The thermodynamic integration method has been incorporated into the tight-binding molecular-dynamics scheme to compute formation free energies of native point defects in bulk silicon. By combining previous simulated diffusivity data with present free-energy estimates, we present a thorough quantum-mechanical picture of self-diffusion in silicon that is both consistent with the state-of-the-art experimental data and able to predict separately the vacancy and self-interstitial contributions.Peer reviewe

Colombo, Luciano

Jääskeläinen, Anna

Nieminen, Risto M.

Physical Review B

Nieminen, Risto

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Silicon self-diffusion constants by tight-binding molecular dynamics

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This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Jääskeläinen, Anna & Colombo, Luciano & Nieminen, Risto M.
Title: Silicon self-diffusion constants by tight-binding molecular dynamics
Year: 2001
Version: Final published version
Please cite the original version:
Jääskeläinen, Anna & Colombo, Luciano & Nieminen, Risto M. 2001. Silicon
self-diffusion constants by tight-binding molecular dynamics. Physical Review B. Volume
64, Issue 23. 233203/1-3. ISSN 1550-235X (electronic). DOI:
10.1103/physrevb.64.233203.
Rights: © 2001 American Physical Society (APS). This is the accepted version of the following article: Jääskeläinen,
Anna & Colombo, Luciano & Nieminen, Risto M. 2001. Silicon self-diffusion constants by tight-binding
molecular dynamics. Physical Review B. Volume 64, Issue 23. 233203/1-3. ISSN 1550-235X (electronic).
DOI: 10.1103/physrevb.64.233203, which has been published in final form at
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.64.233203.
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
duplication or sale of all or part of any of the repository collections is not permitted, except that material may
be duplicated by you for your research use or educational purposes in electronic or print form. You must
obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or
otherwise to anyone who is not an authorised user.
Powered by TCPDF (www.tcpdf.org)
Silicon self-diffusion constants by tight-binding molecular dynamics
Anna Ja¨a¨skela¨inen,1 Luciano Colombo,2,* and Risto Nieminen1
1Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN–02015 HUT, Finland
2Istituto Nazionale per la Fisica della Materia and Department of Physics, University of Cagliari, Cittadella Universitaria,
I–09042 Monserrato (CA), Italy
~Received 16 May 2001; revised manuscript received 19 July 2001; published 29 November 2001!
The thermodynamic integration method has been incorporated into the tight-binding molecular-dynamics
scheme to compute formation free energies of native point defects in bulk silicon. By combining previous
simulated diffusivity data with present free-energy estimates, we present a thorough quantum-mechanical
picture of self-diffusion in silicon that is both consistent with the state-of-the-art experimental data and able to
predict separately the vacancy and self-interstitial contributions.
DOI: 10.1103/PhysRevB.64.233203 PACS number~s!: 66.30.Hs, 61.72.Ji, 61.72.Bb
Self-diffusion in silicon has been the subject of intense
research since it rules many fundamental materials phenom-
ena, such as equilibrium and nonequilibrium properties of
native defects and processes of matter transport. It also un-
derlies the microstructural evolution of bulk silicon under
many technologically-relevant processes, such as Czochral-
ski crystal growth or ion implantation. It is nowadays well
established that the self-diffusion coefficient DSD(T) follows
an Arrhenius equation
DSD~T !5D0exp~2Ea/kBT !, ~1!
over a wide range of temperatures.1 Recent experimental
measurements of self-diffusion in isotope heterostructures2
have shown a temperature dependence accurately described
over many orders of magnitude by an activation energy of
migration Ea54.7560.04 eV and by a preexponential fac-
tor D0553021701250 cm2 s21.
Although the global picture is clear, there is still uncer-
tainty about the relative contributions to DSD(T) due to the
various elementary mechanisms of diffusion. As a matter of
fact, self-diffusion is influenced by vacancy ~V! and self-
interstitial ~I! migration, as well as by the direct exchange of
two Si atoms, occurring between two neighboring lattice
sites.3 This latter mechanism, however, is usually disre-
garded since no experimental evidence has been found for
it,2 and its contribution to DSD(T) has been theoretically
predicted to be negligibly small.4 Accordingly, the self-
diffusion coefficient can be cast in the form2
DSD~T !5 f ICIeq~T !dI~T !1 f VCVeq~T !dV~T !, ~2!
where CI
eq(T) and CVeq(T) represent, respectively, the ~nor-
malized! equilibrium concentration of V’s and I’s defects,
while dI(T) and dV(T) indicate the corresponding diffusivi-
ties. The f I and f V terms represent the autocorrelation factors
for native defect migration trajectories and are well known
for the diamond lattice.5 The open problem, therefore, con-
sists of providing separately an estimation of
CI
eq
,CV
eq
,dI ,dV and their explicit dependences upon tem-
perature.
By means of metal diffusion experiments6 it has been pos-
sible to characterize the I-contribution to self-diffusion as
CI
eq~T !dI~T !52980 exp~24.95 eV/kBT ! cm2 s21, ~3!
which, combined with the above result for the total DSD(T),
allows to evaluate the V contribution as2
CV
eq~T !dV~T !50.92 exp~24.14 eV/kBT ! cm2 s21. ~4!
Other experimental data7 have been published, according to
which the above diffusion constants should be
CI
eq~T !dI~T !5914 exp~24.84 eV/kBT ! cm2 s21, ~5!
and
CV
eq~T !dV~T !50.6 exp~24.03 eV/kBT ! cm2 s21. ~6!
Both data sets predict a self-diffusion dominated by self-
interstitial ~vacancy! mobility at a high ~low! temperature.
However, the crossover temperature for the dominating
mechanism is different: 890 °C and 1000–1100 °C, for Eqs.
~3!,~4! and ~5!,~6!, respectively.
As for theory, a thorough study of self-diffusion requires
accurate free-energy calculations ~aimed at predicting
temperature-dependent equilibrium concentrations! and ex-
tensive diffusivity simulations ~aimed at computing migra-
tion energies and diffusivity prefactors! both for I and V
defects. Once the formation free energies FI ,V
f 5EI ,V
f
1TSI ,V
f
, as well as migration energies EI ,V
m and diffusivity
prefactors dI ,V
0 are known, Eq. ~2! can be recast in the form
DSD~T !5dI
0 expS 2 EIf2TSIfkBT D exp~2EIm/kBT !
1dV
0 expS 2 EVf 2TSVfkBT D exp~2EVm/kBT !, ~7!
so that a direct theory vs experiment comparison is possible.
At present, quantitative free-energy calculations are still
rare, due to the considerable computational demands of
finite-temperature simulations: an accurate simulation in-
cluding high-temperature anharmonic effects, as well as re-
liable quantum-mechanical description of the bond dynamics
involved in defect formation and migration, is in fact a very
demanding task. A model potential molecular dynamics
~MD! simulation, based on the Stillinger-Weber ~SW! poten-
tial, has been presented by Sinno et al.8 Although the pro-
posed numbers define a self-contained picture, the overall
reliability is questionable, due to the limitation of the SW
PHYSICAL REVIEW B, VOLUME 64, 233203
0163-1829/2001/64~23!/233203~3!/$20.00 ©2001 The American Physical Society64 233203-1
potential, as confirmed by the direct comparison with experi-
mental data reported below. A much more reliable picture has
been found by Blo¨chl et al.9 by means of first-principles cal-
culations. In that work, however, the huge computational
workload prevented from evaluating the temperature depen-
dence of the I contribution. Furthermore, as shown in the
following, room for improvement appears as far as vacancy-
mediated diffusion is concerned. Finally, a semi-empirical
tight-binding molecular dynamics ~TBMD! investigation10
has provided a reliable picture for formation and migration
of native defects, while entropic contributions were fitted on
experimental data or guessed from previous investigations.
Thus, although the agreement between those TBMD data and
experimental data2,7 was good, the theoretical investigation
lacked of internal consistency and, therefore, can hardly be
considered as predictive.
The above scenario points out the need for a new, self-
contained, quantum mechanical and atomistic investigation
on self-diffusion constants, aimed at characterizing indepen-
dently the two major mechanisms of diffusion. This is the
scope of the present work, where TBMD free-energy calcu-
lations have been performed in the framework of the TB
representation by Kwon et al.11 and eventually combined
with previous TBMD diffusion simulations.10
The thermodynamic integration ~TI! method12 was
adopted to evaluate FI ,V
f
. According to the TI formalism, the
free-energy difference DF between two systems has been
obtained by integrating
DF5E
l50
l51 K ]H~l!]l L
l
dl , ~8!
where l acts as disposable parameter coupling a given initial
state H0 ~with a well-known free energy! to the target final
state ~with unknown free energy! H1 : H(l)5lH11(1
2l)H0. In this work the state of reference was chosen to be
an Einstein crystal. During the MD simulations a massive
Nose-Hoover chain was attached to each vibrational degree
of freedom to keep the ensemble canonical.13 The target state
was, in turn, the TBMD Si sample. The ensemble average
appearing in Eq. ~8! was performed during constant volume,
constant-temperature simulations on a 6461 atom periodi-
cally repeated cell. The net force fa acting on the ath atom
was defined as fa5lfa
TBMD1(12l)faEinst . . At each state—
defined by a given value of l—the lattice was first carefully
equilibrated with runs long up to 16 ps ~one time step corre-
sponding to 1 fs! and finally averaged over several more ps.
The thermodynamical integration was performed over 16l
points. Our benchmarks ~performed on 21661 atoms and/or
l55 points! proved that it is more important to include a
higher number of l points in the TI evaluation than to en-
large the size of the simulation box. We estimate the present
energy integrals given in Eq. ~8! to be converged to within
;1024 eV. All free-energy calculations were performed at
four different temperatures: 300, 500, 1000, and 1400 K.
The formation entropy SI
f for the self-interstitial defect
was found almost constant with temperature, the average
value being SI
f511.2kB . First-principles calculations per-
formed in the local harmonic approximation9 predict a value
of SI
f;6kB , which, however, increases up to SI
f;10kB when
including anharmonic terms through TI calculations.14 The
case of a vacancy is more complicated. As a matter of fact,
due to the high mobility of such a defect,10 during the equili-
bration 1 observation runs, the defect was mobile, thus add-
ing a sizeable contribution of migration entropy in the TI free
energy calculations. This is confirmed by the fact that the
computed value of SV
f varied in the range 10.2–11.7kB in the
selected temperature interval, with an average value of SV
f
;10.8kB . In fact, the migration contribution is already ef-
fectively included in TBMD simulations aimed at measuring
dV
0
. To prevent the double counting of this term, we per-
formed short observation runs, taking care to select only
those simulations where V diffusion actually did not take
place. The convergence error thus included in our ensemble
averages results in an entropy overestimation of ;2kB . In
conclusion, we estimate an average formation ~i.e.,
configurational1vibrational! entropy of SV
f ;8.8kB . Our
TBMD results for SI
f and SV
f are in good qualitative agree-
ment with first-principles calculations by Blo¨chl et al.9,14 in
the sense that the difference in the entropies of formation for
I’s and V’s is of the order of 1 –2kB in both studies, which
predict a larger formation entropy for the interstitial.
Diffusivity constants were obtained by using the migra-
tion prefactors and energies computed in a previous work10
by means of the same TBMD scheme adopted here. In par-
ticular, we obtain dI
051.5831021 cm2 s21 and dV0 51.18
31024 cm2 s21, EI
m51.37 eV and EV
m50.1 eV. As for the
formation energies the TBMD results are EI
f53.80 eV and
EV
f 53.97 eV. In this way, we get a self-consistent set of
numbers, characterizing the various physical observables rel-
evant to self-diffusion.
The vacancy contribution to self-diffusion predicted by
present calculations is reported in Fig. 1. Experimental data
by Bracht et al.2 and Go¨sele et al.,7 as well as other SW8 and
first-principles9,14 theoretical results are shown for compari-
son.
It is apparent that our TBMD investigation is in excellent
agreement with state-of-the-art experimental data, providing
FIG. 1. Vacancy contribution to self-diffusion coefficient.
BRIEF REPORTS PHYSICAL REVIEW B 64 233203
233203-2
an overall atomistic picture for V-mediated self-diffusion
much more reliable than model-potential simulations. We be-
lieve that the main source of disagreement between the SW
and experimental data is due to the very low value of the
vacancy formation energy EV
f 52.5 eV, which unfavorably
compares with the 3.8 eV and 3.3–3.7 eV values given by
TBMD and first-principles calculations, respectively.10 The
agreement is better for TBMD even when compared with ab
initio simulations which are principle superior. We guess that
this is due to technical details of the calculations, both per-
formed within the same formal framework, but with a differ-
ent choice for the number of l points over which operate the
TI integration ~only three points were used in Ref. 14!. Un-
der this respect, the semi-empirical character of TBMD has
proved to be an advantage: thanks to the reduced computa-
tional workload it is possible to perform TI simulations in a
fully-converged regime, still keeping a quantum-mechanical
description of atom interactions.
As for self-interstitials, Fig. 2 confirms the predictivity
and reliability of the present results. We note that in this case
model-potential data are not so bad, while first-principles
calculations were performed at one temperature only, so that
no estimation for the activation energy of I diffusion can be
obtained.
Finally, we remark that our investigation predicts the I
defects to dominate self-diffusion at high temperatures, as
proved experimentally and confirmed by first-principles cal-
culations. Furthermore, the V-mediated and I-mediated
mechanisms give equal contributions at T¯ ;1030 °C. This
result is again consistent with available experiments.2,7
In conclusion, we propose the set of TBMD values re-
ported here for the diffusion constants ~which frequently may
individually carry some model-related or method-related er-
ror! as the more complete and consistent set of values pro-
viding a correct global picture for self-diffusion in bulk Si.
The degree of reliability of present results is summarized in
Fig. 3 where the silicon TBMD total self-diffusion coeffi-
cient DSD5DI1DV is compared with state-of-the-art experi-
mental data.2,7 The present numbers should be, therefore,
useful in modeling Si bulk processing.
One of us ~L.C.! acknowledges support by CNR under
project ‘‘5%—Microelettronica.’’ This research has been
supported by Academy of Finland through its Center of Ex-
cellence Program 2000–2006.
*Corresponding author.
Email address: luciano.colombo@dsf.unica.it
1 W. Frank, U. Go¨sele, H. Meher, and A. Seeger, in Diffusion in
Crystalline Solids, edited by G. E. Murch and A. Nowick ~Aca-
demic Press, New York, 1984!, p. 31.
2 H. Bracht, E.E. Haller, and R. Clark-Phelps, Phys. Rev. Lett. 81,
393 ~1998!.
3 K.C. Pandey, Phys. Rev. Lett. 57, 2287 ~1986!.
4 K.C. Pandey and E. Kaxiras, Phys. Rev. Lett. 66, 915 ~1990!.
5 K. Compaan and Y. Have, Trans. Faraday Soc. 52, 786 ~1956!;
ibid. 54, 1498 ~1958!.
6 H. Bracht, N.A. Stolwijk, and H. Mehrer, Phys. Rev. B 52, 16 542
~1995!.
7 T.Y. Tan and U. Go¨sele, Appl. Phys. A: Solids Surf. 37, 1 ~1985!.
8 T. Sinno, Z.K. Jiang, and R.A. Brown, Appl. Phys. Lett. 68, 3028
~1996!.
9 P.E. Blo¨chl, E. Smargiassi, R. Car, D.B. Lacks, W. Andreoni, and
S.T. Pantelides, Phys. Rev. Lett. 70, 2435 ~1993!.
10 M. Tang, L. Colombo, J. Zhu, and T. Diaz de la Rubia, Phys. Rev.
B 55, 14 279 ~1997!.
11 I. Kwon, R. Biswas, C.Z. Wang, C.T. Chan, K.M. Ho, and C.M.
Soukoulis, Phys. Rev. B 49, 4242 ~1994!.
12 D. Frenkel and B. Smit, Understanding Molecular Simulations
~Academic Press, New York, 1996!.
13 G.J. Martyna, M.L. Klein, and M. Tuckerman, J. Chem. Phys. 97,
2635 ~1992!.
14 R. Car, P. Blo¨chl, and E. Smargiassi, Mater. Sci. Forum 83-87,
433 ~1992!.
FIG. 2. Interstitial contribution to self-diffusion coefficient.
FIG. 3. Total self-diffusion coefficient in silicon.
BRIEF REPORTS PHYSICAL REVIEW B 64 233203
233203-3


English

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Jääskeläinen, Anna & Colombo, Luciano & Nieminen, Risto M.Title: Silicon self-diffusion constants by tight-binding molecular dynamicsYear: 2001Version: Final published versionPlease cite the original version:Jääskeläinen, Anna & Colombo, Luciano & Nieminen, Risto M. 2001. Siliconself-diffusion constants by tight-binding molecular dynamics. Physical Review B. Volume64, Issue 23. 233203/1-3. ISSN 1550-235X (electronic). DOI:10.1103/physrevb.64.233203.Rights: © 2001 American Physical Society (APS). This is the accepted version of the following article: Jääskeläinen,Anna & Colombo, Luciano & Nieminen, Risto M. 2001. Silicon self-diffusion constants by tight-bindingmolecular dynamics. Physical Review B. Volume 64, Issue 23. 233203/1-3. ISSN 1550-235X (electronic).DOI: 10.1103/physrevb.64.233203, which has been published in final form athttp://journals.aps.org/prb/abstract/10.1103/PhysRevB.64.233203.All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)Silicon self-diffusion constants by tight-binding molecular dynamicsAnna Ja¨a¨skela¨inen,1 Luciano Colombo,2,* and Risto Nieminen11Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN–02015 HUT, Finland2Istituto Nazionale per la Fisica della Materia and Department of Physics, University of Cagliari, Cittadella Universitaria,I–09042 Monserrato (CA), Italy~Received 16 May 2001; revised manuscript received 19 July 2001; published 29 November 2001!The thermodynamic integration method has been incorporated into the tight-binding molecular-dynamicsscheme to compute formation free energies of native point defects in bulk silicon. By combining previoussimulated diffusivity data with present free-energy estimates, we present a thorough quantum-mechanicalpicture of self-diffusion in silicon that is both consistent with the state-of-the-art experimental data and able topredict separately the vacancy and self-interstitial contributions.DOI: 10.1103/PhysRevB.64.233203 PACS number~s!: 66.30.Hs, 61.72.Ji, 61.72.BbSelf-diffusion in silicon has been the subject of intenseresearch since it rules many fundamental materials phenom-ena, such as equilibrium and nonequilibrium properties ofnative defects and processes of matter transport. It also un-derlies the microstructural evolution of bulk silicon undermany technologically-relevant processes, such as Czochral-ski crystal growth or ion implantation. It is nowadays wellestablished that the self-diffusion coefficient DSD(T) followsan Arrhenius equationDSD~T !5D0exp~2Ea/kBT !, ~1!over a wide range of temperatures.1 Recent experimentalmeasurements of self-diffusion in isotope heterostructures2have shown a temperature dependence accurately describedover many orders of magnitude by an activation energy ofmigration Ea54.7560.04 eV and by a preexponential fac-tor D0553021701250 cm2 s21.Although the global picture is clear, there is still uncer-tainty about the relative contributions to DSD(T) due to thevarious elementary mechanisms of diffusion. As a matter offact, self-diffusion is influenced by vacancy ~V! and self-interstitial ~I! migration, as well as by the direct exchange oftwo Si atoms, occurring between two neighboring latticesites.3 This latter mechanism, however, is usually disre-garded since no experimental evidence has been found forit,2 and its contribution to DSD(T) has been theoreticallypredicted to be negligibly small.4 Accordingly, the self-diffusion coefficient can be cast in the form2DSD~T !5 f ICIeq~T !dI~T !1 f VCVeq~T !dV~T !, ~2!where CIeq(T) and CVeq(T) represent, respectively, the ~nor-malized! equilibrium concentration of V’s and I’s defects,while dI(T) and dV(T) indicate the corresponding diffusivi-ties. The f I and f V terms represent the autocorrelation factorsfor native defect migration trajectories and are well knownfor the diamond lattice.5 The open problem, therefore, con-sists of providing separately an estimation ofCIeq,CVeq,dI ,dV and their explicit dependences upon tem-perature.By means of metal diffusion experiments6 it has been pos-sible to characterize the I-contribution to self-diffusion asCIeq~T !dI~T !52980 exp~24.95 eV/kBT ! cm2 s21, ~3!which, combined with the above result for the total DSD(T),allows to evaluate the V contribution as2CVeq~T !dV~T !50.92 exp~24.14 eV/kBT ! cm2 s21. ~4!Other experimental data7 have been published, according towhich the above diffusion constants should beCIeq~T !dI~T !5914 exp~24.84 eV/kBT ! cm2 s21, ~5!andCVeq~T !dV~T !50.6 exp~24.03 eV/kBT ! cm2 s21. ~6!Both data sets predict a self-diffusion dominated by self-interstitial ~vacancy! mobility at a high ~low! temperature.However, the crossover temperature for the dominatingmechanism is different: 890 °C and 1000–1100 °C, for Eqs.~3!,~4! and ~5!,~6!, respectively.As for theory, a thorough study of self-diffusion requiresaccurate free-energy calculations ~aimed at predictingtemperature-dependent equilibrium concentrations! and ex-tensive diffusivity simulations ~aimed at computing migra-tion energies and diffusivity prefactors! both for I and Vdefects. Once the formation free energies FI ,Vf 5EI ,Vf1TSI ,Vf, as well as migration energies EI ,Vm and diffusivityprefactors dI ,V0 are known, Eq. ~2! can be recast in the formDSD~T !5dI0 expS 2 EIf2TSIfkBT D exp~2EIm/kBT !1dV0 expS 2 EVf 2TSVfkBT D exp~2EVm/kBT !, ~7!so that a direct theory vs experiment comparison is possible.At present, quantitative free-energy calculations are stillrare, due to the considerable computational demands offinite-temperature simulations: an accurate simulation in-cluding high-temperature anharmonic effects, as well as re-liable quantum-mechanical description of the bond dynamicsinvolved in defect formation and migration, is in fact a verydemanding task. A model potential molecular dynamics~MD! simulation, based on the Stillinger-Weber ~SW! poten-tial, has been presented by Sinno et al.8 Although the pro-posed numbers define a self-contained picture, the overallreliability is questionable, due to the limitation of the SWPHYSICAL REVIEW B, VOLUME 64, 2332030163-1829/2001/64~23!/233203~3!/$20.00 ©2001 The American Physical Society64 233203-1potential, as confirmed by the direct comparison with experi-mental data reported below. A much more reliable picture hasbeen found by Blo¨chl et al.9 by means of first-principles cal-culations. In that work, however, the huge computationalworkload prevented from evaluating the temperature depen-dence of the I contribution. Furthermore, as shown in thefollowing, room for improvement appears as far as vacancy-mediated diffusion is concerned. Finally, a semi-empiricaltight-binding molecular dynamics ~TBMD! investigation10has provided a reliable picture for formation and migrationof native defects, while entropic contributions were fitted onexperimental data or guessed from previous investigations.Thus, although the agreement between those TBMD data andexperimental data2,7 was good, the theoretical investigationlacked of internal consistency and, therefore, can hardly beconsidered as predictive.The above scenario points out the need for a new, self-contained, quantum mechanical and atomistic investigationon self-diffusion constants, aimed at characterizing indepen-dently the two major mechanisms of diffusion. This is thescope of the present work, where TBMD free-energy calcu-lations have been performed in the framework of the TBrepresentation by Kwon et al.11 and eventually combinedwith previous TBMD diffusion simulations.10The thermodynamic integration ~TI! method12 wasadopted to evaluate FI ,Vf. According to the TI formalism, thefree-energy difference DF between two systems has beenobtained by integratingDF5El50l51 K ]H~l!]l Lldl , ~8!where l acts as disposable parameter coupling a given initialstate H0 ~with a well-known free energy! to the target finalstate ~with unknown free energy! H1 : H(l)5lH11(12l)H0. In this work the state of reference was chosen to bean Einstein crystal. During the MD simulations a massiveNose-Hoover chain was attached to each vibrational degreeof freedom to keep the ensemble canonical.13 The target statewas, in turn, the TBMD Si sample. The ensemble averageappearing in Eq. ~8! was performed during constant volume,constant-temperature simulations on a 6461 atom periodi-cally repeated cell. The net force fa acting on the ath atomwas defined as fa5lfaTBMD1(12l)faEinst . . At each state—defined by a given value of l—the lattice was first carefullyequilibrated with runs long up to 16 ps ~one time step corre-sponding to 1 fs! and finally averaged over several more ps.The thermodynamical integration was performed over 16lpoints. Our benchmarks ~performed on 21661 atoms and/orl55 points! proved that it is more important to include ahigher number of l points in the TI evaluation than to en-large the size of the simulation box. We estimate the presentenergy integrals given in Eq. ~8! to be converged to within;1024 eV. All free-energy calculations were performed atfour different temperatures: 300, 500, 1000, and 1400 K.The formation entropy SIf for the self-interstitial defectwas found almost constant with temperature, the averagevalue being SIf511.2kB . First-principles calculations per-formed in the local harmonic approximation9 predict a valueof SIf;6kB , which, however, increases up to SIf;10kB whenincluding anharmonic terms through TI calculations.14 Thecase of a vacancy is more complicated. As a matter of fact,due to the high mobility of such a defect,10 during the equili-bration 1 observation runs, the defect was mobile, thus add-ing a sizeable contribution of migration entropy in the TI freeenergy calculations. This is confirmed by the fact that thecomputed value of SVf varied in the range 10.2–11.7kB in theselected temperature interval, with an average value of SVf;10.8kB . In fact, the migration contribution is already ef-fectively included in TBMD simulations aimed at measuringdV0. To prevent the double counting of this term, we per-formed short observation runs, taking care to select onlythose simulations where V diffusion actually did not takeplace. The convergence error thus included in our ensembleaverages results in an entropy overestimation of ;2kB . Inconclusion, we estimate an average formation ~i.e.,configurational1vibrational! entropy of SVf ;8.8kB . OurTBMD results for SIf and SVf are in good qualitative agree-ment with first-principles calculations by Blo¨chl et al.9,14 inthe sense that the difference in the entropies of formation forI’s and V’s is of the order of 1 –2kB in both studies, whichpredict a larger formation entropy for the interstitial.Diffusivity constants were obtained by using the migra-tion prefactors and energies computed in a previous work10by means of the same TBMD scheme adopted here. In par-ticular, we obtain dI051.5831021 cm2 s21 and dV0 51.1831024 cm2 s21, EIm51.37 eV and EVm50.1 eV. As for theformation energies the TBMD results are EIf53.80 eV andEVf 53.97 eV. In this way, we get a self-consistent set ofnumbers, characterizing the various physical observables rel-evant to self-diffusion.The vacancy contribution to self-diffusion predicted bypresent calculations is reported in Fig. 1. Experimental databy Bracht et al.2 and Go¨sele et al.,7 as well as other SW8 andfirst-principles9,14 theoretical results are shown for compari-son.It is apparent that our TBMD investigation is in excellentagreement with state-of-the-art experimental data, providingFIG. 1. Vacancy contribution to self-diffusion coefficient.BRIEF REPORTS PHYSICAL REVIEW B 64 233203233203-2an overall atomistic picture for V-mediated self-diffusionmuch more reliable than model-potential simulations. We be-lieve that the main source of disagreement between the SWand experimental data is due to the very low value of thevacancy formation energy EVf 52.5 eV, which unfavorablycompares with the 3.8 eV and 3.3–3.7 eV values given byTBMD and first-principles calculations, respectively.10 Theagreement is better for TBMD even when compared with abinitio simulations which are principle superior. We guess thatthis is due to technical details of the calculations, both per-formed within the same formal framework, but with a differ-ent choice for the number of l points over which operate theTI integration ~only three points were used in Ref. 14!. Un-der this respect, the semi-empirical character of TBMD hasproved to be an advantage: thanks to the reduced computa-tional workload it is possible to perform TI simulations in afully-converged regime, still keeping a quantum-mechanicaldescription of atom interactions.As for self-interstitials, Fig. 2 confirms the predictivityand reliability of the present results. We note that in this casemodel-potential data are not so bad, while first-principlescalculations were performed at one temperature only, so thatno estimation for the activation energy of I diffusion can beobtained.Finally, we remark that our investigation predicts the Idefects to dominate self-diffusion at high temperatures, asproved experimentally and confirmed by first-principles cal-culations. Furthermore, the V-mediated and I-mediatedmechanisms give equal contributions at T¯ ;1030 °C. Thisresult is again consistent with available experiments.2,7In conclusion, we propose the set of TBMD values re-ported here for the diffusion constants ~which frequently mayindividually carry some model-related or method-related er-ror! as the more complete and consistent set of values pro-viding a correct global picture for self-diffusion in bulk Si.The degree of reliability of present results is summarized inFig. 3 where the silicon TBMD total self-diffusion coeffi-cient DSD5DI1DV is compared with state-of-the-art experi-mental data.2,7 The present numbers should be, therefore,useful in modeling Si bulk processing.One of us ~L.C.! acknowledges support by CNR underproject ‘‘5%—Microelettronica.’’ This research has beensupported by Academy of Finland through its Center of Ex-cellence Program 2000–2006.*Corresponding author.Email address: luciano.colombo@dsf.unica.it1 W. Frank, U. Go¨sele, H. Meher, and A. Seeger, in Diffusion inCrystalline Solids, edited by G. E. Murch and A. Nowick ~Aca-demic Press, New York, 1984!, p. 31.2 H. Bracht, E.E. Haller, and R. Clark-Phelps, Phys. Rev. Lett. 81,393 ~1998!.3 K.C. Pandey, Phys. Rev. Lett. 57, 2287 ~1986!.4 K.C. Pandey and E. Kaxiras, Phys. Rev. Lett. 66, 915 ~1990!.5 K. Compaan and Y. Have, Trans. Faraday Soc. 52, 786 ~1956!;ibid. 54, 1498 ~1958!.6 H. Bracht, N.A. Stolwijk, and H. Mehrer, Phys. Rev. B 52, 16 542~1995!.7 T.Y. Tan and U. Go¨sele, Appl. Phys. A: Solids Surf. 37, 1 ~1985!.8 T. Sinno, Z.K. Jiang, and R.A. Brown, Appl. Phys. Lett. 68, 3028~1996!.9 P.E. Blo¨chl, E. Smargiassi, R. Car, D.B. Lacks, W. Andreoni, andS.T. Pantelides, Phys. Rev. Lett. 70, 2435 ~1993!.10 M. Tang, L. Colombo, J. Zhu, and T. Diaz de la Rubia, Phys. Rev.B 55, 14 279 ~1997!.11 I. Kwon, R. Biswas, C.Z. Wang, C.T. Chan, K.M. Ho, and C.M.Soukoulis, Phys. Rev. B 49, 4242 ~1994!.12 D. Frenkel and B. Smit, Understanding Molecular Simulations~Academic Press, New York, 1996!.13 G.J. Martyna, M.L. Klein, and M. Tuckerman, J. Chem. Phys. 97,2635 ~1992!.14 R. Car, P. Blo¨chl, and E. Smargiassi, Mater. Sci. Forum 83-87,433 ~1992!.FIG. 2. Interstitial contribution to self-diffusion coefficient.FIG. 3. Total self-diffusion coefficient in silicon.BRIEF REPORTS PHYSICAL REVIEW B 64 233203233203-3

JAASKELAINEN A.

NIEMINEN R.

COLOMBO, LUCIANO

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Silicon self-diffusion constants by tight-binding molecular dynamics

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