Some Practical Considerations for Density Functional Theory Studies of Chemistry at Metal Surfaces - Metallic Systems: A Quantum Chemist's Perspective - page 187

(a) (100) (b) (110) (c) (111)

(d) (e) (f)

FIGURE 6.5 Low-index. surfaces. of. a. simple. fcc. crystal.. The. top. view. of. a. (100).
surface.(a),.(110).surface.(b),.and.(111).surface.(c),.and.a.side.view.of.a.segment.of.the.cor-
responding.slab.unit.cells.are.shown.in.(d).through.(f).

Construction.of.the.slab.requires.2D.periodicity.within.the.plane..For.the.(100).
surface,.the.unit.cell.is.a.square.and.can.take.two.orientations:.a.larger.face-centered.
square.of.length. a .on.each.side,.or.the.smaller.primitive.cell.rotated.by.45°.enclosed.
by.the.dashed.lines.of.length. a /⎦.2.on.a.side.(see.Figure.6.5)..The.next.layer.of.atoms.
lies. a /2.below.the.top.layer,.between.the.top.layer.atoms,.and.the.third.layer.is.aligned.
with. the. top. layer.. Further. layers. repeat. this. pattern,. to. create. a. slab. as. thick. as.
desired..The.surface.primitive.cell.for.the.(110).surface.is.a.rectangle.of.side. a .along.
one.edge.and. a /⎦.2.along.the.other,.with.the.next.layer.of.atoms.in.the.center.of.the.
rectangle. a /2⎦.2.below.the.top.layer,.and.the.third.layer.directly.below.the.top.layer..
This.pattern.then.repeats.to.ill.the.desired.volume..The.(111).surface.is.a.closed-
pack.triangular.lattice.having.hexagonal.lattice.vectors.of.length. a /⎦.2.at.60°.to.one.
another..The.repeat.sequence.is.three.layers,.the.layers.separate.by. a /⎦.3..The.second.
layer.lies.at.a.distance. a /⎦.3.below.the.top.layer.and.offset.to.the.center.of.every.sec-
ond.triangle..The.third.layer.is.offset.further.and.occupies.the.other.triangle.
The.hcp.(0001).surface.of.the.Er.above.is.closely.related.to.the.(111).surface.of.the.
fcc.crystal.except.the.repeat.sequence.is.only.two.layers..The.third.layer.returns.to.
sit.directly.underneath.the.top.layer,.in.an. ABABAB .stacking,.instead.of.an. ABCABC .
stacking.of.the.fcc.crystal.(the. A ,. B ,.and. C .refer.to.the.three.unique.positions.in.the.
2D.hexagonal.lattice)..Higher.index.surfaces.are.similarly.constructed,.but.the.sur-
face.primitive.cells.get.larger.and.more.complicated.