Structure of Surfaces

Stepped surface

Interference of two waves

Bragg s law Path difference = AB+BC =2dsin ( =glancing angle) If, n =2dsin, constructive interference Ex) in a cubic lattice of unit cell a sin = /2d = (h 2 +k 2 +l 2 ) ½ /2a determination of a

de Broglie relation in 1924 Any particle traveling with a linear momentum p has a wavelength Matter wave: p = mv = h/λ

Δk = 2π/d n = G

radius 00

a b

Park & Madden : Matrix- notation b 1 = m 11 a 1 + m 12 a 2 b 2 = m 21 a 1 + m 22 a 2 ( b = M a M = m 11 m 12 ) m 21 m 22 - Area of unit cell A = l a 1 a 2 l B = l b 1 b 2 l - B = (m 11 m 22 m 12 m 21 ) A = A det M

Substrate lattice a 1, a 2 Surface structure b 1, b 2 b = M a Then, b 1 = m 11 a 1 + m 12 a 2 & b 2 = m 21 a 1 + m 22 a 2 In a similar way, reciprocal lattice b* can be expressed in terms of reciprocal lattice a*. b 1 * = m 11 * a 1 * + m 12 * a 2 * b 2 * = m 21 * a 1 * + m 22 * a 2 * M* : inverse transposed matrix of M ~ M* = M -1 ~ and M = M* -1

A = a 11 a 12 a 21 a 22 -a 21 a 11 A-1 = 1/detA a 22 -a 12 A T = a 11 a 21 a 12 a 22 (A T ) -1 =1/det A a 22 -a 21 -a 12 a 11

i.e. m 11 = 1/(det M*) m 22 * m 12 = -1/(det M*) m 21 * m 21 = -1/(det M*) m 12 * m 22 = 1/(det M*) m 11 * From reciprocal lattice (i.e. LEED pattern), real surface lattice can be induced.

Interference conditions and the Ewald construction 2 /a k 2 1 k a 2 1 Scattering angle

LEED (low energy electron diffraction) -display type (1) Heated cathode emits electrons (2) Electrons are collimated by a lens system and finally leave the drift tube with energy U (20-500 ev). (3) Drift tube and sample: grounded. - electrons go through the field free space to sample. (4) Electron beam diameter= 1 mm at crystal surface due to finite size of cathode. - energy spread of electrons = ~1eV due to thermal energy distribution - emission current = ~ 1 A vary with energy U

(5) back-scattered electrons pass the 1 st grid which is grounded: Scattered electrons are not electrostatically deflected in the field free region between sample and grid. (6) 2 nd grid is at the negative potential slightly lower than the primary electron energy: prohibits the passage of inelastically scattered electrons. (7) Addition of extra grid between the 2 nd grid and the fluorescent screen to improve the LEED pattern. (Since the high voltage of the screen induces the field inhomogeneity in the 2 nd grid mesh) (8) 4 th grid improves the resolution. (9) Elastically scattered electrons are accelerated by positive potential of a few kv to fluorescent screen. (10) diffraction spots at the positions of interference maxima. (11) Observation of LEED pattern through the window of vacuum chamber or taking a picture of the pattern.

(1) Periodicity along the surface normal is lost in 2D arrangement of atoms: no constructive interference of scattered waves. (2) Reciprocal lattice: distance 1/(distance of real space) in z-direction, c* reciprocal lattice has a set of infinitely long rods normal to the plane of atoms. a i * a j * = 2 ij ij = 1, i=j or =0, i j a i *, a j * : unit mesh of the reciprocal lattice a i, a j : unit mesh of the real lattice E = (h 2 /2m)k 2, k = 2 / (3) In 2D, Ewald sphere construction : reciprocal rods are labeled with only 2 indices h, k General reciprocal lattice vector G g hk = h a 1 * + k a 2 *

(4) Diffraction occurs everywhere the Ewald sphere cuts a reciprocal lattice rod & the diffracted beam is labeled with (h, k) of the rod. k ll = k 0ll + g hk ( K ll = G hk ) k = k 0 (energy conservation) (5) In 1D k ll = 2 / cos (90- ) k 0ll = 2 / sin o 2 / (sin - sin o ) = 2 h/a a (sin - sin o ) = h k 0 o k

(6) In LEED, scattered beam k direction comes from the diffraction spots. (intersection of scattered beam & fluorescent screen) Observed LEED diffraction pattern = direct description of surface reciprocal lattice. (7) As k (1/ ), increase in the radius of Ewald sphere. More diffraction spots which move in continuously toward (0,0) spot. (8) (0,0) spot: no change in position with E (since direct reflection of primary beam at surface without diffraction.)

Ex) fcc(100) surface LEED Pattern (01) (11) a 2 a 2 * (00) (10) a 1 adsorbate a 1 * b 2 b 2 * b 1 * b 1 b 1 = 2 a 1 b 2 = 2a 2 b 1 * = ½ a 1 = ½ a 1 * b 2 * = ½ a 2 = ½ a 2 * b* = ½ 0 a* 0 ½ Det (M*) = ¼ m 11 =1/(¼ ) x ½ = 2 m 22 =1/(¼ ) x ½ = 2

Interpretation of LEED pattern (1)Diffraction pattern (2) c(2x2) a 2 * b 2 * b 1 * a 1 * b 2 a 1 a 2 b1 b 1 * = ½ a 1 * + ½ a 2 * b 2 * = -½ a 1 * + ½ a 2 * b 1 = a 1 + a 2 b 2 = -a 1 + a 2 M*= ½ ½ -½ ½ det M* = ½ Then, m 11 = 2x1/2 =1, m 12 = -2 x (-1/2)=1 m 21 = -2x1/2 =-1, m 22 = 2 x (1/2)=1

Electron Diffraction and Surface Defect Structures (1)Perfect crystal with a perfect instrument - infinitely sharp spots with zero intensity between (2) Finite instrumental limitations - finite width of spots & possibly some background (3) Deviations from periodicity by defects in structure or varying composition - alter spot shape & background dramatically (4) Inelastic scattering - may change the pattern

Kinds of surface defects (1)zero-dimensional: point defects due to contamination atoms, missing substrate atoms, or atoms which are displaced temporarily or permanently out of their regular lattice positions (2)1D defects: atomic steps or superstructure domain boundaries (3)2D defects: facets or amorphous top layers

<Kinds of information in the diffraction pattern> (1) Spot separation: always reciprocal to a frequently occurring distance - maybe regular atomic distance due to substrate structure or to superstructure or it may be the average distance between groups of atoms (2) Spot shape: reflects the range of coherent or in-phase scattering - If this range is smaller than the coherent range due to instrumental limitation, it represents the average size of regular atomic position (ex) domain size of superstructure or terrace width of step structure (3) Background - Contains information on the # and correlations of point defects - even in the case of ideal lattice, background intensity due to inelastic process - difficult to distinguish from the elastic process

Vibrations at surfaces - In each solid, the atoms are oscillating around their equilibrium positions due to their thermal energy. -Phonon: energy quantum of a lattice vibration (a) considered as a quasi particle with energy h (b) surface phonons influence specific heat, surface free energy, mean square displacement of surface atoms, thermal expansion (c) phonons may be localized or propagate parallel to the surface -Variation of LEED intensities : I of diffraction spots gradually decrease with increasing T & finally disappear in the background which become continuously brighter ln I -T

Debye spectrum D LEED intensity I = I 0 e -2W, 2W = 12h 2 /mk (cos / ) 2 T/ D 2 e- 2W = Debye-Waller factor Debye frequency: maximum possible frequency in Debye spectrum D = h /k : Debye temperature m= mass of atoms k=boltzman s constant = scattering angle of electrons = wavelength D = Debye temp.

Mean square displacement = 3h 2 /(4 2 m k) (T/ D2 ) ln I vs. T determines D or

(1) Energy dependence D vs. E p -With increase in Ep, D approaches the D bulk -Since increase in penetration depth with increase in Ep, increase in the detected volume -i.e. D surface << D bulk

(2) (00) beam intensity vs. T with variation of -Decrease in D when =0 compared to with =70 (grazing incidence) in surface normal > in surface parallel -Calculations for fcc(100) surfaces of (Cu, Ag, Au, Al, Pb, Ni, Pd, Pt) surface ~ 2.0 bulk // surface ~ 1.2-1.5 bulk