united nation, 354115educational, scientific

and culturalorganization

thesinternational centre for theoretical physics

international atomicenergy agency

SMR/1310-12

SPRING COLLEGE ONNUMERICAL METHODS IN ELECTRONIC STRUCTURE THEORY

(7 - 25 May 2001)

"Physics of matter at extreme conditions: ultra-high pressures and temperatures,ultra-thin metal nanowires"

(Part II)

presented by:

E. TOSATTI

S.I.S.S.A. and The Abdus Salam I.C.T.P.

TriesteItaly

These are preliminary lecture notes, intended only for distribution to participants.

STRUCTURE ANDPROPERTIES OF TIP-SUSPENDED GOLD

NANOWIRES

E. TOSATTI

SISSA, TriesteICTP, Trieste

INFM, UdR Trieste SISSA

S. PRESTIPINO, S. KOESTLMEIER,A. DAL CORSO , F. DI TOLLA

E. Tosatti, S. Prestipino, S. Koestlmeier, A. Dal Corso, and F. DiTolla, Science, 12 January 2001

E. Tosatti and S. Prestipino, Science 289, 561 (2000)

Y. Kondo and K. Takayanagi, Science 289, 606 (2000)

Nanowire References —• _ -\

o.rofrAo*/^,^ F.jE<c«trss?r e .i***-rn*. S.v*rM£ set. i8*/z*ft 10^1"Premelting of thin wires", 0. Gulseren, F. Ercolessi, andE. Tosatti, Phys.Rey B(RC)51# 7377 (1995).

"Non-crystalline structures of Ultra-Thin Unsupported Nanowires",0. Gulseren, F. Ercolessi, and E. Tosatti, Phys. Rev. Lett. 80,3775 (1998).

"The puzzling stability of monatomic gold wires", J.A. Torres,E. Tosatti, A. Dal Corso, F. Ercolessi, J.J. Kohanoff, andJ.M. Soler, Surface Sci. 426, L441 (1999).

"Electronic Properties of Ultra-Thin Aluminum Nanowires"F. Di Tolla, A. Dal Corso, J, A. Torres, E. Tosatti, SurfaceSci. 454-456, 947 (2000).

"Weird Gold Nanowires" , E. Tosatti and S. PrestipinoScience 289, 561 (2000) (in Perspectives).

"String Tension and Stability of Magic Tip-SuspendedNanowires1*, E. Tosatti, S. Prestipino, S. Kostlmeier, A. DalCorso, and F, Di Tolla, Science 291, 288 (2001).

"Structure and evolution of a metallic nanowire-tip junction"E. A. Jagla and E. Tosatti, to be published (2001).

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FIG. 2. Transmission electron micrographs of an NB 2 nmthick; obtained at the focuses of (a) 65 nm and (b) 55 nm.Note the square lattice in (a) and hexagonal one in (b).

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String Tension and Stability ofMagic Tip-Suspended Nanowires

E. Tosatti,1'2'3* S. Prestipino,1'2 S. Kostlmeier,12 A. Dal Corso,1-2

and F. D. Di Tolla1-2

12 January 2001, Volume 291, pp. 288-290

Copyright © 2001 by the American Association for the Advancement of Science

String Tension and Stability ofMagic Tip-Suspended NanowiresE. Tosatti,1-2'3* S. Prestipino,1'2 S. Kostlmeier,1-2 A. Dal Corso,1'2

F. D. Di Tolla1-2

Multishell helical gold nanowires were recently imaged by electron microscopy.We show theoretically that the contact with the gold tips at either end of thewire plays a crucial role and that local minima in the string tension rather thanthe total wire free energy determine the nanowire stability. Density functionalelectronic structure calculations of the simplest and thinnest coaxial gold andsilver wires of variable radius and chirality were carried out. We found a stringtension minimum for a single-tube gold nanowire that is chiral and consists ofseven strands, in striking agreement with observation. In contrast, no suchminimum was found for silver, where the s-d competition leading to surfacereconstruction is lacking.

Recent transmission electron microscope ex-periments have provided unprecedented de-tail about the structure and evolution ofnecks, bridges, and wires formed betweengold tips (1-4). It was found that gold (110)nanobridges, which possess a regular crystal-line face-centered cubic (fee) structure whensufficiently thick (1) or short (4), can trans-form at diameters below ~15 A into regularbut noncrystalline wires several nanometerslong, hanging between tips. These nanowiresappear to form mostly discrete, multishellstructures of specific "magic" sizes, and eachtube consists of a triangular sheet folded cy-lindrically onto itself (3). These structures areexciting for many reasons. Wires represent anovel organization of matter, similar but notidentical to clusters, which is yet to be fullyunderstood (5). The structural transition fromcrystalline (fee) to noncrystalline nanowires,including chiral ones for decreasing radius,confirms an earlier theoretical prediction (6).Nonetheless, the detailed stabilization mech-anism leading to the observed chiral goldstructures is unknown. Also, whereas non-equilibrium necks and their nanosecond evo-lution have been relatively well explored,particularly through simulation (7-10), thelong-lived magic wires (which last for sec-onds at room temperature) are novel and noteasily within the reach of simulation. Theconnection between thermodynamical, geo-metrical, mechanical, electronic, and ballisticconductance properties in these nanowirescalls for a fresh approach.

The equilibrium state of an.iV-atom isolat-ed cluster minimizes conventional free ener-gy F (at zero temperature, this equals the total

11nternational School for Advanced Studies (SISSA);2!stituto Nazionale di Fisica delta Materia, UnitaSISSA; ^International Centre for Theoretical Physics,34014 Trieste, Italy.

*To whom correspondence should be addressed. E-mail: tosatti@sissa.it

energy E). A free wire is a cluster of verylarge extension along one direction only andgenerally does not represent an equilibriumstate because F can be lowered by transfor-mation to a roughly spherical cluster, whosesurface area is smaller by a factor of the orderof (9RI2L)m <$C 1, where L is the wire lengthand R is its radius [we will henceforth con-ventionally take the wire radius to be R —(N/Lirp)1/2, where N is the number of atomsand p is the bulk atom density]. However, ouraim here is not to describe a free wire, butrather to describe a tip-suspended wire, thatis, a long wire (R « L) in contact withbulk-like tips, with which it exchanges atoms.Imagine drawing a nanowire of constant ra-dius R by pulling tips apart from length zeroto L. If JX is the bulk chemical potential and Fis the wire free energy, the positive workdone in drawing the wire out of the tips is (F- \xN), or per unit wire length

F-(1)

/ i s a force that describes the generalized wirestring tension that a tip-suspended wire willseek to minimize. To get a feel for this quan-tity, we can set roughly F = uJV + y2irRLf

where y is a surface tension, leading to / =2TrR-y; the wire tension f(R) generally in-

magicnanowires

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Fig. 1. String tension of a tip-suspendednanowire as a function of radius (schematic).Local minima signal long-lived magic nano-wires. The wire disappears (R = 0) at trueequilibrium.

creases with radius and has its absolute min-imum at R — 0. The decrease in tension withshrinking radius is the driving force behindspontaneous wire thinning (11), which takesplace by depletion of atoms from the nano-wire to the tips, eventually leading to wirebreaking. However, the tension/(#) may alsopossess local minima, corresponding to fa-vorable nanowire structures (Fig. 1). Theseminima signify long-lived metastable states,and we propose that they correspond to themagic observed nanowires. We thus exploreda wide spectrum of free wire structures andcalculated f(R) for each, seeking the localminima. These minima should correspond toobserved magic structures in all cases wherean exchange of atoms between the nanowireand the tip is effectively realized within thetime scale of the experiment.

For a first implementation of this method,we chose the thinnest and therefore simplestamong the magic, noncrystalline gold nano-wires observed in (3). According to Kondoand Takayanagi (5), magic nanowires consistof successive tubes, where each tube isformed by a triangular (111) lattice sheet thatis folded onto itself to form a cylinder, similarto carbon nanotubes (12). The thinnest magicnanowire consists of a single tube and acentral strand. We denote by (m, n) a tubeconsisting of m close-packed strands forminga maximal angle ranging from 30° (n = 0) to0° (n — m/2) with the tube axis. As shown inFig. 2, the tube unit cell is given by thevectors (orthogonal and parallel to the wireaxis) (mal H- na2) and (pal + qa2) with

(P,q)

Fig. 2. Cylindrical folding of a triangular latticefor an [m, n) tube, with views of several coaxialtube nanowires. Each atom is pictured as asphere of atomic radius. The (7, 3) gold nano-wire (note its chirality) was reported to bemagic in (3).

288 12 JANUARY 2001 VOL 291 SCIENCE www.sciencemag.org

qlp = (2m - n)l(m - In) and n < mil. All tubesother than (m9 0) and (m, m/2) are chiral, and(m, «) and (m, m - n) are mirror images of oneanother. At constant m, the wire radius, propor-tional to (m2 + n2 - mn)l/2

9 shrinks forincreasing n9 as strands progressively alignwith the axis. The total atom number per cellis N = 2(m2 + n2 - mn) + q (q atoms makeup the central strand).

We considered infinitely long (m, n)nanowires (the effect of tips is included byEq. 1) ranging from (5, n) to (10, n) andcarried out the following protocol: (i) roughstructural optimization with Lennard-Jones(LJ) potential; (ii) further structural refine-ment with a realistic embedded atom poten-tial (73); (iii) full density functional calcula-tion of electronic structure, total energy E,and string tension / (zero temperature); (iv)final structural refinement to minimize thecalculated string tension; and (v) plotting theresulting optimal tension / against radius Rand searching for magic nanowires as localminima off(R).

The result of this procedure for gold isshown in Fig. 3 (for each m value, only thelargest and smallest n values are shown, forsimplicity). The chiral (7, 3) nanowire (m - 7,n = 3;p = l}q= 11; L = 28.54 A; N = 85;R = 3.92 A; and the actual tube radius is 2.78A) (14) exhibits the lowest tension and shouldthus be magic. The agreement of this theoreticalresult with the smallest observed magic nano-wire (3) suggests that this theory should beextendable to the thicker magic nanowires. Itshould be stressed that minimizing tension in-stead of energy is crucial. In the present case, Ewould be lowest for the (8, 4) wire (which is,moreover, unstable against a crystalline formmarked "rhombic" in Fig. 3), whereas/is lowerfor the (7, 3) wire because of a smaller radius

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Fig. 3. Calculated {22) string tension (1 eV/A =1.6 nN) of tip-suspended gold nanowires atzero temperature (only the largest and smallestn values are shown). The minimum demon-strates why the (7, 3) nanowire is magic. Thecalculations were carried out for infinite tip-free wires, with structure relaxed to minimizestring tension (Eq. 1), starting from initial wiregeometries obtained by Voter's potential;|JL(AU) = -4.401 eV was obtained from a sep-arate bulk calculation.

and larger length L [length is proportional to

tf + f-ptf*}.Our approach also allows a full electronic

characterization of the nanowires. The bandstructure may give important clues about the.stability of magic wires, which could, forexample, be connected to electronic shellclosing, as in the case of sodium wires (75);the band structure also provides the numberof conducting channels for ballistic transport.As Fig. 4 shows, there is no electronic shellclosing in any of the (6, 0), (7, 3), and (8, 4)nanowires. The Fermi level is crossed by six,six, and eight conduction bands, respectively,as expected for cylindrical free-electron ("jel-lium") tubes of increasing radii. Moreover,all chiralities are equally metallic, unlike car-bon nanotubes (12). We are thus left with thenontrivial task of uncovering the microscopicphysical explanation for the stability of themagic nanowires.

We considered packing. For hard spheres,only 5 ^ m ^ 8 may provide good contact.We find that the optimal LJ nanowire is infact (5, 0), the same wire as B4, found inearlier simulations for aluminum (5) and asort of analog of the 13-atom icosahedralcluster (16). Why does the gold wire include,in the outside shell, two more strands andbecome chiral on top of that? A clue may besurface reconstruction: the (111) topmost lay-er in fee gold has a 4.5% higher lateral den-sity (atom density on the tube) than all otherlayers (17). We find that the lateral density is0.151 A" 2 for the (7, 3) wire, 2.9% higherthan 0.147 A"2 of the (6, 0) wire and 11.9%higher than 0.133 A"2 of the (5, 0) wire.Thick, crystalline gold (110) nanowires havealso been shown to reconstruct (1). It is thustempting (5) to envisage the magic (7,3) wireas the result of a kind of "wire reconstruc-tion." Gold surface reconstruction (17) isrelated to s-d competition in bulk cohesion(18), where a large positive 6s Fermi pressurecompensates a corresponding negative 5dpressure. At the surface, the 6s pressure canbe released otherwise, for example, by relax-ation of the first layer, leaving the 5d negative

pressure uncompensated and free to force anincrease of the lateral density. We testedwhether this scenario is pertinent to thenanowires in two ways. First, we comparedthe calculated 6s Fermi energies of the threeprototype wires [(6, 0), (7, 3), and (8, 4)](Fig. 4) and obtained 9.12, 8.58, and 8.18 eV,respectively, indicating a clear decrease, asexpected in the reconstruction picture. Sec-ond, we repeated the whole calculation pro-tocol for another metal that does not recon-struct, namely silver (18). The (7, 3), (6, 0),(6, 3), and (5, 0) tensions are now indistin-guishable in silver (Fig. 5), in line with ex-pectations based on silver's weaker recon-struction tendency. In conclusion, both testssupport the reconstruction mechanism behindthe stabilization of the (7,3) gold nanowire. Itseems likely that this kind of physics will alsobe relevant to the thicker magic wires (3) notaddressed here.

One last feature to be discussed is nano-wire handedness and chirality. The possibleoccurrence of nanowire chirality was high-lighted in earlier work (6), where, incidental-ly, a (7, 3) tube was first found by simulation(the inner tube of lead wire B7). In gold's (7,

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Fig. 4. Electronic structure of se-lected gold nanowires. There isfull metallicity in each case, andthe decreasing width of the 6s(or s and p) bands from (6, 0) to(7, 3) to (8, 4). Solid (open) cir-cles mark single (doubly degen- .•erate) Fermi crossings, indicating;six, six, and eight conductingchannels, respectively. Upper in-set in the (7, 3) panel is a mag-nified detail of the bands cross-ing the Fermi level (the verticalscale is from -0.5 to 0.125 eV).

2.69 A

www.sciencemag.org SCIENCE VOL 291 12 JANUARY 2001 289

3) wire, chirality appears as a by-product ofreconstruction (requiring a large number m =7 of strands per tube) and of tight fitting ofthe central strand inside the tube, which re-quires that the radius (m2 + n2 - mn)l/2 beminimized and thus that n be as high aspossible (here, n = 3). It seems again likelythat this mechanism could carry over to thethicker magic nanowires; because of highstring tension, the wires with the largest n andwith small R and large L values should againprevail. The largest n for odd m is (m - l)/2,and this leads to chirality, in agreement withmodel structures presented for outer tubeswithm = 11, 13, and 15 (3).

Experimental consequences of nanowirechirality should be interesting to pursue.Properties attributed to nanowire chiralitywere pointed out in (19) and (20). Directcalculation of the helical current density thatis connected with each of the conductingchannels of the (7,3) wire, based on our wavefunctions, is possible. Whether the joint ef-fect of all channels would be such as to yielda measurable inductance as in a kind of nano-solenoid (21) is an open question, presentlyunder consideration.

ed pseudopotentials [local s potential and core radii(in bohr), respectively, for 4d (1.8 and 2.4 ultrasoftradius) and for 5p (2.5)] derived as in work by G.Kresse and J. Hafner [J. Phys. Condens. Matter 6,8245 (1994)]. Kinetic energy cutoffs for wave func-tions and charge density is 25 and 200 rydbergs,respectively, for silver and gold. Spacing betweenone-dimensional kz points is 0.147 A""1 (kz is thewave vector along the wire axis).

23. We acknowledge discussions and help from F. Ercolessi,K. Takayanagi, G. Santoro, G. Scoles, J. A. Torres, and A.Delin and support from Ministero dell'Universita e dellaRicerca Scientifica e Tecnologica (Cofinanziamento '99and "Iniziativa trasversale calcolo parallelo"), Trainingand Mobility of Researchers (FULPROP), and IstitutoNazionale di Fisica della Materia.

2 October 2000; accepted 28 November 2000

References and Notes1. Y. Kondo, K. Takayanagi, Phys. Rev. Lett 79, 3455

(1997).2. H. Ohnishi, Y. Kondo, K. Takayanagi, Nature 395, 780

(1998).3. Y. Kondo, K. Takayanagi, Science 289, 606 (2000).4. V. Rodrigues, T. Fuhrer, D. Ugarte, Phys. Rev. Lett. 85,

4124 (2000).5. E. Tosatti, S. Prestipino, Science 289, 561 (2000).6. O. Gulseren, F. Ercolessi, E. Tosatti, Phys. Rev. Lett.

80, 3775 (1998).7. R. N. Barnett, U. Landman, Nature 387, 788 (1997);

U. Landman, Phys. World 11, 18 (July 1998).8. M. R. Soerensen, M. Brandbyge, K. W. Jacobsen, Phys.

Rev. B 57, 3283 (1998).9. O. Tomagnini, F. Ercolessi, E. Tosatti, Surf. Sci. 287/

288, 1041 (1993); unpublished note.10. G. Bilalbegovic, Solid State Commun. 115, 73 (2000).11. J. A. Torres et al, Surf. Sci. 426, L441 (1999).12. N. Hamada, S. Sawada, A. Oshiyama, Phys. Rev. Lett.

68,1579(1992).13. A. F. Voter, Technical Report No. LA-UR-93-3901 (Los

Alamos National Laboratory, Los Alamos, NM, 1987).14. Coordinates of the (7, 3) nanowire are available at

www.sciencemag.org/cgi /content/ ful l /291/5502/288/DC1.

15. A. I. Yanson, I. K. Yanson, J. M. van Ruitenbeek, Phys.Rev. Left 84, 5832 (2000).

16. F. C Frank, Proc. R. Soc. London Ser. A 215, 43(1952).

17. E. Tosatti, F. Ercolessi, Int. J. Mod. Phys. B 5, 413(1991).

18. N. Takeuchi, C. T. Chan, K. M. Ho, Phys. Rev. B 40,1565 (1989).

19. V. N. Bogomolov, Yu. A. Kuzmerov, V. P. Petranovskii,A. V. Fokin, Kristallografiya 35, 197 (1990).

20. T. G. Schaaff, R. L Whetten, J. Phys. Chem. B 104,2630 (2000).

21. Y. Miyamoto, A. Rubio, S. G. Louie, M. L Cohen, Phys.Rev. B 60, 13885 (1999).

22. In our density functional calculations, we used thelocal density approximation. Selected trials using ageneralized gradient approximation (which we do notfavor because it yields surface energies that are toolow) gave similar results. For gold, we used planewaves and Vanderbilt ultrasoft pseudopotentials, asin work by A. Dal Corso et al. [Phys. Rev. B 56,R11369 (1997)], and for silver, we used self-generat-

290 12 JANUARY 2001 VOL 291 SCIENCE www.sciencemag.org