Surface Structure and Morphology 2D Crystallography

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1 Surface Structure and Morphology 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) SOS or TLK-model Terrace-Ledge-Kink Warning: There may be cases where neither long-range nor shortrange periodicities are given (e.g. amorphous solids, liquid surfaces, )

2 Crystal planes and surface modifications (100) (110) (111) bcc - body-centred cubic fcc - face-centred cubic hcp - hexagonal close packed

3 Crystal planes and surface modifications Notations Square brackets, "[100]", denote a particular crystallographic direction A bar over a number denotes a negative value. < > Angular brackets denote equivalent directions. ( ) denotes a particular plane or surface (hkl) where h, k and l are the Miller indexes for the plane. In a cubic system these are also the co-ordinates of the vector normal to the plane. That is, the (111) surface of an fcc crystal is normal to the [111] direction in the crystal. { } denote equivalent planes of one particular type.

4 Crystal planes and surface modifications Relaxation Relaxation is a small and subtle rearrangement of the surface layers which may nevertheless be energetically significant. It involves adjustments in the interlayer spacing perpendicular to the surface, while there is no change both for the periodicity parallel to the surface or for the surface symmetry. Unrelaxed Surface Relaxed Surface (d 1-2 < d bulk ) The driving force of relaxation is the smoothening of the electric charge density at the surface and to the formation of a surface electric field which displaces the positive outermost ionic core layer towards the bulk (Smoluchowski effect). The expansion of the third layer is due to the oscillations of the screening charge.

5 2D Crystallography Bravais lattices in 2D are called Bravais nets Unit 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

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7 2D Crystallography: 2D Point Groups

8 2D Crystallography: 2D Space Groups The 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 only translation matters Translation + rotation Centered structure Translation + mirroring Translation + gliding

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

10 Equivalent positions, symmetry operations and long and short International notations for the 2D space groups Translation + rotation (6 fold axis) + two inequivalent mirror planes

11 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 c 1 and c 2 of the surface mesh may be expressed in terms of the reference net a 1 and a 2 by a matrix operation (P) c c 1 2 a P a 1 2 P P a a Since the area of the 2D substrate unit mesh is a 1 xa 2, det P is the ratio of the areas of the two meshes 11 Based on the values of det P and P ij, systems are sorted out along the following classification: 1) det P integral and all P ij integral The two meshes are simply related with the adsorbate mesh having the same translational symmetry as the whole surface 21 P P

12 2D Crystallography: Relation between Substrate and Selvage 2) det P a rational fraction (or det P integral and some P ij rational) The two meshes are rationally related. The 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 structures. If d 1 and d 2 are the primitive vectors of the true surface mesh, we have d d 1 2 a R a 1 2 c Q c det R and det Q are chosen to have the smallest possible integral values and: det G det R det Q 3) det P irrational The 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 (e.g. rare gases) or if the adsorbed species are too large and they do not feel the periodicity of the substrate 1 2

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14 Ricostruction and superlattices

15 2D Crystallography: Relation between Substrate and Selvage Shorthand 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 c 1 =p a 1 and a 2 =q c 2 with a unit mesh rotation of f, the structure is referred to as X{hkl}p x q-rf -A or often X{hkl}(p x q)rf -A Cu(100)p4mm O Cu(100)(22)-R45 -O 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.

16 Structure Factor, Form Factor and Reciprocal Space The Crystalline structure is usually investigated in scattering experiments. The impinging particles are described by plane waves 0 e ikr and the scattered waves by the set of functions: j = 0 e ik R /R f j (k 0, k) e i(k-k o ) R j where f j (k 0, k) is the atomic scattering factor, determined by how the wave is reflected by the atoms within the unit cell, and e i(k-k o ) R j is the phase difference between waves reflected at the origin (zero) and at position R j For a crystal with M 1 M 2 unit cells the scattered wavefunction is obtained by summing over all scattering centers: example f(electrons) = j j j f j (k 0,k) e i(k-k o ) R j=fg The sum can be partitioned into two independent parts: the form factor F k f j exp ik r j j depending only on the internal disposition of the atoms within the unit cell (determining the scattering intensities ). The structure factor, G, determined only by the spatial arrangement of the unit cells, and giving the diffraction patterns.

17 Given R j =(n 1 a 1,n 2 a 2 ) with a 1 a 2 the unit vectors of the cell: And taking the square of the wavefunction to compute the scattering intensity we obtain the so called interference function: which has maxima for a (k-k o )=n i.e. when the momentum exchange is equal to a reciprocal lattice vector G=(2n/a 1, 2m/a 2 ) The position of the maxima in the diffraction pattern tells us about the crystal lattice symmetries, while the intensity in the different channels tells us about the atomic arrangement inside the unit cell ))] ( 2 1 [sin( ))] ( 2 1 [sin( )] ( 2 1 [sin( ))] ( 2 1 [sin( M M k k a k k a M k k a k k M a G ) ( ) ( M n M n k k a in k k a in e e G

18 In elastic scattering we have the conservation of energy (ħk) 2 /2m and of momentum p=mv=(ħk). In the scattering process we thus have: 2mE k 2 k ' Ewald Construction in 3 Dim space k ' k g hkl k 2 Energy conservation Momentum conservation where g hkl is a reciprocal lattice vector (2 indexes for 2 Dim, 3 for 3 Dim). In order diffraction to be possible k - k must be comparable or larger than g. 2 1) A vector k is drawn terminating at the origin of the reciprocal space 2) A sphere of radius k is constructed about the beginning of k 3) For any point at which the sphere passes through a reciprocal lattice point, a line to this point from the center of the sphere represents a diffracted beam k 4) Notice the reciprocal lattice vector g hkl

19 Diffraction in 2D 2 k 2 k ' k ' k g hk k 2 k 2 k ' 2 k ' 2 Conservation of Momentum Conservation of Energy k is not conserved since the translational symmetry normal to the surface is now broken The indexing of the diffracted beams is, by convention, referenced to the substrate real and reciprocal net. If the selvedge or adsorbate structures have larger periodicities, the surface reciprocal net is smaller than that of the substrate alone. The extra reciprocal net points and associated diffracted beams are denoted by fractional rather than integral indices.

20 Ewald Sphere Construction in 2D Reciprocal Space Notice that the reciprocal lattice is now replaced by infinite reciprocal lattice rods perpendicular to the surface and passing through the reciprocal net points 1) At surfaces 2D translational symmetry holds thereby only. 2) The wave vector parallel to the surface is conserved with the addition of a reciprocal net vector 3) The dashed scattered wave vectors propagate into the solid and are not observable

21 2D Crystallography: Surface Reciprocal Lattice The reciprocal net vectors A and B of the surface mesh are defined, in analogy to the 3 Dim case, as b n a A = b B = 0 A 2 a A = b B = 2π a b n 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. Example: rectangular mesh

22 fcc or bcc (100) c(2x2) superlattice

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25 bcc fcc

26 Relation between 3D and 2D Brillouin zones

27 Exercise: bcc -crystal

28 W(100) c(2x2) reconstruction Tungsten reconstructs 2x2 below 150 K. This reconstruction has c(2x2) symmetry and is therefore difficult to distinguish from the c(2x2)-h superstructure with which is was erroneously confused for a long time. Note that two domains are present. Above 150 K the surface disorders and the reconstruction is lifted so that the pattern goes back to the (1x1) determined by the underlying lattice.

29 H small coverage.. large Coverage Surface reconstructs by shifting the first W layer to the right

30 Fcc surfaces Ir(100)

31 Au(111) (22x3) or herringbone reconstruction Main effect : uniaxial compression Pt(100) Au(111) The reconstruction causes a 4.5% shrinking of the surface layer It occurs in 3 domains due to the 3 fold symmetry axis of the lattice

32 fcc (110) surfaces of Ir, Pt, Au

33 Diamond structure (Si, Ge) and Zinc blende (III-V) structures

34 Diamond structure: Si, Ge Jahn Teller effect Two dangling bonds per unit cell imply two surface states, one empty and one full. The pairing of the bonds and the associated lattice distortion imply a gain in energy

35 Si(001) 2x1 Si(100) non reconstructed Observed structure: Asymmetric dimers (2x1) symmetric dimers

36 Si(001) 2x1

37 Asymmetric dimers (2x1) c(4x2) and p(2x1) are nearly isoenergetic and can coexist

38 Imaging surface states by STM Scanning Tunneling microscopy gives a topographical image in real space for constant current measurements. The images depend on the tip bias and voltage sinced different states are recorded. When measuring the current vs voltage one obtains the local conductivity determined by the local density of states

39 STM: Theory Let s consider a one-dimensional barrier (we neglect the electric field at the junction and assume that the work function f is the same for both electrodes). Given a certain barrier thickness s one obtains: e 0 e 0 kz k ( sz) 2mf with k 1.1 Å -1 for f=4.5 ev 2 The matrix element is then M I 2 2m ds 2k( ) e where the sum extends over all states involved in the tunneling process 0 2 e 2ks e 0 2 represent the surface state density -I depends exponentially on barrier thickness. -For k=1.1 Å -1, I varies by one order of magnitude per Å. - High sensibility to surface corrugation ks

40 STM images of Si(001) 2x1

41 (2x1) and c(4x2) structures coexist on Ge(100) due to the energy gain associated with stress relaxation

42 Reconstruction of Si(100) and Steps Step down direction S a step more energetic than S b step

43 S b steps are rougher at non zero T

44 Si (111) (2x1) The cleaved surface exhibits a (2x1 reconstruction). The reconstruction involves displacements down to the second crystal layer Because of the Jahn Teller effect the pairing of the dimers implies a semiconducting surface.

45 Occupied surface state Empty surface state Tip bias +1V -1V

46 Diamond structure: Si(111) The dots correspond to electronic states on the adatoms

47 Diamond structure: Ge(111) c(2x8)

48 Other compounds Zincblende structure (two displaced fcc lattices 8 atoms per unit cell) and Wurtzite structures (two displaced hcp lattices 4 atoms per unit cell) (111) and (100) planes are polar and different from (-1,-1,-1) and (-1,0,0) planes.

49 III-V Semiconductors : GaAs Clivage surface : (110) Riconstruction by atom displacements within the unit cell Small changes in the bond length Ga As Covalent Bonds inclined by 27 The sp 3 bond dehybridises to a quasi sp 2 (planar geometry).

50 III-V Semiconductors : GaP basically identical to GaAs..

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53 Polar surfaces: (111) vs (-1-1-1) charge neutralization problem

54 Domain walls e.g. between fcc and hcp regions on Au(111)

55 The domain boundary: Frenkel Kontorova model chain of atoms connected by springs in a sinusoidal potential a and b are the natural lattice spacings of the chain and of the potential, respectively. Differentiating with respect to x n we obtain the equilibrium condition a Considering 0.5 ML coverage, assuming unloaded springs if the atom distance a=2b and a soft spring constant (½kb 2 << U), we get for the energy variation: when expanding for small displacements Leading to a minimum for with The displacement vanishes indeed when k tends to zero and the associated energy reads: 2

56 In the limit of strong springs the overlayer becomes incommensurate, but in the limit of displacements u nearly uniform from one atom to the next gives the sine Gordon equation in which u n+1 +u n-1-2u n reduces to the second derivative of the continuous function u(n) Which may be solved by multiplying by du/dn and by partial integration K=U/k in the limit of u(n)0 Domain wall width 4r Integrating again yields

57 Stereogram of bcc and fcc surfaces achiral chiral

58 Steps and defects

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60 Kinks at steps For Cu with α=0.3 one obtains ε=134 mev/kink in fair agreement with experiment

61 Interfaces

62 Dislocations Notice: at the dislocation the steps are straight

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