Surface Science 74 (1978) 2133 @NorthHolland Publishing Company
THEORY OF RECONSTRUCTION APPLICATION
INDUCED SUBSURFACE STRAIN 
TO Si( 100)
Joel A. APPELBAUM and D.R. HAMANN Bell Laboratoties, Murray Hill, New Jersey 0 79 74, USA
Received 21 October 1977; manuscript received in final form 13 December 1977 The stress produced by surface layer reconstruction on semiconductors can produce sizable elastic distortions of deeper layers. These are calculated using a standard bulk model of interatomic force constants for the pairing model of 2 X 1 reconstruction on the (100) surface of C, Si and Ge. Kinematic analysis for the predicted geometry with subsurface distortions is shown to give good agreement with LEED intensity data for Si. This reconciles the LEED results, for which the simple pairing model fails completely, with spectroscopic measurements, which are best fit by the pairing model.
1. Introduction
Surface reconstruction is a widely observed phenomena on many semiconductor surfaces [ 1,2] While the reduced coordination of the surface atoms is undoubtedly the underlying cause of clean surface reconstruction, the mechanisms which push the atoms away from their bulk geometries and the new geometries they attain are poorly understood, and have been subjects of intensive investigation in recent years [3l 51. The purpose of this paper is to point out that motions of the surface layer atoms can produce substantial motions in subsurface layers, despite the fact that the chemical environment and bonding topology of these atoms are unchanged from the bulk. While the mechanisms driving the surface layer are presumably diverse, we believe that deeper subsurface motions can be understood from simple bond force models for most if not all cases. To illustrate the importance of these considerations, we have calculated subsurface distortions for the (100) surface of the elemental semiconductors C, Si and Ge, and demonstrate that these can resolve apparently irreconcilable inconsistencies in recent studies. In contrast to the (110) surfaces of C, Si and Ge, which exhibit very large and different surface periodicities [l], the (100) surfaces of all three elemental semiconductors exhibit a “simple” 2 X 1 unit cell [ 15,161. This suggests a single underlying mechanism, not too sensitive to detailed material differences, and therefore a good candidate for thorough investigation. Three chemically different structures have been proposed which are consistent with the 2 X 1 symmetry (as well as more complex modifications of these basic three). The first, the vacancy model, assumes 21
22
J.A. Appelbaum,
D.R. Hamann /Reconstruction
induced subsurface strain
that half the surface atoms are missing [4,15,16], the second that surface dimerization has occurred [9,16], and the third that a complex conjugated chainlike structure is formed at the surface [ 181. Selfconsistent calculations of the electronic structure of models 1 and 2 were carried out by Appelbaum, Baraff and Hamann (ABH) [ 191, selfconsistent calculations of model 3 by Kerker et al. (KLC) [20]. From these electronic structure studies it was concluded that the surface density of states for the vacancy and conjugated chain models were not consistent with ultraviolet photoemission (UPS) data [21]. By contrast, the dimer or pairing model resulted in a surface electron state density in good agreement [ 191 with UF’S data. A very large effort has been mounted to analyze the arrangement of atoms within the surface unit cell with LEED [ 1l 131. Both dynamic studies [ 131 and kinematic studies using “averaged data” have been undertaken [12,22]. It was found by all LEED studies that a simple model involving only surface vacancies or dimers was not consistent with LEED intensity data. Jona et al., using a dynamical analysis, found that a reasonable accounting of the LEED data could be given within the framework of the conjugated chain model [ 181. White and Woodruff (WW), using constant momentum data averaging techniques, were unable to make a structural determination. Rather, they concluded that either there were many layers of subsurface reconstruction present on Si(lO0) 2 X 1, or there was a breakdown of their averaging techniques. They favored the latter, being unable to reconcile deep reconstruction with studies which indicated that H chemisorbed on Si(100) produces a 1 X 1 structure amenable to kinematic analysis and which showed no subsurface distortions. While favoring the multiple scattering analysis of Jona et al., WW were disturbed by the apparent success of kinematic analysis for H on Si( 100) 1 X 1 [ 1 I] and its apparent breakdown on Si(lO0) 2 X 1 [ 121. The conclusion, then, from LEED analysis is that if one explores only one or two layer reconstruction, the conjugated chain model best fits the data. Unfortunately, this is not consistent with the analysis of spectroscopic data on the basis of electronic structure studies. On the other hand, the simple dimer model, whose electronic surface spectrum is consistent with UPS data, is inconsistent with LEED analysis. The principal results of this paper lead to a simple resolution of this dilemma. We propose that the essential nature of the 2 X 1 reconstruction is, in fact, dimerization. However, surface dimerization introduces sizable subsurface angular strains which drive substantial and deep reconstruction of the subsurface region. Since this reconstruction is an elastic deformation, it readily disappears when the “driving force” of surface layer dimerization is removed by H adsorption. To test this proposal, we construct a model of the surface elastic energy based on the assumption that surface dimerization has occurred and that bond bending and bond stretching forces are the same in the surface region as in the bulk. With this model we calculate the position of all the atoms in the selvedge region, which we
J.A. Appelbaum,
D.R. Hamann /Reconstruction
induced subsurface strain
23
permit to extend arbitrarily deep into the crystal. We find important subsurface reconstruction extending five atomic layers into the crystal. Having determined a structure based on minimizing the surface elastic energy, we then show that a simple kinematic analysis of the fractional order LEED intensities at normal incidence yields a remarkably good fit to the data with no adjustable structural parameters. The remainder of the paper will be organized as follows: Section 2 will contain a discussion of the model for the surface and subsurface elastic energy and the techniques for determining the minimum energy configuration. With this geometry determined we will calculate kinematically the LEED intensity data for four fractional order beams and compare them with experiment in section 3. Section 4 will summarize our conclusions and suggest additional experimental work.
2. Elastic surface energy The geometry of the surface atoms for the simple dimer model [9] of surface reconstruction is shown in figs. la and lb. The formation of the dimer bond leaves the surface atoms with a single dangling bond, and introduces no additional broken bonds within the selvedge. From a study of the electronic charge distribution within the dimer bond Appelbaum, Baraff and Hamann [ 191 concluded that it was nearly the same as the bonds within the bulk. The formation of the dimer bond by simply rotating pairs of surface atoms in the ideal structure toward each other results in nearly perfect tetrahedral coordinations of the bond angles about the surface atoms but significant angular strains about atoms in the second layer [9]. It is the elastic energy stored in these angular distortions which drives the subsurface reconstruction. The model we use to describe this elastic energy is a simple adaptation of Keating’s model for bulk semiconductors [23]. There are two kinds of shortrange elastic forces. These are bond stretching, having the form cy(Xij *Xii  3a2/l 6)2 ,
(2.1)
and bond bending, having the form P(X, . Xik + a’/1 6)2 3
(2.2)
where Xij = Xi  Xj ) and Xi locates the position of atom i, a near neighbor of atoms j and k. The size of the crystallographic unit cell is denoted by a. Eqs. (2.1) and (2.2) are manifestly invariant with respect to rotations and translation of the crystal as a whole, and vanish in the bulk when the atoms attain their nominal lattice positions. While (2.1) is strictly a bond stretching nearneighbor interaction, (2.2) contains not only bond bending forces but also second neighbor interactions. In addition,
24
J.A. Appelbaum,
D.R. Hamann /Reconstruction
induced subsurface strain
(a)
Fig. 1. A top of view (a) and side view (b) of a dimerized (100)2 X 1 surface of the diamond lattice. Atoms in the surface plane are labelled 1, those in the second atomic plane are labelled 2, and so forth. Arrows in the side view indicate the directions in which the atoms move away from the ideal dimer model to lower the strain energy.
both (2.1) and (2.2) contain terms beyond the harmonic approximation. As we will see, these play no significant role here. The elastic properties of bulk C, Si and Ge are well described with the use of (2.1) and (2.2) [23]. In particular, the three linearly independent elastic constants, and cd4 are fit very well with the two parameters CYand 0. In addition to CllJ Cl2 serving as a model of the static elastic properties, (2.1) and (2.2) also describe well the acoustical and optical phonons of these semiconductors. The only qualitative defect of the Keating model is its failure to predict the flattening of the transverse acoustic branch in Si and Ge near the zone boundary [24], In applying (2.1) and (2.2) to the surface one now has an additional bond parallel to the surface, the dimer bond, which introduces bond stretching forces between surface atoms, not present without dimerization, and concomitant bond
J.A. Appelbaum,
D.R. Hamann /Reconstruction
induced subsurface strain
25
bending forces. We assume these are the same as in the bulk, based on the near identity of dimer and bulk bond charges. We construct an elastic energy function, E, dependent on all atomic positions, by summing (2.1) and (2.2) over all bonds and bond pairs in the surface and subsurface E=CY g
(Xii.Xii3/16a2)*
bonds +fl
C
(Xii.Xik
+a*/16)*.
(2.3)
all bond pairs’
We make the plausible assumption that the set Xi which minimizes E must retain the same point group symmetry as the simple dimerized surface, Czmm. This constrains atoms within the surface unit cell to have coordinates which are given in table 1. The actual minimization of (2.3) is carried out two different ways. In the first, we make a quadratic expansion of (2.3) in the relative displacements of the selvage atoms about the ideal dimerized surface geometry. The minimization of this quadratic function is a simple exercise in linear algebra. The displacements of the atoms in the first five layers from their ideal positions are listed in table 2 for C, Si and
Table 1 The positions of atoms within the 2 X 1 reconstructed unit cell are defined layer by layer in terms of a displacement field Xi, Zi or Zo, ZF; the origin of the coordinate system, shown in fig. 1, is within the surface plane and positioned above the fourth atomic layer; note that the crystallographic unit cell has dimensions a Layers n = 0, 1, ... 4n + 1
4n + 2
Coordinates
4n + 4
____ Z
0
na+z4n+l
0
na + Z4n+ 1
ai2JZ:
(n + $2 + Z4n+*
al2a
(n + $a
0
al2J3
(n+ $a + Z&3
ald
a/M
(n + $a + Z&+g
0
0
(n + $k + Z&+3
44
0
(n + $a + Z&+3
al&h + X4n+l WG + X4n+l) al2ti
[email protected]/zfi
4n+3
Y
X
+ X4n+* + X4n+2)
+ Z4n+*
26
J.A. Appelbaum,
D.R. Hamann /Reconstruction
induced subsurface strain
Table 2 The actual displacement field, defined by table 1, is listed for the first five layers of C, Si and Ge within the harmonic approximation and for Si (indicated by an asterisk), including anharmanic terms; the displacement fields have been scaled by the crystallographic unit cell dimension a, to exhibit the similarity between C, Si and Ge; the ratio y, of the bond bending to bond stretching force constants used for C, Si and Ge are also listed (from ref. [23]) Layer
c (y = 0.66) ___
Si (y = 0.29)
Si* (y = 0.29)
Ge (y = 0.32) 
Xla
Xla
Xla
Xl0
Zla
1
0.120
0.010
0.128
2
0.023
0.0
0.022
Zla 0.017 0.001
Z/a
0.129
0.023
0.128
0.022
0.002
0.022
Zla 0.016 0.001
3

0.024 0.024

0.024 0.024

0.025 0.022

0.024 0.024
4

0.012 0.012

0.014 0.014

0.015 0.012

0.013 0.013
5
0.002
0.0
0.006
0.0
0.005
0.0
0.005
0.0
Ge, and shown schematically in fig. 1b. These results were obtained allowing atoms in the first ten layers to vary. Studies allowing 16 layers of atoms to vary showed convergence to much better than the accuracy indicated in the tables. In the second procedure, carried out for Si, the higher order, anharmonic terms implied by (2.3) are retained and E is minimized by nonlinear iterative techniques. The positions of the Si atoms within the frist five layers following this nonlinear procedure are shown in table 2. The differences are relatively minor, This is reassuring since it must be emphasized that the anharmonic terms implied by (2.3) are believed to be correct only in order of magnitude. Other purely anharmonic terms, constructed from higher order scalar invariants of the space group of the crystal, are omitted from (2.3) [23]. An analysis of the displacement fields tabulated in table 2 indicates that with the exception of the dimer bond, little (1%) bond stretching or compression is introduced into the structure. Rather, the substantial angular distortions between the bonds about the second layer are distributed more evenly throughout the selvedge region. For example, second layer bond angles that were 91’ in the ideal dimer model become 99” with subsurface reconstruction, while bond angles about first layer atoms that were 108’ become 104’. (Note that perfect tetrahedral coordination yields bond angles of 109”.) This accounts for the insensitivity of the displacement field to the relative strength of the bond bending to bond stretching forces in going from C to Ge and Si. The detailed displacement fields we have calculated have depended on the assumption that bond bending and stretching force constants are the same in the
LA. Appelbaum, D.R. Hamann /Reconstruction induced subsurface strain
21
bulk as in the selvedge region. For subsurface atoms it would seem to us that this is an extreme good approximation. For bonds involving surface atoms it is undoubtedly not as good. To assess how crucial this assumption is we have calculated the strain field allowing either the dimer bond to become stronger or weaker by 20% or for the bond bending forces to become weaker by 20%. None of these made any substantive change in the displacment fields previously calculated. One would have to make massive changes in the surface force constants before the qualitative picture that emerges breaks down.
3. LEED analysis Having determined a possible structure, the problem of validating it with LEED data remains. A fully dynamic calculation of LEED intensities is clearly desirable, but given the depth of reconstruction, this is presently beyond the capabilities of most existing computational schemes. Instead, we adopt the simplest of all analysis schemes, that which ignores all multiple scattering, i.e., a kinematic analysis [25]. This can establish whether the phase interference between successive layers of reconstructed Si (the only elemental semiconductor for which data are presently available) is sufficient to at least qualitatively account for the position, shape, and number of features seen in the fractional order beams. We have not focused on the primary beam intensities, since experimentally these are qualitatively different from the fractional order beams and show clear and strong evidence for multiple scattering effects. An examination of fig. 6 of ref. [26] shows that the normal incidence fractional order beams have at most four features between 30 eV and 140 eV, the primary beams have typically eight or nine features. If in our kinematic analysis we ignore the weak energy dependence of the Si atom scattering factor, the intensity 1 is proprotional to the surface structure factor 2 lal
2 n=l
c i=l
exp{iGtI*Pi,t
[2(E+~e)G~]‘/2Z~n}
fn12,
(3.1)
where n labels the atomic layer, and i the two Si atoms associated with that layer, whose coordinates are (Pin, Zin). The beams of interest are specified by the reciprocal lattice vector Gr r = (2&r/a) (m r, m2), where m 1is half integer and m2 integer. The energy is E, the inner potential of the crystal surface is Ve, and atomic units are assumed. From selfconsistent calculations this inner potential is the order of 0.5 (we assume 14 eV) for Si. For a we use 10.265 Bohr. In general, the beam will attenuate due to inelastic processes as it passes into the crystal. This is accounted for by the factor fi. With a meanfree path of 5 A for the electron beam, which includes the first five layers, there should be little need for this factor, since beyond the fifth layer there is negligible fractional order kine
28
LA. Appelbaum,
D.R. Hamann /Reconstruction I
SB
I
I
induced subsurface strain I
I
I
I
 0.22
1
0.10
7 \ o
FskK
0.11
0
0.05 M
s
L
0
W
Si {001}2xl
20
40
5 0 0
s
n
60
80
ENERGY
100
120
140
160
(eV)
Fig. 2. The experimentally measured (l/2, 0) and (l/2, 1) normal incidence LEED intensities versus energy by White and Woodruff Cw), Ignatiev and Jona (S) and Debe and Johnson (M) are compared. (From Ignatiev et al. [ 261, with permission.)
scattering. In addition, there is no atomic shadowing until the sixth layer. We therefore set fi equal to 1 for the results shown here [27]. We further restrict ourselves to normal incidence. Four fractional order beams have been measured, (l/2, o), (l/2, I), (312, o), and (3/2, l), and these are reproduced for the reader’s convenience in figs. 2 and 3. Figs. 2 and 3 represent data from three experimental
matic
0
20
40
60
80
ENERGY
100
120
140
’
(eV)
Fig. 3. The experimentally measured (312, 0) and (3/2, 1) normal incidence LEED intensities versus energy taken by White and Woodruff (W), Ignatiev and Jona (S) and Debe and Johnson (M) are compared. (From Ignatiev et al. [ 261, with permission.)
J.A. Appelbaum, D.R. Hamann /Reconstruction induced subsurface strain
0
L
0
29
I
20
40
60
60 ENERGY
100
120
140
(eV)
Fig. 4. Theoretically calculated (from eq. (3.1)) normal incidence LEED intensities for the Si(100) 2 X 1 surface are plotted versus energy for the (l/2,0) and (3/2,0) beams. The surface and subsurface deformations assumed are tabulated in column 3 of table 2.
groups and with the exception of the (3/2, 1) beam agree very well [26]. For this beam, there are experimental discrepancies in the location of the peak near 120 eV of the order of 10 eV. The corresponding theoretically calculated spectra [eq. (3.1)] for these beams are shown in figs. 4 and 5. Considering the level of the theory, and that there are no adjustable structural parameters, we consider the agreement between theory and experiment quite good. It must be emphasized that without the multiple layer interference which involves four of five layers, the narrow features seen in the experiment would not be obtained. For example, we have plotted in fig. 6 the (3/2, 1) beam intensity allowing the electron beam to see two, three and four layers of Si. It requires a full four layers of reconstruction to develop the comparatively narrow peaks and position them at the experimentally correct positions. Two layers are completely inadequate, and one layer, the simple dimer
30
J.A. Appelbaum,
D.R. Hamann 1 Reconstruction
induced subsurface strain
4
3
;; t 5
2
&’ z m a0 a
ENERGY (eV)
Fig. 5. Theoretically calculated (from eq. (3.1)) normal incidence LEED intensities for the Si(100) 2 X 1 surface a.re plotted versus energy for the (3/2, 0) and (3/2, 1) beams. The surface and subsurface deformations assumed are tabulated in column 3 of table 2.
ized surface, would yield no variation of I with incident beam energy at all within our approximation, It is clear that comparatively deep reconstruction plays an essential role in understanding the LEED intensity data. Before concluding we would iike to address one final point. How sensitive are the LEED intensity curves to alternative forms of subsurface reconstruction? TO examine this we change the sign of the displacement field within the third and fourth layers, leaving everything else fmed. The LEED intensity curves for the (3/2, 1) beam is shown for this geometry and compared with the previously calculated results in fig. 7. It is clear that the detailed form of the subsurface displacement field is quite import~t in determining a LEED spectrum.
J.A. Appelbaum,
D.R. Hamann /Reconstruction
induced subsurface strain
31
./. I
20
I
40
60
80
ENERGY
100
120
I'$0
(a.u.1
Fig. 6. Theoretically calculated (from eq. (3.1)) normal incidence LEED intensities for the Si(100) 2 X 1 surface are plotted versus energy for the (3/2, 1) beam assuming only two layers of reconstruction (curve 2), three layers (curve 3), etc. The deformations assumed are those tabulated in column 3 of table 2. Note that only one layer of reconstruction produces an energy independent (horizontal) curve and that five layers (see fig. 5) and four layers have similar LEED intensities.
40
60
80 ENERGY
100
120
140
(eV)
Fig. 7. Theoretically calculated (from eq. (3.1)) normal incidence LEED intensities for Si(100) 2 x 1 surface are plotted versus energy for the (3/2, 1) beam. The dashed curve corresponds to deformations tabulated in column 3 of table 2, the solid line to a deformation identical to that in column 3 of table 2 except that the sign of the vertical displacement in the third and fourth layers have been reversed.
32
J.A. Appelbaum,
D.R. Hamann 1 Reconstruction
induced subsurface strain
4. Conclusions The structural picture that emerges for the (100) surface of the elemental semiconductors is one in which dimerization, driven by the chemical instability of the ideal 1 X 1 surface layer, introduces significant angular distortions about the second layer atoms. To partially relieve these angular strains the subsurface distorts, propagating the basic 2 X 1 surface reconstruction at least four more atomic layers into the crystal. We expect that the electronic structure of the dimerized surface will be relatively insensitive to the subsurface strain field. On the other hand, as we have seen, LEED intensities crucially depend on the existence of this strain. While the simple kinematic analysis strongly supports the subsurface reconstruction we have proposed, it is desirable that dynamic calculations be carried out allowing many layers to distort. In addition, the kinematic analysis of the data using averaging techniques should be reexamined to see if the structure proposed here can be verified. We expect both C(100) and Ge(100) to behave as Si(lO0); the strain fields for these are tabulated in table 2. Comparable LEED studies for these materials are urgently needed. The H chemisorption studies on Si(lO0) of White and Woodruff fit very nicely into the framework of the subsurface strain model. Once the dimers are broken down chemically by the H, the subsurface strain is relieved and the entire surface reverts to a nearly ideal 1 X 1 structure. It is clear that the stress produced by surface reconstruction will induce subsurface reconstruction and relaxation in a large class of semiconductor surfaces. Stress produced by surface layer bonding to ordered chemisorbed overlayers may have similar effects. Fortunately, it should be possible to calculate at least the deeper portions of the subsurface strain using bulk elastic models. This would in turn reduce the number of structural parameters that had to be explored in carrying out a LEED analysis.
Acknowledgements We would like to thank Dr. W. Weber for discussions of the elastic properties of bulk semiconductors and Professors F. Jona and M.B. Webb for discussions of LEED analysis applied to semiconductors and the Si( 100) surface in particular.
References [l] J.J. Lander, in: Progress in Solid State Chemistry (Pergamon Press, Oxford, 1965) p. 26. [2] C.B. Duke, Crit. Rev. Solid State Sci., Proc. 3rd Intern. Summer Inst. in Surface Science (CRC Press, Cleveland, in press). [3] J.A. Appelbaum and D.R. Hamann, Crit. Rev. Solid State. Sci. 6 (1976) 357. [4] W.A. Harrison, Surface Sci. 55 (1976) 1.
J.A. Appelbaum,
D.R. Hamann /Reconstruction
[5] J.V. Florio and W.D. Robertson,
induced subsurface strain
33
Surface Sci. 24 (1971) 173. [6] J.E. Rowe, M.M. Traum and N.V. Smith, Phys. Rev. Letters 33 (1974) 1333. [7] K.C. Pandey and J.C. Phillips, Phys. Rev. Letters 34 (1975) 1450. [8] D. Haneman, Surface Physics of Phosphors and Semiconductors (Academic Press, London, 1975) p. 1. [9] J.P. Levine, Surface Sci. 34 (1973) 90. [lo] C.B. Duke, J. Vacuum Sci. Technol. 14 (1977) 870. [ 1 l] S.J. White and D.P. Woodruff, Surface Sci. 63 (1977) 254. [12] S.J. White and D.P. Woodruff, Surface Sci. 64 (1977) 131. [ 131 F. Jona, H.D. Shih, A. Ignatiev, D.W. Jepsen and P.M. Marcus, J. Phys. Cl0 (1977) L67. [14] P. Mark, J.D. Levine and S.H. McFarlane, Phys. Rev. Letters 38 (1977) 1408. [15] J.C. Phillips, Surface Sci. 40 (1973) 459. [ 161 R.E. Schlier and H.E. Farnsworth, Semiconductor Surface Physics (Univ. of Pennsylvania Press, Philadelphis, 1957) p. 3. [ 171 J.B. Marsh and H.E. Farnsworth, Surface Sci. 1 (1964) 3. [18] R. Seiwatz, Surface Sci. 2 (1964) 473. [ 191 J.A. Appelbaum, G.A. Baraff and D.R. Hamann, Phys. Rev. Letters 35 (1975) 729; Phys. Rev. B14 (1976) 588. [20] G. Kerker, S.G. Louie and M.L. Cohen, Phys. Rev., to be published. [21] J.E. Rowe, Phys. Letters 46A (1974) 400. [22] M.B. Webb, T.D. Poppendieck, T.C. Ngoc and T. Tommet, in: Bulletin of the 36th Annual Conf. on Physical Electronics (Madison, Wisconsin, 1976)) M. 2.4. [23] P.N. Keating, Phys. Rev. 145 (1966) 637. [24] W. Weber, Phys. Rev. B15 (1977) 4789. [25] M.B. Webb and M.G. Lagally, in: Solid State Physics, Vol. 28 (Academic Press, New York, 1973) p. 302. 1261 A. Ignatiev, F. Jona, M. Debe, D.E. Johnson, S.J. White and D.P. Woodruff, J. Phys. Cl0 (1977) 1109. [27] Allowing the beam to attenuate by 15% per atomic layer, corresponding to a meanfree path of 5 A, caused no significant changes in the results.