Nanowires are formed by indenting and subsequently retracting two pieces of
sodium metal. Their cross-section gradually reduces upon retraction and the
diameters can be obtained from the conductance. In previous work we have
demonstrated that when one constructs a histogram of diameters from large
numbers of indentation-retraction cycles, such histograms show a periodic
pattern of stable nanowire diameters due to shell structure in the conductance
modes. Here, we report the observation of a modulation of this periodic
pattern, in agreement with predictions of a supershell structure.Comment: Phys. Rev. Lett., in prin

A. I. Yanson

B. Ludoph

C. A. Stafford

C. Bréchignac

C. Höppler

C. J. Muller

C. Yannouleas

E. Scheer

F. Kassubek

H. E. van den Brom

H. Nishioka

I. K. Yanson

J. A. Torres

J. C. Cuevas

J. M. Krans

J. M. van Ruitenbeek

J. Pedersen

M. Brack

M. Brandbyge

M. Pellarin

R. Balian

S. Bjornholm

W. A. de Heer

English

arXiv

Crossref

Supershell Structure in Alkali Metal Nanowires

Yanson, A.I.

Yanson, I.K.

Ruitenbeek, J.M. van

Leiden University Scholary Publications

VOLUME 84, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 19 JUNE 20005832Supershell Structure in Alkali Metal NanowiresA. I. Yanson,1 I. K. Yanson,1,2 and J. M. van Ruitenbeek11Kamerlingh Onnes Laboratorium, Universiteit Leiden, Postbus 9504, NL-2300 RA Leiden, The Netherlands2B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences,310164, Kharkiv, Ukraine(Received 31 January 2000)Nanowires are formed by indenting and subsequently retracting two pieces of sodium metal. Theircross section gradually reduces upon retraction and the diameters can be obtained from the conductance.In previous work we have demonstrated that when one constructs a histogram of diameters from largenumbers of indentation-retraction cycles such histograms show a periodic pattern of stable nanowire di-ameters due to shell structure in the conductance modes. Here, we report the observation of a modulationof this periodic pattern, in agreement with predictions of a supershell structure.PACS numbers: 73.40.Jn, 61.46.+w, 73.20.Dx, 73.23.–bMetallic nanowires clearly exhibit quantum propertiesin their conductance and structure. At the level of a fewatoms in cross section, the conductance through metallicnanowires can be described in terms of a finite numberof quantum modes, and it has been shown that this num-ber is determined by the number of valence orbitals of themetal atoms involved [1–3]. For monovalent free-electronmetals such as gold and, in particular, the alkali metals,the conductance in the smallest contacts evolves roughlythrough the successive opening of distinct conductancechannels as the contact size increases [4–6]. This quantumcharacter of electrical transport was already inferred fromhistograms of conductance values for atomic-size pointcontacts, which show that the contacts have a preferencefor multiples of the conductance quantum G0  2e2h[7,8], after correction for a small series resistance.It had been suggested that the formation of quantummodes in nanowires should not only determine the con-ductance but also the cohesive energy [9–11]. Recently,we have shown that the stability of nanowires in the rangeof cross sections up to about 130 atoms is determined byelectronic shell structure for quantum modes [12]. Theshell structure was observed in conductance histograms,which were measured by many times indenting and retract-ing two metal electrodes by means of a mechanically con-trollable break junction (MCBJ) [13]. The key evidence forshell structure is a regular spacing of diameters for wireswith enhanced stability, where the diameters were obtainedfrom the semiclassical expression for the conductance [14].The shell structure observed here is a close analog of theshell structure observed for alkali metal clusters [15,16],and it is the same principle that also applies to electrons inatoms and to protons and neutrons in nuclei.For metal clusters produced in vapor jets in vacuum, andanalyzed by mass selection, it was observed that clusterswith certain “magic numbers” of atoms, 8, 20, 40, 58, etc.,are more abundant than others. This was explained by theirenhanced stability due to the closing of the shells of elec-tronic states, modeled as free electron waves confined to aspherical potential well. The magic numbers can even be0031-90070084(25)5832(4)$15.00obtained from a semiclassical expansion for the oscillat-ing part of the density of single-particle levels [15,17,18],where the stable clusters are determined by those diame-ters for which a bouncing electron wave traveling along aclosed classical path inscribed inside the spherical clusterwalls obeys the Bohr-Sommerfeld quantization conditionwith the bulk metal Fermi wave vector, kF . The possibletrajectories are illustrated in Fig. 1. It was shown thatfor spherical clusters the triangular and square orbits, withindices 3, 1 and 4, 1, dominantly determine the magicnumber series.Since the two dominant orbits, triangle and square, leadto a slightly different series of stable diameters, the inter-ference between the two series gives rise to a beat pattern,known as supershell structure [15,17,19]. For alkali metalclusters, a single beat due to this effect has indeed beenreported [20,21]. Here, we report the observation of su-pershell structure in sodium nanowires. We show that, incontrast to clusters but in agreement with the theory fornanowires, the diametric orbit [labeled 2, 1 in Fig. 1] hasa strong contribution. Because of the larger separation be-tween the periods for diametric and higher orbits, severalbeat minima of the supershell structure can be observed.FIG. 1. Series of semiclassical orbits inscribed inside a circularcross section, which are applicable to spheres (clusters) andcylinders (nanowires) alike. The orbits are labeled as M, Q,where M is the number of vertices and Q is the winding number.© 2000 The American Physical SocietyVOLUME 84, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 19 JUNE 2000The experiment is performed using an MCBJ, modifiedto accommodate the very reactive alkali metals [8]. A rect-angular piece of sodium metal is cut, while immersed inparaffin oil for protection against oxidation, and fixed upona phosphor bronze bending beam with four 1 mm screws.A notch is cut in the center of the sodium sample, and theassembly is taken out of the paraffin and quickly mountedin a sample holder in a three-point bending configuration.Current and voltage leads are connected to both sides of thenotch and the sample holder is evacuated and cooled downto liquid helium temperature. By applying force to thebending beam, the sodium sample is broken at the notch,by which two clean fracture surfaces are exposed. Atomic-size contacts can then be established by relaxing the force,using a piezoelectric element for fine control. A heaterand thermometer permit controlling the temperature from4.2 K to above 100 K, while the vacuum can remains im-mersed in liquid helium. The conductance of the contactsis measured with a four-point dc-voltage bias circuit. Thesignal goes via a current-to-voltage converter through a16 bit, 105 sampless analog-to-digital converter to a pc-based controller. The software also drives the two halvesof the sample into and out of contact by controlling thepiezovoltage. Conductance values are automatically ac-cumulated into histograms for typically over 104 contactbreaking cycles, with a resolution of about 10 bins perG0. In order to reduce digitization noise, a three bin widesmoothing function is applied to the histograms.At low temperatures and low voltage bias, only fourpronounced peaks at low conductance are seen in the his-togram, near 1G0, 3G0, 5G0, and 6G0, which have beenattributed to the successive occupation of distinct quantummodes [8]. At higher conductance, the histogram shows anumber of rather wide hills which grow into sharp peaksas we increase the temperature [12]. Figure 2a shows anexample of a histogram for sodium at a sample tempera-ture of 90 K. The histogram is similar to the one presentedin Ref. [12], but the modulation of the peak intensities ismore pronounced here [22]. The radius R of the nanowirescan be obtained from the semiclassical expression for theconductance [14],GG0µkFR2∂2µ1 22kFR∂. (1)The inset of Fig. 2a shows the radii, in units k21F , corre-sponding to the first six maxima against their sequentialnumber. Indeed, for shell structure we expect regularlyspaced peaks as a function of the radius.The modulation of the peak intensities and their peri-odicity is more clearly illustrated by subtracting a smoothbackground and plotting the histogram as a function of theradius (see Fig. 2b). Two beat minima are clearly visibleat about kFR  7 and 11.5, and a third can be seen on theexpanded scale in the inset. Similar modulations have beenobserved in histograms for potassium and lithium, whichwill be discussed elsewhere.0 20 40 60 80 100 12001234561 2 3 4 5 6468105 10 15 20024acounts     [arb. un.]Conductance      [2e2/h]21.816.111.78.745.042.85  k FRPeak or shell numberb counts    [arb. un.]kFR15 200.00.20.40.60.81.01.2FIG. 2. Histogram of the number of times each conductanceis observed. (a) Data taken for sodium at T  90 K and biasvoltage 100 mV, constructed from over 13 800 individual inden-tation cycles. The inset shows a plot of the radii of the first sixmaxima versus the peak number, illustrating the regular spacingof the peak positions (the numbers represent conductance valuesfor each peak in quantum units). (b) The conductance histogramfor sodium from (a) with a smooth background subtracted, plot-ted against the radius. In the inset a magnified portion at highconductances is displayed. The full curve envelops peaks in thehistogram and the arrows point to positions of minimal ampli-tude. These are the nodes of the supershell structure.The components in the periodic structure are separated bymaking a Fourier transform of the curve in Fig. 2b, whichis shown in Fig. 3a. We observe two frequency compo-nents, one at 0.6–0.65 and the other at 0.8 0.9 kFR21.We will argue that these are the components of the super-shell structure.The quantum modes in a nanowire give rise to an oscil-lating contribution in the density of electronic states as afunction of electron energy at constant radius or as a func-tion of the radius of the wire at the Fermi energy. Themodes each form a one-dimensional band, with quantumnumbers determined by the confinement in the two trans-verse dimensions. This leads to an oscillating structure inthe thermodynamic potential [9–11,23–25]. The latter de-termines the stability of a particular structure, which leads5833VOLUME 84, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 19 JUNE 20000.0 0.5 1.0 1.5 2.0012340.0 0.5 1.0 1.5 2.001234Thermodynamic    potentialFrequency       [1/kFR]Fourier transform   [arb. un.] (4,1) (3,1) (2,1)ExperimentalabFrequency       [1/kFR]Fourier transform   [arb. un.] (4,1) (3,1) (2,1)FIG. 3. (a) Fourier transform of the histogram of Fig. 2(b) forsodium. Two main peaks at 0.6–0.65 and 0.8–0.9 demonstratethe importance of the distinct contribution of the diametric orbit.(b) Fourier transform of the calculated oscillatory part of thethermodynamic potential, Eq. (2), including terms Q  1 andM  2, 3, . . . , 7. The shape of the nanowire was set by the pa-rameters kFL0  200 and kFA  50. The range of the variabler is chosen to be approximately the same as for the experimen-tal histograms. The three main orbits, diametric, triangle, andsquare, are indicated.to an oscillating probability for nanowire diameters. Alsothe Sharvin conductance of the nanowires is modified byan oscillating contribution, but this contribution appears tobe of secondary importance to the experiment [12]. Theleading terms in a semiclassical expansion of the oscillat-ing part of thermodynamic potential are [23]Vosc 2´FkFp2pr 00 rX̀M2M2XQ11M72∑sinµpQM∂∏2323 cos∑2Mr sinµpQM∂1p2µM 112∂∏, (2)where r  kFR, r 00  ≠2r≠z2 is the curvature along thedirection of the wire axis, and r and r 00 are taken at the nar-rowest cross section, z  0. The number M correspondsto the number of vertices in the semiclassical orbit (see5834Fig. 1) and Q is the winding number. In order to evalu-ate this expression we need to make some assumptionabout a smooth evolution of the curvature of the wire, butthis will affect only the prefactor and not the oscillatingstructure itself. Following Ref. [23] we assume a para-bolic wire shape, r  r0 1 4A 2 r0 zL2. This shapegrows smoothly from an initial cylindrical wire of lengthL0 and radius A while maintaining a constant volume, sothat the radius at the narrowest point can be expressed interms of its length L as r0  A4 p30L0L 2 5 2 1.Substituting these expressions into Eq. (2), we calculatethe Fourier transform of Voscr in the same range of r asfor the experimental data. The result is given in Fig. 3b,where we have included terms up to M  7 for Q  1 (thehigher Q give rise to lower amplitude harmonics and weconcentrate on the fundamental components). The domi-nant peak is due to the diametric orbit M, Q  2, 1and the peaks for higher M rapidly decrease in amplitudeand their frequency converges at 1, marking the end of thefirst series (Q  1). In the Fourier transform we can re-solve only two peaks in addition to the one for M  2,corresponding to the triangular (M  3) and square orbits(M  4).The experiment clearly shows the same groups of fre-quencies, one corresponding to the diametric orbit andone peak at the position of the triangular and square or-bits, which cannot be individually resolved. Some of thehigher frequency maxima with small intensities are prob-ably partly due to harmonics of the main bands. The rela-tive intensities of three calculated peaks cannot be directlycompared to the theoretical spectrum since the latter re-sults from the thermodynamic potential, which we do notmeasure directly. The experimental spectrum is derivedfrom the conductance histogram and is expected to reflectthe same components as the thermodynamic potential, butthe intensities depend, among others, on kinetic factorsfor surface diffusion of atoms and the time available forreaching the proper minima in the potential. Further dif-ferences may arise from deviations in cylindrical symmetryof the wire and scattering of the electrons on residual sur-face roughness. In particular, the later mechanism wouldfavor the higher order orbits, since specular reflection in-creases for smaller angle of incidence with the surface.In contrast to cylindrical systems, for spherical systemsthe diametric orbit (M  2) is negligible as it has a sig-nificantly smaller degeneracy compared to the triangular(M  3) and squared (M  4) ones [17]: For a givenposition of one vertex, the triangle and square have an ad-ditional degree of freedom being the rotation around thenormal to this point. The contribution of the higher orderorbits decays as a high power of the index M of the or-bit. The closeness of the frequencies for the triangular andsquare orbits leads to a long beating period. The first nodeof the shell structure amplitude should be observed afterapproximately 12 periods of principal oscillations. Thenatural limit on the number of chemical elements makesVOLUME 84, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 19 JUNE 2000the number of shells in the periodic table too small forsupershell structure to be observable. The same applies toshell structure in atomic nuclei. Although the amplitude ofsuch a high number of oscillations is greatly diminished,the first node of the beating pattern in cluster abundancespectra was observed in Refs. [20,21,26].For nanowires, the degeneracy of the diametric, triangle,and square orbits is of the same order, and the largerseparation between the M  2 frequency and the higherones make the beating pattern more readily visible in thenanowire shell structure as compared to the metal clusterexperiments. Thus, for the first time a direct Fourier trans-form of the experimental data yields proof for the existenceof semiclassical orbits responsible for the oscillations inthe thermodynamic potential of the system as a functionof the radius.[1] E. Scheer, P. Joyez, D. Esteve, C. Urbina, and M. H. De-voret, Phys. Rev. Lett. 78, 3535 (1997).[2] E. Scheer, N. Agraït, J. C. Cuevas, A. Levy Yeyati, B. Lu-doph, A. Martín-Rodero, G. Rubio Bollinger, J. M. vanRuitenbeek, and C. Urbina, Nature (London) 394, 154(1998).[3] J. C. Cuevas, A. Levy Yeyati, and A. Martín-Rodero, Phys.Rev. Lett. 80, 1066 (1998).[4] H. E. van den Brom and J. M. van Ruitenbeek, Phys. Rev.Lett. 82, 1526 (1999).[5] B. Ludoph, M. H. Devoret, D. Esteve, C. Urbina, and J. M.van Ruitenbeek, Phys. Rev. Lett. 82, 1530 (1999).[6] B. Ludoph and J. M. van Ruitenbeek, Phys. Rev. B 61,2273 (2000).[7] M. Brandbyge, J. Schiøtz, M. Sørensen, P. Stoltze, K. W.Jacobsen, J. Nørskov, L. Olesen, E. Lægsgaard, I. Stens-gaard, and F. Besenbacher, Phys. Rev. B 52, 8499 (1995).[8] J. M. Krans, J. M. van Ruitenbeek, V. V. Fisun, I. K. Yanson,and L. J. de Jongh, Nature (London) 375, 767 (1995).[9] C. A. Stafford, D. Baeriswyl, and J. Bürki, Phys. Rev. Lett.79, 2863 (1997).[10] J. 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B 47, 2271 (1993).[22] The details of the histograms are, among others, influencedby the sample temperature and the bias voltage. Also, weexpect that the crystal structure of the banks at either side ofthe nanowire will be of importance. We have no possibilityto control this orientation, but it may be responsible forthe slight variation in appearance of the histograms. Thebeat minima are not always equally pronounced, but theirpositions have been reproduced for several samples, and thetwo main frequency components can always be recognizedin the Fourier spectrum.[23] C. Yannouleas, E. N. Bogachek, and U. Landman, Phys.Rev. B 57, 4872 (1998).[24] C. Höppler and W. Zwerger, Phys. Rev. B 59, R7849(1999).[25] F. Kassubek, C. A. Stafford, and H. Grabert, Phys. Rev. B59, 7560 (1999).[26] M. Pellarin, B. Baguenard, C. Bordas, M. Broyer, J. Lermé,and J. L. Vialle, Phys. Rev. B 48, 17 645 (1993).5835

Supershell structure in alkali metal nanowires

Observation of Supershell Structure in Alkali Metal Nanowires

Observation of Supershell Structure in Alkali Metal Nanowires

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