Fullerene-like III-V Binary Compounds
Giancarlo Cappellini 1,2
1
Dipartimento di Fisica, Università degli Studi di Cagliari, Strada Provinciale Monserrato-Sestu Km 0.700, I-09042 Monserrato (Cagliari), Italy
2
INFM-Sardinian Laboratory for Computational Materials Science (SLACS), Strada Provinciale Monserrato-Sestu Km 0.700, I-09042 Monserrato (Cagliari), Italy
email:
[email protected] Tel: +39 070 675 4935 Fax: +39 070 510171
Giuliano Malloci3,4 and Giacomo Mulas3 3
INAF-Osservatorio Astronomico di Cagliari, Strada n. 54, Località Poggio dei Pini, I-09012 Capoterra (Cagliari), Italy 4
Laboratorio Scienza, via Salvemini 4a, I-09028 Sestu (Cagliari), Italy email:
[email protected],
[email protected]
INTRODUCTION
In Physics the term cluster denotes a group of atoms or molecules. Large clusters with stable structures are very interesting systems both by themselves and as building blocks for nanostructures, i.e., objects of nanometer (1 nm = 10-9 m) sizes, intermediate between molecular and microscopic (micrometer-sized, 1 µm =10-6 m) structures. In particular, the discovery of buckminsterfullerene (Kroto et al. 1985) and carbon nanotubes (Iijima 1991), sketched in Figure 1, opened a completely new field of research: nanotechnology. Dealing with structures of typical sizes of 100 nm or smaller, nanotechnology involves the control of matter on an atomic and molecular scale, developing materials or devices and leading to many applications, such as in medicine, electronics, and energy production.
Since the discovery of carbon fullerene and nanotubes, the search for new nano-scaled materials based on chemical elements other than carbon has received much attention (e.g., Buda and Fasolino 1995, Trave et al. 1996, Wang 1996). Given the similarities between boron-nitrogen and carbon-carbon bonds, most studies have focused on boron nitride (BN), which exists in nature in the hexagonal (graphitic-like) structure (e.g, Mårlid et al. 2001, Alexandre et al. 2001, Wurtz et al 2006). These efforts, indeed, resulted in an experimental procedure able to synthesise pure BN nanotubes (Chopra et al. 1995), although the observed BN cages and wires do not clearly show the characteristic pentagonal rings of carbon fullerenes (Goldberg et al. 1998, Parilla et al. 1999), showing instead a sequence of squares and hexagons. In addition, clusters and nanotubes based on other materials such as GaSe (Côté et al. 1998), GeTe (Natarajan and Öğüt 2003), AlN (Zope et Dunlap 2005),and GaN (Lee et al. 1999, Wang et al. 2006, Song et al. 2006) have been predicted
theoretically to be stable. Finally, fullerene-like CdSe nanoparticles with interesting optical properties have been recently studies and synthesised (see e.g., Botti and Marques 2007, and references therein). Nowadays, semiconductors electronic devices mostly use cubic Si crystals (a cubic system, in which the unit cell is in the shape of a cube, is one of the most common and simplest shapes found in crystals and minerals). This material, however, shows an indirect energy gap of ~1.1700 eV between the electronic occupied states (collectively referred as valence bands) and empty states (conduction bands). This hinders its wide application in the fields of optoelectronics and photonics, since electron–hole recombination must always involve also one or more phonons (quantized modes of vibration occurring in a rigid crystal lattice), making cubic bulk Si inefficient at emitting light. On the contrary, several direct-gap III-V systems in the bulk phase are very efficient at emitting light and are thus of outstanding interest in the above mentioned applicative fields. In fact, some of these compounds had, and still have, a prominent position in electronic and optical devices production, e.g., as reported in Madelung 1996:
GaAs in the bulk cubic phase, with a direct gap of 1.51914 eV (near infrared, wavelength of about 0.816 µm), is widely used in infrared light-emitting diodes, laser diodes, solar cells, microwave frequency integrated circuits;
InAs in the bulk cubic phase, with a direct gap of 0.4180 eV (infrared, wavelength of about 2.966 µm), is largely used in infrared detectors and laser diodes;
InSb in the bulk cubic phase, with a direct gap of 0.2363 eV (infrared, wavelength of about 5.247 µm), is mainly used in infrared detectors, thermal imaging cameras, infrared astronomy, infrared military applications, photo-diodes;
GaN in the bulk wurtzitic phase (wurtzite is a member of the hexagonal crystal family), with a direct gap of 3.503 eV (near-ultraviolet, ~ 0.354µm), is mostly used in bright lightemitting diodes production, and is ideal for power amplifiers at microwave frequencies (much higher working temperature and voltages with respect to GaAs).
One interesting question related to future developments of nanoscience and nanotechnology is whether typical semiconductors of the III-V family such as GaP or GaAs, which do not possess a graphitic bulk phase, are able to form hollow fullerene-like structures. Along these lines, following the theoretical prediction that BN cages with six pairs of adjacent pentagonal rings lead to stable clusters with the general formula BxNx±4 (Fowler et al. 1998), ab initio molecular dynamics simulations (Tozzini et al. 2000) showed the spontaneous formation of GaP fullerene-like clusters with 20 and 28 atoms starting from a bulk GaP fragment. These clusters, possessing the same topology as C20 and C28 fullerenes, were predicted to be characterised by stable structures with very symmetric equilibrium geometries (Th and Td symmetry point group), high thermal stability up to a temperature of 1500-2000 K, high chemical stability, expressed by large highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) energy gaps and closedshell electronic structures, and stability against positive and negative ionizations, e.g., the addition or removal of electrons to/from the system. The above properties were later shown to be common to other III-V compounds, namely GaAs, AlAs, and AlP (Tozzini et al. 2001). These pioneering studies therefore opened the real possibility to find an experimental procedure apt to prepare these new nano-scaled materials in macroscopic amounts.
Based on the above discussion about the wide technological applications of bulk III-V systems, it is clear that the knowledge of the electronic excitations of nano-scaled III-V systems has important implications in the light of their potential technological applications in the fields of photonics and optoelectronics (light emitting diodes, lasers, transistors,...).
BACKGROUND
Optical spectroscopy, the study of the interaction between visible light and radiation as a function of wavelength, is known to be a useful tool to characterise cluster geometries (Rubio et al. 1996, Marques and Botti 2005). This technique has been proposed to unambiguously discriminate the ground state of the different isomers of C20, the smallest carbon fullerene (Castro et al. 2002), and has been recently applied to confirm the synthesis of fullerene-like CdSe nano-particles (Botti and Marques 2007). Should an experimental procedure be found which may synthesise small fullerenelike III-V binary clusters, optical absorption spectra could likewise be used to distinguish between different newly formed isomers. The above idea above has been the starting point of a theoretical study (Malloci et al. 2004), aimed at obtaining the optical properties and quasiparticle effects (quasiparticle energies are associated with the addition or removal of an electron in a neutral system) of fullerene-like GaP clusters. Extending our previous study we include in this work other clusters with the stoichiometry IIIxVx±4 predicted by Tozzini et al. 2000 and Tozzini et al. 2001: Ga8As12, Ga12As8, Al8P12, Al12P8, Al8As12, and Al12As8 with Th symmetry, and Ga12As16, Ga16As12, Al12P16, Al16P12, Al12As16, and Al16As12 which belong to the Td symmetry point-group. We recall that point groups are mathematically defined as the sets of geometric symmetry transformations of space leaving at least one point fixed (e.g., reflection with respect to a plane or rotation around an
axis). Since any chemical system is invariant with respect to one well-defined set of such transformations, point group theory is heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming chemical bonds. We will discuss in detail the geometric and electronic properties of each cluster, analysing the behaviour of the relevant physical observables as a function of the size and composition of the III-V fullerene considered. The ground state geometries of the clusters studied in this work are shown in Figs. 2a, 2b, 2c, 2d.
STATE OF THE ART
The accurate determination of the fundamental energy gap in many-electron systems, such as molecules and solids, by means of computationally inexpensive methods, remains an open problem in computational materials science. Density Functional Theory (DFT, Hohenberg and Kohn 1964, Kohn and Sham 1965) has been extremely successful in the calculation of ground state electronic properties of a large class of materials. In this method, the state of the system is completely determined by the electronic density, and all the other physical observables expressed as functionals of it. This leads to an equivalent system of electrons with no explicit interactions, which has the same electronic density and energy as the original, fully interacting system, and whose ground state is determined via a self-consistent procedure. Its success is largely due to the extremely good ratio between the accuracy and the computational cost of this method, which is very fast and simple to implement for most available exchange-correlation (XC) functionals. The XC functional in DFT includes all the exchange and correlation effects between the electrons of the system (Jones and Gunnarsson 1989). Unfortunately, DFT is not designed to give addition and
removal energies in a single calculation: the Kohn-Sham eigenvalues have no direct physical interpretation except for the ones associated with the HOMO; none of the individual Kohn-Sham orbitals has any known physical meaning; virtual (unoccupied) Kohn-Sham orbitals do not even enter the self-consistent cycle and play no role in the theory. It is therefore well recognised that the use of Kohn-Sham eigenvalues to determine the quasiparticle properties of many-electrons systems yields, by and large, results in disagreement with experiments (Jones and Gunnarson 1989). For example, the well-known band-gap problem for bulk semiconductors arises from a severe underestimate (even exceeding 50% in both finite and extended systems) of the electronic excitation energies with respect to available experimental results (optical absorption, direct and inverse photo-emission, Bechstedt et al. 1992). In optical absorption experiments, in particular, an electron excited into a conduction state interacts with the resulting hole in the previously occupied state and two-particles (called excitonic) effects must be properly considered (Onida, Reining and Rubio 2002).
For finite systems, it is possible to obtain accurate excitation energies using the delta-selfconsistent-field (ΔSCF) approach (Jones and Gunnarson 1989). This method, successfully applied to obtain the quasi-particle energies for several clusters, for example C60 (Cappellini et al. 1997) and oligoacenes (Malloci et al. 2007), consists in evaluating total energy differences between the self-consistent field calculations performed for the system with N and N±1 electrons, respectively, to obtain: (i) the vertical electron affinities and the first ionisation energies; (ii) the quasiparticle correction to the HOMO-LUMO energy gap. This quantity is related to molecular hardness, the analogue of the band gap of solids, defined as half the difference between the ionization potential and the electron affinity, which is a key property characterising the chemical behaviour and
reactivity of a molecule. Based on the calculation of the total energy of the electronic ground state, in this work we evaluated the quasi-particle spectra for the sixteen IIIxVx±4 clusters considered, including the correction to the HOMO-LUMO energy gap.
More recently, optical absorption spectra of many different species have been obtained in the framework of Time–Dependent Density Functional Theory (TD-DFT) (Runge and Gross 1984). This approach, the theory for electronic excited-state properties corresponding to the traditional ground state DFT formulation, was applied to a large number of systems in the last years (see e.g., Onida, Reining and Rubio 2002), providing results in very good agreement with experiments for several compounds. Hence we used this approach to compute the low-energy optical absorption spectra of the clusters under investigation. The comparison between the optical gap, i.e. the lowest singlet-singlet excitation energy obtained with TD-DFT, and the quasiparticle–corrected HOMOLUMO gap obtained via ΔSCF, enabled us to estimate the excitonic effects (due to the electronhole interaction) for the clusters under investigation. Our results for the quasi-particle HOMOLUMO gaps support previous results predicting the high stability of these clusters (Tozzini et al. 2000, and Tozzini et al. 2001). Moreover the present optical absorption spectra, calculated by us for the first time, provide reliable, additional tools for the identification of these species via optical absorption experiments.
The calculation of the excitation energies and of the electronic absorption spectra required the previous calculation of the ground-state optimised geometries. For this part of the work we used the Gaussian-based DFT module of the NWCHEM package (Straatsma et a. 2004). Following previous works (Tozzini et al. 2000, and Tozzini et al. 2001) and starting from the ground state
equilibrium geometries given in these papers, we fully optimised the geometries. In particular, we employed a widely used exchange-correlation functional in the generalised gradient approximation (Becke 1988, Perdew 1986) in conjunction with the Gaussian basis set 3-21G to expand the molecular orbitals (Frish et al. 1984). The choice of such a relatively small basis set is a good compromise between computational costs and accuracy, as shown in Tables 1a-1b, where bond lengths and internuclear angles of the small cluster Ga8P12, and Ga8As12 are compared for three different basis sets, ranging in size from 3-21G to 6-311G* (Frish et al. 1984). The agreement between our results and those of Tozzini et al. 2000 indeed improves using the larger 6-311G and 6-311G* basis sets; however we obtained percentile changes of order 5% and 1% respectively for internuclear distances and angles, with respect to the results of the smaller 3-21G basis set, in the face of an increase of one order of magnitude in computational cost.
To confirm the geometry considered to be indeed a local minimum on the potential energy hypersurface, for the smallest clusters III8V12 and III12V8 we also computed the harmonic vibrational frequencies at the optimised geometry, obtaining real values for all of the frequencies of vibration. For all of the sixteen clusters considered the structural parameters obtained at the GGA/3-21G level of theory are given in Tables 2a-2d. Good agreement is found between our data and the geometries given in Tozzini et al. 2000, and Tozzini et al. 2001. Please notice that, in any case, the electronic excitation properties are found to be only weakly dependent upon different choices of the values used for the structural parameters.
At the ground-state geometries optimised as discussed above, we evaluated also the vertical electron affinity (EAv) and the vertical first ionisation energy (IEv). This enabled us to determine
the quasiparticle (QP)-corrected highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) gaps via total-energy differences of charged and neutral clusters. These values are an indicator of stability of the cluster and are used here to evaluate the excitonic effects. In the ΔSCF scheme, the QP-corrected HOMO-LUMO gap of a N-electron system is rigorously defined as:
QPgap(1) = IEv -EAv = EN+1 + EN-1 - 2 EN,
(1)
EM being the total energy of the system with M electrons. We used also the following approximate expression, which is exact in the N → ∞ limit (Godby et al. 1988):
QPgap(2) = εN+1N+1 – εNN,
(2)
where εij is the ith eigenvalue of a j-electron system. The results obtained using the above Eqs. (1) and (2) tend to coincide as the system gets larger and the orbitals more delocalised.
Thanks to the good compromise between accuracy and computational costs, compared to manyelectron wavefunction-based ab initio methods, TD-DFT is the most widely used approach to compute the excitation energies of such complex molecules as the ones in the present investigation. To obtain the optical spectra we used two different implementations of TD-DFT in the linear response regime, in conjunction with different representations of the wave-functions:
1. the frequency-space implementation (Hirata et al. 1999) based on the linear combination of localised orbitals, as given in the NWCHEM package (Straatsma et al. 2004);
2. the real-time propagation scheme using a grid in real space (Yabana and Bertsch 1999), as implemented in the OCTOPUS computer program (Marques et al. 2003).
In scheme 2, the time-dependent Kohn-Sham equations are directly solved in real time, and the wavefunctions are represented by their discretised values on a uniform spatial grid. The static Kohn-Sham wavefunctions are perturbed by an impulsive electric field and propagated for a given finite time interval. In this way, all of the frequencies of the system are simultaneously excited. The whole absolute absorption cross-section σ(E) then follows from the dynamical polarizability α(E), which is related to the Fourier transform of the time-dependent dipole moment of the molecule. The absorption cross section, which has dimensions of an area and is used to express the likelihood of interaction between particles, measures the probability for the molecule to absorb a photon of a particular wavelength; absorption cross-sections are conveniently expressed in Mega barns (1 Mb = 10-18 cm2). The molecular polarizability is defined as the electric dipole moment induced in a molecule by an external electric field, divided by the magnitude of the field. The quantitative relation between absorption cross-section and molecular polarizability is:
σ(E) = (8 π2 E)/(h c) Im[α(E)],
(3)
where h is Planck's constant (h ~ 6.626 × 10-34 J s), Im[α(E)] is the imaginary part of the dynamical polarizability, and c is the velocity of light in vacuum (c ~ 299.792.458 m/s). The dipole strengthfunction S(E) is related to σ(E) by the equation:
S(E) = (me c) / ( h e2) σ(E),
(4)
me and e being, respectively, the mass and the charge of the electron (me ~ 9.109382 × 10-31 Kg, e ~ 1.60217646 × 10-19 C). The dipole strength-function S(E) has the dimensions of oscillator strength (the oscillator strength of a transition is a measure of its intensity) per unit energy and satisfies the Thomas-Reiche-Kuhn dipole sum-rule Ne = ∫ dE S(E), where Ne is the total number of electrons in the molecule. With this approach one obtains the whole absorption spectrum of a given molecule in a single calculation.
In the most widely used frequency-space TD-DFT implementation 1, based on the linear response of the density-matrix, the poles of the linear response function correspond to vertical excitation energies and the pole strengths to the corresponding oscillator strengths (Casida 1995). Poles are found iteratively, starting from zero energy and proceeding upwards. With this method, therefore, computational costs scale steeply with the number of required transitions, and electronic excitations are thus usually limited to the low-energy part of the spectrum. To use an analogy easy to understand for experimental spectroscopists, roughly speaking the difference between the two approaches is intuitively similar to the difference between a frequency-scanning (approach 2) and a Fourier-transform spectrometer (approach 1). From a computational point of view the advantages of the real-time propagation method are discussed, e.g., in Lopez et al. 2005. On the other hand,
the main drawbacks of the real-time approach are that: (i) no information is given by Eq. (3) on dipole-forbidden singlet-singlet transitions (even if this could in principle be retrieved from the time evolution of electronic density by applying a similar analysis on higher order terms of the electric multipole expansion) and on spin–forbidden singlet-triplet transitions, and (ii) one does not obtain independent information for each excited state, such as the irreducible representation of the point group of the given molecular system, and the description of the excitations in terms of promotion of electrons in an orbital picture.
We performed the OCTOPUS calculations in the adiabatic local-density approximation,using the exchange-correlation energy density of the homogeneous electron gas (Ceperley and Alder 1980) parametrised by Perdew and Zunger 1981, which ensures good numerical stability. In addition, nuclei and core electrons are represented by norm-conserving pseudo-potentials (Troullier and Martins 1991), so that only valence electrons are explicitly considered in the DFT calculations. Concerning specific details, we already mentioned that wave-functions are represented by the values they take on a uniform spatial grid in a finite volume. Two important parameters of the calculations are therefore the volume of the box in which the molecule is represented, and the spatial resolution Δs of the grid. The choice of the spatial resolution Δs plays a role similar to the choice of the basis set in methods which represent real or Kohn–Sham wavefunctions as linear combinations of some finite set of appropriately chosen functions (i. e., Gaussians or atomic wavefunctions centered on the nuclei, plane waves etc.): Δs must be small enough to properly sample the wavefunctions of the explicitly represented electrons, but still large enough, to produce a practically tractable size of the resulting uniform grid. In this respect, including the representation of the core electrons together with the nuclei using pseudopotentials is a great
advantage, since explicitly representing the tight oscillations of the wavefunctions of the core electrons would mandate the use of an extremely dense grid, thus increasing computational cost by more than an order of magnitude. With pseudopotentials, we used a grid spacing of 0.3Å and determined the box size by requiring each nucleus to be at least 4Å away from box edges. We used a time integration length of 20 (h/2π)/eV, which corresponds to an energy resolution of (h/2π)/T = 0.05 eV. The time evolution was performed using a time step of 0.001 (h/2π)/eV, which ensured energy conservation with extremely good numerical accuracy, namely better than one part in a million. A thorough description of the program and the specific numerical implementations it uses can be found in the dedicated web-site http://www.tddft.org/programs/octopus/.
In the TD-DFT calculations with NWCHEM we used the frozen core approximation for the 1s electrons and restricted ourselves to the lowest lying excited electronic states up to 3eV. We recall that typical electronic transitions are of the order of the electronvolt (eV), 1eV being the the amount of energy gained by a single electron when it accelerates through an electrostatic potential difference of one Volt; in SI units, 1eV=1.60×10−19 J. Remember that the visible part of the electromagnetic spectrum ranges between about 1.7 eV and 3.2 eV. Our calculations were performed at the same level BP86/3-21G used to obtain the ground-state optimised geometries. Although basis set convergence is not yet expected at this level, our results using the two TD-DFT implementations are almost coincident within numerical errors; we thus believe our theoretical predictions to be sufficiently accurate for the purposes of this work.
CRITICAL DISCUSSION
Table III shows the HOMO-LUMO gap values as obtained using Eq. (1) and (2). The use of the ΔSCF scheme to correct the HOMO-LUMO gap of the clusters considered here is completely justified relatively to their size (Godby and White 1997). The last two columns of Table III report, respectively, the TD-DFT result of NWChem for the corresponding HOMO-LUMO electronic transition, ETD-DFT, and the estimate of the excitonic effects occurring in these molecules, as expressed by the exciton binding energy Ebind = QP1 - ETD-DFT. As an example, we report in Figs. 3 the optical gap ETD-DFT, the QP-corrected HOMO-LUMO gap, and the exciton binding energy Ebind as a function of molecular size, for the GaP clusters (Fig. 3a) and the AlAs clusters (Fig. 3b).
For comparison, in the case of the GaP clusters only, we also computed the quantity E2p, which is the energy difference between the total energy EN of the neutral cluster and the total energy EN* of the excited state obtained by placing an electron in the LUMO state and a hole in the HOMO:
E2p = EN – EN*
(4)
We found that, ranging from Ga8P12, Ga12P8, Ga12P16 to Ga16P12, the E2p values are 2.09, 1.14, 1.56, 1.13, with a deviation from the corresponding TD-DFT results in the range 10-130 meV. The good agreement found between E2p and the corresponding ETD-DFT shows that the excited states corresponding to the HOMO-LUMO transitions in all the GaP molecules considered are well described by a single determinant electronic wave-function.
Comparing the self-energy corrected single-particle HOMO-LUMO gaps and the corresponding TD-DFT results, we obtain strong excitonic effects with binding energies up to a maximum of about 3.5 eV. This result is comparable to what is found for other isolated systems such as the smallest carbon fullerene C20 , or buckminsterfullerene C60 (Cappellini et al. 1997). This is shown in the last row of Table I, which reports our GGA/3-21G results for C60 and compares them with those of Cappellini et al. 1997, obtained with LDA using a discrete variational approach. Both sets of data compare well with the difference between the experimental vertical ionization energy and electron affinity, which falls in the range 4.70-5.20 eV (Cappellini et al. 1997) including error bars. This is an interesting issue, because the similarity in QP energy gaps and excitonic effects with carbon fullerenes takes place in systems made of fundamental components of nowadays technology, such as Ga, As, P, and Al.
The optical absorption spectra obtained with OCTOPUS in the energy range up to about 5 eV for the four families considered are shown in Figs. 4. All plot clearly show that the strongest absorption features corresponding to the different clusters in the family differ in position and strength; as such, these spectroscopic features can be used as a fingerprint of each specific molecule in each class. Note that, in the absence of experimental results to compare our simulations with, the reliability of our OCTOPUS spectra is validated by the good agreement with results obtained using the different TD-DFT frequency space implementation in NWCHEM. This is shown as an example in Table IV, where the first intense electric dipole-permitted transition (marked by an asterisk in Figs. 4) appears to be reproduced by the two simulations within about 0.1 eV in position E and to be almost coincident in the corresponding oscillator strength f. In the case of the largest cluster Ga16P12, only the onset at 1.73 eV in the optical absorption spectrum has
been estimated in the OCTOPUS result, due to the crowding of electronic transitions starting from that energy; such estimate compares well with the prediction of 1.72 eV of NWCHEM for the first intense absorption peak. The small discrepancy shown in Table IV can be ascribed to the small basis set used in NWCHEM calculations and to the different exchange-correlation functionals used (LDA for OCTOPUS and GGA for NWCHEM).
SUMMARY
This work presents the electronic properties of 16 fullerene-like III-V binary compounds in the families GaP, GaAs, AlAs, AlP. Some of these clusters have been predicted to be stable on the basis of first-principles calculations. We computed their optical absorption spectra and the excitonic effects for their first excited states. The quasiparticle corrections we obtained confirm the high stability previously found for these clusters. Our results on both single-particle and twoparticle excitations may play an important role to unambiguously detect the successful synthesis of such molecules.
FUTURE PERSPECTIVE
Using a simple compendium of DFT and TD-DFT calculations with well-known and widely used exchange-correlation functionals, we show in this work the power of theoretical optical spectroscopy to characterise novel synthesised structures. Since the nowadays technology (synthesis, production of devices, characterization, ...) of III-V compounds is widely affirmed, it permits fruitful extension to future nano-scaled systems (fullerenes, nanotubes, ...). Therefore, our
method could be easily extended and applied in a systematic way to other III-V binary compounds, including larger and more complicated structures, of potential interest for optoelectronic applications based on nano-scaled systems.
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FIGURE CAPTIONS
Figure 1. Schematic view of buckminsterfullerene C60 and a semiconducting (10,0) single-wall carbon nanotube. Figure 2. Ground state geometries of four representative clusters of the sixteen of the form III xVx±4 studied in this work. The symmetry of Ga8P12 and Ga12P8, respectively (a) and (b) in the figure, is Th, while Ga12P16 and Ga16P12, respectively (c) and (d), belong to the Td symmetry point group.
Figure 3a. Top panel: optical gap ETDDFT as obtained via TD-DFT (black triangles) and ∆SCF QPcorrected HOMO-LUMO gap QP1 computed via Eq. (1) (blue diamonds), as a function of GaP cluster size. Bottom panel: exciton binding energy Ebind estimated as QP1 - ETDDFT (red squares).
Figure 3b. Top panel: optical gap ETDDFT as obtained via TD-DFT (black triangles) and ∆SCF QPcorrected HOMO-LUMO gap QP1 computed via Eq. (1) (blue diamonds), as a function of AlAs cluster size. Bottom panel: exciton binding energy Ebind estimated as QP1 - ETDDFT (red squares).
Figure 4a. Optical absorption cross-section σ(E) (expressed in megabarns, 1 Mb = 10-18cm2) of the four GaP clusters considered; in each box the asterisk marks the first intense electric dipolepermitted electronic transition.
Figure 4b. Optical absorption cross-section σ(E) (expressed in megabarns, 1 Mb = 10-18cm2) of the four GaAs clusters considered; in each box the asterisk marks the first intense electric dipolepermitted electronic transition.
Figure 4c. Optical absorption cross-section σ(E) (expressed in megabarns, 1 Mb = 10-18cm2) of the four AlP clusters considered; in each box the asterisk marks the first intense electric dipolepermitted electronic transition.
Figure 4d. Optical absorption cross-section σ(E) (expressed in megabarns, 1 Mb = 10-18cm2) of the four AlAs clusters considered; in each box the asterisk marks the first intense electric dipolepermitted electronic transition.
TABLES
Table Ia. Structural parameters of Ga8P12 obtained using different basis sets (with increasing number of Gaussian functions in each atom) to expand the molecular orbitals; the ab initio results of Tozzini et al. 2000 are given in last row for comparison.
Gaussian basis set
Number of functions Ga P 3-21G 23 13 6-311G 41 21 6-311G* 47 27 Tozzini et al. 2000
Internuclear distances (Å) Ga-P P-P 2.390 2.425 2.389 2.363 2.390 2.299 2.288 2.262
GaPP 99.3 99.7 100.6 100.3
Internuclear angles (degrees) GaPGa PGaP 83.9 119.6 84.9 119.5 84.2 119.4 84.7 119.5
Table Ib. Structural parameters of Ga8As12 obtained using different basis sets (with increasing number of Gaussian functions in each atom) to expand the molecular orbitals; the ab initio results of Tozzini et al. 2000 are given in last row for comparison.
Number of functions Ga As 3-21G 23 23 6-311G 41 41 6-311G* 47 47 Tozzini et al. 2000
Gaussian basis set
Internuclear distances (Å) Ga-As As-As 2.476 2.594 2.471 2.569 2.473 2.516 2.389 2.505
Internuclear angles (degrees) GaAsAs GaAsGa AsGaAs 97.4 81.4 120.0 98.5 83.7 119.8 99.0 83.4 119.7 98.2 83.6 119.8
Table IIa. Structural parameters of the four GaP clusters considered; the corresponding ab initio results of Tozzini et al. 2001 are given in parentheses.
Cluster Ga8P12 Ga12P8 Ga12P16 Ga16P12
Internuclear distances (Å) Ga-P 2.390 (2.288) 2.386 (2.294) 2.354, 2.372 2.402, 2.347
P-P 2.425 (2.262)
Ga-Ga -
GaPP 99.3 (100.3)
-
2.363 (2.383)
-
2.453
-
99.6
-
2.347
-
Internuclear angles (degrees) GaPGa PGaP 83.9 119.6 (84.7) (119.5) 87.2 125.5 (88.6) (126.8) 114.2, 87.2, 96.6 131.6 90.7, 120.0, 110.6 129.8
PGaGa 113.3 (112.0) 112.7
Table IIb. Structural parameters of the four GaAs clusters considered; the corresponding ab initio results of Tozzini et al. 2001 are given in parentheses.
Cluster Ga8As12 Ga12As8 Ga12As16 Ga16As12
Internuclear distances (Å) Ga-As 2.476 (2.389) 2.484 (2.396) 2.448, 2.454 2.440, 2.520
As-As 2.594 (2.505)
Ga-Ga -
GaAsAs 97.4 (98.2)
-
2.372 (2.401)
-
2.619
-
97.9
-
2.350
-
Internuclear angles (degrees) GaAsGa AsGaAs 81.4 120.0 (83.6) (119.8) 81.4 126.6 (85.3) (127.0) 114.5, 84.7, 93.8 130.3 85.9, 120.0, 88.9 130.8
AsGaGa 114.6 (113.2) 114.4
Table IIc. Structural parameters of the four AlP clusters considered; the corresponding ab initio results of Tozzini et al. 2001 are given in parentheses. Cluster Al8P12 Al12P8 Al12P16 Al16P12
Internuclear distances (Å) Al-P 2.409 (2.301) 2.413 (2.306) 2.364, 2.400 2.427, 2.373
P-P 2.460 (2.270)
Al-Al -
AlPP 99.3 (100.1)
-
2.556 (2.510)
-
2.478
-
100.0
-
2.527
-
Internuclear angles (degrees) AlPAl PAlP 84.4 119.6 (84.2) (119.5) 89.7 127.1 (88.8) (128.9) 89.0, 114.0, 97.2 131.9 93.2, 120.0, 103.6 131.0
PAlAl 111.4 (111.0) 110.7
Table IId. Structural parameters of the four AlAs clusters considered; the corresponding ab initio results of Tozzini et al. 2001 are given in parentheses. Cluster Al8As12 Al12As8 Al12As16 Al16As12
Internuclear distances (Å) Al-As 2.500 (2.405) 2.520 (2.409) 2.465, 2.488 2.476 2.546
As-As 2.623 (2.521)
Al-Al -
AlAsAs 97.4 (97.8)
-
2.563 (2.522)
-
2.642
-
98.3
-
2.534
-
Internuclear angles (degrees) AlAsAl AsAlAs 81.6 120.0 (82.5) (119.9) 83.6 128.5 (85.0) (129.2) 85.9, 114.3, 93.8 130.9 88.3, 119.9, 94.5 133.1
AsAlAl 113.0 (112.3) 112.1
Table III. HOMO-LUMO gap (expressed in eV) of the sixteen GaP, GaAs, AlP, AlAs clusters considered, as obtained by GGA/3-21G calculations using Eqs. (1) and (2). Last two columns report, respectively, the TD-DFT result and the excitation energy evaluated using Eq. (3). Last row gives the GGA/3-21G results for C60, comparing them with those reported in Cappellini et al. 1997. Cluster
KohnSham gap
IEv
EAv
QP1
QP2
ETD-DFT
Ebind
5.56 4.26 4.60 3.88
5.44 4.21 4.56 3.83
2.08 1.27 1.63 1.19
3.48 2.99 2.97 2.69
4.67 3.64 4.08 3.06
4.64 3.60 4.08 3.03
1.61 0.67 1.44 0.44
3.06 2.93 2.64 2.62
5.52 4.37 4.35 4.14
5.39 4.32 4.37 4.06
2.17 1.40 1.70 1.33
3.35 2.97 2.65 2.81
4.66 3.64 4.07 3.38
1.65 0.82 1.46 0.86
3.04 2.86 2.61 2.57
5.60 4.31 5.04 (5.20)
/ /
/ /
/
/
GaP Ga8P12 Ga12P8 Ga12P16 Ga16P12
2.06 1.12 1.54 1.12
8.15 7.14 7.45 6.85
2.59 2.88 2.85 2.97 GaAs
Ga8As12 Ga12As8 Ga12As16 Ga16As12
1.60 0.60 1.39 0.37
7.43 6.83 7.07 6.52
2.75 3.18 2.99 3.46
Al8P12 Al12P8 Al12P16 Al16P12
2.15 1.30 1.58 1.44
8.30 7.14 7.46 6.87
2.78 2.76 3.11 2.73
Al8As12 Al12As8 Al12As16 Al16As12
1.63 0.68 1.40 0.79
7.57 6.84 7.14 6.55
C20 C36
0.77 0.40 1.81 (1.79)
7.43 7.43
AlP
C60
7.73
AlAs 2.87 4.69 3.16 3.68 3.07 4.07 3.13 3.43 C clusters 1.83 5.60 3.13 4.30 5.06 2.67 (5.21)
Table IV. Energies (eV) and oscillator strengths (in parentheses) of the first intense dipole-allowed electronic transition of the four GaP clusters considered, as obtained by the different TD-DFT implementations of OCTOPUS and NWCHEM. Last column shows the character of the excitation in terms of the molecular orbitals involved in the transition (H denotes the HOMO orbital, H-1 the one just below it, L is the LUMO and L+1 represents the orbital just above it). Cluster Ga8P12 Ga12P8 Ga12P16 Ga16P12
OCTOPUS 2.46(0.08) 1.35(0.03) 2.01(0.05) 1.73(/)
NWCHEM 2.44(0.07) 1.27(0.03) 1.93(0.06) 1.72(0.01)
Transition H-1(tg) → L+1(tu) H(tu) → L(ag) H(t1) → L+1(t2) H(e) → L+1(t2)
ACKNOWLEDGMENTS G. Malloci acknowledges financial support by Regione Autonoma della Sardegna. The authors acknowledge financial support by MIUR under project PON-CyberSar. We thank the authors of OCTOPUS for making their code available under a free license. We acknowledge the High Performance Computational Chemistry Group for the use of NWCHEM, A Computational Chemistry Package for Parallel Computers, Version 4.6 (2004), PNNL, Richland, Washington, USA. Part of the simulations were carried out at CINECA (Bologna).
FIGURES
