1. surfaces - crcwschnei1/courses/cbe_547/lectures/13surfaces/... · lecture 13 surfaces ... fall...

Lecture 13 Surfaces

© Prof. W. F. Schneider CBE 60547 – Computational Chemistry 1/14 12:21:00 AM 12/8/2009 University of Notre Dame Fall 2009

1. Surfaces Supercell models very handy for modeling surfaces, their structure, their energy, adsorption, reaction,… For our models, can think of the surface as arising from cleaving a crystal (Bravais lattice) along some plane. Here’s a couple examples from the metal and oxide worlds: FCC metal and some cleavage planes:

Surface planes can be described by Miller indices. Miller plane label has format (ijk), where the three integers give the direction of the vector perpendicular to the plane, in lattice vector coordinates. The definition thus depends on the choice of cell vectors. For FCC above, for instance, planes are described with respect to the conventional cell.

Lecture 13 Surfaces


(111) termination particularly common, lowest energy for most (maybe all?) FCC metals. Gives abcabc type stacking of hexagonal layers: Rock salt oxide:

Could be something like NaCl or MgO. Crystal structure contains two interpenetrating FCC lattices. Can be described using primitive FCC cell with a basis of two or as a conventional FCC with a basis of eight. Latter is shown above.

Last is an example of a polar surface; the charges on the ions alternate between positive and negative perpendicular to the plane. Typically unstable to reconstructions. Tasker has categorized ionic surfaces: Tasker type I: individual layers are charge neutral, no dipole [MgO(001)] Tasker type II: individual layers are charged, but groups of layers are charge neutral and non-

polar [Al2O3(0001); TiO2(110)] Tasker type III: groups of layers have net dipole [MgO(111); SrTiO3(100)]

2. Surface slab models How to construct a supercell model of a surface? Construct a slab, infinite in two dimensions and with “vacuum” spacing between successive slabs.

Lecture 13 Surfaces


(100) slab

(103) supercell, illustrating a stepped surface:

Key parameters that describe slab model:

(1) Cleavage plane (2) Thickness (number of layers) of slab (3) Thickness of “vacuum” (4) Lateral dimensions of periodic unit

Like all supercells, the slab supercell must be a Bravais lattice, and generally speaking it will be a different Bravais lattice than that of the parent crystal. For instance, parent MgO is FCC, put (001) slabs are described in orthorhombic or tetragonal cells. Rocksalt conventional FCC cell POSCAR: simple cubic

Lecture 13 Surfaces


4.2112 1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 4 4 direct 0.5 0.5 0.5 0.5 0. 0. 0. 0.5 0. 0. 0. 0.5 0. 0. 0. 0.5 0. 0.5 0. 0.5 0.5 0.5 0.5 0. Two-layer-thick tetragonal (001) slab (double the c vector and halve the z fractionals): tetragonal 4.2112 1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 2.0 4 4 direct 0.5 0.5 0.25 0.5 0. 0. 0. 0.5 0. 0. 0. 0.25 0. 0. 0. 0.5 0. 0.25 0. 0.5 0.25 0.5 0.5 0. Simplest case possible. In general, cell axes may need to rotate from bulk to slab, will want to vary slab thickness, vacuum…. FCC metal surfaces:

Example of a √3×√3, 3 layer (111) slab: (111) FCC slab, three layers thick

Lecture 13 Surfaces


4.8057764 .8660254037839999 -.5000000000000000 .0000000000000000 .0000000000000000 1.0000000000000000 .0000000000000000 .0000000000000000 .0000000000000000 3.0 9 Direct 0.000000000000000 0.0000000000000000 0.0000000000000000 0.666666666666666 0.3333333333333333 0.0000000000000000 0.333333333333333 0.6666666666666666 0.0000000000000000 0.333333333333333 0.3333333333333333 0.1666666666666667 0.000000000000000 0.6666666666666666 0.1666666666666667 0.666666666666666 0.9999990000000000 0.1666666666666667 0.666666666666666 0.6666666666666666 0.3333333333333333 0.333333333333333 0.0000000000000000 0.3333333333333333 0.000000000000000 0.3333333333333333 0.3333333333333333

3. Surface energy How to do a calculation on a slab? One common question to address is the surface energy, i.e. energy to generate surfaces from bulk, energy/area (always a positive quantity, or material is unstable!!!). Will differ from surface plane to surface plane.

Because this involves comparing energy from supercells of different shapes and sizes, particular care has to be taken use equivalent lattice constants, computational parameters, energy cutoffs, and k point samples. Results can of course depend on thickness of slab and vacuum. Typically look for convergence with respect to this. Remember, the size of the plane wave basis scales with the box size, so more vacuum will be computationally more expensive. k point sampling need only be down in the plane of the slab; the interactions between slabs is (or should be!) negligibly small, and the results should be insensitive to the k points in the vacuum direction. In general set number of k points in that direction to 1. Typical KPOINT file format, appropriate for an oxide slab like that above: 2x2x1 surface KPOINT file 0 Monkhorst-Pack 2 2 1 0. 0. 0. Conventional MgO (4 formula units): -48.756 eV

Lecture 13 Surfaces


2 layer (100) slab (4 formula units): -46.663 eV area 2 sides * (4.2112 Å)2 = 17.7342 Å2 Surface energy: 0.11 eV/Å2 = 1.9 J/m2. Some well-converged results (see W. F. Schneider, J. Li, and K. C. Hass, “Combined Computational and Experimental Investigation of SOx Adsorption on MgO,” J. Phys. Chem. B, 2001, 105, 6972–6979): What’s missing in our calculation? Slab is too thin, and we did not allow the surface to relax!

Common practice to fix one side of the slab at the bulk positions and relax the other. Have to use “Selective Dynamics” in POSCAR. Rationale is that “fixed” side replicates constraints that would be imposed by a truly semi-infinitely deep slab. Defensible for metals, no worries for rock salt, but can be problematic in other cases, especially non-type-I surfaces, where unrelaxed surface can leave unphysical dangling states. For polar surface, dipole corrections (LDIPOL) can be important to get reasonable energies. Places a charge sheet in the vacuum that compensates the dipole generated by an adsorbate, magnitude calculated self-consistently. Most important in highly dipolar situations.

4. Surface potentials and Fermi energies Calculations give distribution of charge in and near surface. Recall that in a supercell calculation the absolute electrostatic potential is not well defined, so the absolute eigenvalues are not

Lecture 13 Surfaces


meaningful. One of the advantages of having a vacuum space in the cell is that it provides a reference potential for the vacuum, i.e., if we make the vacuum spacing “thick enough” to be truly vacuum, can set the electrostatic potential in that region to zero and use that offset to reference the band energies and Fermi energy to. Set LVTOT=.TRUE., prints LOCPOT file which contains the local potential energy on a grid filling the supercell. Have to plot, average in plane. Example of Ni(111) surface from Vasp hands-on:

work-function

usage of simple utility vtotav gives planaraverage of the potential

vacuum-potential Evac 5 46 eV

Fermi-level !F 0 225 eV(from OUTCAR)

" Evac !F 5 24 eV

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5. Surface adsorption Often interested in how strongly an adsorbate binds at a surface. Figured out using geometry optimizations, as we’ve seen many times before. Challenge is that adsorption geometries can be hard to predict; optimizations can be time consuming.

Lecture 13 Surfaces


Adsorption on oxides Show example of adding O to our MgO slab above. SO2 on MgO another example. Would first calculate SO2 in a vacuum, then add to surface and relax. (At least) two types of adsorbate geometries found:

Calculate adsorption energy Eads = Eslab+ads – Eslab – Eads. Note this convention makes adsorption exothermic; opposite convention is also common. Lateral dimensions can matter a lot; too small a cell and adsorbates will feel the influence of their neighbors, altering adsorption energies. Example above is a c(2×2) cell, with 16 atoms per slab. Typically would test for sensitivity to cell dimensions. How to analyze adsorption?

• Site-projected DOS • Charge density difference: ∆ρ = CHGCARslab+ads – CHGCARslab – CHGCARads. Utilities

available to calculate this difference. Must be done at constant atom positions. • Can calculate vibrational spectra of adsorbates using Selective Dynamics.

Adsorption on metals Metal surfaces can present a number of adsorption site types. Consider (111) surface:

Compare adsorption of an O atom in each location, on Pt(111):

Lecture 13 Surfaces


O prefers FCC site. Could use binding energy to calculate a Langmuir adsorption isotherm, for instance. DFT is generally pretty reliable for calculating adsorption site preferences; adsorption energies have the typical errors of the GGA. CO on Pt is a notable exception; the GGA infamously predicts CO to prefer FCC sites, where they bind to three surface Pt; reality is atop Pt. Consequence of GGA overestimating the importance of π backbonding.

6. Coverage-dependent adsorption Surface adsorbates can interact with (attract or repel) one another, and this can have an effect on adsorption energies, isotherms, saturation coverages. Representative O–O interactions on Pt(111):

3.01 Å

1NN 4.91 Å

2NN 5.63 Å

3NN

Relaxed: 0.20 eV Rigid: 0.24

0.10 eV 0.015

0.01 eV -0.02

Formation energy a common way to compare these:

Some sample surface configurations:

Generate lots of these, plot of formation energies vs. composition shows which arrangements are stable, those that lie along lowest “convex hull.”

0 ML ! ML " ML

Pt(111) p(2!2)-O p(2!1)-O

# ML

p($3%$3)-2O

-1.26 eV/1/2 O2

-0.66 eV/1/2 O2

-0.17 eV/1/2 O2

>0 eV/1/2 O2

Lecture 13 Surfaces


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Slope of this curve gives 0 K binding energy per adsorbate. Binding energy is not a single number, but varies with coverage. Note at 2/3 coverage, binding energy goes to 0; highest coverage accessible with O2. To calculate isotherms can either create interaction model and simulate with Monte Carlo, or can “discretize,” estimate from slopes chemical potential ranges that each configuration is stable:

Plot of γ vs. µ gives a set of lines with slopes of Nads; stability ranges are ranges that given line are lowest:

chemical potential µO in Figure 8. Increasing O potentialminimizes the surface energy of progressively more highlyO-covered surfaces. To place the R-PtO2 overlayer on this samescale, we assume that Pt in the overlayer is in equilibrium witha bulk reservoir, i.e., that Pt can freely exchange between surfaceand the bulk, and transform !XO

(1)(0 K, µO) with respect to thechanges in the number of Pt at the surface, !NPt:

!(2)(0 K, µO, µPt)) !(1)(0 K, µO)- (!NPt/A)µPt (21)

Taking µPt ) EbulkDFT + Ebulk

ZP yields the result shown in Figure 8.Zero-point vibrational contributions to the energy are computedbased on the DFT calculated vibrational spectrum of just theR-PtO2 overlayer.

The R-PtO2 thin film indeed becomes stable at O potentialssimilar to those in equilibrium with the moderate coverageregime, i.e., in the region where the 1/4 and 1/2 ML linesintersect. One cannot rule out the existence of other, even lowerenergy surface oxide films. Thus, based on these DFT results,one can conclude that any surface O coverage greater than 1/2ML is meta-stable to formation of a surface oxide, and any suchcoverages that do form owe their existence to kinetic resistanceof the Pt(111) surface to the formation of such an oxide. Incontrast, the R-PtO2 surface oxide is metastable to surfaceoxygen at lower oxygen chemical potentials.

An oxidized Pt surface has been observed to be formed duringdosing of O atoms to the Pt(111) surface.66,71 The transitionfrom chemisorbed oxygen to an oxidized surface around 0.75ML is characterized by a shift of the O2 TPD peak to highertemperature, suggesting that oxygen is more strongly bound inthe oxide than on the Pt(111) surface. Consistent with this, wecalculate the energy to remove one O atom from the Pt(111)-R-PtO2 overlayer to be 1.435 eV, or at least 0.18 eV greaterthan any surface chemisorbed O. This oxide is observed tocoexist with chemisorbed oxygen down to a coverage of 0.5ML,66 also consistent with the crossover between oxide andsurface oxygen shown in Figure 8. While we have notinvestigated the transformation from the oxide to chemisorbedoxygen, it seems quite likely from the different structures andcompositions of the oxide and chemisorbed oxygen phases thatsuch a transition is activated and contributes to the “explosive”transformation of the oxide to an oxygen-covered surface atconditions at which the oxide is meta-stable to surface oxygen.

4. Analysis and Discussion

Given this catalog of oxygen coverages and binding energies,the limiting equilibrium surface oxygen coverages under various

reaction assumptions and conditions can be determined. To doso, it is useful to imagine a metal surface in a well-mixed reactorin which an oxidizing gas or gas mixture is fed at some fixedinlet concentrations, and to further imagine that the surfacecomes to an equilibrium with oxygen through some limited andprescribed reactions. The conceptually simplest case is that inwhich the only source of surface O is dissociated O2 (g), butthe addition of reductants or alternative oxygen sources canmodify the equilibrium O coverage. Figure 9 illustrates severalrelevant equilibrium scenarios. As we show below, in each casethe oxygen potential can be written in terms of the thermo-chemical properties and composition of the oxygen source usingstandard thermodynamic relationships. This chemical potentialdefinition is used with eq 12 to determine equilibrium coverages.

4.1. Chemisorbed Oxygen from O2. In the case in whichthe metal surface is exposed only to a gas of O2 and inerts,corresponding to 1 in Figure 9, the steady-state, equilibriumsurface oxygen coverage is given by

12

O2 (g)+MhM-O(") (22)

µM-O )12

µO2

0 (23)

The surface free energy of any particular surface coverage,!XO

(1)(T, µO), is thus determined by the gas-phase oxygenpotential, which itself is related to the O2 (g) pressure P andtemperature T through an equation of state. Assuming ideal gasbehavior, as is appropriate for conditions of catalytic relevance,the integrated form of the Gibbs-Duhem equation gives78

µO2

0 )EO2

DFT +EO2

ZP +!GO2(T, 1 bar)+ kBT ln(PO2

/1 bar)

(24)

where !GO2(T, 1 bar) ) GO2(T, 1 bar) - GO2(0 K, 1 bar) andis calculated in the same manner as !HO2(T, 1 bar) in eq 20.Using this method, we find !GO2(298 K, 1 bar) to be 0.0975eV, which is in excellent agreement with the GO2(298 K, 1 bar)- GO2(10 K, 1 bar) of 0.0974 eV determined using the HSCdatabase.79 Notice that µO2

0 ) GO20 , as is the thermodynamic

definition of chemical potential.In Figure 10a we plot the calculated O configurations that

minimize the surface free energy !XO(1)(T, µO) as a function of T

greater than 300 K and P from UHV to ambient, conditionsover which O2 is expected to extensively dissociate over thePt(111) surface. This stability diagram is drawn to include twotypes of regions: ordered configurations and intermediate regionsin which one ordered configuration evolves into the next higheror lower coverage. In the latter we include the configurationsthat lie along the linear portions of the formation energy curvewhich have nearly constant oxygen binding energies and thusare unlikely to be distinguishable as unique phases. The formerare the end points of those regimes, where oxygen bindingenergies change distinctly. Within this model the 1/4 ML

Figure 8. 0 K surface free energies of the stable chemisorbed Oconfigurations and of Pt(111)-R-PtO2 as a function of O potential.

Figure 9. Possible reactions between an oxygen-covered surface andgas-phase oxidants and reductants, modeled in a well-mixed flowreactor.

9566 J. Phys. Chem. C, Vol. 112, No. 26, 2008 Getman et al.

For O2, for instance, have

µO =12

EO2

DFT + EO2

ZP + ΔGO2

(T ) + kBT lnPO2

1 bar

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7. Reaction Barriers Can calculate transition states for surface reactions, e.g. to parameterize kinetic models. Simple example, adsorbate diffusion on a surface, can guess TS and calculate energy and partition functions.

Lecture 13 Surfaces


For more complicated cases, “nudged elastic band” methods are popular. Don’t require knowledge of Hessian, just require initial and final states and some (pretty good!) guess of intermediate “states” or beads. Algorithm works by putting springs between adjacent beads that supplement the true forces and keep the beads separate. Relaxation under combination of true and spring forces causes beads to move to minimum energy surface. Example of O2 dissociation on Pt(111):

From Rachel Getman: Elastic Band Methods for finding the MEP between two minima on a PES

• Class of methods deriving from the “chain-of-states” method that finds MEP between two minima using the gradient of the PES but not the Hessian

o Does not require “good” estimate of MEP/TS geometries • Method:

o Locate minima on PES o Draw a path between them

Most common way to do this: linear interpolation o Discretize the path into a finite # of points (called “images”)

Each geometry is a linear interpolation of the two minima Optimization is performed at each point by analyzing the forces on the

geometry. Goal is to converge each image to the same MEP. • What is the MEP? On the MEP, the forces along the direction parallel to the reaction

pathway are finite, but they are 0 in all other directions

Lecture 13 Surfaces


o • Complex surfaces contain many reaction pathways and thus many MEPs. We want to

make sure we converge all images to the same MEP, so we connect them with a theoretical spring:

o o Spring force on each image is:

• Since we want to converge to the MEP and not necessarily to a stationary point, we need to project the component of the total force perpendicular to the reaction pathway out of the total force:

o

o This way, the reaction path is updated @ each step (or every 10 steps, or …, depends on program, defaults, user inputs, etc.)

o Elastic Band method is converged when = 0 (or goes below some specified convergence criterion.)

If all spring constants are the same, the spring force is minimized when images are equally spaced

Perpendicular component of the molecular force zero-ed when image is on the MEP

• Relaxation in the direction perpendicular to the reaction path stretches images over the PES, much like pulling a rubber band over a surface (hence “Elastic Band” Method)

• Nudged Elastic Band method: o Removes perpendicular component of spring force:

o

F:

rxn coordinate

I 1 2 3 4 F

x

E

Lecture 13 Surfaces


o This way, spring forces do not interfere with relaxation to MEP Takes away unnatural forces perpendicular to the reaction path This clean version of the EB force brings about an optimization algorithm

referred to as “nudging,” hence “Nudged” Elastic Band • EB methods do not necessarily find the “TS”

o Sometimes require interpolation between the two highest energy images • Cure: Climbing Image Elastic Band Methods

o After a few geometry convergence steps in the EB method, locates image with the highest E

This image is now denoted MAX o Spring forces are removed from max, CI-NEB force becomes:

MAX moves down toward MEP and up to TS Upward movement denoted “climbing,” hence “Climbing Image”-Nudged

Elastic Band Relevant literature:

• Plain Elastic Band of Elber and Karplus: Chem. Phys. Lett., 139, 1987, 375. o No projection of forces: PEB force is a sum of spring and total molecular forces.

• Elastic Band Method of Ulitsky and Elber: J. Chem. Phys. Lett., 92, 1990, 15. o Projects out parallel component of molecular force

• Self-Penalizing Walk method of Czerminski and Elber: International Journal of Quantum Chemistry: Quantum Chemistry Symposium 24, 1990, 167.

o Instead of a spring force uses a repulsive force and an attractive force couple to maintain connection but not allow images to get too close. Projects out parallel component of molecular force.

• Locally-Updated Plains method of Choi and Elber: J. Chem. Phys., 94, 1990, 751. o No spring force, i.e. images are not connected. Parallel component of molecular

force is projected out. • Plain Elastic Band method of Gillian and Wilson: J. Chem. Phys., 97, 1992, 3.

o For molecular dynamics. • Nudged Elastic Band Method of Jónsson, Mills, and Jacobsen:

o Projects out parallel component of molecular force and perpendicular component of spring force.

• Improved Tangent Nudged Elastic Band Method of Henkelman and Jónsson: J. Chem. Phys., 113, 2000, 9978.

o Improves how the reaction path is calculated:

o Ensures equi-spacing even in regions of large curvature, gets rid of “kinkiness” of path

• Climbing Image Nudged Elastic Band Method of Henkelman, Uberuaga, and Jónsson: J. Chem. Phys., 113, 2000, 9901.

Lecture 13 Surfaces


o Introduces climbing image: spring forces removed from NEB force of image with highest energy so that it can move along the elastic band in search of the maximum. Searches to minimize total molecular force.

1. surfaces - crcwschnei1/courses/cbe_547/lectures/13surfaces/... · lecture 13 surfaces ... fall...

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