Tetramers on diamond, Si, and Ge„113… surfaces: Ab initio studies

PHYSICAL REVIEW B 68, 205306 2003
<strong>Tetramers</strong> on diamond, Si, and Ge„113… surfaces: Ab initio studies
A. A. Stekolnikov, J. Furthmüller, and F. Bechstedt
Institut für Festkörpertheorie und Theoretische Optik, Friedrich-Schiller-Universität, Max-Wien-Platz 1, 07743 Jena, Germany
Received 4 July 2003; revised manuscript received 5 September 2003; published 14 November 2003
We present ab initio studies of 21 reconstructions of the diamond, Si, and Ge113 surfaces. For all
group-IV elements tetramers stabilize the 113 surface due to the tendency for opening of a surface-state gap.
In the case of diamond, the symmetric tetramer structure with semiconducting properties is shown to be the
most stable reconstruction among all studied geometries. Additional interstitial atoms or adatoms on the C113
surface do not lower the surface energy. Si and Ge113 surfaces show a considerable asymmetry, where the
electron transfer from lower to upper atoms of the tetramers leads to a further band-gap opening. For these
systems, however, 21 reconstructions are less stable than 32 reconstructions with interstitials. Nevertheless,
tetramers are also important reconstruction elements for Si and Ge surfaces.
DOI: 10.1103/PhysRevB.68.205306
PACS numbers: 68.35.Bs, 68.35.Md, 73.20.At
I. INTRODUCTION
Diamond, Si, and Ge113 surfaces are stable against
faceting as shown experimentally and theoretically. 1–4 A detailed
description of these surfaces and the status of their
investigations is given in Ref. 4. Si and Ge113 surfaces are
in general stabilized by adatom-dimer-interstitial ADI or
adatom-interstitial AI reconstructions with 32 translational
symmetry. 4–6 On the contrary, interstitial atoms are not
reconstruction elements of the diamond113 C113 surfaces.
In the diamond case the 31 symmetric adatomdimer
AD model gives the lowest surface energy of all
31 and 32 test geometries but is accompanied by a metallic
surface behavior. So far, there are no experimental results
reported for the atomic structure of the clean C113
surface. However, it is known from the low-index surfaces,
i.e., C111 and C001, that only short-range symmetric
21 reconstructions occur in contrast to the Si(111)77,
Ge(111)c(28) or Si/Ge(001)c(42) superstructures. 7 At
least experimentally the low-index diamond surfaces with
21 translational symmetry clearly show an insulating behavior.
The question arises if the diamond113 surface can give
rise to an electronic structure with nonmetallic character. A
good candidate for its geometry could be a 21 reconstruction.
The bulk-terminated 113 surfaces consist of alternating
rows of twofold-coordinated 001-like atoms and
threefold-coordinated 111-like atoms with two and one
dangling bonds, respectively Fig. 1a. The at least partial
dimerization of the neighbored 001-like atoms along the
11¯0 direction leads to a 21 translational symmetry Fig.
1b. Together with the two 111-like atoms in 332¯ direction
the four atoms form a tetramer. 8,9 Such tetramers have
been found on 32 ADI Si and Ge surfaces. In addition to
their symmetric forms also puckered tetramers have been
discussed within the 32/31 reconstruction models. 4,10
With an additional interstitial atom the tetramers change over
into pentamers. 5 Stable tetramers have been also predicted
for reconstructed Si114 and Si110 surfaces. 11,12
In this paper we present results of first-principles calculations
for the 21-reconstructed C, Si, and Ge113 surfaces.
Energetical, structural and electronic properties are studied to
understand the differences between the group-IV elements
and the physical/chemical trends of the tetramer structures.
The favorization of these reconstruction elements for diamond113
is derived.
FIG. 1. a Top view of a bulk-terminated 113 surface. The
dangling bonds are indicated. b Top view of a (113)21 surface
with symmetric tetramers as reconstruction elements. A possible
unit cell is indicated by thin lines. Filled open circles indicate
atoms in the top second bilayer. Dots represent atoms in the third
bilayer.
0163-1829/2003/6820/2053065/$20.00
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©2003 The American Physical Society

A. A. STEKOLNIKOV, J. FURTHMÜLLER, AND F. BECHSTEDT PHYSICAL REVIEW B 68, 205306 2003
TABLE I. Surface energies of diamond, Si, and Ge113 surfaces
per unit cell (E surf ) or per unit area (). Values for 31
adatom-dimer AD, 32 AD oppositely puckered op, 32
adatom-interstitial AI, and 32 adatom-dimer-interstitial ADI
surface reconstructions are taken from Ref. 4.
Reconstruction E surf (eV/21 cell (J/m 2 )
C Si Ge C Si Ge
FIG. 2. Brillouin zone of the 21-reconstructed 113 surface.
II. COMPUTATIONAL METHOD
Our calculations are performed within the density functional
theory DFT in the local density approximation
LDA. The interaction of the electrons with the atomic cores
is treated by nonnormconserving ab initio ultrasoft
pseudopotentials. 13 Explicitly we use the VASP code. 14 In the
bulk case the DFT-LDA yields cubic lattice constants of a 0
3.531, 5.398, and 5.627 Å and indirect fundamental energy
gaps E g 4.15, 0.46, and 0.00 eV for diamond, Si, and
Ge, respectively. As a consequence of the local density approximation
for exchange and correlation the lattice constants
are slightly underestimated by about 1.0 C, 0.6 Si,
or 0.6 Ge% with respect to the experimental values. In the
ground-state calculations within DFT-LDA the excitation aspect
is not included and, hence, the gaps are also underestimated.
Quasiparticle corrections of about 1.35 C, 0.66 Si,
or 0.66 Ge eV have to be added to obtain the experimental
gap values.
The surfaces are modeled by repeated slabs. Each slab
consists of 11 double layers and the same amount of vacuum
layers. The bottom sides of the slabs are passivated by hydrogen
atoms and kept frozen during the surface optimization.
The topmost six double layers of the slab are allowed to
relax. Eighteen k points are used in one half of the Brillouin
zone BZ for surfaces with 21 translational symmetry. A
symmetric slab is used to calculate the band structure of the
C(113)21 surface to avoid hydrogen-induced states within
the fundamental gap. For the purpose of comparison with
results obtained for other reconstructions with varying numbers
N of atoms in the slab, the surface energy E surf E tot
N is studied instead of the total energy E tot of the slab. 7
The chemical potentials of the atoms have been obtained
from the total energies of the completely filled slabs. Division
of E surf by the area of the studied surface results in ,
the surface excess free energy per unit area at low temperature.
Explicitly, we use the procedure described in Ref. 15.
III. RESULTS AND DISCUSSION
Bulk terminated 10.76 6.67 5.28 8.34 2.21 1.61
21 symmetric 6.59 4.82 4.22 5.11 1.60 1.28
21 asymmetric 4.41 3.37 1.46 1.03
31 AD 7.10 4.50 3.53 5.50 1.49 1.08
32 ADop 4.45 3.47 1.48 1.06
32 AI 4.37 3.24 1.45 0.99
32 ADI 4.23 3.26 1.40 0.99
A top view on a group-IV(113)21 surface with symmetric
tetramers as reconstruction elements is presented in
Fig. 1b. The two-dimensional lattice accompanying the
21 reconstruction is oblique. The corresponding BZ is
shown in Fig. 2 together with the most important highsymmetry
points. We studied a variety of symmetric and
asymmetric tetramers as starting configurations during the
total-energy minimization. For diamond(113)21 a symmetric
tetramer is clearly identified as the most stable reconstruction
element. Also the distances of the tetramer atoms
with respect to the substrate remain almost unchanged. In the
case of Si and Ge(113)21 the low-energy structures with
21 translational symmetry are given by puckered tetramers.
The tetramer atoms belonging to a diagonal pair go up or
down with respect to the substrate in agreement with findings
for puckered Si and Ge11331 and 32 models. 4,10 The
occurrence of asymmetric reconstructions is similar to the
observations for the dimer reconstruction of the 001 surfaces
and the -bonded chain reconstruction in the case of
the 111 surfaces. 7 Because of a stronger bonding strain in
the subsurface layers asymmetric reconstructions are forbidden
for diamond surfaces.
The resulting absolute surface energies and surface energies
per unit area are summarized in Table I. They are compared
with results obtained for 31 and 32 translational
symmetries of AD, AI, or ADI models. 4 The 21 reconstruction
gives rise to a substantial energy gain of 4.17 C,
2.26 Si, and 1.91 (Ge) eV/21 cell with respect to bulk
terminated surfaces. The comparison of 21 results with the
energies obtained for 31 and 32 reconstructions shows
clearly that the 21 reconstruction gives the most stable
structure of the diamond113 surface. With respect to the
31 AD structure, there is an additional decrease of the
surface energy by about 0.4 J/m 2 . The formation of a symmetric
tetramer per three 21 cells instead of two adatoms
in the case of two 31 cells lowers the energy of the diamond
surface by 0.51 eV per 21 unit cell. This is expected
for diamond since adatoms also do not lead to a stable reconstruction
of the C111 surface. 7
The situation is different for the Si and Ge113 surfaces.
Interestingly see Table I the asymmetric tetramers also stabilize
the Si and Ge113 surfaces more than the variety of
originally suggested AD models with 31 or32 translational
symmetries. 4 Only the coverage by additional interstitial
atoms makes the AI 31/32 or ADI 32 reconstructions
of Si and Ge113 surfaces energetically more favorable
than the asymmetric-tetramer 21 reconstruction. This
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TETRAMERS ON DIAMOND, Si, AND Ge113 . . .
PHYSICAL REVIEW B 68, 205306 2003
FIG. 3. Band structures of C, Si, and Ge(113)21 surfaces.
The band structures in the upper panels are calculated for symmetric
tetramers, while the lower ones represent asymmetric tetramers.
FIG. 4. Wave-function squares of the lowest unoccupied c 2 a,
c 1 b and highest occupied v 1 c, v 2 d surface bands at P for the
C(113)21 reconstruction. In a and c cases side views are additionally
plotted in the left panels in order to show the contributions
below the tetramer plane.
seems to be in complete agreement with the experimental
findings. However, we mention that for the layered system
Ge/Si113 a22 reconstruction has been observed. 16 Strain
may be important for the stabilization of other reconstructions
than 31 and 32 on Si and Ge113 surfaces.
Another interesting aspect of the surface energies in Table
I is obvious comparing with the corresponding values for
other surface orientations and their lowest-energy reconstructions,
111 4.06 (C-21), 1.36 (Si-77), and 1.01 J/m 2
Ge-c(28) or 001 5.71 (C-21), 1.41 Si-c(42),
and 1.00 Ge-c(42) J/m 2 . 15 The values for the 113 surface,
even considering only a 21 reconstruction, are inbetween
the values for the low-index surfaces. Not only for
Si and Ge, rather also for diamond the 113 surface should
occur on the equilibrium crystal shape and, hence, represents
a stable surface, although it is difficult to prepare experimentally.
Due to the pronounced insulating character see below
the C(113)21 surface is energetically more favorable than
the clean C(001)21 surface which indeed has been
prepared. 17
In order to understand the driving forces of the tetramer
surface reconstructions of diamond, Si, and Ge(113)21
the corresponding resulting band structures are plotted in
Fig. 3 versus the variation of the k points between highsymmetry
points in the surface BZ Fig. 2. In contrast to the
C(113)31 surface with an AD reconstruction, the
diamond(113)21 surface with tetramers shows a significant
insulating behavior. Two neighbored empty surface
bands in the upper part of the projected fundamental gap are
well separated from the two completely filled surface bands
somewhat below the valence-band maximum VBM of the
bulk band structure. The indirect energy gap is related to a
transition from the P line to the J point and amounts to 2.1
eV within the DFT-LDA, i.e., half the value of the bulk gap.
Surprisingly, already the symmetric tetramers give rise to the
opening of a surface gap for Si(113)21, though the gap is
small with 0.4 eV. This is not the case for the symmetricdimer
reconstruction of the Si(001)21 surface 15 for reasons
which will be seen discussing the nature of the electronic
states below. A gap opening does not occur for
symmetric tetramers on Ge(113)21. However, the prediction
of the surface band structure is difficult due to the remarkable
band-gap underestimate within the DFT-LDA already
for bulk Ge. In any case, for Si and Ge there is a
tendency to open the surface gap by puckering of the tetramers.
The occupied bands, in particular for Ge, are shifted into
the projected valence bands and only an empty surface state
band remains in the projected fundamental gap. However,
due to its strong dispersion the Ge(113)21 surface with
asymmetric tetramers remains metallic. The low position and
the strong dispersion of the lower surface band for
Ge(113)21 is a consequence of the bulk electronic structure.
The bulk conduction-band states arising not far from the
L point in the fcc BZ repel the surface band at J which is in
contrast empty for C and Si towards lower energies even
below the VBM. The general tendency of shifting the surface
bands and opening gaps between surface states with the
asymmetry of the tetramers is in agreement with similar findings
for the 111 and 001 low-index surfaces of Si and
Ge. 18,19
The physical/chemical origin of the surface bands is
indicated in Figs. 4 and 5. The squares of the wave functions
are plotted for the two lowest empty surface bands and
the two highest occupied surface bands for symmetric
tetramers C(113)21, Fig. 4 and asymmetric tetramers
Si(113)21, Fig. 5. The character of the four surface
bands within the fundamental gap is dominated by the six
dangling-bond orbitals situated at the four tetramer atoms
one on each 111-like atom or edge atom and two on each
001-like atom or dimer atom of the tetramer as indicated
in Fig. 1a for the surface atoms in bulk positions. Their
linear combinations form the surface-state wave functions.
Even for the symmetric case of diamond, the two sp 3 orbitals
of each 001-like atom cannot be dehybridized in such a
way forming and orbitals with respect to the directions
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A. A. STEKOLNIKOV, J. FURTHMÜLLER, AND F. BECHSTEDT PHYSICAL REVIEW B 68, 205306 2003
FIG. 5. Wave-function squares of the lowest unoccupied c 2 a,
c 1 c and highest occupied v 1 c, v 2 d surface bands at P for the
asymmetric tetramer reconstruction of the Si113 surface. The
band v 2 d occurs in the projected band structure of the valence
bands and, hence, is not indicated in Fig. 3. In a and c side views
are also plotted in the left panels.
113 and 11¯0 as in the classical dimer picture of 001
surfaces. 18 In contrast to 111 and 001 surfaces with exact
and orbitals, six sp 3 -like dangling bonds have to be
discussed to explain the electronic structure of the tetramer.
As a consequence, for diamond only one bond along the
11¯0 direction is nearly formed by two sp 3 -like orbitals
located at 001-like atoms. The remaining dangling bonds
strongly interact with the dangling bonds of the neighbored
111-like edge atoms of the tetramer. In the case of
diamond(113)21 the distance of the two 001-like atoms
in the tetramer amounts 99% of a bulk bond length d bulk .
This is in contrast to a dimer bond length 0.90 d bulk calculated
for C(001)21. 15,18 Interestingly the two bond lengths
between edge and dimer atoms in the tetramer are reduced to
0.89 d bulk . The reason is a dimerlike behavior of two such
atoms. Already in the bulk case a bond exists. An additional
bond is formed by the two dangling hybrids of the
edge and dimer atoms. No asymmetry occurs in the tetramer
because of the strong bonding between the carbon atoms
which takes place due to the lack of p and d electrons in the
atomic core. The tetramers nearly show a mirror symmetry
along 332¯. They are almost planar. Only a small difference
of 0.22 Å remains between dimer and edge atoms along the
z direction as an artifact of the atomic positions in the bulkterminated
surface. As a consequence of the weak bonding of
the so-called dimer atoms, the occupied tetramer states in
Fig. 4d are governed by antibonding bonding combinations
of dangling orbitals at dimer atoms and bonding antibonding
combinations of dangling orbitals located at edge
and dimer atoms. However, there are also contributions from
the orbitals around the central atom below the tetramer. The
unoccupied states are dominated by antibonding linear combinations
of dangling bonds located at the dimer and edge
atoms.
Because of the most favorable asymmetric puckered tetramer
structures the surface states are completely changed
for the Si and Ge(113)21 surfaces see Fig. 5. A tetramer
does not anymore form a planar structure. Starting from the
left edge atom, approaching the two so-called dimer atoms,
and going around to the second edge atom one finds the
following variations z in the vertical positions: 0, 0.91,
0.29, and 0.77 Å 0, 0.98, 0.33, and 0.86 Å for Si Ge. The
dimer bond length between two 001-like atoms amounts to
1.01 0.99 d bulk , whereas the lengths between 111 and
001 atoms are 0.97 0.95 d bulk for Si113 Ge113. For
Si(113)21 the wave-function squares of the surface states
are shown in Fig. 5. Because of their similarity those for Ge
are not plotted. The consequence of the asymmetry of the
tetramers is obvious. The different surface states at the P
point in the BZ are mainly due to orbitals located at one
tetramer atom. The two lowest unoccupied states are localized
at the two down atoms forming a diagonal of the tetramer.
The two considered occupied states are mainly localized
at the two upper atoms. Energetically the lowest one lies
already in the projected bulk band structure see Fig. 3. The
situation shows similarities to the asymmetric-dimer reconstruction
of the Si(001)21 surface. The outgoing atoms
become a more s-like dangling orbital, whereas the atoms
displaced towards the substrate show a more p z -like character
of the dangling orbital. Electron transfer from the more
p z -like orbitals into the more s-like orbitals is expected to
lower the total energy of the system.
The metastability of the symmetric tetramers, at least for
Si(113)21, and the accompanying semiconducting surface
behavior gives additional insight in the occurrence of more
or less symmetric tetramers in the case of the 32 ADI
reconstruction of both Si and Ge113 surfaces. The experimental
and theoretical STM images of empty and occupied
states of 32 structures show contributions from the tetramer
atoms. 4–6 This means that electronic states should behave
as is shown in Fig. 3 upper panel for the symmetric
21 reconstruction. Of course, in the case of 32 unit cells
the surfaces are stabilized by the adsorption of interstitial
atoms and by the presence of adatoms and, hence, is accompanied
by a more complicated electron distribution. The tetramer
states are pushed into the bulk projected band structure,
and consequently are seen in images with not too high
bias voltages.
One interesting point should be discussed here. While the
metastable (111)21 surfaces represent low-temperature
cleavage faces, the more complicated Si(111)77 or
Ge(111)c(28) surfaces occur after annealing at elevated
temperatures. 7 This may suggest to try to prepare the metastable
Si and Ge(113)21 surfaces at lower temperatures
not using the standard procedure 5 leading to a 32 reconstruction
with interstitials. In the case of diamond adatom
and interstitial atoms are unfavorable. Consequently, the surface
structures with the lowest energies are C(111)21 and
C(113)21. Asymmetries are not allowed, neither in the
-bonded chains on 111 nor in the tetramers on 113.
IV. SUMMARY
In summary, using ab initio total energy and electronicstructure
calculations we have demonstrated that tetramers
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TETRAMERS ON DIAMOND, Si, AND Ge113 . . .
are important reconstruction elements for diamond, Si, and
Ge113 surfaces. The 21 translational symmetry with
symmetric tetramers has been shown to represent the most
stable diamond113 surface. In contrast to the C(113)31
surfaces with adatoms and tetramers its band structure exhibits
a clear insulating character. The surface energy is even
lower than that of the C(001)21 surface. Also, at least in
the Si113 case, the formation of tetramers tends to give rise
to a semiconducting surface. In particular, asymmetries due
to the puckering of the tetramers support the opening of an
energy gap between surface states. Because of the resulting
low surface energy we suggest the observation of tetramers
on Si and Ge113 also with 21 translational symmetry.
The appearance of reconstruction elements, such as tetramers
PHYSICAL REVIEW B 68, 205306 2003
or interstitials, depends remarkably on the group-IV element
C, Si, or Ge. We found that the general tendency for the
occurrence of an asymmetry in the reconstruction elements is
widely independent of the surface orientation 113, 111, or
001 but is clearly related to the substrate, diamond, or silicon
and germanium.
ACKNOWLEDGMENTS
We acknowledge financial support from the Deutsche Forschungsgemeinschaft
Project No. Be1346/12-1 and the European
Community in the framework of the Research and
Training Network NANOPHASE Contract No. HPRN-CT-
2000-00167.
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