Decay dynamics of ultraviolet photoluminescence in ZnO nanocrystals

ARTICLE IN PRESS

Journal of Luminescence 126 (2007) 257–262 www.elsevier.com/locate/jlumin

Decay dynamics of ultraviolet photoluminescence in ZnO nanocrystals Sekika Yamamoto, Hiroyuki Yano, Tomobumi Mishina, Jun’ichiro Nakahara Division of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan Received 15 September 2005; received in revised form 27 July 2006; accepted 8 August 2006 Available online 22 September 2006

Abstract Time resolved photoluminescence (PL) measurements at low temperature are performed on colloidal ZnO nanocrystals dispersed in t-butanol. Considering the particle size dependence of the decay times we conclude that the luminescence is composed of two trap related emissions one of which undergoes lifetime shortening due to a non-radiative process. Initial fast shift of the spectrum within 30 ps is observed and it is interpreted as a fast hole cooling just after the excitation. r 2006 Elsevier B.V. All rights reserved. PACS: 78.67.n; 78.67.Bf Keywords: ZnO; Nanocrystal; Lifetime; Surface modification

1. Introduction Semiconductor nanocrystals are of much interest over two decades because of their unique physical properties resulting from the modification of the electronic states due to the quantum confinement effect [1–5]. Their properties have been found to be greatly affected by surface or interfacial properties such as trapping centers. Among the nanocrystal systems chemically prepared colloidal nanocrystals are of very much importance because their surface properties can be modified through surface passivation by a chemical process or by simply changing the solvent. Modifying surface or interfacial condition is crucial to understand the effect of surface states on the nanocrystal properties. On the other hand, ZnO is a very important material as a possible candidate for UV emitting devices because of its large band gap ð3:37 eVÞ and large exciton binding energy ð60 meVÞ [6]. Meulenkamp reported on the preparation of ZnO nanocrystals in ethanol with narrow size dispersion and intense bandgap UV emission [7], and discussed the size dependence of photo-dissolution of the particles [8]. In addition to the bandgap UV emission the ZnO nanocrystals show a broad green luminescence. Corresponding author. Tel.: +81 11 706 4428; fax: +81 11 706 4926.

E-mail address: [email protected] (S. Yamamoto). 0022-2313/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2006.08.055

The green luminescence is also observed in bulk ZnO but its intensity is emphasized in nanocrystals probably because of its large surface to volume ratio. Dijken et al. investigated the size dependence of these luminescence behaviors and concluded that the broad green luminescence is associated with the electronic transition from shallowly trapped electron to deeply trapped hole [9,10]. To elucidate the carrier dynamics in ZnO nanocrystals Cavaleri et al. measured the transient absorption by the photo-generated carriers in ZnO nanocrystals dispersed in ethanol and reported that the absorption rise time inversely depends on the particle size [11]. Bauer et al. reported on the transient absorption of ZnO nanocrystal thin film and revealed the ultrafast carrier dynamics. They observed multi-exponential absorption decay and assigned each part to carrier relaxation, exciton recombination, and trap emission [12]. However, to our knowledge, there are only few reports on the band gap luminescence decay at low temperatures. In this paper, we will report the result for the time resolved luminescence properties of colloidally prepared ZnO nanocrystals at low temperatures. 2. Experimental procedure Zinc oxide nanocrystals were prepared following the method reported in the reference without modification [7].

ARTICLE IN PRESS S. Yamamoto et al. / Journal of Luminescence 126 (2007) 257–262

ments determined by measuring a time profile of the excitation pulse was 15 ps. The samples were cooled to 30 K by using a helium gas flow type cryostat. For continuous wave PL measurements, a He–Cd laser of 3.815 eV was used for the excitation and a 150 W Xe-lamp was used as a light source for the absorption measurements. 3. Experimental results 3.1. CW measurements at room temperature Fig. 1 shows a typical PL spectrum at 295 K for the sample with average diameter of 3.2 nm excited by He–Cd laser at 3.815 eV. An absorption spectrum of the same sample is shown together. On PL spectrum we see small peaks at 3.67 and 3.74 eV. These peaks are not seen in the spectrum excited at around 4.7 eV and the separations of these peaks from the excitation energy are multiples of LO phonon energy ð72 meVÞ. Therefore, these peaks are attributed to resonant Raman scattering or resonant luminescence. Also seen on the spectrum is a small peak at 3.45 eV, which we assign to a Raman scattering from t-butanol [20,21] because this peak is also observed in a neat t-butanol liquid. We see a Stokes shift of 160 meV between the luminescence peak and the absorption peak. The Stokes shift for the samples we measured did not significantly depend on average particle size. A considerable number of studies have been made on the Stokes shift in quantum confined nanocrystals such as CdSe, CdS, GaN, CuCl and

Optical Density (arb. units)

The size of the ZnO crystals was determined from optical absorption edge using an empirical formula reported in Ref. [7]. The shape of the nanocrystals prepared by essentially the same method has been reported to be spherical or somewhat prolate [12–14]. Once the ZnO nanocrystals grew to a desired size, we kept the colloidal suspension at 5  C by which the particle growth was negligibly slowed. We prepared several specimens of the average diameter of 2.9, 3.2, 3.5, and 5.0 nm. From X-ray diffraction results we confirmed that the nanocrystals have a wurtzite structure. Zinc oxide is known as one of most efficient photocatalytic materials. Dijken et al. reported on the photocatalytic properties of ZnO nanocrystal [15]. After an electron–hole pair is generated in a ZnO nanocrystal by illumination, the photo-generated hole is consumed in oxidizing surrounding molecules such as solvent or remaining reaction byproducts. The remaining electron can occupy specific kind of surface state, which leads to the modification of the luminescence properties such as quenching of the visible green emission or blue shift of the UV spectrum. Such charging effects have been observed in other nanocrystal systems such as CdS and CdSe [16–19]. To suppress the charging effects we chose 2-methyl-2-propanol (t-butanol) as a solvent because it was known to be tolerant for oxidization. After the preparation, the ZnO nanocrystals were precipitated by mixing 4 ml of colloidal solution with 8 ml heptane followed by centrifugation. The supernatant was decanted and the precipitated nanocrystals were redispersed in 8212 ml t-butanol. After changing the solvent from ethanol to t-butanol, the luminescence intensity of the visible luminescence was increased and the tolerance for the degradation induced by illumination was improved. The sample dispersed in t-butanol kept its optical properties at low temperatures under the laser irradiation for over 30 min while those dispersed in ethanol changed its PL decay time after few minutes irradiation. The redispersed suspension became a little blurred which meant some kind of aggregations remained in the suspension. These aggregations could not be precipitated by 3000-rpm centrifugation. However, the luminescence properties were not changed from those dispersed in ethanol except for the improvement of the resistivity against the laser irradiation. The excitation source used in the time resolved PL measurements was a Ti:Sapphire laser of 80 MHz repetition which has a pulse duration of 100 fs. The laser output of 1.57 eV was focused onto a 0.5 mm thick BBO crystal to generate frequency tripled light pulse used for the excitation. The excitation power was 10 mW and it was focused to a spot of 100 mm diameter. By this weak excitation, the expectation number of electron hole pair generated in a single particle by a single pulse is estimated to be 105 . A synchroscan streak camera was used for the time resolved spectroscopy. The luminescence was focused onto the entrance slit of a 30 cm polychromator equipped with the streak camera. Time resolution of our measure-

PL Intensity (arb. units)

258

= 3.2 nm T = 295 K

3.5

4.0 Photon Energy (eV)

4.5

Fig. 1. Room temperature photoluminescence and absorption spectrum for the sample with hDi ¼ 3:2 nm. The small peaks seen on the PL spectrum are Raman scattering of ZnO and t-butanol. There can be seen a Stokes shift of 160 meV.

ARTICLE IN PRESS S. Yamamoto et al. / Journal of Luminescence 126 (2007) 257–262

3.2. Time resolved measurements at low temperature Fig. 3 shows the time resolved PL spectra for the samples with average diameter hDi of 3.5 nm at 30 K. The spectra are taken at t ¼ 0, 17.2, 34.4, and 292 ps in order indicated

(K)

Intensity (logscale)

100

50

30

Ea = 120 meV

10

20

30

40

1000/T (1/K) Fig. 2. Temperature dependence of PL intensity. The PL intensity is almost constant below 100 K. The solid line is a exponential fitting with an activation energy of 120 meV.

= 3.5 nm T = 30 K

PL Peak Energy (eV)

3.58

Normalized Intensity

so on. In these studies the Stokes shifts were attributed to phonon coupling [22–25], polaron shift [26], surface traps [27,19], dark exciton [28], particle size distribution [28]. Although we have not clarified the cause of the Stokes shift, we think that it is related to some kinds of trap exist at the particle surface. In Fig. 2, temperature dependence of spectrally integrated luminescence intensity is shown. We obtain 120 meV as an activation energy for the thermal quenching of the luminescence as shown in the figure. The luminescence intensity is almost constant at low temperatures below 100 K. This shows that the nonradiative process which causes the thermal quenching is negligible at low temperatures. The PL line width is 80 meV at 30 K and the spectral shape is almost identical at low temperatures. If we take this broadening as a direct reflection of the particle size distribution, it leads to a size dispersion of 20%. This dispersion agrees with that reported for the ZnO samples prepared in the same way [7]. Therefore, we consider that the line width at low temperatures mostly comes from the particle size distribution.

259

3.57

3.56

3.55

3.54 0

200

400

600

Time (ps)

3.4

3.5

3.6 Photon Energy (eV)

3.7

Fig. 3. Time resolved and normalized spectra for hDi ¼ 3:5 nm sample. The spectra are taken at t ¼ 0, 17.2, 34.4, and 292 ps in order indicated by an arrow. The spectrum narrows at high-energy side within 30 ps and then shifts to lower energy. The inset shows the time evolution of the PL peak energy. The peak shifts linearly with time until the luminescence vanishes.

by an arrow. The high-energy component seen from 3.6 to 3.7 eV rapidly decays within 30 ps which results in spectrum narrowing. After the initial fast narrowing, the spectrum gradually shifts to lower energy which appears in the same amount of red shift at high-energy side and low-energy side of the spectrum. The time evolution of the spectrum shift is shown in the inset. The spectrum shifts almost linearly with time and the shift continues until the PL intensity becomes so weak that we cannot evaluate its peak position any more. We think that the spectral diffusion caused by the energy transfer between nanocrystals are negligible because the transfer probability depends on the particle separation as R6 and it is expected to be small in a sparse solution [29]. Fig. 4 shows typical decay curves detected at the same energy for the samples with different average diameters. The detection energy 3.509 eV is higher than the spectrum peak energy for the sample with hDi ¼ 5:0 nm and lower than that for the sample with hDi ¼ 3:5 nm as indicated by an arrow in the inset. The relative intensities of these curves are properly adjusted for clarity. The time dependence of the two decays almost coincide except for the existence of the fastest component in hDi ¼ 5:0 nm sample. This component is observed in all samples at higher energy side of the spectrum and corresponds to the spectrum narrowing seen in Fig. 3. Except for this fastest component two curves have almost the same bi-exponential decay profile. To estimate decay times for the bi-exponential components the curves are fitted by a double or a triple exponential curve depending on the energy position relative to the spectrum peak.

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PL Intensity (logscale)

Normalized PL Intensity

260

=5.0 nm

component before 30 ps is almost at the time resolution limit of our detection system and is not shown in the figure. The decay times for the samples with different average size are shown by different symbols. It is seen in the figure that t1 is almost constant and t2 monotonically decreases with energy. The decay times detected at the same energy for the samples with different average size almost coincide. As mentioned above, the spectral shape at low temperatures reflects the particle size distribution and the selection of the detection energy is equivalent to the selection of particle size. Therefore, Fig. 5 is interpreted as the particle size dependence of the decay times. As a result, the slow spectrum shift seen in Fig. 3 should be interpreted as a reflection of the particle size dependent decay time of the luminescence, not as an energy relaxation of electron hole pair.

=3.5 nm

Edet 3.4

3.5

3.6

Energy (eV)

=5.0 nm

T=30 K Edet=3.509 eV

0

4. Discussions

=3.5 nm

200

4.1. Particle size dependent decay time

400

Time (ps)

Fig. 4. Decay curves at the same energy for different samples. Two decay curves are almost identical except for the steep decay seen in hDi ¼ 5:0 nm sample. This indicates that the selection of the detection energy is equivalent to the selection of particle size.

300

Decay Time (ps)

250

T = 30 K

Fig. 6 shows the decay time t2 plotted against the particle diameter. The diameters are estimated from the emission energies under the assumption that the spectrum peak energies correspond to the emission energies of the particles with average diameter for each sample and the confinement energy of the carriers has a D2 dependence on particle size [3,4]. It is known that the radiative lifetime of an exciton in bulk semiconductors is expressed as t / a3B =V ,

τ2

1000 T = 30 K

200

150

τ2 (ps)

100 τ1 50

0 3.4

3.5

3.6

3.7

3.8

100

Energy (eV) Fig. 5. Decay times for slower two components obtained by the fitting. The average diameters are &:2.9, :3.2, 4:3.5, 5:5.0 nm. The fast decay time t1 is almost constant while t2 monotonically decreases with energy.

2

3

4

5

10

Particle Diameter (nm)

The obtained decay times for the bi-exponential components are shown in Fig. 5. The notation t1 and t2 indicate the fast and slow component of the bi-exponential curve after 30 ps, respectively. The decay time of the fastest

Fig. 6. Particle size dependence of t2 . The notation of the symbols is same as Fig. 5. The particle diameters are estimated from the detection energies under the assumption that the spectrum peak energies correspond to the emission energy of the average diameter for each sample. The dashed line shows D3 dependence.

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where aB is the exciton Bohr radius and V the coherence volume of the exciton motion [3,30–32]. The cubic Bohr radius a3B represents the exciton volume which is inversely connected to the probability that the electron and the hole come to the same position. The coherence volume V is related to so-called giant oscillator strength effect [31,33]. In the case where the nanocrystal size is far larger than aB , as in the case known as the weak confinement regime, the exciton lifetime will change with V 1 . This effect has been observed in CuCl nanocrystals [26,34]. On the other hand, in the strong confinement regime where the nanocrystal radius is smaller than or comparable to the exciton Bohr radius, the lifetime is less sensitive to the particle size because a modification of wave function overlap of electron and hole cancels the volume change [3]. It is reported that the exciton Bohr radius in ZnO is 1:8 nm [35–37] and hence the particle size range we measured is in the strong or intermediate confinement regime. Therefore, the radiative lifetime should be rather size independent. If one of the charge carriers is captured to a surface trap, however, the wave function of the trapped carrier will be somewhat localized and the wave function overlap decreases with increasing size if the spread of the wave function of the trapped carrier is independent of the particle size. In this case, the overlap integral of electron and hole wave function will vary as D3 and therefore, the lifetime will increase with particle size as D3 . The D3 dependence is shown as a dashed line in Fig. 6. It agrees with the experimental values at smaller particle size. The deviation at larger particle size can be understood as the carrier wave function cannot spread beyond the exciton size which is 3:6 nm in ZnO. In other words, the electron hole pair is more like a shallowly trapped exciton rather than a trapped electron and an extended hole at these size range. Considering the particle size dependence of t2 , we assign the slowest part to a trap emission in which the wave function of one carrier is localized and the other is extended in the particle. In ZnO nanocrystals, visible broad luminescence is observed in addition to the UV luminescence and it is assigned to the transition between shallowly trapped electron and a hole deeply trapped to interior oxygen vacancy [9,38]. Cavaleri et al. performed the transient infrared absorption measurements on ZnO nanocrystals dispersed in ethanol and observed efficient sub-ps electron trapping [11]. Bauer et al. also observed the electron trapping to the shallow center in 20 nm size ZnO particles [12]. Therefore, we think that the candidate for the trapped carrier should be an electron. There is some possibility that the decay time t2 is determined by the nonradiative process despite the invariance of the PL intensity at low temperatures. The probability for non-radiative recombination is usually written in the form   DE P ¼ s exp  , kB T where DE is an activation energy. If the non-radiative process dominate the carrier recombination the activation

261

energy for such a process must be very small and s must be very large because the PL intensity is constant below 100 K. We think that it is somewhat improbable but the possibility cannot be excluded. In this case, however, s is expected to be proportional to the surface to volume ratio, and therefore the decay time should have a linear dependence to the particle size [39]. In our result, the size dependence of t2 shows clear D3 dependence at small sizes even though the size range is not so wide. Therefore, we deduce that the decay time t2 mostly reflects the radiative recombination time of the initial state of the UV luminescence. The fast decay time t1 is almost constant at every detection energy and it is around 50 ps. This decay time is comparable to the intrinsic radiative lifetime recently calculated for ZnO nanocrystals [40,41]. However, the spectrum of the fast component has about the same Stokes shift to the slow component, indicating that the fast component is also trap related. Therefore, we tentatively attribute the fast component to another type of nanocrystals of which decay time is restricted by other processes such as Auger recombination. As mentioned above, ZnO is one of very efficient photocatalytic materials and that is why we choose t-butanol as a solvent. If there exist some residual materials such as ethanol, heptane or reaction byproducts which are easily oxidized, they will act as hole scavenger, makes some of ZnO particles negatively charged after initial photo-excitation. The excited state of charged nanocrystals is known to decay by Auger process [42–44]. 4.2. Fast energy relaxation of electron hole pair The fastest decay component before 30 ps is seen in all samples at high-energy side of the spectrum and corresponds to the fast spectrum narrowing seen in Fig. 3. The magnitude of the observed spectrum narrowing is 30 meV which is about the same with the energy splitting of the A, B and C valence band of bulk ZnO [6]. Recently, Xu et al. reported the hole relaxation in CdSe nanocrystals [45]. They observed that the relaxation time of a photogenerated hole in the valence band sub-levels is prolonged due to a phonon-bottleneck effect. They estimated the energy loss rate for CdSe nanocrystal to be 0:2 eV=ps. On the other hand, electron relaxation is considered to be very fast. A similar hole relaxation in the valence band within several ps has been observed in CdSe nanocrystals [46]. Therefore, the fast spectrum narrowing may be attributed to the cooling of the photo generated holes initially distributed in the valence band levels due to the excess energy obtained from the large energy difference between the excitation ð4:6 eVÞ and the emission energy ð3:5 eVÞ. 5. Conclusion The time resolved luminescence measurements reveal the spectrum narrowing and succeeding red shift of UV emission of ZnO nanocrystals. We tentatively assign the spectrum narrowing to the hole cooling in the valence band

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states. The slow decay time t2 follows D3 at particle size smaller than exciton size which leads to the interpretation that the emission is trap related and the slow spectrum shift is a reflection of the particle size dependent radiative lifetime of trap emission. The fast decay time t1 is almost independent of the particle size and we assigned it to the emission from other type of particles which decay time is determined by non-radiative process such as Auger recombination. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

A.L. Efros, A.L. Efros, Sov. Phys. Semicond. 16 (1982) 772. R. Jain, R. Lind, J. Opt. Soc. Am. 73 (1983) 647. L.E. Brus, J. Chem. Phys. 80 (1984) 4403. Y. Kayanuma, Phys. Rev. B 38 (1988) 9797. J.Z. Zhang, J. Phys. Chem. B 104 (2000) 7239. K. Hu¨mmer, Phys. Status Solidi (b) 56 (1973) 249. E.A. Meulenkamp, J. Phys. Chem. B 102 (1998) 5566. E.A. Meulenkamp, J. Phys. Chem. B 102 (1998) 7764. A. van Dijken, E.A. Meulenkamp, D. Vanmaekelbergh, A. Meijerink, J. Lumin. 87–89 (2000) 454. A. van Dijken, E.A. Meulenkamp, D. Vanmaekelbergh, A. Meijerink, J. Lumin. 90 (2000) 123. J.J. Cavaleri, D.E. Skinner, D.P. Colombo Jr., R.M. Bowman, J. Chem. Phys. 103 (1995) 5378. C. Bauer, G. Boschloo, E. Mukhtar, A. Hagfeldt, Chem. Phys. Lett. 387 (2004) 176. A. Wood, M. Giersig, M. Hilgendorff, A. Vilas-Campos, L.M. Liz-Marza´n, P. Mulvaney, Australas J. Chem. 56 (2003) 1051. L. Guo, S. Yang, C. Yang, P. Yu, J. Wang, W. Ge, G.K.L. Wong, Appl. Phys. Lett. 76 (2000) 2901. A. van Dijken, E.A. Meulenkamp, D. Vanmaekelbergh, A. Meijerink, J. Phys. Chem. B 104 (2000) 4355. D.J. Norris, A. Sacra, C.B. Murray, M.G. Bawendi, Phys. Rev. Lett. 72 (1994) 2612. O. Cherniavskaya, L. Chen, M.A. Islam, L. Brus, Nano Lett. 3 (2003) 497. A. Javier, D. Magana, T. Jennings, G.F. Strouse, Appl. Phys. Lett. 83 (2003) 1423. M. Shim, S.V. Shilov, M.S. Braiman, P. Guyot-Sionnest, J. Phys. Chem. B 104 (2000) 1494.

[20] J.L. Green, A.R. Lacey, M.G. Sceats, Chem. Phys. Lett. 137 (1987) 537. [21] P.K. Kipkemboi, A.J. Easteal, Can. J. Chem. 80 (2002) 789. [22] V. Spagnolo, G. Ventruti, G. Scamarcio, Superlattices Microstructures 18 (1995) 113. [23] M. Nirmal, C.B. Murray, M.G. Bawendi, Phys. Rev. B 50 (1994) 2293. [24] M.G. Bawendi, W.L. Wilson, L. Rothberg, P.J. Carroll, T.M. Jedju, M.L. Steigerwald, L.E. Brus, Phys. Rev. Lett. 65 (1990) 1623. [25] S. Kalliakos, X.B. Zhang, T. Taliercio, Appl. Phys. Lett. 80 (2002) 428. [26] T. Itoh, M. Nishijima, A.I. Ekimov, C. Gourdon, A.L. Efros, M. Rosen, Phys. Rev. Lett. 74 (1995) 1645. [27] Y. Masumoto, J. Lumin. 70 (1996) 386. [28] A.L. Efros, M. Rosen, M. Kuno, M. Nirmal, D.J. Norris, M. Bawendi, Phys. Rev. B 54 (1996) 4843. [29] S.A. Crooker, J.A. Hollingsworth, S. Tretiak, V.I. Klimov, Phys. Rev. Lett. 89 (2002) 186802. [30] J. Bellessa, V. Voliotis, R. Grousson, X.L. Wang, M. Ogura, H. Matsuhata, Phys. Rev. B 58 (1998) 9933. [31] G.W. ’t Hooft, W.A.J.A. van der Poel, L.W. Molenkamp, Phys. Rev. B 35 (1987) 8281. [32] B. Gil, A.V. Kavokin, Appl. Phys. Lett. 81 (2002) 748. [33] C.H. Henry, K. Nassau, Phys. Rev. B 1 (1970) 1628. [34] T. Itoh, T. Ikehara, Y. Iwabuchi, J. Lumin. 45 (1990) 29. [35] H. Zhou, H. Alves, D.M. Hofmann, Phys. Status Solidi (b) 229 (2002) 825. [36] M.R. Phillips, O. Gelhausen, E.M. Goldys, Phys. Status Solidi (b) 201 (2004) 229. [37] T. Makino, Y. Segawa, M. Kawasaki, H. Koinuma, Semicond. Sci. Technol. 20 (2005) S78. [38] A. van Dijken, E.A. Meulenkamp, D. Vanmaekelbergh, J. Phys. Chem. B 104 (2000) 1715. [39] A. Nakamura, H. Yamada, T. Tokizaki, Phys. Rev. B 40 (1989) 8585. [40] V.A. Fonoberov, A.A. Balandin, Phys. Rev. B 70 (2004) 195410. [41] V.A. Fonoberov, A.A. Balandin, Appl. Phys. Lett. 85 (2004) 5971. [42] V.I. Klimov, D.W. McBranch, C.A. Leatherdale, M.G. Bawendi, Phys. Rev. B 60 (1999) 13740. [43] V.I. Klimov, J. Phys. Chem. B 104 (2000) 6112. [44] A.L. Efros, M. Rosen, Phys. Rev. Lett. 78 (1997) 1110. [45] S. Xu, A.A. Mikhailovsky, J.A. Hollingsworth, V.I. Klimov, Phys. Rev. B 65 (2002) 045319. [46] R.G. Ispasoiu, J. Lee, F. Papadimitrakopoulos, Chem. Phys. Lett. 340 (2001).