ARTICLE IN PRESS
Physica B 340–342 (2003) 570–574
Prediction of XPS spectra of silicon self-interstitials with the
all-electron mixed-basis method
Takeshi Nishimatsua,*, Marcel Sluitera, Hiroshi Mizusekia, Yoshiyuki Kawazoea,
Yuzuru Satob, Masayasu Miyatab, Masamitsu Ueharab
a
Institute for Materials Research (IMR), Tohoku University, Sendai 980-8577, Japan
b
Seiko Epson Corporation, 3-3-5, Owa, Suwa-shi 392-8502, Japan
Abstract
Silicon self-interstitials have been investigated by local-density approximation-based ab initio all-electron
calculations using a mixed-basis of atomic orbitals and planewaves for the wave functions. The results of the
numerical calculations show that the bond-center, hexagonal, tetrahedral and split-/1 1 0S interstitials have deeper
characteristic 2p Kohn–Sham levels than those of perfect host silicon, but the split-/0 0 1S interstitial does not. Hence,
we predict that, within the frozen-orbital approximation, i.e., excluding the effect of electronic relaxation due to the
ejected electron, the X-ray photoelectron spectroscopy spectra for the bond-center, hexagonal, tetrahedral and split/1 1 0S interstitials will have binding energies 0.4–1:1 eV higher than perfect host silicon.
r 2003 Elsevier B.V. All rights reserved.
PACS: 71.15.Ap; 71.55.i; 82.80.Pv
Keywords: Ab initio; X-ray photoelectron spectroscopy; Defect formation energy; Amorphous silicon
1. Introduction
It is well known that self-interstitials in silicon
crystal determine self- and impurity-diffusion [1].
Self-interstitials are one of the thermal equilibrium
defects, but direct detection of self-interstitials has
not been reported. Not only from the viewpoint of
condensed matter physics, but also from that of
LSI-industrial demand, it is desirable to know
what kind of self-interstitials exist dominantly and
how they affect diffusion.
*Corresponding author. Tel.: +81-22-215-2053; fax: +8122-215-2052.
E-mail address: [email protected] (T. Nishimatsu).
X-ray photoelectron spectroscopy (XPS) has the
potential to determine the species of atoms.
Recently, resolution of XPS has improved so
much that in the near future XPS can directly
detect the self-interstitials. In this paper, to clarify
the one-to-one correspondences between XPS
spectra and structures of silicon self-interstitials,
we report ab initio all-electron mixed-basis calculations and predict the XPS spectra of 2p electrons
for five possible self-interstitial configurations, i.e.,
bond-center interstitial (point symmetry: D3d ),
hexagonal interstitial (D3d ), tetrahedral interstitial
ðTd Þ; split-/0 0 1S interstitial ðD2d Þ; and split/1 1 0S interstitial ðC2v Þ as shown in Fig. 1. The
predicted XPS spectra may be informative also for
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2003.09.133
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T. Nishimatsu et al. / Physica B 340–342 (2003) 570–574
Bond-center (D3d)
Split-(001) (D2d)
Hexagonal (D3d)
Split-(110) (C2v)
571
Tetrahedral (Td)
Conventional unit cell for reference
Fig. 1. Structures of silicon self-interstitials. Interstitial atoms are indicated with white spheres. Nearest neighbors to the defect atoms
are shown in gray. For the split-/0 0 1S and split-/1 1 0S structures, the interstitial atom cannot be distinguished from one of the host
atoms. Point symmetry of each structure is indicated in parentheses.
XPS analysis of amorphous silicon because
amorphous silicon has locally such interstitial-like
structures.
2. Calculation methods
We use supercells containing 65 silicon atoms
(64 host atoms and 1 interstitial atom) that have
volume 2a 2a 2a where a is the lattice constant
of the conventional unit cell of the silicon crystal.
( is used in all
The lattice constant a ¼ 5:3894 A
calculations. It has been determined by minimizing
the total energy of the perfect crystal.
The all-electron mixed-basis method can accurately estimate forces exerted on atoms, but it is
still time-consuming. Therefore, the fully relaxed
defect structures are obtained by the pseudopotential method, then the core electron levels of the
relaxed structures are recalculated using the allelectron mixed-basis method. We confirmed that,
using the all-electron mixed-basis method, the
calculated forces exerted on the atoms of the final
(
structures are smaller than 0:1 eV=A:
Both in the pseudopotential method and the allelectron mixed-basis method, the calculations are
performed within the local density approximation
(LDA) [2,3]. Details of both methods are explained in Sections 2.1 and 2.2.
2.1. Pseudopotential method
Electronic wave functions are expanded in
planewaves with cutoff kinetic energy ecut ¼
290 eV using the ultrasoft pseudopotentials [4,5]
as implemented in the VASP package [6,7]. A 8 8 8 grid of k-points are sampled in the first
Brillouin zone of the supercells to determine the
self-consistent charge densities. The fully relaxed
defect structures are obtained with the conjugate
gradient method so that forces exerted on atoms
(
become smaller than 0:001 eV=A:
2.2. All-electron mixed-basis method
The all-electron mixed-basis method includes
not only the valence electrons but also the core
electrons such as the 2p electrons of silicon.
Electronic wave functions are expanded into
1s; 2s; 2p; 3s; and 3p atomic orbitals and planewaves with cutoff energy ecut ¼ 570 eV: A cutoff
( is used for the non-overradius rcut ¼ 1:10 A
lapping atomic spheres associated with the silicon
atoms. Bloch functions at the G-point are sampled.
We did not include spin–orbit coupling, although
it causes a small 2p3=2 –2p1=2 splitting (B0:5 eV;
Ref. [8]) in the XPS spectrum of silicon, because
the chemical shifts are significantly larger. More-
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572
over, spin–orbit coupling does not shift the center
of the 2p peak.
3. Results of calculations and prediction of XPS
spectra
Defect formation energies EF of all self-interstitials calculated with the pseudopotential method
are shown in Fig. 2. In uncharged condition, the
hexagonal ðEF ¼ 3:45 eVÞ and split-/1 1 0S ðEF ¼
3:42 eVÞ interstitials are almost energetically degenerate locally stable defects. The bond-center
5.0
4.5
EF [eV]
4.0
3.5
3.0
2.5
(a)
5.0
4.5
4.5
3.5
3.0
2.5
2.5
(c)
conduction band
σ∗(+ −)
σ(+ +)
px
σ3(+ − +)
σ2(+ 0 −)
py
pz
3pSi
bonding
split-<001> (0)
split-<001> (++)
split-<110> (0)
2.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
µ [eV]
Fig. 2. Formation energies of silicon self-interstitials calculated
with the pseudopotential method as function of the chemical
potential m of the electrons measured from the valence band
maximum. Thick horizontal lines correspond to neutral states
and thin oblique lines correspond to 2+charged states. (a) the
bond-center and hexagonal interstitials, (b) the tetrahedral
interstitial, and (c) the split-/1 1 0S interstitials.
band
gap
3s Si
3.5
3.0
T (0)
T (++)
2.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
(b)
µ [eV]
antibonding
4.0
EF [eV]
EF [eV]
2.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2
µ [eV]
5.0
4.0
BC (0)
BC (++)
Hex (0)
ðEF ¼ 4:82 eVÞ and split-/0 0 1SðEF ¼ 4:67 eVÞ
interstitials have higher defect formation energies.
These results are in good agreement with earlier
LDA results [9,10].
For bond-center, tetrahedral, and split-/0 0 1S
interstitials, respectively 1-, 3-, 2-fold degenerated
mid-gap levels occupied by two electrons arise.
Consequently, due to the Jahn–Teller effect,
tetrahedral and split-/0 0 1S interstitials are unstable structures and symmetry reductions from Td
and D2d ; respectively, are expected. Defect formation energies of the 2+-charged states of these
three interstitials are also calculated and shown in
Fig. 2.
We compared valence band Kohn–Sham levels
at the G point calculated with the pseudopotential
method and with the all-electron mixed-basis
method and confirmed that they are identical.
As shown in Fig. 3, the bonding structure of a
silicon atom (Si) accommodated in the center of a
bond of two Si’s is more complicated than other
interstitials but can be described as two hybrid
orbitals on two Si’s and 3s and 3p orbitals on the
centered Si. This results in three occupied molecular orbitals s1 ; s2 ; and s3 : The elongation of Si–
Si bond reduces the split of the bonding and
σ1(+ + +)
(b)
(a)
normal elongated
Si-Si
Si-Si
valence
band
(c)
Si interstitial
at the bond
center site
(d)
Si atom
Fig. 3. Schematic bonding model of a silicon self-interstitial at
a bond-center site that has D3d symmetry in crystal; (a) bonding
and antibonding levels of a silicon-silicon pair, (b) bonding and
antibonding levels of an elongated silicon-silicon pair, (c)
impurity levels of the bond-center interstitial, and (d) the
atomic levels of a silicon atom.
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-90.5
energy [eV]
-91.0
-91.5
-92.0
-92.5
perfect
Bond-center
Hexagonal
Tetrahedral
Split-(001)
Split-(110)
Fig. 4. Kohn–Sham levels at the G-point of 2p electrons of silicon self-interstitials calculated with the all-electron mixed-basis method
and predicted spectra of XPS. The zero of the scale is placed at the valence band maximum.
Table 1
Madelung potentials in eV for electrons at the interstitial site ðfi Þ; at the nearest neighbor site ðfn:n: Þ; and at sites other than the
interstitial and its nearest neighbor sites ðfavg: ; averaged value) for each interstitial structure
Interstitial structure
Perfect
Bond-center
Hexagonal
Tetrahedral
Split-/0 0 1S
Split-/1 1 0S
fi
fn:n:
favg:
—
—
57.571
57.763
52.123
58.033
42.649
52.052
58.568
42.727
52.142
58.363
55.637
55.771
58.013
48.927
—
58.030
antibonding levels (Figs. 3a,b). With in-phase
overlap, the hybridization between the Si–Si
bonding level sðþþÞ and the 3s orbital of the
centered Si makes a hyper deep level s1 in the
valence band. The molecular orbital s2 is a result
of hybridization between s ðþÞ; the antibonding
level of Si–Si, and the 3pz orbital of the centered
Si. The s3 molecular orbital is the antibonding
level between the sðþþÞ of Si–Si and the 3s orbital
of the centered Si. The weakly bonding 3px and
3py orbitals of the centered Si make an unoccupied
2-fold degenerated mid-gap level.
Finally, we show the Kohn–Sham levels at the
G-point and the predicted XPS spectra of 2p
electrons calculated with the all-electron mixedbasis method in Fig. 4. XPS spectra were
generated by applying 0:1 eV full width at half
maximum Gaussian broadening for each level. For
the bond-center, hexagonal, tetrahedral, and split/1 1 0S interstitials, calculated levels belonging to
the self-interstitials have lower eigenvalues than
those of the normal sites. This is due to the lower
Madelung potentials for electrons at the self-
interstitial sites or at their nearest neighbor sites
than at normal sites. The Madelung potential
fi ðRi Þ that is felt by electrons at the Ri site is
defined in atomic unit as
Z
X
Zj
jn0 j
þ
dr0 ;
fi ðRi Þ ¼ 0
jR
jR
R
j
i
j
i rj
all space
jai
ð1Þ
where Zj ¼ 4 is the charge of the Si4þ ion, Rj ’s
Pare
the coordinates of Si4þ ions, the summation jai
runs over all sites j in the real space except j ¼ i;
and jn0 j is the average density of the valence
electrons. Madelung potentials calculated with the
Ewald sum technique for each site are listed in
Table 1. It can be seen that the hexagonal,
tetrahedral, and split-/1 1 0S interstitials have
lower potentials at the interstitial sites and the
bond-center interstitial has lower potentials at the
two nearest neighbor sites than normal sites,
because they are surrounded by a larger number
of Si’s than normal sites. For the split-/0 0 1S
interstitial, lowering of the potential at the site and
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T. Nishimatsu et al. / Physica B 340–342 (2003) 570–574
nearest neighbor sites are smaller than for other
interstitial structures. These values of the Madelung potentials correspond well with calculated
Kohn–Sham levels. Hence, we predict that, within
the frozen-orbital approximation, i.e., excluding
the effect of electronic relaxation due to the ejected
electron, XPS spectra for the bond-center, hexagonal, tetrahedral and split-/1 1 0S interstitials
have binding energies 0:4 1:1 eV higher than
perfect host silicon. It should be noted that the
zero of the scale of energy in Fig. 4 is placed at the
calculated valence band maximum and that the
binding energy u in XPS can be related to the
corresponding Kohn–Sham level e in Fig. 4 as u ¼
jej þ x; where the offset x includes the vacuum level
measured from the valence band maximum and
constant errors in the employed approximations.
The experimental value of the binding energy in
XPS for 2p electrons of perfect silicon crystal is
100 eV [8].
4. Summary
We investigated silicon self-interstitials with the
ab initio all-electron mixed-basis method. For the
bond-center, hexagonal, tetrahedral, and split/1 1 0S interstitials, calculated levels belonging
to the self-interstitials have lower eigenvalues than
those of the normal sites. Within the frozen-orbital
approximation, the characteristic XPS spectra for
the bond-center, hexagonal, tetrahedral, and split-
/1 1 0S interstitials are expected to have binding
energies 0.4–1:1 eV higher than perfect host
silicon.
Acknowledgements
HITACHI SR8000/G1-64 supercomputer resources were provided by the Center for Computational Materials Science, Institute for Materials
Research (CCMS-IMR), Tohoku University. We
thank the staff at CCMS-IMR for their constant
assistance. We also thank Dr. Taizo Sasaki for
valuable discussion.
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