Introduction

The period for more than three decades has drawn significant attention to explore the physics and chemistry of II–VI semiconductor nanoclusters (Alivisatos 1996; Murray et al. 2000; Trindade et al. 2001; Efros and Rosen 2000; Hagfeldt and Gratzel 1995; Brus 1984). There exists a continuous transition from molecular electronic structure to that of the bulk. A nanocluster is an intermediate phase between the molecule and bulk, whose electronic and other properties may be exotic. However, two fundamental facts are attributable to the small size of these particles instead of that their physical properties are very different from those of the bulk material. The first fact is that the excitons are confined in a dimension to the order of the particle size, resulting in a compression of the bulk exciton and a corresponding blue shift in the first excited state with decreasing size (Rosetti et al. 1984). This property is known as the quantum confinement effect. According to the technological point of view, the quantum confinement effect has created a great interest due to the unique ability to tune the optical properties by changing particle size. The second important fact of these quantum-confined nanoparticles (quantum dots or QDs) is that the surface to volume ratio is much greater than that of the bulk semiconductor, and thus the nature of the surface influences more on the physical properties of these particles.

Among II–VI semiconducting materials, ZnSe is a direct band gap semiconductor with room temperature band gap energy and an emission at 2.80 eV, which suggests that ZnSe is a potentially good material for short-wavelength lasers and other photoelectronic devices. In addition, ZnSe is of special interest as it exhibits tunable-ultraviolet (UV) luminescence via quantum confinement effects. ZnSe is one of the promising materials for fabrication of light emitting devices, such as blue–green laser diodes and tunable mid-IR laser sources for remote sensing applications (Hasse et al. 1991; Jeon et al. 1991; Tawara et al. 1999; Sarigiannis et al. 2002; Rho et al. 2000; Xiang et al. 2003; Karanikolos et al. 2005; Cumberland et al. 2002).

A number of experimental (Kukreja et al. 2004; Zhu et al. 2000; Kumbhojkar et al. 1998; Reiss 2007; Nikesh et al. 2006) and theoretical studies (Matxain et al. 2001; Deglmann et al. 2002; Goswami et al. 2006, 2007; Sanville et al. 2006) for the physical properties of both the surfaces and the bulk phases have been performed. However, a study of the small size nanoclusters of these selenides is still lacking for their better understanding in terms of the size dependence of the properties. Matxain et al. (2001) has calculated ground state geometries of small size Zn n Se n (n = 1–9) nanoclusters using B3LYP gradient-corrected density functional method. Deglmann et al. (2002) have obtained the atomic structure of Zn n Se n nanoclusters up to heptamers (n = 7). Using time-dependent (TD) density functional response theory (DFRT) within the tight-binding approach, the structural and electronic properties of unpassivated Zn m Se n (m + n = 200) clusters have been calculated by Goswami et al. (2006, 2007). Most recently, Sanville et al. (2006) produced the ZnSe nanoclusters by direct laser ablation method and analyzed in a time of flight mass spectrometer. They predicted only few physical properties like atomization energy, HOMO–LUMO gap and vertical ionization potential of small stoichiometric Zn n Se n (n = 1–16) nanocluster at B3LYP level of theory using SKBJ (d, 2df) basis set.

Earlier, we have performed an ab initio study for Ga x N y and Zn x S y nanoclusters (Yadav et al. 2007, 2010) and the study has been extended to Zn m Se n (m + n = 2–4) small nanoclusters with unequal number of n and m atoms. This study covers the other’s study of Zn m Se n nanoclusters with equal number of n and m atoms by different method to obtain the most stable structures up to four atoms. Also, in the experimental study (Kukreja et al. 2004), most of the nanoclusters with unequal m and n atoms are found to be stable. In this paper, we report the results of a theoretical study of the structural, stabilities, HOMO–LUMO gap, adiabatic and vertical ionization potential (IP) and electron affinity (EA), charge on atoms and dipole moment of small size Zn m Se n (m + n = 2–4) nanoclusters by using the B3LYP-DFT/6-311G(3df) method. We present the method used in the investigations in second section. Third section contains the calculation and results and the last section contains the conclusions.

Methods

All the geometrical structures are fully optimized using the hybrid gradient-corrected functional (B3LYP) (Becke 1988, 1993; Lee et al. 1988) within density functional theory frame (Hohenberg and Kohn 1964; Miehlich et al. 1989) in the Gaussian-03 code (GAUSSIAN 03 et al. 2003). The harmonic vibrational frequencies of each optimized structures are determined by analytical differentiation of gradients. The polarizable triple split valance basis set, 6-311G(3df) is used as the basic basis set in this study for the most precise calculation after selecting a large number basis set for each atom. The main advantage of this split valance basis set in comparison to others is that the orbital is allowed to change its size without making any change in its shape. The three d electrons and one f electron of Zn and Se atoms are included with the valence electrons as a polarized function due to their importance for the description of the ground state of each atom in three dimensions.

Calculation and results

Stability of structures

All the possible two and three-dimensional structures have been considered for optimization to obtain a stable ground state geometry for ZnSe nanoclusters. Minimum energy for each structure is achieved by relaxing the atomic positions. In all fully optimized structures, the convergence of the system energy is obtained up to 10−7 meV and the forces of 10−3 eV/Å on each atom are achieved. We have obtained the most stable structure for each Zn m Se n nanocluster. The stability of nanocluster is considered on the basis of final binding energy (FBE). Quantum mechanically it is necessary to understand the zero point energy for the stability of a system. In actual calculation of binding energy of a system, it includes the zero point energy also. Therefore, a more accurate value of binding energy of a system may be obtained by subtracting the zero point energy from the calculated binding energy. Thus, for a more precise calculation, the harmonic vibrational frequencies and the corresponding zero point energy (ZPE) for all the fully optimized structures have also been calculated. The value of FBE is obtained as,

$$FBE = \left[ {\left( {E_{{\rm{i}}} - E_{{\rm{t}}} } \right)/p} \right] - ZPE$$
(1)

where, Ei,Et and p stand for the sum of the energies of all the isolated atoms present in the nanocluster, total energy of the nanocluster and the total number of atoms present in the corresponding nanocluster, respectively. All the most stable structures are presented in Fig. 1, while the symmetry, multiplicity of the ground state and the FBEs for all the optimized structures are presented in Table 1. The most stable structures have been depicted by bold letters. For the most stable structures, the calculated Se–Se, Zn–Se and Zn–Zn bond lengths are presented in Table 2. Now, we discuss each nanocluster individually as below:

Fig. 1
figure 1

Most stable structures of Zn m Se n (m + n = 2–4) nanocluster. (All the bond lengths are in Å)

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Table 1 Symmetry, multiplicity of the ground state (GS), binding energy per atom (BE), zero point energy (ZPE), final binding energy (FBE) and HOMO–LUMO gap for all the configurations of Zn m Se n (m + n = 2–4) nanoclusters
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Table 2 Bond lengths (Å) for all the most stable configurations of Zn m Se n (m + n = 2–4) nanoclusters
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ZnSe: the ground state of ZnSe nanocluster is the singlet state. The present FBE (0.52 eV) is lower than the experimental value (0.89 eV) (Weast et al. 1990). Our computed value for Zn–Se bond length of 2.22 Å is very close to the values reported by others (Matxain et al. 2001; Sanville et al. 2006) as shown in Table 2.

Zn m Se n (m + n = 3): the ground state of three atom nanoclusters is either singlet or triplet. For ZnSe2 configuration, the ground states of the linear and triangular structures are triplet and singlet, respectively, where as the reverse is true for Zn2Se configuration.

ZnSe2: we have considered the two linear structures and a triangular structure. Among them, the linear ZnSeSe has maximum FBE (1.24 eV), but it is unstable due to its imaginary frequency as shown in Table 3. The other linear SeZnSe structure having D∞h symmetry has FBE of 1.22 eV and is found to be most stable. The bond length of Zn–Se of the linear SeZnSe structure is same as in linear ZnSe structure.

Table 3 The calculated vibrational frequencies (cm−1), infrared intensities (IR Int. in km mol−1), relative IR intensities (Rel. IR Int.) and Raman scattering activities (Raman activity in A4/amu) of Zn m Se n (m + n = 2–4) nanoclusters
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Zn2Se: the investigated structures are similar to ZnSe2. Among them the linear ZnSeZn structure possessing D∞h symmetry has maximum FBE of 0.60 eV and is found to be most stable. For the linear ZnSeZn structure, the Zn–Se bond length is 2.49 Å as shown in Table 2. The other two structures, triangular Zn2Se and linear ZnZnSe have lower FBEs in comparison to linear ZnSeZn structure.

Zn m Se n (m + n = 4): all the four atom structures have the singlet and triplet ground states except the pyramidal ZnSe3 structure which has pentalet ground state.

ZnSe3: we have investigated five different structures like two linear chains SeZnSeSe and ZnSeSeSe, rhombus, triangular planar and pyramidal ones. The rhombus geometry having C2V symmetry has maximum final binding energy of 1.72 eV and is most stable. The calculated values of Zn–Se and Se–Se bond lengths for most stable structure are 2.29 and 2.47 Å, respectively.

Zn2Se2: we have studied three linear chains (ZnSeZnSe, SeZnZnSe and ZnZnSeSe) and a rhombus structure. Among them, rhombus geometry having D2h symmetry has maximum FBE of 1.48 eV and is found to be most stable. Sanville et al. (2006) have also reported rhombus geometry as a most stable one with binding energy of 0.68 eV. The computed Zn–Se bond length of rhombus structure is very much close to that of others (Matxain et al. 2001; Sanville et al. 2006) as shown in Table 2.

Zn3Se: here, we have considered the three different geometries as linear chain (ZnZnSeZn), trigonal and rhombus. All the geometries have very low FBE in comparison to other structures. Among them linear ZnZnSeZn structure is most stable one with FBE of 0.45 eV and has C∞V symmetry. The calculated and predicted values of Zn–Zn and Zn–Se bond lengths of linear ZnZnSeZn nanocluster are presented in Table 2.

We observe that the nanocluster containing maximum number of Se atoms for same value of ‘m’ and ‘n’ is most stable in comparison with others. The study also reveals that the FBE of most stable nanoclusters increases with the size of the nanocluster.

Electronic structure

The calculated energy difference between highest occupied molecular orbital and lowest unoccupied molecular orbital (HOMO–LUMO gap) for all the studied structures is presented in Table 1. The predicted HOMO–LUMO gap of ZnSe nanocluster is lower than the others (Weast et al. 1990; Luo et al. 2002) but for the rhombus Zn2Se2 structure, it has higher value than others (Sanville et al. 2006) as shown in Table 1. The value of HOMO–LUMO gap increases with number of Se atoms, except for the two-atom ZnSe nanocluster.

Ionization potential and electron affinity

The ionization potential (IP) is defined as the amount of energy required to remove an electron from a nanocluster. We determine the adiabatic IP by evaluating the energy difference between the neutral and the ionized nanoclusters after finding the most stable state for the ionized nanoclusters using the optimization procedure. The electron affinity (EA) is defined as the energy released when an electron is added to a neutral nanocluster. We have determined the adiabatic EA by finding the energy difference between the neutral and the anionic nanocluster. The anionic nanocluster has been relaxed to its most stable state. The adiabatic and vertical IPs and EAs for the most stable ones are included in Table 4.

Table 4 Adiabatic and vertical ionization potential (IP) and electron affinity (EA) in eV for all the most stable configurations of Zn m Se n (m + n = 2–4) nanoclusters
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The linear SeZnSe (Fig. 1b) geometry has the highest value of adiabatic IP and vertical IP as well as adiabatic and vertical EA. (Sanville et al. (2006) have reported the vertical IP for rhombus Zn2Se2 structure (Fig. 1e) as 8.11 eV, which is very close to our calculated value of 8.03 eV. The linear Zn3Se configuration (Fig. 1f) has the lowest value of adiabatic and vertical IP, while linear ZnSeZn structure (Fig. 1c) has the lowest value of adiabatic EA and the lowest value of vertical EA with linear Zn3Se (Fig. 1f) configuration. A zigzag behavior in the variation of both IP and EA with nanocluster size is observed. Neither experimental data nor earlier calculation is available for comparison of IP and EA.

Charge on atoms and dipole moment of nanoclusters

The charge on atoms of the most stable geometries of Zn m Se n nanoclusters and their dipole moments are depicted in Table 5. The two-atom linear ZnSe nanocluster (Fig. 1a) has highest dipole moment of 5.95 Debye due to large charge transfer between the atoms. The linear SeZnSe (Fig. 1b), linear ZnSeZn (Fig. 1c), and Zn2Se2 rhombus (Fig. 1e) structures have zero dipole moment due to their symmetry. The linear ZnZnSeZn (Fig. 1f) structure has quite small dipole moment. No experimental data for charge on atoms and dipole moment are available for comparison.

Table 5 Charge on atoms of the most stable configurations of Zn m Se n (m + n = 2–4) nanoclusters and their dipole moment (in Debye units)
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Conclusions

The occurrence of the most stable configurations of the various ZnSe nanoclusters has been established in the present investigations. For Zn m Se n nanoclusters, we have predicted the bond lengths, binding energies, HOMO–LUMO gaps, adiabatic and vertical ionization potentials and electron affinities, charge on atoms and dipole moments.

We observe that the nanocluster containing maximum number of Se atom for the same value of ‘m’ and ‘n’ is most stable in comparison with others. The FBEs of most stable nanoclusters also increase with the nanocluster size. The nanoclusters with high (low) FBEs have large (small) number of Se atoms. The value of HOMO–LUMO gap increases with number of Se atoms, except for the two-atom ZnSe nanocluster. We observe that the IP and EA both show zigzag behavior. The imaginary frequency causes a linear ZnSeSe structure as unstable structure, although it has the highest FBE among the ZnSe2 nanoclusters. The studied most stable nanoclusters need to be grown experimentally.