disordered, ordered and coherent with the substrate, and ordered but incoherent with the substrate.

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1 5. Nomenclature of overlayer structures Thus far, we have been discussing an ideal surface, which is in effect the structure of the topmost substrate layer. The surface (selvedge) layers of the solid however differ structurally from this ideal surface or substrate structure, due to the reconstruction of the selvedge, and of an adsorbate (inducing structural changes) or both. Several overlayer structures are possible: disordered, ordered and coherent with the substrate, and ordered but incoherent with the substrate. The first case is hard to deal with in surface crystallography (though it is very important and interesting, e.g. amorphous Si). For the other two cases where order exists in the selvedge, we can designate it with reference to the substrate order. By convention, we always denote the substrate surface as (1 1), with the understanding that we know the type of Bravais net that it belongs to. We then refer the overlayer region (selvedge or adsorbate) to this substrate mesh. 3 Surface%20Structure%20III.htm 1/24

2 Take: (b 1, b 2 ) to be unit basis (or mesh) vectors of the overlayer; and (a 1, a 2 ) to be the unit basis (or mesh) vectors of the substrate. By convention, a 1 < a 2 and b 1 < b 2. Wood notation (E.A. Wood, 1964) Assume the angles between the basis vectors are the same for the overlayer and the substrate, i.e. a1,a 2 = b1,b 2. Let b 1 / a 1 = m and b 2 / a 2 = n (where m and n are integers), and be the angle of rotation required to bring the two meshes to coincide with each other. Case (a) If = 0, the unit mesh is denoted as (m n) or p(m n) for primitive mesh; and c(m n) for centered mesh. Case (b) If 0, the unit mesh is denoted as (m n)r. NOTE: The Wood notation only works for a limited number of overlayer structures and it is not general enough to describe all types of periodic surface structures. EXAMPLES: Ni (100) (1 1) Ni (100) (2 2) O Si (100) (2 1) Si (100) (1 1) H 3 Surface%20Structure%20III.htm 2/24

3 Clean Ni(100) is the unreconstructed 1 1 surface while clean Si(100) is reconstructed 2 1. The adsorption of H on Si(100) de reconstructs the surface back to 1 1, by proper termination of the surface dangling bonds. Matrix notation (Park and Madden, 1968) A more general notation involves a simple vectorial construction, where we expand the overlayer basis (b 1,b 2 ) in terms of the substrate basis (a 1,a 2 ). Or, in matrix notation, b1 = m 11 a 1 + m 12 a 2 b2 = m 21 a 1 + m 22 a 2 We can show that the determinant of the transformation matrix M (or det M = m 11 m 22 m 21 m 12 ) is given by the 3 Surface%20Structure%20III.htm 3/24

4 ratio of the area of the overlayer (or B = b 1 b2 ) to that of the substrate (or A = a 1 a2 ), i.e. det M = B/A det M provides a convenient property to classify the structure of the overlayers. 3 Surface%20Structure%20III.htm 4/24

5 Case (a) det M integral and all matrix elements m ij integral. The two meshes are simply related with the adsorbate mesh having the same translational symmetry as the surface. All commensurate layers (phases in which the adsorbates have the same local structure, i.e. adsorption sites). EXAMPLES: Overlayers that can be denoted by Wood notation: p(2 2), c(2 2), ( 3 3)R30, etc. Case (b) det M a rational fraction (or det M integral and some matrix elements rational). 3 Surface%20Structure%20III.htm 5/24

6 The two meshes are rationally related with the meshes coming into coincidence at regular intervals (coincidence net structures). Adsorbed layers are incommensurate. Case (c) det M irrational. No common periodicity between the meshes exists. The surface is termed an incoherent structure. The substrate provides a flat surface on which the adsorbates form their own 2D structure. The adsorbate layers are incommensurate. Incoherent structure occurs, for instance, if the adsorbate adsorbate interaction is stronger than adsorbate substrate bonding or if the adsorbed species are too large to fit into the substrate grid. NOTE: Substrate is a confusing word! When we discuss clean surface (i.e. no adsorbate), the word substrate refers to the part of the solid with complete 3D periodicity. The word overlayer and the word selvedge are used interchangeably to denote the part with a changing depth periodicity. However, when we discuss adsorbates, we refer both the selvedge and the substrate as the substrate. This switch in definition is commonly encountered in surface science literature, but it is usually clear from the context. Stepped surfaces (Lang, Joyner and Somorjai, 1972) Thus far, we have been discussing adsorbate/substrate or selvedge/substrate structures with low Miller indices. 3 Surface%20Structure%20III.htm 6/24

7 Structures with high indices are found to be highly reactive (See Somorjai s books). A high index surface is broken up into terraces of low index planes (h t k t l t ) with constant widths and steps of monatomic height with planes (h s k s l s ). Then, a normal high index surface can be denoted as: n (h t k t l t ) (h s k s l s ) where n is the number of parallel atom rows forming the terrace. 3 Surface%20Structure%20III.htm 7/24

8 EXAMPLES (7 7 5) is equivalent to 6(111) (11 1) (10 8 7) is equivalent to 7(111) (310) 3 Surface%20Structure%20III.htm 8/24

9 3 Surface%20Structure%20III.htm 9/24

10 6. Electron diffraction as a means for surface structure determination We will give an overview of Low Energy Electron Diffraction (LEED) as a lead in to our next general topic of surface structure determination. LEED is as important to surface crystallography as X ray diffraction to solid crystallography. We will again introduce the important concepts first. 3 Surface%20Structure%20III.htm 10/24

$Diffraction from a surface net Like X ray diffraction, we are concerned mainly with elastic scattering of low energy electrons.$

11 Diffraction from a surface net Like X ray diffraction, we are concerned mainly with elastic scattering of low energy electrons. We can treat the basic physics of this process as wave interference (diffraction) off a lattice net due to electrons manifesting as de Broglie matter waves. As in normal diffraction phenomena, we consider the path length difference (PLD) between the rays scattered off neighbouring surface lattice net points. We can show that PLD = a (sin reflected sin incident) = n where a is the distance between the two scattering centers (two atoms on the space net) and is the wavelength of the de Broglie matter wave. s are measured w.r.t. the surface normal. 3 Surface%20Structure%20III.htm 11/24

12 EXAMPLE: Suppose incident = 0 (i.e. normal incidence case), then sin reflected = n / a (n / a) (1.5 /V) 1/2 where V is the kinetic energy of the incident electron in ev and a is in nm. Now, if we have reflected = 90, then the energy corresponds to the n th reflected ray from a particular point (h,k) at a distance d hk away from the scattering center. If we take n = 1, then V = 1.5/d hk 2 gives the energy that the (h,k) reflex first appears. As we increase V, the diffraction pattern (with n sets of spots) corresponding to a space lattice with d hk will be expanded (i.e. we observe a larger set of spots). 3 Surface%20Structure%20III.htm 12/24

$Although we can obtain the lateral arrangement of the surface net from a LEED diffraction pattern, the relative position of the surface net with respect to the substrate net cannot be determined from$

13 For a constant V, a large lattice with a big d hk will have a lot of spots (i.e. many different n). Although we can obtain the lateral arrangement of the surface net from a LEED diffraction pattern, the relative position of the surface net with respect to the substrate net cannot be determined from this simple consideration based upon Bragg law. In other words, we cannot tell the local structure of the surface atoms by just observing the LEED pattern. We need to measure the intensity variation as a 3 Surface%20Structure%20III.htm 13/24

14 function of V (this is called a I V curve) and then compare this with various theories. Reciprocal net Given that the lattice vectors of a space net is a and b, we can define a complementary reciprocal net as follows: a* = 2 (b n) / [a b n] b* = 2 (n a) / [a b n] where n is the unit vector normal to the surface. Note that we have exactly the same definition as the reciprocal lattice (i.e. 3D case) if we replace n by c. Again, we get the reciprocal relations: a a* = 2 b a* = 0 a b* = 0 b b* = 2 Any general reciprocal lattice net vector on the reciprocal surface net can then be denoted as: ghk = h a* + k b* 3 Surface%20Structure%20III.htm 14/24

15 Diffraction conditions Recall that this is elastic scattering, i.e. the (kinetic) energy before and after the collision with the surface is conserved. i.e. k 2 = k 2 or k 2 + k 2 = k 2 + k Surface%20Structure%20III.htm 15/24

16 In the case of the surface, only the parallel component of the momentum is conserved. i.e. k = k + g hk where g hk is the reciprocal vector of the surface net. Note that the perpendicular component is not conserved. Ewald sphere construction is a convenient graphical representation of the diffraction condition in reciprocal space. It can be used to identify diffracted beams. Case A Bulk or 3D (1) Draw a vector k to terminate at the origin of the reciprocal lattice. (2) Construct a sphere of radius k. (3) Identify the points of the reciprocal lattice with which the surface of this sphere intercepts. (4) The vectors from the centre of the sphere to these points correspond to the diffracted beams with vectors k (and the momentum transfer will be G hkl ). Recall that the diffraction condition for the bulk is: Case B Surface or 2D k = k = k + G hkl 3 Surface%20Structure%20III.htm 16/24

17 (1) Replace the reciprocal lattice with a set of reciprocal lattice rods constructed by extending lines perpendicularly (w.r.t. the surface) from the reciprocal lattice net. (2) Follow the above steps, except here the diffracted beams are identified by crossing the surface of the Ewald sphere with these reciprocal lattice rods. In the case of the bulk, a change in the electron energy or direction of k will cause the loss of reflected beams and the creation of new ones. In the surface case, these changes only cause a slight movement of the diffracted beams. Furthermore, there is a second set of diffracted beams that corresponds to forwardly scattered electrons (i.e. electrons 3 Surface%20Structure%20III.htm 17/24

18 that propagate into the crystal and therefore cannot be observed). Note also that indexing of the diffracted beams is referenced to the corresponding substrate space and reciprocal lattice nets. In other words, individual spots on a reciprocal net as indicated by a LEED pattern are given by the indices (h, k) corresponding to diffracted beams with a momentum transfer of reciprocal lattice net vector g hk. In the special case of a (non primitive) centred rectangular Bravis net, we will have missing net points. In the case of adsorbates, additional half order or fractionalorder spots are usually found (in the reciprocal or LEED pattern) because the space net of the adsorbates is normally bigger than that of the substrate. Low Energy Electron Diffraction (LEED) REF: web The experiment is simple: A low energy electron beam is elastically scattered off the surface and the diffraction spots are displayed using a fluorescent screen or other electron multiplier devices. The wavelength of the electron matter 3 Surface%20Structure%20III.htm 18/24

pattern recognition only e gun, (2 to 4) grids, flourescent screen.

19 waves can be changed by the accelerating potential of the electron beam. Common commercial instruments: Display type: Reciprocal lattice net or pattern recognition only e gun, (2 to 4) grids, flourescent screen. 3 Surface%20Structure%20III.htm 19/24

20 REF: Video type: Pattern recognition and IV curve measurements (quantitative analysis) same as display type but with an add on video camera; or with the fluorescent screen replaced by a Faraday cup to collect the current, and/or with the grids replaced by more sophisticated electron optics. 3 Surface%20Structure%20III.htm 20/24

REF: C.S. Ri et al. J. Vac. Sci. Tech. B 15 (1997) 1623. See examples of structural determination by LEED and IV curve measurement here.

21 REF: C.S. Ri et al. J. Vac. Sci. Tech. B 15 (1997) See examples of structural determination by LEED and IV curve measurement here. SPA LEED: Spot Profiling Analysis LEED e gun, octupole deflection optics and electron detector. This is designed to measure the intensity of a spot and the 3 Surface%20Structure%20III.htm 21/24

22 background intensity to learn about the defects of the surfaces, among other things. 3 Surface%20Structure%20III.htm 22/24

23 3 Surface%20Structure%20III.htm 23/24

24 3 Surface%20Structure%20III.htm 24/24

2. Diffraction as a means to determine crystal structure

$2. Diffraction as a means to determine crystal structure$ 2. Diffraction as a means to determine crystal structure Recall de Broglie matter waves: He atoms: [E (ev)] 1/2 = 0.14 / (Å) E 1Å = 0.0196 ev Neutrons: [E (ev)] 1/2 = 0.28 / (Å) E 1Å = 0.0784 ev Electrons: