morse in- Fig. 1, and the gamma point and [0 0 1.5], shown in 2 2 gamma point and the wave vector [½ 0 2], as given in teraction. For the Morse interathe ctionsametheCOAA anr3 rdgroup0 D valuesact Fig.on the2. Althooxygeughnthecore.dispersionThesecurvarees were not used U r D 1 eÀ ð À Þ 1 : 3 are optimized by fitting,ðran0Þ ¼is O–Othe predetermineÀBuckinghamdÀbondinteraduringctionthe fitting,and aðtheyÞC–OstillMorseshow thein-basic featuresFig. 1,ofandthe the gamma point and [0 0 1.5], shown in h i High-pressure phase transition length and r is the distanceTheteractiomobetwedelenalson.atoms.containsFor thea Morsethree-bodexperiminteray term,ctionentalto maintaincurvtheesAwithanda sDmallvalusystemates Fig.ic error.2. Although the dispersion curves were not used o the2 2 120 bond angle in the CO group. K is optimized A r r0 are optimized by fittin3 g, r is2 the predetermineExperimdentbondshows thatduringthe highthe-pressfitting,ure,theylow-temstillpera-show the basic features of the U r D 1 eÀ ð À Þ 1 : 3 0 ð Þ ¼ The modelÀ canduring getÀ complicatedfitting, h is forthe bond angleð Þ . ture, transition from rhombohedal calcite I to the length and r is the distanceHigh-pressurebetween phaseatoms.transition experimental curves with a small systematic error. h“pencil and paper”i treatment monoclinic calcite II polymorph results in two dis- The model also contains a three-bodK2 y term, 2to maintain o o U h h 120 : 2 2 4 placements. First, there is an 11 rotation in the opposite the 120 bond angle in the CO group. K is optimiA rzedr0 Experiment shows that the high-pressure, low-tempera- ð UÞ ¼r32 ð ÀD 21Þ eÀ ð À Þ 1 : ð Þ direction to the3 adjacent carbonate group along the c during fitting, h is the bond angleð Þ ¼. À ture,À transition from rhombohedðal Þcalcite I to the The potential energy associated with theout-of-plane axis and second smallHigh-pressureanti-parallel displphaseacementtransitionof ad- h imonoclinic calcite II jacentpolymorphCa ionsresultsoccurin(Redftwo erndis-and Angel 1999). The K2 2 displTheacementmodelof Calsowithincontainsthe CO3 agroupthree-bodhas alsoybeenterm, to maintaino U h h 120 : 4 placements. First, therestructurais an 11 lrotatnatureionofinthethefurtheoppositer high-pressure transition ð Þ ¼ 2 ð À Þ incorpotherated120oinbonthedmodel.angleThisðinÞ thedependdirectCOions otogroup.n thethe im-adjacentK is carbonateoptimizedgroupExperimalong theentc shows that the high-pressure, low-tempera- proper torsional angle /, between OCO¢3and OCO¢¢ 2 to calcite III and its structure are unknown. The potential energy associatedduringwithfitting,the out-of-ph is thelanebonaxisd angleand seco. nd small anti-paraThe behavioullel displracementunderture, pressuroftransiad-etionof thefrommodelsrhombohedpro- al calcite I to the planes, the K parameter is determinedjacentduringCa fittionsing.occur (Redfern and Angel 1999). The displacement of C within the CO3 group has also been duced by Dove et al.mono(1992clinicb) andcalcitPavesee etIIal. poly(1996)morph results in two dis- incorporated in the model.U / ThisK 1dependcos 2s/on: the im- structural nature of5 the further high-pressure transition ð Þ ¼ ½ ÀK2 ð Þ 2 ð Þ and the potential generated in this study were analyzed o proper torsional angle /,Ubetweh en OCOh¢ an120d OCO ¢¢: to calcite III and its structure are unknown.4 placements. First, there is an 11 rotation in the opposite The values of the2 empirical coefficients and the charges using a large supercell. Each model has at least three planes, the K parameterwereis determdetermið Þ ¼inednedduringusingð Àthefitting.‘‘relaxedÞ fitting’’The algobehaviourithmrofunderstablepressurphasee ðs:ofÞcalcitthedirectemodelI, calciteions pro-toII andthe aadjacenthigh-pressurecarbonate group along the c GULP. The model was fitted to theducedstructuresby Dovof earag-et al. (1992phase.b)InandrhombPaveseohedralaxiset al.caland(1996)citesecoI thendcarbsmallonate anti-paragroups llel displacement of ad- U / K 1 cos 2/ : The potential energy5 associatedand the potentiawithl thegeneratedout-of-pin thislanestudy were analyzed ð Þ ¼ ½ À ð Þonite and calcite, the elastic, ðstaticÞ and high-frequency are aligned in rowjacents with Caall theionscarboccuronate (Redfgroupsern and Angel 1999). The The values of the empiricadielecdispll coetricffiacementconstcientsantsandofofthebotCchargeshwithinpolymorphsusingthe a,COthelarge3 phogroupsupercelnon hasl.pointEachalsoingmodelinbeenthehassameat leastdirectiothreen in each row. Adjacent were determined using thefrequ‘‘relaxedincorpoencies fitting’’atratedthe [algo½in0rithmthe2] anofmodel.d [0stable0 0] Thispointsphases:dependincalcitthee sIrows, ocalciten ofthecarbII im-andonatea groupsstructurahigh-presslie urewithl naturethe carbofonatethe groupfurther high-pressure transition GULP. The model wasBrillofittedpropeuinto zonether torsstruof cturescalcitionaleofandanarag-glevibrationphase., albetwefrequIn rhombenenciesOCOohedralof rotatancaldedcitebyOCOI60theo. Thecarbtostructureonatecalcitgroupsofe calIIIciteandII isitsderivestrud cturefrom are unknown. onite and calcite, the elastic, static and high-frequency /are aligned in rows ¢with all the ¢¢carbonate groups the plancarbonatees, thedefoKrmatioparametern modesisofdetermcalcite.inedDuriduringng calcitfitte Iing.by rotating eachThecarbonatebehaviougroupr underby a smpressurall e of the models pro- dielectric constants of fitting,both polit waymorphss necessary, the tophoweightnon thepointexperimeing in thentalsamere- directioamountn(upin toeacharoundrow.11Adjaceo at thent transition from calcite frequencies at the [½ 0 2] and [0 0 0] points in the rows of carbonate groupsI) inliethewithoppositethe carbdirducedonateectiongroupbyto itsDovadjacente et groupsal. (1992. Theb) and Pavese et al. (1996) sults.UThe/ experiKmental1 cosdata2were/ weigh: ted duringo fitting 5 Brillouin zone of calcittoe andtakeðvibrationintoÞ ¼accoual½ frequntÀ experienciesð mentalofÞ rotaterrorsed forby 60each. Thepa-structureresultsoffromcalciteðsimuÞIIlationisandderives usithedngfrompotentiaall modelsl showgeneratedthat thein this study were analyzed the carbonate deformatiorametnTheer,modestakingvaluesof carecalcite.of notthetoDuriempiricaproduceng calcital fewcoee Ihighffibycientlyrotataccsingu-andeachrotatthecarbonateionchargescontinuesgrouptousingincreaseby a asmaslargeallpressuresupercelincreasel.sEachuntil model has at least three fitting, it was necessary to weight the experimental re- amount (up to around 11o at the transition from calcite ratewereresultsdetermiat the nedexpenusingse of theother‘‘relaxedparameters.fitting’’For aalgothirdrithmstableofphasestableis observephased wheres: calcitthe ecarbI, onatecalcite II and a high-pressure sults. The experimental examdata wereple, weweighfoutedndduringthat heavilyfitting weighI) intingthe oppositethe crystaldirectiongroupsto pointits adjacentin thegroupssame .dirTheection to their adjacent to take into account experistructuresGULP.mentalwoulderrorsThebeformodelat theeachexpenwaspa- sefittedresulof thets tofromphothenonsimustrufre-lationcturesgroupss usingof(Fig.allarag-models3). showphase.thatIntherhombohedral calcite I the carbonate groups rameter, taking care notquencitooniteproducees, soantheadweightifewcalcite,highnglyontheacctheu-elastic,strurotatcturesionstaticwascontreduceinuesandd.tohighincrease-frequencTo seeas thepressurebehay viourincreaseare ofalignedthes untilmodelins throughrowsthewithphase all the carbonate groups rate results at the expenInternsedielecof alothertricenergyparametconstminimiantsers.zationsForof botawerethirdh polpestablerformedymorphsphaseat ,ischanges,theobservephothed nonwheremodelsthepointwerecarbingallowedonatein totherelaxsameat a randirectioge of n in each row. Adjacent example, we found thatconstheavilyant pressurweighetingallowingthe crystalall individualgroups ionicpointcoordin thei- samepressurdiresectionstartingto theirfrom adjacentcalcite I, calcite II and the structures would be at thefrequexpenenciesse of theatphothenon fre-[½ 0groups2] an(Fig.d [03).0 0] points in the rows of carbonate groups lie with the carbonate group nates and lattice parameters to vary, while constraining highest-pressure phase. The angle oof the CO3 group quencies, so the weightingtheonBrillospacthee-groupstruuinctureszonesymmetwasofry.reducecalcitThed.chare andgeToof thevibrationseeCatheionbehawasalviourfrequrelativeof theenciesmodelto its oforientations throughrotattheinedphasecalciteby 60I is. usedTheasstructurethe order of calcite II is derived from Internal energy minimiheldthezationsfixedcarbonateatwere2+ topeallowrformeddefoforrmatioatlaterchanges,defectn modessubstitutiothe modelsof n.calcite.wereparametallowedDurier totongshowrelaxcalcittheat aphaserane geI changesbyof rotatiningFig. each4. Whencarbonate group by a small constant pressure allowingAlthofitting,allughindividualthermait wal eionicsffectnecessarys coordare noti- takentopressurweightintoes accoustartingthent experimeinfromstartedcalcitentalfromI, re-calcitdiffereentamoIIstruauntnctures,d the(up theto aroundtransitions11ofteno at the transition from calcite nates and lattice paramettheersmodel,to varythe, whilteme constperaturerainingof thehighsysteest-pressurm is biasede phaseoccur. The atangledifferentof thepressurCO3es.groupThis is due to an optimized the space-group symmetry.sults.The charThege ofexperithe Camentalion was datarelativewereto weighits orientationted duringin calcitefittinI isgusedI)asinthetheorderopposite direction to its adjacent groups. The held fixed at 2+ to allow tofor takelater defectinto substitutioaccount n.experiparametmentaler to errorsshow theforphaseeachchangespa- in resulFig. 4.tsWhenfrom simulations using all models show that the Although thermal effects arerametnot takener, takinginto accoucarent notin startedto producefrom diafferfewent struhighctures,ly acctheu-transitionsrotationoftencontinues to increase as pressure increases until occur at different pressures. This is due to an optimized the model, the temperaturerateof resultsthe systeatm ithes biasedexpense of other parameters. For a third stable phase is observed where the carbonate example, we found that heavily weighting the crystal groups point in the same direction to their adjacent structures would be at the expense of the phonon fre- groups (Fig. 3). quencies, so the weighting on the structures was reduced. To see the behaviour of the models through the phase Internal energy minimizations were performed at changes, the models were allowed to relax at a range of constant pressure allowing all individual ionic coordi- pressures starting from calcite I, calcite II and the nates and lattice parameters to vary, while constraining highest-pressure phase. The angle of the CO3 group the space-group symmetry. The charge of the Ca ion was relative to its orientation in calcite I is used as the order held fixed at 2+ to allow for later defect substitution. parameter to show the phase changes in Fig. 4. When Although thermal effects are not taken into account in started from different structures, the transitions often the model, the temperature of the system is biased occur at different pressures. This is due to an optimized Molecular dynamics simulation
Alder+Wainwright (1956) Hard-sphere liquid
Tremendous growth
Exploration, validation of theories, and checks of interaction potentials Ising Model
Monte Carlo Simulation Emergent properties: Not evident just by looking at the equations
The use of the computer is essential for exploration of models Spring constant
Basic idea: Vibrations around an equilibrium point
!(k)
Parameters can be fitted to experiment Cyrille Barreteau INTRODUCTION
History and vocabulary CEA Grenoble, DSM/DRFMC/SP2M Tight-binding : Solid State Physics point of view Mott and Jones, Slater and the pionneering works of Friedel in the 60’s. Huckel,(LCAO) : The chemists world
Semi-empirical method TIGHT-BINDING METHODS SIZE 1.000.000 EMPIRICAL POTENTIALS large systems low transferability SIMPLE CONCEPTS no electronic structure AND 10.000 EXAMPLES TIGHT-BINDING (SEMI-EMPIRICAL) transferability depends on the system and on the parametrization 1000 ‘‘reasonable size’’ electronic structure
100 AB-INITIO good transferability small systems electronic structure TRANSFERABILITYTRANSFERABILITY Most tight-binding methods are parametrized. The pa(accuracy)rameters of the model are fitted either on experimental or on ab-initio results (the most popular procedure nowadays).
The jungle There is a huge number of tight-binding methods depending on : parametrization. level of approximation : orthogonal or non-orthogonal, environment dependance.. degree of “ab-initio”.