V. Kuzmin
Vast variety of applications ranging from
microelectronics to aerospace and automotive
industries and also biomedical technologies
uses ion beam techniques for modification of
different materials. Amongst other carbon and
carbon materials take a special place, being of
great importance for modern as well as for
future technologies and in particularly for
nanotechnology. Considerable interest in producing nanoscale objects make increasing
demands of accuracy of theoretical predictions. However, considerable differences
(more than 40%) between experimental data13
and the values given by Ziegler, Biersack
and Littmark (ZBL) theory4 for range parameters of heavy ions in light targets at energies
of about 1-1000 keV were revealed in recent
years. These discrepancies were explained by
a correlation between the nuclear and electronic energy losses1,2. In addition to the quasielastic description of atomic scattering in
those works it was proposed to transform the
universal ZBL potential with the ZBL electronic stopping power. It should be stressed,
that the universal ZBL potential is an average
of a large number of repulsive potentials, calculated with an approximate method4. Therefore, the large discrepancies between the experimental and theoretical values for range
parameters can indicate the lack of accuracy
of the ZBL potential and/or the ZBL electronic stopping for ion-target combinations in
In the present work interatomic potentials
for Pb – C diatomics have been calculated
with Hartree-Fock and density functional theory (DFT) methods by GAMESS program
system5 including relativistic corrections by
RESC scheme6, and compared with the universal ZBL potential. GAMESS realizes finite
basis set approach using Gaussian basis sets.
The package as well as basis sets were developed for problems of quantum chemistry i.e.
for a consideration of systems nearby (quasi)equilibrium state. Therefore great care
should be taken to select proper basis sets.
Fig. 1. Normalized difference of interatomic
potentials of Pb – C system calculated with
finite basis sets (uWTBS) and fully numerical
(2D) methods at Hartree-Fock level of theory.
Large 20s13p for C and 28s24p18d12f for Pb
well-tempered basis sets7,8 (WTBS), augmented with polarization and diffuse orbitals,
have been employed in the present work.
Completeness and flexibility of the basis sets
were tested against fully-numerical restricted
Hartree-Fock calculations with computer code
by Kobus et al.9. Electronic configurations of
neutral diatomic with lowest spin multiplicity
were considered. As it is seen from Fig. 1,
normalized difference of the potentials calculated with the two methods at Hartree-Fock
level do not exceed 1.5% in the range of interatomic distances of 0.08 – 1.2 Å. For Pb
ions with energies from a few keV to almost 1
MeV bombarding a carbon target these separations are determining in a description of
(quasi)elastic scattering.
It is well known that relativistic effects are
important for heavy atoms. Using the uncontracted well-tempered basis sets (uWTBS),
screening functions have been calculated with
Hartree-Fock method including relativistic
corrections by RESC scheme. Similar calculations with DFT at local spin density approximation (LSDA) level have been performed.
The results of the calculations are shown in
Fig. 2. Also, screening functions for the uni-
Fig. 2. Screening function for Pb – C diatomic
calculated with DFT and HF methods including
relativistic corrections by RESC scheme using
uncontracted WTBS basis sets (see text). Those
for the ZBL and Molière potentials are given
for comparison. The inset shows a behavior of
the screening functions at larger distances.
versal ZBL and the Molière potentials are
given for comparison. It is interesting to note
that in accordance with Ref. 4 the average of
the relative errors squared for the Molière potential is 237% whereas for the ZBL potential
that is only 5%. The difference between the
DFT and the ZBL potentials reaches 11% for
separations r ≤ 0.529 Å (Bohr radius) and rapidly increases with interatomic distance up to
46% at 1.2 Å. Relativistic corrections yield of
Fig. 3. The nuclear stopping powers of Pb ions in
C as determined by DFT potential (filled circles),
by Molière-like fit to HF potential (dash-dotted
line) and that of SRIM2003 (solid line). Also, the
electronic stopping powers given by SRIM2003
(dashed line) and by Land and Brennan tables10
(dots) are plotted for comparison.
about of 3% at the former separation and of
13 % at the latter one.
The differences in the potentials lead to notable changes in the nuclear energy loss. In
Fig. 3 the nuclear stopping powers as determined by the DFT, by Molière-like fit to HF
and by the ZBL potentials are presented. Also, the electronic stopping powers from
SRIM2003 package and by Land and Brennan10 are given for comparison. One can see
that the nuclear stopping power as determined
by the ZBL potential is approximately 10%
larger than that of first principles calculation
in a wide interval of energies above 200 keV.
At energies below 1 keV the differences exceed 20% increasing up to 40% at 100eV.
However at energies below a few hundreds
eV one needs also to take into account manybody character of the interactions.
We conclude that in order to obtain a more
accurate description of range and damage distributions induced by a heavy ion implantation in carbon materials one needs to use the
correct interatomic potentials, which can be
derived from the first principles calculations
taking into account relativistic corrections.
1) M. Behar et al., Nucl. Instrum. Meth.,
B59/60 (1991), 1.
2) P.L. Grande et al., Nucl. Instrum. Meth.,
B61 (1991), 282.
3) E. Friedland et al., Nucl. Instrum. Meth.,
B136-138 (1998), 147.
4) J.F. Ziegler, J.P. Biersack, and U.
Littmark, “The stopping and range of ions
in solids,” New York: Pergamon, 1985.
5) M.W. Schmidt, K.K. Baldridge, J.A.
Boatz., J. Comp. Chem., 14 (1993), 1347.
6) T. Nakajima, K. Hirao, Chem. Phys. Lett.,
302 (1999), 383.
7) S. Huzinaga and B. Miguel, Chem. Phys.
Lett., 175 (1990), 289.
8) S. Huzinaga and M. Klobukowski, Chem.
Phys. Lett., 212 (1993), 260.
9) J. Kobus, L. Laaksonen, D. Sundholm,
Comput. Phys. Comm., 98 (1996), 346.
10) D.J. Land, J.G. Brennan, Atomic Data and
Nucl. Data Tables, 22 (1978), 235.
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