Fitting of the TB-SMA interatomic potentials for Pt/Cu(111) surface alloy
S. A. Dokukin S. V. Kolesnikov A. M. Saletsky A. L. Klavsyuk





kolesnikov@physics.msu.ru



Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russian Federation

	
In this paper we present new parameters of the TB-SMA interatomic potentials for the Pt/Cu(111) surface alloy. The parameters are fitted using both the experimental and ab initio data. The potentials reproduce not only the bulk properties of copper and platinum, but also the energy characteristics of the Pt/Cu(111) surface alloy. The potentials can be used for the simulations of the growth of the Pt/Cu(111) surface alloy on the atomic scale.
	

02.70.Ns, 68.35.bd.




	
Fitting of the TB-SMA interatomic potentials for Pt/Cu(111) surface alloy
    A.L. Klavsyuk
    December 30, 2023
=========================================================================



Theoretical methods for studying the formation of surface alloys can be divided into several groups. Most precise and reliable are the ab initio methods. They are often based on the density functional theory <cit.>. Unfortunately, such methods are too complicated. They can only be used for the calculation of the typical energies or to describe several initial steps of the system's evolution. Another approach is using of the semiempirical potentials for the investigation of the atomic properties of alloys on the long-time scale. For instance, the Bozzolo-Ferrante-Smith (BFS) potentials were used to describe the Cu_3Pt alloy formation during the annealing of the Cu(100) substrate covered with the Pt monolayer <cit.>. The BFS potentials are based on the values of the experimental and ab initio parameters of pure materials and, therefore, can describe some of their properties. However, they have only one fitting parameter for interaction of different types of atoms. Thus, it is almost impossible to fit this parameter in order to describe all the properties of the surface alloy. The embedded-atom method (EAM) and modified embedded-atom method (MEAM) potentials were also proposed for this system <cit.>. Such potentials are quite popular because they can precisely reproduce the typical energies and the bulk characteristics of the pure materials <cit.>. They can also be used to evaluate the parameters of bulk or dilute alloys <cit.>. Unfortunately, the available potentials <cit.> for the Pt/Cu system neglect the bulk properties of pure Cu and Pt. Incorrect values of the lattice constant and elastic constants of the substrate will lead to an additional energy of elastic deformation. Therefore, an accurate calculation of the adsorption energies and diffusion barriers of Pt atoms on a Cu substrate is unlikely. Another type of interatomic potentials are based on the tight binding method in the second moment approximation (TB-SMA) <cit.>. This approximation was used to simulate the formation of various heterogeneous alloys: Co/Cu <cit.>, Fe/Cu <cit.>, Ni/Cu <cit.>, Cu/Ti <cit.>, Co/Pt <cit.>, Co/Au <cit.>, Co/Ag <cit.>, Ag-Au-Pd-Pt <cit.>. The TB-SMA potentials can simultaneously reproduce the atomic adsorption energies and the diffusion barriers, and precisely describe the bulk structure of substrates. Therefore, it is possible to simulate such processes as formation of nanostructures <cit.>, vibration of clusters <cit.>, mesoscopic relaxations <cit.>, phase stability and segregation <cit.>. It was also shown that the TB-SMA potentials are more suitable for the simulation of the nanocontacts than the EAM <cit.>.

We use the interatomic TB-SMA potentials <cit.>. In this approximation, the attractive term E_b^i (band energy) contains the many-body interaction. The repulsive part E_r^i is described by pair interactions (Born-Mayer form). The cohesive energy E_C is the sum of the band energy and the repulsive part:

    E_C=∑_i (E_b^i+E_r^i),


    E_b^i=-√(∑_jξ_αβ^2exp[-2q_αβ(r_ij/r_0^αβ
    -1)]f_c(r_ij)),


    E_r^i=∑_j[A_αβ^1(r_ij/r_0^αβ-1)
    +A_αβ^0]
    exp[-p_αβ(r_ij/r_0^αβ-1)]f_c(r_ij),

where r_ij is the distance between the atoms i and j; α and β are types of the atoms; ξ_αβ is an effective hopping integral; p_αβ and q_αβ describe the decay of the interaction strength with distance between the atoms; and r_0^αβ, A_αβ^0 and A_αβ^1 are adjustable parameters of an interatomic interaction. The cut-off function f_c(r_ij) has the form <cit.>: f_c(r_ij)=0 if r_ij≤ R_on, f_c(r_ij)=1 if r_ij≥ R_off, and

    f_c(r_ij)=(R_off^2-r_ij^2)^2(R_off^2-3R_on^2+2r_ij^2)/(R_off^2-R_on^2)^3

if R_on<r_ij<R_off, where R_off=6.5 Å and R_on=6.0 Å are cut-off distances.



Parameters for Cu-Cu interaction were taken from Ref. <cit.>. The reliability of the Cu-Cu potentials for different atomic structures (supported islands, surface and bulk vacancies, nanocontacts) has been demonstrated <cit.>.

The Pt-Pt and Cu-Pt parameters are optimized simultaneously by including in the fit the experimental data for the bulk properties of Pt (lattice constant a, cohesive energy E_c, and bulk modulus B) <cit.> and the results of the first-principles DFT calculations. The following energies were included in the database (see Fig. <ref> and Table <ref>): the binding energies of adatom E_1on and dimer E_2on1 on the Cu(111) surface, the binding energies of adatom E_1in and dimers on the first E_2in1 and third E_2in3 neighbors in the topmost layer of the Cu(111) surface and the similar energies in the case of the stepped Cu(111) surface. The binding energies of adatoms were calculated from the formula

    E = E_1 - E_0,

where E_1 is the total energy of the calculation cell with the Pt adatom and E_0 is the total energy of the substrate without the adatom. To calculate binding energies of the dimers we used the following formula

    E = E_2 - 2E_1 + E_0,

where E_2 is the total energy of the calculation cell with two Pt adatoms. We also include in our database the following two barriers: the barrier for the jump of Pt adatom embedded in the topmost layer of the Cu(111) surface E_d^surf, and the barrier for the jump of the Pt adatom along the Cu step edge E_d^step (see Fig. <ref>).







All the DFT calculations were performed with the Vienna ab initio simulation package (VASP) <cit.> with the generalized gradient approximation (GGA) <cit.>. The substrate has been modeled as the periodically repeated slabs consisting of the six atomic layers separated by a sufficiently thick vacuum space of 16 Å. Each atomic layer consisted of the 6×4 atoms. Resulting surface of the slab was Cu(111). The positions of the atoms in the four topmost layers of the substrate were optimized using scalar-relativistic calculations until the forces on all unconstrained atoms were converged to less than 0.01 eV/Å. The same structure was used as a substrate for the energies calculations with the TB-SMA potentials. Its relaxation was carried out by the molecular statics method. To calculate the barriers for the fitting procedure we used the same computational cell as for the calculations of the energies.

The set of data used to define the parameters of the interatomic TB-SMA potentials and the corresponding values calculated by means of the optimized potential are given in Table <ref>. The bulk properties, surface properties, and diffusion barriers are well reproduced. We also calculated the same values using the parameters taken form <cit.>: Cu-Cu and Pt-Pt potentials were fitted to the bulk properties of Cu and Pt, respectively, and parameters of the Cu-Pt potential were taken as averaged parameters. In the last case the surface properties and the barriers are reproduced worse. We see that it is critically important to include the surface properties in the fitting procedure. The fitted parameters of interatomic interactions are presented in Table <ref>.

In summary, we have fitted parameters of the interatomic TB-SMA potentials for the Pt-Cu system. These potentials reproduce bulk properties of copper and platinum, and energy characteristics of Pt/Cu(111) surface alloy. We believe that our parameters of the interatomic TB-SMA potentials will be useful for solution of many problems associated with the evolution and the dynamical properties of the Pt-Cu systems.



§ ACKNOWLEDGEMENTS

The research is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University <cit.>.

10

Kohn1996
W. Kohn, A. D. Becke, and R. G. Parr.
Density functional theory of electronic structure.
J. Phys. Chem., 100(31):12974–12980, 1996.

Demarco2003
Gustavo Demarco, Jorge E Garcés, and Guillermo Bozzolo.
Growth of Pt/Cu(100): an atomistic modeling comparison with the
  Pd/Cu(100) surface alloy.
Surf. Sci., 526(3):309 – 322, 2003.

Luyten2007
Jan Luyten, Maarten Schurmans, Claude Creemers, Bouke S. Bunnik, and Gert Jan
  Kramer.
Construction of modified embedded atom method potentials for Cu,
  Pt and Cu-Pt and modelling surface segregation in Cu_3Pt alloys.
Surf. Sci., 601(14):2952 – 2961, 2007.

Mu2017
Haimen Mu, Bo Xu, Chuying Ouyang, and Xueling Lei.
Highly optimized embedding atom method potential for Pt-Cu alloys.
J. Alloys Compd., 696:470 – 480, 2017.

Baskes1992
M. I. Baskes.
Modified embedded-atom potentials for cubic materials and impurities.
Phys. Rev. B, 46:2727–2742, Aug 1992.

Mo2016
Yun-fei Mo, Ze-an Tian, Rang-su Liu, Zhao-yang Hou, Li-li Zhou, Ping
  Peng, Hai-tao Zhang, and Yong-chao Liang.
Molecular dynamics study on microstructural evolution during
  crystallization of rapidly supercooled zirconium melts.
J. Alloys Compd., 688(Part B):654 – 665, 2016.

Zhou2017
Li-li Zhou, Run-yu Yang, Ze-an Tian, Yun-fei Mo, and Rang-su Liu.
Molecular dynamics simulation on structural evolution during
  crystallization of rapidly super-cooled Cu_50Ni_50 alloy.
J. Alloys Compd., 690(Supplement C):633 – 639, 2017.

Cleri1993
Fabrizio Cleri and Vittorio Rosato.
Tight-binding potentials for transition metals and alloys.
Phys. Rev. B, 48:22–33, Jul 1993.

Kolesnikov2009
S. V. Kolesnikov, A. L. Klavsyuk, and A. M. Saletsky.
Atomic-scale self-organization of Co nanostructures embedded into
  Cu(100).
Phys. Rev. B, 79:115433, Mar 2009.

Stepanyuk2008
Oleg V. Stepanyuk, Nikolay N. Negulyaev, Alexander M. Saletsky, and Wolfram
  Hergert.
Growth of Co nanostructures on Cu(110): Atomic-scale simulations.
Phys. Rev. B, 78:113406, Sep 2008.

Negulyaev2008_2
N. N. Negulyaev, V. S. Stepanyuk, W. Hergert, P. Bruno, and J. Kirschner.
Atomic-scale self-organization of Fe nanostripes on stepped
  Cu(111) surfaces: Molecular dynamics and kinetic Monte Carlo
  simulations.
Phys. Rev. B, 77:085430, Feb 2008.

Meyerheim2008
H. L. Meyerheim, D. Sander, N. N. Negulyaev, V. S. Stepanyuk, R. Popescu,
  I. Popa, and J. Kirschner.
Buried Ni/Cu(001) interface at the atomic scale.
Phys. Rev. Lett., 100:146101, Apr 2008.

Qin2010
Bo Qin and Wen Sheng Lai.
Crystalline to amorphous transition and relative stability of
  amorphous phase versus solid solution in the Cu-Ti system studied by
  molecular dynamics simulations.
J. Phys. Soc. Jpn., 79(6):064603, 2010.

Goyhenex2000
C. Goyhenex and G. Tréglia.
Theoretical determination of two critical sizes for strain relaxation
  during Co/Pt(111) heteroepitaxy.
Surf. Sci., 446(3):272 – 282, 2000.

Goyhenex2001
C. Goyhenex and H. Bulou.
Theoretical insight in the energetics of Co adsorption on a
  reconstructed Au(111) substrate.
Phys. Rev. B, 63:235404, May 2001.

Guo2005
H. B. Guo and B. X. Liu.
Structural stability and homogeneity of the nonequilibrium Co-Ag
  alloys.
J. Phys. Soc. Jpn., 74(1):375–381, 2005.

Wu2016
Xia Wu, Zheng Wei, Qiman Liu, Tao Pang, and Genhua Wu.
Structure and bonding in quaternary Ag-Au-Pd-Pt clusters.
J. Alloys Compd., 687(Supplement C):115 – 120, 2016.

Borisova2008
S. D. Borisova, S. V. Eremeev, G. G. Rusina, V. S. Stepanyuk, P. Bruno, and
  E. V. Chulkov.
Vibrations of small cobalt clusters on low-index surfaces of copper:
  Tight-binding simulations.
Phys. Rev. B, 78:075428, Aug 2008.

Stepanyuk2000
V. S. Stepanyuk, D. I. Bazhanov, A. N. Baranov, W. Hergert, P. H. Dederichs,
  and J. Kirschner.
Strain relief and island shape evolution in heteroepitaxial metal
  growth.
Phys. Rev. B, 62:15398–15401, Dec 2000.

Zhao2017
Zheng Zhao, Fang-Hui Wang, Adrian Fisher, Yanchun Shen, and Daojian Cheng.
Phase stability and segregation behavior of nickel-based nanoalloys
  based on theory and simulation.
J. Alloys Compd., 708(Supplement C):1150 – 1160, 2017.

Pu2007
Qing Pu, Yongsheng Leng, Leonidas Tsetseris, Harold S. Park, Sokrates T.
  Pantelides, and Peter T. Cummings.
Molecular dynamics simulations of stretched gold nanowires: The
  relative utility of different semiempirical potentials.
J. Chem. Phys., 126(14):144707, 2007.

Klavsyuk2015
A. L. Klavsyuk and A. M. Saletsky.
Formation and properties of metallic atomic contacts.
Phys. Usp., 58(10):933–951, 2015.

Rosato1989
V. Rosato, M. Guillope, and B. Legrand.
Thermodynamical and structural properties of f.c.c. transition metals
  using a simple tight-binding model.
Philos. Mag. A, 59(2):321–336, 1989.

Levanov2000
N. A. Levanov, V. S. Stepanyuk, W. Hergert, D. I. Bazhanov, P. H. Dederichs,
  A. Katsnelson, and C. Massobrio.
Energetics of Co adatoms on the Cu(001) surface.
Phys. Rev. B, 61:2230–2234, Jan 2000.

Brooks1983
Bernard R. Brooks, Robert E. Bruccoleri, Barry D. Olafson, David J. States,
  S. Swaminathan, and Martin Karplus.
CHARMM: A program for macromolecular energy, minimization, and
  dynamics calculations.
J. Comput. Chem., 4(2):187–217, 1983.

Garbouj2010
H Garbouj, M Said, C Ramseyer, and F Picaud.
Modelling the Cu mono-atomic wire formation on Pt vicinal
  surfaces using kinetic Monte Carlo simulations.
Modelling Simul. Mater. Sci. Eng., 18(8):085009, 2010.

Kresse1993
G. Kresse and J. Hafner.
Ab initio molecular dynamics for open-shell transition metals.
Phys. Rev. B, 48:13115–13118, Nov 1993.

Negulyaev2008_1
N. N. Negulyaev, V. S. Stepanyuk, P. Bruno, L. Diekhöner, P. Wahl, and
  K. Kern.
Bilayer growth of nanoscale Co islands on Cu(111).
Phys. Rev. B, 77:125437, Mar 2008.

Lysenko2002
O. V. Lysenko, V. S. Stepanyuk, W. Hergert, and J. Kirschner.
Mesoscopic relaxation in homoepitaxial metal growth.
Phys. Rev. Lett., 89:126102, Aug 2002.

Kolesnikov2009_2
S. V. Kolesnikov, A. L. Klavsyuk, and A. M. Saletsky.
Vacancy formation on stepped Cu(100) accelerated with STM:
  Molecular dynamics and kinetic Monte Carlo simulations.
Phys. Rev. B, 80:245412, Dec 2009.

Siahaan2016
T. Siahaan, O. Kurnosikov, H. J. M. Swagten, B. Koopmans, S. V. Kolesnikov,
  A. M. Saletsky, and A. L. Klavsyuk.
Co diffusion in the near-surface region of Cu.
Phys. Rev. B, 94:195435, Nov 2016.

Stepanyuk2004
V. S. Stepanyuk, A. L. Klavsyuk, W. Hergert, A. M. Saletsky, P. Bruno, and
  I. Mertig.
Magnetism and structure of atomic-size nanocontacts.
Phys. Rev. B, 70:195420, Nov 2004.

Wang1991
Yue Wang and John P. Perdew.
Correlation hole of the spin-polarized electron gas, with exact
  small-wave-vector and high-density scaling.
Phys. Rev. B, 44:13298–13307, Dec 1991.

Perdew1992
John P. Perdew and Yue Wang.
Accurate and simple analytic representation of the electron-gas
  correlation energy.
Phys. Rev. B, 45:13244–13249, Jun 1992.

NIVC
V. Sadovnichy, A. Tikhonravov, Vl Voevodin, and V. Opanasenko.
"Lomonosov": Supercomputing at moscow state university.
In Contemporary High Performance Computing: From Petascale
  toward Exascale, Chapman & Hall/CRC Computational Science, pages 283–307,
  Boca Raton, United States, 2013. Boca Raton, United States.




