Ch125a-1. copyright 2015 William A. Goddard III, all rights reserved
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1 Lecture, October 28, 205: Si, Ga crystal surfaces Ch 25a: Elements of Quantum Chemistry with Applications to Chemical Bonding and Properties of Molecules and Solids Ch 20a:Nature of the Chemical bond Room 5 BI Hours: -:50am Monday, Wednesday, Friday William A. Goddard, III, wagoddard3@gmail.com 36 Beckman Institute, x3093 Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Teaching sistants: Ch25a Sijia Dong <sdong@caltech.edu> Ch20a Kurtis Carsch < kcarsch@caltech.edu > Ch25a-
2 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 2
3 Examples of special planes c c/l a a/h To denote all equivalent planes we use {h,k,l} so that indicates netive b/k b {,0,0} for cubic includes the 3 cases in the first row) A number with a bar From Wikipedia 3
4 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 4
![The Si Crystal viewed from the [00] direction [00] [00] [0] [00] [00 [00] (00) Surface st](/docs-images/95/123019415/images/5-0.jpg "Layer RED 2 nd Layer GREEN 3 rd Layer ORANGE 4 th Layer WHITE [,-,0] not show bonds 5 to 5 th")
5 The Si Crystal viewed from the [00] direction [00] [00] [0] [00] [00 [00] (00) Surface st Layer RED 2 nd Layer GREEN 3 rd Layer ORANGE 4 th Layer WHITE [,-,0] not show bonds 5 to 5 th layer
6 The Si Crystal (00) surface, unreconstructed Projection of bulk cubic cell Surface zig-zag row Surface unit cell P(x) 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 A state of SiH 2 ) st Layer RED 2 nd Layer GREEN 3 rd Layer ORANGE 4 th Layer WHITE 6
![Si(00) surface (unreconstructed) viewed (nearly) along the [0] direction Each surface](/docs-images/95/123019415/images/7-0.jpg "atom has two dangling bond orbitals pointing to left and right, along [,-,0] direction")
7 Si(00) surface (unreconstructed) viewed (nearly) along the [0] direction Each surface atom has two dangling bond orbitals pointing to left and right, along [,-,0] direction 7
![The (00) Surface Reconstruction viewed (nearly) along the [0] direction](/docs-images/95/123019415/images/8-0.jpg "Spin pair dangling bond orbitals of adjacent atoms in [,-,0] direction")
8 The (00) Surface Reconstruction viewed (nearly) along the [0] direction Spin pair dangling bond orbitals of adjacent atoms in [,-,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) 8
9 Si(00) 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 9
10 Si(00) surface reconstructed (top view) New unit cell reconstructed surface P(2x) Rows of dimer pairs are parallel original unit cell unreconstructed surface P(x) 0
11 Get 2x2 unit cell but atom at center is equivalent to atom at corner, therform c(2x2)
12 Two simple patterns for (00) Surface Reconstruction Dimer rows alternate C(2x2), high energy Dimer rows parallel P(2x), low energy 2
13 P(2x) more stable than c(2x2) by ~ kcal/mol The Sisurf-Si2nd-Sisurf bond for c(2x2) opens up to 20º because the Sisurf move opposite directions 20º 0º 20º 0º For P(2x) the Sisurf move the same directions and Sisurf-Si2nd-Sisurf bond remains at 0º 3
14 Construct () surface using cubic unit cell Start at diagonal atom #0 Go straight down to atom # Atom # bonded to 3 atoms #2 Each #2 is bonded to 3 atoms # in top layer. Get hexagonal double layer Each #2 is bonded straight down to an atom#3 Each atom #3 is bonded to 3 atom# c
15 Si() 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 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(x) 5
16 Reconstruction of Si() 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 2x unit cell. Indeed freshly cleaved Si() at low temperature does show 2x Surface unit cell P(x) 6
17 LEED experiments (Schlier and Farnsworth, 959) observed 7th Order Spots 7x7 unit cell This was one of the first LEED experiments, a breakthrough in surface technology Very surprizing result, no one expected From 959 to 98 many models proposed to fit various experiments or calculations, but interpretations from different calculations and experiments contradicted each other A 2 nd major breakthrough in surface technology occurred in 982 that showed all previous interpretations of theory and experiment were WRONG. What was that? 7
18 Stopped Oct 28 8
19 LEED experiments (Schlier and Farnsworth, 959) observed 7th Order Spots 7x7 unit cell From 959 to 98 many models proposed to fit various experiments or calculations. Binnig et al., 98 did first STM image of Si (7x7) and saw 2 bright spots in 7x7 cell, showed that every previous model was incorrect Takayanagi et al., 985, proposed the DAS Model that explained the experiments No one has yet explained the reaction mechanism 9
20 two 7x7 cells What kind of interactions can go over a 7x7 region, with cell size 26.6 by 26.6 A? 20
21 Origin of complex reconstruction of Si() In 49 surface unit cells have 49 dangling bonds for the unreconstructed unit cell. The cohesive energy of Si crystal is 08 kcal/mol What do you expect the surface energy to be? 2
22 Origin of complex reconstruction of Si() In 49 surface unit cells have 49 dangling bonds for the unreconstructed unit cell. Since cohesive energy of Si crystal is 08 kcal/mol expect average bond energy must be 08/2 = 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 =. ev per surface atom Calculated value =.224 ev snap and.200 ev relaxed. 22
23 Consider bonding an atom on top of 3 dangling bonds T 4 H 3 T 4 T 4 H 3 T 4 Get 3x2 unit cell By adding a cap of one adatom Si per 3 top layer Si, can tie off all original dangling bonds. Thus
24 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) 24
25 T4 versus H3 site bonding to dangling bonds top nodal plane orthogonal to purple-red bonds Same for H 3 & T 4 2 nd Nodal plane orthogonal to green-orange bonds 3 for H 3 & for T 4 Thus T 4 better SOMO H 3 SOMO T 4 bad good 25
26 0 2 H3 reconstruction 0 Top layer labeled 2 nd layer green Addon layer 0, blue Need just /3 Monolayer to tie up bonds. Surface energy increases by 0.3 ev Because get three linear 0--2 links, must orthogonalize Unit cell
27 0 H3 reconstruction, unit cell is 3 x With respect x unit cell the new unit cell is 3 longer Also the unit cell is rotated 30º so we could say new unit cell is ( 3 x 3)R30º 3/
28 T4 reconstruction 3 x 3 2 Top layer labeled 2 nd layer green Addon layer 0, blue Need just /3 Monolayer to tie up bonds. Surface energy decreases by 0.0 ev Because 0--2 ~ 00º Unit cell 28
29 T4 reconstruction 2x2 2 Top layer labeled 2 nd layer green Addon layer 0, blue Need just /3 Monolayer to tie up bonds, leave dangling bond orbital Surface energy decreases by 0.2 ev Per 2x2 cell Unit cell 29
30 Best Si() structure is DAS7x7 with energy 0.56 ev =3.6 kcal/mol lower than x Not much, but per 49 cells this is 76.2 kcal/mol = 7.64 ev Final structure has 2 adatoms out of 49 2*0.7*4=5.62 ev The rest of DAS 2.02eV/cell=0.04eV/atom Solares SD, Dasgupta S, Schultz PA, Kim YH, Musgrave CB, Goddard WA Langmuir 2 (26): (2005) 30
31 The () 7x7 DAS Surface 3
32 The () 7x7 DAS Surface Layers (purple, brown and blue atoms have one dangling bond) Adatoms on Top layer These adatoms protrude from the surface so that they show up prominently in STM Which way will this show up, tunneling from tip to surface or surface to tip? 32
33 The () 7x7 DAS Surface Layers (purple, brown and blue atoms have one dangling bond) st 2 nd red atoms, all bonded to st layer 3 rd 4 th First unreconstructed layer 33
34 The () 7x7 DAS Surface 2-membered ring at corner of cell 34
35 The () 7x7 DAS Surface Side view 35
36 The () 7x7 DAS Surface Cornerhole 36
37 Si() 7x7 37
38 The () 7x7 DAS Layer Positions REF REF REF 39
39 DAS 7x7 (side view) 40
40 DAS 7x7 (top view) 4
41 The () 3x3 DAS Surface Unit Cell ev worse than 7x7 Side view Top view 2-membered rings 42
42 The () 5x5 DAS Surface Unit Cell Side view 43
43 The () 5x5 DAS Surface Unit Cell ev worse than 7x7 Top view 2- and 8-membered rings 44
44 The () 9x9 DAS Surface Unit Cell 0.0 ev worse than 7x7 Side view 45
45 The () 9x9 DAS Surface Unit Cell Top view 2- and 8-membered rings 46
46 DAS Surface Energies (PBE DFT).09 Energy, ev/x Cell Regression Ab Initio DAS Cell Size Unreconstructed relaxed surface:.200 ev/x cell Infinite DAS model:.07 ev/x cell 47
47 DAS Surface Energies 5x5 Surface (9 dangling bonds) singlet pairing of remaining dangling bonds is best (wag does not understand why) Energy, ev/x Cell Spin Polarization 48
48 DAS Reconstruction Driving Force 49 unpaired electrons (/2 Si-Si bond) per 7x7 ev = 58.8 ev/cell DAS 7x7 Surface energy = 5.2 ev/cell (2+=3 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 = 9 7º (Equilibrium 09.4 ) 49
49 The (0) plane (outlined in green, layer ) [00] c [-,,0] [00] [00] [0] Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 50
50 Si(0) surface (top view) Cut through cubic unit cell surface unit cell P(x) Ch20a-Goddard-L09-0 Surface atoms red copyright 200 William A. Goddard III, all rights reserved 5
![Si(0) surface (viewed nearly along [-,,0] direction) One dangling bond electron per surface atom Surface atoms](/docs-images/95/123019415/images/51-0.jpg "red bulk atoms orange [,,0] [00] Ch20a-Goddard-L09-0 copyright 200 William A.")
51 Si(0) surface (viewed nearly along [-,,0] direction) One dangling bond electron per surface atom Surface atoms red bulk atoms orange [,,0] [00] Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 52
52 Reconstruction of (0) surface, surface atoms only side view (along [-,,0]) Showing just 2 dangling bond orbitals 54.7º 54.7º Top view (from [-,-,0]) [00] [,,0] [-,,0] Ch20a-Goddard-L09-0 [00] copyright 200 William A. Goddard III, all rights reserved 53
53 Reconstruction of (0) surface, surface atoms only We have a chain of dangling bond orbitals along the [-,,0] direction, each tilted by 35.3º from the [0] (vertical) axis They will want to tilt toward the vertical axis, reducing their angle from 35.3º). This leads to moving the surface atoms toward the bulk. There could be 2 by 2 pairing to double the surface unit cell in the [-,,0] direction [0] side view (along [-,,0]) Showing just 2 dangling bond orbitals 54.7º 54.7º 54.7º Ch20a-Goddard-L09-0 [00] copyright 200 William A. Goddard III, all rights reserved 54
54 The zincblende or sphalerite structure Replacing each C atom of the diamond structure alternately with Ga and so that each Ga is bonded to four and each is bonded to four Ga leads to the zincblende or sphalerite structure (actually zincblende is the cubic form of ZnS and the mineral sphalerite is cubic ZnS with some Fe) at corners: (0,0,0) at face centers: (a/2,a/2,0), (a/2,0,a/2), (0,a/2,a/2) Ga 4 internal sites: (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 has 4 and 4 Ga. Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 55
55 Bonding in Ga Making a covalent bond between to each atoms, one might have expected tetrahedral to make 3 bonds with a left over lone pair pointing away from the 3 bonds, while Ga might be expected to make 3 covalent bonds, with an empty sp 3 orbital point away from the 3 bonds, as indicated here, where the 3 covalent bonds are shown with lines, and the donor acceptor (DA) or Lewis acid- Lewis base bond as an lone pair coordinated with and empty orbital on Ga Of course the four bonds to each atom will adjust to be equivalent, but we can still think of the bond as an average of ¾ covalent and ¼ DA Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 56
56 Other compounds Similar zincblende or sphalerite compounds can be formed with Ga replaced by B, Al,In and /or replaced by N, P, Sb, or Bi. They are call III-V compounds from the older names of the columns of the periodic table (new UIPAC name 3-5 compounds). In addition a hexagonal crystal, called Wurtzite, also with tetrahedral bonding (but with some eclipsed bonds) is exhibited by most of these compounds. In addition there are a variety of similar II-VI systems, ZnS, ZnSe, CdTe, HgTe, etc Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 57
![2 The (0) plane (outlined in green, layer ) [00] 3 2 0 2 0 0 c [0] [-,,0] [00] Cut through cubic unit cell surface unit cell](/docs-images/95/123019415/images/57-1.jpg "P(x) atoms top layer Ch20a-Goddard-L09-0 Ga atoms top layer [00] [-,,0] copyright 200 William A.")
57 2 The (0) plane (outlined in green, layer ) [00] c [0] [-,,0] [00] Cut through cubic unit cell surface unit cell P(x) atoms top layer Ch20a-Goddard-L09-0 Ga atoms top layer [00] [-,,0] copyright 200 William A. Goddard III, all rights reserved 58
58 Reconstruction of (0) surface, side view along [-,,0] Si has dangling bond electron at each surface atom 54.7º 54.7º Surface has 3 covalent bonds to Ga, with 2 e in 3s lone pair, relaxes upward until average bond angle is 95º Surface Ga has 3 covalent bonds leaving 0 e in 4th orbital, relaxes downward until average bond angle is 9º. Ga angle 0º 26º 54.7º Ga [0] Si (0) Ga (0) [00] Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 59
59 Top view (from [-,-,0]) Reconstruction of Ga(0) surface has 3 covalent bonds, leaving 2 electrons in 3s lone pair, Ga has 3 covalent bonds leaving 0 eletrons in 4 th orbital Ga 54.7º 54.7º [00] [,,0] [-,,0] Ch20a-Goddard-L09-0 [00] copyright 200 William A. Goddard III, all rights reserved side view (along [-,,0]) 60
60 Reconstruction of (0) Ga Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 6
61 III-V reconstruction Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 62
62 Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 63
63 Reconstruction of Ga(0) surface, discussion We consider that bulk Ga has an average of 3 covalent bonds and one donor acceptor (DA) bond. But at the surface can only make 3 bonds so the weaker DA bond is the one broken to form the surface. The result is that Ga cleaves very easily compared to Si. No covalent bonds to break. has 3 covalent bonds, leaving 2 electrons in 3s lone pair. H3 has average bond angle of 92º. At the Ga surface relaxes upward until has average bond angle of 95º Ga has 3 covalent bonds leaving 0 eletrons in 4th orbital. GaH3 has average bond angle of 20º. At the Ga surface Ga relaxes downward until has average bond angle of 9º. This changes the surface Ga- bond from 0º (parallel to surface to 26º. Observed in LEED experiments and QM calculations Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 66
64 Analysis of charges Bulk structure: each has 3 covalent bonds and one Donoraccepter bond(lewis base Lewis acid). This requires 3+2=5 electrons from and 3+0=3 electrons from Ga. We consider that each bulk Ga bond has 5/4 e from and ¾ e form Ga. Each surface has 5/4+++2 = 5.25e for a net charge of each surface Ga has ¾+++0= 2.75 e for a net charge of Thus considering both surface Ga and, the (0) is neutral 5.25e 2.75e Net Q = Ga Ga Ga 3/4 5/4 3/4 5/4 3/4 5/4 5/4 3/4 5/4 3/4 5/4 3/4 5/4 3/4 5/4 3/4 5/4 3/4 a g a g a g 3/4 5/4 3/4 5/4 3/4 5/4 3/4 5/4 3/4 5/4 3/4 67 Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved
65 Ga (00) Start with at surface, denote Ga on 2 nd layer as. Then top layer is pure. Not stable, get net netive charge at surface. If cut off top layer, get pure Ga on surface Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 68
66 The Ga (00) surface, unreconstructed Every red surface atom is bonded to two green 2 nd layer Ga atoms, but the other two bonds were to two Ga that are now removed. This leaves three non bonding electrons to distribute among the two dangling bond orbitals sticking out of plane (like H 2 ) Ch20a-Goddard-L09-0 st Layer RED 2 nd Layer GREEN 3 rd Layer ORANGE 4 th Layer WHITE copyright 200 William A. Goddard III, all rights reserved 69
67 Ga(00) surface reconstructed (side view) For the perfect surface, in top layer, Ga in 2 nd layer, in 3 rd layer, Ga in 4 th layer etc. For the unreconstructed surface each has two bonds and hence three electrons in two nonbonding orbitals. Expect atoms to dimerize to form a 3 rd bond leaving 2 electrons in nonbonding orbitals. Surface - bonds Ga Ga Ga Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 70
68 Charges for 2x Ga(00) 2 nd layer has 3 e 2e - bond 2e LP st layer has 5.5 e 3/4 3/4 Ch20a-Goddard-L /4 5/4 3/4 3/4 3/4 5/4 5/ /4 3/4 5/4 3/4 5/4 3/4 5/4 2e - bond 3/4 3/4 3/4 3/4 3/4 3/4 copyright 200 William A. Goddard III, all rights reserved Top layer, 2 nd layer, 3 rd layer, as Each surface has extra 0.5 e dimer has extra e Not stable 7
69 Now consider a missing row of for Ga(00) Top layer, empty LP st layer has 5.5 e 3/4 3/4 Ch20a-Goddard-L09-0 3/4 5/4 3/ /4 2 nd layer has 2.25e 3/4 5/4 3/4 3/4 3/4 3/4 3/4 3/4 copyright 200 William A. Goddard III, all rights reserved 0 2 nd layer, 3 rd layer, as Each 2 nd layer next to missing is deficient by 0.75e extra 0.5 e 4 are missing 3e 72
70 Consider missing row out of =0 net charge Extra e missing 3e Extra e Thus based on electron counting expect simplest surface reconstruction to be 4x2. This is observed Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved Extra e Extra e missing 3e 73
71 Different views of Ga(00)4x2 reconstruction Ch20a-Goddard-L e Previous page, 3 dimer rows then one missing +.5e Two missing row plus missing Ga row Exposes 3 rd row Agrees with experiment Hashizume et al Phys Rev B 5, 4200 (995) copyright 200 William A. Goddard III, all rights reserved 74
72 Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 75
73 Ga () Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved Start with at surface, denote Ga on 2 nd layer as. Then top layer is pure. Not stable, get net netive charge at surface. Cut off top layer, to get pure Ga on surface, but break 3 bonds. Thus get at front always but back slab is Ga 76
74 summary Postulate of surface electro-neutrality Terminating the bulk charges onto the surface layer and considering the lone pairs and broken bonds on the surface should lead to: the atomic valence configuration on each surface atom. For example with 3 covalent bonds and a lone pair and Ga with 3 covalent bonds and an empty fourth orbital A neutral surface This leads to the permissible surface reconstructions Ch20a-Goddard-L09-0 copyright 200 William A. Goddard III, all rights reserved 78
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