Molecular-dynamics computer simulations which employ the embedded-atom potential show that nanowires of gold exist as multishelled structures. We simulate double-walled gold nanowires and calculate the capacitance of a finite nanometer-size cylindrical capacitor. For the sizes for which multishelled nanowires appear in simulations we find the capacitances below one attofarad.Računalne simulacije molekulskom dinamikom, koje primjenjuju višečestični potencijal, pokazuju da postoje višeslojne nanožice od zlata. Simulirali smo dvoslojne zlatne nanožice i izračunali kapacitet takvog konačnog cilindričnog kapacitora. Za veličine kapacitora koje se nalaze u simulacijama višeslojnih nanožica dobivamo kapacitete ispod jednog atofarada

Bilalbegović, Goranka

Stepanić, Josip

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ISSN1330–0008CODENFIZAE4LETTER TO THE EDITORNANOMETER-SCALE CAPACITORSJOSIP STEPANIĆa and GORANKA BILALBEGOVIĆbaFaculty of Mechanical Engineering and Naval Architecture, University of Zagreb,I. Lučića 5, 10000 Zagreb, CroatiabDepartment of Physics, University of Rijeka, Omladinska 14, 51000 Rijeka, CroatiaDedicated to Professor Boran Leontić on the occasion of his 70th birthdayReceived 11 November 1999; Accepted 3 April 2000Molecular-dynamics computer simulations which employ the embedded-atom po-tential show that nanowires of gold exist as multishelled structures. We simulatedouble-walled gold nanowires and calculate the capacitance of a finite nanometer-size cylindrical capacitor. For the sizes for which multishelled nanowires appear insimulations we find the capacitances below one attofarad.PACS numbers: 68.65.+g,61.46.+w,73.61.Tm,85.40.Ux UDC 537.213Keywords: nanowires of gold, molecular dynamics, embedded-atom potential, capacitanceFinite-size systems of atomic-scale dimensions are attracting a great attention.Interest is focused on fundamental aspects as well as on applications. The remark-able physical properties of nanostructures allow one to design nano-scale electricaland mechanical devices. The continuing miniaturization of engineering devices leadsto a technological revolution. Research on the properties of cylindrical nanostruc-tures of various materials, e.g., nanowires, has been an especially active field.Multishelled nanostructures have been produced of carbon clusters and wires[1, 2], as well as wires of WS2, MoS2, and NiCl2 [3]. Gold nanowires are madeby Scanning Tunneling Microscopy (STM) [4, 5] and electron-beam litography [6].In recent molecular-dynamics (MD) computer simulation, multishelled cylindricalnanostructures of gold were analysed [7]. Wires with radii around a nanometerand of a length/diameter ratio between 1 and 3 were studied. It was found thatmultiwalled structures exist for longer of these nanowires, i.e., for a length/diameterratio of 2 and 3. The formation of shells has also been found in a jellium-modelcalculation for finite sodium nanowires [8]. This was confirmed by the conductanceFIZIKA A (Zagreb) 8 (1999) 4, 261–266 261stepanić and bilalbegović: nanometer-scale capacitorsmeasurements [9]. We simulate here cylindrical double-walled gold nanowires andstudy the capacitance for this geometry.It is well known that an application of a many-body potential is necessary foran accurate description of metallic bonding by the MD simulation method [10]. Weused (as in Ref. [7]) an embedded-atom potential for gold which produces a goodagreement with the available experimental results for bulk, surfaces and nanopar-ticles [11]. A time step of 7.14 × 10−15 s was employed in the simulation. Thetemperature was controlled by rescaling the particle velocities. We started froman ideal fcc (111) structure at T = 0 K, and included in the cylindrical MD boxall particles whose distance from the nanowire axis was smaller than 1.2 nm. Thelength of the sample of 689 atoms was 6 layers. We also studied a nanowire of9 layers and 1032 particles. The samples were first relaxed, then annealed andquenched. To prevent melting and collapse into a drop, instead of usual heating to≈ 1000 K used in MD simulation of gold nanostructures, our finite nanowires wereheated only to 600 K. Such a procedure, as in Ref. [7], gives the atoms a possi-bility to find local minima and models a constrained dynamical evolution presentin fabricated nanowires. The structures were analyzed after a long MD run atT = 300 K. Figure 1 shows the shape of the MD box for a nanowire of 689 atomsafter 3.55 ns of simulation. The top view of the whole cylindrical MD box for thisFig. 1. Atomic positions (side view) for a nanowire with length 2.6 nm and anexternal radius of 1.2 nm.nanowire is shown in Fig. 2. While the presence of a double-walled and internallyfilled structure is obvious, the walls are still not completely homogeneous. Severalatoms remain close to the walls. Simulation shows that an almost identical structureexists in up to 106 time steps, i.e., 7.1 ns [12]. We also found a double-walledstructure for nanowire of 1032 atoms. These structures suggest an application ofgold nanowires as cylindrical capacitors. Therefore, we calculated the capacitanceof a nanometer-scale capacitor for the geometry shown in the inset of Fig. 3.262 FIZIKA A (Zagreb) 8 (1999) 4, 261–266stepanić and bilalbegović: nanometer-scale capacitorsFig. 2. MD trajectory plot of a top view for a gold nanowire shown in Fig. 1. Thewhole thickness of the wire along its axis is shown after 3.55 ns of simulation. Thisnanowire consists of the two coaxial near walls and a large filled core.0.4 0.7 1.0r1 (nm)0.00.40.8C (aF)rrL12Fig. 3. Capacitance of a capacitor as a function of an internal radius r1, for L = 2.6nm, and an external radius r2 = 1.2 nm. Inset: the geometry of a nanometer-scalecylindrical capacitor.The calculation starts from the electrical potential φ(r, z) in the space betweentwo coaxial cylinders of radii r1 and r2 (r1 < r2), of a finite length L in the zFIZIKA A (Zagreb) 8 (1999) 4, 261–266 263stepanić and bilalbegović: nanometer-scale capacitorsdirection [13]φ(r, z) = V4π∑k=1,3,5,...sin νkzkI0(νkr)K0(νkr1)− I0(νkr1)K0(νkr)I0(νkr2)K0(νkr1)− I0(νkr1)K0(νkr2) , (1)where νk = kπ/L and 0 ≤ z ≤ L. The parameter V is the potential differenceapplied between the two cylinders, whereas Ii(x) and Ki(x) (here and below) arethe modified Bessel functions of the order i [14]. The capacitance C = (r1, r2, L) ofthe structure is obtained fromC =2πr1ε0VL∫0(∂φ∂r)r=r1dz, (2)where ε0 = 8, 854× 10−12 Fm−1 is the electric permittivity of vacuum. The resultisC = 16r1ε0∑k=1,3,5,...1kI0(νkr1)K1(νkr1) + I1(νkr1)K0(νkr1)I0(νkr1)K0(νkr2)− I0(νkr2)K0(νkr1) . (3)Using Eq. (3), we calculated the capacitance for several sizes typical for structuresobtained in MD simulations. The results are presented in Fig. 3 and in Table 1.Table 1. Capacitance C of a nanometer-scale capacitor with an external radiusr2 = 1.2 nm, an internal radius r1 = 0.9 nm, for several lengths L.L (nm) 2.0 2.2 2.4 2.6 2.8 3.0C (aF) 0.336 0.374 0.413 0.452 0.490 0.529In summary, molecular-dynamics computer simulation based on the well-established embedded-atom potential [7] shows that gold wires of nanometer di-mensions form double-walled structures. Motivated by this result, we calculatedthe capacitance of a finite nanometer-scale cylindrical capacitor. Our calculationsof the capacitance is based on the classical electrodynamics. However, on very smallscales the quantum corrections appear. This is the point we are pursuing now [15].It is known that typical inhomogeneous electric fields used in the polarizability mea-surements for metallic clusters are of the order of 105 Vcm−1 and field gradientsare ≈ 105 Vcm−2 [16]. These fields induce a dipole moment and slightly deformthe clusters. Therefore, in order to strictly preserve the shape of a nanometer-scalecapacitor weaker and homogeneous fields should be applied in an electronic device.An STM could be used to manipulate gold atoms and assemble small nanowires[4, 5]. The STM experiments show that gold nanostructures can sustain large fieldsof up to 1 V [5]. Wires produced by electron-beam litography will approach thesmallest nanometer-scales discussed here. It is possible to fill the space between264 FIZIKA A (Zagreb) 8 (1999) 4, 261–266stepanić and bilalbegović: nanometer-scale capacitorstwo cylindrical electrodes with various insulators and obtain nanometer-scale ca-pacitors with different capacitances. Similar filling of carbon nanotubes with metalswas recently realized [17, 18]. Moreover, fabrication of metallic nanowires using car-bon multishelled nanotubes as templates could be used to produce nanometer-scalecylindrical capacitors not only of gold, but also of other metals.AcknowledgementsThis work has been carried out under the HR-MZT project 119206 - “DynamicalProperties of Surfaces” and the EC Research Action COST P3 - “Simulation ofPhysical Phenomena in Technological Applications”.References1) S. Iijima, Nature 354 (1991) 56; MRS Bull. 19 (1994) 43;2) D. Ugarte, Chem. Phys. Lett. 198 (1992) 596;3) R. Tenne, L. Margulis, M. Genut and G. Hodes, Nature 360 (1992) 4444; L. Mar-gulis, G. Salitra, R. Tenne and M. Tallenker, ibid 365 (1993) 113; Y. R. Hacohen, E.Grunbaum, R. Tenne, J. Sloan and J. L. Hutchison, ibid 395 (1998) 336;4) H. Ohnishi, Y. Kondo and K. Takayanagi, Nature 395 (1998) 780;5) A. I. Yanson, G. Rubio Bollinger, H. E. van den Brom, N. Agrait and J. M. vanRuitenbeek, Nature 395 (1998) 783;6) H. Hegger, B. Huckestein, K. Hecker, M. Janssen, A. Freimuth, G. Reckziegel and R.Tuzinski, Phys. Rev. Lett. 77 (1996) 3885;7) G. Bilalbegović, Phys. Rev. B 58 (1998) 15412;8) C. Yannouleas and U. Landman, J. Phys. Chem. B 101 (1997) 5780;9) A. I. Yanson, I. K. Yanson and J. M. van Ruitenbeek, Nature 400 (1999) 144;10) M. S. Daw, S. M. Foiles and M. I. Baskes, Mater. Sci. Rep. 9 (1993) 251;11) F. Ercolessi, M. Parrinello and E. Tosatti, Philos. Mag. A 58 (1988) 213;12) G. Bilalbegović, Molecular Simulations, to be published;13) L. E. Cirlin, Selected Problems in Calculations of Electric and Magnetic Fields (inRussian), Moskva, Sovetskoe Radio (1977);14) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, AcademicPress, New York (1965);15) J. Stepanić and G. Bilalbegović, unpublished;16) W. A. de Heer, Rev. Mod. Phys. 65 (1993) 611;17) P. M. Ajayan and S. Iijima, Nature 361 (1993) 333;18) N. Demoncy, O. Stephen, N. Brun, C. Colliex, A. Loiseau and H. Pascard, Europhys.Journ. B 4 (1998) 147.FIZIKA A (Zagreb) 8 (1999) 4, 261–266 265stepanić and bilalbegović: nanometer-scale capacitorsNANOMETARSKI KAPACITORIRačunalne simulacije molekulskom dinamikom, koje primjenjuju višečestični po-tencijal, pokazuju da postoje višeslojne nanožice od zlata. Simulirali smo dvoslojnezlatne nanožice i izračunali kapacitet takvog konačnog cilindričnog kapacitora. Zaveličine kapacitora koje se nalaze u simulacijama višeslojnih nanožica dobivamokapacitete ispod jednog atofarada.266 FIZIKA A (Zagreb) 8 (1999) 4, 261–266

Nanometarski kapacitori

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