Electronic structure of some semiconductor fullerenes

NANoSTRUCTURED MATERIALS MOL. 3, PP. 469-477, 1993 COPYRIGHT ~ 1 9 9 3 PERGAMONPRESS L'ro. ALL RIGHTSRESERVED.

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ELECTRONIC STRUCTURE OF SOME SEMICONDUCTOR FULLERENES F. Aguilera-Granja, J. Dorantes.D~ivila and J.L. Mor~in-L6pez Instituto de Ffsica "Manuel Sandoval Vallarta" Universidad Aut6noma de San Luis Potosi 78000 San Luis Potosi, S.L.P., Mexico

J. Ortiz-Saavedra Escuela de Fisica Universidad Auton6ma de Zacatecas Zacatecas, Zac., Mexico

Abstract---The electronic structure of small germanium, silicon, and carbon clusters is calculated. The calculation is carried out within a Hubbard-like Hamiltonian in which s- and pelectrons are taken into account. Charge transfer is allowed between the various atomic sites in order to achieve global charge neutrality. The local electronic density of states is calculated by means of the recursion method. The results for atomic aggregates with 20, 60 and 70 atoms are presented and compared with those obtained with other methods and with experimental data. INTRODUCTION In the last few years, a large number of experimental (1-10) and theoretical (11-17) studies have been carried out on semiconductor neutral and charged nanostructures. The main interest is to know when and how the properties of the nanostructures approach those in the bulk material as the cluster size is increased. Due to the covalent nature of their bonding with sp3 hybridization directional bonds, one would expect that they would condense in clusters with open structures, similar to those of bulk materials. However, it has been found that the unsatisfied bonds at the surface force closed packed structures for systems with a small number of atoms. Several models for the mechanism of cluster formation and cluster structures have been proposed (12-14). The geometrical structure and stability of small silicon clusters Sin (n < 12), has been the subject of various theoretical studies (11-17). Originally an open structure for the Sil0 called adamantane was proposed (11). However, it was shown later that closed structures, the capped octahedron and the distorted tetracapped triangular prism (12-15) are more stable. More recently, based on the observation that there is a dramatic variation on the reactivity of sificon clusters with ammonia and methanol as a function of clusters size with an apparent periodicity in units of six atoms (4), led to the proposition that clusters in the range of 20"A_n < 60 are arranged in stacked, six membered rings (16,17). 469

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Initially, the high reactivity of semiconductor surfaces motivated the study of silicon and germanium clusters. However, the synthesis of macroscopic quantifies of carbon bucky balls (cage-shaped carbon molecules) (6), lead to an intense study of their physical and chemical properties in the last years. Their potential application as new materials is the major driving force. Studies of the optical properties of interstellar matter motivated the laboratory production of small carbon molecules (7). It was observed that only even-atom clusters could be produced, and simple physicochemical considerations led to the conclusion that hollow cages formed by pentagons and hexagons were the best candidate structures. These hollow cages, called now bucky balls or fullerenes, can be formed by 20, 24, 28, 32, 36, 50, 60 and 70 atoms (8). The most stable of all seems to be C6o and has been obtained both, in molecular form and in macroscopic crystals with face-centered-cubic structure. We show in Figure 1 the 20, 60 and 70-atom Fullerenes. Experiments on the C6o solid reveal that is a non-conductor, and band structure calculations performed within the local-density approximation in the density functional theory indicate that the minimum bandgap is about 1.5 eV. (9). This system, when doped with some alkali atoms and with particular stoichiomelries, turns the solid into a superconductor with transition temperatures as high as 33K in CsRb2C6o (10).

(al

(b)

(c)

e

d

e

Figure 1. The dodecahedron (a), the truncated icosahedron (b), and the 70-atom Fullerene structures. The five inequivalent sites in the 70-atom cluster are denoted by a - e .

ELECTRONICSTRUCTUREOFSEMICONDUCTORFULLERENES

471

Here we are interested in calculating the electronic structure of the 20, 60 and 70 bucky balls by means of real space solid state methods and compare the results obtained with other more elaborate methods and with experimental data. Areasonable agreementwill give us confidence to apply this method to the giant Fullerenes and tubulae, systems difficult to treat by means of ab initio theories.

THEORY We consider a Hubbard-like Hamilton[an that takes into account s- andp-electrons given by n , l = ot,a.a X ti~ otlJciaacYfla + + Hr.

[11

i.j

Here, c~a(ciaa) refers to the creation (annihilation) operator of an electron with spin c at atomic site i in the orbital ot (or = s, p) and t~j[3 to the hopping integral between sites i andj. The interaction Hamilton[an 1-1/in the unrestricted Hartree-Fock approximation is given by: otza - Edc, HI = E eior~*i6 iota

[2]

with,

[31 (~,o-)~¢e,o') In equation [2],

_ 1 X "~ Edc - 2 iab ( e ~ - eOa),

[4]

refers to the correction due to double counting, and ~ to the electron number operator. The intraatomic Coulomb interactions Ua~" between electrons can be written in terms of direct coulomb integrals, Ua# = ( U ~ + U ~ ) / 2 . Notice that we neglect exchange for the considered s-and pelectrons. The atomic level e°a can be separated into atomic and neighbor contributions, - eia

+ Aeia,

[51

where ~/~ refers to the bare level of the atom and AEictto the overlap interaction, which takes into account effects of non-orthogonality of the basis orbitals (18,19).

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Ii

20.0

Ii Ii

Chox

"Ii

I1,, III

el

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I Oq

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ii ii

ft

i

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:"I

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'

J -1.5

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-1.0

-0.5

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1 0.5

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ENERGY(Ry) Figure 2. The total electronic density of states per atom in the infinite carbon hexagonal lattice. The top figure shows (solid line) the photoelectron energy distribution curve at ~ = 122 eV (22). The dashed line is the local density of states as obtained from band structure calculations (23).

i" I I

"!

II

"

;:-, " %/

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i|

11

,

~l

,," ';.

l

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~1 ~I II I I l,.I I ii

i! ;",

" i "1 t l'

20.0I I , 'iI t ,l, ~I

O. + In I In 0,m

10.0

0.0

--15

-5

5

ENERGY(eV) Figure 3. Comparison between the measured photoemission and inverse photoemission dam (27,28) and the calculated total electronic density of states per atom of the C60 with truncated icosahedral structure. The Fermi energy is marked by the dashed line.

ELECTRONICSTRUCTUREOF SEMICONDUCTORFULLERENES

The number of s-, and p-electrons, ha(i),

473

(¢t = s, p), is given by the sum: > + <

>.

[6]

In general, in small systems where not all the geometrical sites are equivalent, this number is location dependent. As a consequence, charge transfer between different atomic sites i will result, and the atomic levels are fixed by imposing global charge neutrality: 1

q=n~i qi = ln lot ~,na(i).

[7]

The local density of states p/~ is calculated by using the recursion method (20). For the parametrized Slater-Koster hopping integrals

ti7lJ. up to third neighbors, we take the values chosen

to fit the bulk band structure of the semiconductor materials as reported by Papaconstantopoulos (21). To guarantee a good accuracy in the continuous fraction expansion of the Green function, we compute up to the level N, such that the result becomes independent of N. Empirically. we have found that this criterion is achieved for N = 60. The value we use for the intra-atomic Cotdombic terms U ~ is 5.0 eV. Here we point out that the results are not very sensitive to the U ~ value, since we take the same value for the various U's. It is worth noticing that we are neglecting the overlap effects on the hopping integrals and in the on-site energies.

RESULTS First we calculated the electronic structure of the infinite hexagonal lattice of carbon, and the infinite diamond lattice in the case of Si and Ge. We show in Figure 2 our results and compare them with the experimental photoelectron energy distribution curve obtained at Ptto= 122 eV (22). The histogram shows the electronic density of states as obtained by band-energy calculations (23). One observes a small density of states for the p-type at the Fermi energy, in contrast to the diamond structure. Our results reproduce fairly well the experimental observation and agree with the band structure calculations. The results for bulk silicon and germanium are very similar to those obtained by pseudopotential methods (24). We calculated the electronic structure of three bucky balls: the dodecahedron, the truncated icosahedron, and the 70-atom complex. These structures are shown in Figure 1. The dodecahedron contains 20 vertices, 30 edges and 12 pentagonal faces. On the other hand, the truncated icosahedron contains 60 vertices, 90 edges, 12 pentagonal faces and 20 hexagonal faces. Assuming that the semiconductor atoms A, are located in the vertices, these structures correspond to A20 and A6o. All the atomic sites are equivalent and are coordinated to three nearest neighbors. In the dodecahedron each atom belongs to three pentagonal faces and the dihedral angle is of 116034 '. In the truncated icosahedron each atom belongs to one pentagonal and two hexagonal faces. In this case, the dihedral angles between pentagonal and hexagonal faces and between two hexagonal

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:.[i ,;

I

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.... ] ...... "'" " -

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•

20.0.

0,'i

! ~1 | I .11. | I I I1%| I

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+

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tj.

I

o 10.0.

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-5

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ENERGY(eV) Figure 4. Comparison between the measured photoemission and inverse photoemission spectra (31) and the calculated total electronic density of states per atom of the C70 molecule with the FuUerene structure. The Fermi energy is marked by the dashed line. faces are 142037 ' and 138 ° 11', respectively. The 70-atom Fullerene is also a network of pentagons and hexagons. The number of pentagonal faces is also 12 but the number of hexagonal faces is 25. The total density of states (s + p-contributions) for the C6o truncated icosahedral cluster is shown in Figure 3. The Fermi energy is marked with a dashed line. For this geometry, all the sites are equivalent. One can observe that the Fermi energy falls in a well defined gap. This feature is what makes the C6o Fullerene very stable. From the partial s- andp-local density of states (25) one can see in a neater way that the electronic states near the Fermi energy are of p-type. The bottom of the band is populated mainly with s-electrons. These characteristics coincide with photoemission experiments performed in graphite and diamond (22,26), in which the authors concluded that the electronic states that fall within ~ 10 eV of the Fermi energy are of p-type and that deeper states have s-character. We obtain that the electronic states occupy an energy range of approximately 2.5

Ry. The broken line shows the experimental photoemission and inverse photoemission spectra (27,28). We can notice that the main characteristics of the experimental data are well reproduced. The calculated gap, close to -3.5 eV, is clearly seen. We have also compared our results with calculations performed by more sophisticated methods (29,30) finding a very good agreement.

ELECTRONIC STRUCTUREOF SEMICONDUCTORFULLERENES

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0.0 -2.0

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-1.5

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ENERGY(Ry) Figure 5(a). The total electronic density of states per atom of C20, Si20, and Ge20 with dodecahedral structure. The Fermi energy is marked by a dashed line. We show in Figure 4 the total density of states per atom for the CT0-atom bucky ball. In this case there are five inequivalent sites, and the self-consistent results are obtained under the constraint of global charge neutrality. The partial s- and p-local density of states will be published elsewhere. One can notice that the Fermi energy falls also in a minimum. Here we show also the photoemission and inverse photoemission spectra (31). One also sees that in this case our results

476

F AGUILERA-GRANJAET At.

agree well with the experiment. Our results are also in good agreement with more elaborated methods (29,30). We show in Figure 5 the electronic density of states for the dodecahedral clusters. The results for carbon, silicon, and germanium are presented in Figures 5a, 5b, and 5c, respectively. The Fermi energy is marked with a dashed line. In this case all the sites are also equivalent. The energy range between the lowest and highest energy levels (band-width) W~t(A-- C, Si, Ge) is wider for carbon that for the other elements. One can observe that the Fermi energy in the case of carbon falls in a gap. On the other hand the density of states close to EF in Si and Ge is more dense. In all the cases the low energy part is mainly of s-character and that thep-electronic states occupy the high energy region (25). Although no C20 has been found experimentally, it might be interesting to show the electronic structure of these molecule to compare to other structures. Finally, we would like to note that although the parametrization of carbon Fullerenes based on the graphite structure seems to be appropriate, it might not be suitable to parametrize the Ge and Si clusters with bulk structures. Based on the satisfactory results obtained for the small bucky balls, the application of this method to larger systems is justified and is in progress. ACKNOWLEDGEMENTS This work was partially supported by Direcci6n General de Investigaci6n Cientffica y Superaci6n Acad6mica de la Secretarfa de Educaci6n Pfiblica through Grants C910724-001-268 and C910724-001-963, and by Consejo Nacional de Ciencia y Tecnologfa through Grants 0932E9111 and 1774-E9210.

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