Lecture February 13-15, Silicon crystal surfaces

Transcription

1 Lecture February 13-15, 2012 Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy Course number: Ch120a Hours: 2-3pm Monday, Wednesday, Friday William A. Goddard, III, 316 Beckman Institute, x3093 Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Teaching Assistants: Caitlin Scott Hai Xiao Fan Liu Silicon crystal surfaces Ch120a-1

2 Diamond Replacing all H atoms of ethane and with methyls, leads to with a staggered conformation Continuing to replace H with methyl groups forever, leads to the diamond crystal structure, where all C are bonded tetrahedrally to four C and all bonds on adjacent C are staggered A side view is This leads to the diamond crystal structure. An expanded view is on the next slide 2

3 Infinite structure from tetrahedral bonding plus staggered bonds on adjacent centers 2 nd layer st layer nd layer c 1 1st layer nd layer Chair configuration 1st layer of cylcohexane Not shown: zero layer just like 2 nd layer but above layer 1 3 rd layer just like the 1 st layer but below layer 2 3

4 The unit cell of diamond crystal An alternative view of the diamond structure is in terms of cubes of side a, that can be translated in the x, y, and z directions to fill all space. Note the zig-zag chains c-i-f-i-c and cyclohexane rings (f-i-f)-(i-f-i) There are atoms at all 8 corners (but only 1/8 inside the cube): (0,0,0) all 6 faces (each with ½ in the cube): (a/2,a/2,0), (a/2,0,a/2), (0,a/2,a/2) plus 4 internal to the cube: (a/4,a/4,a/4), (3a/4,3a/4,a/4), (a/4,3a/4,3a/4), (3a/4,a/4,3a/4), Thus each cube represents 8 atoms. All other atoms of the infinite crystal are obtained by translating this cube by multiples of copyright a in 2011 the William x,y,z A. Goddard directions III, all rights reserved c c f i c f c i f f i c f c i f c c 4

5 4 b 2 b Diamond Structure 5 a 3 a 1 a b 4 a 2 a 5 b 3 b 1 c 7 Start with C1 and make 4 bonds to form a tetrahedron. Now bond one of these atoms, C2, to 3 new C so that the bond are staggered with respect to those of C1. Continue this process. Get unique structure: diamond Note: Zig-zag chain 1 b Chair cyclohexane ring: b -7-1 c 5

6 Properties of diamond crystals 6

7 Properties of group IV molecules (IUPAC group 14) There are 4 bonds to each atom, but each bond connects two atoms. Thus to obtain the energy per bond we take the total heat of vaporization and divide by two. Note for Si, that the average copyright 2011 bond William is A. much Goddard III, different all rights reserved than for Si H 7

8 Comparisons of successive bond energies SiH n and CH n p lobe lobe p lobe lobe p p 8

9 Miller indices A 3D crystal is characterized by a unit cell with axes, a, b, c that can be translated by integer translations along a, b, c to fill all space. The corresponding points in the translated cells are all equivalent. Passing a plane through any 3 such equivalent points defines a plane denoted as (h,k,l). An equally space set of parallel to (h,k,l) pass through all equivalent points, which the l,m,n correspond to the reciprocal intersections on the unit cell when one plane passes through the origin. These are called Miller indices c a c/l a/h b/k b 9

10 Examples of special planes c c/l a a/h To denote all equivalent planes we use {h,k,l} so that indicates negative b/k b {1,0,0} for cubic includes the 3 cases in the first row) A number with a bar From Wikipedia 10

11 Crystallographic directions A lattice vector can be written as Rmnp = m a + n b + p c where m,n,p are integers. This is denoted as [m,n,p] The set of equivalent vectors is denoted as <m,n,p> Examples are shown here. From Wikipedia 11

12 The Si Crystal viewed from the [001] direction [001] [010] [110] [100] [010 [100] (001) Surface 1 st Layer RED 2 nd Layer GREEN 3 rd Layer ORANGE 4 th Layer WHITE [1,-1,0] not show bonds 12 to 5 th layer

13 The Si Crystal (100) surface, unreconstructed Projection of bulk cubic cell Surface zig-zag row Surface unit cell P(1x1) Every red surface atom is bonded to two green 2 nd layer atoms, but the other two bonds were to two Si that are now removed. This leaves two non bonding electrons to distribute among the two dangling bond orbitals sticking out of plane (like the 1 A 1 state of SiH 2 ) 1 st Layer RED 2 nd Layer GREEN 3 rd Layer ORANGE 4 th Layer WHITE 13

14 Si(100) surface (unreconstructed) viewed (nearly) along the [110] direction Each surface atom has two dangling bond orbitals pointing to left and right, along [1,-1,0] direction 14

15 The (100) Surface Reconstruction viewed (nearly) along the [110] direction Spin pair dangling bond orbitals of adjacent atoms in [1,-1,0] direction (originally 2 nd near neighbors Get one strong σ bond but leave two dangling bond orbitals on adjacent now bonded atoms (form weak π bond in plane) 15

16 Si(100) surface reconstructed (side view) Surface atoms now bond to form dimers (move from 3.8 to 2.4A) Get row of dimes with doubled surface unit cell One strong σ bond, plus weak π bond in plane orginal cell New cell Surface length length bond Lateral 7.6A 3.8A 2.4A displacements 0.7A 0.7A 16

17 Si(100) surface reconstructed (top view) New unit cell reconstructed surface P(2x1) Rows of dimer pairs are parallel original unit cell unreconstructed surface P(1x1) 17

18 Get 2x2 unit cell but atom at center is equivalent to atom at corner, therform c(2x2) 18

19 Two simple patterns for (100) Surface Reconstruction Dimer rows alternate C(2x2), high energy Dimer rows parallel P(2x1), low energy 19

20 P(2x1) more stable than c(2x2) by ~ 1kcal/mol The Sisurf-Si2nd-Sisurf bond for c(2x2) opens up to 120º because the Sisurf move opposite directions 120º 110º 120º 110º For P(2x1) the Sisurf move the same directions and Sisurf-Si2nd-Sisurf bond remains at 110º 20

21 stop 21

22 Si(110) surface (top view) Cut through cubic unit cell surface unit cell P(1x1) Surface atoms red 22

23 One dangling bond electron per surface atom Si(110) surface (side view) Surface atoms red 23

24 The (111) 1x1 Surface Unit Cell Unpaired electron

25 Construct (111) surface using cubic unit cell Start at diagonal atom #0 Go straight down to atom #1 Atom #1 bonded to 3 atoms #2 Each #2 is bonded straight down to an atom#3 Each atom #3 is bonded to 3 atom#4. Atoms 2 form a red (111) plane atoms #4 form a green (111) plane c

26 Si(111) surface (alternate construction) Start with red atom on top, bond to 3 green atoms in 2 nd layer Each green atom is bonded to 2 other 1 st layer atoms plus a 3 rd atom straight down (not shown) The 3 rd layer atoms bond to 3 4 th layer atoms in orange Surface unit cell P(1x1) 26

27 Reconstruction of Si(111) surface Each surface atom has a single dangling bond electron, might guess that there would be some pairing of this with an adjacent atom to form a 2x1 unit cell. Indeed freshly cleaved Si(111) at low temperature does show 2x1 Surface unit cell P(1x1) 27

28 LEED experiments (Schlier and Farnsworth, 1959) observed 7th Order Spots 7x7 unit cell From 1959 to 1981 many models proposed to fit various experiments or calculations. Binnig et al., 1981 did first STM image of Si (7x7) and saw 12 bright spots in 7x7 cell, showed that every previous model was incorrect Takayanagi et al., 1985, proposed the DAS Model that explained the experiments 28

29 two 7x7 cells What kind of interactions can go over a 7x7 region, with cell size 26.6 by 26.6 A? 29

30 Origin of complex reconstruction of Si(111) In 49 surface unit cells have 49 dangling bonds. Since cohesive energy of Si crystal is 108 kcal/mol expect average bond energy must be 108/2 = 54 kcal/mol (each atom has 4 bonds, but double count the bonds) (H3Si-SiH3 bond energy is 74 kcal/mol) Thus each dangling bond represents ~ 27 kcal/mol of surface energy = 1.1 ev per surface atom Calculated value = ev snap and ev relaxed. 30

31 Consider bonding an atom on top of 3 dangling bonds Two ways to do this. T 4 and H 3 T 4 (observed) H 3 (not observed) Bond angle strain (H3) Pauli repulsion (H3) Bond alignment/linear dependence (T4) *HOMO delocalization (T4) 31

32 T4 versus H3 site bonding to dangling bonds 32

33 33

34 The (111) 7x7 DAS Surface 34

35 The (111) 7x7 DAS Surface Layers (purple, brown and blue atoms have one dangling bond) 1 st 2 nd 3 rd 4 th First unreconstructed layer 35

36 The (111) 7x7 DAS Surface 12- and 8-membered rings 36

37 The (111) 7x7 DAS Surface Side view 37

38 The (111) 7x7 DAS Surface Cornerhole 38

39 The (111) 7x7 DAS Layer Positions REF REF REF 39

40 The (111) 7x7 DAS Surface Adatoms T 4 (observed) H 3 (not observed) Bond angle strain (H3) Pauli repulsion (H3) Bond alignment/linear dependence (T4) *HOMO delocalization (T4) 40

41 The (111) 7x7 DAS Surface Adatoms 41

42 The (111) 3x3 DAS Surface Unit Cell Side view Top view 12-membered rings 42

43 The (111) 5x5 DAS Surface Unit Cell Side view 43

44 The (111) 5x5 DAS Surface Unit Cell Top view 12- and 8-membered rings 44

45 The (111) 9x9 DAS Surface Unit Cell Side view 45

46 The (111) 9x9 DAS Surface Unit Cell Top view 12- and 8-membered rings 46

47 DAS Surface Energies (PBE DFT) 1.09 Energy, ev/1x1 Cell Regression Ab Initio DAS Cell Size Unreconstructed relaxed surface: ev/1x1 cell Infinite DAS model: ev/1x1 cell 47

48 DAS Surface Energies 5x5 Surface (9 dangling bonds) Energy, ev/1x1 Cell Spin Polarization 48

49 DAS Reconstruction Driving Force 49 unpaired electrons (1/2 Si-Si bond) per 7x7 1.2 ev = 58.8 ev/cell DAS 7x7 Surface energy = 51.2 ev/cell (19 unpaired electrons) Energy reduction due to reconstruction = 7.6 ev Difference is due to strain Bond length range = Å (equilibrium 2.35 Å) Bond angle range = º (Equilibrium ) 49

50 DAS Surface Energy Contributions 1.2 Energy, ev/1x1 Cell (DAS Model Cell Size) -1 1x1 T4 8R 12R F D TOTAL 50

51 DAS Surface Energies: Sequential Size Change Model 5 Energy, ev/16x16 Cell SSC Irregular-odd and even SSC regular-odd -20 SSC Cell Size Real-time STM by Shimada & Tochihara,

52 DAS Surface Energies: Origin of a finite cell size Energy, ev/1x1 Cell SSC Irregular-odd and even SSC regular-odd DFT Cell Size 52

53 DAS 3x3 (side view) 53

54 DAS 3x3 (top view) 54

55 DAS 5x5 (side view) 55

56 DAS 5x5 (top view) 56

57 DAS 7x7 (side view) 57

58 DAS 7x7 (top view) 58

59 DAS 9x9 (side view) 59

60 DAS 9x9 (top view) 60

61 stop 61