fulltext

Role of the surface in the optical properties of finite systems:

from exotic nuclei to atomic quantum wires

(

 ) R. A. BROGLIA( 1 )( 2 )( 3 ), A. LORENZONI( 1 )( 2 ), H. E. ROMAN( 4 )and F. ALASIA( 2 ) ( 1

) Dipartimento di Fisica, Universit`a di Milano - Via Celoria 16, I-20133 Milano, Italy (

2

) INFN, Sezione di Milano - Via Celoria 16, I-20133 Milano, Italy (

3

) The Niels Bohr Institute, University of Copenhagen - 2100 Copenhagen, Denmark (

4

) Institut f¨ur Theoretische Physik III, Universit¨at Giessen - Heinrich-Buff-Ring 16 35392 Giessen, Germany

(ricevuto il 18 Giugno 1997; approvato il 15 Ottobre 1997)

Summary. — Making use of knowledge and techniques acquired in the study of the

electromagnetic response of atomic nuclei, the electronic properties of atom quantum wires are discussed.

PACS 31.15 – Calculations and mathematical techniques in atomic and molecular physics (excluding electron correlation calculations).

PACS 61.46 – Clusters, nanoparticles, and nanocrystalline materials. PACS 79.70 – Field emission, ionization, evaporation, and desorption. PACS 01.30.Cc – Conference proceedings.

1. – Introduction

The properties of finite many-particle systems do not depend so much on the nature of the particles themselves or the forces acting among them, but on the fact that they are confined and that they are many (cf., e.g., [1] and references therein). In particular, it is well known that the electromagnetic response of the atomic nucleus is strongly influenced by the shape of the system, and by the spill-out of the nucleons from the nuclear surface. In fact, one has observed a conspicuous enhancement of the long-wavelength photoab-sorption cross section in the case of strongly deformed nuclei and of halo nuclei, as com-pared to the corresponding quantity associated with spherical nuclei lying along the stabil-ity valley (cf., e.g., [2, 3]). Because metals tend to be highly absorbing at long-wavelengths (visible and infrared), the above results suggest that nanometer wires have to be searched among finite atomic systems where electrons feel a strongly deformed mean-field, which allows for a conspicuous spill-out of the particles from its surface. Among the systems

( 

)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

1158 R. A. BROGLIA, A. LORENZONI, H. E. ROMANandF. ALASIA

satisfying these requirements, single-wall nanotubes [4] and linear carbon chains [5, 6] seem to be particularly promising. In fact we have found from ab initio calculations that they behave as metallic needles when they are subjected to an electromagnetic field. We have furthermore observed that, under standard bias conditions, linear Carbon chains are prolific emitters of electrons, the associated currents versus voltage curves displaying a behavior typical of metallic systems. Single-wall nanotubes and linear carbon chains are thus likely to constitute the ultimate atomic-scale quantum wires. While providing information from worlds separated by five-to-six orders of magnitude in dimensions and in energy, the electromagnetic reponses associated with atomic clusters and with atomic nuclei have surprisingly similar properties, once the proper scalings are done, and testify to the many parallels which can be drawn between these two classes of finite many-body systems. Nanostructured materials (cf., e.g., [7, 8] and references therein) are made out of atoms as their more common forms, but the atoms are arranged in nanometer or sub-nanometer-size clusters, which become the constituent grains or building blocks of these new materials. Because these tiny grains respond to light, mechanical stress and electric-ity quite differently from micron- to millimeter-size grains, nanostructured materials on the whole are expected to display an array of novel attributes.

Following ref. [9], one can hardly ”...doubt that when we have some control of the ar-rangement of things on a small scale we will get an enourmously greater range of possible properties that substances can have, and of different things that we can do... When we get to the very, very small world — say circuits of seven atoms — we have a lot of things that would happen... Atoms on a small scale behave like nothing on a large scale, for they satisfy the laws of quantum mechanics”.

It would have amused Feynman to know that while seven atom wires can actually be made, some aspects of their behaviour do not differ much from that displayed by classical, metallic wires of similar shape. This result is to be expected from the study of the elec-tromagnetic reponse of atomic nuclei, paradigm of finite quantal systems of correlated fermions (cf., e.g., [10] and references therein).

2. – Atomic wires

In studying the field emission of electrons from individually mounted carbon nan-otubes, Smalley and collaborators have found [5] a conspicuous increase of the yield when the nano-tube tips were opened by laser evaporation, an increase which became accentu-ated when the open tips cooled down to room temperature. This behaviour led the authors to conclude that the emitting structure was an ”atomic wire” of 10 to 100

sp

-bonded carbon atoms pulled from the open graphene sheet of the nanotube by the electric field.

Stimulated by these results, we have studied the quantal properties of Cn chains

(

n

=3–11), making use of ab initio methods. We have found that when the linear chains of carbon atoms, which display an electronic structure very close to the cumulenic form (


= C = C = C = C :), are subject to a bias field of intensity 0.6–0.7 V/ ˚A, corresponding

to bias potentials of the order of 30–50 V, currents of the order of1



A are obtained , in

overall agreement with the experimental findings (cf. [6] and [11] and references therein). Shining photons on the Cnchains leads to a frequency-dependent photoabsorption

cross-section which essentially coincides with that of a classical metallic needle of the same shape. These results lend strong support to the conjecture [5] that linear chains of carbon atoms can be viewed as atomic-scale metallic wires.

The electronic structure calculations of the Cn chains have been carried out in the

Fig. 1. – (A) Electronic density of C8calculated in the local density approximation as described in the text. (B) Difference
U between the total potential felt by the electrons when the C

8-chain is subject to an electric fieldF =0:8V/ ˚A, and when the external field is switched off. To be noted that the bias potential U leading to such a field isU 20V, in keeping with the relationF =U=kR, wherekis a constant of the order of20, andRis the tip radius of the chain(R1:2A).˚

the parametrization of Perdew and Zunger [12], while the role of the carbon atoms were taken into account in terms of norm-conserving pseudopotentials ( [13], cf. also [14]). The resulting bond length is constant and equal to1

:

31A for˚

n

-odd chains, while it alternates

by less than2%around this value in the case of

n

-even chains (cf. also [15]).The constancy

of the bond length can be seen from fig. 1 (A), where the electronic density of the linear chain C8is displayed. It is also found an almost complete screening of the field along the

1160 R. A. BROGLIA, A. LORENZONI, H. E. ROMANandF. ALASIA

Fig. 2. – Single-particle energy levels for linear carbon chains: (A) odd-nchains, (B) even-nchains. The length of the lines indicates the degeneracy of the levels (2 or 4). The thick lines indicate in (A) the HOMO and LUMO levels, while in (B) indicate the last partially filled level which has degeneracy 4 and occupancy 2.

Fig. 3. – Energy centroidsh!(Cn)for linear carbon chains withn=3–11, calculated in the time-dependent LDA (solid dots), in comparison with experimental findings [19] (open squares). The continuous curve displays the results of the relationh!

s =

p L h!

0[20] (cf. text).

The single-particle levels associated with different linear chains are shown in fig. 2. Odd-

n

chains are closed shell systems, while in the case of even-

n

chains the HOMO state is partially filled. This is the reason why the equilibrium configuration of even-

n

chains are closed rings [17, 18]. In what follows, we shall make the hypothesis than even-

n

chains subject to an external bias are linear.

We have subjected the Cn-chains to a time-dependent electromagnetic field. The

asso-ciated longitudinal photoabsorption cross section calculated in the time-dependent LDA (cf. ref. [14]) essentially displays a single peak which collects 90% of the oscillator

strength. The corresponding energy centroids

h!

(C n

)are displayed in fig. 3. The theory

provides an excellent account of the experimental findings [19] (cf. also ref. [18]). The values of

h!

(C

n

)shown in fig. 3 can be compared with the results of the relation 

h!

s =

p

L



h!

0(cf. [4] and [18]). This expression is a generalization of the Mie resonance

expression, and describes the surface plasmon of elongated macroscopic metallic particles in terms of the bulk plasmon frequency

!

0 [20]. The quantity

L

= ((1,

e

2 )

=e

2 )(,1+ 1 2e log( 1+e 1,e

)) is the depolarization factor for vibrations along the symmetry axis of the

system.The quantity

e

is related to the ratio of short to long axis

R

?

=R

k according to

e

2 =1,(

R

?

=R

k ) 2 . The quantity

h!

sprovides a good fit to the centroid energies 

h!

(C n ) with

h!

0

=24eV (cf. fig. 3). This value is quite close to the one obtained by inserting the

density of graphite in the standard plasmon relation

!

2 0

=4

e

2

n=m

.

As learned from the study of the atomic nucleus, when a large fraction of the fermions move in phase under the influence of an external field, the semiclassical approximation becomes applicable. This is the case for the collective motion like, e.g., for rotations and vibrations (nuclear structure, cf. [21] and references therein) as well as in the case of

rel-1162 R. A. BROGLIA, A. LORENZONI, H. E. ROMANandF. ALASIA

Fig. 4. – Polarizability for carbon chains withn=3–11 atoms, calculated in the long wavelength limit of the time-dependent LDA. The continuous curve displays the results of the relations = v[(,1)=(1+L(,1))]with=0:34N

1:24

+1(cf. text and [20]).

ative motion between two ions (cf., e.g., [22] and references therein). The origin of the classical behaviour in the case of nuclear structure can be traced back to the existence of a mean field, the most collective phenomenon displayed by a nucleus, which defines a sur-face (vibration) or a privileged direction (rotation) in the variety of spaces ((

x;y;z

)-space, gauge space, isospace, etc.). The spontaneous symmetry breaking mechanism which is at the basis of these phenomena, reflects the fact that for

h

! 0, the solution of a

Hamil-tonian needs not to respect the symmetries of this HamilHamil-tonian. The origin of classical behaviour in the case of reactions among heavy ions is associated with the short wave-length associated with the relative motion of large mass objects [22].

We have also calculated the polarizability of the system and displayed it in fig. 4. These results can again be compared with the polarizability associated with a macroscopic con-ductor of the same shape, that is



s

=

v

[(



,1)

=

(1+

L

(



,1))](cf. [20]). In this expression

v

and



are the volume and the dielectric constant of the system, respectively. The long wavelength limit of the time-dependent LDA results is accurately reproduced by the func-tion



=0

:

34

N

1:24

+1, which already for a10nm linear chain leads to a dielectric constant

of the order of10 2

, indicating the conducting properties of the system.

Linear carbon chains are like to be the ultimate atomic scale wires. In fact, we have shown that these quantal systems, also a seven-atoms chains, behave as a macroscopic metallic particle of the same shape, that is a truly metallic wire.

  

Financial support by NATO under grant CRG 940231 is gratefully acknowledged. We are also indebted for financial support to INFM Advanced Research Project Class.

REFERENCES

[1] BROGLIAR. A., Cont. Phys., 35 (1994) 95.

[2] GALLARDOM., DIEBELM., DØSSINGT. and BROGLIAR. A., Nucl. Phys. A, 443 (1985) 415.

[3] GHIELMETTIF., COLO`G., VIGEZZIE., BORTIGNONP. F. and BROGLIAR. A., Phys. Rev. C, 54 (1996) R2143.

[4] ROMANH. E., COL`OG., ALASIAF. and BROGLIAR. A., Chem. Phys. Lett., 251 (1996) 111. [5] RINZLER A. G., HAFNER J. H., NIKOLAEV P., LOU L., KIM S. G., TOMANEK´ D.,

NORDLANDERP., COLBERTD. T. and SMALLEYR. E., Science, 269 (1995) 1550.

[6] LORENZONIA., ROMANH. E., ALASIAF. and BROGLIAR. A., Chem. Phys. Lett., 276 (1997) 237.

[7] SCHAEFERH. E., W ¨USCHUMR., GLEITERH. and TSAKALOKOST. (Editors), Proceedings of the Second International Conference on Nanostructured Materials, in Nanostructured materials, Vol. 6 (1995), Numbers 1-4.

[8] HADGIPANAGI G. G. and SIEGEL R. W. (Editors), Nanophase Materials: Synthesis, Properties and Applications (Kluwer Academic Press Publishers) 1994.

[9] FEYNMANR. P., Science, 254 (1991) 1300.

[10] BERTSCHG. F. and BROGLIAR. A., Oscillations in Finite Quantum Systems (Cambridge University Press, Cambridge) 1994.

[11] ROMANH. E., LORENZONIA., BREDAN., BROGLIAR. A. and ONIDAG., this issue, p. 1165.

[12] PERDEWJ. P. and ZUNGERA., Phys. Rev. B, 23 (1981) 5048. [13] TROULLIERN. and J.L. MARTINS, Phys. Rev. B, 43 (1991) 1993.

[14] ALASIAF., BROGLIAR. A., ROMANH. E., SERRAL., COL`OG. and PACHECOJ. M., J. Phys. B, 27 (1994) L643.

[15] RAGHAVACHARIK. and BINKLEYJ. S., J. Chem. Phys., 87 (1987) 2191. [16] LOUL., NORDLANDERP. and SMALLEYR. E., Phys. Rev. B, 52 (1995) 1429. [17] LOUL. and NORDLANDERP., Phys. Rev. B, 54 (1996) 16659.

[18] YABANAK. and BERTSCHG. F., to be published in Z. Phys. D.

[19] FORNEYD., FREIVOGELP., GRUTTERM. and MAIERJ. P., J. Chem. Phys., 104 (1995) 4954.

[20] BOHRENC. F. and HAUFFMAND. R., Absorption and Scattering of Light by Small Particles (Wiley, New York) 1983.

[21] BOHR A. and MOTTELSON B. R., Nuclear Structure, Vol II (Addison-Wesley, Reading, Massachusetts) 1975.

[22] BROGLIA R. A. and WINTHER A., Heavy Ion Reactions (Addison-Wesley, Reading, Massachusetts) 1991.