HAL Id: jpa-00249718
https://hal.archives-ouvertes.fr/jpa-00249718
Submitted on 1 Jan 1997
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Dislocation and Grain Boundary Energies in Si and Ge from an Anharmonic Bond Charge Model
H. Teichler, J. Wilder
To cite this version:
H. Teichler, J. Wilder. Dislocation and Grain Boundary Energies in Si and Ge from an Anhar- monic Bond Charge Model. Journal de Physique III, EDP Sciences, 1997, 7 (12), pp.2281-2292.
�10.1051/jp3:1997259�. �jpa-00249718�
Dislocation and Grain Boundary Energies in Si and Ge IfoIn an Anharlnonic Bond Charge Model
H. Teichler (*) and J. Wilder (**)
Institut fir Metallphysik, Universitat G6ttingen, 37073 G6ttingen, Germany
(Received 3 October 1996, revised 16 September 199( accepted 16 Septembre 1997)
PACS 61.72 -y Defects and impurities in crystals
Abstract. The paper presents calculated line energy values for the reconstructed 60° and
90° glide-set partial dislocations in Si and Ge, formation and migration energy for the recon- structed kink
on the reconstructed 90° partials, and energy data for the symmetric
E
= 9 < 011 > and for two variants of the symmetric E
= 11 < 011 > tilt grain boundaries. Cri- teria are formulated to identify interatomic force field models which are able to provide reliable
energy estimates The anharmonic bond charge (a.bc) model is introduced as an example that
approximately fulfills the basic criteria, i.e., describes well the second and third order elastic
constants and the phonon dispersion curves Deviations between energy estimates from the a.bc
model and less reliable approaches are discussed. It is shown that in case of the E
= 11 < 011 >
tilt grain boundary the a.bc model gives different energetical ranking for the sc-called E
= llA
and the E
= llB variants in Si and Ge, in agreement with the experimental observations.
1. Introduction
The present paper is concerned with energy estimates for extended defects, such as dislocations and grain boundaries, in semiconductors by computer modelling. (For reviews on dislocations in semiconductors see, e. g., [I], on grain boundaries, e. g., [2]). The paper provides, in particular,
for Ge and Si calculated energy data for the 60° and 90° glide-set partial dislocations, estimates of the formation and migration energy of kinks on the 90° partial, and values for the stored
energy of the symmetric E = 9 (221) [i10] and different variants of the E
= 11 (332) [i10] tilt
grain boundaries. Computer modelling of the structure and estimation of the related energies
is a question of special importance for the extended lattice defects since for these, in contrast to the majority of point defects, for a nominally given defect in many cases a variety of competing
structural variants can be imagined which can differ in important properties, like the bound electron and phonon states at the defect, carrier and matter transport accross and along the
defect, its ability to act as sink or source for vacancies and interstitials, or impurity segregation.
In these cases energy estimates have to be used to identify the energetically most favourable,
and hence by nature rather probably realized, defect configuration. Independent need for
correct estimation of extended defect energies comes from situations when the absolute value
of the defect formation energy matters, e.g., in the field of dislocations in case of epitaxial
* Author for correspondence (e-mail: teichler©umpa06.gwdg de) (**) Now at the MPI fbr Polymerforschung, D-55128 Mainz, Germany
@ Les #ditions de Physique 1997
hetero structures where the dislocation core energy enters the'technological significant critical thickness for misfit dislocation generation.
Despite the importance of the problem, reliable energy estimates for extended defects are rather scarce and those from (ifferent modelling approaches are in some important cases con-
tradicting. This point was a(dressed in our recent studies [3,4]. It also was considered by Duesbery et al. [5] who tested the applicability of some modern empirical interaction poten- tials for Si, such as Tersoff's approach [6] and the Stillinger-Weber potential [7], by comparing total energy values of atomic clusters with (dipoles of partial) dislocations. They find that the
different potentials yield rather different values for the stored energy, and the Stillinger-Weber potential provides a wrong result about the symmetry of the stable variant for the 90°-partial.
The above stressed difficulties indicate the particular need for criteria to identify interatomic valence force fields (VFF'S) that provide reliable energy estimates for extended lattice defects.
Below, in Section 2, we present some arguments and formulate criteria which may help with this respect, and we introduce the anharmonic bond charge model as one realization that
approximately fulfills the criteria. Section 3 reports results from this anharmonic model. As already mentioned, we shall present estimates of the line energies of the "reconstIucted" 90°- and 60°-partial dislocations, of the formation energy for the isolated, reconstructed kink on the 90°-partial, and an upper limit of the migration energy for this kink. Furthermore we provide results for the energies of the symmetric E = 9 and E =11 < i10 > tilt grain boundaries in Ge and Si. Section 4 is devoted to some concluding remarks.
2. The Model
2 1. CRITERIA FOR SELECTINp A "WELL-SUITED" INTERATOMIC MODEL. The scattering
of estimated energy values obiaiqed for one and the same extended defect by use of different interatomic models is due to tie,fact that the defect energy contributions stem from different regions aIound the defect which are differently well descIibed by different models. As in [3,4]
we may distinguish I) the elastic far field
it) the intermediate "harmonically" deformed lattice region iii) the region of anharmomc deformations
iv) the central core
where the latter is the region of "broken bonds" and "twin boundary" elements (i.e., elements of a different phase, like the wurtzite structure). The labelling of the regions indicates that
in most of the space the stored energy can be ascribed to local, deformed covalent bonds.
f~om identifying the "regions" it is obvious [3] thjt a model aimed at describing extended defect energies can be characterized by quoting its ability to describe the elastic far field, which demands reproduction of the elastic constants, the harmonilally deformed region where
the atomistic structure of matter enters, and which demands modelling of the wavelengths dependence of deformation energies, i. e., the phonon dispersion curves, the anharmonic region which reflects the third older elastic constants and phonon-phonojl interactions, and the core
region, which might be characterized by the need to use full ab mitio electron theory for its reliable modelling and which is the region of possible "broken" bonds.
According to our knowledge none of the presently existing approaches was pIoved to be able to model correctly the energy in all of the regions. As discussed in [3,4] the phenomenological
valence force models m many cases aren't even able to simultaneously reproduce correctly the
energy in the elastic far field and in the harmonically deformed intermediate region. These difficulties arise, in particular, for VFF models with short ranged nearest neighbour couplings like, e-g-, Keating's potential [8j, Tersoff's model [6j, or the Stillinger-Weber potential [7j.
In the harmonic limit the short ranged potentials cannot describe well the elastic constants
and, simultaneously, the flattening of the transversal acoustic (TA) phonon branch outside the
r-point, which is responsible for a strong angular stiffness reduction against short wavelengths deformations. As was realized by Cochran [9j as early as in 1959 the long-ranged forces required
to describe the flattening of the TA branch indicate the incomplete screening of local charges in the semiconducting materials and the therefrom resulting long-ranged ion-ion and electron-ion
interactions. These difficulties of short ranged VFF models are well known in the literature
(e.g. [10j). In the context of Keating's model, e-g, Baraff et al. [10j took into account this
angular stiffness reduction in Si by reducing the angular stiffness force constant by nearly a
factor of three, on account of significant deviations in the elastic constants, i.e., in the energy of the elastic far field. For Ge a similar "modified Keating potential" was proposed by Mauger
et al. ill].
First-principles ab mit~o electron theoretical calculation at present cannot help to overcome the above sketched difficulties. At present,, due to computational limitations, the ab mitio
calculations can treat only small systems of some hundreds of atoms at most (see, e.g., [12j).
This makes it necessary to use phenomenological, empirical potentials at least for the far field and the intermediate region of harmonic and anharmonic deformations of the defect. Combinig
electron theory and phenomenological descriptions then raises additional problems. E.g., in
case of core structures with broken bonds, which demand proper treatment of the central region
by electron theory, one has the problem of matching correctly the energies (and forces) from the core region and the outer space, a problem considered, e.g., in [13,14j within the LCAO approach for "broken bond" core dislocations and for vacancies trapped by dislocations resp.
grain boundaries.
In order to overcome the computational limitations, simplified electron theoretical descrip-
tions have been developed aimed at treating larger ensembles of atoms by an approximated ab initio scheme One of the early approaches in this field is described by Paxton and Sutton [lsj,
more recent are those by Sankey and Niklewski [16j and by Hansen et al. [17j. But systematic
studies are missing so far on the abilities of these models to simultaneously reproduce cor- rectly the elastic constants, the phonon dispersion including the TA branch flattening, and the
anharmonic properties of the materials.
2.2. BASIC FEATURES OF THE ANHARMONIC BOND CHARGE MODEL. Firm electron
theoretical treatments of the dislocation cores (e.g., [17,18j) and the earlier estimates ill of the bond breaking energy there is good evidence that the cores of most of the low energy
dislocation don't contain any broken bonds. Also most of the high symmetric, low energy
grain boundaries seem free of broken bonds. There is hope that for these defects the stored energy can be estimated from phenomenological, empirical potentials, provided the latter yield
correct energy contributions for the above mentioned regions ii) to (iii) and the "twin element"
problem can be settled. If broken bonds cannot be ruled out from the beginning, e.g., in case of
extreme grain boundaries like the high E
= 5 twist boundary, an electron theoretical treatment may be necessary as carried out in'[19j. In our previous studies [3,4j we estimated the energies of extended defects by use of Weber's adiabatic bond charge model [20j. This model is able to
reproduce the elastic constants and the phonon dispersion curves and thus describes well the energy stored in the far field and in the harmonically deformed region. In order to estimate the magnitude of the anharmonic corrections we have generalized Weber's model to take into
account the anharmonicities reflected by the third order elastic constants.
Vbd
v~~
Fig. I. Interactions taken into account m the anharmomc bond charge model (open circles atoms,
full circles: bond charges).
The anharmonic model applies to defects with fourfold coordinated atoms. It takes into
account the third order elastic constants and is an anharmonic generalization of Weber's [20j
bond charge model in a similar way as the treatment by Koizumi and Ninomija [21j is an extension of Keating's harmonic model [8j to an anharmonic one. (Note in this context the recent anharmonic generalization of Keating's model by Riecker and Methfessel [22j and the
recent strain dependent bond charge model [23j). Weber introduced in his model the positions
of "bond charges" (bc's) as additional degrees of freedom, based on Cochran's explanation [9j
that the flattening of the TA phonon branch is caused by incomplete screening of local charges
in the semiconductors In the bc model it is assumed that the bc's respond adiabatically to lattice deformations as to minimize the energy of the atomic arrangement. The interactions
considered in the model are sketched in Figure I. It takes care of energy contributions from bond stretchings, Vbs, from bond deformations, Vdb, which depend on the bond charge positions relative to the adjacent atoms, and bond-bond (angle) deformations Vbb. In our anharmonic model Vbs and Iib include the anharmomcities. (For details see [24j) In the harmonic limit our
anharmonic model reduces to Weber's [20j model. Our approach thus describes the second order elastic constants and the phonon dispersion curves as Weber's model does. In the elastic limit
our model turns into Keating's anharmonic model and it thus describes the third order elastic
constants like Keating's anharmonic model does. Table I provides the properties reproduced
by our model. In the table we included, too, the cohesion energy and the internal displacement parameter ( [25j. It should be stressed that the deduced (-values are in excellent agreement
with the current interpretation of x-ray scattering results, which according to [25j now are in
favour of ( = 0 54 + 0.04 for Si and Ge after elimination of a systematic error in the earlier
analysis.
3. Energy Estimates for Extended Defects in Si and Ge
3.1. 90° GLIDE-SET PARTIAL DISLOCATIONS. Here we present results for the 90° glide-set partials. Regarding the 30° partials we refer to the harmonic bc model calculations in [4j, the 60° partials are considered in the next subsection. The partial dislocations result from
dissociation of complete screw, 60°
,
and edge dislocations which from the most frequent dislocation types in deformed semiconductors with diamond structure. As mentioned above,
we consider primarily the "reconstructed-core" variants. For the 30° partials (in Si and Ge)
and for 90° partial in Si these variants have been shown to be the stable ones (comp. ill).
There also is good evidence that the reconstructed variants of the 60° partials in Si and Ge and the 90° in Ge are the stable configurations ill, but in these cases the data at present are
Table I. Compartson of experimental data and anharmomc bond charge model results.
Si Ge
th. exp. th. exp.
1.56 1.66 1.33 1.29
C44(10~~dyn/cm~j 0 78 0.79 0.65 0.67
C12(10~~dyn/cm~j 0.58 0.64 0.52 0.48
Ciii(10~~dyn/cm~j -8.57 -8.25 -7.44 -7.10
Ci12(10~~dyn/cm~j -4.52 -4.51 -3.44 -3.89
C123(10~~dyn/cm~j -0.66 -0.64 -0.18 -0.18 C144(10~~dyn/cm~j 0.12 0.12 -0.07 -0.23
C166(10~~dyn/cm~j -3.51 -3.10 -2 96 -2.92 C456(10~~dyn/cm~j -0.27 -0.64 -0.18 -0.53
((bond bending param.) 0.50 0 54 0.52 0.54
Ecoh(evlatomj 4.64 4.64 3.88 3.88
not so complete as for the former partials.
Our calculation method closely follows Marklund's [26j early Keating model calculations.
The method was outlined m some details in the context of the harmonic bc calculations [4j. In
the treatment a cluster of atoms including a dislocation is considered with boundary conditions
that describe a periodic atomic pattern along the straight lined dislocation and positions of
the atoms from linear elasticity theory [27j at the surface of an outer cylinder around the
dislocation. Atomic positions and bc positions are varied as to minimize the total energy of the cluster.
In case of a 90° partial dislocation with Burger's vector bp and a stacking fault attached, it follows from linear elasticity theory [27j that outside the dislocation core the energy (per line
length) stored in a cylindrical region with radius R increases like
FIR) = Gbi/141rll v))lnlR/rc) + Ec(rc) + ~fsfR Ii) (G: shear modulus, v: Poisson ratio, ~tsf: stacking fault energy), where rc is an inner cut-off
radius and E(rc) the energy stored in this inner cylinder. The here considered interatomic models of bc land Keating) type with only nearest neighbour interactions yield vanishing specific stacking fault energy, so that this contribution has to be added by hand. Below we
neglect this contribution at all.
The linear dependence on In(R) predicted by elasticity theory when subtracting the stacking
fault energy is clearly reflected by the bc data shown m Figures 2a and 2b, which display the energy (per line length) of atoms (and bond charges) in a cylindrical region around the dislocation from the harmonic and anharmonic bc model. For comparison also Keating model data are provided in Figure 2, as resulting from Keating's original parameter sets adapted to the elastic constants, and the modified parameter sets [10,llj aimed at taking into account the TA phonon branch flattening. The line energies E(R) deduced from original Keating potential and from the models exhibit the same slope as function of In(R). This indicates that these models involve similar shear modulus values, as intended. The slope of the values from the
a
(
~
~~
$ ~ o ##
~ o ~++
~'
Q~ o ~_+ /
~ o ~ al
o +/
o
~~++&
o ~
+
~ ~
~&AA
~
~ 8
~
~
D
°
-2 -1 0
a~
~
'n(Rinmi)
6 b
/
~m( I
R j°°
a 4 ]
+4~~
~s o ~##~
o$ o ~+
~ O »~
~ ~
o +
o ~~~++
~~
~ o
a
+
l~~"**
» a
D d#
D
-2 -1 0
~) ln(R(nmJ)
Fig. 2. Line energies of the "reconstructed" 90° partial dislocations against In(R) from the harmonic bc model (crosses), from the anharmonic bc model (triangles), original Keating potential (diamonds),
and modified Keating potential (squares). (a): silicon, (b): germanium.
modified Keating potentials [10, llj is smaller by a factor 2.4 ip Si (2.7 in Ge), which reflects the reduction of the shear modulus due to the reduced bond binding stiffness assumed in this model. The absolute values of line energy deduced from~ Keating's original parameters
are significantly larger than those from the bc mode)s; the line energy from the modified parameters is significantly lower. The former is due to )he fact that Keatinj's potential is not able to properly model the reduction of bond bending stiffness for deformations with short
wavelengths as occurring in the dislocation core. The latter reflects that the energy in the dislocation core is stored to a significant amount in deformations other than short-wavelength
shear deformations, where the energy is markedly underestimated by Keatings's potential with
/
,
'
' J
a) ," b)
Fig. 3. Schematic plot of the atomic pattern and bonds in the glide plane of the "reconstructed"
60° partial dislocation. (a): variant A; (b). variant B
modified parameters. These results indicate that estimates from Keating's models should be descarded whith regard to their obvious failures. The energy difference between harmonic and anharmomc bc model turns out 0.58 eV/nm in Si, 1.15 eV/nm in Ge. It reflects that in the anharmonic model, perhaps due to easier bond elongations, the dislocation core energies can
be reduced.
3. 2. 60° GLIDE-SET PARTIAL DISLOCATIONS. Figures 3 and 4 present results concerning
the line energy of the reconstructed 60° glide-set partials in Si and Ge A variety of config-
urations has been firoposed for this partial [28-30j. We here consider the fully reconstructed variant [29,30j, below named A, and a new one with modified core, named B (also presented by M. Heggie at the Conference on Extended Defects in Semiconductors, Giens, FIance, 1996).
Figure 3 shows the atomic positions in the dislocations cores of variants A and B projected
on the [lllj glide plane. The corresponding cumulated deformation energies in a cylinder of radius R around the dislocations are displayed in Figure 4, where data from the harmonic and
anharmonic bc model are provided. In this figure the results for the variant A are indicated
by squares (harmonic model) and by diamonds (anharmonic model), those for variant B by
crosses (harmonic model) and by "plus" signs (anharmonic model) Apparently the core struc-
ture of variant B is energetically more favourable than that of variant A, but at larger distances variant B costs more energy. For Ge at larger R the (more reliable) anharmonic model yields slightly lower energy values for variant A than for B. In Si closer inspection indicates in case of the anharmonic model slightly lower energy values for the B variant within the sketched radial
regime, but the slope of the energy with In(R) turns out larger for B than for A. This indicates
a cross-over between the predicted total energy values at larger R outside the range of Figure 4
This finding reflects a larger energy for the B variant in the elastic far field than for the A variant. It clearly demonstrates that the question of the relative stability of different defect variants demands proper estimation of the defect energy from all three regimes mentionend in the introduction.
For the A variants the core energy values in the anharmonic case turn out lower by 0.63 eV/nm
in Si (0.87 eV/nm in Ge) than the harmonic results. Harmonic and anharmonic model differ
in the deduced atomic pattern. The harmonic model for Ge and Si yields a maximum bond
elongation of 5.9~, maximum bond shortening of 6.4~, a largest bond-bond angle of136°,
and a smallest one of 91°. The anharmonic model gives a maximum bond elongation of 7.8i~
for Si (6.5~ for Ge), a maximum bond shortening of 3~ (for Si and Ge), a largest bond-bond
