2D Crystallography

Selvage (or selvedge (it. cimosa)): Region in the solid in the vicinity of the mathematical surface

Surface = Substrate (3D periodicity) + Selvage (few atomic layers with 2D periodicity)

Warning: There may be cases where neither long-range nor short-range periodicity are given

TLR-modelTerrace-Ledge-Kink

2D CrystallographyBravais lattices in 2D are called Bravais netsUnit cells in 2D are called unit meshes

There are just 5 symmetrically different Bravais nets in 2D

The centered rectangular net is the only non-primitive net

2D Bravais Nets andUnit Meshes

Rectangular (c) net|a1|≠|a2| =90°

Oblique (p) net|a1|≠|a2| ≠90°

Rectangular (p) net|a1|≠|a2| =90°

Square (p) net|a1|=|a2| =90°

a1

a2

a1

a2

a1

a2

Hexagonal (p) net|a1|=|a2| =120°

a1

a2

a2’

a1’a1

a2

Primitive cell

Unit cell|a1’|≠|a2’| ≠90°

2D Crystallography: 2D Point Groups

2D Crystallography: 2D Space GroupsThe combination of the 5 Bravais nets with the 10 different point groups leads to 17 space groups in 2D (i.e. 17 surface structures)

Equivalent positions, symmetry operations and long and short “International” notations for the 2D space groups

Equivalent positions, symmetry operations and long and short “International” notations for the 2D space groups

Equivalent positions, symmetry operations and long and short “International” notations for the 2D space groups

2D Crystallography: Relation between Substrate and Selvage

Whenever there is a selvage (clean surface or adsorbate) the surface 2D-net and 2D-mesh are referred to the substrate 2D-net and 2D-mesh

The vectors c1 and c2 of the surface mesh may be expressed in terms of the reference net a1 and a2 by a matrix operation (P)

c1

c2

G

a1

a2

G11 G12

G21 G22

a1

a2

Since the area of the 2D substrate unit mesh is |a1xa2|, det G is the ratio of the areas of the two meshes

2D Crystallography: Relation between Substrate and SelvageBased on the values of det G and Gij, systems are sorted out along the following classification:1) det G integral and all Gij integralThe two meshes are simply related with the adsorbate mesh having the same translational symmetry as the whole surface2) det G a rational fraction (or det G integral and some Gij rational)The two meshes are rationally relatedThe structure is still commensurate but the true surface mesh is larger than either the substrate or adsorbate mesh. Such structures are referred to as coincidence net structuresNow, if d1 and d2 are the primitive vectors of the true surface mesh, we have

d1

d2

P

a1

a2

Q

c1

c2

2D Crystallography: Relation between Substrate and Selvage2 continued) det G a rational fraction (or det G integral and some Gij rational)det P and det Q are chosen to have the smallest possible integral values and they are related by

detG det Pdet Q

3) det G irrationalThe two meshes are now incommensurate and no true surface mesh exists.This might be the case if the adsorbate-adsorbate bonding is much stronger than the adsorbate-substrate bonding or if the adsorbed species are too large and they do not “feel” the periodicity of the substrate

2D Crystallography: Relation between Substrate and SelvageShorthand notation (E. A. Wood, 1964)It defines the ratio of the lengths of the surface and substrate meshes along with the angle through which one mesh must be rotated to align the two pairs of primitive translation vectors.If A is the adsorbate, X the substrate material and if|c1|=p|a1| and |a2|=q|c2|with a unit mesh rotation of , the structure is referred to as

X{hkl}p x q-R °-Aor often

X{hkl}(p x q)R °-A

Warning: This notation is less versatile. It is suitable for systems where the surface and substrate meshes have the same Bravais net, or where one is rectangular and the other square. It is not satisfactory for mixed symmetry meshes.

2D Crystallography: Surface Reciprocal LatticeThe reciprocal net vectors c1* and c2* of the surface mesh are defined as

c1 • c2* = c2 • c1* = 0c1 • c1* = c2 • c2* = 2π (or 1)

The reciprocal net points of a diperiodic net may be thought of (in 3D space) as rods.

The rods are infinite in extent and normal to the surface plane where they pass through the reciprocal net points.Imagine a triperiodic lattice which is expanded with no limit along one axis, thus the lattice points along this axis are moved altogether and in the limit form a rod.

Ricostruzioni e superreticoli

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