Surface Effects and Quantum Confinement in Nanosized GaN Clusters

13516

J. Phys. Chem. C 2008, 112, 13516–13523

Surface Effects and Quantum Confinement in Nanosized GaN Clusters: Theoretical Predictions Barbara Brena† and Lars Ojama¨e*,‡ Department of Physics, Uppsala UniVersity, Box 518, SE-751 20 Uppsala, Sweden and Department of Chemistry, IFM, Linko¨ping UniVersity, SE-581 83 Linko¨ping, Sweden ReceiVed: May 31, 2008

The structure and the electronic properties of stoichiometric (GaN)n clusters (with 6 e n e 48) were investigated by means of quantum-chemical hybrid density functional theory (DFT) using the B3LYP functional. Particular emphasis was put on the investigation of the evolution of the physical properties of the clusters as a function of their size. Two types of model clusters were studied. Cage-type structures were found to be the most stable for smaller cluster sizes, whereas for larger sizes conformations cut out from the GaN wurtzite crystal were favorable. The study of the electronic structure shows that the energy gap of the clusters tends to become larger as the dimensions of the clusters increase. The vertical electronic absorption energies were calculated by means of time-dependent (TD) DFT. For such small clusters, probably due to the predominant amount of surface atoms, well-defined quantum confinement effects, as commonly observed in crystalline quantum dots, are not apparent. Introduction Low-dimensional structures such as quantum wires, quantum wells, quantum dots (QD), and nanoparticles of gallium nitride (GaN), a group III-V nitride semiconductor, have attracted a growing interest in recent years due to a vast range of promising applications as novel optical and electronic devices.1 The small dimensions strongly affect the electronic properties of these nanostructures, which exhibit novel behavior compared to the corresponding bulk material. Their physical properties are moreover dependent on their size and structure and generally can be opportunely tuned. QDs are usually considered zero dimensional structures, where the electrons are confined in three dimensions, and are characterized by a discrete and atomiclike density of states.2 GaN QDs with dimensions on the order of only few nanometers, have been successfully produced in recent years by several methods1,3-6 and have been proposed for a plethora of technological applications. The latter vary from blue-light emitting diodes,7 to photodetectors and quantum dot lasers, to fluorescent bioprobes for biological applications.8 There is still a lack of knowledge about the evolution of the properties of the nanoparticles as a function of their size, by passing from the molecular limit of aggregates composed by only few atoms to the crystalline QD. In this context, theoretical investigations can play a crucial role. QDs are nanoparticles that preserve the crystalline structure of the bulk materials. By reducing the magnitude, the atoms lying on the surface of the particle become a more and more significant fraction of the total atoms, and in the smallest QD the fraction of surface atoms can, for example, be about 20%. In smaller particles, the amount of surface atoms can even constitute the major part, and generally the morphology of the cluster is expected to differ from the bulk material.9 The transition from a cluster-type structure to a crystalline ordering depends on the size of the * Corresponding author phone: +46-13-281380; fax: +46-13-281399, e-mail: [email protected]. † Uppsala University. ‡ Linko ¨ ping University.

nanostructure, and in general the physical properties of the clusters can be different from those of the QD. The purpose of this work is to study, by means of calculations based on hybrid density functional theory (DFT), the geometrical order and the evolution of the physical properties of isolated stoichoimetric (GaN)n clusters (where n indicates the number of GaN units that compose each cluster) as a function of their size and conformation. Bulk GaN presents two distinct crystal arrangements, depending on the growth conditions: the hexagonal wurtzite, which is the most commonly grown, and the cubic zinc blende. QDs have been fabricated in recent years with both types of crystal structures.6 Some theoretical works have been dedicated in recent years to the determination of the structure and of the properties of very small (GaN)n clusters, reaching, to the best of our knowledge, a maximum of 12 GaN units.10-15 In this work we have focused on model clusters derived from a wurtzite type GaN, with n varying from 6 to 48, to be able to explore the change from the molecular level almost to the crystalline ordering. The larger clusters we have studied, such as (GaN)32, (GaN)36, and (GaN)48, measure more than 1 nm, approaching the dimensions of some of the smallest experimentally synthesized QDs (for instance, in ref 3, QDs with a diameter of 30 ( 12 Å were synthesized; and in ref 16, QDs with heights on the order of a few nanometers were grown). All the clusters derived from the wurtzite crystal were compared to more open, surface dominated clusters, so-called cage-type structures. The electronic structures of the larger clusters were finally compared to the GaN bulk. Computational Details The cluster calculations were carried out using the Gaussian 03 code17 at the DFT level and employing the hybrid functional B3LYP.18 The Lanl2DZ19,20 basis set was used for the geometry optimizations of clusters sizes up to (GaN)32. The larger clusters (GaN)36 and (GaN)48 were optimized using a different basis set. It consisted of large core pseudopotentials for Ga20 and N21,22

10.1021/jp8048179 CCC: $40.75  2008 American Chemical Society Published on Web 08/12/2008

Nanosized GaN Clusters: Theoretical Predictions and a split-valence basis set for N.23 To obtain a split-valence basis set for Ga of a similar efficient type (sp) to that of N, we started from the two valence sp shells of the 3-21G24 basis set and reoptimised the exponents of the innermost and the outermost primitive gaussians by minimizing the energy of a (GaN)6 cluster at the Hartree-Fock level. The so-obtained basis had the exponents (0.9627, 0.1884) with contraction coefficients, (-0.5319, 0.1251) (s-type), and (-0.1028, 0.1044) (p-type) for the innermost contracted Gaussian and the exponent 0.1095 for the outermost uncontracted Gaussian. This basis set is referred to as the “HW” basis set in the following, and the corresponding type of basis set has been used in many previous studies of semiconductors.25-28 The electronic structures of the geometry-optimized clusters were computed by employing the 6-31G(d,p)24 basis set. The crystal and electronic structure of the bulk phase was computed by periodic B3LYP computations using the CRYSTAL03 program.22 The cell axes and fractional coordinates were optimized in the P63mc space group using the HW basis set, and the experimental29/computed values for the cell axes a and c were 3.189/3.216 Å and 5.186/5.190 Å, respectively. The optimized fractional z-coordinates were -0.0001 (Ga) and 0.3790 (N), which corresponds to bonded Ga-N distances of 1.960 (near ab plane) and 1.968 Å (parallel to c-axis), to be compared to the experimental values 1.946 and 1.965 Å.29 The time-dependent DFT (TD-DFT) method30 with the HW basis set was used for the clusters to calculate the optical absorption spectra, that is, the excited energies and oscillator strengths. For each cluster, we have chosen two different types of conformations. The first type, which we refer to in this work as crystal-cut and which is presented in Figure 1, was obtained by cutting out the clusters from a gallium nitride bulk crystal of the wurtzite structure. The structural unit of the crystal-cut clusters consists of (GaN)6, a twelve-atom structure. On the (0001) plane of the bulk ordering, (GaN)6 presents a hexagonal section whose corners are alternating Ga and N atoms. These are arranged in a buckled way, as can be seen in Figure 1. The distance between adjacent Ga and N atoms in two different layers varies between 1.97 and 3.22 Å, as indicated in Figure 1. For simplicity, we define this cluster as being composed by two layers each containing three Ga and three N atoms, see Figure 1, where we also indicate the distance between Ga and N atoms belonging to the two different layers. All the crystalcut models we have studied are built by combining more (GaN)6 units. We have selected seven structural groups having different symmetry and different shapes, as shown in Figure 1. The two main conditions we applied to design the clusters were to choose structures with low dipole moment, which generally also implies highly symmetrical shapes, and to minimize the surface area. The second type of conformation we have taken into account is cage-like, that is, all the atoms are situated at the surface. These clusters were chosen to have the same size, that is, they were formed by the same amount of Ga and N atoms, as the crystal-cut models. For the cage-type clusters of sizes 6 to 36, we have used as starting points the most stable (AlN)n cages published by Wu et al.,31 who had designed and characterized (AlN)n cages with n varying between 2 and 41. The most stable cages have alternate Al-N bonding and are formed exclusively by four- and six-atom rings,31 in analogy to the five- and sixatom rings composing the fullerene molecules. We have substituted Ga atoms to the Al atoms in the cages of ref 31 and have geometry-optimized these new structures. The cage 48 was not obtained from ref 31 but was designed in the present work following the same structural criteria. We have compared the

J. Phys. Chem. C, Vol. 112, No. 35, 2008 13517

Figure 1. Crystal-cut clusters before geometry optimization. Clusters are represented in top view, or parallel to the 0001 plane of the original wurtzite crystal, and in side view, or orthogonal to the same plane. The chosen shapes are gathered in seven groups. Each cluster is composed by a number of layers varying from 2 to 4.

energetic stabilities and the geometric and electronic properties of the two types of structures. The (GaN)6 cluster, which can be considered as the building block of our crystal-cut conformations, can be viewed simultaneously as a crystal-cut model and as a cage model. Test calculations were performed for the crystal-cut and cage clusters with 10 formula units in order to validate the quality of the basis sets used. Comparing the difference in B3LYP energy between the (GaN)10 crystal-cut and cage structures, the cage was lower in energy by 55 kJ/mol using the LanL2DZ basis set and by 52 kJ/mol using the HW basis set at the LanL2DZ-optimized energy (if the geometry was optimized using the HW basis then the B3LYP/HW energy difference shifted only slightly to 53 kJ/mol). The 6-31G(d,p) energy difference was lower, 29 kJ/mol, whereas the difference using the 6-311++G(2d,2p) basis set was 54 kJ/mol. The geometries were very similar irrespective of whether the LanL2DZ or HW basis was used, for example, average Ga-N distances of 1.90 or 1.92 Å in the crystal-cut structures. The computed first excitation energies in the cage/crystal-cut structure were 1.37/ 1.48 eV (HW), 1.25/1.36 eV (LanL2DZ), 1.05/1.04 eV (6-31G(d,p)), and 1.06/1.05 eV (6-311++G(2d,2p)); thus, HW is likely to overestimate the excitation energy. For comparison, the first excitation energy obtained using the CIS(D)32 method and 6-311++G(2d,2p) basis set was 1.23/ 1.19 eV for the cage/crystal-cut structure.

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Figure 3. Side and top view of the larger optimized crystal-cut models: n ) 22; 16, 32; 24, and 36, 48.

Figure 2. Crystal-cut model clusters, after geometry optimization: smaller clusters with n ) 6, 9, 12; 10, 15, 20; 14, 21; and 13, 26 in top and side views.

Results and Discussion Model Clusters. The geometry-optimized structures of the crystal-cut model clusters are shown in Figures 2 and 3. In general, the precise wurtzite arrangement of the crystal-cut conformations was lost during the geometry optimization, although to a different extent for different cluster sizes. Besides the variation from the original shape, the hexagonal arrangement in the plane of the layers was left unaltered, with the exception of clusters 14 and 21 (Figure 2), where the central part assumes a rectangular, almost cubic, form. It is possible to point out some trends for the geometryoptimized structures. The clusters formed by two layers, such as (GaN)10, (GaN)13, (GaN)16 (Figure 2), (GaN)22, and (GaN)24 (Figure 3), evolve into more open or swollen conformations, where the internal volume of the cluster is slightly expanded. These model clusters, together with the (GaN)14 and (GaN)21 clusters discussed above, are the ones that become more different from the original wurtzite shape. Structures with three or four layers, instead, become more compact, in the sense that the distance between atoms belonging to different layers shortens and the layers lose the buckling of the Ga and N atoms characteristic of the crystal form. Examples are the geometryoptimized clusters 6, 9, and 12, clusters 15 and 20, and cluster 21, all shown in Figure 2. A similar tendency is also observed for the large clusters in Figure 3, such as 32, 36, and 48 and in cluster 26 of Figure 2, but the final structure of these larger clusters preserved a degree of buckling in the central part. One important structural difference among the various crystalcut models, which is expected to influence the properties of the clusters, is the ratio between atoms that face the surface (or

surface atoms, for which bonds were broken when the cluster was cut out from the crystal structure) and atoms that lie inside the cluster (or bulk atoms, atoms that preserve their coordination from the bulk). For these clusters the overall percentage of surface atoms is less than 100%. The bulk atoms are all fourcoordinated, whereas the atoms in the cage clusters are, in general, three-coordinated. However, the coordination, that is, the number of the nearest neighbors, is not an unambiguous way to distinguish the two types of atoms. In fact, in several of our crystal-cut model clusters, there are, after structure relaxation, atoms lying on the surface having three, four, or even five neighbors in close contact. The crystal-cut structures with n equal to 26, 32, 36, and 48 contain gallium and nitrogen atoms of distinguishable bulk type. In the clusters 26, 32, and 48, which are formed by four layers, and after the geometry optimization, the bulk type atoms preserve an arrangement of crystal type, whereas the surface atoms assume the same compact arrangement as discussed above for the three- and four-layer clusters. In cluster 26 there are two bulk-type atoms in adjacent layers at a distance of 2.84 Å from each other, whereas surface atoms belonging to adjacent layers are at a distance of about or less than 2 Å from each other. (There are also three pairs of Ga and N atoms of bulk type with interatomic distances of 1.93 Å.) The distance 2.84 Å should be compared to the distances along the c-axis in the bulk crystal structure between Ga and N atoms in adjacent ab sheets, which are 3.22 Å (nonbonded) and 1.97 Å (bonded and nearest neighbors). The same behavior appears in cluster 32, where there are four bulk atoms at 2.87 Å from each other. In cluster 48 there are 12 atoms lying at about 2.51 Å apart of bulk type. Cluster 36, formed by three layers, is extremely flat in the second layer, but a mild undulation is seen in the central part of the external layers. In this case there are six bulk atoms in total. The observed overall behavior could be due to an increased tendency of the bulk atoms to arrange in a way where the coordination geometry becomes more tetrahedral and that

Nanosized GaN Clusters: Theoretical Predictions

J. Phys. Chem. C, Vol. 112, No. 35, 2008 13519

Figure 5. Cohesive energy in eV per GaN unit, as a function of n, the number of GaN units and relative to an isolated GaN unit. Crystal-cut clusters are represented by squares, and cage clusters by circles. Inset: differences in total energy between crystal-cut and cage models.

Figure 4. Geometry-optimized cage structures.

locally resembles the hexagonal crystal structure with buckled alternating ab sheets stacked along the c-axis. The optimized structures of the cage models are shown in Figure 4. These are considerably “open” structures, where all the atoms are three-coordinated. The geometry-optimized GaN cages are slightly larger than the AlN cages31 but basically retain the same shape. Energetic Aspects. The relative stabilities of the model clusters were investigated. The binding energy was computed as the difference between total energy of the (GaN)n cluster divided by n and the total energy of a GaN monomer, which provides an estimation of the cohesive energy per GaN unit. The optimization of an isolated GaN monomer, performed with the same methods employed for the GaN clusters, has evidenced that the most stable structure has a triplet spin configuration, in agreement with ref 13. The binding energy is expected to depend on both size and shape. The results for the relative binding energy are plotted in Figure 5. There is a general tendency toward lower relative binding energies for larger clusters (the absolute value increases with the cluster size), and this is evident in both types of model structures. The crystal-cut and the cage relative binding energies are very close for all values of n lower than 24. In particular, Figure 5 indicates that below this size the crystal-cut structures have a slightly lower relative binding energy, apart from the three cases n ) 9, 10, and 20, where the values for the crystalcut and for the cage are similar. For values of n larger than 24 this trend changes, and for the larger clusters it is evident that the bulk models have a lower relative binding energy. In the inset of Figure 5 we have plotted the difference in total energy between cage and crystal-cut models of equal size, as a function of n. In this diagram, positive values show at what sizes the cage models are more stable than the crystal-cut, and negative

values indicate where the crystal-cut models are more stable. It is interesting to observe that all the crystal-cut GaN clusters with n higher than 24, where the crystal-cut configurations become more stable than the cage ones of the same size, are characterized by the presence of the bulk atoms. Although in the case of (GaN)n clusters the theoretical investigations have been limited so far to rather small structures, there are studies on stoichiometric clusters of aluminum nitride, (AlN)n, a group III-V semiconductor like GaN, where larger clusters were taken into account.31,33,34 In their global geometry optimization of (AlN)n clusters with 1 e n e 100 performed by the basin-hopping (BH) method, Costales et al.33 have found that conformations characterized by a cage morphology are energetically most stable at certain cluster sizes. Moreover Costales et al. have found that the most-stable conformations for the larger clusters (those with n g 35) present a mixture of bulk and surface elements.33 Electronic Properties: Density of States and Band Gap. The evolution of the electronic properties as a function of the cluster size was analyzed, taking into consideration the influence from their different shapes. The total density of states (DOS) spectra of all crystal-cut models were calculated, and the results are shown in Figure 6. Occupied and unoccupied levels are reported over an energy range between -15.0 and 0.0 eV, which includes the valence region. On top of the DOS of each cluster, two vertical lines indicate, for comparison, the positions of the HOMO (highest occupied molecular orbital) at -6.0 eV and of the LUMO (lowest unoccupied molecular orbital) at -3.6 eV of (GaN)48, which is the largest cluster we have considered. The energies of HOMO and LUMO of small clusters with 6 < n < 21 and n ) 26 lie almost all at higher and, respectively, lower energy than those of (GaN)48. The DOS values of the larger cage clusters containing a fraction of bulk atomsswith n ) 26, 32, 36, and 48sare compared with the DOS of the cage models of equal size, plotted in the upper part of the diagram of Figure 6. The positions of the HOMO and LUMO of all these cage structures is nearly constant for the considered sizes, their LUMOs lying at higher energy than that of (GaN)48; these clusters have, consequentially, a larger energy gap than the crystal-cut models. The “band gap” of our model structures was calculated in this work as the difference between the energies of the HOMO and LUMO. The band gaps for each cluster and the related values of the HOMO and LUMO energies are reported in Figure 7. In all cases, as stated above for the larger cage models, the calculated LUMO energies for the cage-type structures are

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Figure 8. Band gap of the crystal-cut clusters as a function of the number of GaN units per cluster; the clusters are grouped according to their shapes, following the scheme of Figure 1.

Figure 6. DOS of the crystal-cut structures (for all considered n values between 6 and 48) and the DOS of cage clusters for n equal to 26, 32, 36 and 48. The MO levels are convoluted with Gaussian curves of 0.3 eV of full width at half-maximum (fwhm) in the construction of these spectra. The vertical lines reported in each diagram indicate the positions of the HOMO and the LUMO in the crystal-cut cluster of size 48.

Figure 7. The HOMO and LUMO energies for each cluster. Inset: band gap (enegy difference between HOMO and LUMO) for crystalcut and cage clusters.

slightly higher, and those of the HOMO slightly lower, than those obtained for the crystal-cut models. The resulting energy gaps for the cage structures are generally larger than those of the crystal-cut models. It is possible to observe that the energy gap increases with the size of the clusters in both types of models, as is evidenced by the insert in Figure 7, which shows the variation of the band gap as a function of n. To further analyze the dependence on the specific morphology of the crystal-cut clusters, the band gaps gathered according to the different shapes (see Figure 1) are plotted in Figure 8. One can observe in the groups of clusters such as 10, 15, and 20, or 24, 36, and 48, consisting, respectively, of two, three, and four

Figure 9. The DOS of bulk GaN was computed using the modified 6-31G(d,p)′ basis set and compared with the DOS of the crystal-cut (GaN)32 cluster computed using the 6-31G(d,p)′ and the ordinary 6-31G(d,p) basis sets. The molecular orbital energy levels of the 32 cluster were convoluted with Gaussian curves of fwhm of 0.5 eV.

layers, that the band gap of the model formed by three layers is lower than that of the other two. In QDs the energy difference between occupied and unoccupied electronic levels, or the band gap, is expected to undergo considerable variations due to the quantum confinement, and specifically it has been shown in several works that in QD the band gap increases when the dimensions of the QD decrease (see, for example, ref 35). This behavior is not necessarily maintained when the clusters are so small that they do not exhibit the same crystalline arrangement as the bulk material. We have also compared the DOS of the crystal-cut cluster of size 32 with the DOS of the bulk GaN in the wurtzite crystal (Figure 9). The calculation of the bulk DOS was performed using the CRYSTAL03 program22 and the 6-31G(d,p) basis set. The basis set had to be slightly modified in order to achieve SCF convergence in the bulk calculation, in that the exponent of the most diffuse s basis function of Ga was increased from 0.072 to 0.100. The modified 6-31G(d,p) basis set was also used to calculate a DOS of the crystal-cut (GaN)32 cluster and was compared with the DOS obtained using the ordinary 6-31G(d,p) basis set. Both curves are plotted in Figure 9, where they were aligned with the HOMO of the GaN bulk at about -4.5 eV. The DOS obtained with the two basis sets for the cluster are practically indistinguishable in the valence band, and they differ only at energies above 0 eV. The calculated band gap of the bulk (3.75 eV) is much larger that that of the crystalcut cluster, but the overall shape of the occupied DOS is generally similar. The experimental energy gap of the wurtzite crystal of GaN has been estimated to be about 3.50 or 3.60 eV.36 A possible explanation for the smaller energy gap in the

Nanosized GaN Clusters: Theoretical Predictions

J. Phys. Chem. C, Vol. 112, No. 35, 2008 13521 TABLE 2: Excitation Energies of the (GaN)n Clusters (n ) 6, 9, 10, 12, 20, 26, 32, 36, 48) Calculated with TD-DFT and the HW Basis Seta crystal-cut

cage

excitation energy eV

nm

f

eV

nm

f

9

1.47 3.18 3.49 3.56 3.56 4.02 1.48 2.41 2.44 2.70 2.82 3.41 1.73 3.20 3.20 3.72 3.95 3.95 1.75 2.48 3.35 3.44 3.47 3.49 1.79 2.27 2.28 2.35 3.11 3.29 2.01 2.53 2.81 3.14 3.37 3.60 1.92 2.27 2.29 2.99 3.27 3.27 2.33 2.45 2.45 2.48 3.01 3.41

842 390 355 349 349 309 836 515 509 460 440 364 716 387 387 333 313 313 708 500 369 360 357 354 693 528 528 527 398 376 616 490 441 394 367 343 645 546 541 414 380 380 532 508 508 599 412 364

37 220 948 240 239 112 15 52 52 65 78 166 17 139 139 79 259 259 0 204 256 133 123 169 0 80 80 138 160 125 0 199 266 151 159 160 0 246 271 229 515 515 0 208 206 194 115 250

1.26 2.98 3.00 3.00 3.36 3.55 1.37 2.59 2.59 3.10 3.31 3.47 2.44 4.06 4.06 4.06 4.24 4.24 2.40 3.30 3.50 3.60 3.65 3.99 2.77 3.63 3.65 3.82 4.10 4.07 2.86 3.67 3.70 3.84 3.88 3.99 3.06 3.75 3.75 3.75 4.39 4.39 2.86 3.66 3.65 3.82 4.05 4.05

986 416 414 414 369 349 907 478 478 400 375 357 508 305 305 305 293 293 517 376 354 347 339 311 447 342 340 325 302 304 433 337 334 323 319 311 405 330 330 330 282 282 434 339 339 324 306 306

0 67 158 160 117 195 4 49 49 58 47 282 3 111 106 113 54 54 1 42 21 44 38 58 0 18 19 29 98 22 0 44 51 24 24 25 0 91 91 91 385 385 0 95 95 41 345 344

10

Figure 10. Absorption spectra obtained from TD-DFT excitation energies (eV and nm) and oscillator strengths for selected crystal-cut and cage clusters. Each peak was convoluted with a Gaussian curve of 0.1 eV fwhm. The absorption curves were normalized to set the maximum intensity to unity.

12

TABLE 1: Band Gap of (GaN)n Clusters Calculated with the 6-31G(d,p) and with the HW Basis Sets

20

n

crystal-cut/6-31G (d.p) (eV)

cage/6-31G (d.p) (eV)

crystal-cut/HW (eV)

cage/HW (eV)

6 9 10 12 13 14 15 16 20 21 22 24 26 32 36 48

1.33 1.76 1.78 1.86 1.89 1.93 1.66 2.06 1.87 1.89 2.24 2.29 1.91 2.08 2.04 2.38

1.84 1.78 2.58 1.97 2.28 2.59 2.47 2.45 2.67 2.48 2.64 2.76 2.79 2.97 2.80

2.09 2.24 2.25 2.44 2.23 2.36 2.11 2.41 2.37 2.41 2.64 2.76 2.31 2.54 2.46 2.82

2.10 2.11 3.14 2.38 2.68 3.15 3.00 3.00 3.27 3.08 3.22 3.35 3.42 3.60 3.39

cluster is probably to be found in the large influence of the surface and in the surface states. Absorption Spectra. TD-DFT has been used to calculate the vertical electronic absorption energies at optical wavelengths for the GaN model clusters (Figure 10). TD-DFT is an established method for computing the electronic valence excitation energies and the related oscillator strengths for small and large molecules,30,37,38 and it is widely used to compute optical properties of nanostructures.39,40 For example, absorption spectra have been computed with this method in the case of silanebridged silicon nanoclusters41 and in boron-nitride fullerenelike clusters, which are structures that are analogous to our cage clusters.42 In the present study the excitation energies were calculated using the HW basis set; the use of a small basis set allowed us to extend our calculation to a large number of excitations for all the clusters (at least 100 excited states of singlet character were considered for each cluster, reaching in every case at least 4.2 eV in excitation energy). A comparison with the TD-DFT spectra obtained for crystal-cut clusters with n equal to 6, 9, and 12, with the 6-31G(d,p) basis set, show that there is an energy shift of about one-half of an electronvolt between these results and those obtained with the HW basis

excitation energy

n

26

32

36

48

a The first excitation energy corresponding to the HOMO f LUMO transition and the five lowest excitation energies in the interval 1.5 to 4.2 eV with high oscillator strength (f) are listed. f was multiplied by a factor of 104. For the (GaN)6 cluster, which is simultaneously of the crystal-cut and cage types, the first excitation energy is 1.41 eV (873 nm) with f ) 0, and the five lowest excitations energies with nonzero f are two peaks at 2.23 eV (f ) 127), two peaks at 3.17 eV (f ) 233), and one peak at 3.56 eV (f ) 505).

set, where the HW results have the higher energies. The first excitation energies for these clusters are 1.33, 1.76, and 1.78 eV, respectively, obtained with the 6-31G(d,p) basis set, and 2.09, 2.24, and 2.25 eV, obtained with the HW basis set. This indicates that the absorption spectra we present here should be shifted toward lower energies by about one-half of an electron-

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Figure 11. Isodensity surfaces of the squares of the third highest occupied molecular orbital (HOMO - 3) (left) and of the lowest unoccupied molecular orbital (LUMO) (right), which are mainly responsible for the lowest nonzero intensity peak, in the (GaN)48 cluster.

volt when compared with the experimental results. The same difference is observed in the band gap obtained for the same clusters with the two different basis sets (Table 1). The simulated spectra obtained for cage and crystal-cut type clusters of different sizes (6, 9, 10, 12, 20, 26, 32, 36, and 48) are presented in Figure 10. The main excitation energies and the related oscillator strengths are also listed in Table 2. We report all excitations falling in the energy interval in terms of wavelengths between 700 and 250 nm (corresponding to 1.8-4.9 eV). This energy interval includes the visible spectrum (400-700 nm). The absorption spectra of the crystal-cut clusters and those of the cage clusters present evident differences. The larger crystal-cut clusters are characterized by absorption features at lower energies that do not show up for the cage clusters. These peaks are located at about 2.4-2.8 eV, which means at about 500 and 450 nm, in the visible region between the blue and the green light. Further high intensity peaks are present for all these clusters between 3 and 4 eV, which is between 400 nm (violet light) and 300 nm (out of the visible window). For the very small crystal-cut clusters (for n between 6 and 12) the most significant absorption lines are found at energies higher than 3 eV. These are very similar to the smaller cage clusters. In the cage models for n above 12, the whole spectra begin above or close to 4 eV (or 310 nm), outside of the visible region. To understand if excitations in specific regions of the clusters are responsible for the different results obtained for the larger crystal-cut clusters, the electronic energy levels that are involved in the most intense absorption peaks within the visible window were identified. With reference to the results shown in Figure 10, the findings are as follows. The spectrum of (GaN)26 presents two peaks at 2.35 and 3.56 eV that correspond to transitions from states located mostly at the surface into other surface states. (GaN)32 has two main peaks below 3 eV. The peak at 2.53 eV is due in large percentage to a transition between surface states (mostly from the HOMO - 7 to the LUMO energy levels, which the computations show to be located mostly on surface atoms). Another peak at 2.81 eV corresponds to a transition from a bulk state (HOMO - 11) to the surface state LUMO. In (GaN)36 the low energy peak at 2.29 eV is mostly due to transitions from surface states into the LUMO, which has components spread all over the cluster but predominantly on surface atoms. Also at higher energy, for the peaks at 2.9 and at 3.27 eV, the transitions mostly take place between states localized on the surface of the particle. In (GaN)48 the lowest lying peak at 2.45

eV is formed by excitations between surface states (HOMO-3 to LUMO). The orbitals of (GaN)48 involved in these transitions are shown in Figure 11. The analysis shows that most of the absorption excitations in these nanostructures are dominated by transitions between surface states. However, the more compact conformations of the crystal-cut particles, as compared to the cage structures, and the smaller band gap favor the appearance of lower-energy absorption peaks. Several luminescence measurements have been carried out on GaN nanoparticles (powders and quantum dots). The emission spectra can be expected to present significant differences as compared to the absorption spectra. In general, the TD-DFT calculations in the ground-state can only approximately represent the emission spectra, since excited-state relaxation effects are neglected in the calculations. However, interestingly, in a photoluminescence study on GaN nanopowders, Kudrawiec et al. found emission peaks at 2.8 and 2.0 eV for GaN grains of average size of 11 nm,43 whereas Nyk et al., for GaN grains of average size of 15 nm, found that a peak at 2.2 eV dominates the emission spectrum.44 Similar results were obtained by Gao et al. for small spherical wurtzite nanoparticles with diameter of 5-8 nm, which showed blue-emission photoluminescence peaks between 2.62 and 2.85 eV.6 These photoluminescence intensities in the range between 2 and 3 eV could be related to the absorption peaks we have obtained for our larger clusters in the same interval. It is argued by the authors of these studies that surface states can have a large influence in the optical properties of the small particles, and therefore that the outcome of the PL experiments can deviate from the typical quantum confinement of the QD. Photoluminescence spectra of GaN QD are generally characterized by a peak corresponding to the band gap of the particle, and shifts in the position of these peaks give information about quantum confinement effects in relation to the size of the particles; an increase of the band gap magnitude with decreasing size of the QD is generally observed.1,3,6 Instead, what we find is that, for the nanometer and sub-nanometer particles considered in this work, the band gap is considerably smaller than the one of the bulk GaN structure (both wurtzite and zinc blende), which opposes the typical quantum confinement phenomena of the QD, in agreement with the observations in ref 6. We found absorption peaks in the range of 2.4-2.8 eV not directly related to the HOMO f LUMO excitations, but involving mostly surface states. The surface states seem to dominate the optical properties of the clusters we have considered. In addition, it follows from

Nanosized GaN Clusters: Theoretical Predictions the computations that the excitation energies of the cage structures differ significantly from the experimental measurements for the GaN nanoparticles. In the larger cages, with n above 12, they are all at wavelengths well below the visible edge of 390 nm. It can therefore be concluded that GaN nanoparticles are not likely to assume cage-type conformations. Summary Two types of (GaN)n stoichiometric model clusters with 6 e n e 48 were considered in this work, either cut out from a wurtzite-type GaN (crystal-cut clusters) or in a cage-like shape (cage clusters). Their electronic properties were studied as a function of their structures and of their sizes. The considerable variation in size of our clusters allowed us to explore the cluster behavior reaching almost the crystalline ordering. The total energies of the particles indicate that smaller clusters (up to n ) 24) prefer a cage-type arrangement, whereas the larger clusters favor a more compact crystal-cut structure, with a mixture of surface and bulk atoms. In the study of the electronic structure, we have not observed the typical quantum confinement effects that characterize the QD. The electronic structure of the clusters is influenced by the surface atoms, of which all the cage, and the smaller of the crystal-cut clusters, are composed. The clusters with n equal to 26, 32, 36, and 48, which possess more typical bulk atoms, arrange themselves locally in a way that is reminiscent of the hexagonal ordering after the geometry optimization. However, even in these structures the total number of bulk atoms still represents only a small fraction of the total atoms. The investigation of computed absorption spectra for the present clusters may have bearings on the analysis of photoluminescence spectra of GaN powders, where the surface effects are dominant. Acknowledgment. Support from the Swedish Foundation for Strategic Research SSF, the Swedish Research Council VR, and the Swedish Supercomputer Centres SNIC is gratefully acknowledged. Supporting Information Available: The Cartesian coordinates of the clusters in Figures 2-4. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Ramvall, P.; Tanaka, S.; Nomura, S.; Riblet, P.; Aoyagi, Y. Appl. Phys. Lett. 1998, 73, 1104. (2) Gammon, D. Nature 2000, 405, 899. (3) Micic, O. I.; Ahrenkiel, S. P.; Bertram, D. A. J.; Nozik, A. J. Appl. Phys. Lett. 1999, 75, 478. (4) Arakawa, Y.; Kako, S. Phys. Stat. Sol. A 2006, 203, 3512. (5) Qiu, H.; Cao, C.; Zhu, H. Mater. Sci. Eng. B 2007, 136, 33. (6) Gao, Y. G.; Chen, X. L.; Li, J. Y.; Lan, Y. C.; Liang, J. K. Appl. Phys. A: Mater. Sci. Process. 2000, 71, 229. (7) Tanaka, S.; Lee, J.-S.; Ramvall, P.; Okagawa, H. Jpn. J. Appl. Phys. 2003, 42, L885–L887. (8) Neogi, A.; Li, J.; Neogi, P. B.; Sarkar, A.; Morkoc, H. Electron. Lett. 2004, 40, 1605. (9) Schmitt-Rink, S.; Miller, D. A. B.; Chemla, D. S. Phys. ReV. B 1987, 35, 8113. (10) BelBruno, J. J. Heteroatom Chem. 2000, 11, 281. (11) Kandalam, A. K.; Blanco, M. A.; Pandey, R. J. Phys. Chem. B 2001, 105, 6080.

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